Post on 15-Aug-2020
Research ArticleSample-Data Modeling of a Zero Voltage Transition DC-DCConverter for On-Board Battery Charger in EV
Teresa R Granados-Luna1 Ismael Araujo-Vargas2 and Francisco J Perez-Pinal3
1 Coacalcos Institute of Tecnological Studies 16 de Septiembre Avenue No 54 Col Cabecera Municipal55700 Coacalco de Berriozabal MEX Mexico
2 School of Mechanical and Electrical Engineering National Polytechnic Institute of Mexico ESIME CulSanta Ana Avenue No 1000 Col San Francisco Culhuacan 04430 Coyoacan DF Mexico
3 Automotive Mechanical Engineering Department Polytechnic University of Pachuca Ex Hacienda de Santa BarbaraCarretera Pachuca Cd Sahagun Km 20 43830 Zempoala HGO Mexico
Correspondence should be addressed to Ismael Araujo-Vargas iaraujoipnmx
Received 30 November 2013 Accepted 5 February 2014 Published 2 June 2014
Academic Editor Sheldon S Williamson
Copyright copy 2014 Teresa R Granados-Luna et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Battery charger is a key device in electric and hybrid electric vehicles On-board and off-board topologies are available in themarket Lightweight small high performance and simple control are desired characteristics for on-board chargers Moreoverisolated single-phase topologies are the most common system in Level 1 battery charger topologies Following this trend this paperproposes a sampled-data modelling strategy of a zero voltage transition (ZVT) DC-DC converter for an on-board battery chargerA piece-wise linear analysis of the converter is the basis of the technique presented such that a large-signal model and therefore asmall-signal model of the converter are derived Numerical and simulation results of a 250W test rig validate the model
1 Introduction
Advanced vehicular systems are based on the more electricsystems (MES) concept MES is the intensive application ofpower electronic converters (PEC) electric machines (EM)and advanced embedded control systems to aeronauticalautomotive and maritime systems MES was initially appliedto aeronautical systems toward the reduction andor sub-stitution of mechanical pneumatic and hydraulic systemsthat is the more electric aircraft (mea) and totally integratedmore electric systems (TIMES) [1] MES are more efficientcompared to their counterparts due to (i) small utilizationof electric energy (ii) high energy efficiency (iii) reducedweight and (iv) low maintenance [2] After that MES wasimplemented in automotive sector resulting in the moreelectric vehicle (MEV) MEV includes electric vehicles (EV)hybrid electric vehicles (HEV) and plug-in hybrid electricvehicles (PHEV) [3] In particular MES applied to vehicularsystems has become popular due to the market introduction
of theHEVToyota Prius in 1997 [4]HEVare being developedby companies like BMW Chrysler Daimler AG GeneralMotors PSA Peugeot Citroen Suzuki Motor Corp ToyotaandVolkswagenMotivations to develop EVHEV and PHEVare based on economic environmental and energetic factsRegardless of these kinds of configurations at least twodifferent sources of energy are needed to achieve the sameperformance compared to an internal combustion engine(ICE) Indeed at least one EM and PEC are needed in thepropulsion stage at any EV HEV and PHEV configurationSeries parallel seriesparallel and integrated starter alter-nator (ISA) with its optional plug-in capability are typicalconfigurations available in the market
PHEV uses an off-board or on-board charger similar toEV The standard SAEJ1772 is used in North America andcomprises three charge methods AC level 1 (supply voltagevaries from 120VAC 1-phase) AC level 2 (208V to 240VACand 600V DC maximum with a maximum current (amps-continuous) from 12A 32A and 400A) and DC charging
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 712360 15 pageshttpdxdoiorg1011552014712360
2 Mathematical Problems in Engineering
AC supply
Rectifier ZVT DC-DC converter Battery
Battery charger
+ PFC
(a)
RCf
ic Io
IZ
Lf
D3 D4
D2D1
isum
isec
C iABA
B
secprim
AN
BN
s AB
iD2iD1
iD4iD3
DD
TA2
TA1TB1
TB2
o
iL119878
L119878
iL119891
L119891LS
(b)
Figure 1 System configuration (a) block diagram and (b) phase-shift controlled ZVT DC-DC converter
Additionally SAEJ1772 provides a guide to the AC level-3 vehicle an on-board charger capable of accepting energyfrom an AC supply source at a nominal voltage of 208Vand 240VAC and a maximum current of 400A In additionSAEJ1772 provides information about the coupler require-ments general electric vehicle supply equipment (EVSE)requirements control and data and general conductivecharging system description [5]
Single- and three-phase isolated and nonisolated andunidirectional or bidirectional configurations have beenproposed in literature as battery chargers such as reportedin [6] Methods to improve their performance are usingone or several combinations of the following techniquespower factor correction (PFC) interleaved multicell andresonant configurations softhard switching zero voltageswitching (ZVS) and zero current switching (ZCS) More-over the control algorithms include proportional integral(PI) proportional-integral-derivative (PID) sliding modesfuzzy logic and adaptive neural network Following thistrend this paper proposed a sample-data modeling strategyof aDC-DCZVT to understand its dynamic characteristics asan on-board battery charger In this topology the switches areturned on during zero voltage reducing the switching lossesas a result a compact lightweight systemwith high switchingfrequency can be designed A typical peak current method isused in this work for control purposes resulting in a simpleand inexpensive control law
This paper is organized as follows The principle ofoperation of the converter is described in Section 2 usingidealized waveforms Then a mathematical analysis basedon a piece-wise linear analysis is provided in Section 3where a phase control strategy is modeled to obtain a large-signal model of the converter Using this model a half-cyclesample-data linear model is obtained which helps providethe final small-signal transfer functions of the converterNumerical and simulation results of a 250W prototype arepresented to validate the model obtained Final conclusionsare summarized in Section 5
2 Principle of Operation of the Converter
21 Circuit Description A typical system configuration for anEV battery charger is shown in Figure 1(a) which normallyconsists of a boost power factor corrected (PFC) rectifierconnected to an AC supply and a high frequency (HF)DC-DC converter to regulate the load of the batteriesThe topology of the DC-DC ZVT converter is shown in
Figure 1(b) which has a full bridge inverter supplied witha DC voltage source a HF transformer with a turns ratioof 1 N to generate a quasisquare phase-controlled wave astray inductance connected in series to the inverter outputmainly formed by the leakage inductance of the transformera full-bridge rectifier connected to the transformer secondaryside and then a LC filter to smooth the pulsating rectifiedvoltage waveform of the output of the rectifier The modelalso considers a disturbance current source in parallel withthe load
The left-hand leg of the inverter denoted by leg 119860 isused as the reference to describe the converter operationTheswitches of leg 119860 operate complementarily with fixed dutyratios of 50 at high frequency The switches of the right-hand inverter leg leg 119861 also operate complementarily withfixed duty ratios of 50 but the operation of leg 119861 is delayedby 120575T2 respective to leg 119860 where 119879 is the switching periodand 120575 is the phase control variable which ranges from 0 to 1
The steady state operation of the circuit of Figure 1 maybe explained with the steady state voltage and currentwaveforms of Figure 2
The first four waveforms shown in Figure 2 are the statesof the switches of the inverter leg 119860 Then the third andfourth waves show the states of the leg 119861 switches which aredelayed by 1205751198792 respective to the first and second waves ofFigure 2 The fifth and sixth waveforms of Figure 2 are theinverter output voltage V
119860119861= V1198602
minus V1198612
and output current119894119860119861
(also 119894119871119878
) The next two waveforms of this figure are therectifier output voltage V
119863119863 and the filter inductor current
119894119871119891
119894119871119891
is a continuous wave with a small ripple componentwhich is also present in 119894
119871119878
but amplified by the turns ratioN and reverted during the negative semicycle of V
119860119861 The
last three waveforms of Figure 2 are the current of diodes1198631to 1198634 1198941198631
to 1198941198634
and the supply current waveform 119894sumWhen V
119860119861= 119881in the current 119894119871
119878
is positive and flows through1198631and 119863
4 whereas when V
119860119861= minus119881in the current 119894
119871119878
isnegative and flows through 119863
2and 119863
3since these diodes
are positively biased When the V119860119861
waveform changes fromzero to plusmn119881in the diodes1198631 and119863
4are naturally commutated
short-circuiting the transformer secondary winding due tothe overlapped operation of the diodes The duration of thediodes overlap 119879
119874119871 causes a fast reversal of the inverter
primary current 119894119871119878
being limited by the inductance 119871119878
which prevents a short circuit of the inverter outputThe production of 119879
119874119871may be described using the waves
1198941198631
to 1198941198634
of Figure 2 When V119860119861
changes from zero to119881in thecurrents 119894
1198631
and 1198941198634
rise from zero to the output current level
Mathematical Problems in Engineering 3
T
T
T
T
T
T
T
T
T
T
T
T
T
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
120575T2
Io
Io
Io
NIo
NIoisum
AB
BN
AN
TOL
MI
MII
MII
IM
IV MV
MV
I
s
s
s
s
NsDD
1 iD4
iD
iD2 iD3
gsTA1
g
sTA2
gsTB2
g
sTB1
t
t
t
t
t
t
t
t
t
t
t
t
t
iL119878
iL119891
Figure 2 Ideal waveforms of the converter
and 1198941198632
and 1198941198633
fall to zero whereas when V119860119861
changes fromzero to minus119881in 119894119863
2
and 1198941198633
rise from zero to the output currentlevel and 119894
1198631
and 1198941198634
fall to zero During the 119879119874119871
period agradual current transfer is effectuated from one diode pair tothe other in such a way that 119894
119871119891
continues the slight currentslope which feeds the load The steady state value of 119879
119874119871may
be calculated assuming that the rate of change of the currentreversal of 119894
119860119861 119889119894119860119861
119889119905 only depends on the supply voltage
and the amplitude of the DC output filter current IO [3]which may be expressed as
119879119874119871
=2119873119871119878119868119874
119881119904
(1)
22 Piece-Wise Analysis of the ZVTConverter FromFigure 2119894119871119878
presents six different behaviour intervals which may betermed operating modes I to VI
For each mode of operation a different circuit configu-ration may be obtained which is shown in Figure 3 Theseequivalent circuits may be described using the state-spaceequation (2) and the state-space output expression (3)
= 119860119899119883 + 119861
119899119880 (2)
119884 = 119865119899119883 + 119866
119899119880 (3)
where 119883 = [119894119871119878
119894119871119891
V119900]119879
is the state vector 119880 = [119881119904 119868119885]119879 is
the input vector 119868119885is the output current disturbance 119884 =
[119894sec 1198941198631
1198941198632
119894sum]119879 is the output vector 119894sum is the supply
current and 119860119899 119861119899 119865119899 and 119866
119899are the state matrixes of the
six operating modes being 119899 = 1 2 6Mode I is formed when 119879
1198601
and 1198791198612
are in the on state1198791198602
and 1198791198611
are in the off state and 1198631to 1198634are conducting
due to the overlap rectifier phenomenaThe equivalent circuitof Mode I is shown in Figure 3(a) and the equations thatdescribe this mode are shown in (2) and (3) with 119899 = 1 Thematrixes A1 B1 F1 and G1 are listed in Table 5
In Mode II the state of the inverter switches is exactlyas that of Mode I but 119863
1and 119863
4are conducting and 119863
2
and 1198633are off The equivalent circuit of Mode II is shown in
Figure 3(b) and again (2) and (3) describe Mode II but with119899 = 2 The matrixes 119860
2 1198612 1198652 and 119866
2are shown in Table 5
In Mode III 1198791198601
and 1198791198611
are in the on state 1198791198602
and 1198791198612
are in the off state 1198631and 119863
4are conducting and 119863
2and
1198633are off The equivalent circuit of Mode III is shown in
Figure 3(c) being (2) and (3) with 119899 = 3 the mathematicalmodel of this mode Again the matrixes 119860
3 1198613 1198653 and 119866
3
are shown in Table 5Mode IV is a mirror of Mode I but with 119879
1198601
and 1198791198612
inthe off state and 119879
1198602
and 1198791198611
in the on state whilst Modes Vand VI are mirrors of Modes II and III respectively since thestate of the switches and diodes is complementary to that ofModes II and III Again (2) and (3) describeModes IV V andVI but with 119899 = 4 5 and 6 respectively The correspondingmatrixes to these operating modes are shown in Table 5
23 Current Control Loop Description The circuit shownin Figure 4 is a DC-DC ZVT converter with peak currentcontrol loop which has a current transducer with gain119877
119904that
senses 119894sum one SR flip-flop and two D flip-flops a clock sig-nal VCLK a sawtooth generator VSAW and the reference cur-rent level ViREF which is provided by an outer voltage loop
The operation of the circuit of Figure 4 may be explainedwith the state voltage and current waveforms of Figure 5Thefirst waveform shown in Figure 5 is the clock signal of systemVCLK The second waveform is 119894sum plotted together with the
4 Mathematical Problems in Engineering
Cf R
isum
ic Io
C IZ
Lf
D1 D2
D3 D4
iAB
secprims
isec
AB o
iD2iD1
iD3iD4
DD
TA1
TB2
iL119878L119878
iL119891
L119891
LS
(a)
isum
isec
ic Io
C Cf R IZ
Lf
D1
D4
iAB
secprims AB o
iD1
iD4
DD
TA1
TB2
iL119878
L119878
iL119891
L119891
LS
(b)
Cf R
isumic Io
C IZ
Lf
D1
D4
iAB
isec
secprims oAB
iD1
iD4
DD
TA1TB1 iL119878
L119878
iL119891
L119891
LS
(c)
isec
isum
C Cf R IZ
ic Io
Lf
D1 D2
D4D3
iAB
secprims AB o
iD2iD1
iD3iD4
DD
TA2
TB1 iL119878
L119878
iL119891
L119891
LS
(d)
RCf
ic Io
IZ
Lf
D3
D2
isum
C iAB
sec
sec
prims ABoiD2iD3
DD
TA1
TA2
TB1 iL119878
L119878
iL119891
L119891
LS
(e)
isec
isum
C Cf R IZ
ic
Lf
D2
D3
IoiAB
secprims AB oiD2iD3
DDTA2
TB2
iL119878
L119878
iL119891
L119891
LS
(f)
Figure 3 Equivalent circuits formed from the operation ZVT DC-DC converter (a) Mode I (b) Mode II (c) Mode III (d) Mode IV (e)Mode V and (f) Mode VI
deference of ViREF with VSAW where VSAW is a negative slopesynchronized with VCLK while ViREF is the current referencethat regulates the peak level of 119894sum VCOMP is the state of theoutput comparator which is the thirdwaveformof this figureThe fourth and fifth waveforms are the SR flip-flop outputs119876119860and 119876
119861 with 119876
119860set to the on state by VCLK and to the
off state when VCOMP switches to the on state The fifth andsixthwaveforms are the outputs of the first flip-flop VgsTA1
andVgsTA2
which are controlled by the rising edge of119876119860 whereas
VgsTB1and VgsTB2
the outputs of the second 119863 flip-flop whichare the last waves of this figure are controlled by the risingedge of 119876
119861
24 Numerical Estimation of the 119879119874119871
and 120575 Periods 119879119874119871
defines the duration of Modes I and IV and may be numer-ically estimated by determining the instant when either 119894
1198632
or 1198941198631
reaches zero 119879119874119871
may also be calculated using theNewton-Raphson method [4] This numerical method maybe implemented using
119879119874119871 (119899 + 1) = 119879
119874119871 (119899) minus119891 (119879119874119871
)
1198911015840 (119879119874119871
) (4)
where
119891 (119879119874119871
) = 1198651198631
(1205931(119879119874119871
)119883 (1199051))
+ (1198651198631
119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611+ 1198661198631
)119880
1198911015840(119879119874119871
) = 1198651198631
11986011198901198601119879119874119871119883(119905
1)
+ 1198651198631
[
[
infin
sum
119895
(119895 + 1)119860119895
1119879119895
119874119871
(119895 + 1)
]
]
1198611119880
(5)
Equations of (5) use the output equation that includes1198941198631
which determine the duration of Mode I whereas theduration of Mode IV is determined by 119894
1198632
such that (6) maybe rewritten as follows
119891 (119879119874119871
) = 1198651198632
(1205934(119879119874119871
)119883(119879
2))
+ (1198651198632
119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614+ 1198661198634
)
1198911015840(119879119874119871
) = 1198651198632
11986041198901198604119879119874119871119883(
119879
2)
+ 1198651198632
[
[
infin
sum
119895
(119895 + 1)119860119895
4119879119895
119874119871
(119895 + 1)
]
]
1198614119880
(6)
120575 defines the duration of Modes II and V and maybe numerically estimated by determining the instant whenthe equation ViREF minus VSAW is equal to 119877
119904119894sum 120575 may also
Mathematical Problems in Engineering 5
sumsum
Sawtooth
Peak current comparatorwith latch (RS Flip-Flop)
D Flip-Flopfor inverter leg A
D Flip-Flopfor inverter leg B
generator
Rsisum
Ls Cf R
isum
ic Io
C
isec
Lf
D1 D2
D3D4
iAB
H(s)+ +
+
minus minusminus
R FF
SQD
Q998400
Q998400
CLK
CLK
Q
Q
D
Q998400
CLK
QA
QB
middot middot middot
middot middot middot
seco
o
iREF
SAW
CLK
primABs
Rsisum
COMP
oREF
iD2iD1
iD4iD3
DD
TA2
TA1TB1
TB2
iL119878
L119878
iL119891
L119891
g119904T1198601
g119904T1198601
g119904T1198602
g119904T1198602
g119904T1198611
g119904T1198611
g119904T1198612
g119904T1198612
Figure 4 DC-DC ZVT converter with current control loop
be calculated using the Newton-Raphson method [4] Thisnumerical method may be implemented using
120575 (119899 + 1) = 120575 (119899) minus1198942sum (120575)
1198941015840
2sum (120575) (7)
where
i2sum (120575) = minus(
1
(119877119904)) [(VV minus V
119901) (
1205751198792
1198792) + V119901]
1198911015840(120575)
= (VV minus V119901) 120575
minus 1198652sum (119860
2
119879
2)
times (1205932(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
))
+ 1198652sum
119879
2
[
[
infin
sum
119895
(119895 + 1)119860119895
2((1205751198792) minus 119879
119874119871)119895
(119895 + 1)
]
]
119880
(8)
Equations of (8) use the output equation that includes1198942sum and the expression ViREF minus VSAW = 119877
119904119894sum which
determines the duration of Mode II whereas the duration of
Mode V is determined by 1198945sum and the expression ViREF minus
VSAW = 119877119904119894sum such that (9) may be rewritten as follows
1198945sum(120575) = minus
(VV minus V119901) ((1205751198792) (1198792)) + V
119901
(119877119904)
1198941015840
5sum (120575) = (VV minus V119901) 120575
minus 1198655sum (119860
5
119879
2)
times (1205935(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
))
+ 1198655sum
119879
2
[
[
infin
sum
119895
(119895 + 1)119860119895
5((1205751198792) minus 119879
119874119871)119895
(119895 + 1)
]
]
119880
(9)
3 Modeling of the ZVT Converter withCurrent Control Loop
31 Piece-Wise Linear Model Equation (2) may be used todevelop a piece-wise linear model of the converter of Figure 1
6 Mathematical Problems in Engineering
isum
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
CLK
NIo
120575T
2
TOL
QA
COMP
QB
120575T
2
iREF
g119904T1198601
g119904T1198602
g119904T1198611
g119904T1198612
Figure 5 Ideal waveform DC-DC ZVT converter with currentcontrol loop
throughout all the modes of operation The solution of (2)may be expressed as
119883(119905119899minus1
+ 119905119899) = 120593119899(119905119899)119883 (1199051) + 119860minus1
119899[120593119899(119905119899) minus 119868119889] 119861119899119880
(10)
where
120593119899(119905119899) = 119890119860119899119905119899 (11)
which defines in the first term of (10) the natural responseof the system along the period of time 119905
119899minus 119905119899minus1
with theinitial condition 119883(119905
119899minus1) at Mode n The second term of
(10) is the steady state response which is obtained by usingthe convolution integral Therefore using (10) for the timeinterval of 119905
1le 119905 lt 119905
1+ 119879119874119871 together with the matrixes of
Mode I the solution of the state vector forMode I is obtainedas follows
119883(1199051+ 119879119874119871
) = 1205931(119879119874119871
)119883 (1199051) + 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611119880
(12)
In a similar way the solutions for Modes II to VI at thetime intervals 119905
1+ 119879119874119871
le 119905 lt 1199051+ 1205751198792 119905
1+ 1205751198792 le 119905 lt
1199051+ 1198792 119905
1+ 1198792 le 119905 lt 119905
1+ 1198792 + 119879
119874119871 1199051+ 1198792 + 119879
119874119871le 119905 lt
1199051+ (1 + 120575)1198792 and 119905
1+ (1 + 120575)1198792 le 119905 lt 119905
1+ 119879 become
119883(1199051+
120575119879
2) = 120593
2(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
)
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612119880
(13)
119883(1199051+
119879
2) = 120593
3(119879
2minus
120575119879
2)119883(119905
1+
120575119879
2)
+ 119860minus1
