(a) *c<0

(b) v(0

nC ЛС

0 dT f

Fig, 7.31 Capacitor waveforms for the flyback example: (a) capacitor currenl, (b) tupacjtor vollage.

{.,(0), = dO)[(v,W)-{/(f)),/. By inserting tiiis result into Eq. (7.20), we obtain the averageii intiuctor equation,

v / Ч /л (*))r

I- = CO {v,(()) -d(t) {iit)) R , - dC)

The capacitor waveforms are cotistructed in Fig. 7.21. The average capacitor current is Ibis leads to the averaged capacitor equation

(7.55)

(7.56)

(7.57)

The averaged converter equations (7.56), (7.58) and Fig. 7.22 Input source current waveform, (7.59) are collected below: flyback example.

7.2 The Ba.iic Modeling Appwach

Лт).

- = d\t)

-n---T~

(7.60)

{i,<.t))=d(t)iiit)).,

This is a noiiliiieai set of differential equatioiLS, and hence the next step is to peiturb and linearize, to construct the converter small-signal ac equations. We assume that the converter input voltage v(() and duty cycle i/(f) can be expressed as qitiescent values plits small ac variations, as follows:

dtt)D + d(t)

(7.61)

In response to these inputs, and after all tian.sients have decayed, the average convertei waveforms can also be expressed as quiescent values plus small ac variations:

(4i))j. = V-H!(()

With tliese substitutions, the birge-.signal averaged inductor equation becomes Upon multiplying this expression out and collecting terms, we obtain

(7.62)

(7.63)

§ + \= [DV-iy-DRj) + I flyf)-D + (v, + -/* )(0-DfiJ(l)

Detenus

1 order ac terms (linear) J(f)yt)-K/(t)-(t)f(OR

2 order ac tetnas (nonlinear)

<7.64)

As usual, this equation contains three types of terms. The dc terms contain no time-varying quantities. The first-order ac terms are linear functions ofthe ac variations in the circuit, while the second-order ac terms arc functions of the products of the tie variations. If the small-signal assumptions of Eq. (7.32) arc satisfied, then the sccond-oider terms arc much smaller in magnitude diat the tirst-order terms, and hence can be neglected. The dc terms must satisfy

0 = DV- - DRJ

(7.6S)

This result could also be derived by applying the principle of inductor volt-second balance to the steady-state inductor voltage waveform. The first-order ac terms must satisfy

(7.66)

This is the linearized equation that describes ac variations in the inductor current.

Upon substitution of Eqs. (7.61) aud (7.62) into the averaged capacitor equation (7,60), one

obtains

div + m]

(7.67)

By collecting terms, we obtain

(7.68)

Dc terms

1 order ac terms 2 order ac lertn (Unear) (nonlinear)

We neglect the second-order tenns. The dc terms ofEq. (7.68) must satisfy

Dl V > Rj

This result could also be obtained by use of the principle of capacitor charge balance on the steady-state capacitor current waveform. The first-order ac terms ofEq. (7.68) lead to the small-signal ac capacitor equation

dvU) Dt(t) Kt) Idjt) dt R ~ n

Substitution of Eqs. (7.61) and (7.62) into the averaged input current equation (7.60) leads to Upon collecting terms, we obtain

(7.70)

(7.71)

= Df *

Di(t) + Idtj)

Dc term Г order ac term Dc term Г order ac terms 2 order ac term

(linear) (nonlinear)

(7.72)

The dc tenns must satisfy