8/9/2019 Analysis of an Ac to Dc Voltage Source Converter Using Pwm With Phase and Amplitude Control
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ANALYSIS
O F
AN AC
TO
DC VOLTAGE SOURCE CONVERTER
USING PWM WITH PHASE AND AMPLITUDE CONTROL
Rusong Wu, S.B.Dewan,
G.R.
Slemon
Depar tment of Electrical Engineering
University of Toronto, Canada , M5S 1A4
Abstract
This paper presents a comprehensive analysis of
a
pulse-
width modulated AC to DC voltage source convecter under
phase and ampl itude control. A general mathematica l model
of the converter, which is dicontinuous, time-variant, and non-
linear, is first established. To obt ain closed-form solutions, the
following three techniques are used: Fourier analysis, transfor-
mation of reference frame and small signal linearization. Th ree
models, namely,
a
steady-state DC model,
a
low frequency
small signal AC model and a high frequency model, are con-
sequently developed. Finally, three solution sets, namely, the
steady-state solution, various dynamic transfer functions and
the high frequency harmonic components, are obtained from
the th ree models. The theoretical results are verified experi-
mentally.
1.
Introduction
Increasingly, AC to DC converters a re required t o provide
good input power factor, low line current distortion and regen-
eration. The pulse-width modulated AC to DC voltage source
converter has such good features. Several control strategies
of this kind of converter have been proposed [1]-[9]. One of
them is th e phase and amplitude control (PAC)
[1)-[4],
hich
has
a
simple structure and provides a good switching pattern.
This paper will present
a
comprehensive analysis
of
the con-
verter under this control, including steady-state, dynamic and
harmonic aspects.
The procedure of analysis is shown diagrammatically in Fig.
1. A
general mathematical model
of
the AC to DC voltage
source converter has been derived in a previous paper[9]. It is
useful in computer simulation to get
a
detailed time response of
the converter. But this model is time-variant, non-linear and
includes switching functions. It is therefore difficult to get ana-
lytical closed-form solutions. To solved it, three techniques are
applied. The first one is the Fourier analysis to get rid of dis-
continuities. After that, the general model is divided into two
models, the low frequency model and the high frequency model.
Both a re continuous and the boundary separating them is the
switching frequency. The second technique applied is the tr ans-
formation to a rotating frame of reference, synchronized with
the utility frequency, making the system time-invariant. T he
third technique is small signal linearization to linearize within
a
small area around the DC operating point. The system is fur-
ther divided into two parts, t he steady-state DC model and the
small-signal AC model. From them, the steady-stat e opera ting
point and various dynamic responses can be solved seperately.
Finally, the input current harmonic and the output voltage
ripple can be calculated from the high frequency model.
2. General Model of the AC t o DC Voltage Source
Converter
Th e main circuit of an AC to DC voltage source converter
This circuit will be analyzed under the
(1) the utility is
a
three phase balanced, sinusoidal voltage
is shown in Fig.2.
following assumptions:
source;
G e n e r a l M o d e l
E s t a b l i s h m e n t
E q u a t i o n s w i t h
Low F r e q u e n c y S w i t c h i n g F u n c t i o n
High Frequency
N o n l i n e a r
T r a n s f o rm a t i o n I n p u t o u t p u t
C u r r e n t V o l t a g e
o a R o t a t i o n
Frame of Rip p le
armonic P@ '
A n a l y s i s A n a l y s i s
e f e r e n c e
T i m e -
1
n v a r i a n t
E q u a t i o n s ,
S w i t c h i n g
P a t t e r n
s m a l l - S ig n a l C r e a t e d b y
L i n e a r i z a t i o n PAC C o n t r o l
L i n e a r
E q u a t i o n s
S t e a d y - S t a t e S m a l l - S i g n a l
DC Model AC m o d e l
s o l u t i o n
Fig.1: Procedure of analysis
(2) the filter inductors L are linear; saturation is not con-
sidered;
(3)
the DC load is equivalent to
a
resistance ro in series
with a electromotive force eh.
This load model can represent
a
wide variety of loads. Th e
converter can work either in the rectifying mode
(
e t
< V d
)
or in the regenerating mode
(
e h
>
Vd ). The series ro,
e t
equivalent load can simulate current source loads by use of large
TO and
e t
values. I t can also represent a voltage source load,
if
ro
is set low, or represent
a
pure resistance load by setting
e L
zero. When t he equivalent load includes some inductive
or
capacitive elements, t he whole system will have a higher order.
