You are currently viewing Transfer Function Of Closed Loop System | Derivation
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“.

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.

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