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58 Asian Journal of Control, Vol. 1, No. 1, pp. 58-65, March 1999

MODELING OF A THREE-PHASE STEP UP/DOWN

AC/DC CONVERTER

Jeng-Yue Chen, Ching-Tsai Pan and Yi-Shuo Huang

ABSTRACT

In this paper, analytic duty cycle control laws for a three-phase step up/ down converter containing seven active switches are first derived to achieve aclean sinusoidal input current, adjustable power factor, bidirectional powerflow capability and clean output DC voltage. Then a DC model and a smallsignal model are derived by means of coordinate transformation, perturbationand linearization. Finally, some simulation results are also presented to verifythe validity of the theoretical results.

KeyWords: Modeling, three-phase, AC/DC- converter.

I. INTRODUCTION

Recent AC/DC converter researches have focusedon providing a good input power factor and low linecurrent distortion in order to satisfy different harmonicstandards, along with a possible regeneration capability[1-8]. However, most of the existing results are related

to boost type converters. There are very few papers onthree-phase step up/down converters. Basically, somebuckboost type three-phase AC/DC converters [e.g., 6,7]have been proposed. However, the disadvantages of pulsating input and output currents still exist. Hence,recently, Pan and Shieh [9] proposed new space vectorcontrol strategies for a three-phase step up/down AC/DCconverter in order to overcome the above disadvantages.They proposed an equivalent duty cycle for the general-ized zero voltage space vector such that control of the ACand DC parts of the converter circuit can be integrated inorder to achieve the ideal characteristics. In other words,the input current can be made purely sinusoidal with aunity power factor, and the output can be stepped up/ down with a clean DC voltage. However, the hardwarecircuit is implemented by using an EPROM without adynamic model. Since the six active switches normallyoperate randomly so as to reduce the resulting error, it isvery difficult to find an analytic dynamic model. However,based by the equivalent DC duty cycle for the generalizedzero voltage space vector presented in [9], it is conjec-tured that it may be possible to derive the close form dutycycles of the active switches as in three-phase boost

AC/DC converters [5]. It turns out that the answer ispositive if the unidirectional diode in [9] is replaced with abidirectional active switch.

The remaining contents of this paper may beoutlined as follows. In Section II, open-loop closedform duty cycle control laws for the seven active switchesare first derived by using the familiar state averaging

technique. Then the small signal model of the proposedconverter is derived in Section III by means of perturbationand linearization after coordinate transformation. Also,some numerical results are presented in Section IV for thepurpose of verification. Finally, some conclusions aremade in the last section.

II. DERIVATION OF THE CLOSED FORMDUTY CYCLE CONTROL LAWS

Figure 1 shows the proposed three-phase step up/ down AC/DC converter, where S 1, S 2, , S 7 are the activeswitches and R1 is the ESR of L1. Unlike the six switchesin three-phase boost AC/DC converters, where the upper

Manuscript received March 2, 1999; Accepted April 14, 1999.

The authors are with the Dept. of Electrical Engineering,National TsingHua University, Hsinchu 300, Taiwan. Fig. 1. Power circuit of the proposed step up/down AC/DC converter.

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J.-Y. Chen, et al.: Modeling of A Three-Phase Step Up/Down AC/DC Converter 59

and the lower switches of each bridge arm are notallowed to be shorted, the first six active switches in Fig. 1can be operated independently. Based on the equivalentDC duty cycle presented in [9], Fig. 2 shows the concep-tual gating signals of the seven switches for one switchingperiod T s. From Fig. 2, one can see that during eachswitching period, there is one perod of time, namelyd 0T s, when all six active switches of the bridge are closedsuch that vC 1 in Fig. 1 can be discharged in order to supplyan output and enable the DC stage to work as a Cuk converter. The purpose in turning on six switches, whichcorresponds to applying the V 77 space voltage vector in[9], is to minimize the conduction loss. Also, to retain thefunction of the Cuk converter, S 7 should operate in acomplementary manner during the remainder of theswitching period, namely (1 d 0)T s, when all six switchesof the bridge function basically as a boost type converter

[5] in order to charge C 1.Assume that the switching frequency is much higher

than the AC line frequency. Then, one can use the statespace averaging technique [10] to get the following aver-aged equation:

di adt di bdt

di cdt di L2dt

dvC 1dt dvodt

=

R1 L 1

0 0 0 d 1 L 1

0

0 R1 L 1

0 0 d 2 L 1

0

0 0 R

1 L 1 0 d

3 L 1 0

0 0 0 01 d 7 L 2

1 L 2

d 1C 1

d 2C 1

d 3C 1

( 1 d 7)C 1

0 0

0 0 0 1C 00 1 RC 0

iai

bici L2vC 1v0

+ 1

L 1

e ae be c

000

1

L 1

111

000

v NO . (1)