Th e analysis will be more complicated. bu t t he procedures to
be shown are still applicable.
Fig.2:
Main circuit of the AC to DC voltage source
converter
89CH2792-0/89/0000-11S6$01.00 0 1989 IEEE
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A' =
L O O 0
O L O O
O O L O
o o o c
Z =
(1
- d; T
(1 + d , ) r
0 -R 0
-(d:- ;Ed:)
Fig.3: Symmetrical double-edge modulated switching
3)
0 0
-R
-(d:-
E d ; )
function
d;
d; d; -1lro
1 0 0 0
0 1 0 0
0 0 1 0
B =
where
d,
is the average value (or duty rat io) of the switching
function
dt
within one switching period. Therefore
(4)
a.
=
LJ l+ds)r
1 d(w,t) = d,
277 l - d , ) n
a, = 0
b,
= -sin(nd;r)r
2
(9)
(10)
2
nr
m
(5)
df = di
+
c(-l) .sin(nd,a) cos nw,t
n =l
where
R
=
RL
+
Rs
is the total series resistance in one phase,
and df i = 1, 2 or 3) is the switching function of the switching
device Si. When the Si is on, d: = 1. Otherwise df = 0.
Because no specific restriction was imposed on the switch-
ing function dr during the derivation, this mathematical model
is a general one, and universally applicable to six-step, var-
ious forms of pulse-width modulation or to other switching
strategies. It provides an exact solution at any moment if the
switching function d: is defined, and it is especially useful in
a computer simulation to obtain a detailed waveform in the
time domain. The problem with this model is that it can give
only a piece-wise solution instead of a continuous closed-form
one owing to the existence of the switching function. I t is diffi-
cult to use this model to evaluate the steady-state or dynamic
performance of the system analytically.
3. Fourier Analysis Applied to the Converter Model
To obtain continuous equations to describe the converter,
The Fourier series of
a
periodical time function is
the Fourier analysis can be applied to the model.
m m
f ( w t ) =
a0 +
ansinnwt
+
b, cos nut (7)
n =l n = l
For a natural sampling sinusoidal pulse-width modulation,
the switching points within one switching period are not sym-
metrical. However, when the switching frequency is much
higher then the utility frequency, the modulating wave can be
regarded as
a
constant within each switching period. Therefore
the switching pattern is close to a symmetrical one, as shown
in Fig.3. The switching funtion df can be expressed as follows:
3 3
2
n r
C O
Ed:
= E d ; +
c[c(-l) .
sin(nd,r)]cosnw,t (11)
Substitution of eq.10 and 11 into the matrix
A*
(eq.3) yields:
r l i=l n = l i=l
A ' = A + A h (12)
where
-R 0 0
- ( d l E:=i)
and
0 -R 0
- (& -+E d ; )
0 0
-R
-(d3-:Edi)
n=l
,--,
2
O
A4k = [(-1) . ~sin(ndka)cosnwst] I =
1,
2, or3(16)
n l
The matrix A describes the low frequency property of the
converter and the matrix Ah gives the characteristics in the
range equal to and higher than the switching frequency. The
variable vector x also can be partitioned into two parts corre-
spondingly,
2 = 2
+ xh
(17)
and Therefore eq.1 becomes:
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Z(xr + h ) = ( A + Ah)(Xi+ X h ) B e
(18)
It can be regarded as
a
combination of two models: one is
the low frequency model in the range lower than the switching
frequency F,,
Z i l = Ax , +
B e (19)
The other is the high frequency model in the range of
F
1 ,,
Z X h
=
AhXl
+
AXh -kAhXh
(20
1
Eq.19 can also be derived using the state-space averag-
ing technique [lo] which ignores high frequency components.
However, through the use of Fourier analysis, one can get bo th
the low and the high frequency models.
For simplicity, the subscript
I
in all low frequency equa-
tions will be omitted henceforth, i.e.
Zx = Ax + B e
(21)
The corresponding variables, i l ,
2 2 ,
iJ,
vd
represent only the
respective low frequency components. Th e high frequency com-
ponents, i.e. the harmonics, will be expressed with a subscript
h.
