Transfer Function Of Closed Loop System Derivation

## Transfer Function Of Closed Loop System | Derivation

Hello friends, welcome to webenotes. In this article, we will study about the “transfer function of closed loop system“.

Contents

## Transfer Function

The transfer function is the ratio of the Laplace transform of the output signal to the Laplace Transform of the Input Signal, keeping initial conditions at zero.

## Closed-loop Control System

The control system in which control action depends upon the output is called a closed loop or feedback control system.

### Block-diagram Of Closed Loop System

Fig. block diagram of closed loop system

Block-diagram Of Closed Loop System

Above fig. shows the Block Diagram of Closed Loop Control System, where

E(s) = Actuating or Error Signal

X(s) = Reference Input Signal.

G(s) = Forward Path Transfer Function.

Y(s) = Output Signal.

H(s) = Feedback Transfer Function.

B(s) = Feedback Signal.

So, the transfer function of the closed-loop system is Y(s)/X(s).

From the block diagram,

Y(s) = G(s).E(s) ………1

B(s) = H(s).Y(s) ………2

E(s) = X(s) + B(s) ………3a (For positive feedback)

= X(s) – B(s) ……….3b   (For negative feedback)

### For Negative Feedback (using eq.3b)

Put the value of E(s) from eq.3b in eq.1
Y(s) = G(s).[X(s)-B(s)]

Y(s) = G(s).X(s) – G(s).B(s) ………4

Put the value of B(s) from eq.2 in eq.4
Y(s) = G(s).X(s) – G(s).H(s).Y(s)

Y(s) + G(s).H(s).Y(s) = G(s).X(s)

Y(s){1 + G(s).H(s)} = G(s).X(s)

Y(s)/X(s) = G(s) / {1 +G(s).H(s)} ………5

#### For Positive Feedback(using eq.3a)

For a positive feedback system, we will use eq.3a and repeat all the same steps and we will get the transfer function as;

Y(s)/X(s) = G(s) / {1 – G(s).H(s)} ………6

### For Unity Feedback

For unity feedback H(s) = 1, the eq.5 & eq.6 will become

Y(s)/X(s) = G(s)/{1 + G(s)} …….for negative unity feedback

Y(s)/X(s) = G(s)/{1 – G(s)} ……..for positive unity feedback

### Aanchal Gupta

Welcome to my website! I'm Aanchal Gupta, an expert in Electrical Technology, and I'm excited to share my knowledge and insights with you. With a strong educational background and practical experience, I aim to provide valuable information and solutions related to the field of electrical engineering. I hold a Bachelor of Engineering (BE) degree in Electrical Engineering, which has equipped me with a solid foundation in the principles and applications of electrical technology. Throughout my academic journey, I focused on developing a deep understanding of various electrical systems, circuits, and power distribution networks.