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Stability Analysis of a Switched ModeInverter using 'Cuk Converters

Chin - Yuan Hs uE.E. Department, National Kaohsiung Insti tue of Technelogy

415 Chien-Kung R d. KaohsiungTaiwan R.O.C.

Abstract - A four-quadrant DC to ACswitched-mode inverter, using 'Cuk con-verter, is analyzed. Also, a feedback loop isused to regulate the output voltage and pro-duce purely sinusoidal voltage. First, thesmall signal dynamic model of the inverter isderived using state-space averaging method[1,2]. Then, based on this model, threeapproaches are used fo r the stability an alysis:(1) Frequency do main analysis; Bode-diagramis used to design th e compensator and to deter-mine the relative stability and bandwidth ofthe closed-loop system. (2 ) Root locus analy-sis; According to the perturbation of the DutyCycle, D, the root locus are plotted. Usingthese plots, the behavior of the system isinvestigated. (3) Step-response; This test isused to illustrate the relative stability andbandwidth of the system. Finally, some com-puter simulation and experimental results arepresented.

Lfst of symbols

= quiescent DC output voltage= DC voltage source= quiescent duty ratio= load resistance= inductor coil resistance= inductance= mutual inductance= instantaneous current= instantaneous voltage= instantaneous duty ratio= perturbation of duty ratio

f,Ae

trXXAX

perturbation of control voltageperturbation of output voltageinstantaneous output voltagequiescent output voltagepeak value of the ramp voltageLaplace transform variablegain crossover agnular frequencygain crossover frequencyperturbation of error amplifieroutput voltagerise time of the step responseinstantaneous state space vectorquiescent state space vectorperturbation of state space vector

I - IntroductionHigh quali ty switching inverters are desir-

able for many ind ustrial applications, like com-puter systems and aerospace power supplysystems. For the conventional DC to ACinverters, there are some drawbacks amongthem. For example,the undesirable disadvan- tages of the conventional inverters used inmotor drives [3], uninterruptible power supplysystems [4 ] and the power conditioners thatinterface with photovoltaic arrays and utilities[ 5 , 6 ] are harmonic heating, requiring a largechoke inductor, and severe electromagneticinterference problems. Hence, a four-quadrantDC to AC switched mode inverter, using 'Cukconverter [ 7.8 1, is proposed and analyzed.

In order to improve the quali ty of the out-put voltage, a feedback loop is used. Hence,

0-7803-1859-5/94/$4.00 a 1994 IEEE78 5

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this inverter not only has all the advantages of'Cuk converters [8], such as small size, hghtweight, high efficiency and the excellentnonpulsating current characterisics in both theinput and output ports bu t also can providepurely sinusoidal voltageT he main pur- of this paper is toanalyze the stab ility of the feedback loop,basedon the perturbation of the duty cycle Thissmall signa l dyn amic models is derived usingthe state space averaging technique [l, 21. Allthe analyses are performed in s ta tespacematrix equations And the approaches [9 ]include : 1) Freq uen cy domain analysis, i. e .Bode diagram method, (2) Root-ocus analy-sis (3 ) Step response tes t Then some resultsfrom computer simulation are given for com-pariso n Finally, some conclusions are offered,

tliV b e t t

11. Configuration of power stage

- -V b u rI2I"02 01" 02 2-0

Th e power stage is configured on the' Cuk converter [1. he 'Cuk converter isshown in Fig.1

The problem of generation of bipolarvoltage at output is solved by the symm etricalpush-pull method as shown in Fig. 3.

Fig. 3 the four-quadrant Cuk converter.The voltage gain ca n be derived as follows

Assume that the two converters are operatedout of phase Th en the output voltage can beobtained as

Th e duty ratio to output characteristics isshown in Fig.4. It ca n be seen from Fig.4 thatif the duty ratio is varied around the D=O.5point, i e. D=O. 5+d( t) , then there will be anA.C. output voltage across the outpu t termina ls

Fig. 1 The basic 'Cuk converter.By using voltage-second balance method,

one can get the following voltage relation forfor the continuous conduction mode:

t1 0. 1 6. 2 1.3 8.1 1. 5 0 . b 8 . 7 8.0 8. 9 IB

Fig. 2 The tw oqua dran t 'Cuk Converter.