In the high frequency model, eq.20, the harmonic compo-
nent
x h
is usually much smaller than the low frequency compo-
nent x , consequently, the second and third items on th e right
side of eq.20 can be neglected to get an approximation,
Z X h
N
Ah2 (22)
Two separate models obtained in eq.21 and 22 pave the
way for solving the system analytically both in the low and
high frequency ranges.
4. State-Space Equations
in
Rotation Frame of
Reference
To solve eq.21, the function of the duty ratio d; in matrix
A
should be known. For the PAC control, d; is controlled by the
phase shift
11
and modulation index
m.
From Fig.4 it is seen
th at , if the modulating wave of phase
1
is
m
cos(wt- ), and
the switching frequency is much higher than the modulating
frequency, the du ty ratio
di
can be expressed for phase i
as:
e =
2T 1
d;
=
-
os
w t -
1
-
2
-
1)-3
+
-
[
3 2
e ,
cos wt
e ,
cos(wt
-
9)
e , cos(wt
9)
el
e2
-
e3
eL eL
Because d ; is a function of time, matrix A of eq.13 is time-
variant. Fortunately, the duty ratio d; of PAC control is
a
cosine function of time synchronized with the utility frequency.
It is therefore possible to transform the system
to a
rotating
frame of reference, in which it appea rs time-invariant.
First, apply a transformation to the voltage vector e .
e
= T e ,
or
e, = T - e
(24)
where the subscript r represents the variable, vector or ma-
trix in
a
rotating frame
of
reference. The transformation ma-
trix
T
and its inverse matrix
T-
re:
1.0
0
-1.0
u
Fig.4: Switching function d; and duty ratio
dl
in the
phase and ampl itude control
l o 0
For a three phase balanced system,
After transformation, the voltage vector in the rotating frame
of reference
e,
becomes:
(27)
Now, the voltage vector does not change with time in the rotat-
ing frame of reference. Th e zero sequence component eo equals
zero owing to the balanced condition.
Both the forward and
backward components e and eb equal &e,/2. Th e dc side
electromotive force eL is not affected by the transformation.
The next step is applying the transformation to the state
variable vector,
x
=
T x ,
(28)
(29)
or
x , = T- x
i.e.
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L&
Lib
CCd
La f
39)
r
- -
- R 0 0 0 io
0
-
%A
( 3 3 )
0 0 - R - j w L i b
+=
- R + j w L
0 e e j * if
e,-,
eJ
1
V d
9
0
-
70
-
5. Small Signal Linearization and DC,
AC
Models
After rotating of the reference frame, the converter repre-
sented by
eq.34
becomes a time-invariant system. But it is
still a nonlinear one. Small signal linearization around its DC
operating point can be applied for solution. Let:
x = x .rr
(41)
O =
i.e.
- R + ] R L
0 -Fe, f
*
0
- R - ] R L
- F e - , *
Ib
+
*
E L
Ra - -
d
R a -
Fe-,*FeJ* _ -
Arr
=
1159
- ~ + j w ~ o + e ] +
0
- R - ~ ~ Li e - j + (38)
-1
+ z e - j + + , j + -
o -
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In the lossless situation
( R
= 0 r
= 7r/2) ,
L i L 0 0 1
I
:
:L
:J
, =
0 0 0
(53)
6. Steady S t a t e So lut ions
It is important to obtain steady-state solutions not only
because they give knowledge abou t t he relationsh ip between
state variables and system parameters in a steady-state oper-
ation, but also because the dynamic response of this nonlinear
converter is related to the steady -state operating point.
From the
D C
model of eq.44, the steady- stat e solutions can
be obtained:
where
E
= E , / 4
(57)
and
r =
tan-'
F)
Two other imp ortant variables to be observed are the power
factor angle 9 nd th e peak AC line current
I,.
They can be
derived from the forward and backward curren t components
I1
and Ib as
E s i n r - % si n( *+ +)
E c o s r - %cos(* + r)
Arg I1)=
tan-'
(59)
and the
RMS
value of th e stead y-s tate line current from eq.60
is
I
=
I m / 2 = d M
sin*
(for R =
0,9
= 0) (63)
2&RL
Th e function M 9 ) or keeping
9
= 0 is shown in Fig.5. It
is seen that the M Q ) unction is symmetrical with respect to
M axis assuming no losses. When th e resistance of the main
circuit is taken into account, th e modulation index M to keep
9
=
0
is nearly a const ant in the rectifying area, bu t
a
larger
variation is needed in the regenerating area. Therefore, in pa-
rameter selection, a sufficient design margin of the modulation
index has to be provided for regenerating operation.
