It can be verified that Et]s. (12.77) and (12.78) are equivident tti the trmsfer functittn.s derived in Section 12.2.

When an artificial ramp is present, then the gain is reduced to a finite value, The ctirrent-pro-grammed controller no longer perfectly regulates the inductorcurrent 4, and the terras on the right-hand side ofEq. (12.75) do not add to zero. In the extrerae case of a very large artificial ramp (large M and hence small F ), the current-programmed controller degenerates to duty-cycle control. The artificial ramp and analog comparator [)f Fig. 12.8 then function as a pulse-width modulator similar to Fig. 7.63, with small-signal gain F . For small F and for -* Q, * 0, the control-to-output transfer function (12.73) reduces to

lim G,.is)FJJ,As) .ГП.11Л , (12.79)

which coincides with conventional duty cycle control. Likewise, Eq. (12.74) reduces to

lim G,. ,. , (v) = 0,.

(12,80)

f>-i(l

which is the lirre-to-output transfer furrction for conventional duty cycle control.

12.3.4 Current-Programmed Transfer Functions of the CCM Buck Converter

The control-to-output transfer function fr ,/.) and line-to-output tran.sfer function C{s) of the CCM buck converter with duty cycle control are tabulated in Chapter 8, by analysis of the equivalent circuit model in Fig. 7.17(a). The results are:

where the denominator polynomial is

den(s)[+.4 + s-LC (12.ИЗ)

The inductorcurrent transfer functions C;,;(i) and G.C*) defined by Eqs. (12.68) and (12.69) are also found by solution of the equivalent circuit model in Fig. 7.17(a), with the following results:

(12.8S)

where den(s) к again given by Et]. (12.83).

With no artificial rainp and negligible ripple, the controi-to-output transfer function reduces to the ideal expression (12.77). Snbstitution of Eqs. (12.81) and (12.84) yields

(12.86)

Under the same conditions, the iine-to-output transfer function reduces to the ideal expression (12.78). Substitution of Eqs. (12.81) to (12.85) leads to

, . с; ()0.-,(.о-о.,()С (л)

.lim C., (.T) =-= 0

(t2.R7)

Equations (12.86) and (12.S7) coincide with the expressions derived in Section 12.2 forthe CCM buck converter.

For arbitrary F F and/-, the controi-to-output transfer function is given by Eq. (12.73). Substitution of Eqs. (12.81) to (12.85) into Eq. (12.73) yields

У 1

D denU)

. + F.

V 1 + sRC , ... DR iffH(Y) l -

D del,(s)

(12.RR)

Simplification leads to

(12,89)

dm.i)+{i+sRC) + F, t-\ Finally, the control-to-output transfer function can be written in the following normalized foim:

(12.90)

where

(12.91)

1 Г F]J FJ-V

(12.92)

PR RCK.V

In the above equations, the salient features C,.( tO., and are expressed as tlte duty-ratio-eontrol value, multiplied by a factor that accounts for the effects of current-programmed control.

It can be seen from Eq. (12.93) that current programming tends to reduce the Q-factor of the

poles. For large F, Q. varies as ; consequently, the poles become real and well-separated in magnitude. The Iow-2 approximation of Section S.l,7 then predicts that the low-frequency pole becomes

* DR* D

RCty

(12.94)

For large F and small F, this expression can be furdier approximated as

(12,9S)

which coincides with the low-frequency pole predicted by the simple model of Section 12,2. The low-Q approximatiou also predicts that the high-frequency pole becomes

, CF y 1 .

(12.96)

For large f this expression tan be further approximated as

-dl-dm:.

(12.97)

The high-frequency pole is typically predicted to lie near to or greater than the switching frequency It should be pointed out that the converter switching and modulator .sampling processes lead to discrete-time phenomena that affect the high-frequency behavior of the converter, and that are not predicted by the continuous-time averaged analysis employed here. Hence, the averaged model is valid only at frequencies sufficiently less than one-half of the .switching frequency.

For arbitrary F, F and F, the cunent-piogrammed line-to-output transfer function Cji.., (i) is given by Eq. (12.74). This equatitm is most easily evaluated by first finding the ideal transfer function, Eq. tl2.78), and then using tlte result to simplify Eq. 0274). In the case of the buck converter, Eq. (12.87) shows that thequantity (GGj- GjGj) is equal to zero. Hence, Eq. (12.74) becomes

(12.98)

Substitution of Eqs. (12.81) to (12.85) into Eq. (12.98) yields