Fig.4 duty ratio to output voltagechara cteristics of th e four quad rant' Cuk converter. (0.356DS .65 fo rstable operation)

A small-signal model is derived for thispower stage in the following section using State-Space method Based on this small-signal model,a stability analysis, including the coinpensatordesign of th e feedback system , is performed.

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111. Small signal dynamic model ofthe inverter s y h mFig.5 shows the block diagram of the inver-The detailed configuration of com-er system.

pensator and PWM circuit are given in Fig.6.And the power stage is a four-quadrant 'CukConveter as shown in Fig.7.

F i g . 5 Regulated DGto- AC Inverter system

V - 2 . 8 vf r = Z B K H zi i i ' HzS I N Y R V EFig.6 Compensator and PWM Circuit

F i g . 7 The power stage of the regulated( 1) Modelling of the power stageoff, the sta te equations are:

inverter system.During the period of Sl, S4 on and S2, S3

L PI00 -R u 0

0 0 -10 1 00 -(R+2R,

LA,

Y C x EI U (5)Simililarly, during the period of S2, S3 onan d Sl, S4 off, th e state equations are:

Az X- -vo=R&=[O 0 RL 0 01 I! 1 + 0

VIPJY GT X Ea U (7 )

Because of th e curren t- bidirectionality, th epower stage is operated in the continuous con-duction mod e Hence, the averag e behavior ofthe inverter system can be obtained, using theState - Space averaging method , as folows[lo-141 :

Pk= A x+ Buy =CTx+ Euwhere

P=Pid+P2(1-d)A=Aid+A?( 1-d)B=Bd+E$(l-d)CT=C:d+GT( 1-d)E=Eid+ 1- d)To obtain the small-signal model, on need

to perturb the above equations around thequiescent operating wint(D, , ,, Vo), . e.d= D +a

x=x+^x h (v#=Va+GR u= U+u )v.=V.+v. (y=Y+yA )h

A h h hWhere d, x, v, and vo represent thecorresponding small perturbation% respectively.For th e simplicity of analysis, one can assum eVFO. By sub stit utin g EQs lo) , (U ) into Eq. (8)and recognizing that in quiescent steady state,X=O , one can obtainP&AX+ BV, + A? + [ ( AI - &) X+ BI - &) V, ] ~

h

+terms containing products of ^x an d a ( tobe neglected) (12)where

A=AID+A?(l-D)

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whereA= AID+ 1-D)B=&D+&(l - D) (13)The steady-state equation can be obtained

from a.12) by setting all the time derivativesand the perturbation terms to zero. Therefore,the quiescent steady- state equation is

(14)0 = AX + BV,

and therefore, in Eq. (12)

Similarly, substituting Eqs ( lo) , (U ) intoEq. (9) results in

whereIn quiescen t steady- state,

CT=C,'D+GT(l-D)

Y=CTXand therefore,

where

Taking Laplace transformation of both Eqs(E)nd (20), on e can obtain the control - to -output transfer function,

hTd s)= =CT( P-A)-lK+ (n)d( s)

The transfer function Tm(s) of the PWMmodulator can be derived as [15]

where Vr," is the peak value of the rampvoltage(3 ) Modelling of the Compensator C16-181

A simple cornyensator using an operationalampliiier and an RC network is shown in Fig.8.

R 3 + V c 2 -e R 1

.$Z f

6Fig.8 The circuit diagram of the comwnsator.

The dynamic equations are

r - 7

h E].=-v,=[O 0 -11and the corresponding transfer function is

h

Tc (s)=-%-

where

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(4 ) Cascading of tw o blocks into oneblock Bystem

The dynamic equations can be obtainedusing the principle of casca ding of two blocks,as shown in Fig. 9 [19,20].

