1. INTRODUCTION

Frequency-Domain Analysisof Switching Closed-LoopRegulators

$ERBAN BiRCA-GALATEANUPolytechnical InstituteBucharest, Romania

An accurate analysis of a switching voltage regulator power

stage is performed by eliminating the mathematical unrigorousness

of the averaging technique. The differential equations are solved for

the ON and OFF intervals to obtain the exact mean values in

function of the values at the beginning and at the end of the

intervals. The accurate relation is used for the integral of the

derivative forming the left member of some equations. A flyback

regulator power stage is analyzed as an example.

Some considerations are made concerning the regulator

compensation using a two-loop control circuit.

Manuscript received April 8, 1986.

Author's address: Institutul Politehnic, Facultatea ElectronicS, SplaiulIndependenlei 313, 77206 Bucurelti 15, Romania.

0018-9251/87/0300-0240 $1.00 ) 1987 IEEE

The averaging technique of analysis of the switchingclosed-loop regulators [1-4] is simple, but inaccurate,especially for modulation frequencies comparable to halfthe switching frequency. The inaccuracy is due to thelack of mathematical rigorousness: in averaging, theintegration and differentiation have been interverted,without the valuability conditions being checked. On theother hand, one-cycle average value x was obtained, ineach equation, as

(dA, + (1 - d)A2)x = dAIx1 + (1 - d)A2x2

where xl, x2 are the mean values on the dT and (1 - d)Tintervals, respectively, and A,, A2 are the coefficients inthe state-space equation considered for the dT and(1 - d)T intervals. In fact, the addition is accurate only ifx1 = x2(--x)orAl = A2 (then x = dx1+ (1-d)x2); thecondition A, = A2 is fulfilled only for the buckconverter, in continuous inductor-current operation mode.For this circuit, the state-space equations are (13), (14),(15)-the same for both the ON and OFF intervals, withthe only difference that in the ON interval a, term Eappears. For the buck converter in discontinuousinductor-current mode and for any other converter, noneof the abovementioned conditions is fulfilled.

An accurate analysis may be performed by solving thedifferential equations system in the dT and (1 - d)Tintervals. In consideration of the subsequent one-cycleaveraging, the input voltage and the load are to beconsidered constant during one cycle. In the solution ofthe differential equations, the initial or final values areused, according to the simplicity of the equations, and totheir physical significance; e.g., 0min = 0 is theboundary condition of the continuous/discontinuousmagnetomotive force (MMF) modes and 0max isnecesssary for designing the transformer core.

The method offers a reasonable compromise betweenthe averaging technique simplicity and the discretetechnique accuracy.

A two-windings flyback regulator power stage(Fig. 1) is analyzed as an example. The linear equivalentcircuit models for the transistor ON and OFF intervals inthe continuous MMF operating mode are shown in Figs.2(a) and (b). Load variations are represented by a currentsource io which injects a disturbance to the converteroutput.

11. THE ON INTERVAL

All the equations and values referring to this intervalhave the index "1.,"

When the transistor switch is ON, the circuitoperation (Fig. 2(a)) is described [3] by the equations:

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL AES-23, NO 2 MARCH 1987240

1 diT

d T Vdt = RLio + (RL±RC)C[V(0)-RLio]

dT1 - exp(R-- R)CFig. 1. Flyback regulator power stage circuit.

V-.-e+tE+ LLPI

(a)

(b)Fig. 2. Equivalent circuits for (a) ON and (b) OFF intervals.

(di RP 4) + 1 Edt LP NPdV B B.dt RLC C

where B = RL

RL + RC

U = BV + RCBio.

