(b) 1,(0

v,{0

l2(0

/. -b ?,

1 :D

/л + t.

d (0Q Уг-

Fig, 7,50 Three basic switch networks, antl their CCM dc and small-signal ac averaged awiicli models: (я) the buck switch network, (b) the boost switch network, and (c) the general two-switch network.

switch network in Fig. 7.48(a) with the averaged switch model of Fig. 7.49(a) leads to the converter averaged circuit model of Fig. 7.49(b). The circuit model of Fig. 7.49(b) reveals that the switch network performs the functions of: (/) transformation of dc and small-signal ac voltage and current levels according to the 1:D conversion ratio, and (;/) introduction of ac voltage and current variations into the converter circuit, driven by the control input i/(r). The model is easy to solve for both dc conversion ratio and small-signal freqtiency responses. It is identical to the model shown in Fig. 7.17(a).

The three basic switch networks-the buck switch network, the boost switch network, and the general two-switch network-together with the corresponding averaged switch models are shown in Fig. 7.50. Averaged switch mtKlels can be refined to incltide conduction and .switching losses. These models can then be used to predict the voltages, currents, and efficiencies of nonideal converters. Two examples of averaged switch models that include losses are described in Sectitms 7.4.5 and 7.4.6.

7.4.5 Example: Averaged Switch Modeling of Conduction Losses

An averaged switch model can be refined to include switch coiidtiction losses. Consider again the SEPIC of Fig. 7.37. Suppose that the transistor on-resistance is and the diode forward voltage drop Vj are approximately constant. In this example, all other conduction or switching losses are neglected. Our objective is to derive an averaged switch model that includes conduction losses caused by the voltage drops across R and Кд, Let us define the switch network as in Fig. 7.39(a). The waveforms of the switch network terminal currents are the same as in Fig. 7.38, hut the voltage waveforms are affected by the voltage drops across R and V,-, as shown in Fig. 7.51. We select i,(() and vif) as the switch network independent inputs, as in Section 7.4.1. The average values of V](f) and WjfOcan he found as follows:

(7.152)

-i-rf(i)(- V )

(7.153)

Next, we proceed to eliminate Oi.i())7j, (и№)г,. (с\1))т, (аФт о write the above equations in terms of the averaged independent terminal currents and voltages of the switch network. By combining Eqs. (7.152) and (7.153), we obtain:

Since the current waveforms are the same as in Fig. 7.38, Eq. (7.134) can be used here:

(7.154)

(7155)

vjd)

Fig. 7.51 The switcli network tcfininal voltages v[(t) and vit) for the case when the transistor on-resistauce is /t and the diode forward voltage drop is V,y

7.4 Circdi! kveraging and Avmijed Switch Modding 143

-W(-1 I-\+Л-

Fig. 7.52 Large-signal averaged switch model tor the general two-switch network of Fig. 7.50, This model includes conduction losses due to the transistor on-resistauce R atid the diode forward voltiigc drop V.

Substitution of Eqs. (7.154) and (7.155) mto Eq. (7.152) results in:

Equation (7Л56) Ctin be solved for the voltage {v{t))\

The expression for the averagedcurrent (<5(0)г is given by Eq. (7.137) derived in Section 7.4.2:

(7.156)

(7.157)

(7.158)

Equations (7.157) and (7.15S) constitute the averaged terminal relations of the switch network. An equivalent circuit corresponding to these relationships is shown in Fig. 7.52. The generators that depend on the transistor duty cycle d{t) are combined into an ideal transformer with the turns ratio d(ty.d(t). This part of the model is the same as in the averaged switch model derived earlier for the switch network with ideal switches. The elements R/dand Vj model the conduction losses in the switch network. This is a large-signal, nonlinear model. If desired, this mtxiel can be perturbed and linearized in the usual manner, to obtain a small-signal ac switch model.

The mtxiel t>f Fig. 7.52 is also well suited ft)r ctmputer simulations. As an example t>f this application, ctmsider the btick-boost ctmverter in Fig 7.53(a). In this converter, the transistor on-resistance is Rf = 30 m£i, while the diode forward voltage dropis Vp= 0.Й V. Resistor = 100 mii models the copper loss of the inductor. All other losses are neglected. Figure 7.53(b) shows the averaged circuit model of the converter obtained by replacing the switch network with the averaged switch model of Fig. T52.

Lets investigate how the ctmverter t>utput vtdtage reaches its steady-state value, starting frtnn zero initial conditions. A transient simulation can be used to generate converter waveforms during the start-up transient. It is instructive to compare the responses obtained by simulation of the converter switching circtiit shown in Fig. 7.53(a) against the responses obtained by simulation of the averaged circuit inodel shown in Fig. 7.53(b). Details of how these simulations are performed can be found in Appendix B.l. Figure 7.54 shows the start-up transient waveforms ofthe inductor current and the output voltage. In the waveforms obtained by simulation ofthe averaged circuit nwdel, the switching ripple is removed, but other features of the converter transient responses match very closely the responses