A NOVEL THREE-PHASE THREE-LEVEL ZVS PWM DC-DC CONVERTER

Eloi Agostini Jr., Ivo Barbi Instituto de Eletrônica de Potência – INEP

Universidade Federal de Santa Catarina Florianópolis, Brasil

eloi@inep.ufsc.br, ivobarbi@inep.ufsc.br

Abstract – A novel three-phase dc-dc converter based on the three-phase neutral point clamped (NPC) commutation cell is proposed. A static analysis is made for a particular mode of operation, allowing the development of a design procedure for the power stage. The small-signal analysis based on the phasor transformation is also proposed, providing fundamental knowledge for a satisfactory compensation in closed-loop operation. From the theoretical analysis carried out, a design procedure is elaborated, providing the values of all power stage components. Finally, the simulation of a 10kW TPTL-PWM converter is performed to validate all theoretical considerations made in the paper.

Keywords – Three-phase, NPC, dc-dc converter, ZVS,

phasor transformation.

I. INTRODUCTION

Recently, it has been observed an increased interest in three-phase dc-dc conversion for high power applications. The major reason is that three-phase dc-dc converters can achieve lower power component current stresses and also drastically reduce input and output filter requirements, when compared to single-phase topologies. Moreover, high frequency three-phase power transformer can handle higher power levels than single-phase one, when both have the same size.

Another concern regarding high power dc-dc conversion is related to equipment size and weight. It is well known that frequency is a major parameter on determining the amount of material necessary for magnetics construction. In the case that frequency values are within the range of some decades of kilohertz, an increase in this parameter would certainly represent a reduction in magnetics size and weight. For hard-switched topologies it would not be convenient since the reduction achieved in magnetics should not compensate the need for more heat sink material to dissipate higher levels of switching losses [1], [2]. To overcome this issue, an effort has been carried out on researching soft-switched topologies that would allow frequency increase without compromising thermal management and efficiency [3], [4], [5], [6].

In this paper, a novel three-phase three-level soft-switched PWM (TPTL-PWM) dc-dc converter (Fig. 1) is proposed, with some important characteristics:

• Switch voltage stresses are half input voltage value.

• Possibility of achieving ZVS in all switches. • Output stage has voltage source characteristics. • PWM employs symmetrical duty cycle.

Fig. 1 Proposed TPTL-PWM converter.

II. PRINCIPLE OF OPERATION

The proposed TPTL-PWM converter has thirteen operation modes: six in continuous conduction mode (CCM) and seven in discontinuous conduction mode (DCM). A given operation mode is classified as discontinuous if there is no current flowing through the inductor Lin during at least one operation stage. Otherwise, the mode is classified as continuous. Each operation mode is composed by eighteen operation stages, if commutation stages are not considered.

In this work, only one continuous operation mode will be described since the paper size is limited. The other twelve modes can be described in a similar manner. Fig. 2 shows nine operation stages corresponding to the mode in which the converter detailed in section VI operates under rated conditions. The other nine stages are not depicted but they are symmetrically equivalent, simply permute inner and outer switches states in each converter leg, from the first nine stages.

III. STATIC ANALYSIS

The TPTL-PWM converter static analysis will be made to provide the fundamental equations that are going to be used as a basis in a design procedure elaboration. Thus, all components in the power stage can be correctly chosen in order to meet all design specifications.

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Fig. 2 Operation stages.

The voltages van, vbn, and vcn are imposed by the three-phase NPC commutation cell, and are graphically represented in Fig. 3.

Since voltages van, vbn, and vcn have the same waveforms, except for the fact that they are phase-shifted of 120o, the analysis can be performed for one of the three phases (here was chosen phase “a”), and the corresponding results of the other two phases are the same, if considered the mentioned phase-shift. It can be shown that the fundamental component of voltage van, here defined by van1, is given by equation (1). Its angular frequency is ω and its amplitude is controlled by duty cycle D. The angle φ represents the phase difference

between the voltages vkn and vRkn, where k = a, b, c. By convention, the voltage vRan fundamental component phase angle is set to 0o.