3[1205933(119879
2minus
120575119879
2) minus 119868119889]1198613119880
(14)
119883(1199051+
119879
2+ 119879119874119871
) = 1205934(119879119874119871
)119883(1199051+
119879
2)
+ 119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614119880
(15)
119883(1199051+
(1 + 120575) 119879
2) = 120593
5(120575119879
2minus 119879119874119871
)119883(1199051+
119879
2+ 119879119874119871
)
+ 119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615119880
(16)
119883(1199051+ 119879) = 120593
6((1 minus 120575) 119879
2)119883(119905
1+
(1 + 120575) 119879
2)
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(17)
32 Large-Signal Model The large-signal model of the ZVTconverter may be obtained by substituting (12) in (13) (13) in(14) (14) in (15) (15) in (16) and (16) in (17) in such a waythat a single expression is obtained as shown in
119883(1199051+ 119879)
= 1205936((1 minus 120575) 119879
2)1205935(120575119879
2minus 119879119874119871
)1205934(119879119874119871
) 1205933
times ((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
Mathematical Problems in Engineering 7
+ [1205936((1 minus 120575) 119879
2)
times [1205935(120575119879
2minus 119879119874119871
)
times [1205934(119879119874119871
) [1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)
times 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
)
minus119868119889]1198612]
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]
+119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614]
+119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615]
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(18)
33 Half-Cycle Model The waveform 119894119871119878
of Figure 2 showsthat operation modes IV V and VI are mirrors of Modes III and III respectively therefore the first three operationmodes are sufficient to describe the function of the converterA particular matrix titled as W may relate the modes I IIand III with the modes IV V and VI which satisfies thecondition119882119882 = 119868
119889 the identity matrix Therefore the half-
cycle model of the ZVT converter may be obtained replacingthe termsA4 A5 A6 B4 B5 and B6 by expressionsWA1WA2WA3 WB1 WB2 andWB3 respectively in (18) such that thehalf-cycle model is
119883(1199051+
119879
2)
= 1205933((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
+ [1205933((1 minus 120575) 119879
2)12059321205933((1 minus 120575) 119879
2)1205932
times (120575119879
2minus 119879119874119871
)119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 1205933((1 minus 120575) 119879
2)119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]119880
+ 119882(1205933((1 minus 120575) 119879
2)(1205932(120575119879
2minus 119879119874119871
)
times 1198821205931(119879119874119871
)119883 (1199051)
+ 1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)119860minus1
1
times [1205931(119879119874119871
) minus 119868119889] 1198611)
+ 1205933((1 minus 120575) 119879
2)119860minus1
2
times [1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+ 119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198821198613)119880 (119905
1)
(19)
The previous equation may be rewritten as119883(1199051+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) for practical purposes
34 Sample-Data Small-Signal Linear Model of the Converterin Open-Loop Conditions The equation 119883(119905
1+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) may be used as a half-cycle discrete
model of the ZVT converter which may be written as
1198831015840
119870+1= 119860MC119883119870 + 119861MC119880119870 (20)
where1198831015840119870+1
= 119883(1199051+ 1198792)119883
119870= 119883(119905
1) and 119880
119870= 119880(119905
1)
A sample-data small-signal model may be obtained byusing the Taylor series (21) and using small-signal perturba-tions as 120575xK 120575UK 120575119879119874119871
119896
and 120575120575K One has
f (119909) = f (1199090) +J (119909 minus 119909
0) (21)
Using this equation the sample-data small-signal linearmodel becomes
120575119909119896+1
asymp120597119909119896+1
120597119909119896
120575119909119896+
120597119909119896+1
120597119880119896
120575119880119896
+120597119909119896+1
120597120575120575120575119896+
120597119909119896+1
120597119879119874119871
120575119879119874119871119896
(22)
The solution of the partial derivatives is 120597119909119896+1
120597119909119896
=
119860MC 120597119909119896+1120597119880119896 = 119861MC 120597119909119896+1120597120575 = 119862120575 and 120597119909
119896+1120597119879119874119871
=
119863119879119874119871
120575119879119874119871119896
may be determined utilizing the restrictionequation of119879
119874119871 whereas 120575120575K is obtained using the restriction
equation of 120575
35 Restriction Equations of the Control Loop The restrictionequations for 119879
119874119871may be obtained analyzing the waveforms
119894119871119878 11989411986311198633
and 119894sum which are shown in Figure 2 during theMode I The slope of 119894sum during Mode I is named 119892
1and is
determined by the rate of change of 119894119871119878
that is 1198921= 119881119904119896
119871119878
8 Mathematical Problems in Engineering
while the slopes of 11989411986311198633
during the Mode I are named 1198923
which are contrary and of lower amplitude than 1198921 that
is g3
= minusg12N The restriction equation of 119879
119874119871may be
determined by integrating the waveforms 11989411986311198633
duringModeI
119894119871119878119896
2119873+
119894119871119891119896
2+
119881119904119896
119871119878
119879119874119871119896
= 0 (23)
The previous equation is not linear therefore it isnecessary to use theTaylor series to obtain a linearizedmodel
120575119879119874119871119896
=2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
(24)
The restriction equation of 120575 may be obtained analyzingthe waveforms 119894
119871119878
and 119894sum with ViREF minus VSAW during Mode II(Figures 2 and 5) VSAW is determined by 119872(120575
1198961198792) and the
slope of 119894sum duringMode II named 1198922 and may be obtained
by integrating again 119894sumwhen 119877119904119894sum = ViREF minus VSAW
119877119904(119894119871119878119896
+ 1198921119879119874119871119896
+ 1198922(120575119896119879
2minus 119879119874119871119896
)) = 119881119894REF minus 119872
120575119896119879
2
(25)
The Taylor series is used to linearize the previous equa-tion and therefore the restriction equation of 120575 becomes
120575120575119896=
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
(26)
where Δ120575= minus(2119879)(1198721198792119877
119878+ 1(119871
119878+ (1119873
2)119871119891))(V119904CD
minus
(1119873)V0CD
) 1198861205751
= 120597119894119871119878119896
120597119894119871119878119896
1198861205752
= 120597119894119871119878119896
120597119894119871119891119896
1198861205753
=
120597119894119871119878119896
120597119881119878119896
1198861205754
= 120597119894119871119878119896
120597119881iREF119896
and 1198861205755
= 120597119894119871119878119896
120597119881119900119896
36 Sample-Data Small-Signal Linear Model of the Converterin Closed-Loop Conditions The sample-data small-signallinear model may be obtained by substituting 120575
119879119874119871119896
and 120575120575K(24) and (26) respectively in (22)
120575119909119896+1
= 119860MC[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
+ 119861MC [120575119881119904119896
120575119868119885119896
]
+ 119862120575(
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
)
+ 119863119879119874119871
(2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
)
(27)
Table 1 Operating parameters
Supply voltage 200V plusmn 20Output voltage 48VMaximum power 250WMinimum power 50WOutput voltage ripple 25mVOutput current ripple 300mAMaximum phase 50Switching frequency 50 kHz
such that the small-signal model becomes
120575119909119896+1
= 119860119888119897120575119909119896+ 120596119888119897120575119908119888119897119896
(28)
where 120575119909119896+1
= [1205751198941015840
119871119878119896
1205751198941015840
119871119891119896
1205751198811015840
119904119896
]119879
120575119909119896
=
[120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896]1015840 and 120575119908
119888119897119896
= [120575119881119904119896
120575119881iREF119896
120575119868119885119896]119879
Equation (28) may be solved by using the 119885 transformsuch that 120575119909
119870becomes
120575119909119896= (119868119889minus 119860119888119897)minus1119885120596119888119897120575119908119888119897119896
(29)
37 Transfer Function To verify the dynamic characteristicsof the converter is necessary to analyze the transfer functionsthat relate V
119900with 119881
119904 ViREF and 119868
119911 which are the throughput
input-to-output DC voltage transfer function 119867V(119911) =
V119900(119911)V119904(119911) the control-to-output transfer function119879
119874119871(119911) =
V119900(119911)ViREF(119911) and the output impedance transfer function
119885119900(119911) = V
119900(119911)119868119911(119911) respectively
The magnitude and phase of each transfer function maybe obtained using a Bode diagram whereas the root locustechnique may be employed to describe the behaviour of thepoles and zeros of (30) One has
119867V (119911) =119911 + 1198861
1199112 + 1198871119911 + 1198881
119879119874119871 (119911) =
11988621199112+ 1198872119911 + 1198882
1199112 + 1198871119911 + 1198881
119885119900 (119911) =
11988631199112+ 1198873119911 minus 1198883
1199112 + 1198871119911 + 1198881
(30)
4 Verification of the Proposed Model
41 Prototype Operating Parameters A 250W ZVT DC-DC prototype converter was designed under the analysisdescribed in [7] to verify the large-signal model of (14)Table 1 shows the operating parameters of the converter
The steady-state output voltage 119881119874 may be calculated as
the average of the rectified voltage V119863119863
such that at full load
119881119900= 120575max119873119881in minus
41198732119868119900(max)119871119878
119879 (31)
Taking the assumption shown in (31) the converter com-ponent values must comply with the zero-voltage switching
Mathematical Problems in Engineering 9
012345
(ms)(ms)
Piece-wise analysisPiece-wise model
Simulation circuitPiece-wise analysisPiece-wise model
Simulation circuit
01234
0 01 02 03 04 05 06 07 08 09 01minus5
minus4
minus3
minus2
minus1
minus4
minus3
minus2
minus1A A
09
0902
0904
0906
0908
091
0912
0914
0916
0918
092
iL119878iL119878
Figure 6 119894119871119878
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
1
0
1
2
3
4
5
6
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
09
902
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
0
1
2
3
4
5
6
7
(ms)
A
0 01 02 03 04 05 06 07 08 09minus1
(ms)
A
iL119891iL119891
Figure 7 119894119871119891
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
0
10
20
30
40
50
0
10
20
30
40
50
60
minus10
A
1
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
0 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
6
090
8
091
091
4
091
6
091
8
092
(ms)
A
o o
Figure 8 vo voltage waveform obtained with the piece-wise linear model and a Micro-Cap simulation
10 Mathematical Problems in Engineering
012345
0
2
4
6
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
minus6
minus4
minus2
minus4
minus3
minus1
minus2
A A
10 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
(ms)
isum and SAW-iREF isum and SAW-iREF
Figure 9 Waveforms of 119894sum and VSAW minus ViREF obtained with the piece-wise linear model and a Micro-Cap simulation
Mag
nitu
de (d
B)
0
Phas
e (de
g)
101 102 103 104
Frequency (rads)
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus225
minus180
minus135
minus90
minus45
Bode Diagram (os)
Figure 10 Bode plot of Hv(z)
05
10152025303540
Mag
nitu
de (d
B)Ph
ase (
deg)
101 102 103 104minus240minus195minus150minus105minus80minus15
Frequency (rads)
Bode Diagram (oiREF)
Figure 11 Bode plot of 119879119874119871(z)
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
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2 Mathematical Problems in Engineering
AC supply
Rectifier ZVT DC-DC converter Battery
Battery charger
+ PFC
(a)
RCf
ic Io
IZ
Lf
D3 D4
D2D1
isum
isec
C iABA
B
secprim
AN
BN
s AB
iD2iD1
iD4iD3
DD
TA2
TA1TB1
TB2
o
iL119878
L119878
iL119891
L119891LS
(b)
Figure 1 System configuration (a) block diagram and (b) phase-shift controlled ZVT DC-DC converter
Additionally SAEJ1772 provides a guide to the AC level-3 vehicle an on-board charger capable of accepting energyfrom an AC supply source at a nominal voltage of 208Vand 240VAC and a maximum current of 400A In additionSAEJ1772 provides information about the coupler require-ments general electric vehicle supply equipment (EVSE)requirements control and data and general conductivecharging system description [5]
Single- and three-phase isolated and nonisolated andunidirectional or bidirectional configurations have beenproposed in literature as battery chargers such as reportedin [6] Methods to improve their performance are usingone or several combinations of the following techniquespower factor correction (PFC) interleaved multicell andresonant configurations softhard switching zero voltageswitching (ZVS) and zero current switching (ZCS) More-over the control algorithms include proportional integral(PI) proportional-integral-derivative (PID) sliding modesfuzzy logic and adaptive neural network Following thistrend this paper proposed a sample-data modeling strategyof aDC-DCZVT to understand its dynamic characteristics asan on-board battery charger In this topology the switches areturned on during zero voltage reducing the switching lossesas a result a compact lightweight systemwith high switchingfrequency can be designed A typical peak current method isused in this work for control purposes resulting in a simpleand inexpensive control law
This paper is organized as follows The principle ofoperation of the converter is described in Section 2 usingidealized waveforms Then a mathematical analysis basedon a piece-wise linear analysis is provided in Section 3where a phase control strategy is modeled to obtain a large-signal model of the converter Using this model a half-cyclesample-data linear model is obtained which helps providethe final small-signal transfer functions of the converterNumerical and simulation results of a 250W prototype arepresented to validate the model obtained Final conclusionsare summarized in Section 5
2 Principle of Operation of the Converter
21 Circuit Description A typical system configuration for anEV battery charger is shown in Figure 1(a) which normallyconsists of a boost power factor corrected (PFC) rectifierconnected to an AC supply and a high frequency (HF)DC-DC converter to regulate the load of the batteriesThe topology of the DC-DC ZVT converter is shown in
Figure 1(b) which has a full bridge inverter supplied witha DC voltage source a HF transformer with a turns ratioof 1 N to generate a quasisquare phase-controlled wave astray inductance connected in series to the inverter outputmainly formed by the leakage inductance of the transformera full-bridge rectifier connected to the transformer secondaryside and then a LC filter to smooth the pulsating rectifiedvoltage waveform of the output of the rectifier The modelalso considers a disturbance current source in parallel withthe load
The left-hand leg of the inverter denoted by leg 119860 isused as the reference to describe the converter operationTheswitches of leg 119860 operate complementarily with fixed dutyratios of 50 at high frequency The switches of the right-hand inverter leg leg 119861 also operate complementarily withfixed duty ratios of 50 but the operation of leg 119861 is delayedby 120575T2 respective to leg 119860 where 119879 is the switching periodand 120575 is the phase control variable which ranges from 0 to 1
The steady state operation of the circuit of Figure 1 maybe explained with the steady state voltage and currentwaveforms of Figure 2
The first four waveforms shown in Figure 2 are the statesof the switches of the inverter leg 119860 Then the third andfourth waves show the states of the leg 119861 switches which aredelayed by 1205751198792 respective to the first and second waves ofFigure 2 The fifth and sixth waveforms of Figure 2 are theinverter output voltage V
119860119861= V1198602
minus V1198612
and output current119894119860119861
(also 119894119871119878
) The next two waveforms of this figure are therectifier output voltage V
119863119863 and the filter inductor current
119894119871119891
119894119871119891
is a continuous wave with a small ripple componentwhich is also present in 119894
119871119878
but amplified by the turns ratioN and reverted during the negative semicycle of V
119860119861 The
last three waveforms of Figure 2 are the current of diodes1198631to 1198634 1198941198631
to 1198941198634
and the supply current waveform 119894sumWhen V
119860119861= 119881in the current 119894119871
119878
is positive and flows through1198631and 119863
4 whereas when V
119860119861= minus119881in the current 119894
119871119878
isnegative and flows through 119863
2and 119863
3since these diodes
are positively biased When the V119860119861
waveform changes fromzero to plusmn119881in the diodes1198631 and119863
4are naturally commutated
short-circuiting the transformer secondary winding due tothe overlapped operation of the diodes The duration of thediodes overlap 119879
119874119871 causes a fast reversal of the inverter
primary current 119894119871119878
being limited by the inductance 119871119878
which prevents a short circuit of the inverter outputThe production of 119879
119874119871may be described using the waves
1198941198631
to 1198941198634
of Figure 2 When V119860119861
changes from zero to119881in thecurrents 119894
1198631
and 1198941198634
rise from zero to the output current level
Mathematical Problems in Engineering 3
T
T
T
T
T
T
T
T
T
T
T
T
T
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
120575T2
Io
Io
Io
NIo
NIoisum
AB
BN
AN
TOL
MI
MII
MII
IM
IV MV
MV
I
s
s
s
s
NsDD
1 iD4
iD
iD2 iD3
gsTA1
g
sTA2
gsTB2
g
sTB1
t
t
t
t
t
t
t
t
t
t
t
t
t
iL119878
iL119891
Figure 2 Ideal waveforms of the converter
and 1198941198632
and 1198941198633
fall to zero whereas when V119860119861
changes fromzero to minus119881in 119894119863
2
and 1198941198633
rise from zero to the output currentlevel and 119894
1198631
and 1198941198634
fall to zero During the 119879119874119871
period agradual current transfer is effectuated from one diode pair tothe other in such a way that 119894
119871119891
continues the slight currentslope which feeds the load The steady state value of 119879
119874119871may
be calculated assuming that the rate of change of the currentreversal of 119894
119860119861 119889119894119860119861
119889119905 only depends on the supply voltage
and the amplitude of the DC output filter current IO [3]which may be expressed as
119879119874119871
=2119873119871119878119868119874
119881119904
(1)
22 Piece-Wise Analysis of the ZVTConverter FromFigure 2119894119871119878
presents six different behaviour intervals which may betermed operating modes I to VI
For each mode of operation a different circuit configu-ration may be obtained which is shown in Figure 3 Theseequivalent circuits may be described using the state-spaceequation (2) and the state-space output expression (3)
= 119860119899119883 + 119861
119899119880 (2)
119884 = 119865119899119883 + 119866
119899119880 (3)
where 119883 = [119894119871119878
119894119871119891
V119900]119879
is the state vector 119880 = [119881119904 119868119885]119879 is
the input vector 119868119885is the output current disturbance 119884 =
[119894sec 1198941198631
1198941198632
119894sum]119879 is the output vector 119894sum is the supply
current and 119860119899 119861119899 119865119899 and 119866
119899are the state matrixes of the
six operating modes being 119899 = 1 2 6Mode I is formed when 119879
1198601
and 1198791198612
are in the on state1198791198602
and 1198791198611
are in the off state and 1198631to 1198634are conducting
due to the overlap rectifier phenomenaThe equivalent circuitof Mode I is shown in Figure 3(a) and the equations thatdescribe this mode are shown in (2) and (3) with 119899 = 1 Thematrixes A1 B1 F1 and G1 are listed in Table 5
In Mode II the state of the inverter switches is exactlyas that of Mode I but 119863
1and 119863
4are conducting and 119863
2
and 1198633are off The equivalent circuit of Mode II is shown in
Figure 3(b) and again (2) and (3) describe Mode II but with119899 = 2 The matrixes 119860
2 1198612 1198652 and 119866
2are shown in Table 5
In Mode III 1198791198601
and 1198791198611
are in the on state 1198791198602
and 1198791198612
are in the off state 1198631and 119863
4are conducting and 119863
2and
1198633are off The equivalent circuit of Mode III is shown in
Figure 3(c) being (2) and (3) with 119899 = 3 the mathematicalmodel of this mode Again the matrixes 119860
3 1198613 1198653 and 119866
3
are shown in Table 5Mode IV is a mirror of Mode I but with 119879
1198601
and 1198791198612
inthe off state and 119879
1198602
and 1198791198611
in the on state whilst Modes Vand VI are mirrors of Modes II and III respectively since thestate of the switches and diodes is complementary to that ofModes II and III Again (2) and (3) describeModes IV V andVI but with 119899 = 4 5 and 6 respectively The correspondingmatrixes to these operating modes are shown in Table 5
