Transient Response Counts When Choosing Phase Margin

Nov 1, 2008 12:00 PM
By Christophe Basso, Applications Manager, ON Semiconductor, Toulouse, France

The crossover frequency versus the closed-loop quality coefficient were already defined in Eq. 15. To capitalize on the definition in Eq. 20:

The next step is to extract the closed-loop quality coefficient from Eq. 21 and simplify the result:

This means there is now a relationship between the main design criterion, the open-loop phase margin and the quality coefficient the loop will exhibit once closed. The best thing to do is to explore the various quality coefficients that different phase margin choices will bring (Fig. 5).

If one wants to combine speed and a lack of overshoot, Fig. 2 suggests a quality coefficient of 0.5. Reading the corresponding phase margin in Fig. 5, it can be seen that a design criterion of 76 degrees satisfies this request for such a quality coefficient, far away from the 45 degrees recommended in the majority of textbooks.

What does it mean then? In the response to a load step, once the loop is closed, the open-loop phase margin mostly affects the recovery shape and a little of the undershoot depth. Therefore, it really depends on the kind of response a designer is looking for or what the customer specifications impose on a design.

If a designer needs a fast recovery and a little overshoot to be acceptable, then reducing the phase margin can be an option. On the contrary, if absolutely no overshoots are tolerated, the designer has no choice but to increase the phase margin to the detriment of the recovery speed.

Whatever solution designers selects, they have to make sure that — whatever the operating conditions, input/output, temperature and normal parametric variations (ESRs for instance) — the phase margin never goes below 45 degrees. In other words, shooting for a typical value around 70 degrees should become a good design practice.

Transient Response and Phase Margin

The buck converter in this example uses one of the automated simulation platforms described in another paper.^[1] The technique allows designers to keep the same crossover frequency while only working on the phase margin. The overall shape is the same as that presented in Fig. 4 with a 10-kHz crossover frequency. The output is subjected to step ranging from 1 A to 2 A in 1 µs. The results appear in Fig. 6. The 76-degree phase margin gives a little overshoot of 0.05%, whereas the 45-degree margin triples that overshoot, still reasonable though given the vertical-axis scale of 20 mV/division.

However, one can observe a faster recovery in the 45-degree phase case (70 µs) versus the 76-degree case (227 µs). Why do designers still have overshoot with the 76 degrees when theory states there should be none? It is because Eq. 8 is a simplified view of the transfer function in the vicinity of the crossover frequency. If a designer has three or more poles installed near the crossover frequency, the quality coefficient factor approximation done here does not work anymore and extra work will be required.^[2] Nevertheless, as exemplified by Fig. 6, a small phase margin leads to a peaky closed-loop response.

References

1. Basso, C. Switch Mode Power Supplies: SPICE Simulations and Practical Designs, McGraw-Hill, 2008.

2. Erickson, R. and Maksimovic, D. Fundamentals of Power Electronics, Kluwers Academic Press, 0-7923-7270-0.

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