11

4 ( )2an

V Dv sen sen tπ ω φπ

⎛ ⎞= +⎜ ⎟⎝ ⎠

(1)

The definition of duty cycle D is given by (2).

tD ω

πΔ

= (2)

Equation (3) presents the phasor representation of voltage van1.

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11

42an

V DV sen π φπ

⎛ ⎞= ⎜ ⎟⎝ ⎠

(3)

Fig. 3 Waveform of voltages van, vbn and vcn.

The voltage vRan waveform is shown in Fig. 4, and its fundamental component is given by (4).

12 ( )o

RanVv sen tωπ

= (4)

Fig. 4 Waveform of voltage vRan.

The phasor representation of voltage vRan1 is given by (5).

12 0oo

RanVVπ

= (5)

As proposed in [7], the three-phase bridge rectifier can be represented as a resistive load, when considering only the fundamental components of voltages and currents. Equation (6) presents the value of the three-phase bridge rectifier equivalent resistance.

1

2 2

1 1

in Raneq

an Ran

L VR

V V

ω=

− (6)

In case that only the fundamental components are considered, the TPTL-PWM converter can be represented by the equivalent circuit presented in Fig. 5, for static analysis purpose.

Fig. 5 Equivalent circuit for TPTL converter analysis.

It can be derived from Fig. 5 circuit analysis that the fundamental component phasor representation of current ia is given by (7).

2 21 11

1 2 20 0

( )

an oan o oa

ineq in

V VVI

LR L ωω

−= =

+ (7)

The output current mean value is given by (8), in the case that the currents flowing through inductors Lin have sinusoidal waveform.

13

o aI Iπ

= (8)

Let the static gain be defined by (9) and the output parameterized current by (10).

12

= onVqV

(9)

2

16in

o oLI I

Vπ ω

= (10)

Where: - n: turns ratio between transformer’s primary and

secondary. Using the several equations previously determined, it can

be shown that equation (11) gives the relation between static gain and parameterized current, for a given duty cycle value.

2 212 4 o

Dq sen Iπ⎛ ⎞⎟⎜= −⎟⎜ ⎟⎜⎝ ⎠ (11)

The TPTL-PWM converter’s output characteristic graph can be derived from equation (11), and it is presented in Fig. 6. It is important to notice that the theoretical analysis considering only the fundamental components of voltages and currents provides good results only when the converter operates in continuous conduction mode. Otherwise, harmonics present in voltages and currents will play an important role in converter operation, and since their influences are not considered in the proposed analysis, the

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results become compromised. Observing Fig. 6 one can notice that continuous and dotted lines are almost coincident only in a determined region of TPTL-PWM converter output characteristic, which precisely corresponds to CCM operation.

Fig. 6 TPTL-PWM converter output characteristic: analytical

(continuous lines); simulated (dashed lines).

IV. DYNAMIC ANALYSIS

Many applications require a converter capable of maintaining output voltage within certain limits. Looking at converter’s output characteristic, it can be observed that output voltage value is dependent on load conditions, and therefore there is a need for output voltage closed-loop compensation. Thus, a controller has to be proposed in order to meet several design requirements, as load transients, closed-loop stability, response time, and so on. Basically, the main task in the dynamic analysis is to determine the transfer function that relates output voltage with duty cycle, obtained from a linearization performed on a given operating point of the converter.

Classical averaging approaches would certainly fail when applied to this kind of converter, since there is a large ripple in many important variables, as for example, in the current through inductors Lin. The methodology used to determine the control-to-output transfer function is based on the phasor transformation technique [8].

The differential equation (12) can be derived from Fig. 5 circuit analysis. The index “k” can assume the value “a”, “b” or “c”, depending on which system phase such equation is intended to refer to.

11 1 1 0k

kn in k eq k indiv j L i R i Ldt

ω− + + + = (12)

Applying the Laplace transformation to (12) and with some algebra, equation (13) can be determined.