23 Current Control Loop Description The circuit shownin Figure 4 is a DC-DC ZVT converter with peak currentcontrol loop which has a current transducer with gain119877
119904that
senses 119894sum one SR flip-flop and two D flip-flops a clock sig-nal VCLK a sawtooth generator VSAW and the reference cur-rent level ViREF which is provided by an outer voltage loop
The operation of the circuit of Figure 4 may be explainedwith the state voltage and current waveforms of Figure 5Thefirst waveform shown in Figure 5 is the clock signal of systemVCLK The second waveform is 119894sum plotted together with the
4 Mathematical Problems in Engineering
Cf R
isum
ic Io
C IZ
Lf
D1 D2
D3 D4
iAB
secprims
isec
AB o
iD2iD1
iD3iD4
DD
TA1
TB2
iL119878L119878
iL119891
L119891
LS
(a)
isum
isec
ic Io
C Cf R IZ
Lf
D1
D4
iAB
secprims AB o
iD1
iD4
DD
TA1
TB2
iL119878
L119878
iL119891
L119891
LS
(b)
Cf R
isumic Io
C IZ
Lf
D1
D4
iAB
isec
secprims oAB
iD1
iD4
DD
TA1TB1 iL119878
L119878
iL119891
L119891
LS
(c)
isec
isum
C Cf R IZ
ic Io
Lf
D1 D2
D4D3
iAB
secprims AB o
iD2iD1
iD3iD4
DD
TA2
TB1 iL119878
L119878
iL119891
L119891
LS
(d)
RCf
ic Io
IZ
Lf
D3
D2
isum
C iAB
sec
sec
prims ABoiD2iD3
DD
TA1
TA2
TB1 iL119878
L119878
iL119891
L119891
LS
(e)
isec
isum
C Cf R IZ
ic
Lf
D2
D3
IoiAB
secprims AB oiD2iD3
DDTA2
TB2
iL119878
L119878
iL119891
L119891
LS
(f)
Figure 3 Equivalent circuits formed from the operation ZVT DC-DC converter (a) Mode I (b) Mode II (c) Mode III (d) Mode IV (e)Mode V and (f) Mode VI
deference of ViREF with VSAW where VSAW is a negative slopesynchronized with VCLK while ViREF is the current referencethat regulates the peak level of 119894sum VCOMP is the state of theoutput comparator which is the thirdwaveformof this figureThe fourth and fifth waveforms are the SR flip-flop outputs119876119860and 119876
119861 with 119876
119860set to the on state by VCLK and to the
off state when VCOMP switches to the on state The fifth andsixthwaveforms are the outputs of the first flip-flop VgsTA1
andVgsTA2
which are controlled by the rising edge of119876119860 whereas
VgsTB1and VgsTB2
the outputs of the second 119863 flip-flop whichare the last waves of this figure are controlled by the risingedge of 119876
119861
24 Numerical Estimation of the 119879119874119871
and 120575 Periods 119879119874119871
defines the duration of Modes I and IV and may be numer-ically estimated by determining the instant when either 119894
1198632
or 1198941198631
reaches zero 119879119874119871
may also be calculated using theNewton-Raphson method [4] This numerical method maybe implemented using
119879119874119871 (119899 + 1) = 119879
119874119871 (119899) minus119891 (119879119874119871
)
1198911015840 (119879119874119871
) (4)
where
119891 (119879119874119871
) = 1198651198631
(1205931(119879119874119871
)119883 (1199051))
+ (1198651198631
119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611+ 1198661198631
)119880
1198911015840(119879119874119871
) = 1198651198631
11986011198901198601119879119874119871119883(119905
1)
+ 1198651198631
[
[
infin
sum
119895
(119895 + 1)119860119895
1119879119895
119874119871
(119895 + 1)
]
]
1198611119880
(5)
Equations of (5) use the output equation that includes1198941198631
which determine the duration of Mode I whereas theduration of Mode IV is determined by 119894
1198632
such that (6) maybe rewritten as follows
119891 (119879119874119871
) = 1198651198632
(1205934(119879119874119871
)119883(119879
2))
+ (1198651198632
119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614+ 1198661198634
)
1198911015840(119879119874119871
) = 1198651198632
11986041198901198604119879119874119871119883(
119879
2)
+ 1198651198632
[
[
infin
sum
119895
(119895 + 1)119860119895
4119879119895
119874119871
(119895 + 1)
]
]
1198614119880
(6)
120575 defines the duration of Modes II and V and maybe numerically estimated by determining the instant whenthe equation ViREF minus VSAW is equal to 119877
119904119894sum 120575 may also
Mathematical Problems in Engineering 5
sumsum
Sawtooth
Peak current comparatorwith latch (RS Flip-Flop)
D Flip-Flopfor inverter leg A
D Flip-Flopfor inverter leg B
generator
Rsisum
Ls Cf R
isum
ic Io
C
isec
Lf
D1 D2
D3D4
iAB
H(s)+ +
+
minus minusminus
R FF
SQD
Q998400
Q998400
CLK
CLK
Q
Q
D
Q998400
CLK
QA
QB
middot middot middot
middot middot middot
seco
o
iREF
SAW
CLK
primABs
Rsisum
COMP
oREF
iD2iD1
iD4iD3
DD
TA2
TA1TB1
TB2
iL119878
L119878
iL119891
L119891
g119904T1198601
g119904T1198601
g119904T1198602
g119904T1198602
g119904T1198611
g119904T1198611
g119904T1198612
g119904T1198612
Figure 4 DC-DC ZVT converter with current control loop
be calculated using the Newton-Raphson method [4] Thisnumerical method may be implemented using
120575 (119899 + 1) = 120575 (119899) minus1198942sum (120575)
1198941015840
2sum (120575) (7)
where
i2sum (120575) = minus(
1
(119877119904)) [(VV minus V
119901) (
1205751198792
1198792) + V119901]
1198911015840(120575)
= (VV minus V119901) 120575
minus 1198652sum (119860
2
119879
2)
times (1205932(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
))
+ 1198652sum
119879
2
[
[
infin
sum
119895
(119895 + 1)119860119895
2((1205751198792) minus 119879
119874119871)119895
(119895 + 1)
]
]
119880
(8)
Equations of (8) use the output equation that includes1198942sum and the expression ViREF minus VSAW = 119877
119904119894sum which
determines the duration of Mode II whereas the duration of
Mode V is determined by 1198945sum and the expression ViREF minus
VSAW = 119877119904119894sum such that (9) may be rewritten as follows
1198945sum(120575) = minus
(VV minus V119901) ((1205751198792) (1198792)) + V
119901
(119877119904)
1198941015840
5sum (120575) = (VV minus V119901) 120575
minus 1198655sum (119860
5
119879
2)
times (1205935(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
))
+ 1198655sum
119879
2
[
[
infin
sum
119895
(119895 + 1)119860119895
5((1205751198792) minus 119879
119874119871)119895
(119895 + 1)
]
]
119880
(9)
3 Modeling of the ZVT Converter withCurrent Control Loop
31 Piece-Wise Linear Model Equation (2) may be used todevelop a piece-wise linear model of the converter of Figure 1
6 Mathematical Problems in Engineering
isum
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
CLK
NIo
120575T
2
TOL
QA
COMP
QB
120575T
2
iREF
g119904T1198601
g119904T1198602
g119904T1198611
g119904T1198612
Figure 5 Ideal waveform DC-DC ZVT converter with currentcontrol loop
throughout all the modes of operation The solution of (2)may be expressed as
119883(119905119899minus1
+ 119905119899) = 120593119899(119905119899)119883 (1199051) + 119860minus1
119899[120593119899(119905119899) minus 119868119889] 119861119899119880
(10)
where
120593119899(119905119899) = 119890119860119899119905119899 (11)
which defines in the first term of (10) the natural responseof the system along the period of time 119905
119899minus 119905119899minus1
with theinitial condition 119883(119905
119899minus1) at Mode n The second term of
(10) is the steady state response which is obtained by usingthe convolution integral Therefore using (10) for the timeinterval of 119905
1le 119905 lt 119905
1+ 119879119874119871 together with the matrixes of
Mode I the solution of the state vector forMode I is obtainedas follows
119883(1199051+ 119879119874119871
) = 1205931(119879119874119871
)119883 (1199051) + 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611119880
(12)
In a similar way the solutions for Modes II to VI at thetime intervals 119905
1+ 119879119874119871
le 119905 lt 1199051+ 1205751198792 119905
1+ 1205751198792 le 119905 lt
1199051+ 1198792 119905
1+ 1198792 le 119905 lt 119905
1+ 1198792 + 119879
119874119871 1199051+ 1198792 + 119879
119874119871le 119905 lt
1199051+ (1 + 120575)1198792 and 119905
1+ (1 + 120575)1198792 le 119905 lt 119905
1+ 119879 become
119883(1199051+
120575119879
2) = 120593
2(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
)
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612119880
(13)
119883(1199051+
119879
2) = 120593
3(119879
2minus
120575119879
2)119883(119905
1+
120575119879
2)
+ 119860minus1
3[1205933(119879
2minus
120575119879
2) minus 119868119889]1198613119880
(14)
119883(1199051+
119879
2+ 119879119874119871
) = 1205934(119879119874119871
)119883(1199051+
119879
2)
+ 119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614119880
(15)
119883(1199051+
(1 + 120575) 119879
2) = 120593
5(120575119879
2minus 119879119874119871
)119883(1199051+
119879
2+ 119879119874119871
)
+ 119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615119880
(16)
119883(1199051+ 119879) = 120593
6((1 minus 120575) 119879
2)119883(119905
1+
(1 + 120575) 119879
2)
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(17)
32 Large-Signal Model The large-signal model of the ZVTconverter may be obtained by substituting (12) in (13) (13) in(14) (14) in (15) (15) in (16) and (16) in (17) in such a waythat a single expression is obtained as shown in
119883(1199051+ 119879)
= 1205936((1 minus 120575) 119879
2)1205935(120575119879
2minus 119879119874119871
)1205934(119879119874119871
) 1205933
times ((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
Mathematical Problems in Engineering 7
+ [1205936((1 minus 120575) 119879
2)
times [1205935(120575119879
2minus 119879119874119871
)
times [1205934(119879119874119871
) [1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)
times 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
)
minus119868119889]1198612]
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]
+119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614]
+119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615]
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(18)
33 Half-Cycle Model The waveform 119894119871119878
of Figure 2 showsthat operation modes IV V and VI are mirrors of Modes III and III respectively therefore the first three operationmodes are sufficient to describe the function of the converterA particular matrix titled as W may relate the modes I IIand III with the modes IV V and VI which satisfies thecondition119882119882 = 119868
119889 the identity matrix Therefore the half-
cycle model of the ZVT converter may be obtained replacingthe termsA4 A5 A6 B4 B5 and B6 by expressionsWA1WA2WA3 WB1 WB2 andWB3 respectively in (18) such that thehalf-cycle model is
119883(1199051+
119879
2)
= 1205933((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
+ [1205933((1 minus 120575) 119879
2)12059321205933((1 minus 120575) 119879
2)1205932
times (120575119879
2minus 119879119874119871
)119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 1205933((1 minus 120575) 119879
2)119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]119880
+ 119882(1205933((1 minus 120575) 119879
2)(1205932(120575119879
2minus 119879119874119871
)
times 1198821205931(119879119874119871
)119883 (1199051)
+ 1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)119860minus1
1
times [1205931(119879119874119871
) minus 119868119889] 1198611)
+ 1205933((1 minus 120575) 119879
2)119860minus1
2
times [1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+ 119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198821198613)119880 (119905
1)
(19)
The previous equation may be rewritten as119883(1199051+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) for practical purposes
34 Sample-Data Small-Signal Linear Model of the Converterin Open-Loop Conditions The equation 119883(119905
1+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) may be used as a half-cycle discrete
model of the ZVT converter which may be written as
1198831015840
119870+1= 119860MC119883119870 + 119861MC119880119870 (20)
where1198831015840119870+1
= 119883(1199051+ 1198792)119883
119870= 119883(119905
1) and 119880
119870= 119880(119905
1)
A sample-data small-signal model may be obtained byusing the Taylor series (21) and using small-signal perturba-tions as 120575xK 120575UK 120575119879119874119871
119896
and 120575120575K One has
f (119909) = f (1199090) +J (119909 minus 119909
0) (21)
Using this equation the sample-data small-signal linearmodel becomes
120575119909119896+1
asymp120597119909119896+1
120597119909119896
120575119909119896+
120597119909119896+1
120597119880119896
120575119880119896
+120597119909119896+1
120597120575120575120575119896+
120597119909119896+1
120597119879119874119871
120575119879119874119871119896
(22)
The solution of the partial derivatives is 120597119909119896+1
120597119909119896
=
119860MC 120597119909119896+1120597119880119896 = 119861MC 120597119909119896+1120597120575 = 119862120575 and 120597119909
119896+1120597119879119874119871
=
119863119879119874119871
120575119879119874119871119896
may be determined utilizing the restrictionequation of119879
119874119871 whereas 120575120575K is obtained using the restriction
equation of 120575
35 Restriction Equations of the Control Loop The restrictionequations for 119879
119874119871may be obtained analyzing the waveforms
119894119871119878 11989411986311198633
and 119894sum which are shown in Figure 2 during theMode I The slope of 119894sum during Mode I is named 119892
1and is
determined by the rate of change of 119894119871119878
that is 1198921= 119881119904119896
119871119878
8 Mathematical Problems in Engineering
while the slopes of 11989411986311198633
during the Mode I are named 1198923
which are contrary and of lower amplitude than 1198921 that
is g3
= minusg12N The restriction equation of 119879
119874119871may be
determined by integrating the waveforms 11989411986311198633
duringModeI
119894119871119878119896
2119873+
119894119871119891119896
2+
119881119904119896
119871119878
119879119874119871119896
= 0 (23)
The previous equation is not linear therefore it isnecessary to use theTaylor series to obtain a linearizedmodel
120575119879119874119871119896
=2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
(24)
The restriction equation of 120575 may be obtained analyzingthe waveforms 119894
119871119878
and 119894sum with ViREF minus VSAW during Mode II(Figures 2 and 5) VSAW is determined by 119872(120575
1198961198792) and the
slope of 119894sum duringMode II named 1198922 and may be obtained
by integrating again 119894sumwhen 119877119904119894sum = ViREF minus VSAW
119877119904(119894119871119878119896
+ 1198921119879119874119871119896
+ 1198922(120575119896119879
2minus 119879119874119871119896
)) = 119881119894REF minus 119872
120575119896119879
2
(25)
The Taylor series is used to linearize the previous equa-tion and therefore the restriction equation of 120575 becomes
120575120575119896=
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
(26)
where Δ120575= minus(2119879)(1198721198792119877
119878+ 1(119871
119878+ (1119873
2)119871119891))(V119904CD
minus
(1119873)V0CD
) 1198861205751
= 120597119894119871119878119896
120597119894119871119878119896
1198861205752
= 120597119894119871119878119896
120597119894119871119891119896
1198861205753
=
120597119894119871119878119896
120597119881119878119896
1198861205754
= 120597119894119871119878119896
120597119881iREF119896
and 1198861205755
= 120597119894119871119878119896
120597119881119900119896
36 Sample-Data Small-Signal Linear Model of the Converterin Closed-Loop Conditions The sample-data small-signallinear model may be obtained by substituting 120575
119879119874119871119896
and 120575120575K(24) and (26) respectively in (22)
120575119909119896+1
= 119860MC[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
+ 119861MC [120575119881119904119896
120575119868119885119896
]
+ 119862120575(
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
)
+ 119863119879119874119871
(2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
)
(27)
Table 1 Operating parameters
Supply voltage 200V plusmn 20Output voltage 48VMaximum power 250WMinimum power 50WOutput voltage ripple 25mVOutput current ripple 300mAMaximum phase 50Switching frequency 50 kHz
such that the small-signal model becomes
120575119909119896+1
= 119860119888119897120575119909119896+ 120596119888119897120575119908119888119897119896
(28)
where 120575119909119896+1
= [1205751198941015840
119871119878119896
1205751198941015840
119871119891119896
1205751198811015840
119904119896
]119879
120575119909119896
=
[120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896]1015840 and 120575119908
119888119897119896
= [120575119881119904119896
120575119881iREF119896
120575119868119885119896]119879
Equation (28) may be solved by using the 119885 transformsuch that 120575119909
119870becomes
120575119909119896= (119868119889minus 119860119888119897)minus1119885120596119888119897120575119908119888119897119896
(29)
37 Transfer Function To verify the dynamic characteristicsof the converter is necessary to analyze the transfer functionsthat relate V
119900with 119881
119904 ViREF and 119868
119911 which are the throughput
input-to-output DC voltage transfer function 119867V(119911) =
V119900(119911)V119904(119911) the control-to-output transfer function119879
119874119871(119911) =
V119900(119911)ViREF(119911) and the output impedance transfer function
119885119900(119911) = V
119900(119911)119868119911(119911) respectively
The magnitude and phase of each transfer function maybe obtained using a Bode diagram whereas the root locustechnique may be employed to describe the behaviour of thepoles and zeros of (30) One has
119867V (119911) =119911 + 1198861
1199112 + 1198871119911 + 1198881
119879119874119871 (119911) =
11988621199112+ 1198872119911 + 1198882
1199112 + 1198871119911 + 1198881
119885119900 (119911) =
11988631199112+ 1198873119911 minus 1198883
1199112 + 1198871119911 + 1198881
(30)
4 Verification of the Proposed Model
41 Prototype Operating Parameters A 250W ZVT DC-DC prototype converter was designed under the analysisdescribed in [7] to verify the large-signal model of (14)Table 1 shows the operating parameters of the converter
The steady-state output voltage 119881119874 may be calculated as
the average of the rectified voltage V119863119863
such that at full load
119881119900= 120575max119873119881in minus
41198732119868119900(max)119871119878
119879 (31)
Taking the assumption shown in (31) the converter com-ponent values must comply with the zero-voltage switching
Mathematical Problems in Engineering 9
012345
(ms)(ms)
Piece-wise analysisPiece-wise model
Simulation circuitPiece-wise analysisPiece-wise model
Simulation circuit
01234
0 01 02 03 04 05 06 07 08 09 01minus5
minus4
minus3
minus2
minus1
minus4
minus3
minus2
minus1A A
09
0902
0904
0906
0908
091
0912
0914
0916
0918
092
iL119878iL119878
Figure 6 119894119871119878
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
1
0
1
2
3
4
5
6
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
09
902
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
0
1
2
3
4
5
6
7
(ms)
A
0 01 02 03 04 05 06 07 08 09minus1
(ms)
A
iL119891iL119891
Figure 7 119894119871119891
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
0
10
20
30
40
50
0
10
20
30
40
50
60
minus10
A
1
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
0 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
6
090
8
091
091
4
091
6
091
8
092
(ms)
A
o o
Figure 8 vo voltage waveform obtained with the piece-wise linear model and a Micro-Cap simulation
10 Mathematical Problems in Engineering
012345
0
2
4
6
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
minus6
minus4
minus2
minus4
minus3
minus1
minus2
A A
10 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
(ms)
isum and SAW-iREF isum and SAW-iREF
Figure 9 Waveforms of 119894sum and VSAW minus ViREF obtained with the piece-wise linear model and a Micro-Cap simulation
Mag
nitu
de (d
B)
0
Phas
e (de
g)
101 102 103 104
Frequency (rads)
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus225
minus180
minus135
minus90
minus45
Bode Diagram (os)
Figure 10 Bode plot of Hv(z)
05
10152025303540
Mag
nitu
de (d
B)Ph
ase (
deg)
101 102 103 104minus240minus195minus150minus105minus80minus15
Frequency (rads)
Bode Diagram (oiREF)
Figure 11 Bode plot of 119879119874119871(z)
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
T
T
T
T
T
T
T
T
T
T
T
T
T
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
T2
120575T2
Io
Io
Io
NIo
NIoisum
AB
BN
AN
TOL
MI
MII
MII
IM
IV MV
MV
I
s
s
s
s
NsDD
1 iD4
iD
iD2 iD3
gsTA1
g
sTA2
gsTB2
g
sTB1
t
t
t
t
t
t
t
t
t
t
t
t
t
iL119878
iL119891
Figure 2 Ideal waveforms of the converter
and 1198941198632
and 1198941198633
fall to zero whereas when V119860119861
changes fromzero to minus119881in 119894119863
2
and 1198941198633
rise from zero to the output currentlevel and 119894
1198631
and 1198941198634
fall to zero During the 119879119874119871
period agradual current transfer is effectuated from one diode pair tothe other in such a way that 119894
119871119891
continues the slight currentslope which feeds the load The steady state value of 119879
119874119871may
be calculated assuming that the rate of change of the currentreversal of 119894
119860119861 119889119894119860119861
119889119905 only depends on the supply voltage
and the amplitude of the DC output filter current IO [3]which may be expressed as
119879119874119871
=2119873119871119878119868119874
119881119904
(1)
22 Piece-Wise Analysis of the ZVTConverter FromFigure 2119894119871119878
