Idle Interval: Ts < t < Ts
When the previous interval ends at 7¾, both is and vjj are zero. The resonant switch has returned to its idle state. This state persists until the switch S is again turned ON, initiating the next cycle. The duration of the idle interval during any switching period is Ts — Ts, where l/Ts is the switching frequency, F⅛. Control over the converter is exercised by varying Ts and hence changing the length of this idle interval.
Table 3.2 summarizes the equations describing is and υp during each of the four intervals for the zero-current resonant switch. The corresponding waveforms are shown in Fig. 3.6. The equations of Table 3.2 were derived using only Definition 1 concerning the structure of a PWM converter, Definition 2 describing the topology of the resonant switch, and Theorems 1 through 4 derived from those two definitions. No reference was made to sinusoid al-wave or square-wave switch types, and the quantities V<ιc and ∕<jc (related to the topological positions of Lr and Cr) do not appear in any of the equations of Table 3.2. One must therefore conclude that the waveforms is and v∑> are the same for any converter using a zero-current resonant switch, regardless of underlying PWM topology or location of Lr and Cr. Only the type of switch S has an effect on the waveforms, the half-wave switch cutting off the current is as it passes through zero from above and the full-wave switch interrupting the current as it reaches zero from below.
Interval
Switch Current
*s(0∕ Λ>n
Diode Voltage vp(t)∕⅛ff
Duration of Interval
u>o∆t 0 < t < T1 1
-w0tP 0 P
Tι<t<T2 1 + i sin u>o(t - Tι)
P 1 — cos ωo (i — Tι) -
' --- π + sin-1 p (half-wave)
2τr — sin-1 p (full-wave) T2<t<Ts
half-wave:
full-wave:
0 0
1 + √1 - p2 - pωo(t - T2) 1 - √1 - p2 - pωo(t - T2)
i(ι + √Γ^7) i(ι-√Γ^7) τ3<t<τs
(Idle)
0 0 —
Table S.2: Summary of the equations describing the behavior of the zero-current resonant switch.
unusual to find either kind of converter operating with large ripple.
Large voltage ripple on internal capacitors increases the peak voltage stress, forcing the capacitor to have a higher rating than if the voltage ripple were small. (Capacitors at the input or output have small ripple by nature of the dc input and output voltages.) Inductors with large ripple currents suffer unnecessarily large power loss in their parasitic resistances. And perhaps most important, excessive ripple leads to higher voltage and current stresses on the semiconductor devices.
For these reasons, converters are usually designed with small to moderate ripple waveforms on reactances; values of 5-20% are common. The small-ripple approximation is not grossly violated by ripple of this magnitude, but neither is it well-satisfied.
The fact that the small-ripple approximation is often satisfied is, however, only one of three reasons why it is applied here. Another reason is that the experimental results of Chapter 7 indicate that the ac analysis of Chapter 5 is not particularly sensitive to ripple in ∕on and ½,ff∙ With ripple as high as 21% in Ion, the small-signal frequency response of a boost converter still agrees very well with the response predicted using the small- ripple approximation. At least in some cases, then, the small-ripple approximation can accurately predict the behavior of a converter even when that converter has appreciable ripple.
The third and most important reason for using the small-ripple approximation has nothing to do with whether the approximation is valid. The approximation is applied here because it paves the way for clear, simple, and general dc and ac analyses of quasi- resonant converters. Without the small-ripple approximation, Ion and Vojf could not be considered dc or even slowly-varying. This would make the resonant switch a system of fourth or higher order, and the equations describing the resonant waveforms would be unmanageable. Even worse, the resonant switch in each converter would have to be analyzed separately because the resonant waveforms would be functions of the mag
nitude and shape of the ripple, as well as of the average values of Ion and Voff∙ The assumption of small ripple may be justified, then, even if the results are not entirely accurate. The approximate information it provides allows simultaneous treatment of all quasi-resonant converters, and this may be more useful—for designing and understanding the converters—than accurate results requiring computer solution.
Small ripple on ∕on and ½,ff is achieved in the same way in PWM and in quasi- resonant converters—by making the inductors and capacitors providing Jon and Voff
“stiff.” These reactances are part of the linear network in Figs. 3.1 and 3.4. In a PWM converter, the linear network is driven by rectangular waveforms is and υp. These wave
forms are quasi-sinusoidal in a quasi-resonant converter, but because they are periodic at the switching frequency they still have no frequency components below the switching frequency. The sizes of the reactances needed to attenuate these driving waveforms to produce small ripple in Ion and Voff are therefore comparable between PWM and quasi- resonant converters operating at the same power level, load, and switching frequency.
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.