Eliminate the Guesswork in Selecting Crossover Frequency

Aug 1, 2008 12:00 PM
By Christophe Basso, Application Manager, ON Semiconductor, Toulouse, France

Rather than arbitrarily choosing a converter’s crossover based on the switching frequency, you can analytically determine the relationship between its crossover and its undershoot in response to a load step.

In most power-supply design examples it is common to arbitrarily place the crossover frequency at one-fifth or one-tenth the switching frequency. However, the crossover frequency actually affects the converter's output impedance and there is a true relationship between these two parameters. Therefore, once a designer selects the output capacitor based on its operating parameters such as rms current, temperature or acceptable voltage ripple, the designer can analytically select the crossover frequency to match the desired output undershoot.

This discussion shows how to derive the relationship that links crossover frequency and undershoot and describes how to tailor the bandwidth to fit exactly the converter's design requirements.

Simplified Buck Converter

Fig. 1 shows a simplified buck converter represented by a square-wave generator driving a low-pass filter. Both the inductor and the capacitor have ohmic losses. The output impedance of such a network can be derived with the input source shorted:

where Z_OUT equals the output impedance in ohms, L_OUT equals the output inductance in henrys, C_OUT equals the output capacitance in farads, r_Lf equals the inductor resistance in ohms and r_Cf equals the C_OUT equivalent series resistance (ESR) in ohms.

Through inspection, the designer can see that the inductor's resistance dominates the output impedance at dc (L_OUT is shorted and C_OUT is open) and that its inductance dominates as the frequency increases. Then the capacitor impedance starts to take over until it becomes a short circuit and leaves the impedance value to its series loss r_Cf.

By connecting a 1-A ac source to the output, the designer has the ability to quickly plot the output impedance versus frequency using a SPICE simulator. Fig. 2 portrays the obtained results. As can be seen, a peaking occurs at the resonant frequency (f₀). The maximum of this peaking can be analytically derived^[1]:

where Z_OUTMAX equals the maximum output impedance in ohms and Z₀ = equals the characteristic impedance of the filter in ohms.

Such peaking is typical of a buck output impedance behavior where the LC filter has been optimized to minimize the losses. This situation induces a high-quality coefficient, hence a severe peaking in the impedance graph. One of the feedback aims is to minimize the output impedance to reduce as much as possible the voltage drop due to a load step. On an amplitude versus frequency plot, the natural output impedance of the filter dramatically peaks at the resonant frequency. Therefore, if a crossover frequency below the LC filter resonance is selected, there will not be enough of a gain to get rid of the resonance and, despite a good phase margin, the system will oscillate. If the designer wants to obtain a good transient response, he or she has to make sure the loop gain remains high enough to tame the peaking when it occurs. In other words, the crossover frequency (f_C) must be selected at least three to five times above f₀.

In Fig. 2, if a crossover region is selected beyond the resonance, the designer can see an impedance graph dominated by the output capacitor impedance (C_OUT). At the crossover frequency, this impedance is:

where f_C equals the crossover frequency and Z_OUTOL equals the open-loop output impedance.

Above the crossover frequency, the capacitor's ohmic losses dominate the network's output impedance. To ensure Eq. 3 rules the output impedance alone at the crossover point, the capacitor ESR must be much smaller than the output impedance at the crossover frequency. Mathematically, the following condition must be met:

In choosing the final capacitor, besides ripple current and temperature considerations, the designer must also consider the capacitor ESR at the selected crossover frequency.

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