presents six different behaviour intervals which may betermed operating modes I to VI
For each mode of operation a different circuit configu-ration may be obtained which is shown in Figure 3 Theseequivalent circuits may be described using the state-spaceequation (2) and the state-space output expression (3)
= 119860119899119883 + 119861
119899119880 (2)
119884 = 119865119899119883 + 119866
119899119880 (3)
where 119883 = [119894119871119878
119894119871119891
V119900]119879
is the state vector 119880 = [119881119904 119868119885]119879 is
the input vector 119868119885is the output current disturbance 119884 =
[119894sec 1198941198631
1198941198632
119894sum]119879 is the output vector 119894sum is the supply
current and 119860119899 119861119899 119865119899 and 119866
119899are the state matrixes of the
six operating modes being 119899 = 1 2 6Mode I is formed when 119879
1198601
and 1198791198612
are in the on state1198791198602
and 1198791198611
are in the off state and 1198631to 1198634are conducting
due to the overlap rectifier phenomenaThe equivalent circuitof Mode I is shown in Figure 3(a) and the equations thatdescribe this mode are shown in (2) and (3) with 119899 = 1 Thematrixes A1 B1 F1 and G1 are listed in Table 5
In Mode II the state of the inverter switches is exactlyas that of Mode I but 119863
1and 119863
4are conducting and 119863
2
and 1198633are off The equivalent circuit of Mode II is shown in
Figure 3(b) and again (2) and (3) describe Mode II but with119899 = 2 The matrixes 119860
2 1198612 1198652 and 119866
2are shown in Table 5
In Mode III 1198791198601
and 1198791198611
are in the on state 1198791198602
and 1198791198612
are in the off state 1198631and 119863
4are conducting and 119863
2and
1198633are off The equivalent circuit of Mode III is shown in
Figure 3(c) being (2) and (3) with 119899 = 3 the mathematicalmodel of this mode Again the matrixes 119860
3 1198613 1198653 and 119866
3
are shown in Table 5Mode IV is a mirror of Mode I but with 119879
1198601
and 1198791198612
inthe off state and 119879
1198602
and 1198791198611
in the on state whilst Modes Vand VI are mirrors of Modes II and III respectively since thestate of the switches and diodes is complementary to that ofModes II and III Again (2) and (3) describeModes IV V andVI but with 119899 = 4 5 and 6 respectively The correspondingmatrixes to these operating modes are shown in Table 5
23 Current Control Loop Description The circuit shownin Figure 4 is a DC-DC ZVT converter with peak currentcontrol loop which has a current transducer with gain119877
119904that
senses 119894sum one SR flip-flop and two D flip-flops a clock sig-nal VCLK a sawtooth generator VSAW and the reference cur-rent level ViREF which is provided by an outer voltage loop
The operation of the circuit of Figure 4 may be explainedwith the state voltage and current waveforms of Figure 5Thefirst waveform shown in Figure 5 is the clock signal of systemVCLK The second waveform is 119894sum plotted together with the
4 Mathematical Problems in Engineering
Cf R
isum
ic Io
C IZ
Lf
D1 D2
D3 D4
iAB
secprims
isec
AB o
iD2iD1
iD3iD4
DD
TA1
TB2
iL119878L119878
iL119891
L119891
LS
(a)
isum
isec
ic Io
C Cf R IZ
Lf
D1
D4
iAB
secprims AB o
iD1
iD4
DD
TA1
TB2
iL119878
L119878
iL119891
L119891
LS
(b)
Cf R
isumic Io
C IZ
Lf
D1
D4
iAB
isec
secprims oAB
iD1
iD4
DD
TA1TB1 iL119878
L119878
iL119891
L119891
LS
(c)
isec
isum
C Cf R IZ
ic Io
Lf
D1 D2
D4D3
iAB
secprims AB o
iD2iD1
iD3iD4
DD
TA2
TB1 iL119878
L119878
iL119891
L119891
LS
(d)
RCf
ic Io
IZ
Lf
D3
D2
isum
C iAB
sec
sec
prims ABoiD2iD3
DD
TA1
TA2
TB1 iL119878
L119878
iL119891
L119891
LS
(e)
isec
isum
C Cf R IZ
ic
Lf
D2
D3
IoiAB
secprims AB oiD2iD3
DDTA2
TB2
iL119878
L119878
iL119891
L119891
LS
(f)
Figure 3 Equivalent circuits formed from the operation ZVT DC-DC converter (a) Mode I (b) Mode II (c) Mode III (d) Mode IV (e)Mode V and (f) Mode VI
deference of ViREF with VSAW where VSAW is a negative slopesynchronized with VCLK while ViREF is the current referencethat regulates the peak level of 119894sum VCOMP is the state of theoutput comparator which is the thirdwaveformof this figureThe fourth and fifth waveforms are the SR flip-flop outputs119876119860and 119876
119861 with 119876
119860set to the on state by VCLK and to the
off state when VCOMP switches to the on state The fifth andsixthwaveforms are the outputs of the first flip-flop VgsTA1
andVgsTA2
which are controlled by the rising edge of119876119860 whereas
VgsTB1and VgsTB2
the outputs of the second 119863 flip-flop whichare the last waves of this figure are controlled by the risingedge of 119876
119861
24 Numerical Estimation of the 119879119874119871
and 120575 Periods 119879119874119871
defines the duration of Modes I and IV and may be numer-ically estimated by determining the instant when either 119894
1198632
or 1198941198631
reaches zero 119879119874119871
may also be calculated using theNewton-Raphson method [4] This numerical method maybe implemented using
119879119874119871 (119899 + 1) = 119879
119874119871 (119899) minus119891 (119879119874119871
)
1198911015840 (119879119874119871
) (4)
where
119891 (119879119874119871
) = 1198651198631
(1205931(119879119874119871
)119883 (1199051))
+ (1198651198631
119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611+ 1198661198631
)119880
1198911015840(119879119874119871
) = 1198651198631
11986011198901198601119879119874119871119883(119905
1)
+ 1198651198631
[
[
infin
sum
119895
(119895 + 1)119860119895
1119879119895
119874119871
(119895 + 1)
]
]
1198611119880
(5)
Equations of (5) use the output equation that includes1198941198631
which determine the duration of Mode I whereas theduration of Mode IV is determined by 119894
1198632
such that (6) maybe rewritten as follows
119891 (119879119874119871
) = 1198651198632
(1205934(119879119874119871
)119883(119879
2))
+ (1198651198632
119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614+ 1198661198634
)
1198911015840(119879119874119871
) = 1198651198632
11986041198901198604119879119874119871119883(
119879
2)
+ 1198651198632
[
[
infin
sum
119895
(119895 + 1)119860119895
4119879119895
119874119871
(119895 + 1)
]
]
1198614119880
(6)
120575 defines the duration of Modes II and V and maybe numerically estimated by determining the instant whenthe equation ViREF minus VSAW is equal to 119877
119904119894sum 120575 may also
Mathematical Problems in Engineering 5
sumsum
Sawtooth
Peak current comparatorwith latch (RS Flip-Flop)
D Flip-Flopfor inverter leg A
D Flip-Flopfor inverter leg B
generator
Rsisum
Ls Cf R
isum
ic Io
C
isec
Lf
D1 D2
D3D4
iAB
H(s)+ +
+
minus minusminus
R FF
SQD
Q998400
Q998400
CLK
CLK
Q
Q
D
Q998400
CLK
QA
QB
middot middot middot
middot middot middot
seco
o
iREF
SAW
CLK
primABs
Rsisum
COMP
oREF
iD2iD1
iD4iD3
DD
TA2
TA1TB1
TB2
iL119878
L119878
iL119891
L119891
g119904T1198601
g119904T1198601
g119904T1198602
g119904T1198602
g119904T1198611
g119904T1198611
g119904T1198612
g119904T1198612
Figure 4 DC-DC ZVT converter with current control loop
be calculated using the Newton-Raphson method [4] Thisnumerical method may be implemented using
120575 (119899 + 1) = 120575 (119899) minus1198942sum (120575)
1198941015840
2sum (120575) (7)
where
i2sum (120575) = minus(
1
(119877119904)) [(VV minus V
119901) (
1205751198792
1198792) + V119901]
1198911015840(120575)
= (VV minus V119901) 120575
minus 1198652sum (119860
2
119879
2)
times (1205932(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
))
+ 1198652sum
119879
2
[
[
infin
sum
119895
(119895 + 1)119860119895
2((1205751198792) minus 119879
119874119871)119895
(119895 + 1)
]
]
119880
(8)
Equations of (8) use the output equation that includes1198942sum and the expression ViREF minus VSAW = 119877
119904119894sum which
determines the duration of Mode II whereas the duration of
Mode V is determined by 1198945sum and the expression ViREF minus
VSAW = 119877119904119894sum such that (9) may be rewritten as follows
1198945sum(120575) = minus
(VV minus V119901) ((1205751198792) (1198792)) + V
119901
(119877119904)
1198941015840
5sum (120575) = (VV minus V119901) 120575
minus 1198655sum (119860
5
119879
2)
times (1205935(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
))
+ 1198655sum
119879
2
[
[
infin
sum
119895
(119895 + 1)119860119895
5((1205751198792) minus 119879
119874119871)119895
(119895 + 1)
]
]
119880
(9)
3 Modeling of the ZVT Converter withCurrent Control Loop
31 Piece-Wise Linear Model Equation (2) may be used todevelop a piece-wise linear model of the converter of Figure 1
6 Mathematical Problems in Engineering
isum
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
CLK
NIo
120575T
2
TOL
QA
COMP
QB
120575T
2
iREF
g119904T1198601
g119904T1198602
g119904T1198611
g119904T1198612
Figure 5 Ideal waveform DC-DC ZVT converter with currentcontrol loop
throughout all the modes of operation The solution of (2)may be expressed as
119883(119905119899minus1
+ 119905119899) = 120593119899(119905119899)119883 (1199051) + 119860minus1
119899[120593119899(119905119899) minus 119868119889] 119861119899119880
(10)
where
120593119899(119905119899) = 119890119860119899119905119899 (11)
which defines in the first term of (10) the natural responseof the system along the period of time 119905
119899minus 119905119899minus1
with theinitial condition 119883(119905
119899minus1) at Mode n The second term of
(10) is the steady state response which is obtained by usingthe convolution integral Therefore using (10) for the timeinterval of 119905
1le 119905 lt 119905
1+ 119879119874119871 together with the matrixes of
Mode I the solution of the state vector forMode I is obtainedas follows
119883(1199051+ 119879119874119871
) = 1205931(119879119874119871
)119883 (1199051) + 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611119880
(12)
In a similar way the solutions for Modes II to VI at thetime intervals 119905
1+ 119879119874119871
le 119905 lt 1199051+ 1205751198792 119905
1+ 1205751198792 le 119905 lt
1199051+ 1198792 119905
1+ 1198792 le 119905 lt 119905
1+ 1198792 + 119879
119874119871 1199051+ 1198792 + 119879
119874119871le 119905 lt
1199051+ (1 + 120575)1198792 and 119905
1+ (1 + 120575)1198792 le 119905 lt 119905
1+ 119879 become
119883(1199051+
120575119879
2) = 120593
2(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
)
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612119880
(13)
119883(1199051+
119879
2) = 120593
3(119879
2minus
120575119879
2)119883(119905
1+
120575119879
2)
+ 119860minus1
3[1205933(119879
2minus
120575119879
2) minus 119868119889]1198613119880
(14)
119883(1199051+
119879
2+ 119879119874119871
) = 1205934(119879119874119871
)119883(1199051+
119879
2)
+ 119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614119880
(15)
119883(1199051+
(1 + 120575) 119879
2) = 120593
5(120575119879
2minus 119879119874119871
)119883(1199051+
119879
2+ 119879119874119871
)
+ 119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615119880
(16)
119883(1199051+ 119879) = 120593
6((1 minus 120575) 119879
2)119883(119905
1+
(1 + 120575) 119879
2)
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(17)
32 Large-Signal Model The large-signal model of the ZVTconverter may be obtained by substituting (12) in (13) (13) in(14) (14) in (15) (15) in (16) and (16) in (17) in such a waythat a single expression is obtained as shown in
119883(1199051+ 119879)
= 1205936((1 minus 120575) 119879
2)1205935(120575119879
2minus 119879119874119871
)1205934(119879119874119871
) 1205933
times ((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
Mathematical Problems in Engineering 7
+ [1205936((1 minus 120575) 119879
2)
times [1205935(120575119879
2minus 119879119874119871
)
times [1205934(119879119874119871
) [1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)
times 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
)
minus119868119889]1198612]
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]
+119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614]
+119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615]
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(18)
33 Half-Cycle Model The waveform 119894119871119878
of Figure 2 showsthat operation modes IV V and VI are mirrors of Modes III and III respectively therefore the first three operationmodes are sufficient to describe the function of the converterA particular matrix titled as W may relate the modes I IIand III with the modes IV V and VI which satisfies thecondition119882119882 = 119868
119889 the identity matrix Therefore the half-
cycle model of the ZVT converter may be obtained replacingthe termsA4 A5 A6 B4 B5 and B6 by expressionsWA1WA2WA3 WB1 WB2 andWB3 respectively in (18) such that thehalf-cycle model is
119883(1199051+
119879
2)
= 1205933((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
+ [1205933((1 minus 120575) 119879
2)12059321205933((1 minus 120575) 119879
2)1205932
times (120575119879
2minus 119879119874119871
)119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 1205933((1 minus 120575) 119879
2)119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]119880
+ 119882(1205933((1 minus 120575) 119879
2)(1205932(120575119879
2minus 119879119874119871
)
times 1198821205931(119879119874119871
)119883 (1199051)
+ 1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)119860minus1
1
times [1205931(119879119874119871
) minus 119868119889] 1198611)
+ 1205933((1 minus 120575) 119879
2)119860minus1
2
times [1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+ 119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198821198613)119880 (119905
1)
(19)
The previous equation may be rewritten as119883(1199051+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) for practical purposes
34 Sample-Data Small-Signal Linear Model of the Converterin Open-Loop Conditions The equation 119883(119905
1+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) may be used as a half-cycle discrete
model of the ZVT converter which may be written as
1198831015840
119870+1= 119860MC119883119870 + 119861MC119880119870 (20)
where1198831015840119870+1
= 119883(1199051+ 1198792)119883
119870= 119883(119905
1) and 119880
119870= 119880(119905
1)
A sample-data small-signal model may be obtained byusing the Taylor series (21) and using small-signal perturba-tions as 120575xK 120575UK 120575119879119874119871
119896
and 120575120575K One has
f (119909) = f (1199090) +J (119909 minus 119909
0) (21)
Using this equation the sample-data small-signal linearmodel becomes
120575119909119896+1
asymp120597119909119896+1
120597119909119896
120575119909119896+
120597119909119896+1
120597119880119896
120575119880119896
+120597119909119896+1
120597120575120575120575119896+
120597119909119896+1
120597119879119874119871
120575119879119874119871119896
(22)
The solution of the partial derivatives is 120597119909119896+1
120597119909119896
=
119860MC 120597119909119896+1120597119880119896 = 119861MC 120597119909119896+1120597120575 = 119862120575 and 120597119909
119896+1120597119879119874119871
=
119863119879119874119871
120575119879119874119871119896
may be determined utilizing the restrictionequation of119879
119874119871 whereas 120575120575K is obtained using the restriction
equation of 120575
35 Restriction Equations of the Control Loop The restrictionequations for 119879
119874119871may be obtained analyzing the waveforms
119894119871119878 11989411986311198633
and 119894sum which are shown in Figure 2 during theMode I The slope of 119894sum during Mode I is named 119892
1and is
determined by the rate of change of 119894119871119878
that is 1198921= 119881119904119896
119871119878
8 Mathematical Problems in Engineering
while the slopes of 11989411986311198633
during the Mode I are named 1198923
which are contrary and of lower amplitude than 1198921 that
is g3
= minusg12N The restriction equation of 119879
119874119871may be
determined by integrating the waveforms 11989411986311198633
duringModeI
119894119871119878119896
2119873+
119894119871119891119896
2+
119881119904119896
119871119878
119879119874119871119896
= 0 (23)
The previous equation is not linear therefore it isnecessary to use theTaylor series to obtain a linearizedmodel
120575119879119874119871119896
=2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
(24)
The restriction equation of 120575 may be obtained analyzingthe waveforms 119894
119871119878
and 119894sum with ViREF minus VSAW during Mode II(Figures 2 and 5) VSAW is determined by 119872(120575
1198961198792) and the
slope of 119894sum duringMode II named 1198922 and may be obtained
by integrating again 119894sumwhen 119877119904119894sum = ViREF minus VSAW
119877119904(119894119871119878119896
+ 1198921119879119874119871119896
+ 1198922(120575119896119879
2minus 119879119874119871119896
)) = 119881119894REF minus 119872
120575119896119879
2
(25)
The Taylor series is used to linearize the previous equa-tion and therefore the restriction equation of 120575 becomes
120575120575119896=
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
(26)
where Δ120575= minus(2119879)(1198721198792119877
119878+ 1(119871
119878+ (1119873
2)119871119891))(V119904CD
minus
(1119873)V0CD
) 1198861205751
= 120597119894119871119878119896
120597119894119871119878119896
1198861205752
= 120597119894119871119878119896
120597119894119871119891119896
1198861205753
=
120597119894119871119878119896
120597119881119878119896
1198861205754
= 120597119894119871119878119896
120597119881iREF119896
and 1198861205755
= 120597119894119871119878119896
120597119881119900119896
36 Sample-Data Small-Signal Linear Model of the Converterin Closed-Loop Conditions The sample-data small-signallinear model may be obtained by substituting 120575
119879119874119871119896
and 120575120575K(24) and (26) respectively in (22)
120575119909119896+1
= 119860MC[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
+ 119861MC [120575119881119904119896
120575119868119885119896
]
+ 119862120575(
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
)
+ 119863119879119874119871
(2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
)
(27)
Table 1 Operating parameters
Supply voltage 200V plusmn 20Output voltage 48VMaximum power 250WMinimum power 50WOutput voltage ripple 25mVOutput current ripple 300mAMaximum phase 50Switching frequency 50 kHz
such that the small-signal model becomes
120575119909119896+1
= 119860119888119897120575119909119896+ 120596119888119897120575119908119888119897119896
(28)
where 120575119909119896+1
= [1205751198941015840
119871119878119896
1205751198941015840
119871119891119896
1205751198811015840
119904119896
]119879
120575119909119896
=
[120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896]1015840 and 120575119908
119888119897119896
= [120575119881119904119896
120575119881iREF119896
120575119868119885119896]119879
Equation (28) may be solved by using the 119885 transformsuch that 120575119909
119870becomes
120575119909119896= (119868119889minus 119860119888119897)minus1119885120596119888119897120575119908119888119897119896
(29)
37 Transfer Function To verify the dynamic characteristicsof the converter is necessary to analyze the transfer functionsthat relate V
119900with 119881
119904 ViREF and 119868
119911 which are the throughput
input-to-output DC voltage transfer function 119867V(119911) =
V119900(119911)V119904(119911) the control-to-output transfer function119879
119874119871(119911) =
V119900(119911)ViREF(119911) and the output impedance transfer function
119885119900(119911) = V
119900(119911)119868119911(119911) respectively
The magnitude and phase of each transfer function maybe obtained using a Bode diagram whereas the root locustechnique may be employed to describe the behaviour of thepoles and zeros of (30) One has
119867V (119911) =119911 + 1198861
1199112 + 1198871119911 + 1198881
119879119874119871 (119911) =
11988621199112+ 1198872119911 + 1198882
1199112 + 1198871119911 + 1198881
119885119900 (119911) =
11988631199112+ 1198873119911 minus 1198883
1199112 + 1198871119911 + 1198881
(30)
4 Verification of the Proposed Model
41 Prototype Operating Parameters A 250W ZVT DC-DC prototype converter was designed under the analysisdescribed in [7] to verify the large-signal model of (14)Table 1 shows the operating parameters of the converter
The steady-state output voltage 119881119874 may be calculated as
the average of the rectified voltage V119863119863
such that at full load
119881119900= 120575max119873119881in minus
41198732119868119900(max)119871119878
119879 (31)
Taking the assumption shown in (31) the converter com-ponent values must comply with the zero-voltage switching
Mathematical Problems in Engineering 9
012345
(ms)(ms)
Piece-wise analysisPiece-wise model
Simulation circuitPiece-wise analysisPiece-wise model
Simulation circuit
01234
0 01 02 03 04 05 06 07 08 09 01minus5
minus4
minus3
minus2
minus1
minus4
minus3
minus2
minus1A A
09
0902
0904
0906
0908
091
0912
0914
0916
0918
092
iL119878iL119878
Figure 6 119894119871119878
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
1
0
1
2
3
4
5
6
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
09
902
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
0
1
2
3
4
5
6
7
(ms)
A
0 01 02 03 04 05 06 07 08 09minus1
(ms)
A
iL119891iL119891
Figure 7 119894119871119891
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
0
10
20
30
40
50
0
10
20
