Transient Response Counts When Choosing Phase Margin

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

There is now an equation that describes the approximate closed-loop response of the dc-dc converter and it includes a quality coefficient. The next step is to establish a relationship between the closed-loop quality coefficient and the key design parameter, the open-loop phase margin. First, based on Eq. 8, calculate the crossover frequency determined by the location of the origin pole and its associated high frequency pole. At the crossover point, it is known that the T(s) module equals 1; therefore:

Extracting ω_C and rearranging this equation gives:

If Eq. 12 is substituted into Eq. 14, a quality coefficient-dependent crossover frequency can be obtained:

Eq. 15 shows how the closed-loop quality coefficient and the open-loop crossover frequency are linked. It is important for this remark to be well understood: Q represents the resulting closed-loop response quality coefficient based on the open-loop pole/zero arrangement describing the approximated open-loop compensated transfer function in Eq. 8.

To continue further with this analysis, evaluate the phase rotation of T(s) at the crossover frequency:

The phase margin represents the distance between the total phase rotation at the crossover frequency as given by Eq. 16 and the -180-degree limit. In this case, the phase reversal brought by the operational amplifier is purposely neglected. Hence:

Recalling those “far, far away” trigonometric classes, this means:

Thanks to Eq. 19, Eq. 16 can be updated as:

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