NThe superfluous variable i = -p 4 [3] was not use

By solving these equations, one obtains

E ~ ~ ( RP

V = RLio [ -exp (R +R)C-

(RL + RC)C

U = B(V+Rcio) = RLio [1 Bexp (RL+RC)CJ

+ BV(0) exp

(RL + RC)C

In order to obtain averaged values, each term is tointegrated and ratioed to the dIT interval, T being theperiod of the switching cycle and d, the duty ratio. Usithe (4) and (5) forms, one obtains

1 d1T ELo LPR[Pdt= + 0

~~~NpRP R

ex[-( dp )]Td1T LP

(8)

By considering E and io constant during one cycle,the terms containing E or io are the one-cycle meanvalues; consequently they are the mean values on the d1Tinterval, too.

1 dT d( '(d1T) - 4(0)dT- -dt =d1To dt T

not equal to d, dt [d fJo dt1.

This also holds true for dV/dt.The equations for the ON interval, integrated and

multiplied by d1, become

4(dlT) -4(0) 1 [ELP -T T LNPRP

[1 exp( Rp dIT)(2)

(3) V(d1T) V(0)= 1 LiO- V(0)]

Li[ x(RL+RC)Cl

(4)

d1U1 = dlRLio - T

x RLiO - V(O)J

[ P(RL+Rc)Cl

(9)

(10)

(1 1)

(12)

where U1 is U mean value on the ON interval.(5)

Ill. THE OFF INTERVAL

All the equations and values referring to this intervalwill have the index "2." The circuit (Fig. 2(b))

(6) operation is described [3] by the equations:B B

--A-)V--RRcio,be dt Ns N s

ing ) where A = ' R(13)

dt - L C4 - V+ C io (14)dt L sC RLC C

(7) U=B 4 + BV+BRcio. (15)

BIRCA-GALATEANU: ANALYSIS OF SWITCHING REGULATORS 241

By solving these equations, one obtains

exm()exp - 2 tJ

[ ( a 7 bm sinw t bx a -- - - cosWt

2n W n

g + exp-n

x [( _gm sinwt

-g .,cssWtV 2nX W n

(16)

(17)

B RCRsNs RL + RS

V(T) - V(dT) _ mAdT NS amTB/C = Tfexp- 2d2! lLL 2

sin.)d2Tgj

+ (Ns a + c) coswd2T}

RL + Rs 1n =

RL + Rc LsC

)22W2 = n -4

a = Ai)(d1T) + [Rc io + V(d, T)];N5

b = i0 + n(dldT)NsC

B _V(d, T) +Nsc = - io R + Ls 4(dlT)i;

cL- RL L,

BRSg =L-C 0 - nV(dlT)

LsC

if n-m2/4 > 0. From the design conditions andrelations for the power stage [7], one can see thatn -m2/4 is positive for flyback converters.

U RcNsU-= R + V + RcioB L,

-= L )(d1T) -- + V(dT) + + RcioLSn

m fRcNs bmA- exp(- t (a

- -egm) sinwt (RcNs bgcosWt}2n J C-O LS n-J

(19

After calculations similar to the ON interval, theequations for the OFF interval, integrated and multipliedby d2, become

4)(T) - 4(dl T)T

1 m ~ am

Tn p( 22! 1[A 2 b

_ B (cm - sinwd2T

Ns( 2 c)

+ (Aa ~- c) coswd2T}

B Tn exp2 L, 2

-cm sinwd2T2 W

Jr RcNs a _-C) coswd2T)

RL + RC+ RRd2iR

RL + RS(22)

d2T being the OFF interval duration. In the continuous(never-zero) MMF operation mode, 4)min = 4)(T) ' 0;

therefore d1 + d2 = 1.

The equations were not expressed in matrix form,because the coefficients would have been very clumsyand because the duty ratio appears as a linear term, as an

(18) exponent, and as an argument.

IV. THE AVERAGED EQUATIONS

The left members of (10) and (20) are (1IT)[4(d1T)-4)(0)I and (1IT) [4)(T)-4(d,T) ],respectively. By adding them, one obtains (lIT)[o)(T) - 4)(0)]. The notation d4)Idt [3] may be adopted,but in this notation 4) is the value at the beginning of thecycle (4)(0)), not the mean value, and the derivative isdefined at discrete moments in time (t = kT). The same

considerations are taken for (11), (21), and the variableV.

From now on, and V mean 4)(kT) and V(kT). Thecycle is considered to begin with the ON interval. Forsimplicity, the notation 4)* = 4)(kT+ dT) is also used.

In order to calculate the small-signal circuit behavior,all the variables are assigned small variations. Capitalletters are used for the steady-state (dc) components andsmall ones for variations, regardless of whether a capitalor a small letter was used previously for the givenvariable. Current io signifies load variations; therefore itssteady-state value is zero. Continuous MMF operationmode is considered; therefore d, = D + d, d2 = D2-d= 1-D -d. The other variables become