30
40
50
60
minus10
A
1
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
0 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
6
090
8
091
091
4
091
6
091
8
092
(ms)
A
o o
Figure 8 vo voltage waveform obtained with the piece-wise linear model and a Micro-Cap simulation
10 Mathematical Problems in Engineering
012345
0
2
4
6
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
minus6
minus4
minus2
minus4
minus3
minus1
minus2
A A
10 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
(ms)
isum and SAW-iREF isum and SAW-iREF
Figure 9 Waveforms of 119894sum and VSAW minus ViREF obtained with the piece-wise linear model and a Micro-Cap simulation
Mag
nitu
de (d
B)
0
Phas
e (de
g)
101 102 103 104
Frequency (rads)
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus225
minus180
minus135
minus90
minus45
Bode Diagram (os)
Figure 10 Bode plot of Hv(z)
05
10152025303540
Mag
nitu
de (d
B)Ph
ase (
deg)
101 102 103 104minus240minus195minus150minus105minus80minus15
Frequency (rads)
Bode Diagram (oiREF)
Figure 11 Bode plot of 119879119874119871(z)
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Cf R
isum
ic Io
C IZ
Lf
D1 D2
D3 D4
iAB
secprims
isec
AB o
iD2iD1
iD3iD4
DD
TA1
TB2
iL119878L119878
iL119891
L119891
LS
(a)
isum
isec
ic Io
C Cf R IZ
Lf
D1
D4
iAB
secprims AB o
iD1
iD4
DD
TA1
TB2
iL119878
L119878
iL119891
L119891
LS
(b)
Cf R
isumic Io
C IZ
Lf
D1
D4
iAB
isec
secprims oAB
iD1
iD4
DD
TA1TB1 iL119878
L119878
iL119891
L119891
LS
(c)
isec
isum
C Cf R IZ
ic Io
Lf
D1 D2
D4D3
iAB
secprims AB o
iD2iD1
iD3iD4
DD
TA2
TB1 iL119878
L119878
iL119891
L119891
LS
(d)
RCf
ic Io
IZ
Lf
D3
D2
isum
C iAB
sec
sec
prims ABoiD2iD3
DD
TA1
TA2
TB1 iL119878
L119878
iL119891
L119891
LS
(e)
isec
isum
C Cf R IZ
ic
Lf
D2
D3
IoiAB
secprims AB oiD2iD3
DDTA2
TB2
iL119878
L119878
iL119891
L119891
LS
(f)
Figure 3 Equivalent circuits formed from the operation ZVT DC-DC converter (a) Mode I (b) Mode II (c) Mode III (d) Mode IV (e)Mode V and (f) Mode VI
deference of ViREF with VSAW where VSAW is a negative slopesynchronized with VCLK while ViREF is the current referencethat regulates the peak level of 119894sum VCOMP is the state of theoutput comparator which is the thirdwaveformof this figureThe fourth and fifth waveforms are the SR flip-flop outputs119876119860and 119876
119861 with 119876
119860set to the on state by VCLK and to the
off state when VCOMP switches to the on state The fifth andsixthwaveforms are the outputs of the first flip-flop VgsTA1
andVgsTA2
which are controlled by the rising edge of119876119860 whereas
VgsTB1and VgsTB2
the outputs of the second 119863 flip-flop whichare the last waves of this figure are controlled by the risingedge of 119876
119861
24 Numerical Estimation of the 119879119874119871
and 120575 Periods 119879119874119871
defines the duration of Modes I and IV and may be numer-ically estimated by determining the instant when either 119894
1198632
or 1198941198631
reaches zero 119879119874119871
may also be calculated using theNewton-Raphson method [4] This numerical method maybe implemented using
119879119874119871 (119899 + 1) = 119879
119874119871 (119899) minus119891 (119879119874119871
)
1198911015840 (119879119874119871
) (4)
where
119891 (119879119874119871
) = 1198651198631
(1205931(119879119874119871
)119883 (1199051))
+ (1198651198631
119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611+ 1198661198631
)119880
1198911015840(119879119874119871
) = 1198651198631
11986011198901198601119879119874119871119883(119905
1)
+ 1198651198631
[
[
infin
sum
119895
(119895 + 1)119860119895
1119879119895
119874119871
(119895 + 1)
]
]
1198611119880
(5)
Equations of (5) use the output equation that includes1198941198631
which determine the duration of Mode I whereas theduration of Mode IV is determined by 119894
1198632
such that (6) maybe rewritten as follows
119891 (119879119874119871
) = 1198651198632
(1205934(119879119874119871
)119883(119879
2))
+ (1198651198632
119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614+ 1198661198634
)
1198911015840(119879119874119871
) = 1198651198632
11986041198901198604119879119874119871119883(
119879
2)
+ 1198651198632
[
[
infin
sum
119895
(119895 + 1)119860119895
4119879119895
119874119871
(119895 + 1)
]
]
1198614119880
(6)
120575 defines the duration of Modes II and V and maybe numerically estimated by determining the instant whenthe equation ViREF minus VSAW is equal to 119877
119904119894sum 120575 may also
Mathematical Problems in Engineering 5
sumsum
Sawtooth
Peak current comparatorwith latch (RS Flip-Flop)
D Flip-Flopfor inverter leg A
D Flip-Flopfor inverter leg B
generator
Rsisum
Ls Cf R
isum
ic Io
C
isec
Lf
D1 D2
D3D4
iAB
H(s)+ +
+
minus minusminus
R FF
SQD
Q998400
Q998400
CLK
CLK
Q
Q
D
Q998400
CLK
QA
QB
middot middot middot
middot middot middot
seco
o
iREF
SAW
CLK
primABs
Rsisum
COMP
oREF
iD2iD1
iD4iD3
DD
TA2
TA1TB1
TB2
iL119878
L119878
iL119891
L119891
g119904T1198601
g119904T1198601
g119904T1198602
g119904T1198602
g119904T1198611
g119904T1198611
g119904T1198612
g119904T1198612
Figure 4 DC-DC ZVT converter with current control loop
be calculated using the Newton-Raphson method [4] Thisnumerical method may be implemented using
120575 (119899 + 1) = 120575 (119899) minus1198942sum (120575)
1198941015840
2sum (120575) (7)
where
i2sum (120575) = minus(
1
(119877119904)) [(VV minus V
119901) (
1205751198792
1198792) + V119901]
1198911015840(120575)
= (VV minus V119901) 120575
minus 1198652sum (119860
2
119879
2)
times (1205932(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
))
+ 1198652sum
119879
2
[
[
infin
sum
119895
(119895 + 1)119860119895
2((1205751198792) minus 119879
119874119871)119895
(119895 + 1)
]
]
119880
(8)
Equations of (8) use the output equation that includes1198942sum and the expression ViREF minus VSAW = 119877
119904119894sum which
determines the duration of Mode II whereas the duration of
Mode V is determined by 1198945sum and the expression ViREF minus
VSAW = 119877119904119894sum such that (9) may be rewritten as follows
1198945sum(120575) = minus
(VV minus V119901) ((1205751198792) (1198792)) + V
119901
(119877119904)
1198941015840
5sum (120575) = (VV minus V119901) 120575
minus 1198655sum (119860
5
119879
2)
times (1205935(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
))
+ 1198655sum
119879
2
[
[
infin
sum
119895
(119895 + 1)119860119895
5((1205751198792) minus 119879
119874119871)119895
(119895 + 1)
]
]
119880
(9)
3 Modeling of the ZVT Converter withCurrent Control Loop
31 Piece-Wise Linear Model Equation (2) may be used todevelop a piece-wise linear model of the converter of Figure 1
6 Mathematical Problems in Engineering
isum
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
CLK
NIo
120575T
2
TOL
QA
COMP
QB
120575T
2
iREF
g119904T1198601
g119904T1198602
g119904T1198611
g119904T1198612
Figure 5 Ideal waveform DC-DC ZVT converter with currentcontrol loop
throughout all the modes of operation The solution of (2)may be expressed as
119883(119905119899minus1
+ 119905119899) = 120593119899(119905119899)119883 (1199051) + 119860minus1
119899[120593119899(119905119899) minus 119868119889] 119861119899119880
(10)
where
120593119899(119905119899) = 119890119860119899119905119899 (11)
which defines in the first term of (10) the natural responseof the system along the period of time 119905
119899minus 119905119899minus1
with theinitial condition 119883(119905
119899minus1) at Mode n The second term of
(10) is the steady state response which is obtained by usingthe convolution integral Therefore using (10) for the timeinterval of 119905
1le 119905 lt 119905
1+ 119879119874119871 together with the matrixes of
Mode I the solution of the state vector forMode I is obtainedas follows
119883(1199051+ 119879119874119871
) = 1205931(119879119874119871
)119883 (1199051) + 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611119880
(12)
In a similar way the solutions for Modes II to VI at thetime intervals 119905
1+ 119879119874119871
le 119905 lt 1199051+ 1205751198792 119905
1+ 1205751198792 le 119905 lt
1199051+ 1198792 119905
1+ 1198792 le 119905 lt 119905
1+ 1198792 + 119879
119874119871 1199051+ 1198792 + 119879
119874119871le 119905 lt
1199051+ (1 + 120575)1198792 and 119905
1+ (1 + 120575)1198792 le 119905 lt 119905
1+ 119879 become
119883(1199051+
120575119879
2) = 120593
2(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
)
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612119880
(13)
119883(1199051+
119879
2) = 120593
3(119879
2minus
120575119879
2)119883(119905
1+
120575119879
2)
+ 119860minus1
3[1205933(119879
2minus
120575119879
2) minus 119868119889]1198613119880
(14)
119883(1199051+
119879
2+ 119879119874119871
) = 1205934(119879119874119871
)119883(1199051+
119879
2)
+ 119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614119880
(15)
119883(1199051+
(1 + 120575) 119879
2) = 120593
5(120575119879
2minus 119879119874119871
)119883(1199051+
119879
2+ 119879119874119871
)
+ 119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615119880
(16)
119883(1199051+ 119879) = 120593
6((1 minus 120575) 119879
2)119883(119905
1+
(1 + 120575) 119879
2)
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(17)
32 Large-Signal Model The large-signal model of the ZVTconverter may be obtained by substituting (12) in (13) (13) in(14) (14) in (15) (15) in (16) and (16) in (17) in such a waythat a single expression is obtained as shown in
119883(1199051+ 119879)
= 1205936((1 minus 120575) 119879
2)1205935(120575119879
2minus 119879119874119871
)1205934(119879119874119871
) 1205933
times ((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
Mathematical Problems in Engineering 7
+ [1205936((1 minus 120575) 119879
2)
times [1205935(120575119879
2minus 119879119874119871
)
times [1205934(119879119874119871
) [1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)
times 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
)
minus119868119889]1198612]
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]
+119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614]
+119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615]
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(18)
33 Half-Cycle Model The waveform 119894119871119878
of Figure 2 showsthat operation modes IV V and VI are mirrors of Modes III and III respectively therefore the first three operationmodes are sufficient to describe the function of the converterA particular matrix titled as W may relate the modes I IIand III with the modes IV V and VI which satisfies thecondition119882119882 = 119868
119889 the identity matrix Therefore the half-
cycle model of the ZVT converter may be obtained replacingthe termsA4 A5 A6 B4 B5 and B6 by expressionsWA1WA2WA3 WB1 WB2 andWB3 respectively in (18) such that thehalf-cycle model is
119883(1199051+
119879
2)
= 1205933((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
+ [1205933((1 minus 120575) 119879
2)12059321205933((1 minus 120575) 119879
2)1205932
times (120575119879
2minus 119879119874119871
)119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 1205933((1 minus 120575) 119879
2)119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]119880
+ 119882(1205933((1 minus 120575) 119879
2)(1205932(120575119879
2minus 119879119874119871
)
times 1198821205931(119879119874119871
)119883 (1199051)
+ 1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)119860minus1
1
times [1205931(119879119874119871
) minus 119868119889] 1198611)
+ 1205933((1 minus 120575) 119879
2)119860minus1
2
times [1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+ 119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198821198613)119880 (119905
1)
(19)
The previous equation may be rewritten as119883(1199051+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) for practical purposes
34 Sample-Data Small-Signal Linear Model of the Converterin Open-Loop Conditions The equation 119883(119905
1+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) may be used as a half-cycle discrete
model of the ZVT converter which may be written as
1198831015840
119870+1= 119860MC119883119870 + 119861MC119880119870 (20)
where1198831015840119870+1
= 119883(1199051+ 1198792)119883
119870= 119883(119905
1) and 119880
119870= 119880(119905
1)
A sample-data small-signal model may be obtained byusing the Taylor series (21) and using small-signal perturba-tions as 120575xK 120575UK 120575119879119874119871
119896
and 120575120575K One has
f (119909) = f (1199090) +J (119909 minus 119909
0) (21)
Using this equation the sample-data small-signal linearmodel becomes
120575119909119896+1
asymp120597119909119896+1
120597119909119896
120575119909119896+
120597119909119896+1
120597119880119896
120575119880119896
+120597119909119896+1
120597120575120575120575119896+
120597119909119896+1
120597119879119874119871
120575119879119874119871119896
(22)
The solution of the partial derivatives is 120597119909119896+1
120597119909119896
=
119860MC 120597119909119896+1120597119880119896 = 119861MC 120597119909119896+1120597120575 = 119862120575 and 120597119909
119896+1120597119879119874119871
=
119863119879119874119871
120575119879119874119871119896
may be determined utilizing the restrictionequation of119879
119874119871 whereas 120575120575K is obtained using the restriction
equation of 120575
35 Restriction Equations of the Control Loop The restrictionequations for 119879
119874119871may be obtained analyzing the waveforms
119894119871119878 11989411986311198633
and 119894sum which are shown in Figure 2 during theMode I The slope of 119894sum during Mode I is named 119892
1and is
determined by the rate of change of 119894119871119878
that is 1198921= 119881119904119896
119871119878
8 Mathematical Problems in Engineering
while the slopes of 11989411986311198633
during the Mode I are named 1198923
which are contrary and of lower amplitude than 1198921 that
is g3
= minusg12N The restriction equation of 119879
119874119871may be
determined by integrating the waveforms 11989411986311198633
duringModeI
119894119871119878119896
2119873+
119894119871119891119896
2+
119881119904119896
119871119878
119879119874119871119896
= 0 (23)
The previous equation is not linear therefore it isnecessary to use theTaylor series to obtain a linearizedmodel
120575119879119874119871119896
=2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
(24)
The restriction equation of 120575 may be obtained analyzingthe waveforms 119894
119871119878
and 119894sum with ViREF minus VSAW during Mode II(Figures 2 and 5) VSAW is determined by 119872(120575
1198961198792) and the
slope of 119894sum duringMode II named 1198922 and may be obtained
by integrating again 119894sumwhen 119877119904119894sum = ViREF minus VSAW
119877119904(119894119871119878119896
+ 1198921119879119874119871119896
+ 1198922(120575119896119879
2minus 119879119874119871119896
)) = 119881119894REF minus 119872
120575119896119879
2
(25)
The Taylor series is used to linearize the previous equa-tion and therefore the restriction equation of 120575 becomes
120575120575119896=
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
(26)
where Δ120575= minus(2119879)(1198721198792119877
119878+ 1(119871
119878+ (1119873
2)119871119891))(V119904CD
minus
(1119873)V0CD
) 1198861205751
= 120597119894119871119878119896
120597119894119871119878119896
1198861205752
= 120597119894119871119878119896
120597119894119871119891119896
1198861205753
=
120597119894119871119878119896
120597119881119878119896
1198861205754
= 120597119894119871119878119896
120597119881iREF119896
and 1198861205755
= 120597119894119871119878119896
120597119881119900119896
36 Sample-Data Small-Signal Linear Model of the Converterin Closed-Loop Conditions The sample-data small-signallinear model may be obtained by substituting 120575
119879119874119871119896
and 120575120575K(24) and (26) respectively in (22)
120575119909119896+1
= 119860MC[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
+ 119861MC [120575119881119904119896
120575119868119885119896
]
+ 119862120575(
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
)
+ 119863119879119874119871
(2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
)
(27)
Table 1 Operating parameters
Supply voltage 200V plusmn 20Output voltage 48VMaximum power 250WMinimum power 50WOutput voltage ripple 25mVOutput current ripple 300mAMaximum phase 50Switching frequency 50 kHz
such that the small-signal model becomes
120575119909119896+1
= 119860119888119897120575119909119896+ 120596119888119897120575119908119888119897119896
(28)
where 120575119909119896+1
= [1205751198941015840
119871119878119896
1205751198941015840
119871119891119896
1205751198811015840
119904119896
]119879
120575119909119896
=
[120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896]1015840 and 120575119908
119888119897119896
= [120575119881119904119896
120575119881iREF119896
120575119868119885119896]119879
Equation (28) may be solved by using the 119885 transformsuch that 120575119909
119870becomes
120575119909119896= (119868119889minus 119860119888119897)minus1119885120596119888119897120575119908119888119897119896
(29)
37 Transfer Function To verify the dynamic characteristicsof the converter is necessary to analyze the transfer functionsthat relate V
119900with 119881
119904 ViREF and 119868
119911 which are the throughput
input-to-output DC voltage transfer function 119867V(119911) =
V119900(119911)V119904(119911) the control-to-output transfer function119879
119874119871(119911) =
V119900(119911)ViREF(119911) and the output impedance transfer function
119885119900(119911) = V
119900(119911)119868119911(119911) respectively
The magnitude and phase of each transfer function maybe obtained using a Bode diagram whereas the root locustechnique may be employed to describe the behaviour of thepoles and zeros of (30) One has
119867V (119911) =119911 + 1198861
1199112 + 1198871119911 + 1198881
119879119874119871 (119911) =
11988621199112+ 1198872119911 + 1198882
1199112 + 1198871119911 + 1198881
119885119900 (119911) =
11988631199112+ 1198873119911 minus 1198883
1199112 + 1198871119911 + 1198881
(30)
4 Verification of the Proposed Model
41 Prototype Operating Parameters A 250W ZVT DC-DC prototype converter was designed under the analysisdescribed in [7] to verify the large-signal model of (14)Table 1 shows the operating parameters of the converter
The steady-state output voltage 119881119874 may be calculated as
the average of the rectified voltage V119863119863
such that at full load
119881119900= 120575max119873119881in minus
41198732119868119900(max)119871119878
119879 (31)
Taking the assumption shown in (31) the converter com-ponent values must comply with the zero-voltage switching
Mathematical Problems in Engineering 9
012345
(ms)(ms)
Piece-wise analysisPiece-wise model
Simulation circuitPiece-wise analysisPiece-wise model
Simulation circuit
01234
0 01 02 03 04 05 06 07 08 09 01minus5
minus4
minus3
minus2
minus1
minus4
minus3
minus2
minus1A A
09
0902
0904
0906
0908
091
0912
0914
0916
0918
092
iL119878iL119878
Figure 6 119894119871119878
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
1
0
1
2
3
4
5
6
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
09
902
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
0
1
2
3
4
5
6
7
(ms)
A
0 01 02 03 04 05 06 07 08 09minus1
(ms)
A
iL119891iL119891
Figure 7 119894119871119891
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
0
10
20
30
40
50
0
10
20
30
40
50
60
minus10
A
1
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
0 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
6
090
8
091
091
4
091
6
091
8
092
(ms)
A
o o
Figure 8 vo voltage waveform obtained with the piece-wise linear model and a Micro-Cap simulation
10 Mathematical Problems in Engineering
012345
0
2
4
6
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
minus6
minus4
minus2
minus4
minus3
minus1
minus2
A A
10 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
(ms)
isum and SAW-iREF isum and SAW-iREF
Figure 9 Waveforms of 119894sum and VSAW minus ViREF obtained with the piece-wise linear model and a Micro-Cap simulation
Mag
nitu
de (d
B)
0
Phas
e (de
g)
101 102 103 104
Frequency (rads)
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus225
minus180
minus135
minus90
minus45
Bode Diagram (os)
Figure 10 Bode plot of Hv(z)
05
10152025303540
Mag
nitu
de (d
B)Ph
ase (
deg)
101 102 103 104minus240minus195minus150minus105minus80minus15
Frequency (rads)
Bode Diagram (oiREF)
Figure 11 Bode plot of 119879119874119871(z)
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
sumsum
Sawtooth
Peak current comparatorwith latch (RS Flip-Flop)
D Flip-Flopfor inverter leg A
D Flip-Flopfor inverter leg B
generator
Rsisum
Ls Cf R
isum
ic Io
C
isec
Lf
D1 D2
D3D4
iAB
H(s)+ +
+
minus minusminus
R FF
SQD
Q998400
Q998400
CLK
CLK
Q
Q
D
Q998400
CLK
QA
QB
middot middot middot
middot middot middot
seco
o
iREF
SAW
CLK
primABs
Rsisum
COMP
oREF
iD2iD1
iD4iD3
DD
TA2
TA1TB1
TB2
iL119878
L119878
iL119891
L119891
g119904T1198601
g119904T1198601
g119904T1198602
g119904T1198602
g119904T1198611
g119904T1198611
g119904T1198612
g119904T1198612
Figure 4 DC-DC ZVT converter with current control loop
be calculated using the Newton-Raphson method [4] Thisnumerical method may be implemented using
120575 (119899 + 1) = 120575 (119899) minus1198942sum (120575)
1198941015840
2sum (120575) (7)
where
i2sum (120575) = minus(
1
