Transfer function of Close loop system


Transfer Function

The transfer function of a linear, time-invariant, differential equation system is defined as the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function) under the assumption that all initial conditions are zero.



Above fig. shows the Block Diagram of Closed Loop Control System, where all variables are in laplace form i.e.

E(s) = Error Signal

G(s) = Forward Path Transfer Function.

Y(s) = Output Signal

X(s) = Reference Input Signal             

H(s) = Feedback Transfer Function.

B(s) = Feedback Signal.

Transfer function of system is
 
From the block diagram,       

Y(s) = G(s).E(s) ........1

B(s) = H(s).Y(s) ........2


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


        = X(s) - B(s) .........4   (For negative feedback)


FOR NEGATIVE FEEDBACK 

Put the value of E(s) from eq.4 in eq.1

Y(s) = G(s).[X(s)-B(s)]

Y(s) = G(s).X(s) - G(s).B(s) 


Put the value of B(s) 

Y(s) = G(s).X(s) - G(s).H(s).Y(s)

Y(s) + G(s).H(s).Y(s) = G(s).X(s)




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;



FOR UNITY FEEDBACK 
For unity feedback H(s) = 1




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