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I need help finding the gain \$K\$ so that the phase margin for the system equals 50 degrees.

$$F(s) = K$$

$$G(s) = \dfrac{1}{(s+1)^2}$$

\$v\$ is a process disturbance sinusoid with amplitude 2.5 and freq 0.5 rad/s. Image below.

Closed Loop Transfer Function

Now I need to find K so that the phase margin = 50 deg. I tried:

\$\varphi_m = 180\$ deg \$+ \arg G(i\omega_c) + \arg F(i\omega_c) = 50\$ deg

and solve for \$\omega_c\$ but I can't really figure out how to. Do I have to account for the disturbance as well? If so, why? why not?

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    \$\begingroup\$ No - the disturbance has nothing to do with stability margins. Are you able to find the phase response for G(s)? \$\endgroup\$ – LvW Aug 19 '20 at 9:32
  • \$\begingroup\$ @LvW I got: arg G(jw) = arctan(0) - arctan(2w/(1-w)) = -arctan(2w/(1-w)). I know that arg F + arg G = 50-180 deg. \$\endgroup\$ – Mati Aug 19 '20 at 11:33
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    \$\begingroup\$ PM only depends on the OLTF. Signals don't matter. \$\endgroup\$ – Chu Aug 19 '20 at 12:27
  • \$\begingroup\$ This is somewhat like 2 non-inverting Op Amps with negative feedback and choose gain so PM 50deg, at what gain. Or in other words , what is the GBW? what is BW and PM with a closed loop gain of 1 and at f-3dB \$\endgroup\$ – Tony Stewart EE75 Aug 19 '20 at 12:29
  • \$\begingroup\$ Dont use the phase equation to find \$\omega_c\$. Use the gain equation (i.e. the definition of gain cross over frequency) to find \$omega_c\$. \$|G(\omega_c)\cdot F(\omega_c)| = 1\$. Solve for \$\omega_c\$ from this. Then use that in the phase equation to find K. \$\endgroup\$ – AJN Aug 19 '20 at 14:29
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From you question I understand that you have written one equation and have unknown variables in it; viz, \$\omega_c\$ and K. If you write one more equation, you may be able to solve for K. To that end,

Definition of gain cross over frequency is

\$ \begin{align} |GF|_{s=j\omega_c} ={}& 1\\ \frac{K}{|1+j\omega_c|^2} ={}& 1\\ K^2={}& (1+\omega_c^2)^2 \\ \end{align}\$

Solve for \$\omega_c\$ in terms of K.

In the phase equation you already have with you, substitute \$\omega_c\$ with the expression containing K. Now that equation has only one unknown, viz. K. Solve!

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