For simplicity, ESR of C 1, C o and L2 are neglectedin the above derivation. From Kirchhoffs current law,ia + ib + ic = 0, and the upper half of equation (1), it isstraightforward to get

v NO = 13 (d 1 + d 2 + d 3)vC 1 . (2)

As can be observed from Figs. 1 and 2, in order topreserve the dc Cuk converter operation principle, thefull bridge is allowed to be short circuited with an equiva-lent DC duty cycle, 1 d 7. Thus, for convenience, onecan define the duty cycles as follows:

d 0 = 1 d 7,

d k =7 + mk (t )

2 ,

d k + 3 =7 mk (t )

2 , k = 1, 2, 3, (3)

where mk (t ), k = 1, 2, 3 are the time varying part modula-tion indices to be decided, and

0 mk (t ) 1

d 0 + d k + d k +3 = 1, k = 1, 2, 3. (4)

Then, by substituting equations (2) and (3) into (1),one has the following form:

dX dt =

A 1 A 2 A 3 A 4

X + Bu , (5)

whereFig. 2. Conceptual gating signals of seven switches.

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60 Asian Journal of Control, Vol. 1, No. 1, March 1999

X = [ ia ib ic i L2 vC 1vo]

T , A 1 = diag R1 L 1

R1 L 1

R1 L 1

,

A 2 =

0 m12 L 1

0

0 m22 L 1

0

0 m32 L 1

0

, A 3 =

0 0 0

m12C 1

m22C 1

m32C 1

0 0 0

,

A 4 =

0d 0

L 2 1 L 2

d 0C 1

0 0

1C 0 0

1 RC 0

, B =

1 L 1

0 0

0 1 L 10

0 0 1 L 10 0 00 0 00 0 0

,

U = [e a eb e c]T .

Next, assume that

ea = E m cos t , ia = I m cos( t )

eb = E m cos( t 120 0), ib = I m cos( t 120 0),

ec = E m cos( t + 120 0), ic = I m cos( t + 120 0), (6)

where a phase shift is included in the line currents forthe purpose of generality.

It follows from equations (5)-(6) and under steadystate conditions that one can obtain the following solution:

mk (t )2 =

1vC 1

[ E mcos ( t (k 1)1200)

R1 I mcos ( t ( k 1)1200)

+ L 1 I msin ( t ( k 1)1200)] , k = 1, 2, 3 .

(7)

Therefore, the desired closed form duty functions of (3) can be obtained easily. Also, one can see that by usingthe above algebraic approach, coordination among theseven switches can be achieved automatically. Finally, itmay be worth mentioning that for a desired output voltagethe choice of d 0, and hence d 7, is not unique as will be shownin later sections.

III. LOW FREQUENCY DC ANDAC MODELS

For simplicity, define the following space vectorsfirst :

e s =23 (e a + ae b + a

2e c) = E me j t , (8)

i s =23 (ia + ai b + a

2ic) = I me j( t ) , (9)

d s = 23 (m1 + am 2 + a2m3) = d me j

( t ) , (10)

where

a e j2 3 ,

[e a eb e c] = Re{[1 a2 a ]e s} ,

[ia ib ic] = Re{[1 a2 a ] i s} ,

[m1 m2 m3] = Re{[1 a2 a ]d s} .

Thus, application of the above notation to equation(5) yields

d i sdt =

R1 L 1

i s vC 12 L 1

d s +1

L 1e s , (11)

i L2dt =

0

L 2vC 1

1 L 2

v0 , (12)

dvC 1dt =

34C 1

Re[ i s d s*]

d 0C 1

i L2 , (13)

dv0dt =

1C 0 i L2

1 RC 0 v0 , (14)

where * denotes the complex conjugation and ia + ib + ic =0 has been applied to obtain the above equation.

Next, one can transform e s, i s, d s into the synchro-nously rotating reference frame, i.e., define

i s ( I d + jI q)e j t ,

d s (d d + jd q)e j t ,

where I d + jI q and d d + jd q are phasors in the new coordinate.

Then, one can obtain the following equation:

d dt

I d I 1i L2vC 1v0

=

R1 L 1

0 d d 2 L 1

0

R1 L 1

0 d q2 L 1

0

0 0 0d 0

L 2 1 L 2

3d d 4C 1

3d q4C 1

d 0C 1

0 0

0 0 1C 00 1 RC 0

I d I qi L2vC 1v0

+

1 L 1

0

0

0

0

E m ,

(15)

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where

I d = I mcos , I q = I m sin ,

d d = d m cos , d q = d m sin .

