f(r)/ O.-il)

vw-*-.lv

V) [J

then the nonlinear equations (7.28) ean be linearized. This is done by inserting Eqs. (7.30) and (7.31) into Eq. (7.28). For the inductor equation, one obtains

i -- = [л+m] iy,+v,(o} + (d - d{,)] [v+v(o)

It should be noted that the complement ofthe duty cycle is given by d\() = f 1 -d(t)] = 1 - [d + d(n] = D- dU\

(7.33)

(7.34)

where ГУ = I - £>. The minus sign arises in the expression for f/(f) because a rf(r) variation that causes d(f} to increase will cause йГ(/) to decrease.

By multiplying out Eq. (7.33) and collecting terms, one obtains

[dV+ Zyv) H- Df>,(0 + DiJ(0 [V, - V] i(0 + J(t) [v (0 - v{t)

(7.35)

Ete terms

I order ac terms (linear)

2 order ac terms (nonlinear)

The derivative of/is zero, since / is by definition a dc (constant) tenn. For the puфoses [)f deriving a small-signal ac model, the dc terms can be considered Ituown constant quantities. On the right-hand side of Eq. (7.35), three types of terms arise:

Dc terms: These terms contain dc quantities only.

Firsi-order ac senns: Eath of these terms contains a single ac qtianlily. usually multiplied by a constant coefficient such as adc term. These terms are linear functions of the ac variations.

Second-order ac terms. These terms contain the products of ac quantities. Hence they are nonlinear, because they Involve the multiplication of time-varying signals.

It is desired to neglect the nonlinear ac terms. Provided that the small-signal assumption, Eq. (7.32), is satisfied, then each ofthe sectmd-order ntmlinear terms is much smaller in magnitude that one or more of the linear First-order ac terms. For example, the sectmd-order ac term (/(OP/O is much smaller in magnitude than the first-order ac term dj(l) whenever D. So we can neclect the second-order terms. Also, by definition [or by use of Eq. (7.29)], the dc terms on the right-hand side ofthe equation are equal to the dc terms on the left-hand side, or zero.

We are left with the tirst-order ac temis on b[)th sides [>f the equation. Hence,

= + P(;) - (1/j - t] (?<r)

(7.36)

Tljis is the desired result: the small-signal linearized equation that describes variations in the inductor current.

The capacitor equation can be linearized in a similar manner. Insettinu nf Eqs. (7.30) and (7.31) into the capacitor equation ofEq. (7.28) yields

с= - (D-(0) [ / + f (;)) -

Upon multiplying out Eq. (7.37) and collecting terms, one obtains

(7.37)

dt * dt

-Df(f)- + /

Dc terms

1 order ac terms (linear)

(OkO

2 order ac term (nonlinear)

(7.за)

By neglecting the secoud-order terms, and noting that the dc terms on both sides of the equation are equal, we again obtain a linearized first-order equation, containing only the first-order ac terras of Eq. (7.38):

C = -D7(r)-./J(0

{7.3У}

This is the desired small-signal linearized equation that describes variations in the capacitor voltage.

Finally, the equation of the average input current is also linearized. Insertion of Eqs. (7.30) and (7.31) into the input current equation ofEq. (7.28) yields

+ 1>) = (D 4 (/(()) [/-H(O)

(7.40)

By collecting terms, we obtain

,70 = [Ul] + [Df{0 + J(0) +

Dc term 1 order ac term Dc term 1 order ac terms 2 order ac term

(linear) (nonlinear)

(7.41)

We again neglect the second-order nonlinear terms. The dc terms on both sides of the equation are equal. The remaining first-order linear ac terms are

L(t) = Dht) + /d(r)

(7.42)

This is the linearized small-signal equation that describes the low-frequency ac components of the converter input current.

In summary, the nonlinear averaged equations of a switching convetter can be linearized about a quiescent operating point. The converter independent inputs are expressed as constant (dc) values, plus small ac variations. In response, the converter averaged waveforms assume similar forms. Insertion of Eqs. (7.30) and (7.31) into the converter averaged nonlinear equations yield.s dc terms, linear ac terms, atid nonlinear terms. If the ac variations are sufficiently small in magnitude, then the nonlinear terms are

much smaller than the linear ae terms, and so can be neglected. The remaining linear ae terms cximprise the smidl-signal iie model ofthe converter.

7.2,6 Construction of the Small-Signal Equivalent Circuit Model

Equations (7.ЭЙ), (7.39), and (7.42) iire the sinall-signal ac description ofthe ideal buck-boost conveiter, and are collected below:

L(/)=Df(i) + C()

(7.43)

In Chapter 3, we collected the averaged dc equations of a converter, and reconstructed an equivalent circuit that modeled the dc properties ofthe converter. We can use the same procedure here, to construct averaged small-signal ac models of converters.

The inductor equation of (7.43), or Eq. (7.36), describes the voltages around a loop containing the inductor. Indeed, this equation was derived by finding the inductor voltage via loop analysis, then averaging, perturbing, and linearizing. So the equation represents the voltages mound a loop of the sinall-signal model, which contains the inductor. The loop current is the small-signal ac inductor current [(l). As illustrated in Fig. 7.13, the term Ldi{i)/dt represents the voltage across the inductor L in the small-signal model. This voltage is equal to three other voltage terms. Dv{t) and ТУЩ) represent dependent sources as shown. These terms will be combined into ideal transformers. The term iV-V)<l(l) is driven by the control input t/((), and is represented by an independent source as shown.

The capacitor equation of (7.43), or Eq. (7.39), describes the currents flowing into a node attached to the capacitor. This equation was derived by finding the capacitor current via node analysis, then averaging, perturbing, and linearizing. Hence, this equation describes the currents flowing into a node ofthe small-signal mtxiel. attached to the capacitor. As illustrated in Fig. 7.14, the term CdvUVdt represents the current flowing through capacitor С in the small-signal model. The capacitor voltage is p(/). According to the equation, this current is equal to three other terms. The term - Di{i) represents a dependent source, which w ill eventually be combined into an ideal transformer. The term - iit)IR is rec-

D,(0

Dvit)

Fig, 7.13 Circuit equivalem to the .small-signal ae inductor loop eqttation of Eq, (7,43) or (7.36),