# 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. ### Transfer function of open loop system

The transfer function of open loop sytem is given below ### 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 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 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))