(119877119904)) [(VV minus V
119901) (
1205751198792
1198792) + V119901]
1198911015840(120575)
= (VV minus V119901) 120575
minus 1198652sum (119860
2
119879
2)
times (1205932(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
))
+ 1198652sum
119879
2
[
[
infin
sum
119895
(119895 + 1)119860119895
2((1205751198792) minus 119879
119874119871)119895
(119895 + 1)
]
]
119880
(8)
Equations of (8) use the output equation that includes1198942sum and the expression ViREF minus VSAW = 119877
119904119894sum which
determines the duration of Mode II whereas the duration of
Mode V is determined by 1198945sum and the expression ViREF minus
VSAW = 119877119904119894sum such that (9) may be rewritten as follows
1198945sum(120575) = minus
(VV minus V119901) ((1205751198792) (1198792)) + V
119901
(119877119904)
1198941015840
5sum (120575) = (VV minus V119901) 120575
minus 1198655sum (119860
5
119879
2)
times (1205935(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
))
+ 1198655sum
119879
2
[
[
infin
sum
119895
(119895 + 1)119860119895
5((1205751198792) minus 119879
119874119871)119895
(119895 + 1)
]
]
119880
(9)
3 Modeling of the ZVT Converter withCurrent Control Loop
31 Piece-Wise Linear Model Equation (2) may be used todevelop a piece-wise linear model of the converter of Figure 1
6 Mathematical Problems in Engineering
isum
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
CLK
NIo
120575T
2
TOL
QA
COMP
QB
120575T
2
iREF
g119904T1198601
g119904T1198602
g119904T1198611
g119904T1198612
Figure 5 Ideal waveform DC-DC ZVT converter with currentcontrol loop
throughout all the modes of operation The solution of (2)may be expressed as
119883(119905119899minus1
+ 119905119899) = 120593119899(119905119899)119883 (1199051) + 119860minus1
119899[120593119899(119905119899) minus 119868119889] 119861119899119880
(10)
where
120593119899(119905119899) = 119890119860119899119905119899 (11)
which defines in the first term of (10) the natural responseof the system along the period of time 119905
119899minus 119905119899minus1
with theinitial condition 119883(119905
119899minus1) at Mode n The second term of
(10) is the steady state response which is obtained by usingthe convolution integral Therefore using (10) for the timeinterval of 119905
1le 119905 lt 119905
1+ 119879119874119871 together with the matrixes of
Mode I the solution of the state vector forMode I is obtainedas follows
119883(1199051+ 119879119874119871
) = 1205931(119879119874119871
)119883 (1199051) + 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611119880
(12)
In a similar way the solutions for Modes II to VI at thetime intervals 119905
1+ 119879119874119871
le 119905 lt 1199051+ 1205751198792 119905
1+ 1205751198792 le 119905 lt
1199051+ 1198792 119905
1+ 1198792 le 119905 lt 119905
1+ 1198792 + 119879
119874119871 1199051+ 1198792 + 119879
119874119871le 119905 lt
1199051+ (1 + 120575)1198792 and 119905
1+ (1 + 120575)1198792 le 119905 lt 119905
1+ 119879 become
119883(1199051+
120575119879
2) = 120593
2(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
)
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612119880
(13)
119883(1199051+
119879
2) = 120593
3(119879
2minus
120575119879
2)119883(119905
1+
120575119879
2)
+ 119860minus1
3[1205933(119879
2minus
120575119879
2) minus 119868119889]1198613119880
(14)
119883(1199051+
119879
2+ 119879119874119871
) = 1205934(119879119874119871
)119883(1199051+
119879
2)
+ 119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614119880
(15)
119883(1199051+
(1 + 120575) 119879
2) = 120593
5(120575119879
2minus 119879119874119871
)119883(1199051+
119879
2+ 119879119874119871
)
+ 119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615119880
(16)
119883(1199051+ 119879) = 120593
6((1 minus 120575) 119879
2)119883(119905
1+
(1 + 120575) 119879
2)
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(17)
32 Large-Signal Model The large-signal model of the ZVTconverter may be obtained by substituting (12) in (13) (13) in(14) (14) in (15) (15) in (16) and (16) in (17) in such a waythat a single expression is obtained as shown in
119883(1199051+ 119879)
= 1205936((1 minus 120575) 119879
2)1205935(120575119879
2minus 119879119874119871
)1205934(119879119874119871
) 1205933
times ((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
Mathematical Problems in Engineering 7
+ [1205936((1 minus 120575) 119879
2)
times [1205935(120575119879
2minus 119879119874119871
)
times [1205934(119879119874119871
) [1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)
times 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
)
minus119868119889]1198612]
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]
+119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614]
+119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615]
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(18)
33 Half-Cycle Model The waveform 119894119871119878
of Figure 2 showsthat operation modes IV V and VI are mirrors of Modes III and III respectively therefore the first three operationmodes are sufficient to describe the function of the converterA particular matrix titled as W may relate the modes I IIand III with the modes IV V and VI which satisfies thecondition119882119882 = 119868
119889 the identity matrix Therefore the half-
cycle model of the ZVT converter may be obtained replacingthe termsA4 A5 A6 B4 B5 and B6 by expressionsWA1WA2WA3 WB1 WB2 andWB3 respectively in (18) such that thehalf-cycle model is
119883(1199051+
119879
2)
= 1205933((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
+ [1205933((1 minus 120575) 119879
2)12059321205933((1 minus 120575) 119879
2)1205932
times (120575119879
2minus 119879119874119871
)119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 1205933((1 minus 120575) 119879
2)119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]119880
+ 119882(1205933((1 minus 120575) 119879
2)(1205932(120575119879
2minus 119879119874119871
)
times 1198821205931(119879119874119871
)119883 (1199051)
+ 1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)119860minus1
1
times [1205931(119879119874119871
) minus 119868119889] 1198611)
+ 1205933((1 minus 120575) 119879
2)119860minus1
2
times [1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+ 119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198821198613)119880 (119905
1)
(19)
The previous equation may be rewritten as119883(1199051+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) for practical purposes
34 Sample-Data Small-Signal Linear Model of the Converterin Open-Loop Conditions The equation 119883(119905
1+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) may be used as a half-cycle discrete
model of the ZVT converter which may be written as
1198831015840
119870+1= 119860MC119883119870 + 119861MC119880119870 (20)
where1198831015840119870+1
= 119883(1199051+ 1198792)119883
119870= 119883(119905
1) and 119880
119870= 119880(119905
1)
A sample-data small-signal model may be obtained byusing the Taylor series (21) and using small-signal perturba-tions as 120575xK 120575UK 120575119879119874119871
119896
and 120575120575K One has
f (119909) = f (1199090) +J (119909 minus 119909
0) (21)
Using this equation the sample-data small-signal linearmodel becomes
120575119909119896+1
asymp120597119909119896+1
120597119909119896
120575119909119896+
120597119909119896+1
120597119880119896
120575119880119896
+120597119909119896+1
120597120575120575120575119896+
120597119909119896+1
120597119879119874119871
120575119879119874119871119896
(22)
The solution of the partial derivatives is 120597119909119896+1
120597119909119896
=
119860MC 120597119909119896+1120597119880119896 = 119861MC 120597119909119896+1120597120575 = 119862120575 and 120597119909
119896+1120597119879119874119871
=
119863119879119874119871
120575119879119874119871119896
may be determined utilizing the restrictionequation of119879
119874119871 whereas 120575120575K is obtained using the restriction
equation of 120575
35 Restriction Equations of the Control Loop The restrictionequations for 119879
119874119871may be obtained analyzing the waveforms
119894119871119878 11989411986311198633
and 119894sum which are shown in Figure 2 during theMode I The slope of 119894sum during Mode I is named 119892
1and is
determined by the rate of change of 119894119871119878
that is 1198921= 119881119904119896
119871119878
8 Mathematical Problems in Engineering
while the slopes of 11989411986311198633
during the Mode I are named 1198923
which are contrary and of lower amplitude than 1198921 that
is g3
= minusg12N The restriction equation of 119879
119874119871may be
determined by integrating the waveforms 11989411986311198633
duringModeI
119894119871119878119896
2119873+
119894119871119891119896
2+
119881119904119896
119871119878
119879119874119871119896
= 0 (23)
The previous equation is not linear therefore it isnecessary to use theTaylor series to obtain a linearizedmodel
120575119879119874119871119896
=2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
(24)
The restriction equation of 120575 may be obtained analyzingthe waveforms 119894
119871119878
and 119894sum with ViREF minus VSAW during Mode II(Figures 2 and 5) VSAW is determined by 119872(120575
1198961198792) and the
slope of 119894sum duringMode II named 1198922 and may be obtained
by integrating again 119894sumwhen 119877119904119894sum = ViREF minus VSAW
119877119904(119894119871119878119896
+ 1198921119879119874119871119896
+ 1198922(120575119896119879
2minus 119879119874119871119896
)) = 119881119894REF minus 119872
120575119896119879
2
(25)
The Taylor series is used to linearize the previous equa-tion and therefore the restriction equation of 120575 becomes
120575120575119896=
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
(26)
where Δ120575= minus(2119879)(1198721198792119877
119878+ 1(119871
119878+ (1119873
2)119871119891))(V119904CD
minus
(1119873)V0CD
) 1198861205751
= 120597119894119871119878119896
120597119894119871119878119896
1198861205752
= 120597119894119871119878119896
120597119894119871119891119896
1198861205753
=
120597119894119871119878119896
120597119881119878119896
1198861205754
= 120597119894119871119878119896
120597119881iREF119896
and 1198861205755
= 120597119894119871119878119896
120597119881119900119896
36 Sample-Data Small-Signal Linear Model of the Converterin Closed-Loop Conditions The sample-data small-signallinear model may be obtained by substituting 120575
119879119874119871119896
and 120575120575K(24) and (26) respectively in (22)
120575119909119896+1
= 119860MC[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
+ 119861MC [120575119881119904119896
120575119868119885119896
]
+ 119862120575(
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
)
+ 119863119879119874119871
(2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
)
(27)
Table 1 Operating parameters
Supply voltage 200V plusmn 20Output voltage 48VMaximum power 250WMinimum power 50WOutput voltage ripple 25mVOutput current ripple 300mAMaximum phase 50Switching frequency 50 kHz
such that the small-signal model becomes
120575119909119896+1
= 119860119888119897120575119909119896+ 120596119888119897120575119908119888119897119896
(28)
where 120575119909119896+1
= [1205751198941015840
119871119878119896
1205751198941015840
119871119891119896
1205751198811015840
119904119896
]119879
120575119909119896
=
[120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896]1015840 and 120575119908
119888119897119896
= [120575119881119904119896
120575119881iREF119896
120575119868119885119896]119879
Equation (28) may be solved by using the 119885 transformsuch that 120575119909
119870becomes
120575119909119896= (119868119889minus 119860119888119897)minus1119885120596119888119897120575119908119888119897119896
(29)
37 Transfer Function To verify the dynamic characteristicsof the converter is necessary to analyze the transfer functionsthat relate V
119900with 119881
119904 ViREF and 119868
119911 which are the throughput
input-to-output DC voltage transfer function 119867V(119911) =
V119900(119911)V119904(119911) the control-to-output transfer function119879
119874119871(119911) =
V119900(119911)ViREF(119911) and the output impedance transfer function
119885119900(119911) = V
119900(119911)119868119911(119911) respectively
The magnitude and phase of each transfer function maybe obtained using a Bode diagram whereas the root locustechnique may be employed to describe the behaviour of thepoles and zeros of (30) One has
119867V (119911) =119911 + 1198861
1199112 + 1198871119911 + 1198881
119879119874119871 (119911) =
11988621199112+ 1198872119911 + 1198882
1199112 + 1198871119911 + 1198881
119885119900 (119911) =
11988631199112+ 1198873119911 minus 1198883
1199112 + 1198871119911 + 1198881
(30)
4 Verification of the Proposed Model
41 Prototype Operating Parameters A 250W ZVT DC-DC prototype converter was designed under the analysisdescribed in [7] to verify the large-signal model of (14)Table 1 shows the operating parameters of the converter
The steady-state output voltage 119881119874 may be calculated as
the average of the rectified voltage V119863119863
such that at full load
119881119900= 120575max119873119881in minus
41198732119868119900(max)119871119878
119879 (31)
Taking the assumption shown in (31) the converter com-ponent values must comply with the zero-voltage switching
Mathematical Problems in Engineering 9
012345
(ms)(ms)
Piece-wise analysisPiece-wise model
Simulation circuitPiece-wise analysisPiece-wise model
Simulation circuit
01234
0 01 02 03 04 05 06 07 08 09 01minus5
minus4
minus3
minus2
minus1
minus4
minus3
minus2
minus1A A
09
0902
0904
0906
0908
091
0912
0914
0916
0918
092
iL119878iL119878
Figure 6 119894119871119878
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
1
0
1
2
3
4
5
6
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
09
902
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
0
1
2
3
4
5
6
7
(ms)
A
0 01 02 03 04 05 06 07 08 09minus1
(ms)
A
iL119891iL119891
Figure 7 119894119871119891
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
0
10
20
30
40
50
0
10
20
30
40
50
60
minus10
A
1
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
0 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
6
090
8
091
091
4
091
6
091
8
092
(ms)
A
o o
Figure 8 vo voltage waveform obtained with the piece-wise linear model and a Micro-Cap simulation
10 Mathematical Problems in Engineering
012345
0
2
4
6
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
minus6
minus4
minus2
minus4
minus3
minus1
minus2
A A
10 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
(ms)
isum and SAW-iREF isum and SAW-iREF
Figure 9 Waveforms of 119894sum and VSAW minus ViREF obtained with the piece-wise linear model and a Micro-Cap simulation
Mag
nitu
de (d
B)
0
Phas
e (de
g)
101 102 103 104
Frequency (rads)
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus225
minus180
minus135
minus90
minus45
Bode Diagram (os)
Figure 10 Bode plot of Hv(z)
05
10152025303540
Mag
nitu
de (d
B)Ph
ase (
deg)
101 102 103 104minus240minus195minus150minus105minus80minus15
Frequency (rads)
Bode Diagram (oiREF)
Figure 11 Bode plot of 119879119874119871(z)
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
isum
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
tT2 T
CLK
NIo
120575T
2
TOL
QA
COMP
QB
120575T
2
iREF
g119904T1198601
g119904T1198602
g119904T1198611
g119904T1198612
Figure 5 Ideal waveform DC-DC ZVT converter with currentcontrol loop
throughout all the modes of operation The solution of (2)may be expressed as
119883(119905119899minus1
+ 119905119899) = 120593119899(119905119899)119883 (1199051) + 119860minus1
119899[120593119899(119905119899) minus 119868119889] 119861119899119880
(10)
where
120593119899(119905119899) = 119890119860119899119905119899 (11)
which defines in the first term of (10) the natural responseof the system along the period of time 119905
119899minus 119905119899minus1
with theinitial condition 119883(119905
119899minus1) at Mode n The second term of
(10) is the steady state response which is obtained by usingthe convolution integral Therefore using (10) for the timeinterval of 119905
1le 119905 lt 119905
1+ 119879119874119871 together with the matrixes of
Mode I the solution of the state vector forMode I is obtainedas follows
119883(1199051+ 119879119874119871
) = 1205931(119879119874119871
)119883 (1199051) + 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611119880
(12)
In a similar way the solutions for Modes II to VI at thetime intervals 119905
1+ 119879119874119871
le 119905 lt 1199051+ 1205751198792 119905
1+ 1205751198792 le 119905 lt
1199051+ 1198792 119905
1+ 1198792 le 119905 lt 119905
1+ 1198792 + 119879
119874119871 1199051+ 1198792 + 119879
119874119871le 119905 lt
1199051+ (1 + 120575)1198792 and 119905
1+ (1 + 120575)1198792 le 119905 lt 119905
1+ 119879 become
119883(1199051+
120575119879
2) = 120593
2(120575119879
2minus 119879119874119871
)119883 (1199051+ 119879119874119871
)
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612119880
(13)
119883(1199051+
119879
2) = 120593
3(119879
2minus
120575119879
2)119883(119905
1+
120575119879
2)
+ 119860minus1
3[1205933(119879
2minus
120575119879
2) minus 119868119889]1198613119880
(14)
119883(1199051+
119879
2+ 119879119874119871
) = 1205934(119879119874119871
)119883(1199051+
119879
2)
+ 119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614119880
(15)
119883(1199051+
(1 + 120575) 119879
2) = 120593
5(120575119879
2minus 119879119874119871
)119883(1199051+
119879
2+ 119879119874119871
)
+ 119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615119880
(16)
119883(1199051+ 119879) = 120593
6((1 minus 120575) 119879
2)119883(119905
1+
(1 + 120575) 119879
2)
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(17)
32 Large-Signal Model The large-signal model of the ZVTconverter may be obtained by substituting (12) in (13) (13) in(14) (14) in (15) (15) in (16) and (16) in (17) in such a waythat a single expression is obtained as shown in
119883(1199051+ 119879)
= 1205936((1 minus 120575) 119879
2)1205935(120575119879
2minus 119879119874119871
)1205934(119879119874119871
) 1205933
times ((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
Mathematical Problems in Engineering 7
+ [1205936((1 minus 120575) 119879
2)
times [1205935(120575119879
2minus 119879119874119871
)
times [1205934(119879119874119871
) [1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)
times 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
)
minus119868119889]1198612]
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]
+119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614]
+119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615]
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(18)
33 Half-Cycle Model The waveform 119894119871119878
of Figure 2 showsthat operation modes IV V and VI are mirrors of Modes III and III respectively therefore the first three operationmodes are sufficient to describe the function of the converterA particular matrix titled as W may relate the modes I IIand III with the modes IV V and VI which satisfies thecondition119882119882 = 119868
119889 the identity matrix Therefore the half-
cycle model of the ZVT converter may be obtained replacingthe termsA4 A5 A6 B4 B5 and B6 by expressionsWA1WA2WA3 WB1 WB2 andWB3 respectively in (18) such that thehalf-cycle model is
119883(1199051+
119879
2)
= 1205933((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
+ [1205933((1 minus 120575) 119879
2)12059321205933((1 minus 120575) 119879
2)1205932
times (120575119879
2minus 119879119874119871
)119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 1205933((1 minus 120575) 119879
2)119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]119880
+ 119882(1205933((1 minus 120575) 119879
2)(1205932(120575119879
2minus 119879119874119871
)
times 1198821205931(119879119874119871
)119883 (1199051)
+ 1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)119860minus1
1
times [1205931(119879119874119871
) minus 119868119889] 1198611)
+ 1205933((1 minus 120575) 119879
2)119860minus1
2
times [1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+ 119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198821198613)119880 (119905
1)
(19)
The previous equation may be rewritten as119883(1199051+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) for practical purposes
34 Sample-Data Small-Signal Linear Model of the Converterin Open-Loop Conditions The equation 119883(119905
1+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) may be used as a half-cycle discrete
model of the ZVT converter which may be written as
1198831015840
119870+1= 119860MC119883119870 + 119861MC119880119870 (20)
where1198831015840119870+1
= 119883(1199051+ 1198792)119883
119870= 119883(119905
1) and 119880
119870= 119880(119905
1)
A sample-data small-signal model may be obtained byusing the Taylor series (21) and using small-signal perturba-tions as 120575xK 120575UK 120575119879119874119871
119896
and 120575120575K One has
f (119909) = f (1199090) +J (119909 minus 119909
0) (21)
Using this equation the sample-data small-signal linearmodel becomes
120575119909119896+1
asymp120597119909119896+1
120597119909119896
120575119909119896+
120597119909119896+1
120597119880119896
120575119880119896
+120597119909119896+1
120597120575120575120575119896+
