TRANSFER FUNCTION OF CLOSED LOOP SYSTEM | DERIVATION

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TRANSFER FUNCTION OF CLOSED LOOP SYSTEM | DERIVATION

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

TRANSFER FUNCTION – 

Transfer function is the ratio of Laplace transform of output signal to the Laplace Transform of Input Signal, keeping initial conditions to be zero.

CLOSED LOOP CONTROL SYSTEM –

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

BLOCK-DIAGRAM OF CLOSED LOOP SYSTEM

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 

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 

For positive feedback system we will use eq.3a and repeat the 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 postive unity feedback

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