Transfer Function | Poles and Zeros of a Transfer Function | Control System

Transfer Function: 

The transfer function of a system is defined as the ratio of Laplace transform of output to the Laplace transform of input where all the initial conditions are zero.

Where,

1. T(S) = Transfer function of the system.  

2. C(S) = output.  

3. R(S) = Reference output.  

4. G(S) = Gain.  

Steps to get the transfer function:

Step 1: Write the differential equation.

Step 2: Find out Laplace transform of the equation assuming 'zero' as an initial condition.

Step 3: Take the ratio of output to input.

Step 4: Write down the equation of G(S) as follows -

Here, a and b are constant, and S is a complex variable

Characteristic equation of a transfer function:

  Here, the characteristic equation of a linear system can be obtained by equating the denominator to the polynomial of a transfer function is zero. Thus the characteristic equation of the transfer function of Eq.1 will be:

Poles and Zeros of a transfer function:

  Consider the Eq. 1, the numerator and denominator can be factored in m and n terms respectively:

Where,is known as the gain factor and 's' is the complex frequency.

Poles

  Poles are the frequencies of the transfer function for which the value of the transfer function becomes zero.

Zeros

  Zeros are the frequencies of the transfer function for which the value of the transfer function becomes zero.

We will apply Sridharacharya method to find the roots of poles and zeros -

If any poles or zeros coincide then such poles and zeros are called multiple poles or multiple zeros.

If the poles and zeros do not coincide then such poles and zeros are called simple poles or simple zeros.

For example-

Find the transfer function of the following function

The zeros of the function are S = -3 and the poles of the function are S = 0, S = -2, and multiple poles at S = -4 i.e. the pole of order 2 at S = -4.