(Aktu Btech) Control System Important Unit-4 Frequency Response Analysis

Master Control Systems using Aktu’s Quantum Notes. Access critical insights and frequently asked questions to excel in your B.Tech courses. Your journey to success begins here! Unit-4 Frequency Response Analysis

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Q1. What do you understand by frequency response ? What are its various methods ? Discuss their advantages, disadvantages and limitations.

Ans. A. Frequency response: 

• 1. The frequency response is the magnitude and phase relationship between a system’s sinusoidal input and steady-state output.
• 2. Linear system with sinusoidal input 

B. Various methods of frequency response: 

1. Polar plots                                               2. Inverse polar plots 

3. Nyquist stability criterion                         4. Bode plot 

C. Advantages of frequency domain approach: 

• 1. The frequency response of a stable open loop system can be obtained experimentally without knowing the transfer function.
• 2. These methods are simple to apply for control design and determining absolute and relative system stability.
• 3. Even with substantial nonlinearity, the system can be represented by an approximation transfer function.
• 4. There is a close relationship between a system’s frequency response and its step response.

D. Disadvantages of frequency response: 

• 1. The frequency test is difficult to perform for systems with very large time constants because the time necessary for the output to achieve steady-state for each frequency of the test signal is disproportionately long.
• 2. Non-interruptible systems cannot be subjected to the frequency response test.

E. Limitations of frequency response methods : 

• 1. These methods can be applied only to linear systems. 
• 2. For an existing system, obtaining frequency response is possible only if the time constants are up to few minutes. 
• 3. Obtaining frequency response practically is fairly time consuming. 

Q2. Sketch the polar plot of the following function, also determine Gain Margin, Phase Margin, H(s) = 1

Ans. 

10. As GM and PM are positive, system is stable.

Q3. Sketch the polar plot for the following transfer function 

                                    G(s) = (1 + 4s)/s² (s + 1) (2s + 1)

Ans. 

6. Polar plot makes an intercept on the negative real axis at point X.

7. The intercept OX is calculate as given below :

Equating imaginary part to zero

Q4. Sketch Nyquist plot for G(s) H(s) = 6/s² (s + 2) comment, on stability.

Ans. Given: G(s) H(s) = 6/s²(s +2) 

To Sketch: Nyquist plot. 

1. In frequency domain the given transfer function will become,

2. Magnitude (M), 

                                      Z= 2 (Number of zeros on right-hand side) 

So the system is unstable because its value lies in the right half s-plane. 

Q5. Sketch the Nyquist plot for the system with open loop transfer function

and comment on stability. 

Ans. 

Q6. The forward path transfer function of unity feedback control system is G(s) = 100/s(s + 6.45). Find the resonance peak M_r, resonant frequency ω_r, and bandwidth of the closed loop system.

Ans. 

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