(open- loop sysLem)Fig .9 Cascading of two blocks into one blocksystem.%=A$,+ Baa, Pb=A9,+&Cl,,;.=C&+BG. y,=Cbji.b+Il;;illb (27)

Th e result of the cascading of th e twoblocks is [19,20]

( 5 ) Q" i c model of the cJ&-loopsystem

Fig.10 shows th e block diagram of theclosed-loop system.

Fig.10 Th e block diagram of the closed-loopsystem.

The dynamic equations of the closed-loopsystem can be obtained by using some matrix

algebra as follows [19,201:hx= Af2+E$r( t)y=Gx^+Qr( t )

whereAI=A-BF(I+EF)-'C, B=B-BF(I+EF)-' ECf=( + EF)-IC, E,=( I+EF)-'E (31)

IV Frequency domain analysis of theinverter systema. Bode diagram of th e open-loopuncompemted systemCombining Eqs (22) an d (23), the Bode

diagram of ?,/i? ca n be obtained, usingMATLAB software [211, as shown in Fig. 11

m

. .. . .. .

H I

Fig.ll Bode diagram of th e own-loop system( a t D=O. )

Froin this diagram, the phase margin andgain crossover frequency can be recognized as

P.M =23.423 degreew g =U700 , f,=2 . 9 K H z (32)

b. Bode diagram of th e oompemtorThe Bode diagram of Eq. (26) multiplied

by the feedback gain, 0.16, is plotted as shownin Fig.12. T h e poles, zeros and gain of Eg. (26)ar e calculated as in the following:

21F 27r2 n

- -fd=J'=O Hz f,=--@=3.62X101 Hzf,=-'h =U 8X 101 Hz k( gain)= l. x 106f d = z = 5 8 . 9 5 H z 9 f a = - = W . 8 6 Hz (33)- 21F

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. .llrrltd. r..)onr. - e108, : . . I. . .0 50 ........ ............ ............ ............ .............................. j............. .. . ..

. .A -1 18-1 tee ie1 ss2 101 10' 183 186

HI

.............

............. ............. ........ .......... .-11010-a 10-1 iz 101 181 1113 1st 10s 10'

H

Fig.12 Bode diagram of the compe nsatortranster function multipled by the feed-back gain, 0.16. ( a t D=O. 5)

On the phase plot of the Bode diag ram,the phase lead around 1KHz is used forcompensation, to increase the phase margin ofthe closed loop system.

c . Bode diagram of the loop gainThe Bode diagram of the loopgain isobtained, using MATLAB software, as shownin Fig.13.

nr n l t u d s ma o w e loo 11 " 4-1.5

. ..................... ............ ........ ............. . .........!...........e............. .1. .. .. .

......... . ........... ....... .:.. ................. ............. ... .. .HZ

Fig. 13 Bode diagram of the loop gain ( a t D=O. 5)From the phase plot of Fig.13, It can be

seen that the phase margin is increased by thecompensator.

P.M. =58.7"w p l .X38E+004 , f S = 1 . 8 l KHk (34)

Because of t he increa se of the phase marginfroin 23.4' o 58.7" , the stability of th e closedloop system will be improvedd. M e di agramof the clwed loop

compensafedThe frequency response of the closed loopsystem is given in Fig. 14.

- u o I I10-1 IOs 18' 1 11 IE' 18' 18'Hr

?h.ie n. o w 0 1=0.5, 188 .......... ....,... :. . . . . . . . . . . . . . .................. ."........................., . . . .. ... .. .. .:2111 ..........:............. ......... : ............:............ 1 .......... . . . . . .i ..........

