# How is the slope of the gain curve related to robustness?

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To first approximation, there is a direct relation between the slope of the gain magnitude curve and the amount of phase in a Bode plot:

-2 slope --> -180 deg phase

-1 slope --> -90 deg phse

0 slope --> 0 deg phase

1 slope --> 90 deg phase

2 slope --> 180 deg phase

etc.

An exact relation using a weighting kernel is given in section 9.4 (p. 280) in AM08.

For robustness, you want relatively high phase margin, and that is achieved by minimizing the phase (making it greater than -180 deg). This can be achieved by minimizing the slope of the magnitude plot when it crosses unity.

--Fuller 17:17, 4 December 2007 (PST)