12

1 2 22

1

2

eq

in ink

eq eqk

in in

Rs j

L LiR Rv

s sL L

ω

ω

⎛ ⎞+ −⎜ ⎟

⎝ ⎠=+ + +

(13)

The instantaneous current phasors can be generically represented by (14). 1k kR kIi I jI= + (14)

Where: - IkR: real component of instantaneous phasor 1ki .

- IkI: imaginary component of instantaneous phasor 1ki . Taking current ia1 as reference (0o), the components of 1ki

are: 0kR s kII I and I= = (15)

Considering the reference adopted, it can be shown that the voltages imposed by the three-phase NPC commutation cell are given by (16). 1 1kn kn sv v vφ φ= = (16)

The angle φ can be evaluated by equation (17).

1 in

eq

LtgRωφ −⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

(17)

Equation (16) can be rewritten as:

( )

( )1 22

sk eq in

eq in

vv R j LR L

ωω

= ++

(18)

Substituting (18) into (13) follows that:

( )

( )1 2 22

2 2

2

1

2

eq

eq inin in

k s

eq eqeq in

in in

Rs j

R j LL Li v

R R R Ls sL L

ωω

ωω

+ −+

=++ + +

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

(19)

It can be demonstrated that:

1 11

ˆ ˆˆ kR k R kI k Ik

k

I i I iiI+

= (20)

Where:

( )22

0skR kI

eq in

VI and IR Lω

= =+

(21)

( )

2 2

22

sk kR kI

eq in

VI I IR Lω

= + =+

(22)

( )

( )

22

2

2

22 2 2

2

1

ˆ ˆ2

eq in eq in

in

kR s

eq eq

eq in

in in

R L s R LL

i vR R

R L sL L

ω

ω ω

+ +

=

+ + + +

⎡ ⎤⎣ ⎦

⎛ ⎞⎜ ⎟⎝ ⎠

(23)

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( )2

22 2 2

2

ˆ ˆ2

kI s

eq eq

eq in

in in

si v

R RR L s

L L

ω

ω ω

=

+ + + +⎛ ⎞⎜ ⎟⎝ ⎠

(24)

Substituting equations (21), (22), (23) and (24) into equation (20) yields the relation between 1

ˆki and vs, as

presented in (25).

( )

( )

22

2

2

22 2 2

2

1

ˆ ˆ2

eq in eq in

in

k s

eq eq

eq in

in in

R L s R LL

i vR R

R L sL L

ω

ω ω

+ +

=

+ + + +

⎡ ⎤⎣ ⎦

⎛ ⎞⎜ ⎟⎝ ⎠

(25)

It is also possible to demonstrate the validity of (26).

3 ˆ

o ki iπ

= (26)

Thus:

( )

( )

22

2

2

22 2 2

2

3

ˆ ˆ2

eq in eq in

in

o s

eq eq

eq in

in in

R L s R LL

i vR R

R L sL L

ωπ

ω ω

+ +

=

+ + + +

⎡ ⎤⎣ ⎦

⎛ ⎞⎜ ⎟⎝ ⎠

(27)

Since the desired control variable is duty cycle “d”, there is the need to investigate how variations in this parameter will affect the fundamental component of the voltages generated by the three-phase NPC commutation cell. Such relation is presented in equation (28).

1

1ˆ ˆˆ 2

2s

s

Vv d V cos D d

Dπ∂ ⎛ ⎞= = ⎜ ⎟∂ ⎝ ⎠

(28)

Using equations (27) and (28) the transfer function (29) can be determined.

( )( )

( )

122

2

2

22 2 2

2

62

ˆ

ˆ 2

eq in eq in

o in

eq eq

eq in

in in

V cos DR L s R L

i L

R RdR L s

L L

π

ωπ

ω ω

+ +

=

+ + + +

⎡ ⎤⎣ ⎦

⎛ ⎞⎜ ⎟⎝ ⎠

(29)

The relation between output voltage vo and current io is still remaining to be obtained. It can be done by analyzing the converter’s output filter, which yields the equation (30).

