Many converters with more than one transistor or diode may be converted into quasi- resonant converters. As long as the multiple switches operate with some symmetry, these resonant converters may be analyzed by simple modifications to the process already put forth. In lieu of a general treatment, the principles involved will be demonstrated by an example.
Figure 6.5(a) shows a half-bridge PWM converter. The two large capacitors labeled Coa have small ripple; they are not resonant capacitors. Balanced operation is assumed so that these two capacitors always divide the source voltage Vg exactly between them.
The switching waveforms for this converter are shown in Fig. 6.5(b). The transistors Qχ and Q2 are driven out of phase, alternately applying +Vg∕2 and — Vg∕2 to the transformer primary. The voltage pulses are rectified by the diode bridge and filtered by Lj and Cf to produce a dc load voltage V.
The ON current in each transistor, defined as 7θn, is the reflected inductor current.
The inductor current is therefore Ion∕N. The OFF voltage across the diode bridge is defined as Voff and in this case is the voltage Vi∕2 across one of the C00 capacitors, reflected to the secondary of the transformer. Hence Vg∕2 = Vou(N. The conversion ratio M = V∕Vg for this converter is D∕2N, so that Dp(M∕N) = M∕2N. The factor of N of course comes from the transformer, while the factor of two is a result of the voltage divider formed by the two large input-side capacitors.
Figure 6.6(a) shows one of several ways to make a half-bridge quasi-resonant con
verter. The corresponding switch waveforms are shown in Fig. 6.6(b). The switching frequency is defined according to the waveforms following the rectifier bridge. Thus Sχ and S,2 are each operated at F⅛∕2, but 180 degrees out of phase so that the effective frequency at the output side of the transformer is F⅛.
Since Lr and Cr are on opposite sides of the transformer, Rq and p are defined as usual. The dc gain is found from
Fs Dp{M∕N} M 1
Fo G(MR0∕R) 2NGhw(MR0∕R) l j
where G⅛w is used because the two-quadrant switches in Fig. 6.6(a) form half-wave
iQι
iQt
(b)
Figure 6.5: A half-bridge P WM converter (a) and its waveforms (b).
switches.
The small-signal dynamic model is best presented as separate circuits for the input and output sides of the converter, coupled by dependent sources as shown in Fig. 6.7.
The presence of the resistors indicates the ever-present damping effect of the half-wave switch.
+
V
h---
t∕Fs ---≈∣(b)
Figure 6.6: A half-bridge quasi-resonant converter (a) and its switching waveforms (b)
(a)
Figure 6.7: Small-signal model of the half-bridge quasi-resonant converter: (a) input-side portion, (b) secondary-side portion.
Chapter 7
Experimental Verification
Both the dc and ac analyses of previous chapters relied on assumptions that have not been quantified. The small-ripple and low-frequency approximations are necessary for a clear and simple analysis, but it was noted that these approximations may not always be well-satisfied, and just how “small” or how “low” is needed for good agreement is not clear. The results of experimental work given in this chapter demonstrate that the analyses are valid when the approximations are met, and, surprisingly, often give good results even when the approximations are poorly satisfied. The good agreement between the measurements and the predictions is encouraging, and demonstrates that the models developed here can be useful design tools.
7.1 Dc Measurements
The general dc analysis method of Chapter 3 yields results identical to those found in [4,7,8] for the buck, boost, and buck-boost converters with “sine-wave type” switches.
Adequate agreement between dc analysis and measurements has been demonstrated for these converters in [4,7,8]. The purpose of this section is to demonstrate that the con
version ratio is independent of the placement of the resonant elements.
Chapter 3 showed that, as long as the resonant inductor and capacitor satisfy Rules 1 and 2 of Definition 2, the waveforms is and v∑> are always the same. Consequently, the dc conversion ratio is independent of the placement of the resonant elements. This is reflected in Eq.(4.11). One equation is sufficient to yield the conversion ratio for all variations of a given topology; one does not need separate expressions for sine-wave, square-wave, LI, L2, and M types of circuits.
To verify this result, a resonant-switch buck converter was constructed according to
Figure 7.1: An experimental quasi-resonant buck converter with a half-wave switch. The resonant capacitor and inductor are in the “Ll-sine” position.
Fig. 7.1. (The 1.8 k resistor across Lr was added to reduce ringing of Lr with its parasitic capacitance during the time when the switch S is OFF.) The resonant elements were then moved to two other positions, as shown in Fig. 7.2. The names “LI, sine-wave type,” etc., follow [9]. Figure 7.3 compares the measured conversion ratio M of the Ll-sine converter with the values predicted by Eq.(4.11). The frequency and load dependence are both in good agreement. As might be expected, the conversion ratio is always slightly less than the predicted value as a result of losses in Rι, the resonant tank, and ON losses of the transistor and diodes. All losses were neglected in Eq.(4.1l).
Measurements of the conversion ratio of the other two variations of the converter yielded values nearly identical at every operating point. With fixed input voltage, the output voltage differed by no more than 2 mV between any of the three variations, a margin of less than 3 percent. This result supports the claim in Chapter 4 of a common conversion ratio among all the variations of a quasi-resonant converter topology.
Although, as has been emphasized, all the variations of a given quasi-resonant con
verter have the same conversion ratio, these topological variations are not necessarily equivalent in other respects. Ripple voltages and currents and peak stresses in some components may vary widely between different variations. Some of these effects are cat
alogued in [9]. When losses are considered, the conversion ratio may be sensitive to these
Figure 7.2: Two variations of the quasi-resonant buck converter: (a) aL2-sine* type and (b) “M-square” type.
differences, leading to slightly different conversion ratios between topological variations.
In the Emit of high efficiency and small ripple, however, the conversion ratio is the same for all variations of a given topology, as demonstrated by the measurements made here.