them may be exploited to yield general results.
changes as the load varies. This is in contrast to PWM converters (in the continuous conduction mode) and to full-wave quasi-resonant converters, both of which may be operated open-loop (with constant D or J⅛) while showing little sensitivity to load changes. The load-dependence of half-wave converters is very similar to that of PWM converters operating in the discontinuous conduction mode.
The variation in the output voltage as a function of a change in load current, with the control fixed, is an indication of the output resistance of a converter. The incremental output resistance, Rout is defined by
Rout — ~ ∂V
∂I0ut (4.15)
where the input voltage Vg and the control variable (P for a PWM converter, Fs for a quasi-resonant converter) are held constant. Equation (4.9) and implicit differentiation of Eq.(4.11) can be used to show that
9V G,(p}Dp(M)
°"' ∂Imt κ°βMD'p(M) ,
where the partial derivative is taken with Fs and Vg both held constant. The derivative of a function with respect to its argument is indicated by a prime. (Note that D,p {M} is not 1 — Pp(ΛΓ).) The derivative G,(p) is negative while Dp(Af) is positive, so that Rout is positive as expected.
For example, in the PWM buck converter Mp(D} is simply D, so that Dp{M} — M and Dp(M) = 1. The output resistance of a quasi-resonant buck converter may therefore be expressed in the form
⅞ut _ 2W R pG{p) ∙
This quantity is plotted in Fig. 4.4 for the half-wave switch. The output resistance of a half-wave quasi-resonant buck converter is an appreciable fraction of the load resistance, particularly for low values of MRq∕R, or equivalently p.
For full-wave switches, G' is negligibly small because G is very nearly constant, as seen in Fig. 3.7. Using G, = 0 in Eq.(4.16) reveals that R0ut = 0 for all quasi- resonant converters with full-wave switches. Ideally, full-wave converters have no output resistance, just like their PWM counterparts.
Figure 4 4: Incremental output resistance of the half-wave quasi-resonant buck converter.
The high output resistance of converters with half-wave resonant switches may be considered undesirable, but this feature can be turned to advantage. Unlike PWM and full-wave converters, with their low output resistances, half-wave converters may be connected directly in parallel to the same load. The case of two converters in parallel is shown in Fig. 4.5. All the converters receive the same input voltage and operate at the same switching frequency J⅛, a frequency determined by a single voltage feedback loop regulating the load voltage. Such a scheme has several advantages, including built-in redundancy.
Ideally, each of n paralleled converters supplies an output current of exactly ∕∕n, where I is the total load current, but slight differences in component values inevitably lead to offsets between the various output characteristics of the different converters. For instance, a slightly higher value of resonant inductance in one converter will cause the resonant frequency Fo to be slightly lower. At the same Fg, this converter will have a higher ratio Fs∕Fq and will tend to produce a higher output voltage. The converter will have to provide more than its “fair share” of current in order to produce the same output voltage as all the other converters.
The degree of sensitivity of the output current to these inevitable errors is inversely proportional to the output resistance. Consider the case where two converters operate in parallel at the same switching frequency Fs∙ Figure 4.6(a) shows a portion of the output
v a
Control
+
¼oad
Figure 4-5: Two quαsi-resonαnt converters operating in parallel.
characteristics of the two converters. As a result of component tolerances, converter A has an output voltage curve shifted a small amount ΔV relative to converter B. Because their outputs are tied in parallel, both converters must have the same output voltage ½0αd∙
The voltage error ΔV results in an imbalance 2ΔI between the two output currents.
Converter A supplies a current Iιoadf,2∙ ÷ ∆f while converter B supplies Iιoad∕^ — ΔJ, as shown in Fig. 4.5. The current ΔJr circulates out of converter A and into converter B, never reaching the load. The amount of imbalance is
2Δ∕=--. ΔV (4.18)
Λou(
The voltage error ΔV will usually be small, so any value of Rout that is a significant fraction of the load resistance keeps the current imbalance tolerably small. Paralleling half-wave converters is therefore practical as long as p is not near unity.
PWM and full-wave converters have very small output resistances, however. The out
put characteristics of two such converters operating in parallel are indicated in Fig. 4.6(b).
The low output resistance means the output curves are nearly flat. Even a small shift ΔV results in a large current imbalance Δ∕. For this reason, elaborate precautions must be taken to provide current-sharing when paralleling PWM or full-wave converters.
2 (a)
(b)
Figure 4-6: (o.) Output characteristics of two converters operated in parallel when each converter has an appreciable output resistance, (b) The same characteristics for two converters with small output resistances.
Half-wave topologies in contrast share current naturally, a benefit of their large output resistance.
Chapter 5
Small-Signal Analysis of Quasi-Resonant Converters
The dc analysis of the previous section was based on the single assumption that the quantities Ion and Vroff had small ripple and therefore could be considered constant during each cycle of the resonant switch. This same approximation, along with a restriction to low modulation frequencies, is the basis in this chapter for an ac small-signal model of quasi-resonant converters. In Section 5.1, the goal of an ac model—a low-frequency, time- invariant representation of the resonant switch—is discussed. Section 5.2 then derives a model of the resonant switch which leads to a linear small-signal circuit model in Section 5.3. Section 5.4 discusses the implications of the small-signal circuit model, then, as an illustration of the technique, the small-signal circuit model of a quasi-resonant buck converter is presented in Section 5.5.