I : M(D)

Fig. 7.52 The carnMiital niodel, for ideal CCM converters eotltaiuing a sirtgk JtidiKtor and capacitor.

Table 7.i Canonical model parameters for the ideal buclc, boost and buck-boost conveners

Converter

M(D)

Buck

Boost

Buck-boost

inductor value L, but also on the quiescent duty cycle Z). Furthermore, the current flowing in the effective inductance Ldoes not in general coincide with the physical inductor ctirrent / + l{t).

The model t)f Fig. 7.62 can be solved using conventional linear circuit analysis, to find quantities of interest such as the ctmverter transfer functions, input impedance, and output impedance. Tran.sformer isolated versions of the buck, boost, and buck-boost converters, such as the full bridge, forward, and tfyback converters, can also be modeled using the equivalent circuit of Fig. 7.62 and the parameters of Table 7.1, prtwided that one correctly accounts for the transformer turns ratio.

MODELIN(; THE PULSE-WIDTH MODULAT(}R

We have now achieved the goal, stated at the beginning of this chapter, t)f deriving a useful equivalent circuit model for the switching converter in Fig. 7.1. One detail remains: modeling the piil.se-width modulator. The pulse-width modulator block shown in Fig. 7.1 produces a k)gic signal 5(/) that commands the converter power transistor to switch on and off. The logic signal 5(/) is periodic, with frequency/j and duty cycle d(t). The input to the pulse-width modulator is an analog control signal Vj,(0. The function of the pulse-width modulator is to produce a duty cycle d{f) that is proportional to the analog control volt-

Sawtooth

wave generator

Coinparator

Analog input

vt) o-

PWM waveforrti

Hg. 7.63 A simpli; pulse-width modubtor circuit.

age v,.{0.

A schematic diagram of a simple pulse-width modulator circuit is given in Fig. 7.63. A sawtooth wave generator produces the voltage wavefonn v ; ,(t) illustrated in Fig. 7.64. The peak-to-peak amplitude of this waveform is V . The converter switching frequency/, is determined by and equal to the frequency of v,(0. An analog comparator compares the analog contrt)! voltage vjt) to Vjj,(;). This coraparattff prttduces a logic-level output which is high whenever v(0 is greater than v (r), and is otherwise low. Typical wavefonns are illustrated in Fig. 7.64.

If the sawtooth waveform i,J.t) has minimum value zert), then the duty cycle will be zero whenever v,.(0 is le.ss than or equal to zero. The duty cycle will be D = 1 whenever v,.(0 is greater than or equal to Vj. If, over a given .switching period, ,. ,(0 varies linearly with t, then for 0 < v(t) < V the duty cycle rf will be a linear function of f,.. Hence, we can write

(0 = - for 0 < t,(0 < V

(7.170)

0 dT,

Fig, 7.<i4 Waveforms of the circuit of Fig. 7.63.

7.6 Modeliuji the Fiilse-Viidih Modulmor

Fig. 7.6S Pulse-width modulatof block diagratn.

This equation is the input-output chaiacteiistic of the pulse-width modulator [2,11].

To be consistent with the perturbed-and-linearized converter rat)dels of the previous sections, we can perturb Eq. (7.170). Let

v(0= v, + i;,(o

d{t) = D + J(t)

Insertion of Eq. (7.171) into Eq. (7.170) leads to

0 + (0 =

(7.171)

(7.172)

A block diagram representing Eq. (7.172) is illustrated in Fig. 7.fi5. The pulse-width modulator has lin-eat gain lV/. By equating hke tenns on both .sides of Eq. (7.172), one obtains

(7,173)

So the quiescent value of theduty cycle is determined in practice by

The pulse-width modulator model of Fig. 7.65 is sufficiently accurate for nearly all applications. FIt)wever, it should be pointed out that pulse-width mt)dulators also introduce sampling of the waveform. Although the analog inptit signal vU) is a continuous function t)f time, there can be only one discrete value of the duty cycle during every switching period. Therefore, the pulse-width modulator samples the waveform, with sampling rate equal to the switching frequency j. Hence, a more accurate modulator block diagram is as in Fig. 7.66 [10]. In practice, this sampling restricts the useful frequencies t)f the ac variations to values much less than the switching freqtiency. The designer must ensure that the bandwidth of the control system be sufficiently less than the Nyquist rate fJ2,

Significant high-ficqucncy variations in the control signalvr) can also alter the behavior of the pulse-width modulator. A common example is when v{l) contains switching ripple, introduced by the feedback loop. This phenomentm has been analyzed by several authors [10,19], and effects of indtictor current ripple on the transfer ftinctions of current-programmed converters are investigated in Chapter 12. But it is generally best to avoid the case where v (r) contains significant components at the switching frequency or higher, since the pulse-width modulators of such systems exhibit poor noise immunity.