Transfer function of a system

What you will learn here about transfer function of a system:

  • What is transfer function
  • Transfer function of open loop system
  • Transfer function of closed loop system
    • Transfer function of closed loop system with negative feedback
    • Transfer function of closed loop system with positive feedback

What is transfer function

Transfer function is the ratio of Laplace transform of the output signal to Laplace transform of input signal assuming all the initial conditions to be zero.

What is transfer function

Transfer function of open loop system

The transfer function of open loop sytem is given below

Transfer function of open loop system

Transfer function of closed loop system

The transfer function of closed loop system with the positive and negative system is given below

Transfer function of closed loop system with negative feedback

Transfer function of closed loop system with negative feedback
The transfer function of closed loop system with negative feedback derivation is given below:

C(s)=G(s)*E(s) —— eq(1)

B(s)=H(s)*C(s) —— eq(2)

E(s)=R(s)-B(s) —— eq(3)

Now put eq(2) in eq(3)

E(s)=R(s)-H(s)*C(s) —— eq(4)

Now put eq(4) in eq(1)

C(s)=G(s)*(R(s)-H(s)*C(s))

C(s)=G(s)*R(s)-G(s)*H(s)*C(s)

C(s)+G(s)*H(s)*C(s)=G(s)*R(s)

C(s)(1+G(s)*H(s))=G(s)*R(s)

C(s)/R(s)=G(s)/(1+G(s)*H(s))

Transfer function of closed loop system with positive feedback

Transfer function of closed loop system with positive feedback

The transfer function of closed loop system with positive feedback derivation is given below:

C(s)=G(s)*E(s) —— eq(1)

B(s)=H(s)*C(s) —— eq(2)

E(s)=R(s)+B(s) —— eq(3)

Now put eq(2) in eq(3)

E(s)=R(s)+H(s)*C(s) —— eq(4)

Now put eq(4) in eq(1)

C(s)=G(s)*(R(s)+H(s)*C(s))

C(s)=G(s)*R(s)+G(s)*H(s)*C(s)

C(s)-G(s)*H(s)*C(s)=G(s)*R(s)

C(s)(1-G(s)*H(s))=G(s)*R(s)

C(s)/R(s)=G(s)/(1-G(s)*H(s))

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