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.
Fs∕F0
Figure 7.S: Measured values of the conversion ratio M of the buck converter of Fig. 7.1.
The theoretical conversion ratio is given by the solid curves.
an exciting sinusoid. Automated frequency-response measurement systems simplify this task and allow the results to be stored and presented in graphical form.
Since a power converter output by nature contains “noise” in the form of switch
ing ripple, the output waveforms under modulation contain not only the modulation frequency but also the switching frequency, its harmonics, and “cross-modulation” com
ponents. The frequency response, however, is concerned only with the component of the output waveform at the modulation frequency. The solution to the measurement problem is to use a network analyzer in conjunction with a sine-wave generator, or synthesizer, as shown in Fig. 7.4.
The synthesizer generates a pure sinusoid vc at the desired modulation frequency, fm.
This sinusoid is applied to the modulator which varies the converter control variable, the switching frequency, at fm∙ The waveform vc is also applied to channel A of the ana
lyzer. Channel B measures the converter output. The analyzer filters these waveforms, saving only the components at fm. Comparison of the magnitude and phase of the fm components of the channel A and channel B waveforms yields the frequency response at fm- The synthesizer is stepped through all desired values of the modulation frequency to obtain a frequency-response curve.
Figure 7.4: A network analyzer coupled with a synthesizer measures the frequency re
sponse of a quasi-resonant converter and its modulator.
The modulator used to control the switching frequency of the resonant-switch con
verters appears in Fig. 7.5. The dc switching frequency is set manually, while mod
ulation of the switching frequency is determined by the voltage vc ac-coupled to the voltage-controlled oscillator (VCO) input. The one-shot delivers a pulse of fixed width at the beginning of each switching period. The pulse, amplified by the driver, is applied to the transistor gate through an isolation transformer and level-shifting circuit. For a zero-current-switching (ZCS) converter, the pulse width is manually adjusted so that the pulse ends during the time that diode Dχ is blocking, when the transistor Q is carrying no current. For a ZVS converter, the pulse is set to end when J9χ is ON and Q is under zero voltage.
To obtain the predicted frequency response curves that follow, the VCO was modelled as a constant ratio between vc and fs, with no dynamics. This assumption is justified in Section 7.3, where the internal circuits of different VCO’s and their accompanying dynamics are discussed.
Figure 7.5: Block diagram of the method used to drive the transistor of a quasi-resonant converter. The voltage-controlled oscillator (VCO) modulates the switching frequency under the control of the signal ve.
It should be noted that, while all the frequency-response measurements made here are of the control (∕s) to output (υ) transfer function, any transfer function of interest can be measured in a similar fashion. For example, the input voltage υ9 may be modulated and the input current i9 measured. The ratio v9∣i9 is the small-signal input impedance.
One of the advantages of a small-signal model such as that developed in Chapter 5 is that it gives a circuit from which all transfer functions may be found.
7.2.1 Half-wave Buck Converters
The control-to-output transfer function of a half-wave quasi-resonant buck converter was derived in Chapter 5, resulting in the model of Fig. 5.10 and Eqs.(5.23) through (5.28). To verify the model, a converter was constructed according to Fig 7.1.
(Since a linear resistance was used as the load in the experimental circuits, the dc load R and small-signal resistance Rι, are equal.) When the converter is operated with a load of 10.4 Ω (nearly equal to 2⅛), and a conversion ratio of 0.72, the measured frequency response is as shown in Fig. 7.6. The response predicted by Eqs.(5.23) through (5.28) is shown as a dashed line in the figure. The agreement is very good. Note that the max
imum modulation frequency in Fig. 7.6 is only 1/12 of the switching frequency, so that
Figure 7.6: Magnitude (upper curves) and phase (lower curves) of the control-to-output response of the half-wave buck converter of Fig. 7.1 operating with a relatively large value of p. The solid curve is the measured response and the dashed lines show the response predicted by the small-signal model.
the low-frequency approximation is well-satisfied. The ratio p = MRq/R is 0.71 at this operating point, a fairly high value. The second-order response has a Q of approximately unity. The output filter alone has a Q of 1.6, and it is this Q that would appear if the resonant elements were removed and the circuit operated as a PWM buck converter.
When operated at the lower value MRq/R = 0.15, achieved by increasing the load to 49.7 Ω while keeping the conversion ratio nearly the same, the frequency response changes considerably. In a PWM buck converter, increasing the load resistance would increase the loaded Q of the output filter, leading to an underdamped frequency response. The response of the quasi-resonant buck converter, shown in Fig. 7.7, exhibits the opposite effect. The lower value of p has increased the damping (represented by —Jιυ,∙ in the circuit model of Fig. 5.10) to such an extent that the poles have separated to positions at roughly 100 Hz and 5 kHz. A PWM buck converter with the same element values in the output filter would exhibit an underdamped response with a Q of about 5.6.
Figure 7.7: Frequency response of the half-wave buck converter of Fig. 7.1 operating with a low value of p. The solid curves are the measured response and the dashed curves show the predicted response.
7.2.2 Full-wave Buck Converter
The full-wave resonant-switch contributes no damping to a power converter. To confirm this result, the half-wave resonant switch of Fig. 7.1 was changed to a full- wave switch by moving the diode Dχ to a position across the transistor. The resulting frequency response is shown in Fig. 7.8. Except for the phase deviations near half the switching frequency, the agreement between the measurement and the response predicted using the small-signal model of Chapter 5 is quite good. (Phase discrepancies near half the switching frequency are commonly observed in PWM converters. These deviations are to be expected as the low-frequency approximation breaks down, and neglect of the modulator sampling process becomes invalid.)
