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The given unity feedback system shown in Figure 2 with G(s)=s(s+7)K​ is operating with 15% overshoot. Using frequency response techniques, design a compensator to yield Kv​=50 with the phase-margin frequency and phase margin remaining approximately the same as in the uncompensated system.

User Meriley
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Step-by-step explanation:

To design a compensator that yields a Kv of 50 with the phase-margin frequency and phase margin remaining approximately the same as in the uncompensated system, we can follow these steps:

1. Determine the transfer function of the uncompensated system:

- The transfer function of the given unity feedback system is G(s) = s(s + 7)K.

2. Calculate the open-loop transfer function:

- The open-loop transfer function is GOL(s) = G(s).

3. Find the gain crossover frequency (ωgc) and phase margin (PM) of the uncompensated system:

- Using frequency response techniques, determine the gain crossover frequency (ωgc) and the corresponding gain magnitude (|GOL(jωgc)|).

- Find the phase margin (PM) corresponding to the gain crossover frequency (ωgc).

4. Design a compensator that achieves a Kv of 50:

- The steady-state error constant (Kv) is given by Kv = lim(s→0) [sGOL(s)].

- We need to increase Kv from its current value to 50.

- To achieve this, introduce a compensator C(s) in the feedback path.

5. Adjust the compensator C(s) to achieve the desired Kv:

- The compensator transfer function C(s) can be written as C(s) = (s + α).

- By adjusting the value of α, we can control the steady-state error constant Kv.

- Set α = 50/K - 7 to achieve a Kv of 50, where K is the original gain of the system.

6. Evaluate the phase-margin frequency and phase margin of the compensated system:

- Calculate the new open-loop transfer function GOL_c(s) = C(s) * G(s).

- Determine the gain crossover frequency (ωgc_c) and the corresponding gain magnitude (|GOL_c(jωgc_c)|).

- Find the phase margin (PM_c) corresponding to the gain crossover frequency (ωgc_c).

7. Verify that the phase-margin frequency and phase margin are approximately the same as in the uncompensated system:

- Compare the gain crossover frequency (ωgc) and phase margin (PM) from step 3 with the values obtained in step 6.

- If they are approximately the same, the compensator design is successful.

Remember to perform the calculations and substitute the appropriate values for K, α, ωgc, PM, ωgc_c, and PM_c in the above steps to get the exact compensator design.

User Srivatsa Marichi
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