ˆˆ1

oo o

o o

Rv isR C

⎛ ⎞= ⎜ ⎟+⎝ ⎠

(30)

Finally, the transfer function between output voltage and duty cycle is given by (31).

( )( )

( )

( )

122

22 2

2

2 2

2

62

ˆˆ 2

1

o

eq in eq in

in eq ino

eq eq

o o

in in

R V cos DR L s R L

L R Lv

R RdsR C s

L L

π

ωπ ω

ω

+ ++

=

+ + + +

⎡ ⎤⎣ ⎦

⎛ ⎞⎜ ⎟⎝ ⎠

(31)

A. Model Validation In order to verify the validity of transfer functions (29)

and (31), the TPTL-PWM converter was simulated and the results were compared to models response, as shown in Fig. 7 and in Fig. 8.

Fig. 7 Output current dynamic responses: simulation (continuous

line) and model (dashed line).

Fig. 8 Output voltage dynamic responses: simulation and model.

It can be concluded observing Fig. 7 and Fig. 8 that the transfer functions derived from the proposed theoretical analysis provide a satisfactory representation of TPTL-PWM dynamic behavior.

V. THREE-PHASE HIGH FREQUENCY TRANSFORMER DESIGN CONSIDERATIONS

As previously mentioned, three-phase transformer can handle higher power levels than single phase one. Thus, for the same rated power it is expected that the choice for three-phase transformer provides a reduction in converter’s volume and weight. In this section, the design of three-phase high frequency transformer will be detailed, where the following assumptions are made:

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• The load is balanced. • The analysis is made for steady state operation. • The three-phase transformer has spatial

symmetry. Fig. 9 shows the spatial representation of the three-phase

transformer under analysis.

Fig. 9 Spatial representation of three-phase transformer.

Primary and secondary voltages are given by equations (32) and (33), respectively.

, 1, 2,3.ipi p

dv N idtφ

= = (32)

, 1, 2,3.isi s

dv N idtφ

= = (33)

Considering that the primary windings are identical, and so are the secondary, one can conclude that only half window area will be available to accommodate one primary and one secondary winding. Mathematically, this can be expressed by (34).

, 1, 2,3.2

p pief s siefw w N I N Ik A iJ+

= = (34)

Where: - Np, Ns: primary and secondary windings number of

turns, respectively. - Ipief, Isief: RMS value of primary “i” and secondary “i”

current, respectively. - J: current density. - Aw: window area. - kw: window area utilization factor. The magnetic flux density in the transformer’s leg “i” can

be determined by relation (35). Parameter Ae refers to the cross section area of a transformer leg.

, 1, 2,3.ii

e

B iAφ

= = (35)

From Faraday’s law and equation (35) follows that:

1 ( )p p

max e

N v t dt constantB A

= +∫ (36)

1 ( )s s

max e

N v t dt constantB A

= +∫ (37)

Solving integrals (36) and (37) for the waveform presented in Fig. 4, it is possible to obtain the values of Np and Ns.

3 2

pefp

max e

VN

B A f= (38)

3 2

sefs

max e

VN

B A f= (39)

Where: - Vpef, Vsef: RMS value of primary and secondary

voltages, respectively. - Bmax: maximum magnetic flux density. - f: frequency. From the equations previously presented one can evaluate

the product between Ae and Aw, which provides a good estimation for transformer dimensions.

349 2e w

w max

SA A

k J B fφ= (40)

Where: - 3S φ : total three-phase apparent power. Using the several equations previously determined it is

possible to design the high frequency three-phase transformer of TPTL-PWM converter. Comparing the estimation given by (40) to the dimensions expected from the design of three single phase transformers able to process the same amount of power, it can be concluded that a size reduction of approximately 33% is expected when using three-phase transformer.

