See figure below for the problem statement.
My idea was to trace backwards from the ω→+∞ at the origin until I reach a vector from the origin to a point on the curve with magnitude one. This ω will correspond to the largest gain cross over frequency and provide me with the phase margin.
I could then link this value of the phase margin back to zeta through the following equation,
\(\text{P.M.} = \gamma = tan^{1}\left( \frac{2\zeta}{\sqrt{2\zeta^{2}+\sqrt{1+4\zeta^{4}}}}\right)\)
The first point I could find that would give me a vector of magnitude one resides at (0.6, 0.8) yielding a phase margin of 126.87°. Unfortunately this PM yields a negative value for zeta.
Any idea where I went wrong or an easier way to solve the problem?
My idea was to trace backwards from the ω→+∞ at the origin until I reach a vector from the origin to a point on the curve with magnitude one. This ω will correspond to the largest gain cross over frequency and provide me with the phase margin.
I could then link this value of the phase margin back to zeta through the following equation,
\(\text{P.M.} = \gamma = tan^{1}\left( \frac{2\zeta}{\sqrt{2\zeta^{2}+\sqrt{1+4\zeta^{4}}}}\right)\)
The first point I could find that would give me a vector of magnitude one resides at (0.6, 0.8) yielding a phase margin of 126.87°. Unfortunately this PM yields a negative value for zeta.
Any idea where I went wrong or an easier way to solve the problem?
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