10-1 18-1 iZ 181 181 103 18. le' 11.( . ..-3 s

Ha

Fig.14 T h e frequency response of the closedComparing Fig.11with Fig.14, It can be

shown that the bandwidth is increased frombelow 1 RH z to above 1 K H z

loop system. (at D- 0.5)

V Root locus analysis of the invertersystem

a. Root-locus of the open loop systemUsing duty ratio D as variable parameter,the root locus of the open-loop system,according

to Es. 221, can be plotted as shown in Fig. 5.,e,i , .Root l a e u . ~ a p a n loop , ,

-3 -2.5 -7 -1 .5 -1 -6 .5 0104 I r l 8 ~(a ) Root locus of pole p,

Root L~cut-opsn loop

P=8'99888znaa

I

(b) Ropot locus of polesp ~ bX root positions a t D=O. 5NG root pasitions a t D=O.99

Fig.15 Root locus of the open loop system

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From Fig.15, It can be seen that the openloop system will be stable However, due to thenon-linearity of the 'Cu k converter, as in Fig.4,the output voltage will not be sinusoidal.b. Root locus of th e clased h pmmpemted@

In order to make the output voltage puresinusoidal, an integra tor type comp ensator isintroduced to increase the type of the systemfrom type 0 to type 1 9] . Using the duty ratioD as variable parameter, the root locus of th eclosed loop system, according to I2q. (31), ca nbe obtained as shown in Fig.16.

- 1881

(d) Root locus of pole pl31 ' . ' , ' ' I

-1 - 2 .1 -2 -1.5 -1 -8 .5 B 1 .5 1 1.5 2ra.1 "18'

(a) Root locus of poles p1 & p2X l B ' Root Locur-clod loop

t d . 9 9 I=E 5

i - 1 - B4.:is, ,4

-6-1 -D.5 I 8.5 I ' 1.5 2*I I X I S '

( b ) Root locus of poles p3 & p4

-288' I-sane -4588 -4188 -3515 -3888 -2588 -2883r e a l

(e) Root locus of pole pax root positions at D=O.+ root positions at D=O. 5m root positions a t D=0.99Fig.1 6 Root locus of th e closed loop syste m.

Fo r D greater than 0.65, the root locuswill be in the right 'hal f plane And the system'will become unstable. Therefore, th e duty ratioD had better be limited to 0.351D40.65o ensurethe stability of the system. The maximum allowableoutput voltage can be estimated as, using eq. Z),

c. Dimmiin of the instability of th e cl&l o o p s y S t em f o rD

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-6 -5 -4 -3 -2 -I (I 1re11 xI8'

( a ) Ze r o l oc us o f zl, zz a n d z3

1

I

._=>EDr8 61 D-8.65

14888-ma I-6 -5 -4 -3 -2 -1 1 1 214888-ma I-6 -5 -4 -3 -2 -1 1 1 2r ea Bias

( c ) Ze r o l oc us o f 23Fig. 17 Zero locus of the power stage using 'cuk

convertersFrom Fig.17, It can been Seen that when

D>O.65, th e zero locus of the ope n loop uncom -pensaed system (i. e. power stage) will move intothe righ half plane Hence, the root locus ofthe closed loop system will also ehter into theright half plane, if the loop gain is large enoughAnd the stable operation range must be limited to0.35 4 D d 0.65 in case only a simple classical

compensater is used It is suggested that in orderto ob tain global stability of the closed loop system,the modern control design method known as poleplacement should be adopted [al.

VI Step response testAll the above stability analysis, such as

Bode diagram and root locus, are derived from th etheoretical viewpoint From the experimental pointof view, the step respo nse test can be used to. illus-trate t he relative stability, and bandw idth ( ie. th einverse of rise time ) of the closed loop sysem.

Fig.18 shows the step response of the openloop system (PW M+Inverter). And the steprespon* of the closed loop system (regu latedinverter system) is shown in Fig.19.

I0 # . l E Z # . # E4 1.8116 1.888 1.11 8.012 1.114 1.116 8.118 0.02

.a.Xn(;

tr=O. 8msovershoot=18%

Fig.18 Step response on the open loop system

8' I8 E.882 1 . 1 E 4 1.806 1.188 1 . E I 8.012 1.014 8.01b 0.018 0.82seconds

tr= 0.3msovershoot= 8%Fig.19 Step response of the closed loop system

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T h e rise time, tr, and overshoot of bothsystems can be estimated a s followfor open loop system

t r = O . 8m sovershoot=18%for closed loop systemtr=O. 3msovershoot= 8%

Hence,both the r ise time and overshoot arereduced by the feedback loop.VI1 Some computer simulation andexperhnertal results

Th e simulation results are obtained usingthe simulation software PSPICE [231. Fig. 20and21 show the simulation and experimental resultsfor the open loop system and the closed loopsystem respectively. Th e results show that theoutput voltage of the closed loop system is purelysinusoidal while the output voltage of the openloop system is not

*.U- .....................................................................

