AC Equivalent Circuit Modeling

7.1 INTRODUCTION

Converter systems invariably require feedback. For example, in a typical dc-dc converter application, tlie otitput voltage v(t) must be kept constant, regardless of clianges in tlie input voltage v(t) or in tlie effective load resistance r. Tliis is accomplished by building a circuit that varies the converter control input [i.e., the duty cycle d{t)] in such a way that the output voltage t{r) is regulated to be equal to a desired reference value In inverter systems, a feedback loop causes the output voltage to follow a sinusoidal reference voltage. In modem low-harmonic rectifier systems, a control system causes the converter input current to be proportional to the inptit voltage, such that the input port presents a resistive load to the ac source. So feedback is commonly employed.

A typical dc-dc system incorporating a buck converter and feedback loop block diagram is illustrated in Fig. 7.1. It is desired to design this feedback system in such a way that the output voltage is accurately regulated, and Is insensitive to disturbances in v{t) or in the load current. In addition, the feedback systetn should be stable, imd properties such as transient overhotit, settling tirne, imd steady-state regulation should meet specifications. The ac nitjdeling and design of ctmverters and their ctmtrol systems such as Fig. 7.1 is the subject trfPart II of this book.

To design the system of Fig. 7.1, we need a dynamic model ofthe switching converter. How do variations in the power input voltage, the load current, or the duty cycle affect the output voltage? What are the small-signal transfer ftinctions? To answer these questions, we will extend the steady-state models developed in Chapters 2 and 3 to include the dynamics introduced by the inductors and capacitors of the converter. Dynamics trf converters t>perating in the continuous conduction mode can be modeled using techniques quite similar to those of Chapters 2 and 3; the resulting ac equivalent circuits bear a strong resemblance to the dc equivalent circuits derived in Chapter 3.

Modeling is the representation of physical phenomena by mathematical means. In engineering.

Power input

Switching converter

-nnnr

Load

Transistor gate driver

verj\

t)<R

Compensator

Feedback connection

Voltage reference

Controller

Fig. 7.1 A simple dc-dc regulator system, including a buck converter power stage and (i feedback network.

it is desired to model tlie important dominant behavior of a system, while neglecting other insignificant phenomena. Simplified terminal equations of the component elements ate used, and many aspects of the system resptHLse are neglected altogether, that is, they are unmodeled. The resulting simplified tnodel yields phy.sical insight into the systetn behavior, which aids the engineer in designing the system to operate in a given specified manner. Thus, the modeling prcx;ess involves use uf appiciximations to neglect small hut complicating phenomena, in an attempt to understand what is most important. Once this basic insight is gained, it may be desirable to carefully refine the model, by accounting for some of the previously ignored phenomena. It is a fact of life that real, physical systems are complex, and their detailed analysis can easily lead to an intractable and useless mathematical mess. Approximate mtidels are an important tool for gaining understanding and phy.sical insight.

As discussed in Chapter 2, the switching ripple is small in a well-designed converter operating in continuous conduction mode (CCM). Hence, we should ignore the switching ripple, and model only the underlying ac variations in the converter waveform.4. Ft>r example, suppose that some ac variation is introduced intti the converter duty cycle (f(f), such that

ifCO = 0 + D cosu) ,(

(7.1)

where D and are constants, D, aud the modulation frequency is tnuch smaller than the converter switching frequency ti\ = 271/ The resulting transistor gate drive signal is illustrated in Fig. 7.2(a), and a typical converter output voltage waveform i(f) is illustrated in Fig. 7.2(b). The spectrum of v(f) is iihistrated in Fig. 7.3. This spectrum contains components at the .switching frequency as well as its harmonics and sidebands; these components are small in magnitude if the switching ripple is small. In addition, the spectrutn contains a low-frequency cotnponent at the modulation frequency 0) . The magnitude and phase of this component depend not only on the duty cycle variation, but also on the frequency response of the converter. If we neglect the switching ripple, then this low-frequency compo-

Cote drive

<b)

АсШ1 wavefomi v{t) including ripple

Averaged waveform {v(t))j with ripple neglected

rig, 7.2 At variation of the converter signals: (a) transistor gale drive signal, in which the duly cycle varies slowly, and (b) the resulting converter output voltage waveform. Both the actual waveform v(r) (including high frequency switching ripple) and its averaged, low-frequency componem, {v{t)\, are illusli-aled.

Spectrum of 40

Fig, 7.3 Spectrum of the output voltage waveform of Fig. 7,2,

nent remains [also illustrated in Fig. 7.2(b)]. The objective of our ac modeling efforts is to predict this low-frequency cotnponent.

A simple method for deriving the small-signal model of CCM converters is explained in Section 7.2. The switching ripples in the inductor current and capacitor voltage waveforms are recnoved by averaging over one switching period. Hence, the low-frequency components of the inductor and capacitor wavefonns are modeled by equations of the form

where {x{t))j. denotes the average of л(/) over an interval of length T;

(7.2)

426951162900