M I
\ % = 5.0
1.0I
5.0
-20
-10 10 20
for keeping @
= 0
Fig.5: Modulation index M vcrsus
ijd
phase control Ik
7
Dynamic Response Analysis
The small signal AC model of the converter has been de-
rived in eq.46. Its Laplace transformation is:
?rr(S) =
Szrr
Ass)-I[Amxrr+(s)
+
A+Xrr$(s)+
AWX,SJ( )+B.,e^,(~)+A,oX,,io(
s)+B,oio
s)+B.L~L(
) ]
64)
(65)
and
and
(60)
2 = ( S L + R + j n L )
Z , = ( S L + R - j R L )
[ ( s L
+
R)'
+
R Z L z ]
+
~ ( s LM 2
+
R)(66)
[Ecosr- cos(* r)]'+ Esinr - sin(* + r)]
RZ
+
R2L2)/2
The converter is usually operated at unity power factor.
Th e necessary modulation index
A4
for keeping @ = 0 can be
obtained from Eq. 59,
In eq.64, the steady-state solution Xrrhas been solved in
the previous section. All the matrices are known. Therefore
any kind of transfer functions between the sta te variables
i f ,
~.
; b , t?d and the controls
l i z ,
4 oad disturbance o ,
?L
or input
disturbance
2 , 6,
can be obtained from this equation.
61)
for 0
= 0)
4 E in
I'
v d
sin(@+ I )
M =
I160
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As examples, Fig.6, Fig.7 and-Fig.8 show the dynafnic re-
sponses between the variables
?&, q5
and the controls&, 1c in the
conditions of E = 63.5V,
R =
377, L = 6.43mH, C = 13.7mF,
v d N 197V, and cosq N 1.0. Th e calculated solutions from the
eq.64 are close to the experimental results, especially in the low
frequency area. Based on these dynamic responses obtained,
the regulators of a closed-loop control system can be designed
or adjusted properly.
-450
Fig.6: Frequency response of 6d/& at Ro
=
29.2ohm,
E L = 0, M
=
0.83,
rk
= 20.1 , R E 1.13ohm
- r - I - - l - - - t t t c - - - - c - - t c t ( -
I 0.1 1 10 100
-2701803
-360
EL
=
0,
M =
0.83,
9
= 13.2 ,
R
N
1.220hm
Calculated
Fig.7: Frequency response of / m at
Ro =
42.lohm,
Phase
Gl(s)= 239
(1 -
rn) 1
+A
-90
- 80
-270
. . Experimental
alculated
Fig.8: Frequency response of fd/G at Ro
=
42.lohm,
E L = 0, M = 0.83, \k = 12.6 ,R N 1.220hm
8. Inpu t Cur rent Harmonic Analysis
Previous sections were dedicated to the analysis in the fre-
quency range lower than the switching frequency. This section
and the following one will focus on the harmonic analysis of
the converter.
An approximate high frequency model has been established
in eq.22. For the steady-state harmonics
zxh N AhX (67)
x h = [ I l h r IZh, 13/19
Vdh]
(68)
where
I i h
i = 1, 2 or 3) he steady-state harmonic of the
h e urrent
I;,
and Vdh he steady state ripple of the DC
output voltage
vd.
In a three phase balanced system, each phase current has
the same steady-state harmonic content. Hence, only one phase,
namely phase 1, need be discussed. The first row of the eq.67
is
This equation means that there is a high frequency voltage
A14Vd applied to the inductance L which creates the harmonic
I 1 h . Therefore, the steady-state harmonic current is
(70)
Ai4vd
Impedance of the inductor
l h =
1 3
= n = l
{(-l)n& [sin(ndin) - Csin(ndin)
i=l
Because the amplitude of the harmonic is inversely propor-
tional to the square of the order
n*,
t is reasonable to approx-
imate the harmonic current by the first order component with
a modifying coefficient
kh,
i.e.