Finally, to obtain the DC and small signal models,one can apply the following perturbations:

I d = I d 0 + I d , I q = I q0 + I q ,

vC 1 = V C 1 + vC 1 , i L2 = I L2 + i L2 ,

v0 = V 0 + v0 , E m = E m0 + E m ,

d d = D d + d d , d q = D q + d q ,

d 0 = D 0 + d 0 .

Application of the above equations to equation (15)and neglect of higher order terms yield the following DCand small signal models:

D d =2

V C 1( E m R1 I d + L 1 I q) , (16)

D q =2

V C 1( R1 I q L 1 I d ) , (17)

D 0 =3( D d I d + D q I q)

4 I L2, (18)

V 0 = D 0V C 1 , (19)

I L2 =V 0 R , (20)

d dt

I d I qi L2vC 1v0

=

R1 L 1

0 D d 2 L 1

0

R1 L 1

0 D q2 L 1

0

0 0 0 D 0 L 2

1 L 2

3 D d 4C 1

3 D q4C 1

D 0C 1

0 0

0 0 1C 00 1 RC 0

I d I qi L2vC 1v0

+

V C 12 L 1

0 0

0 V C 12 L 1

0

0 0 V C 1 L 23 I d 04C 1

3 I q04C 1

I L2C 1

0 0 0

d d d qd 0

+ 1 L 1

0 0 0 0T

E m . (21)

From the above DC model, one can understand the

meaning of D0. Basically, it is not only related to V o and V C 1,but also dependent on the AC line currents, I L2 and the dutycycles of the first six active switches. In fact, fromequations (16) to (20), one can obtain the following results:

32 E m I mcos =

V o2

R +32 I m

2 R1 , (22)

D d =2 D 0V 0

( E m R1 I d + L 1 I q) , (23)

D q =2 D 0V o

( R1 I q L 1 I d ) . (24)

Equation (22) actually represents the principle of conservation of power. From equations (23) and (24), onecan see that for a given output voltage V o, D d and D q can beobtained by choosing a proper value of D 0 which satisfiesthe following inequality:

1 D 0 > D m D d 2 + D q

2 (25)

as can be seen from equation (3) in order to guarantee apositive value duty cycle.

Similarly, from the state equation of (21), one can use

Laplace transform to get the desired transfer functions. Forexample, the duty cycle control used to output transferfunctions take the following forms:

T d (s) vo(s)d d (s)

= 1 ( V C 12 L 1

b 1 +3 I do4C 1

b 4) ,

T q(s) vo(s)d q(s)

= 1 ( V C 12 L 1

b 2 +3 I qo4C 1

b 4) ,

T 0(s) vo(s)d 0(s)

= 1 (V C 1 L 1

b 3 + I L2C 1

b 4) ,

where

= s 5 + a 4s4 + a 3s

3 + a 2s2 + a 1s + a 0 ,

a 4 =1

RC 0+

2 R1 L 1

,

a 3 = 1 L 2C 0

+2 R1

RL 1C 0+

R12

L 12 +

2 +3( D d

2 + D q2)

8 L 1C 1+

D 02

L 2C 1,

a 2 =

2 R1 L 1 L 2C 0

+ R12

RC 0 L 12 +

2 RC 0

+3( D d

2 + D q2)

8 RL 1C 1C 0

+3 R1

( D d

2 + D q

2)

8 L 12C 1

+ D 02

RL 2C 1C 0 + 2 R1

2

D 02

L 1 L 2C 1 ,

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a 1 =

3( D d 2 + D q

2)8 L 1 L 2C 1C 0

+3 R1( D d

2 + D q2)

8 RL 12C 1C 0

+ 2

L 2C 0+

R12

L 12 L 2C 0

+2 R1 D 0

2

RL 1 L 2C 1C 0+

R12 D 0

2

L 12 L 2C 1

+ 2 D 0

2

L 2C 1,

a 0 =

3 R1( D d 2 + D q

2)

8 L 12 L 2C 1C 0

+ R1

2 D 02

RL 12 L 2C 1C 0

+ 2 D 0

2

RL 2C 1C 0,

b 1 =3 D 0 D d

4 L 2C 1C 0S +

3 D 04 L 2C 1C 0

( D d R1

L 1 3 D q ) ,

b 2 =3 D 0 D 1

4 L 2C 1C 0S +

3 D 04 L 2C 1C 0

( D q R1

L 1+ 3 D d ) ,

b 3 =

1C 0

S 3 +2 R1

L 1C 0S 2 + (

R12

L 12C 0

+3( D d

2 + D q2)

8 L 1C 1C 0+

2

C 0)S

+

3 R1( D d 2 + D q

2)

8 L 12C 1C 0

,

b 4 = D 0

L 2C 0S 2 +

2 D 0 R1 L 1 L 2C 0

S + D 0

L 2C 0( R1

2

L 12 +

2) .

