The designer should take note, however, that while PWM converters usually operate at constant frequency, quasi-resonant converters are frequency-controlled. The require
ments on the filter reactances become more severe as the switching frequency is reduced to regulate against input voltage or load changes.
Figure S. 7: Plots of the functions G(p) for the half-wave and full-wave resonant switches, denoted Ghw andGfw, respectively.
full-wave switches, respectively, the functions are given by
Ghw(p) = I + * + sin-1 p + ~ (l + γ∕l-p2) Gfw(p) = g? 1.2 + 2π ~ βin
(3.42) (3.43)
1 p +J( i -√ iξ 7)
These functions are plotted in Fig. 3.7. Despite the small difference in the waveforms of Fig. 3.6, the functions Gfιv∣ and Gfw are quite different. Gfιw is strongly dependent on p and in fact grows without limit as p→0. In contrast, Gfw is nearly constant, always being within 1.2% of unity.
3.5.2 Restrictions on the Averages
Note that p is always positive and must be less than unity to be a proper argument for the inverse sine function of Eqs.(3.42) and (3.43). The physical meaning of the restriction p < 1 is apparent from Fig 3.6. The resonating current is a sinusoid of amplitude plon with an offset of Ion. If p > 1, the resonating current will not reach zero and the switch cannot turn off into zero current. Since the zero-current turn-OFF is the chief advantage of the resonant switch, only that case is considered here, and p is restricted to 0 < p < 1.
Figure S.8: Upper limit on the switching frequency in a quasi-résonant converter.
is∕IθN (max) and
vd∕^off (∏>3×)
Figure 8.9: Maximum value of the average ratios is/Ion resonant switch.
and vj}∕Voft for a zero-current
One other physical condition restricts the value of p in Eqs.(3.42) and (3.43). With the discontinuous operation assumed, in which the resonant switch always returns to its idle state before the switch S is turned on, the time 3s must be less than the switching period T⅛. Using the equations of Table 3.2, this implies
fb ≤ G(p) + √4τr , (3 44)
This constraint, as a function of p, is plotted in Fig. 3.8. When the limits of Fig. 3.8 are combined with the curves of Fig. 3.7 for G{ρ), the result is an upper limit on the average values of Eqs.(3.40) and (3.41). These limits, shown in Fig. 3.9, are achieved when the switching frequency is increased to the point that the switch S is turned on
just as soon as the diode voltage vjy reaches zero. If the frequency is increased beyond this point, the switch never reaches its idle state. This represents a different mode of operation of the resonant switch, a “continuous” mode in which the capacitor voltage never reaches its idle value. The initial value of the capacitor voltage for each switching cycle depends upon the conditions during the previous cycle, yielding dynamics in the resonant switch that are missing in the “discontinuous” mode considered so far. The only mode considered here is the one in which the switch does have an idle period. In this mode, the curves of Fig. 3.9 show that must always be less than 7o∏ and that ϋρ cannot exceed Vroff∙
Chapter 4
Dc Analysis of Quasi-Résonant Converters
The average behavior of the resonant switch is used in this chapter to derive ex
pressions for the conversion ratio of quasi-resonant converters. The relation between a quasi-resonant converter and its PWM parent, developed in the preceding chapter, now pays off, allowing the quasi-resonant conversion ratio to be expressed in Section 4.1 in terms of the well-known ratio of the corresponding PWM converter. Following sections explore the implications of the quasi-resonant conversion ratio, showing that some con
verters behave much like their PWM parents, while others have very different behavior.
Some quasi-resonant converters, it will be shown, are particularly well-suited to parallel operation.
4.1 Dc Conversion Ratio
Consider the converter model shown in Fig. 4.1 (a). Just as in Figs. 3.1 and 3.4, this model views the converter as a linear network driven by the switch waveforms ig and
vjj. The dc behavior of the system, and in particular the dc conversion ratio M ≡ V∕Va, involves only the dc value of the sources ig and υ∑>. Hence, for purposes of dc analysis, the model of Fig. 4.l(a) can be replaced by that of Fig. 4.1(b). In this model, the sources is and v∑> are replaced by their average values, ts and ΰρ. In addition, all inductors in the linear network become short circuits, since every inductor has zero average voltage in steady-state. Similarly, all capacitors become open circuits. (Note that it is the dc load resistance R, defined as the ratio of dc load voltage and current, that appears in Fig. 4.1.)
The model of Fig. 4.1(b) applies equally to a PWM converter (in which the transistor current îq is equivalent to the switch current is} and to any quasi-resonant converter
(a)
(b)
Figure 4.I: (a) A converter viewed as a linear network driven by the switch current and diode voltage waveforms, (b) At dc, the average of the switch waveforms drives the dc network.
derived from the PWM topology. The averaged linear network of a quasi-resonant con
verter is the same as that of its PWM parent, because Lr (added in series) is shorted and Cr (added in parallel) is open. Figure 4.1(b) clearly suggests that when a quasi-resonant converter and its PWM parent have the same values of ts and t⅛ the conversion ratio M must be the same.
