rietwork i

- V2 + h +

i-Q-

h + г

V+v

Fig. 7.44 Coiistraction of an averaged tircuiL model for an ideal boost converter example: (a) eoiiverter circuit

with the switch iietwort of Fig. 7.39(o) identified; (b) и de and Smnll-signal aC avej-aged circuit model obtained by

replacing the switch network with the model of Fig, 7,39(d).

pie, Fig. 7.44 shows an averaged circuit model ofthe boost converter obtained by identifying the switch network of Fig. 7.39(a) and replacing the switch network with the iruxJel of Fig. 7,39(d).

In summary, the circuit averaging method involves replacing the switch network with equivalent voltage and current sources, such that a time-invariant network is obtained. The converter wavefonns are then averaged over one switching peritxJ to remove the switching harmonics. The large-signal model is perturbed and linearized about a quiescent operating point, to obtain a dc and a small-signal averaged switch model. Replaceinent ofthe switch network with the averaged switch iruxJel yields a toinplete averaged circuit model of the converter.

7.4.4 Switch Networks

So far, we have described derivatitm ofthe averaged switch model for the general two-switch network where the ports of the switch network coincide with the switch ports. No connections are assumed between the switches within the switch network itself. As a result, this switch network and its averaged model tan be used to easily construct averaged circuit inodels of tnany two-switch converters, as illustrated in Fig. 7.43. It is important to note, however, that the definition ofthe switch network ports is not unique. Different definitions of the switch network lead to equivalent, but not identical, averaged switch inodels. The alternative forms of the averaged switch model may result in simpler circuit mtxJels, or models that provide better physical insight. Two alternative averaged switch models, better suited for analyses of boost and buck converters, are described in this section.

Consider the ideal boost converter of Fig. 7.45(a). The switch network contains the transistor

AC Equivakits Circuir Modeling

iil) i,{t)

v,(0

Swiich network !

Fig. 7.4S An ideal boost coaverter example: (a) converter circuit showing another possible definition of lhe switch network; (b) lerminal waveforms of the switch network.

v,(0

and the diode, as in Fig. 7.44(a), but the switch network ports are defined differently. Let ns proceed with the derivation of the corresponding averaged switch model. The switch network terminal waveforms are shown in Fig. 7.45(b). Since ((t) and vit) coincide with the converter inductor current and capacitor voltage, it is convenient to choose these waveforms as the independent inputs to the switch network. The steps in the derivution of the averaged switch model are illustrated in Fig. 7.46.

First, we replace the switch network with dependent voltage and current generators as illustrated in Fig. 7.46(b). The voltage generator Vj(f) models the dependent voltage waveform at the input port of the switch network, i.e., the transistor voltage. As illustrated in Fig. 7.45(b), V(() is zero when the transistor conducts, and is equal to Vj(f) when the diode conducts:

Vl(() =

0, Q<(<di;

(7.144)

When Vfil) is defined in this manner, the inductor vtdtage waveform is unchanged. Likewise, ((O models the dependent current waveform at port 2 of the network, i.e., the diode current. As illustrated in Fig. 7.45(b), y) is equal to zero when the transistor conducts, and is equal to ([(f) when the diode conducts:

With JjW defined in this manner, the eapaeitor current waveform is unchaiiged. Therefore, the original converter circuit shown in Fig. 7.45(a), and the circuit obtained by replacing the switch network of Fig. 7.4й(а) with the switch network of Fig. 7,4<i(b), are electrically identical. So far, no approximations have been made. Next, we remove the switching harmonics by averaging all signals over one switching period, as in Eq. (7.3). The results are

{т)гГ\ПШг (7 146)

Here we have assumed that the switching ripples of the inductor current and capacitor voltage are small, or at least linear functions of time. The averaged switch model of Fig. 7.46(c) is now obtained. This is a large-signal, nonlinear model, which can replace the switch network in the original ctmverter circuit, for construction of a large-signal nonlinear circuit model of the converter. The switching harmonics have been removed from all converter waveforms, leaving only the dc and low-frequency ac components.

ТЪе model can be linearized by perturbing and linearizing the converter waveforms about a quiescent operating point, in the usual manner. Let

(v,(0)=V, + V)

d(t)=D + dii} d\i) = D-J(l)

-t (7,147)

{KO), = {v,(O),=K+0(t)

{v((f)) = V,+v,(0

The nonlinear voltage generator at port 1 of the averaged switch network has value

(zJ-ff(/)(V + m\ = D{V + m] - vd{r)-v(t)dU) (--s)

The term 9{t)d(i) is nonlinear, and is small in magnitude provided that the ac variations are much smaller than the quiescent values [as in Eq. (7.32)]. When the small-signal assumption is satisfied, this terra can be neglected. The terni VJ(t) is driven by the control input, and hence can be represented by an independent vohage source. The term D{V + v(l)) is equal to the constant value D multiplied by the output voltage (V+ (*(f)). This term is dependent oti the output cacitor voltage; it is represented by a dependent voltage source. This dependent source will become the primary winding of an ideal transformer.

The nonlinear ctirrent generator at the port 2 ofthe averaged switch network is treated in a similar manner. Its current is

(fT d{t)] {{ + Un] = D (/ + ((()) - Jdil) - hDdit) t-*

The term !(t)d(f) is nonlinear, and can be neglected provided that the snudl-signal assumption is satisfied.