E EE+e; :> ()+ ; * 4> * + (p*; U => U+

io = +io; V :> V+v; V* => V* + v*.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-23, NO. 2 MARCH 1987

) =- 4(dlT) - bn

V = V(d1T) +

(20)

- b)

where

mB=A+RL C

(21)

1 cm

RL 2

242

The capacitor voltage and output voltage have thesame one-cycle mean value U. The symbol V means thecapacitor voltage at the beginning of the cycle. Assumingthe disturbances are very small in magnitude, comparedwith the steady-state values, equations may be linearizedby neglecting higher order terms (products of two ormore variations).

By replacing a, b, c, g with their expressions (18),the averaged equations for steady-state values become

0= 4)*X- V)-* = X*[X- exp( fDT)J

NVB

[E

e p ) ] (23)

0=4* Sy- V + V*Z * y + V*S5 S5

LZ exP(RR)

where

1 ( m \X = 1 + exp - -jD2T) [

From equations (23) and (24), 4)* and V* may beeasily obtained, but have very clumsy expressions:

)* = EL, [exp( DT)-l]

X - exp (RL+ Rc)C

X LZ[ exp(RL+ R)C]

X -e ( RDT)] + § Y2} (26)

V* = jexp RP-DTI-1I Y:

RPCNs \LP

X Z exp (RLRc)CJ(24)

X -exp (DT)J

( B2 m A2mL C 2LSC 2 2

+ An)

sinwD2T + B2 AX + V-A2J coswD2T

(exp -D2T) m2

Bmm+ -nRLC 2 I

sinwD2T Bxsn2 + (A + RLC) coswD2T

B2 / m\ m mZ = 1+7 expK- 2D2T) L 2LS C

sinwD2T /1 1XX D +(L RC coswD2T.

n )RLB

B 2 }) (27)

These expressions are to be replaced in the dynamic(ac) equations. Finally, 4)*, up* and V*, v* are expressedin function of 4), (p and V, v, because the equationsdescribing the pulsewidth modulator become much morecumbersome if the cycle is considered to begin with theOFF interval, while the pulsewidth modulation (PWM)ramp voltage begins in the same moment as the ONinterval.

A numerical example is given, deliberately very closeto the one in [3]. Converters parameters are LP =

0.5 mH, LS = 0.1 mH, RP = 1.5 Q, RS = 0.3 fQ.N =100, Ns = 45, C = 500 [iF, Rc = 75 mfl. Operatingconditions are E = 20 V, U = 10 V, D = 0.6, T =25 ps, RL = 15 fQ. A second example is calculated, forcomparison, with the only different parameters RL =1 £Q, U = S V, RS = 0.05 fQ. From these values,

The left members of these equations are zero, becauseonly the dynamic (ac) parts of 4), 4)* and V, V* havevariations.

After linear combination with equations (23) and (24),the expression for U gets a much simpler form,containing only 4)*, V*, D.

=1 RL_ N(5T RL + Rs [NS(4-))-RLC(V*-V)I

=iRL{N(ELp 4*)TRL + R.{SS(NpRP )

X [expRpDTD11

+ RLCV* [exp -T]}.L(RL+Rc)C J (25)

LP_ Ls = 333 [s > 10 TRp Rs

consequently,

exp( -R RI-D71

:- 1 - PDTLP LP

(RL + RC)C : 7.5 ms>> 10 T, 0.5 ms > 10 T.

Therefore

DT DTexp ~ z 1 +

(RL + Rc)C (RL + Rc)CThe conditions Lp/Rp > 10 T and RLC > 10 T are

always to be fulfilled in order to obtain high efficiency,respectively, low ripple.

B = 0.995; 0.925, A = 0.3745; 0.37,

BIRCA-GALATEANU: ANALYSIS OF SWITCHING REGULATORS 243

Bm ~->> A, m = 133; 1850,

RLC

n= 2 X 107; 2.42 X 107, B2/(LSC) >> A.

For flyback converters, 4n >> m2. In forwardconverters, an accumulation/filtering series inductor isused in front of the output capacitor, therefore m2 may belarger than 4n.

The variable Xo 4.47 x 101; 4.38 x 103, thereforeD2T 0.0447; 0.0438. If the output capacitor is largeenough for good filtering, wD2T is always much smallerthan unity; consequently, cos(wD2T) 1, sin(wD2T)wD2T.

D2T = 0.665 x 10 3; 0.925 x 10 2 «1.

Therefore

exp(- 22! 1 -2-D2T.( 2 )2The abovementioned approximations lead to