The averages ts and v∑∣ axe easily found for the PWM converter. Figure 3.2, for example, reveals that
*s — DIon v∑> = DV0ff
(4.1) (4.2) For a quasi-resonant converter, the averages are given by Eqs.(3.40) and (3.41) of Chap
ter 3, repeated here:
*s = Fs
Vd = Fo F^
Fo
G(p)Ioa G(p)Voii .
(4.3) (4.4) Comparison of Eqs.(4.1) and (4.2) with Eqs.(4.3) and (4.4) reveals that the conversion ratios of the PWM converter and its quasi-resonant derivative must be equal whenever
Fs
Fo GW (4.5)
For a PWM converter, the voltage conversion ratio M ≡ V∕Vg is a function only of D, that is
M≡- = Mp{D) (4.6)
The conversion-ratio function Mp(D} is well-known for each PWM converter. Table 4.1 lists Mp(D) for the buck, boost, buck-boost, and Cuk topologies.
Under the equivalence of Eq.(4.5), the conversion ratio M for the quasi-resonant converter is given by
M = Mp ⅞GW (4.7)
This equation is not immediately useful, however, because ρ involves ∕o∏ and Voff, quan
tities that in turn depend on the conversion ratio. A relation between p and M will D
Converter Mp{D) Dp(M)
Buck D M
Boost 1 M-l
1- D M
Buck-boost D M
and Cuk 1-D 1~M
Table f.l: The PWM conversion ratio Mp(D) and its inverse Dp(M) for several com
mon P WM converters.
resolve this problem. For every known converter meeting Definition 1 (see, for example, [14] for a catalog of many such converters), the following is true:
‰ = A
ion
M ’
(4.8)where R is the dc load resistance, the ratio of dc load voltage and current. This expres
sion, when combined with the definition of p in Eq.(3.11), yields the equivalent relation ',=*"⅛=m⅞∙ (-,∙9>
That Eq.(4.9) should hold for such a large class of converters is somewhat surprising.
Such a general result is usually fairly obvious, but must be given here without proof.
Substitution of Eq.(4.9) in Eq.(4.7) yields
M = M,[∣θ(^l)]. (4.10)
The function G is complicated, with combinations of algebraic and trigonometric func
tions, so that Eq.(4.10) cannot be explicitly solved for M. It can, however, be solved explicitly for the control variable, F⅛, by inverting the function Mp(D).
Since Λ4p(D) is always one-to-one for converters meeting Definition 1, the inverse function Dp{M} always exists. Table 4.1 gives the inverse conversion-ratio functions Dp[M} for the most common PWM converter topologies.
When Eq.(4.10) is solved for Fs/Fo, the result is Fs = Dp(M)
⅞ G (4.11)
Fs∕F0
Figure /.S: Conversion ratio M for the buck converter with a zero-current resonant switch.
It is unfortunate that no explicit expression for M can be found, but hardly surprising in light of the complexity of the resonant waveforms. Equation (4.11) is nevertheless very useful. For example, one might step through a range of M values, plotting the required switching frequency versus M and thereby generating a plot of M versus Fs ∕Fq without need of a numerical root finder.
The curves of Figs. 4.2 and 4.3 were generated in just this way. These figures give ex
amples of the dependence of the conversion ratio on the load and the switching frequency.
The curves end where the switching frequency reaches the upper limit of Eq.(3.44) and Fig. 3.8.
Perhaps the greatest advantage of Eq.(4.11) is that it allows one to express Fs ∕Fq as a function of M and R. This in turn allows the average ratios ts ∕Ion and t>p∕Voff to be expressed as functions of M alone:
*s
■ion
Vp _ Dp(M)
V^^t~ G G = Dp{M} (4.12)
In the ac analysis of the following chapter, this will provide a convenient way of writing the small-signal parameters in terms of the dc quantities M and R.
Figure 4.8: Conversion ratio M for the boost converter with a zero-current resonant switch.
Equation (4.12) states a simple concept worth examining. In a PWM converter, the duty ratio D gives the ratios «ς//Οη and υjp∕Voff. The PWM conversion ratio Λf is given by the function Mp(D) of these ratios. In a quasi-resonant converter based on this PWM topology, the conversion ratio M is the same function of the ratios ιs∕Iaa and v∑>fVoκ.
In other words, for the same value of M, a quasi-resonant converter and its PWM parent must have the same ratios *s∕2θn and v∑,fVott, a ratio given by Dp{M} for the PWM converter. It is this equivalence that is expressed in Eq.(4.12).
References [4,7,8] contain analyses of the dc behavior of buck, boost, and buck-boost quasi-resonant converters. Since [4,7,8] make assumptions equivalent to the small-ripple approximation, the results are identical to those predicted by Eq.(4.11) for those three converters. The method used here is more powerful than that of [4,7,8] because in one equation, Eq.(4.1l), it yields results for all zero-current quasi-resonant converters. One needs only the well-known function Mp(D) of the original PWM converter and the appropriate function G⅛w or Gjw for the resonant switch, and Eq.(4.1l) immediately gives the relation from which M can be found. Equation (4.11) is an illustration of how the relation between quasi-resonant converters and the PWM topologies that underlie
them may be exploited to yield general results.