A State-space Modeling of Non_ideal

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V D-1A STATE-SPACE MODELING OF NON-IDEAL DC-DC CONVERTERS

C. T.R im G. B . Joung and G. H . Cho

Dept. of Electrical Engineering, Korea Advanced Insti tute of Science and TechnologyP.O.Box 150 Chongryang, Seoul 131, Korea (Tel. 966-1931 ext. 3723)

ABSTRACTA modeling based on the practical switches with finiteturn-on, turn-off, delay, storage and reverse recovery times isproposed. The switching effects on the sta te equation, systemstability, dc gain and efficiency are shown for the boost, buckand buck-boost converters. The effect of switching time varia-

tion by current on system stability is given as an extension ofthe storage time modulation effect. And the effect of averageduty cycle variation on system stability is newly introduced. DCgains of the boost and buck-boost converters are shown to bemuch deteriorated by switching loss. It is also shown that noadditional resistance is generated by switching in contradictionto previous work. Besides, it is found that buck converter issuperior to others from the viewpoints of the system stabilityand dc gain, and the efficiencies of all the converters aremaximized when the dc gain is close to unity. Previous relatedpapers are improved and unified using a new state-spacemodeling. Analysis results are very simple in spite of the com-plex switching waveforms. This new modeling is believed to bevery useful fo r the high frequency or high power applicationswhere switching effects become dominant.

I. INTRODUCTIONThe modeling of networks which contain switches hasdrawn much attention because of the unusual properties ofswitches in comparison with other circuit elements. This is thereason why there are so many modelings in the power electron-ics which are not found in other fields.The difficulties in the modeling of switched networks aremainly due to the non-linear and time-varying nature ofswitches. Furthermore the control mechanism of switching systems is far different from ordinary systems. The difference isshown in Fig. 1. By switching, the topologies of networks are

changed and the system matrices A , B , C and D are, accord-ingly, changed. So to control the output of switching systems,the system matrices should be varied by switches instead of thesource as done in ordinary systems. This is why switching sys-tems are non-linear and time-varying.Most modelings in power electronics are mainly intendedto convert this non-linear time-varying problem to an easierform [l-51. However they are inadequate for some delicateproblems since they are based on ideal switches. So a fewpapers based on non-ideal switches are proposed [6-91. Focus-ing on the switching effects for the related previous works, the

storage time modulation effect on small signal dynamics isfound in [6]. And switching time effect on efficiency assum-ing ideal filters is investigated in detail in [7]. A substitutionof switching effect to an equivalent resistance is done in [8].Switching effect on dc gain degeneration is discussed in [9] .

PESC '88 RECORD (APRIL 1988)

input k= A ( t) x f B (t) Uduty function1 y=C(t)x+D(t)u 1

Jt disturbance(a) a switching system

input i = A x + B u ( t )- I y

A B disturbancesC , D(b) an ordinary systcm

Fig. 1 Comparison of systems.

However they explain only parts of switching effects withsomewhat rough or complex expressions.This paper is intended to unify and extend the previousworks so that this may be a good summary for switchingeffects on power systems. A new state space modeling is pro-

posed to obtain both exact and simple results, which arebelieved to be the most important features of system model-ings. Conventional state space average modeling is based ontwo or three sets of state equations which are valid fo r eachmode. So this is inadequate for the representation of the inter-val between successive modes which appears in the practicalswitching case. Here a time-varying state equation valid notonly for each mode but also for the interval between modes isset up using switching functions with little effort. Then moreextended and compact state-space averaging and perturbationsare taken for the boost converter example. The state-spaceanalyses for the system stability, dc gain and efficiency forthree converters are done and the results are summarized in atable.

II. MODELING PROCEDURE(BOOST CONVERTER CASE)

943 CH 2523-9/88/0000 -0943 1.00 @ 1988 IEEE

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- 1

- Tnon - ideal switches

Fig. 2 Boost converter.

The modeling procedure is shown for the boost convertercase. It is assumed that all the passive elements are LTI andthat the switches have only switching loss. The conduction lossand parasitic resistance loss will be counted in the Part I1 ofthis paper in later. The converter is assumed to operate in thecontinuous conduction mode. Fig. 2 shows the circuit to bemodeled and Fig. 3 shows the switching waveforms, respec-tively.A. Exact State-Space ModelTo derive a state equation, states x1 and x2 are allocatedto the current of inductor L and the voltage of capacitor C ,respectively. Then the derivatives of the states as functions ofstates, source and circuit parameters are found to be

x2RLL X l = u - R , x l - v , C i 2 = i - - . 1)

Note that this state equation is valid also for the non-ideal switch case where the switching waveform can be arbi-trary. What remains is v and i to be expressed as functions ofalready known variables. From the switching waveforms it canbe seen that v and i are the switched portions of x2 and xl ,respectively.v = Sl(t) x2 , i = %(t) x1 (2)

where sl(t), s2(t) are defined as the switching functionsdetermined by

Sl(t) = 2 , %(t)= . (3)x2 X1The switching functions are shown in Fig. 4. Provided that theswitching times are known then the switching functions arefully determined.