IV. SOME NUMERICAL RESULTS

To aid the readers understanding of the above theo-retical results, some simulation results are given below.The parameters of the converter in Fig. 1 are listed belowfor reference:

L1 = L2 = 7.5 mH ,C 1 = C o = 820 F ,

R1 = 0.45 ,

output power V o2

R = 4 KW,

AC line frequency 60 Hz,

phase voltage amplitude 220 23

volts,

power factor = 1.0.

The seven active switches are considered to be idealin the simulation process. The sawtooth wave frequency is3 kHz. It may be worth mentioning that the duty ratio of d 0is generated through comparison with a negative slopesawtooth wave. Thus, the duty ratio of S 7 is obtained byinverting the output again.

As for the duty ratios of S 1, S 2, , S 6, each one is thesum of two parts. The first part is obtained by comparingmk (t ) with a positive slope sawtooth wave. The second partis the same as that obtained from d 0.

Fig. 3 shows some results obtained using the con-verter for V o = 500V, d 0 = 0.58, and P o = 4KW . In Fig. 3(a),one can see the duty cycle controls. Figs. 3(b) and 3(c)show the waveforms of the a-phase voltage and current as

well as the output voltage. Similarly, Fig. 4 shows thecorresponding step down case, where V o = 48V, d 0 = 0.11

(a)

(b )

(c)

Fig. 3. (a) Waveforms of the gating signals of all the switches under a stepup operation. (b) Waveforms of the phase voltage ea(t ) and phasecurrent 5 ia under a step up operation. (c) Waveforms of the outputvoltage o(t ) under a step up operation.

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Finally, the duty cycle control used to output voltagetransfer functions for the step-up case are given as follows:

T d (s) =

1.332 10 9(s 1431)( s + 60)

u(s),

T q(s) =

5.022 10 11 (s 1431)

u(s),

T 0(s) =

1.402 10 8(s 2.133)( s 2 + 115.6 s + 1.556 10 5)

u(s),

where

u(s) = ( s + 7.75) ( s2 + 111.1 s + 1.488 10 5)

(s2 + 20.73 s + 2.236 105).

Similarly, the corresponding transfer functions forthe step-down case with V o = 48 V , D 0 = 0.11 and P o = 4 KW are listed below:

T d (s) =

2.526 10 8(s 1431)( s + 60)

d (s),

T q(s) =

9.5243 10 10(s 1431)

d (s),

T d (s) =

7.0953 10 7(s 7.038)( s 2 + 101.4 s + 1.84 10 5)

d (s),

where

d (s) = ( s + 2073) ( s2 + 93.1 s + 2657)

(s2 + 106.6 s + 1.854 105).

For convenient reference, Figs. 5 and 6 show theBode plots for the two cases, respectively.

From the above results, it is very interesting to seethat a right half plane zero also exists in the transferfunctions of the proposed AC/DC converter. This impliesthat the converter dynamic has a time delay from thecontrol to the output.

V. CONCLUSIONS

In this paper, closed form duty cycle control laws forthe active switches of a three phase step up/down converterwere first derived by using the familiar state averagingtechnique. These analytic duty cycle controls are veryuseful for analysis and simulation, and for obtaining engi-neering insight into the operation of switched circuits.Application of the control laws to the proposed convertercan achieve a clean sinusoidal input current, controllablepower factor, adjustable DC voltage and bidirectional

power flow capability. Next, after a coordinatetransformation, a low frequency DC model and a small

(a)

(b)

(c)

Fig. 4. (a) Waveforms of the gating signals of all the switches under a stepdown operation. (b) Waveforms of the phase voltage ea(t ) andphase current 5 ia under a step down operation. (c) Waveforms of the output voltage o(t ) under a step down operation.

and P o = 4KW . One can see that application of the derived

open loop duty ratio control laws indeed works very well asexpected.

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(a)

(b)

(c)

Fig.5. (a) Bode plots of T d (s) under a step up operation. (b) Bode plotsof T q(s) under a step up operation. (c) Bode plots of T 0(s) undera step up operation.

Fig.6. (a) Bode plots of T d (s) under a step down operation. (b) Bode plotsof T q(s) under a step down operation. (c) Bode plots of T 0(s) understep down operation.

