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Master Control Systems Using Aktu’s Quantum Notes. Access critical insights and frequently asked questions to ace your B.Tech studies. Your journey to success begins here! **Unit-2 Time Response Analysis**

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**Q1. Explain various standard test signals, and also find relation between them.**

**Ans. A. Standard test signals: **

**1. Unit step signal:**

- i. Signals which start at time t = 0 and have magnitude of unity are called unit step signals.
- ii. They are represented by a unit step function u(t).

- iii. They are defined mathematically as:

**2. Unit ramp signal:**

- i. Signals which start from zero and are linear in nature with a constant slope m are called unit ramp signals.
- ii. They are represented by a unit ramp function r(t).
- iii. They are defined mathematically as:

**3. Unit impulse signal:**

- i. Signals which act for very small time but have large amplitude are called unit impulse functions.
- ii. They are represented by δ(t).
- iii. They are defined mathematically as,

**4. Unit parabolic signal :** The continuous-time unit parabolic function p(t), also called acceleration signal starts at t = 0, and is defined as:

**B. Relation between standard test signals:**

1. Relation between impulse and step signal :

2. Relation between step and ramp Signal:

3. Relation between ramp and parabolic signal:

**Q2. Derive the time response of first order system.**

**Ans. **1. Consider a first order system with unity feedback.

**3. Response to unit step input:**

Taking inverse Laplace transform, we have

**4. Response to unit impulse input :**

**5. Response to unit ramp input: **

Taking inverse Laplace transform, we have

**Q3. The open loop transfer function of a unity feedback control system is given by G(s) = 9/s (s + 3). Find the natural frequency of response, damping ratio, damped frequency and time constant.**

**Ans. **1. Transfer function of closed loop system,

2. Comparing eq. (2.6.1) by standard second order characteristic equation, ‘

5. Damped frequency,

**Q4. Define the following term: **

**i. Rise time **

**ii. Peak time **

**iii. Peak overshoot **

**iv. Settling time**

**Ans. 1. Delay time (t**_{d}**):** It is the time required for the response to reach 50 % of the final value in first time.

**2. Rise time (t**_{r}**):** It is the time required for the response to rise from 10% to 90 % of its final value for overdamped system and 0 to 100 % for underdamped systems.

**3. Peak time (t**_{p}**):** The peak time is the time required for the response to reach the first peak of the time response or first peak overshoot.

**4. Maximum overshoot (M**_{p}**): **It is the normalized difference between the peak of the time response and steady output. “The maximum percent overshoot is defined as

**5. Settling time (t**_{s}**): **The setting time is the time required for the response to reach and stay within the specified range (2 % to 5 %) of its final value.

**Q5. The unity feedback system is characterized by an open loop transfer function is G(S)= K/s(s + 20). Determine the gain K, so that the system will have a damping ratio of 0.6. For this value of K, determine unit step response, time domain specifications: settling time (2 % criterion), peak overshoot, rise time, penk time, delay time for a unit-step input. **

**Ans. **

**Q6. Write a short note on proportional control. Also write its advantages and disadvantages.**

**Ans. A. Proportional control:**

- 1. In proportional control, the error signal serves as the actuating signal for the control action in the control system.
- 2. The error signal is the difference between the feedback signal acquired from the output and the reference input signal.
- 3. The system described in Fig. is a proportional control system because the actuating signal is proportional to the error signal.
- 4. Take into account a second order system where the mistake itself serves as the controller input and the proportional constant is K=1.

**B. Advantages:**

- 1. Steady state error is reduced hence the system becomes more stable.
- 2. Easy to implement.
- 3. Relative stability is improved.

**C. Disadvantages: **

- 1. Due to the presence of these controllers, we get some offsets in the system.
- 2. Proportional controllers also increase the maximum overshoot of the system.

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