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.