(a)

(b)

(c)

signal model are derived by using a small perturbation and

linearization process. Depending on the output variableand input variable, of interest, one can apply Laplace

transform to obtain the desired transfer functions.

Finally, some numerical simulation results were pre-sented to show the validity of the theoretical results. It is

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interesting to see that a right plane zero also exists in theduty cycle control to output transfer function.

REFERENCES

1. Pan, C.T. and T.C. Chen, Modeling and Design of aThree-phase AC-to-DC Converter, J. Chinese Inst.

Electr. Eng. , Vol. 1, No.1, pp. 1-11 (1994).2. Mao, H., F.C. Lee, Y. Jiang and D. Borojevic, Review

of power factor correction techniques, Proc. 2nd Int.Power Electron. Motion Contr. Conf. , Hangzhou, China,pp. 9-21 (1997).

3. Oishi, H., H. Okada, K. Ishizaka and R. Itoh, Single-phase Step Up/down Rectifier with Improved SupplyCurrent Waveform, IEE Proc . Electr. Power Appl .,Vol. 44, No. 1, pp. 6-12 (1997).

4. Shieh, J.J. and C.T. Pan, Rom-based Current Control-

ler for Three-phase Boost-type AC/DC Converter, IEE Proc. Electr . Power Appl ., Vol. 145, No. 6,pp. 544-552 (1998).

5. Pan, Ching-Tsai and Jenn-Jong Shieh, A Family of Closed-form Duty Control Laws for Three-phase BoostAC/DC Converter, IEEE Trans. Ind. Electron ., Vol.45, No. 4, pp. 530-543 (1998).

6. Itoh, R. and K. Ishizaka, Three-phase Flyback AC/DCConverter with Sinusoidal Supply Currents, IEE Proc .Part B , Vol. 138, No 3, pp. 143-151, May (1991).

7. Pan, C.T. and T.C. Chen, Step Up/Down Three-phaseAC to DC Converter with Sinusoidal Input Currentsand Unity Power Factor, IEE Proc. Electr. Power

Appl ., Vol. 141, No. 2, pp. 77-84 (1994).8. Pan, C.T. and J.J. Shieh, A Single Stage Three-phase

Boostbuck AC/DC Converter Based on GeneralizedZero Voltage Space Vectors, accepted, IEEE Trans.Power Electron. , (1999).

9. Pan, Ching-Tsai and Jenn-Jong Shieh, New SpaceVector Control Strategies for Three-phase Step Up/ Down AC/DC Converters, accepted, IEEE Trans.

Ind. Electron. , (1999).10. Middlebrook, R.D. and S.M. Cuk, A General Unified

Approach to Modeling Switching Convertor PowerStages, Proc. IEEE Power Electron. Specialists Conf .,pp. 18-34 (1976).

Jeng-Yue Chen was born in Tainan,Taiwan, Republic of China, in June1963. He received the B.S. degree inelectrical engineering from theNat ional Taiwan Inst i tu te of Technology, Taipei, Taiwan, in 1988,and the M.S. degree from NationalTsing Hua University, Hsinchu,

Taiwan, in 1990. He is currently a Ph.D. student at theNational Tsing Hua University. His research interests arein power electronics and motor control.

Ching-Tsai Pan was born in Taipei,Taiwan, Republic of China, in Octo-ber 1948. He received the B.S. degree

in electrical engineering from the na-tional Cheng Kung University, Tainan,Taiwan, in 1970, and the M.S. and Ph.D. degrees f rom Texas TechUniversity, Lubbock, in 1974 and

1976, respectively, all in electrical engineering. Since1977, he has been with the Department of ElectricalEngineering, National Tsing Hua University, Hsinchu,Taiwan. From 1986 to 1989 and from 1989 to 1992, he wasthe Director of the University Computer Center and theComputer Center of Ministry of Education, respectively.From 1994 to 1997, he served as the chairman of theDepartment of Electrical Engineering. He has been therecipient of outstanding teaching award and outstandingresearch award for several times. He is a member of IEEE,ICE, IEE, ACS, ICS, Phi Tau Phi, Eta Kappa Nu, and Phi.Kappa Phi. His research interests are in the areas of powerelectronics, motor control and numerical analysis.

Yi-Shuo Huang was born in Hualien,Taiwan, Republic of China, in March1974. He received the B.S. degree inelectrical engineering from theNational Sun Yat-Sen University,Kaohsiung, Taiwan, in 1996. He iscurrently working toward the Ph.D.degree at the National Tsing Hua

University. His research interests are in three-phaseAC-DC converters.

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