For phase and amplitude control, the duty ratio di has the
expression shown in eq.23. Therefore eq.71 becomes
I l h
I l h m cos(wst)
(72)
where
The real waveform of the current harmonic is close to a tri-
angular wave instead of a sine wave, so that the coefficient k h
is used to modify the difference of the peak
values
between a
triangular wave and
a
sine wave with equal RMS value,
k h = Z 21 1.23
8
The maximum harmonic value can be calculated by taking the
derivative of the I l h m with respect to
Rt.
At ( R t
-
Q = f n / 2 ,
I l h m
has its maximum value
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Eq.72 to 74 provide mathematica l expressions of th e line cur-
rent harmonic. It is seen from eq.72 tha t the line current har-
monic is predominantly related t o switching frequency F,. The
envelope of th e harmonic amplitude is given in eq.73 and the
maximum amplitude is expressed in eq.74.
At no load operation, the line current equals the harmonic
component. As an example, consider the no load current un-
der the following conditions: vd
=
202b', F,
=
3100Hz, L =
6.43mH, A4 = 0.89, @ = 0 . From eq.74 the maximum peak
harmonic can be obtained
as
I I i h m l m a x = 0 .5444
The envelope of its amplitude is calculated from eq.73 and
plotted in Fig.9. The experimental current waveform is shown
in Fig.10 under the same conditions. Both waveforms are in
good agreement with each other, except that the real no load
current still has a small amount of fundamental component
owing to the losses of t he circuit.
When a load is applied to the converter, the phase con-
trol
Q
increases correspondingly, but the modulation index M
changes very little (see Fig.5). The maximum harmonic in
eq.74 is only related to t he value of M , not the value
of
a.
Therefore the harmonic component superimposed on the fun-
damental component has an amplitude similar to that in the
no load condition. The difference is tha t t he harmonic compo-
nent is phase sh ifted owing to the increase of the phase control
(eq.73). This shift can be seen from the current waveform
shown in Fig.11.
/ -
Envelope of the no load
/
current harmonic //
\
/.
\,-/,/
Line-to-neutral supply
voltage
as
a reference
Fig.9: Envelope of the no load input line current
Fig.11: Waveforms of the input line current an d t he
line-to-neutral utility voltage (t:2ms/div,
e:100
V/div, z:2.5
A/div)
An important specification of the converter is the relative
value of the current harmonic rather th an the absolute value.
This relative harmonic can be defined as the ra tio of maximum
peak harmonic
I I l h m l m a x
to th e peak value of th e nominal cur-
rent I,,,
Substitution of eq.63 and 74 in the above equation yields:
4khR [I
os
(F)]
I;
=
3$F,M sin
a,,
(75)
where
an
s the nominal phase control angle.
It is seen that the ratio of the switching frequency to the
utility frequency
(F,/R)
is the main factor in influencing the
relative harmonic value. The higher the ratio, the lower the
current harmonic. It is also interesting to note that the rela-
tive current harmonic is not directly related to the inductance
L.
The explanation is that, when the value
L
increases, the
absolute current harmonic reduces. But the converter can pro-
vide less nominal line current if the nominal phase control @
is kept constant. Therefore their rati o does not change with
the variation of inductance L.
Eq.76 can be used to determine the necessary switching
frequency Fa, if a certain amount of relative harmonic value
I;
is required,
9. Output Voltage Ripple Analysis
The ou tpu t voltage ripple can be calculated in a similar
way from the high frequency model using the fourth row of the
eq.67,
V d h =
Vd hm
c0sw.t (78)
where
l / d hm = *
cos [+cos [.t -
k
- i - )-]]7f
+F,C
3
x cos R t - i - 1)-
79)
[ 2x }
Fig.10: Waveforms of the no load input line current
and t he lin e-to-neutral utility voltage (t:2 msfdiv, e:100
Vfdiv, i:2.5 Afdiv)
At no load operation and
@
= 0, the maximum voltage ripple
occurs at Rt = fk:, where I is an integer, an d t he maximum
outp ut voltage ripple
is
given by
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Ivdhmlmaz
= t cos ( ) - os (
(80)
Eq.78 to 80 give the expressions of the voltage ripple, including
its envelope Vdhm and the maximum ripple Ivdhmlmaz. The
necessary capacitance of the DC filter can be determined by
the requirement for DC voltage ripple Vdr, which is the double
of
Ivdhmlmar,
Conclusions
From the analysis of the phase and ampli tude controlled
PWM AC to DC converter the following conclusions can be
drown:
1) The general model of an AC to DC voltage source con-
verter gives detailed time response if the switching func-
tion is known and is therefore useful in computer simu-
lations. Analytical solution is however difficult because
of
the discontinuities, time variation and nonlinearity of
the model.