120597119909119896+1
120597119879119874119871
120575119879119874119871119896
(22)
The solution of the partial derivatives is 120597119909119896+1
120597119909119896
=
119860MC 120597119909119896+1120597119880119896 = 119861MC 120597119909119896+1120597120575 = 119862120575 and 120597119909
119896+1120597119879119874119871
=
119863119879119874119871
120575119879119874119871119896
may be determined utilizing the restrictionequation of119879
119874119871 whereas 120575120575K is obtained using the restriction
equation of 120575
35 Restriction Equations of the Control Loop The restrictionequations for 119879
119874119871may be obtained analyzing the waveforms
119894119871119878 11989411986311198633
and 119894sum which are shown in Figure 2 during theMode I The slope of 119894sum during Mode I is named 119892
1and is
determined by the rate of change of 119894119871119878
that is 1198921= 119881119904119896
119871119878
8 Mathematical Problems in Engineering
while the slopes of 11989411986311198633
during the Mode I are named 1198923
which are contrary and of lower amplitude than 1198921 that
is g3
= minusg12N The restriction equation of 119879
119874119871may be
determined by integrating the waveforms 11989411986311198633
duringModeI
119894119871119878119896
2119873+
119894119871119891119896
2+
119881119904119896
119871119878
119879119874119871119896
= 0 (23)
The previous equation is not linear therefore it isnecessary to use theTaylor series to obtain a linearizedmodel
120575119879119874119871119896
=2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
(24)
The restriction equation of 120575 may be obtained analyzingthe waveforms 119894
119871119878
and 119894sum with ViREF minus VSAW during Mode II(Figures 2 and 5) VSAW is determined by 119872(120575
1198961198792) and the
slope of 119894sum duringMode II named 1198922 and may be obtained
by integrating again 119894sumwhen 119877119904119894sum = ViREF minus VSAW
119877119904(119894119871119878119896
+ 1198921119879119874119871119896
+ 1198922(120575119896119879
2minus 119879119874119871119896
)) = 119881119894REF minus 119872
120575119896119879
2
(25)
The Taylor series is used to linearize the previous equa-tion and therefore the restriction equation of 120575 becomes
120575120575119896=
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
(26)
where Δ120575= minus(2119879)(1198721198792119877
119878+ 1(119871
119878+ (1119873
2)119871119891))(V119904CD
minus
(1119873)V0CD
) 1198861205751
= 120597119894119871119878119896
120597119894119871119878119896
1198861205752
= 120597119894119871119878119896
120597119894119871119891119896
1198861205753
=
120597119894119871119878119896
120597119881119878119896
1198861205754
= 120597119894119871119878119896
120597119881iREF119896
and 1198861205755
= 120597119894119871119878119896
120597119881119900119896
36 Sample-Data Small-Signal Linear Model of the Converterin Closed-Loop Conditions The sample-data small-signallinear model may be obtained by substituting 120575
119879119874119871119896
and 120575120575K(24) and (26) respectively in (22)
120575119909119896+1
= 119860MC[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
+ 119861MC [120575119881119904119896
120575119868119885119896
]
+ 119862120575(
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
)
+ 119863119879119874119871
(2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
)
(27)
Table 1 Operating parameters
Supply voltage 200V plusmn 20Output voltage 48VMaximum power 250WMinimum power 50WOutput voltage ripple 25mVOutput current ripple 300mAMaximum phase 50Switching frequency 50 kHz
such that the small-signal model becomes
120575119909119896+1
= 119860119888119897120575119909119896+ 120596119888119897120575119908119888119897119896
(28)
where 120575119909119896+1
= [1205751198941015840
119871119878119896
1205751198941015840
119871119891119896
1205751198811015840
119904119896
]119879
120575119909119896
=
[120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896]1015840 and 120575119908
119888119897119896
= [120575119881119904119896
120575119881iREF119896
120575119868119885119896]119879
Equation (28) may be solved by using the 119885 transformsuch that 120575119909
119870becomes
120575119909119896= (119868119889minus 119860119888119897)minus1119885120596119888119897120575119908119888119897119896
(29)
37 Transfer Function To verify the dynamic characteristicsof the converter is necessary to analyze the transfer functionsthat relate V
119900with 119881
119904 ViREF and 119868
119911 which are the throughput
input-to-output DC voltage transfer function 119867V(119911) =
V119900(119911)V119904(119911) the control-to-output transfer function119879
119874119871(119911) =
V119900(119911)ViREF(119911) and the output impedance transfer function
119885119900(119911) = V
119900(119911)119868119911(119911) respectively
The magnitude and phase of each transfer function maybe obtained using a Bode diagram whereas the root locustechnique may be employed to describe the behaviour of thepoles and zeros of (30) One has
119867V (119911) =119911 + 1198861
1199112 + 1198871119911 + 1198881
119879119874119871 (119911) =
11988621199112+ 1198872119911 + 1198882
1199112 + 1198871119911 + 1198881
119885119900 (119911) =
11988631199112+ 1198873119911 minus 1198883
1199112 + 1198871119911 + 1198881
(30)
4 Verification of the Proposed Model
41 Prototype Operating Parameters A 250W ZVT DC-DC prototype converter was designed under the analysisdescribed in [7] to verify the large-signal model of (14)Table 1 shows the operating parameters of the converter
The steady-state output voltage 119881119874 may be calculated as
the average of the rectified voltage V119863119863
such that at full load
119881119900= 120575max119873119881in minus
41198732119868119900(max)119871119878
119879 (31)
Taking the assumption shown in (31) the converter com-ponent values must comply with the zero-voltage switching
Mathematical Problems in Engineering 9
012345
(ms)(ms)
Piece-wise analysisPiece-wise model
Simulation circuitPiece-wise analysisPiece-wise model
Simulation circuit
01234
0 01 02 03 04 05 06 07 08 09 01minus5
minus4
minus3
minus2
minus1
minus4
minus3
minus2
minus1A A
09
0902
0904
0906
0908
091
0912
0914
0916
0918
092
iL119878iL119878
Figure 6 119894119871119878
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
1
0
1
2
3
4
5
6
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
09
902
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
0
1
2
3
4
5
6
7
(ms)
A
0 01 02 03 04 05 06 07 08 09minus1
(ms)
A
iL119891iL119891
Figure 7 119894119871119891
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
0
10
20
30
40
50
0
10
20
30
40
50
60
minus10
A
1
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
0 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
6
090
8
091
091
4
091
6
091
8
092
(ms)
A
o o
Figure 8 vo voltage waveform obtained with the piece-wise linear model and a Micro-Cap simulation
10 Mathematical Problems in Engineering
012345
0
2
4
6
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
minus6
minus4
minus2
minus4
minus3
minus1
minus2
A A
10 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
(ms)
isum and SAW-iREF isum and SAW-iREF
Figure 9 Waveforms of 119894sum and VSAW minus ViREF obtained with the piece-wise linear model and a Micro-Cap simulation
Mag
nitu
de (d
B)
0
Phas
e (de
g)
101 102 103 104
Frequency (rads)
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus225
minus180
minus135
minus90
minus45
Bode Diagram (os)
Figure 10 Bode plot of Hv(z)
05
10152025303540
Mag
nitu
de (d
B)Ph
ase (
deg)
101 102 103 104minus240minus195minus150minus105minus80minus15
Frequency (rads)
Bode Diagram (oiREF)
Figure 11 Bode plot of 119879119874119871(z)
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
+ [1205936((1 minus 120575) 119879
2)
times [1205935(120575119879
2minus 119879119874119871
)
times [1205934(119879119874119871
) [1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)
times 119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 119860minus1
2[1205932(120575119879
2minus 119879119874119871
)
minus119868119889]1198612]
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]
+119860minus1
4[1205934(119879119874119871
) minus 119868119889] 1198614]
+119860minus1
5[1205935(120575119879
2minus 119879119874119871
) minus 119868119889]1198615]
+ 119860minus1
6[1205936((1 minus 120575) 119879
2) minus 119868119889]1198616119880
(18)
33 Half-Cycle Model The waveform 119894119871119878
of Figure 2 showsthat operation modes IV V and VI are mirrors of Modes III and III respectively therefore the first three operationmodes are sufficient to describe the function of the converterA particular matrix titled as W may relate the modes I IIand III with the modes IV V and VI which satisfies thecondition119882119882 = 119868
119889 the identity matrix Therefore the half-
cycle model of the ZVT converter may be obtained replacingthe termsA4 A5 A6 B4 B5 and B6 by expressionsWA1WA2WA3 WB1 WB2 andWB3 respectively in (18) such that thehalf-cycle model is
119883(1199051+
119879
2)
= 1205933((1 minus 120575) 119879
2)1205932(120575119879
2minus 119879119874119871
)1205931(119879119874119871
)119883 (1199051)
+ [1205933((1 minus 120575) 119879
2)12059321205933((1 minus 120575) 119879
2)1205932
times (120575119879
2minus 119879119874119871
)119860minus1
1[1205931(119879119874119871
) minus 119868119889] 1198611
+ 1205933((1 minus 120575) 119879
2)119860minus1
2[1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198613]119880
+ 119882(1205933((1 minus 120575) 119879
2)(1205932(120575119879
2minus 119879119874119871
)
times 1198821205931(119879119874119871
)119883 (1199051)
+ 1205933((1 minus 120575) 119879
2)
times 1205932(120575119879
2minus 119879119874119871
)119860minus1
1
times [1205931(119879119874119871
) minus 119868119889] 1198611)
+ 1205933((1 minus 120575) 119879
2)119860minus1
2
times [1205932(120575119879
2minus 119879119874119871
) minus 119868119889]1198612
+ 119860minus1
3[1205933((1 minus 120575) 119879
2) minus 119868119889]1198821198613)119880 (119905
1)
(19)
The previous equation may be rewritten as119883(1199051+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) for practical purposes
34 Sample-Data Small-Signal Linear Model of the Converterin Open-Loop Conditions The equation 119883(119905
1+ 1198792) =
119860MC119883(1199051) + 119861MC119880(119905
1) may be used as a half-cycle discrete
model of the ZVT converter which may be written as
1198831015840
119870+1= 119860MC119883119870 + 119861MC119880119870 (20)
where1198831015840119870+1
= 119883(1199051+ 1198792)119883
119870= 119883(119905
1) and 119880
119870= 119880(119905
1)
A sample-data small-signal model may be obtained byusing the Taylor series (21) and using small-signal perturba-tions as 120575xK 120575UK 120575119879119874119871
119896
and 120575120575K One has
f (119909) = f (1199090) +J (119909 minus 119909
0) (21)
Using this equation the sample-data small-signal linearmodel becomes
120575119909119896+1
asymp120597119909119896+1
120597119909119896
120575119909119896+
120597119909119896+1
120597119880119896
120575119880119896
+120597119909119896+1
120597120575120575120575119896+
120597119909119896+1
120597119879119874119871
120575119879119874119871119896
(22)
The solution of the partial derivatives is 120597119909119896+1
120597119909119896
=
119860MC 120597119909119896+1120597119880119896 = 119861MC 120597119909119896+1120597120575 = 119862120575 and 120597119909
119896+1120597119879119874119871
=
119863119879119874119871
120575119879119874119871119896
may be determined utilizing the restrictionequation of119879
119874119871 whereas 120575120575K is obtained using the restriction
equation of 120575
35 Restriction Equations of the Control Loop The restrictionequations for 119879
119874119871may be obtained analyzing the waveforms
119894119871119878 11989411986311198633
and 119894sum which are shown in Figure 2 during theMode I The slope of 119894sum during Mode I is named 119892
1and is
determined by the rate of change of 119894119871119878
that is 1198921= 119881119904119896
119871119878
8 Mathematical Problems in Engineering
while the slopes of 11989411986311198633
during the Mode I are named 1198923
which are contrary and of lower amplitude than 1198921 that
is g3
= minusg12N The restriction equation of 119879
119874119871may be
determined by integrating the waveforms 11989411986311198633
duringModeI
119894119871119878119896
2119873+
119894119871119891119896
2+
119881119904119896
119871119878
119879119874119871119896
= 0 (23)
The previous equation is not linear therefore it isnecessary to use theTaylor series to obtain a linearizedmodel
120575119879119874119871119896
=2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
(24)
The restriction equation of 120575 may be obtained analyzingthe waveforms 119894
119871119878
and 119894sum with ViREF minus VSAW during Mode II(Figures 2 and 5) VSAW is determined by 119872(120575
1198961198792) and the
slope of 119894sum duringMode II named 1198922 and may be obtained
by integrating again 119894sumwhen 119877119904119894sum = ViREF minus VSAW
119877119904(119894119871119878119896
+ 1198921119879119874119871119896
+ 1198922(120575119896119879
2minus 119879119874119871119896
)) = 119881119894REF minus 119872
120575119896119879
2
(25)
The Taylor series is used to linearize the previous equa-tion and therefore the restriction equation of 120575 becomes
120575120575119896=
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
(26)
where Δ120575= minus(2119879)(1198721198792119877
119878+ 1(119871
119878+ (1119873
2)119871119891))(V119904CD
minus
(1119873)V0CD
) 1198861205751
= 120597119894119871119878119896
120597119894119871119878119896
1198861205752
= 120597119894119871119878119896
120597119894119871119891119896
1198861205753
=
120597119894119871119878119896
120597119881119878119896
1198861205754
= 120597119894119871119878119896
120597119881iREF119896
and 1198861205755
= 120597119894119871119878119896
120597119881119900119896
36 Sample-Data Small-Signal Linear Model of the Converterin Closed-Loop Conditions The sample-data small-signallinear model may be obtained by substituting 120575
119879119874119871119896
and 120575120575K(24) and (26) respectively in (22)
120575119909119896+1
= 119860MC[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
+ 119861MC [120575119881119904119896
120575119868119885119896
]
+ 119862120575(
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
)
+ 119863119879119874119871
(2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
)
(27)
Table 1 Operating parameters
Supply voltage 200V plusmn 20Output voltage 48VMaximum power 250WMinimum power 50WOutput voltage ripple 25mVOutput current ripple 300mAMaximum phase 50Switching frequency 50 kHz
such that the small-signal model becomes
120575119909119896+1
= 119860119888119897120575119909119896+ 120596119888119897120575119908119888119897119896
(28)
where 120575119909119896+1
= [1205751198941015840
119871119878119896
1205751198941015840
119871119891119896
1205751198811015840
119904119896
]119879
120575119909119896
=
[120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896]1015840 and 120575119908
119888119897119896
= [120575119881119904119896
120575119881iREF119896
120575119868119885119896]119879
Equation (28) may be solved by using the 119885 transformsuch that 120575119909
119870becomes
120575119909119896= (119868119889minus 119860119888119897)minus1119885120596119888119897120575119908119888119897119896
(29)
37 Transfer Function To verify the dynamic characteristicsof the converter is necessary to analyze the transfer functionsthat relate V
119900with 119881
119904 ViREF and 119868
119911 which are the throughput
input-to-output DC voltage transfer function 119867V(119911) =
V119900(119911)V119904(119911) the control-to-output transfer function119879
119874119871(119911) =
V119900(119911)ViREF(119911) and the output impedance transfer function
119885119900(119911) = V
119900(119911)119868119911(119911) respectively
The magnitude and phase of each transfer function maybe obtained using a Bode diagram whereas the root locustechnique may be employed to describe the behaviour of thepoles and zeros of (30) One has
119867V (119911) =119911 + 1198861
1199112 + 1198871119911 + 1198881
119879119874119871 (119911) =
11988621199112+ 1198872119911 + 1198882
1199112 + 1198871119911 + 1198881
119885119900 (119911) =
11988631199112+ 1198873119911 minus 1198883
1199112 + 1198871119911 + 1198881
(30)
4 Verification of the Proposed Model
41 Prototype Operating Parameters A 250W ZVT DC-DC prototype converter was designed under the analysisdescribed in [7] to verify the large-signal model of (14)Table 1 shows the operating parameters of the converter
The steady-state output voltage 119881119874 may be calculated as
the average of the rectified voltage V119863119863
such that at full load
119881119900= 120575max119873119881in minus
41198732119868119900(max)119871119878
119879 (31)
Taking the assumption shown in (31) the converter com-ponent values must comply with the zero-voltage switching
Mathematical Problems in Engineering 9
012345
(ms)(ms)
Piece-wise analysisPiece-wise model
Simulation circuitPiece-wise analysisPiece-wise model
Simulation circuit
01234
0 01 02 03 04 05 06 07 08 09 01minus5
minus4
minus3
minus2
minus1
minus4
minus3
minus2
minus1A A
09
0902
0904
0906
0908
091
0912
0914
0916
0918
092
iL119878iL119878
Figure 6 119894119871119878
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
1
0
1
2
3
4
5
6
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
09
902
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
0
1
2
3
4
5
6
7
(ms)
A
0 01 02 03 04 05 06 07 08 09minus1
(ms)
A
iL119891iL119891
Figure 7 119894119871119891
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
0
10
20
30
40
50
0
10
20
30
40
50
60
minus10
A
1
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
0 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
6
090
8
091
091
4
091
6
091
8
092
(ms)
A
o o
Figure 8 vo voltage waveform obtained with the piece-wise linear model and a Micro-Cap simulation
10 Mathematical Problems in Engineering
012345
0
2
4
6
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
minus6
minus4
minus2
minus4
minus3
minus1
minus2
A A
10 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
(ms)
isum and SAW-iREF isum and SAW-iREF
Figure 9 Waveforms of 119894sum and VSAW minus ViREF obtained with the piece-wise linear model and a Micro-Cap simulation
Mag
nitu
de (d
B)
0
Phas
e (de
g)
101 102 103 104
Frequency (rads)
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus225
minus180
minus135
minus90
minus45
Bode Diagram (os)
Figure 10 Bode plot of Hv(z)
05
10152025303540
Mag
nitu
de (d
B)Ph
ase (
deg)
101 102 103 104minus240minus195minus150minus105minus80minus15
Frequency (rads)
Bode Diagram (oiREF)
Figure 11 Bode plot of 119879119874119871(z)
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
while the slopes of 11989411986311198633
during the Mode I are named 1198923
which are contrary and of lower amplitude than 1198921 that
is g3
= minusg12N The restriction equation of 119879
119874119871may be
determined by integrating the waveforms 11989411986311198633
duringModeI
119894119871119878119896
2119873+
119894119871119891119896
2+
119881119904119896
119871119878
119879119874119871119896
= 0 (23)
The previous equation is not linear therefore it isnecessary to use theTaylor series to obtain a linearizedmodel
120575119879119874119871119896
=2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
(24)
The restriction equation of 120575 may be obtained analyzingthe waveforms 119894
119871119878
and 119894sum with ViREF minus VSAW during Mode II(Figures 2 and 5) VSAW is determined by 119872(120575
1198961198792) and the
slope of 119894sum duringMode II named 1198922 and may be obtained
by integrating again 119894sumwhen 119877119904119894sum = ViREF minus VSAW
119877119904(119894119871119878119896
+ 1198921119879119874119871119896
+ 1198922(120575119896119879
2minus 119879119874119871119896
)) = 119881119894REF minus 119872
120575119896119879
2
(25)
The Taylor series is used to linearize the previous equa-tion and therefore the restriction equation of 120575 becomes
120575120575119896=
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
(26)
where Δ120575= minus(2119879)(1198721198792119877
119878+ 1(119871
119878+ (1119873
2)119871119891))(V119904CD
minus
(1119873)V0CD
) 1198861205751
= 120597119894119871119878119896
120597119894119871119878119896
1198861205752
= 120597119894119871119878119896
120597119894119871119891119896
1198861205753
=
120597119894119871119878119896
120597119881119878119896
1198861205754
= 120597119894119871119878119896
120597119881iREF119896
and 1198861205755
= 120597119894119871119878119896
120597119881119900119896
36 Sample-Data Small-Signal Linear Model of the Converterin Closed-Loop Conditions The sample-data small-signallinear model may be obtained by substituting 120575
119879119874119871119896
and 120575120575K(24) and (26) respectively in (22)
120575119909119896+1
= 119860MC[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
+ 119861MC [120575119881119904119896
120575119868119885119896
]
+ 119862120575(
1
Δ120575[1198861205751 119886
12057521198861205753
1198861205754
1198861205755]
[[[[[
