Eliminate the Guesswork in Selecting Crossover Frequency

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

Closing the Loop

Picture any voltage generator with an equivalent circuit associating a dc source V_TH and an output resistor R_TH. According to Thévenin's theorem, V_TH is evaluated by measuring the output voltage on an unloaded converter and R_TH is found by measuring the output-voltage difference in two loading current conditions. Imagine that Fig. 3a depicts an open-loop buck converter using the Fig. 1 approach. Once loop control is installed through a compensator, bringing gain and phase boost as in Fig. 3b, the open-loop impedance transforms into a closed-loop output impedance that now obeys Eq. 5 (Fig. 3c):

where Z_OUTCL (s) equals closed-loop output impedance for a loop gain of T(s) = H(s)G(s).

The designer now has an output impedance whose value depends on the open-loop gain. At dc, for s = 0, assume a large loop gain to ensure good dc regulation. In other words, the feedback brings the open-loop impedance to a very low value. On the contrary, when the frequency increases, the gain reduces and when the crossover point is reached, the gain no longer acts upon the output impedance. Mathematically, this is:

If a SPICE average model is used and a voltage-mode buck converter is compensated, there is the possibility to ac sweep its output impedance as was done in Fig. 2. The output impedance in Fig. 4 shows what Eqs. 5 and 6 predicted: Thanks to a high open-loop gain in the low-frequency domain, the output impedance remains extremely small, . But as the frequency increases, inductive behavior can start to be seen. Then, at the crossover point, the loop gain reaches 0 dB and both the open-loop and closed-loop impedances are almost equal to the output capacitor impedance given by Eq. 3.

Approximate Output Impedance

Previously, the term “almost” has been used to compare the open- and closed-loop output impedances at the crossover frequency. However, try to see how close they are in the vicinity of the crossover point. There are several methods to calculate the module of Eq. 5's right term, . One method applies a sinusoidal modulation to the complete chain made of the converter transfer function H(s) followed by the compensator transfer function G(s).

This is exactly what would be done in the laboratory to explore the true open-loop response of the compensated converter. However, in this particular case, rather than expressing the modulation signal through a classical form of sin(wt+j), a phasor notation will be used where φ represents the phase lag brought by the total chain when stimulated at the crossover frequency. Following is what details for a 1-V modulation.

The phasor notation can be update using Euler's formula:

In this equation, the term φ relates to the phase difference between the output signal and the input modulation. A design criteria here is not φ but j_m, the phase margin. To help link both, Fig. 5 shows the contribution of the loop to the total phase lag.

Based on the figure, it can be written:

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