Hello friends, welcome to ELECTRICAL ENCYCLOPEDIA. In this article, we will study about the "transfer function of closed loop system".
Above fig. shows the Block Diagram of Closed Loop Control System, where
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
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| 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 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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