SEL))!.2.w+...................,...................,.............................,ZIV-.....................................................................W61)

-2 ;.....................................................................81 L s l k s 1 % ~ 2k . 1 3 b s 3 L sD -Ut9,15)

T1.e

reference voltageoutput voltage (0.355 S 0.65)experimental output voltage waveform.(2ms/div. , OV/div. ) (0.356 SO. 65)

1. w .......................................................................1-3.u ,.........I.... .... - ........ ..................,........._.........

U1611zu.......................................................................

......,........,.........,........I ........,....... ............I. Smr lb. 159, t b . &r *s 3 h *0 U(3.19)

Fig.2 l Simulation, and exg rim ent al waveform forthe closed loop system.(a) reference voltage.(b) output voltage(c) exprimental output voltage waveform.

(0.35sS .65)(2ins/div. , OV/div. ) (0.35sd 0.65)

For duty ratio D greater than 0.65, herewill be some oscillation phenomenon as shown inFig.22. A t the tip of the sine waveform, someripple voltages, due to oscillation, c an be seen.This oscillation is predicted by the root locusplot of Fig.16. In Fig.16, for the duty ratio D>0.65,the locus will enter the right half plane andthe closed loop system will become unstable

Fig. 20 Simulation, and experimental waveformfor the open loop system.

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Fig.22 Simulation, and experimental waveformof the closed loop system fo r L O. 65.(a) reference voltage (b ) ou tput voltage.(D- =0.68) (c) experimental outputvoltage waveform (Zms/div. , OV/div. )(D m d . 8)

V I1 ConclusionsA four-quadrant switched-mode inverter,

using C uk co nverter, is analyzed By usingst at e space averaging technique, the small signaldynam ic model of the inverter is obtained Afeedback loop is used to improve the quality ofAC output voltage In order to ensu re th estability of the feedb ack loop, a simplecompensator is designed Three systematicapproaches are adopted to analyze the stabilityof the feedback loop, namely (1) Frequencyresponse i e.analysis and (3 ) step response. All th eanalyses are performed in state spa ce matr ix

Bode diagram, (2) Root locus

equations, using the MATLAB softwareT h e analytical results show that for duty

ratio, D(0.35 or D>O.65, th e feedback loop willbecome unstable. Interestingly, both resultsfrom computer simulations using the PSPICEsoftware , and experiments demonstrate thatfor0.35 S D 4 0.65, the output voltage ispure ly sinusoidal, while for D(0. 35 or D>O.65,the tip of the output v oltage has someoscillation phenomena And this confirms theanalytical prediction that for duty ratio D(0.35 orD>O.65, the feedback loop will be unstable.Hence, the stable operation range of th einverter wth feedback loop is limited toduty ratio 0.35SD40.65. nd t h i s meansthat th e peak value of th e output voltage mus tbe limited to = L31Vg where V, is th e DCvoltoge source.

In addition, for DO.5 or D>O. 65, th e ins ta-bility of the closed loop system is due to th emovement of zeros of the C uk converter ( i . th epower stage) into the right half s plane It issuggested that in order to obtain global stabilityof the closed loop system, the mode rn controldesign method known as pole placement or statefeedback control should be adopted

References

R D . Middlebrook and S. Cuk, A Generalunified Approac h to Modelling Switching-Converter Power Stages, 1976 JEEEPower Electronics Syecialists ConferenceRecord PP.1834.Seth R. Sanders, J. Mark Noworolski,Xiaojun 2. Liu, and George C Verghese,Generalized Averaging Method for PowerConversion Circuits, IEEE Transactions onPower Electronics vol.6 No. April 1991pp.251- 259.F. Barzegar and S Cu k, Solid-state drivesfor induction motor : early technology tocurrent research, Proceedings of theRegion 6 Conference, February 15.18, 1982