By applying (2) to (1) and rearranging it, the stateequation and the output equation of the boost converter areobtained.

1x11 I L L 11x11 L I

(4) is of thc form

-t- --t -- -\

Fig. 3 Switching waveforms of boost converter.

l e th , T 1 h7T

I I

Fig. 4 Switching functions sl(t) and %(t)

Note that the exact model of the converter is of time-varyingform, which is the general property of switching systems. Thisnew model is exact enough to be used for simulations.B. State - Space Averaging and PerturbationTo eliminate the time-varying terms in the matricesA (t) ,B t) and C (t) , a generalized state-space averaging is

taken for them as

.__I

where

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In the boost converter case the matrices are evaluated as

The s1 and are the averages of sl(t) and (t), respec-tively. They are evaluated from Fig. 4 as; Ts1 = - sl(t)dr = 1 h+t l f for Hmi o h ( H m a x (9')d o1

T O% = -jk(t)dr = 1 - h + f for Hmi,{h ( H m a X (9b)

whereQ; td: delay time, t,: rise time, 4: all time, ts : storage time,D; tm: reverse recovery time, sp: reverse peak, h : duty factor,

tt2 2h = h i = l - h , , ti =td +t,.+fr-fs , 5 =td +-- 7-6 -- ,1

f = -, ff,in = (td +$+t,) / T H,,, = 1- t, +$-td )/;(lo)Hmi n,H,, , are determined by the turn-on, off overlapconditions of the switches. And T is introduced assuming thatthe diode reverse recovery charge Qm s propodom1 to itsforward current as

e,=-\i,( t)dr = - *trr 2 -x,(t)- sP frtrr0 2

The inductor current x1 (t) has been assumed to be constantand the T is found to be the life time of the minority carrier.Assuming that the switching times are non-linear functionof the inductor current and the C matrix is constant, then (6)

becomesk = A(h , x ) x+ B(h ,x) U , y = C O . (12)

To eliminate the non-linearity in (12), matrices are expandedin power series neglecting higher order terms as

aA(h ,x)8x1 I xl = x , XI -xi = A , + A 1 . ( h H o ) + A , . (xl -X, )(13a)

andB ( h , x ) = B 0 + B 1 . ( h - H o ) + B 2 . ( x l - X 1 ) . (13b)

Applying (13) to (12) and taking the following perturbationsh = H o + L , x = X o + f , y = Y o + ? (14)

the resultant equations become

( x , + t )= ( A , + A , h + A, ; , ) (X ,+ f ) + ( B o + B 1 h + B 2 i 1 ) U,(15)Y , + ~ ) = C , ( X , + % ) .

Ignoring bilinear terms, (15) is separated into the dc and acparts as~ , =A , x , +B , u , , ~ , = c , x , (16)

andt = ( A , + A,X,W + B , U, w ) % + ( A , x,+B, U,)L9 = C , (17)

(18)where W is defined as

W = [ l O ] .From 16),operating point Y o is determined by setting X, tozero as follows:

A o X o + B o U o = O , Y o = - C o A o - l B o U o . (19)From (17), the transfer function G(s) is determined as

System stability can be evaluated by examining the characteris-tic equation

In the boost converter case, the matrices, A, , A l , . . ., C Oare found to be

L1 l - H o + b fI

A , =

1 - H , +tlf

1--R LRL '13 , = B , = O , C , = [ O 1 1 . (22)

C. Pole Frequency Deviation EffectBy applying (22) to (21) poles are found to beR S * 1 ( - H , +tlf ) ( 1-Hg +%*f ) R, / R ,s2+ ( -+-)s +L C R L L C

where

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5

Fig. 5 Pole frequency deviation of boost converter.

It is observed that R , and 3 are replaced by R,* and 3 byswitching time modulation. This means that the poles can becalculated 5s if there is no switching time modulation by sub-stituting R , Note that the switching time modula-tion effect which is undesirable in practice becomes dominantas X,, , are increased. So it is recommanded that the operat-ing point be low to avoid the undesirable effect. It had beendiscussed in [6] that R , is effectively varied by storage timemodulation. That result is in part true, however, it needs to beextended as described here to give more exact solution. Stillthere remains another switching effect to be explained, the dutycycle variation effyt. Duty cycle is moved from H , toN o - lf or H, -3 as shown in (23). It is apparent thatthis effect will be dominant at high switching frequency.

