AC and DC Equivalent Circuit
Modeling of the Discontinuous
Conduction Mode
So far, we have derived equivalent circuit models for dc–dc pulsewidth modulation (PWM) converters
operating in the continuous conduction mode. As illustrated in Fig.
11.1, the basic dc conversion prop
erty is modeled by an effective dc transformer, having a turns ratio equal to the conversion ratio
M
(
D
).
This model predicts that the converter has a voltagesource output characteristic, such that the output
voltage is essentially independent of the load current or load resistance
R.
We have also seen how to
refine this model, to predict losses and efficiency, converter dynamics, and smallsignal ac transfer func
tions. We found that the transfer functions of the buck converter contain two lowfrequency poles, owing
to the converter filter inductor and capacitor. The controltooutput transfer functions of the boost and
buckboost converters additionally contain a right halfplane zero. Finally, we have seen how to utilize
these results in the design of converter control systems.
What are the basic dc and smallsignal ac equivalent circuits of converters operating in the dis
continuous conduction mode (DCM)? It was found in Chapter 5 that, in DCM, the output voltage
becomes loaddependent: the conversion ratio
M
(
D
,
K
) is a function of the dimensionless parameter
which in turn is a function of the load resistance
R.
So the converter no longer has a voltage
source output characteristic, and hence the dc transformer model is less appropriate. In this chapter, the
averaged switch modeling [18] approach is employed, to derive equivalent circuits of the DCM switch
network.
In Section 11.1, it is shown that the
lossfree resistor
model [911] is the averaged switch model
of the DCM switch network. This equivalent circuit represents the steadystate and largesignal dynamic
characteristics of the DCM switch network, in a clear and simple manner. In the discontinuous conduc
tion mode, the average transistor voltage and current obey Ohm’s law, and hence the transistor is mod
ele
d
by an effective resistor
The average diode voltage and current obey a power source
characteristic, with power equal to the power effectively dissipated in
Therefore, the diode is modeled
with a
dependent power source.
11
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AC and DC Equivalent Circuit Modeling of the Discontinuous Conduction Mode
Since most converters operate in discontinuous conduction mode at some operating points,
smallsignal ac DCM models are needed, to prove that the control systems of such converters are cor
rectly designed. In Section 11.2, a smallsignal model of the DCM switch network is derived by linear
ization of the lossfree resistor model. The transfer functions of DCM converters are quite different from
their respective CCM transfer functions. The basic DCM buck, boost, and buckboost converters essen
tially exhibit simple singlepole transfer functions [12, 13], in which the second pole and the RHP zero
(in the case of boost and buckboost converters) are at high frequencies. So the basic converters operating
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 Spring '09
 DariuszCzarkowski
 Switchedmode power supply, DCM, discontinuous conduction mode, DC Equivalent Circuit Modeling

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