VI. CONVERTER DESIGN AND SIMULATION RESULTS

A 10kW TPTL-PWM converter design and simulation are presented in this section. Basically, several calculations are organized and exposed in such a way that a design procedure can be elaborated, allowing the determination of all converter parameters. Therefore, the converter simulation can be performed and the results used to validate the theoretical analysis carried out.

A. Converter Specifications TABLE I contains the main converter specifications that

will be used for converter design and simulation. TABLE I

TPTL-PWM converter specifications Specification Value Rated Power 10kW Input Voltage 1kV

Output Voltage 200V Switching Frequency 40kHz

Output Voltage Ripple < 1%

B. Design Procedure The first task to be performed in the converter design

consists of choosing the desired nominal operation point in the output characteristic graph. Basically, the nominal static gain and duty cycle are chosen in such a way that one or mode converter attributes could be optimized. Here, no optimization will be performed and the operation point will be chosen by inspection of the graph presented in Fig. 6. The operation point mapped by (41) seem to be satisfactory, since

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there is a good duty cycle excursion for control action and a wide load range is covered maintaining the output diode bridge rectifier in CCM. 0, 2 0,6= =q and D (41)

Thus, the parameterized output current is given by (42).

2 22 1,5682o

DI sen qπ⎛ ⎞⎟⎜= − =⎟⎜ ⎟⎜⎝ ⎠ (42)

Based on converter specifications the inductance Lin can be determined, as presented in equation (43).

12

6 37,92oin

o

V IL HI

μπ ω

= = (43)

Where:

10 50200

oo

o

P kWI AV V

= = = (44)

Finally, the capacitance Co is given by equation (45).

2

1

%

121

351,887

6

o s

o o

oo

R IV V

C FR

πμ

ω

⎛ ⎞⎜ ⎟ −⎜ ⎟Δ⎝ ⎠= = (45)

It can be concluded from equation (45) that the value of output capacitance, obtained addressing output voltage ripple restrictions, will be very low considering the amount of power processed. In order to improve the converter response during severe load transients, the value presented in (46) will be chosen for capacitance Co.

1500=oC Fμ (46) Where:

200 450

oo

o

V VRI A

= = = Ω (47)

C. Controller Design Using the parameters previously determined, it is possible

to verify that the converter control-to-output transfer function is given by (48).

7 13

3 3 2 2 8 10

ˆ ( ) 1, 464.10 1, 536.10ˆ( ) 6.10 7, 705.10 4, 038.10 6, 728.10o

v s s

d s s s s−

+=

+ + +(48)

The Bode diagram of transfer function (48) is shown in Fig. 10.

A PI controller is proposed in order to provide a good converter dynamic response in closed-loop operation. The controller’s transfer function is presented in equation (49).

5

0,001 1( )5.10

sC ss−

+= (49)

Fig. 11 contains the Bode diagram of controller’s transfer function.

Considering that voltage sensor and PWM modulator are modeled by simple gains of 0.01 and 1, respectively, the compensated system open-loop transfer function Bode diagram is thus given in Fig. 12.

From Fig. 12 analysis one can observe that compensated system has a phase-margin of approximately 83o. It was chosen not to raise the system gain since the frequency

values of zero dB crossing and converter’s conjugate complex poles are too close. Otherwise, possible non-modeled dynamics could bring instability to the system in a practical implementation.

Fig. 10 TPTL-PWM converter transfer function Bode diagram.

Fig. 11 Controller transfer function Bode diagram.

Fig. 12 Compensated system open-loop transfer function Bode

diagram.

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D. Simulation Results The simulation of the previously designed 10kW TPTL-

PWM converter was performed, and its results will be presented in this section. For a more realistic result the PSpice™ software was used employing commercial semiconductor devices. Dynamic transients to evaluate control response were simulated in PSIM™ software.

Fig. 13 Output voltage vo and voltage generated by the three-phase

NPC commutation cell in phase “a” van waveforms.

Fig. 14 Commutation detail of switches S1 and S2.

The waveforms of converter output voltage vo and the voltage generated by NPC commutation cell in phase “a” van are presented in Fig. 13. Observing this figure one can notice that the simulated converter is capable of regulating its

output voltage in the predetermined value of 200V, which is consequence of closed-loop operation.