maximum errors of 3 percent for the first numericalexample and 4.5 percent for the second. Neglecting(m/2)D2T << 1 in the terms X, Y, Z leads to a furthererror of 0.67 percent for the first example and 9.25percent for the second. Only after all theseapproximations does the factor X contain no more than asecond-order term in D2T. The factor Y may be linearizedonly for large RL values (RL = 15 fQ), and in the factorZ no coefficient may be neglected. Finally, the system ofequations contains second-order terms in D2T.

n ( 2 LsC(2)B2 (1

- T -BRSvi*) ]L+C

NRL +RS eP RLP (30)

dv_ d F DT mndv d- V* exp ±T. MDTV*

dt (RL+RC)C | (RL+Rc)C 2

+ RL (1 - -D2T)

[X D2L(-DT S + V* RSCV*

(RL Ls 2n) LS (RLC 2n)}

D BB m+ (RLio-v*)+ l(-D2T)

(RL +Rc)C TnC \2

X [v*B (A -R2C) ± (p LS (nD2T + RLC)

(Rc D2T 1 N+ i0BL5 LC RLC) (31)

=dBIV*DT

+ m1PDT'

U CxBpV*exp (R +R)C 2/

F(LS RLCLV*N )LS RLC Lji

xB(m + _RN _n L

v*37TB ( RC + B

-jD2T) LV* B -D2Tn)D2T)(28)

l (12 D2T) ( - nD2T), for RL = 15f

(29)

The averaged equations for the dynamic (ac) parts ofthe variables are very clumsy, even after all theabovementioned approximations.

NS*MB . B

+ RLio (D RCRL ± RC

V. THE TWO-LOOP CONTROL CIRCUIT

dt {pLP )1 +-p DT) - (-2 D2T)

B2 mB2_VX[DTC ( RL LS)

L5C 2n Ns (RLC 2n

+ expK- - D2T) N [C( ° L LS )

In a two-loop regulator, the first one is the loopsensing the instantaneous value of the output voltage, theusual dc loop. The output voltage shows an almosttriangular ripple, due to the finite C value and to itsequivalent series resistance RC. Generally, pulsewidthmodulators are disturbed by such ripple; therefore it hasto be filtered out.

The second loop has to sense the current through theoutput filter inductor [2], if there is one (in forwardconverters), or the flux in the transformer core (flybackequals buck-boost converters), because the inductor

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-23, NO. 2 MARCH 1987

+ D2 RL ~ + DBv*.2 RI +Rs((32)

-z:z; -B my - 1 - D2TnRLC 2

244

current or the transformer flux shape is very similar to theoutput voltage shape (ripple). The primary switch current[3] ip equals (Np/Lp) 4) only during the ON interval, butis zero during the OFF interval. The voltage having theinductor current (flux) shape and proper magnitude is tobe subtracted from the output voltage. The ripple at theerror-amplifier input may thus be very much reduced, andtherefore much less attenuation through filtering is thenneeded. Consequently, the error-amplifier transferfunction poles may be placed at much higher frequencies,so that the phase shift produced by these poles becomessignificant at higher frequencies. Improved stability andlower audiosusceptibility may result. Certainly, moredegrees of freedom result for shaping the error-amplifierfrequency response, in order to improve converterstability. In this sense, the second feedback loop may beconsidered to have the main contribution, not anadditional one, in implementing loop compensation.

It is not possible to get directly a voltage proportionalto the core flux or to the inductor current, without largepower loss. The simple and convenient solution is to adda feedback winding on the transformer (inductor) core.This winding gives a voltage proportional to dildt or d4)ldt and, of course, with zero dc component. This voltageis easily integrated through an RiCi cell, then subtractedfrom the output voltage. The time constant v = RiCi iscomparable to the period T and is chosen to give thecompensation voltage a shape as close as possible to theoutput voltage ripple. For that purpose, a resistor shuntedby a diode may be used (in flyback converters) in orderto have different T seconds in the ON and OFF intervals.

