7.5 The Canonkai Circuit Model

1 :D

i V,

Averaged switch model

Fig. 7.57 Dc equivalent circuit tuodel, buck converter switching loss example.

By inserting the .switch model of Fig. 7.56 into the original convertercircuit of Fig. 7.48(a). and by letting all wavefonns be equal to their quiescent values, we obtain the steady-state model of Fig. 7.57. This model predicts that the steiidy-state output voltage is:

(7.162)

To find the efficiency, we must compute the average input and output powers. The converter input power is

The average output power is

Hence the converter efficiencу is

+ 1 + 01г [I, I, j

-1-

Beware, the efficiency is not simply equal to WDV,.

(7.163)

(7.164)

(7.165)

THE CANONICAL CIRCUIT MODEL

Having discussed several methods for deriving the ac equivalent circuit models of switching converter.s, let us now pause to interpret the results. All PWM CCM dc-dc converters perform similar basic functions. First, they transform the voltage and current levels, ideally with 1(X)% efficiency. Second, they contain Itnv-pass filtering ofthe waveforms. While necessiuy to remove the high-frequency switching ripple, this filtering also influences low-frequency voltage and current variations. Third, the converter waveforms can be controlled by variation ofthe duty cycle.

We expect that converters having similar physical properties should have qualitatively similar equivalent circuit models. Hence, we can define a canonicai circuitmodel that correctly accounts for all

of these basie properties [1-3]. The ac equivalent circuit of any CCM PWM de-do converter can be manipulated into this canonical form. This allows us to extract physical insight, and to compare the ac properties of converters. The canonical model is used in several later chapters, where it is desired to analyze converter phenoraena in a general manner, without reference to a specific converter. So the canonical model allows us to define and discuss the physical ac properties of converters.

In this section, the canonical circuit model is developed, based on physical arguments. An example is given which illustrates how to maniptilate a converter eqtiivalent circuit into canonical form. Finally, the parameters of the canonical model are tabulated for several basic ideal converters.

7.5.1 Development of the Canonical Circuit Model

The physical elements of the canonical circuit model are collected, one at a time, in Fig. 7.58. The converter contains a power input port IfO and a control input port d(l), as well as a power output port and load having voltage i(r). As discussed in Chapters, the basic function of any CCM PWM dc-dc converter is the conversion of dc voltage and current levels, ideally with 100% efficiency. As illustrated in Fig. 7.58(a), we have modeled this property with an ideal dc transformer, having effective turns ratio 1 :M{D) where JW is the conversion ratio.This conversion ratio is a function of the quiescent duty cycle D. As discussed in СЬ<ф1ег 3, this model can be refined, if desired, by addition of resistors and other elements that model the converter losses.

Slow variations v/t) in the power input induce ac variations i(r) in the converter output voltage. As illustrated in Fig. 7.58(b), we expect these variations also to be transformed by the conversion ratio M[D).

The converter must also contain reactive elements that filterthe switching harmonics and transfer energy between the power input and power outptit ports. Since it is desired that the output switching ripple be small, the reactive elements should comprise a low-pass filter having a cutoff frequency well below the switching frequency. This low-pass characteristic also affects how ac line voltage variations influence the output vohage. So the model should contain an effective low-pass filter as illustrated in Fig. 7.58(c). This figure predicts that the line-to-output transfer function is

G,.,(i} = = M(D) His) (7.166)

where Н(а) is the transfer function of the effective low-pass filter loaded by resistance R. When the load is nonlinear, R is the incremental load resistance, evaluated at the quiescent operating point. The effective filter also influences other properties of the converter, such as the small-signal input and output impedances. It should be noted that the elemental values in the effective low-pass filter do not necessarily coincide with the physical element values in the converter. In general, the element values, transfer function, and terminal impedances of the effective low-pass filter can vary with quiescent operating point. Examples are given in the following subsections.

Control inptit variations, specifically, duty cycle variations al.so induce ac variations in the converter voltages and currents. Hence, the model should contain voltage and ctirrent sources driven by d{t). In the examples of the previous section, we have seen that both voltage sources and current sources appear, which are distributed around the circuit model. It is possible to manipulute the model such that all of the J(t) sources arc pushed to the input side of the equivalent circuit. In the process, the sources may become frequency-dependent; an example is given in the next subsection. In general, the sources can be combined into a single voltage source l(s)cI(s) and a single current source j(s)(f(s) as shown in

1 :M{Di

Ф M

Fewer Input

Coittmt input

Lead

Power input

CiuiinA input

load

Effective low-pass fitter

Power input

Corarol input

Load

input

Conirot input

Load

Vvg. 7.58 DevelopniciU ofthe caiicaical circuit tnotlef hased on physical arguinents: (a) dc transfortner model, (b) intlHsiwi ol ac variations, (c) reactive elements imrotiuce effective low-pass filler, (d) inclusion of ac duty cycle variations.