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