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Anaheim, CA.H. V. Manjunath, V. T. Ranganathan andB. S. Ramakrishna Iyengar, " Designconsiderations for high switching frequencyPWM inverter for UPS application usinghybrid darlington switches", LEEE PESC,1985, pp.10251032.Robert L. Steigerwald, A. Ferraro andFred G . Turnbull, "Application of powertransistors to residential and intermediaterating photovoltaic array power conditioners' I , E E E Transactions on IndustryApplications, vol. IA- 19, No. 2, March/Apd1983, pp. 254-267.A Khoder, K Al-Haddad, V. Rajagopalan,"Innovative utility interactive D. C. to A. C.power conditioning system", E E E P E ,1985, pp.ll5LL255.N. J. Tsacoumangas, "The implementationof FET four-quadrant 'Cuk converter ,Procewlings of Powercon 17 . F-3, p p 1-12,1984.F. Barzegar and S. 'Cuk, "A new switched-mod e amplifier produces clean three-phasewwer" , The Ninth International Solid-statePower EIectronics Conference July 13-15,1932, Washington DC.Benjamin C. Kua, Automatic Control Systems, Sixth Edition, Prentice Hall, Inc.,Englewood Cliffs, N. J. 07632, 1991

[lo] Middlebrook, R D. "SmallsignalModellingof Pu lse Width Modulated Switched- ModePower Converters", IEEE. Proc voL 76,No.4, pp.343--354, April 1988.[U] . Martinez, A. Poveda and h4 Miguel, "Modelling and analysis of the 'Cukconverter using the discrete impulseresponse m e t h d , EEE Pramding , Pt G,

[E!] . R. K Chetty, "Modellingand design ofswitching regulators", ZEEE Transactionson Aeruspace and Electronics system, voLAES18, No. 3, Ma y 1982, pp.333343.

[13] R. J. Dirkman, "Generalized sta te spaceaveragi&', IEEE PESC, 1983, pp. 285294.

[14] D. 1. Shortt and F. C . Lee, "Extensions ofthe discrete- averag e models for Conve rter

VOL 133, NO.2, April 1986, pp. 77-83.

Power Stages" EE E Transactions onAerospace and Electronic Systems,vol. AES-20, No. 3, May 1984, pp.279291[lq R D. Middlebrook, "PredictingModulatorPhase La g in PWM Converter FeedbackLoops, 8th Znternational Solid-Sta e PowerDallas, TXloop", Unitrcde Switching Regulated PowerSupply Design sermnar IvIanuaJ, 1985, pp.c1-1-c1- 31

[17] H. Dean Venable, "T he K-Factor : A newMathematical Tool for stability Analysisand Synthesis,"Proceedings of Powercon 10March n24, 983, Saa Diego, C A

[181 Frederik E. Thaw " A feedback Compensatordesign Procedure for switching regulator",ZEEE Transactions on Industrial ElectronicsConference, Instrumentation, VO L IECI-26, No. 2, Ma y 1979, pp. 104110.

[19] William L. Brogan, Modern Control Theroy, Quantum Publishers, Inc. 257 PARKAVENUE SOUTH, New Youk, N. Y . ,10010, 1974.

[20] homas KAILATH, Linear Systems,Prentice-Hall, I nc., Englewood Cliffs, N. J.07632, 1980.

[Zl]Charles L. Philips and Royce D. Harbor,Feedback Control Systems, Second Edition,Prentice-Hall Inc. , Eng l e w d Cl i f f s , N. J.07632, 1991, Ch apt er 10.[22] The Student Edition of MATLAEfor M SDOS Personal Computerq Pren tice Hall, Inc. , Englewood Cliffs, N. J. 07632, 1992.

[231 PSPZCE-The Design Center-User's Guide,Microsim Corporation, 20 Fairba nks, Irvine,California 92718 U. S.A . , 1992.

E l ~ t r ~ n i c ~anferenCe, A p d 27-30, 1981[la Lloyd H. Dixon, Jr. ,"Closing the feedback

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