Since the product T has a value comparable to the periodT, the components having frequencies significantly lowerthan the switching frequency are not integrated(averaged). Subtraction of the signals given by the twofeedback loops does not lead to pole-zero cancellation[6]. For accurate modeling, an RiCi cell isnot satisfactory; an Ri(Ci - R') or even more elaboratecircuit is needed. To design it, the solution of thedifferential equations is used for the power stage.

The two-loop regulator may be also approached as aproportional, integral, differential (PID) control loop. Thesecond feedback loop may be regarded as implementing aderivative term, to be optimized for best regulatorstability and fast response.

In the voltage given by the dc loop, it is desirable tocancel, by filtering or subtraction, only the ripple at theswitching frequency, without attenuation and phase shiftfor lower frequency output voltage variations.Unfortunately, it is impossible to separate the switchingripple.

The compensation by subtraction may affect theregulator response fastness less. Only after this maincompensation, the error-amplifier frequency response hasto be shaped to ensure proper gain/phase margin. Theprocessed error signal is then compared with the rampvoltage to obtain pulsewidth modulation. Constant-frequency (constant-period T) control is preferred toconstant-toff, constant-ripple, or other control methods,for easier electromagnetic interference filtering and easierstability assuring.

BiRCA-GALATEANU: ANALYSIS OF SWITCHING REGULATORS 245

REFERENCES

[1] Chetty, P.R.K. (1981)Current-injected equivalent circuit approach to modelingswitching dc-dc converters.IEEE Transactions on Aerospace and Electronic Systems,AES-17, 6 (Nov. 1981), 802-808.

[2] Kelkar, S.S., and Lee, F.C. (1983)A novel feedforward compensation canceling input filter-regulator interaction.IEEE Transactions on Aerospace and Electronic Systems,AES-19, 2 (Mar. 1983), 258-268.

[3] Lee, F.C., Carter, R.A., and Fang, Z.D. (1983)Investigations of stability & dynamic performances of acurrent-injected regulator.IEEE Transactions on Aerospace and Electronic Systems,AES-19, 2 (Mar. 1983), 274-286.

[4] Shortt, D.J., and Lee, F.C. (1983)

Improved switching converter model using discrete andaveraging techniques.IEEE Transactions on Aerospace and Electronic Systems,AES-19, 2 (Mar. 1983), 190-202.

[5] Lee, F.C., Fang, Z.D., and Lee, T.H. (1985)Optimal design strategy of switching converters employingcurrent injected control.IEEE Transactions on Aerospace and Electronic Systems,AES-21, 1 (Jan. 1985), 21-35.

[6] Daly, K.C. (1985)An alternative interpretation of the two-loop, adaptive-control strategy of Lee and Yu.IEEE Transactions on Aerospace and Electronic Systems,AES-21, 4 (July 1985), 584-587.

[7] BircA-GAlaeanu, S. (1985)Flyback converter output voltage stabilization.Presented at the First European Power ElectronicsConference, Brussels, Belgium, Oct. 16-18, 1985.

Serban Birca-GA1Ateanu was born in Urziceni, Romania, on May 21, 1944. Hereceived the Engineering Degree in electronics in 1967, and the Ph.D. degree in 1976,both from the Polytechnical Institute of Bucharest.

Since then, he has been with the Applied Electronics Department of the sameInstitute, as an Assistant, then Assistant Professor (Lecturer) in industrial electronics.

He has authored a number of technical papers and books on voltage stabilizers,wideband low-pass amplifiers, optoelectronics, digital control, and processing circuits.He has been involved in many research and development and design works forindustry and is currently a Technical Assistant. He has some patents on industrialmeasurements and control.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-23, NO. 2 MARCH 1987246