The normalized natural frequency and normalized damping factor which are the ratios of the practical ones and theideal ones with zero duty factor are

and * .

U,,* and 5 are plotted in Fig. 5for a typical power switch-ing transistor whose current level is hundred amperes. In factthe switching times are varied by current and temperature,however, they are fixed for illustration purpose. Typical valuesof them and resistors selected, which are to be used throughoutthis paper, are as follows:td= 8.0 p s , tr=0.6 p s , t,=8.5 p s , t,=S.O p s ,

t,=1.0 p s , 1=11 0 p s , t l= td+ f+ t,- g = 12.1 p s ,tr 4 d tl3=td+-- i-G--=-8.22 p s , -X,=20Xl p s o h m s ,

3 X,=-15 p s , R , =OS0 o h m s , R , =20 ohms, (26)It can be seen that system poles are non-linear to the duty fac-tor and are considerably deviated from the ideal switching caseas the switching frequency is increased. And the open loop sys-tem stabili ty for a typical switching case is improved byswitching.

The switching effect on system stability can be summar-ized as two effects; one is the switching time modulation effect,which results in the variation of R , , and the other is theaverage duty factor modification effect, which results in thevariation of average duty factor.

dx1

D. DC Gain Deviation EffectTo check the switching effect on dc gain, (19) isevaluated using (22). The dc gain G , is thenYO (I-Ho + f )U0 R ,- + ( ~ - H o + t , f ) ( ~ - ~ o + % f )RL

(27)

G, = -= C A -10 0 Bo=

The dc gain is shown in Fig. 6 as a function of duty factorfor several switching frequencies. The switching times andresistances used here are the same as (26). Note that the prac-tical boost converter as the maximum dc gain point whichoccurs at the point, H o given by

3 0 t A\

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c. Switching Effects on Pole Frequency, DC Gain, andEfficiency Degeneration

Characteristic equation is found to beR Sb 1 1 + R , * J R ,s2+ ( s+ = 0 (39)L C R , L C

where

There is pole frequency deviation due to the switckng timemodulation, which results in a variation of R , to R s . How-ever there is no duty factor variation effect on pole frequency.Therefore it can be said that the buck converter is relativelystrong against the switching effect.DC gain is found to be

The efficiency assuming low ripple is given by

Po yoxi RL H o - f ~ f . ( 4 2 )-.-q=-=--Pi U 1 R , + R , Ho-QfThe efficiency considering the switching loss only becomes

H, 1fHo Qf-fl I R = O = ( 4 3 )

It can be seen that the efficiency becomes the maximum whenthe duty factor is unity. G , fl are shown in Fig. 9 and Fig. 10respectively.IV. BUCK - BOOST CONVERTER

A. Exact State - Space ModelThe state equation and output equation are found to be

ori=A(t)x+B(t)u , y = C x . ( 4 5 )

Note that A(t) and B(t) are exactly the same as those of theboost and the buck converters, respectively. Thus it is expectedthat the buck-boost converter is the general converter amongthe dc converters.B. State - Space Averaging and Perturbation

1.0-

15 k

_ _ - -O

0.5

0 0.5 10

Fig. 9 DC gain deviation of buck converter.

0 10 k 20 k61z

Fig. 10 Efficiency degeneration of buck converter.

and

A , =

f

RS 1-H, +t l jL L

C CRL

-- -

1 - H o +t2f 1--A , =

1O F1c 0-

The averaged and perturbed matrices are

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+U I-

I -- - - - Jnon - ideal switches

Fig. 11 Buck-boost converter.

B , = c = = [ O 11 . (47)C. Switching Effects on Pole Frequency, DC Gain, and

Efficiency DegenerationThe characteristic equation is found to be the same asthat of $he boost converter with a slight modification of thevalue R s .

The tendency is much similar to the boost converter case.The efficiency assuming low ripple is given by

I H o -t ,f)(l-H, + f?T = G , .- =IC R s

R[ - -+ l -Ho +t , f ) ( l -H , +t2f)l O -5f)

50)The efficiency considering the switching loss only becomes

H 1f l - H , +zfI R , = O = (51)l - H , + t l f H o -zf

Note that (51) is just the product of the efficiencies of thebuck and boost converters. So the efficiency of the buck-boostconverter is the lowest one. And the maximum efficiency isobtained at H,* given by

Therefore the maximum efficiency is

0 0.5 1 oFig. 12 MJ gain degeneration of buck-boost conver ter.