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
120575119881iREF119896
120575119881119900119896
]]]]]
]
)
+ 119863119879119874119871
(2119873119871119878
119881119878CD
[minus1
2119873minus1
2minus
119879119874119871CD
2119873119871119878
][
[
120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896
]
]
)
(27)
Table 1 Operating parameters
Supply voltage 200V plusmn 20Output voltage 48VMaximum power 250WMinimum power 50WOutput voltage ripple 25mVOutput current ripple 300mAMaximum phase 50Switching frequency 50 kHz
such that the small-signal model becomes
120575119909119896+1
= 119860119888119897120575119909119896+ 120596119888119897120575119908119888119897119896
(28)
where 120575119909119896+1
= [1205751198941015840
119871119878119896
1205751198941015840
119871119891119896
1205751198811015840
119904119896
]119879
120575119909119896
=
[120575119894119871119878119896
120575119894119871119891119896
120575119881119904119896]1015840 and 120575119908
119888119897119896
= [120575119881119904119896
120575119881iREF119896
120575119868119885119896]119879
Equation (28) may be solved by using the 119885 transformsuch that 120575119909
119870becomes
120575119909119896= (119868119889minus 119860119888119897)minus1119885120596119888119897120575119908119888119897119896
(29)
37 Transfer Function To verify the dynamic characteristicsof the converter is necessary to analyze the transfer functionsthat relate V
119900with 119881
119904 ViREF and 119868
119911 which are the throughput
input-to-output DC voltage transfer function 119867V(119911) =
V119900(119911)V119904(119911) the control-to-output transfer function119879
119874119871(119911) =
V119900(119911)ViREF(119911) and the output impedance transfer function
119885119900(119911) = V
119900(119911)119868119911(119911) respectively
The magnitude and phase of each transfer function maybe obtained using a Bode diagram whereas the root locustechnique may be employed to describe the behaviour of thepoles and zeros of (30) One has
119867V (119911) =119911 + 1198861
1199112 + 1198871119911 + 1198881
119879119874119871 (119911) =
11988621199112+ 1198872119911 + 1198882
1199112 + 1198871119911 + 1198881
119885119900 (119911) =
11988631199112+ 1198873119911 minus 1198883
1199112 + 1198871119911 + 1198881
(30)
4 Verification of the Proposed Model
41 Prototype Operating Parameters A 250W ZVT DC-DC prototype converter was designed under the analysisdescribed in [7] to verify the large-signal model of (14)Table 1 shows the operating parameters of the converter
The steady-state output voltage 119881119874 may be calculated as
the average of the rectified voltage V119863119863
such that at full load
119881119900= 120575max119873119881in minus
41198732119868119900(max)119871119878
119879 (31)
Taking the assumption shown in (31) the converter com-ponent values must comply with the zero-voltage switching
Mathematical Problems in Engineering 9
012345
(ms)(ms)
Piece-wise analysisPiece-wise model
Simulation circuitPiece-wise analysisPiece-wise model
Simulation circuit
01234
0 01 02 03 04 05 06 07 08 09 01minus5
minus4
minus3
minus2
minus1
minus4
minus3
minus2
minus1A A
09
0902
0904
0906
0908
091
0912
0914
0916
0918
092
iL119878iL119878
Figure 6 119894119871119878
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
1
0
1
2
3
4
5
6
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
09
902
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
0
1
2
3
4
5
6
7
(ms)
A
0 01 02 03 04 05 06 07 08 09minus1
(ms)
A
iL119891iL119891
Figure 7 119894119871119891
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
0
10
20
30
40
50
0
10
20
30
40
50
60
minus10
A
1
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
0 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
6
090
8
091
091
4
091
6
091
8
092
(ms)
A
o o
Figure 8 vo voltage waveform obtained with the piece-wise linear model and a Micro-Cap simulation
10 Mathematical Problems in Engineering
012345
0
2
4
6
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
minus6
minus4
minus2
minus4
minus3
minus1
minus2
A A
10 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
(ms)
isum and SAW-iREF isum and SAW-iREF
Figure 9 Waveforms of 119894sum and VSAW minus ViREF obtained with the piece-wise linear model and a Micro-Cap simulation
Mag
nitu
de (d
B)
0
Phas
e (de
g)
101 102 103 104
Frequency (rads)
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus225
minus180
minus135
minus90
minus45
Bode Diagram (os)
Figure 10 Bode plot of Hv(z)
05
10152025303540
Mag
nitu
de (d
B)Ph
ase (
deg)
101 102 103 104minus240minus195minus150minus105minus80minus15
Frequency (rads)
Bode Diagram (oiREF)
Figure 11 Bode plot of 119879119874119871(z)
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
012345
(ms)(ms)
Piece-wise analysisPiece-wise model
Simulation circuitPiece-wise analysisPiece-wise model
Simulation circuit
01234
0 01 02 03 04 05 06 07 08 09 01minus5
minus4
minus3
minus2
minus1
minus4
minus3
minus2
minus1A A
09
0902
0904
0906
0908
091
0912
0914
0916
0918
092
iL119878iL119878
Figure 6 119894119871119878
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
1
0
1
2
3
4
5
6
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
09
902
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
0
1
2
3
4
5
6
7
(ms)
A
0 01 02 03 04 05 06 07 08 09minus1
(ms)
A
iL119891iL119891
Figure 7 119894119871119891
current waveform obtained with the piece-wise linear model and a Micro-Cap simulation
0
10
20
30
40
50
0
10
20
30
40
50
60
minus10
A
1
Piece-wise modelPiece-wise analysis
Simulation circuitPiece-wise modelPiece-wise analysis
Simulation circuit
0 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
6
090
8
091
091
4
091
6
091
8
092
(ms)
A
o o
Figure 8 vo voltage waveform obtained with the piece-wise linear model and a Micro-Cap simulation
10 Mathematical Problems in Engineering
012345
0
2
4
6
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
minus6
minus4
minus2
minus4
minus3
minus1
minus2
A A
10 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
(ms)
isum and SAW-iREF isum and SAW-iREF
Figure 9 Waveforms of 119894sum and VSAW minus ViREF obtained with the piece-wise linear model and a Micro-Cap simulation
Mag
nitu
de (d
B)
0
Phas
e (de
g)
101 102 103 104
Frequency (rads)
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus225
minus180
minus135
minus90
minus45
Bode Diagram (os)
Figure 10 Bode plot of Hv(z)
05
10152025303540
Mag
nitu
de (d
B)Ph
ase (
deg)
101 102 103 104minus240minus195minus150minus105minus80minus15
Frequency (rads)
Bode Diagram (oiREF)
Figure 11 Bode plot of 119879119874119871(z)
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
012345
0
2
4
6
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
Piece-wise analysisPiece-wise analysisPiece-wise model
Piece-wise modelSimulation circuitSimulation circuit
minus6
minus4
minus2
minus4
minus3
minus1
minus2
A A
10 01 02 03 04 05 06 07 08 09
(ms)
09
090
2
090
4
090
6
090
8
091
091
2
091
4
091
6
091
8
092
(ms)
isum and SAW-iREF isum and SAW-iREF
Figure 9 Waveforms of 119894sum and VSAW minus ViREF obtained with the piece-wise linear model and a Micro-Cap simulation
Mag
nitu
de (d
B)
0
Phas
e (de
g)
101 102 103 104
Frequency (rads)
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus225
minus180
minus135
minus90
minus45
Bode Diagram (os)
Figure 10 Bode plot of Hv(z)
05
10152025303540
Mag
nitu
de (d
B)Ph
ase (
deg)
101 102 103 104minus240minus195minus150minus105minus80minus15
Frequency (rads)
Bode Diagram (oiREF)
Figure 11 Bode plot of 119879119874119871(z)
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
510152025
Mag
nitu
de (d
B)4590
135180225
Phas
e (de
g)101 102 103 104
Frequency (rads)
oIz)Bode Diagram (
Figure 12 Bode plot of 119885119900(119911)
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus1minus08minus06minus04minus02
0020406081
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 13 Root locus of119867V(119911)
Table 2 Component values used in the 250W prototype
Devices ValueStray inductance (119871
119878) 1635 uH
Filter inductor (119871119891) 528 uH
Filter capacitor (119862119891) 1218 uF
Resistive load 9216ΩTransformer turns ratio119873 0683
Table 3 Verification of the large-signal model
119883(1199051)
large-signalmodel
119883(1199051+ 119879)
large-signalmodel
119883(1199051+ 119879)
verification error
119894119871119878
minus35292 minus3529 minus35304 004119894119871119891
50171 5017 50063 02119881119900 48035 48019 48002 04
phenomena to reduce the transistor switching losses under acertain load range For instance 119871
119878should be large enough
to keep the converter operation with ZVT under a low loadcondition whilst 119873 should be small to maintain regulatedoutput voltage for a maximum input voltage The list ofparameters shown in Table 1 together with (31) defines thecomponent values that may be used to keep the DC-DCconverter operatingwith theZVTeffectwithin a load range of
Table 4 Verification of the half-cycle model
119883(1199051)
half-cyclemodel
119883(1199051+ 1198792)
half-cyclemodel
119883(1199051+ 1198792)
verification error
119894119871119878
minus3415 347 34153 0016119894119871119891
4954 4957 48431 0023119881119900 4853 4853 471245 0029
50W to 250W and the values shown in Table 2 were decidedto be appropriate for the converter design The output filtercomponents of the rectifier were determined with the outputvoltage ripple Δ119881
119874 and the filter inductor current ripple
Δ119868119874 which may be calculated by using
Δ119868119900=
2119881119900
120596119871119891
120587
2sin(cosminus1 ( 2
120587)) minus cosminus1 ( 2
120587)
Δ119881119900=
119879Δ119868119900
8119862
(32)
42 Simulation and Results The piece-wise linear model thelarge-signal model and the half-cycle model of the converterwere verified by iterative program developed in MatLab Thepiece-wise model was solved using the Runge-Kutta numeri-cal method and using a small simulation step time togetherwith 119879
119874119871and 120575 to calculate the duration of each operating
mode whereas the large-signal model and half-cycle model
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
Table 5 Definition of matrix for each mode of operation
Mode I
1198601
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198611
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode II
1198602
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198612
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
1
0
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode III
1198603
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0 minus119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198613
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 1 0
0 1 0
0
0
0
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode IV
1198604
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[
[
0 0 0
0 0 minus1
119871119891
01
119862119891
minus1
119877119862119891
]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[
[
minus1
119871119878
0
0 0
01
119862119891
]]]]]
]
[119881119904
119868119885
]
1198614
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[[[[[[[
[
1
1198730 0
1
2119873
1
20
minus1
2119873
minus1
1
2
0
0
0
]]]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mode V
1198605
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+
[[[[[[[[
[
minus1198732
(1198732119871119878+ 119871119891)
0
119873
(1198732119871119878+ 119871119891)
0
01
119862119891
]]]]]]]]
]
[119881119904
119868119885
]
1198615
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
minus1
1
119873
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
Table 5 Continued
Mode VI
1198606
[[
[
1198941015840
119871119878
1198941015840
119871119891
1198811015840
119900
]]
]
=
[[[[[[[[
[
0 0119873
(1198732119871119878+ 119871119891)
0 0 minus1
(1198732119871119878+ 119871119891)
01
119862119891
minus1
119877119862119891
]]]]]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[[
[
0 0
0 0
01
119862119891
]]]
]
[119881119904
119868119885
]
1198616
[[[[
[
119894sec1198941198631
1198941198632
119894sum
]]]]
]
=
[[[[
[
0 minus1 0
0 0 0
0
0
1
0
0
0
]]]]
]
[[
[
119894119871119878
119894119871119891
119881119900
]]
]
+[[
[
0 0
0 0
0 0
]]
]
[119881119904
119868119885
]
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
Real axis
Imag
inar
y ax
is
010203040506070809
03 120587T
1120587T01 120587T
01 120587T
02 120587T
02 120587T03 120587T
04 120587T
04 120587T
05 120587T
05 120587T
06 120587T
06 120587T
07 120587T
07 120587T
08 120587T
08 120587T
09 120587T
09 120587T1120587T
Figure 14 Root locus of 119879119874119871(z)
Root locus
Real axis08 085 09 095 1 105 11 115 12
minus025minus02minus015minus01minus005
00050101502025
Imag
inar
y ax
is
01020304050607
0809
1120587T
Figure 15 Root locus of Z(z)
were solved by calculating 119879119874119871
and 120575 with the Newton-Raphson method Figures 6 to 9 show a comparison of theresults obtained with the piece-wise model and simulationresults obtained with Micro-Cap The waveforms plotted onthese figures are the transformer primary-side current 119894
119871119878
Figure 6 the filter inductor current 119894
119871119891
Figure 7 the outputvoltage V
119874 Figure 8 and the supply current 119894sum together with
VSAW minus ViREF Figure 9 Tables 3 and 4 show a comparison ofthe instantaneous values of the state vector obtained at theend of a full cycle in steady state conditions which verifiesthe exactitude of the large-signal model whereas Table 4
shows those of the half-cycle model Both tables list resultstogetherwith instantaneous results obtainedwithMicro-CapThe value of 119879
119874119871calculated for Modes I and IV is 0727 120583s
while 120575 is 03998Figures 10 11 and 12 show the bode diagram of the
transfer functions 119867V(119911) 119879119874119871(119911) and 119885119900(119911) calculated with
the symbolic equation tool of MatLab whilst Figures 13 14and 15 show the roots locus of119867V(119911) 119879119874119871(119911) and 119885
119900(119911)
Themagnitude of119867V(119911) at low frequency converges to thesteady-state DC of V
119900minusDC119881119904 while the magnitude of 119879119874119871
(119911)
reveals the gain of the control-to-output under dynamic
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
04120587T05120587T06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02120575min
120575max
Mmin
Mmax
H(z) TOL(z) Z(z)
RSmin
RSmax
002040608
1
Root locusIm
agin
ary
axis
Real axis
0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T04120587T
05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
01120587T
01120587T
02120587T
03120587T04120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T06120587T
07120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
08120587T
09120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus2minus3 minus1 0 1 2 3
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
08120587T
08120587T
09120587T
09120587T
1120587T1120587T
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
1120587T01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
Root locus
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus1
minus05
0
05
1
15
1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T08 120587T
09 120587T
1120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
Imag
inar
y ax
is
minus15
minus1
minus05
0
05
1
15
010203040506070809
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T1120587T
01 120587T
02 120587T
03 120587T
04 120587T05 120587T06 120587T
07 120587T
08 120587T
09 120587T
010203040506070809
01 120587T
02 120587T
03 120587T04 120587T
05 120587T06 120587T
07 120587T
08 120587T
09 120587T
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2minus15
minus1
minus05
0
05
1
15
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15 2
1120587T
01120587T
02120587T03120587T
04120587T05120587T06120587T07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Real axis
minus02 0 02 04 06 08 1 12 14 16 18
010203040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
0 02 04 06 08 1
120587T
120587T
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
minus1minus08minus06minus04minus02
002040608
1
Root locus
Imag
inar
y ax
is
Real axis
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01 120587T
02120587T
03120587T04120587T05120587T
06120587T07120587T
08120587T
09120587T
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1Root locus
Imag
inar
y ax
is
Real axis
minus1 minus05 0 05 1 15
1120587T
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
1120587T
010203040506070809
01120587T
02120587T
03120587T
04120587T05120587T06120587T
07120587T
08120587T
09120587T
minus1 minus08 minus06 minus04 minus02
minus1
minus08
minus06
minus04
minus02
0102
03040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
010203
040506070809
01120587T
01120587T
02120587T
02120587T
03120587T
03120587T
04120587T
04120587T
05120587T
05120587T
06120587T
06120587T
07120587T
07120587T
1120587T1120587T
08120587T
08120587T
09120587T
09120587T
Figure 16 Root locus plots of119867V(119911) 119879119874119871(119911) and 119885119874(119911) obtained by ranging of 120575119872 and 119877
119904
conditions and the magnitude of 119885(119911) converges to smallvalue below and above the resonant frequency determined by1radic119871
119891+ 1198732119871
119878119862119891
The poles of119867119881(119911) are conjugates complex roots whereas
there is a single zero located at minus275 119879119874119871
(119911) is steady with again lower than 0596 dB and its poles are conjugates complexroots with two steady zeros 119885
119900(119911) is steady with a gain lower
than 00682 dB and its poles are conjugates complex rootsand there are two zeros located at minus393 and 093
To design a controller is necessary to determine thebehaviour of the poles and zeros of the transfer functionswithrespect to variations of 120575119872 and 119877
119904 Figure 16 shows diverse
root locus plots of 119867119881(119911) 119879
119874119871(119911) and 119885
119900(119911) by ranging the
values of 120575119872 and 119877119904
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
5 Conclusion
This paper presented the mathematical derivation of asample-data small-signal model for a ZVT DC-DC con-verter The method used a piece-wise linear analysis toobtain a large-signal model which was verified with numer-ical predictions that depend on the integration step sizeto obtain high accuracy A sample-data half-cycle linearmodel was derived using the large-signal model such thata dynamic model of the converter was obtained by usinga linear approximation A comparison of the instantaneousvalues listed in Tables 3 and 4 showed that there is closecorrespondence between the derived models and the circuitsimulation especially with the half-cycle model Also Bodediagrams and a root locus analysis showed that the controlsystem may be steady if 120575 is defined within an interval of 03to 06 119877
119904within 09VA to 12 VA andM within 0044Vs
to 01 Vs Therefore the half-cycle model is more accuratethan the large-signal model and it is useful to determinea small-signal sample-data model of the DC-DC converterfor dynamic studies The presented method may help topower electronic practitioners to derive discrete transferfunctions of soft switched DC-DC converter and understandthe dynamic behaviour of power electronic systems whichserves as a basic principle to design a controller for theconverter outer loop
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the National Council of Scienceand Technology of Mexico (CONACyT) the Postgraduateand Research Department (SEPI) of the School of Mechan-ical and Electrical Engineering Campus Culhuacan of theNational Polytechnic Institute (IPN) of Mexico and theTechnological Studies Superiors of Coacalco for their encour-agement and the realization of the prototype
References
[1] F J Perez-Pinal The Electric Vehicle Design Stages Considera-tions Editorial Academica Espanola Madrid Spain 2011
[2] F J Perez-Pinal C Nunez R Alvarez and M Gallegos ldquoStepby step design of the power stage of a light electric vehiclerdquoInternational Review of Electrical Engineering vol 3 no 1 pp100ndash108 2008
[3] F J Perez-Pinal J C Kota-Renteria J C Nunez-Perez andN Al-Mutawaly ldquoHybrid conversion kit applied to publictransportation a taxi case solutionrdquo International Review onModelling and Simulations vol 6 no 2 pp 554ndash559 2013
[4] J Liu and H Peng ldquoModeling and control of a power-splithybrid vehiclerdquo IEEE Transactions on Control Systems Technol-ogy vol 16 no 6 pp 1242ndash1251 2008
[5] F J Perez-Pinal N Al-Mutawaly and J C Nunez-PerezldquoImpact of plug-in hybrid electric vehicle in distributed gener-ation and smart grid a brief reviewrdquo International Review onModelling and Simulations vol 6 no 3 pp 795ndash805 2013
[6] M Yilmaz and P T Krein ldquoReview of battery charger topolo-gies charging power levels and infrastructure for plug-inelectric and hybrid vehiclesrdquo IEEE Transactions on PowerElectronics vol 28 no 5 pp 2151ndash2169 2013
[7] A Tapia-Hernandez I Araujo-Vargas M Ponce-Silva andM Ponce-Flores ldquoDesign of a ZVT DC-DC converter withstray components integration for a public-transport electricvehiclerdquo in Proceedings of the International Symposium onPower Electronics Electrical Drives Automation and Motion(SPEEDAM rsquo10) pp 478ndash483 June 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of