The commutation process of switches S1 and S2 is depicted in Fig. 14, where it is possible to observe ZVS occurrence for both switches. Commutations of switches S4, S5, S8, S9 and S12, and commutations of switches S4, S5, S8, S9 and S12, are equivalent to S1 and S2, respectively, due to operation symmetry. It is also important to observe the behavior of some currents flowing through the main converter components. Fig. 15 contains the waveforms of currents in inductor Lin, switches S1 and S2, and diodes D19 and D20.

Fig. 15 Waveforms of currents ia, iS1, iS2, iD19, and iD20.

Fig. 16 presents the dynamic behavior of output voltage during a 10kW to 5kW load transient. The analysis of this response demonstrates that the converter dynamic response is satisfactory, providing an excellent output voltage regulation, even under adverse conditions.

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Fig. 16 Output voltage during a 10kW to 5kW load transient.

VII. CONCLUSION

A novel three-phase three-level soft-switched PWM dc-dc converter was proposed. An approximated static analysis was performed, which results were satisfactory for CCM operation. The closed-loop compensation was also implemented, where a transfer function determined by the phasor transformation technique was used as basis for controller designing. For more realistic results, models of commercial semiconductor devices were used when simulating the TPTL-PWM converter.

The main converter advantages observed include: minimum output filter requirements, zero voltage switching, possibility of operation at high frequency values, use of three-phase transformer, voltage-source output, switch voltage stresses are half input voltage value, resulting in a high power density equipment.

VIII. REFERENCES

[1] P. D. Ziogas, A. R. Prasad, and S. Manias, “A Three-Phase Resonant dc/dc Converter”, in Proc. IEEE 22nd Annu. Power Electronics Specialists Conf. (PESC ‘91), Cambridge, MA, Jun. 24-27, pp. 463-473, 1991.

[2] R. W. A. A. De Doncker, D. M. Divan, and M. H. Kheraluwala, “A Three-Phase Soft-Switched High-Power-Density dc/dc Converter for High-Power Applications”, in IEEE Transactions on Industry Applications, vol. 27, no. 1, pp. 63-73, Jan./Feb. 1991.

[3] D. S. Oliveira and I. Barbi, “A Three-Phase ZVS PWM DC/DC Converter With Asymmetrical Duty Cycle for High Power Applications”, in IEEE Transactions on Power Electronics, vol. 20, no. 2, pp. 370-377, March 2005.

[4] H. Cha and P. Enjeti, “A Novel Three-Phase High Power Current-Fed DC/DC Converter with Active Clamp for Fuel Cells”, in Proc. IEEE Power Electronics Specialists Conference, PESC 2007, pp. 2485-2489, June 2007.

[5] T. Song, H. S. H. Chung, S. Tapuhi, and A. Ioinovici, “A High Input Voltage Three-Phase ZVZCS DC-DC Converter with Vin/3 Voltage Stress on Primary

Switches”, in Proc. IEEE Power Electronics Specialists Conference, PESC 2007, pp. 350-356, June 2007.

[6] D. V. Ghodke, K. Chatterjee, and B. G. Fernandes, “Three-Phase Three Level, Soft Switched, Phase Shifted PWM DC-DC Converter for High Power Applications”, in IEEE Transactions on Power Electronics, vol. 23, no. 3, pp. 1214-1227, May 2008.

[7] V. Caliskan, D. J. Perreault, T. M. Jahns, and J. G. Kassakian, “Analysis of Three-Phase Rectifiers With Constant-Voltage Loads”, in IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications, vol. 50, no. 9, pp. 1220-1226, September 2003.

[8] C. T. Rim and G. H. Cho, “Phasor Transformation and its Application to the DC/AC Analyses of Frequency Phase-Controlled Series Resonant Converters (SRC)”, in IEEE Transactions on Power Electronics, vol. 5, no. 2, pp. 201-211, April 1990.

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