-D f 1

den{s) ID rfen(j)

1 + R

V l±jRC. p У 1 DR denis) DdertU)

(12.99)

Simplification leads to

denU)[]..,RC]F ,F,

(12.1Ш)

Finally, ttie current-progiimimed line-to-tmtput transfer function can be written in the following nonnal-ized form:

! +7-

(12.101)

where

F-V FFV

2M,.

(12.102)

The quantities Q.and (it. are given by Eq.s. (12.92) and (12.9. 5).

Equation (12.102) shows how current programming reduces the dc gain ofthe buck converter line-to-output transfer functittn. For duty cycle control (f) ~* 0)> Go is equal to D. Nonzero values of F, reduce the numerator and increase the dentmiinattw ofEq. (12.102), \vhich tends to reduce G.,. We have already seen that, in the ideal case (Г , 0, F 0), becomes гею. Equarion (12.102)

reveals that nonideal current-programmed buck converters can also exhibit zero Gq, if the artificial ramp slope is chosen equal to 0.5A/j, The current programmed controller then prevents input line voltage variatitms from reaching the output. The mechanism that leads to this result is the effective feedforward of Vg, inherent in the current programmed controller via the Fv term in Eq. (12.Й6). It can be seen from Fig. 12.2Й that, when FFfiis) = GJ.s), then the feedforward path from through F induces variations in the tnitput V that exactly cancel the v-induced variations in the direct forward path of the ctmverter tlirough GCi). This cancellation occurs in the buck converter when jVf = O.SAfj.

12,3.5 Results for Basic Converters

The transfer functions ofthe basic buck, boost, and buck-boost converters \vith current-programmed control are summarized in Tables 12.3 to 12.5. Control-to-output and line-to-output transfer functions for both the simple model of Section 12.2 and the more accurate model derived in this sectitm are listed. For completeness, the transfer functions for duty cycle ctmtrol are included. In each case, the salient features are expressed as the corresponding quantity with duty cycle control, multiplied by a factor that accounts forcurrent-programined control.

Ciirrem Fnygraiimied Control

Tible 12.3 Summary of i-esults for the CPM buck Converter

Simple mode! V R

Duty cycle controlled guitis

More accurate model

c = z>

Jahk 13,4 Summary of results tor the CPM boost converter

Simple rrrode!

Duty cycle controlled gains

11 2

More accurate model

5k* (if

L. FV FF]

1 + ЯС

t-F f./ +

L СУ

F v1

1 -rj,v + Ht-

таЫе 12.s Summaiy of results for the CPM buck-boost oonveiter

Simple model

Duty cycle controlled gains

±

More accurate model

t 1 +

OCf= Off

vTtV Off

,/i/-l4(/)~iAM

The two poles of the line-to-output transferfunctions 0,..р, und control-to-output trunsfer functions G,ofull three converters typically exhibit low £i-fuctors in CPM. The iow-Q upproxiniution can be applied, as in Eqs. (12.94) to (12.97). to find the low-freqtiency pciie. The hne-to-output transfer functions of the boost and buck-boost converters exhibit Iwo poles and one zero, wilh substantial dc Eain,

12,3.ft Quantitative ElTetti of Current-Programmed Control on the Converter Transfer FunclJuiiS

The frequency responses of a CCM buck converter, operating with current-programmed control and with duty cycle control, are compaied in Appendix B. Section B.3.2. The btick converler of Fig. B.25 was simulated as described in Appendix B, and the resulting plots are reproduced here.

The magnitude and phase of the control-to-output transfer functions are illustrated in Fig. 12.27. It can be seen that, for duty cycle control, the transfer function G(s) exhibits a resonant two-pole response. The substantial damping introduced by current-programmed control leads to essentially a single-pole response in Ihe current-programmed conlrol-to-output transfer function J.). A second pole appears in the vicinity of КЮ ItHz, which is near the 2(Ю ItHz switching frequency. Because of this effective .iingie-pole response, it is relatively easy to design a controller that exhibits a weii-behaved response,