2) Closed form solution can be obtained using Fourier anal-
ysis, transformation of reference frame and small signal
linearization converting the general model into
a
steady-
state DC model, a small-signal AC model and
a
high
frequency model.
3) The steady-state DC model gives steady-state solutions.
The small-signal model, which is suitable n the frequency
range lower than the switching frequency, provides vari-
ous transfer functions between state variables, controls,
input
or
load disturbances.
The high-frequency model
provides information about input current harmonics and
output voltage ripple. From the solutions, the circuit pa-
rameters and the regulators of a closed-loop control can
be properly designed.
4) When the circuit resistance is taken into account, the re-
quired modulation index
M
to obtain unity power factor
is relatively independent-of load in the rectifying area.
However, a larger change in
M
is required in the regen-
erating area. Therefore enough margin should be left for
a
four quadrant operation.
The relative current harmonic is not directly related to
the value of the filter inductance L .
Both input current harmonic content and output volt-
age ripple are predominantly affected by the r atio of the
switching frequency F, to the utility frequency R. The
higher the ratio, the lower the relative current harmonic
and the DC voltage ripple.
The theoretical results were verified experimentally.
References
[I] Eugenio Wernekinck, Atsuo Kawamura and R.Hoft,
A
High Frequency AC/DC Converter with Unity Power
Factor and Minimum Harmonic Distortion, Conf. Record,
IEEE-PESC, 1987, pp. 264-270
[2] S.Manias, A.R.Prasad, P.D.Ziogas, A 3-Phase Induc-
tor Fed SMR Converter with High Frequency Isolation,
High Power Density and Improved Power Factor, Conf.
Record, IEEE-PESC, 1987, pp. 253-263
[3] S.B.Dewan, Rusong Wu, A Microprocessor-Based Dual
PWM Converter Fed Four Quadrant AC Drive System,
Conf. Record, IEEE-IAS, 1987, pp.755-759
[4] J.W.Dixon, B.T.Ooi, Indirect Current Control
of
a Unity
Power Factor Sinusoidal Current Boost Type Three-phase
Rectifier, IEEE Trans. on Indust rical Electronics, Vo1.35,
No.4, pp.508-515
[5] B.T.Ooi, J.C.Salmon, J.W.Dixon, A.B.Kulkarri, A 3-
Phase Controlled Current PWM Converter with Leading
Power Factor, Conf. Record IEEE-IAS, 1985, pp. 1008-
1014
[6] J.
W.
D xon, A. B. Kulkarni, M. Nichimoto,
B
.T.Ooi, Char-
acteris tics of
a
Controlled-Current PWM Rectifier-Inverter
Link , Conf. Record, IEEE-IAS, 1986, pp.685-691
[7] H.Kohlmeier, D.Schroder, Control of a Double Voltage
Inverter System Coupling a Three Phase Mains with an
AC-Drive Conf. Record, IEEE-IAS, 1987, pp.593-599
181 Hidehiko Sugimoto, Sigeo Morimoto, Masao Yano, A
High Performance Control Method of a Voltage-Type
PWM Converter, Conf. Record, IEEE-PESC, 1988,
pp.36-368
(91 Rusong Wu, S.B.Dewan, G.R.Slemon, A PWM AC to
DC Converter with Fixed Switching Frequency, Conf.
Record, IEEE-IAS, 1988, pp.706-711
[IO] R.D.Middlebrook, S.M.Cuk, A General Unified Approach
to Modelling Switching Converter Power Stages, Conf.
Record, IEEE-PESC, 1976, pp.18-34
[ l l ]
Khai D.T.Ngo, Slobodan Cuk, R.D.Middlebrook,
A
New
Flyback DC-To-Three-phase Converter with Sinsoidal Out-
puts, Conf. Record, IEEE-PESC, 1983, pp.377-388
[ 2] Khai
D.T.Ngo,
LowFrequency Characterization ofPWM
Converters, IEEE Trans. on Power Electronics, Vol.
PE-1, No.4, Oct. 1986, pp.223-230
[13] D.M.Mitchel1, DC-DC Switching Regulator Analysis,
McGraw-Hill, 1988
1163
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