A third curve appears in Fig. 7.8, showing the measured response obtained by shorting Lr, removing Cr, and operating the converter as a PWM converter with a duty ratio D equal to the ratio Fs/Fq of the resonant converter. (The third curve was in fact obtained by perturbing the frequency with the modulator of Fig. 7.5, the same as for the
Figure 7.8: Measured frequency response (A) of a full-wave buck converter (solid line).
Shown by dashed lines is the predicted response for the full-wave converter (B) and the measured response of a PWM converter at the same operating point (C).
resonant-switch version of the converter. However, it can be shown that d _ fs
D~ Fs ’ (7.1)
and hence modulating the frequency is equivalent to modulating the duty ratio.) Again except for the phase deviations at high frequencies, the response of the full-wave switch converter, the PWM converter, and the model from Chapter 5 are all nearly identical.
This demonstrates the claim made in Chapter 5 that full-wave converters behave just like their PWM parents, with the control variable d replaced by ∕s∕Fq∙
7.2.3 Effects of Cr and Lr
Section 5.6 of Chapter 5 pointed out that the resonant elements Lr and Cr were often negligible, but warned that in some cases they might have important effects. Figure 7.9 illustrates such a case, showing the measured response of a buck converter with the topology of Fig. 5.12(a) and the element values of Fig. 7.1. The resonant capacitor Cr appears across L∕ in this example (a topology referred to as the “L2” configuration
Figure 7.9: Frequency response of a quasi-resonant buck converter showing a resonant peak from the interaction of Lj and Cr. The solid curves show the measured response while the dashed curves indicate the prediction from the small-signal model.
in [9]) and the load is 25.1 Ω. The frequency-response curves of Fig. 7.9 contain a high-Q resonance from the parallel-resonant circuit formed by L∕ and Cr. With this configuration, Cr can be “small” relative to other capacitors in the circuit, but if Lj is large, the resonant notch can easily appear within the frequency range of interest, with attendant difficulties in controlling the converter. Thus, while the positions of Cr and Lτ make no difference in the dc conversion ratio, they can make a large qualitative difference in the frequency response.
7.2.4 Zero-voltage-switching Boost Converter
Figure 7.10 shows the circuit of an experimental ZVS boost converter with a half
wave switch. The frequency response predicted by the model of Chapter 5 is compared with the measured response in Fig. 7.11, and the agreement is seen to be excellent.
The presence of a right half-plane zero is evident from the 270 degree phase lag seen in Fig. 7.11. Right half-plane zeros make it extremely difficult to design a high-bandwidth regulator, and one may wonder whether the half-wave switch can ever provide enough
Figure 7.10: Circuit of an experimental zero-voltage-switching boost converter.
Figure 7.11: Measured (solid curves) and predicted (dashed curves) frequency response of the ZVS boost converter of Fig. 7.10.
damping to move the RHP zero out to high frequencies or even into the left half-plane.
From the analytic form of the transfer function, derived using the method of Chapter 5, the (radian) zero frequency is
_ huR/M — R[ — hvi
z ~ Lf , (7-2J
With the use of the parameter expressions from Chapter 6,
The zero frequency is largest when Rι is negligibly small. The percentage ripple current in the input inductor can be shown to be proportional to the ratio of R∕M2Lj to Fs, so the frequency ωz∕2π will always be much less than the switching frequency.
If Rι is larger than R∕M2, the zero can be pushed into the left half-plane. Attempting to stabilize the converter with this approach suffers from two drawbacks, though. First, the power loss in Rι is RιM2 ∕ R times the output power. The zero crosses into the left half-plane only when the efficiency is less than 50%. Second, this method creates a left half-plane zero by moving the RHP zero through zero frequency. The resulting zero will therefore occur at low frequencies and, most important, will move back and forth as the load changes. If the load becomes too large, the zero will cross into the right half-plane at very low frequencies, with disastrous results on the feedback loop.
In conclusion, the damping of the half-wave switch is of no use in removing the right half-plane zero in the ZVS boost converter. The same conclusion holds for zero current-switching boost converters.
7.2.5 Small-ripple Approximation
It is worthwhile to check how well the small-ripple approximation is justified in each of the preceding measurements. Table 7.1 gives the percent ripple in Vroff an^ Λ>n for each of the frequency response curves of this chapter. The percent ripple is defined by
, . , 1 pk-pk excursion percent ripple =--- ---
2 average value (7.4)
A triangular waveform just touching zero therefore has 100% ripple.
Figure
% Ripple Voff Λ>n
7.6 .5 1
7.7 .5 12
7.8 (Resonant) .3 6
7.8 (PWM) .1 5
7.9 .5 4
7.11 .7 21
Table 7.1: Percent ripple in Ion andVo{{ for each of the measurements presented in this chapter.
While the ripple on ½,ff is small in every case, ∕o∏ has a significant ripple, especially in the ZVS boost converter of Fig. 7.10. Although the ripple in ion was highest for the measurement of Fig. 7.11, the agreement with the small-signal model, based on the small- ripple approximation, is best for this measurement! At least is this case, the parameters of the small-signal model are not extremely sensitive to the amount of ripple, and the small-signal approximation is justified.