ViCf)

Fifi. 7.38 Terminal switch network waveforms in the CCM SEP[C.

currents can be taken as indepemient inputs to the switch network. The remaining two voltages and/or currents are viewed as dependent outputs of the switch network. In general, the choice of independent inputs is arbitrary, as long as the inputs can indeed be independent in the given converter circint. For CCM operation, one can chtxtse one terminal current and one terminal voltage as the independent inputs. Let us select jj(t) and V2(() as the switch network independent inputs. In addition, the duty cycle d{<} is the independent control input.

In Fig. 7.39(b), the ports of the switch network are replaced by dependent voltage and current sources. The waveforms of these dependent sources are defined to be identical to the actual dependent outputs Vf(f) and JjCf) given in Fig. 7.3S. Since all waveforms in Fig. 7.39(b) match the waveforms of Figs. 7.39(a) and 7.3Я, the circuits are electrically equivalent. So far, no approximations have been made.

7.4.2 Circuit Averaging

The next step is determination of the average values of the switch network terminal waveforms in terms of the converter state vatiable.s (inductor currents and capacitor voltages) and the converter independent inputs (such as the input voltage and the transistor duty cycle). The basic assumption is made that the natural time constants of the converter network arc much longer that the switching period T-This assumption coincides with the recjuirement for small switching ripple. One may average the waveforms over a titne interval which is short cornpared to the systern natural titne constants, without significantly altering the system response. Hence, when the basic assumption is satisfied, it is a gotxi approximation to average the converter waveforms over the switching period Ту The residting averaged model predicts the low-frequency behavior ofthe system, while neglecting the high-frequency switching harmonics. In the SEPIC example, by use of the usual small ripple approximation, the average values of the switch network terminal waveforms of Fig. 7.38 can be expressed in terms ofthe independent inputs and the state variables as follows:

AC Equivalent Circuit Modeiing

( 1(0

Switch network \

v,(/)

lv,(f)

Switch network i

Averaged switch netvorh

Fig. 7.39 Derivation of lhe averaged switch modci for lhe CCM SEPIC: (a) switcli network; fb) switch network where the switcVies nie replaced with rtepentlciit aimrcen whose waveforms match tlie switch terminal dependent wavcfomis; (c) large-signal, nonlinear averaged switch inodcl obtained by averaging the .switch network termitial waveforms in (b); (й) tk: and ac small-signal averaged switch model.

(7,130)

(7,131)

r3(()) = rft/)j{rc,(0), + (v (0)J

(7,132)

(7,133)

We have selected (iiW)j;. and ((O) as the switch network independent inputs. The dependent outptits ofthe averaged switch network are then IWJtj (0)г,- The next step is to express, if possible, the switch network dependent outputs {(2{/>);г and {*(f))r, as functions lo/e/y of the switch network independent inputs (((0)j;> (*i(0)7:,> and the control input d(r). In this step, the averaged switch outputs should not be written as functions of other converter signals such as (v(t)}j-y (ci(0}r cCO),. (i,())7;, (ii2(i)>,, etc.

We can use Eqs. (7.131) and (7.132) to write

d{l)

(7.134)

(7,135)

Substitution of these expressions into Eqs. (7.130) and (7.133) leads to

(7,136)

(7.137)

The averaged equivalent circuit for the switch network, that conesponds to Ec]s. (7.136) and (7.137), is illustrated in Fig. 7.39(c). Upon completing the averaging step, the switching harmonics have been removed from all converter waveforms, leaving only the dc and low-frequency ac components. This large-signal, nonlinear, time-invariant model is valid for frequencies sufficiently less than the switching freqtiency. Averaging the waveforms of Fig. 7.38 modifies oidy the switch network; the remainder of the converter circuit is unchanged. Therefore, the averaged circuit model of the converter is obtained simply by replacing the switch network with the averaged switch model. The switch network of Fig. 7.39(a) can be identified in any two-switch conveiter. such as buck, boost, buck-boost, SEPIC, or Cuk, If the converter operates in continuous conduction mode, the derivation of the averaged switch model follows the same steps, and the result shown in Fig. 7.39(c) is the same as in the SEPIC example. This means that the model of Fig. 7.39(c) can be used as a general large-signal averaged switch model for all two-switch converters operating in CCM.