Fig, 12.35 Small-signal models of DCM CPM converters, derived hy (icrturbuliciEi and linearization of Figs 12.32 and 12.34: (a) buck, (h) boost, (c) huck-boost.

.Li:m

i + -i

(12.119)

with mdefined as in Eq. (!2.!).

Steady-state characteristics of the DCM CPM buck, boost, and buck-boost converters are summarized in Table 12.6. In each case, the dc load power is Pj, = V/ and P is given by Eq. (12.117). The conditions for operation of a current programmed converter in the discontinuous conduction mode can be expressed as follows:

(12.120)

where / is the dc load current. The critical load current at the CCM-DCM boundary, I. is expressed as a function of and the voltage conversion ratioM = VfV in Table 12.6.

In the discontinuous conduction mode, the inductor current is zero at the beginning and end of

Aablle ил Current programmed DCM smill-sigiifil tiquivillenl eiitoil piinimeters; inpiil p[>it

Convener

Buck

Boost

i + :

m,

2-m , 2mjmi m I

Buck-raost

Tftblc 12.Й CuiTt;n; prfjriiiTUtietl DCM small-sijniil ctuivuleiU 4;ircuU paiuueieifi: output puit

Converter

Buck

r\\-m}

Boost

M- i

Buck-boost

each switching pciind. As a resuit, the current programmed controller does not exhibit the type of instability described in Section 12. L The current programmed controllers of DCM boost and buck-boost converters are stable for ail duty cycles with no artificial ramp. However, the CPM DCM buck converter exhibits a different type of low-frequency instability when M > 2/3 and = 0, that arises because the dc output characteristic is nonlinear and can exhibit two equilibrium points when the converter drives a resistive itjad. The stability range can be extended tt) 0 < D < I by addition of an artificiid riunp have slope ш > 0.086 mj, or by addition of output voltngc feedback.

Smaii-signai models of DCM CPM converters cnn be derived by perturbation and lincnrization of the avcmgcd models of Figs. 12.32 and 12.34. The results ate given in Fig. 12.35. Parameters of the small-signal models are listed in Tables 12.7 and 12.S.

The CPM DCM small-signal models of Fig. 12.35 arc quite .similar to the respective small-signal models of DCM duty-ratio controlled converters illustrated in Figs. 11.15 and 11.17. The .sole differences are the parameter expressions of Tables 12.7 and 12.S. Transfer functions can be determined in a

C - Rf

Pig, 13.36 Simplified small-signal mtirte! obtained by letiing /- hecome zern in Fig. 12.35 (a), fb), or (c).

similar manner. In particuJar, a simple approximate way to determine the low-irequeney small-signal transfer functions of the CPM DCM buek, boost, and buck-boost converters is to simply let the inductance L tend to zero in the equivalent circuits of Fig. 12.35. This appnjximation isjustified for fretjuen-cies sufficiently less than the converter switching frequency, because in the discontinuous conduction mode the value of L is small, and hence the pole and any RHP zero associated with L occur at frequencies near to or greater than the switching frequency. For all three converters, the equivalent circuit of Fig. 12.36 is obtained.

Figure 12.36 predicts that the control-to-output transferfunction ( (-ч) is

<!2.12!)

with

(Ri\r,)C

The line-to-output transfer function is predicted to be

G,.,(.v)= -

with

(12.122)

(1 = .12(111-3)

If desired, more accurate expressions which account for inductor dynamics can be derived by solution of the models of Р1ц. 12.35.

US SUMMARY OF KEY POINTS

In current-programmed cunirol, the peak switch currenl ijj) follows the control input jjf)- This widely used contrt)! scheme has the iidvtinlage t)f a simpler control-ttj-output transfer ftinclitjn. The line-U)-output transfer functions of current-programmed buck converters are also reduced.