1.0

0.5 .

0n 1 k 20 k

H Z

Fig. 13 Efficiency degeneration of buck-boost converter.

V. SUMMARY

Important results are tabulated using the normalized resis-tance r and the average switching functions s1 % ,% and s4only. The results are very simple and regular in spite of thevery complex switching waveforms and the exact description.In a whole, the properties of the buck or boost converters arethe special cases of the general converter, buck-boost converter.The pole frequency, dc voltage gain, dc current gain and effi-ciency of the buck or boost converters are those of the buck-boost converter, whose s1 ,% or s3 ,s4 are set to ls, respec-tively.

Furthcrmore those of the buck-boost converter is just theproducts of those of buck converter and those of boost con-verter if the inductor resistance is neglected.VI. CONCLUSION

The extension and generalization of the state-space aver-age modeling to the non-ideal switching case and the unifica-tion and extension of the previous works have been performed.The switching effects on the system stability, dc gain and effi-

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Table. Summaries of switching effect sI ypc Buck Urmst Buck-bmstitcm

s ta te eq .

1W

F l rz FJA , B ( t ) A ( I ) , B A ( I ) , B ( t )

1

r = R , / R , .I ' = R , I R , , , s l = l - l l o + l l f ,s , = l - l l o + t 2 f , s , ' = l - I I , + t * ' f ,S J = " o - t , f , S 4 = l O - t 2 f

ciency for the buck, boost and buck-boost converters areexplored.The stabilities of all converters are affected by theswitching time modulation effect which results in the variationsof R, and tz . And those of the boost and buck-boost convert-ers are also affected by the average duty cycle variation effectdue to finite switching times. In the sense of stability the buckconverter is strong against switching effect, however the boost

and buck-boost converters are weak in general and thisbecomes dominant when duty cycle approaches to unity.The dc gains of the boost and buck-boost converters aredeteriorated by the inductor resistance and the switching loss.And this effect becomes dominant also when duty cycleapproaches to unity. So it is preferable to use the buck con-verter instead of other converters to avoid serious switchingeffects on dc gain. And it is needed especially to avoid unityduty cycle for the boost and buck-boost converters.The efficiencies of all the converters ar e maximized whentheir dc gains are close to unity. It is recommanded to avoidthe buck-boost converter for higher efficiency.So it can be concluded that the optimal duty factor of thebuck converter is near unity, that of the boost converter is nearzero and that of the buck-boost converter is near half, respec-tively.It is believed that the analysis results are very helpful forthe high frequency or high power applications and for the s u mmary of the switching effects on the system stability, gainand efficiency.

REFERENCES[ l ] R. D. Middlebrook and S . Chc, A general unifiedapproach to modeling switching converter stages, inIEEE Power Electronics Specialists Conf. R e c . , pp. 18-34,1976.[2] P. Wood, General theory of switching power converters,in IEEE Power Electronics Specialists Conf. R e c . , pp. 3-10,1979.[3] G. Vcrghese and U. Mukherji, Extended averaging andcontrol procedures, in IEEEConf. R e c . , pp. 329-336, 1981.[4] G . C. Verghese, M. E. Elbuluk and J. G. Kassakian, Ageneral approach to sampled data modeling for power elec-tronic circuits, IEEE Trans. Power Electronics, vol. PE-1,No. 2, pp. 76-89, April 1986.[SI Khai D. T. Ngo, Low frequency characterization ofPWM converter, IEEE Trans. Power Electronics, vol. PE-1,No. 4, pp. 223-230, October 1986.[6] W. M. Polivka, P. R. K. Chetty and R. D. Middiebrook,State-space average modeling of converters with parasitics andstorage-time modulation, in IEEE Power ElectronicsSpecialists Conf. R e c . , pp. 219- 243, 1980.[7] G . Eggers, Fast switches in linear networks,IEEE Trans. Power Electronics, vol. PB1, No. 3, pp. 129-140, July 1986.[8] R. C. Wong, G. E. Rodriguez, H. A. Owen, Jr. and T.G. Wilson, Application of small signal modeling and meas-urement techniques to the stability analysis of an integrated

switching-mode power system, in IEEE Power ElectronicsSpecialisis Conf. R e c . , pp. 104-118, 1980.[9] J. Sebastian, J. Uceda, The double converter: A fullyregulated two- output DC- DC converter, IEEE Trans.Power Electronics, vol. PE-2, No. 3, pp. 239-246, July1Y87.

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A State-space Modeling of Non_ideal