As transfer function is a ratio of Laplace of output to Laplace of input and hence can be expressed as a ratio of polynomials in 's'.
This can be further expressed as,
The numerator and denominator can be factorized to get the factorized form of the transfer function as,
Where K is called system gain factor. Now if in the transfer function, values of 's' are substituted as s1, s2, s3 ….. sn in the denominator then value of T.F. will become infinity.
1. Poles of a Transfer Function
Definition : The values of 's', which make the T.F. infinity after substitution in the denominator of a T.F. are called 'Poles' of that T.F.
So values s1, s2, s3 …. sn are called poles of the T.F.
These poles are nothing but the roots of the equation obtained by equating denominator of a T.F. to zero.
For example, let the transfer function of a system be,
The equation obtained by equating denominator to zero is,
s(s+4) = 0
... s = 0 and s = 4
s(s+4) = 0
... s = 0 and s = 4
If these values are used in the denominator, the value of transfer function becomes infinity. Hence poles of this transfer function are s = 0 and -4.
If the poles are like s = 0, -4, -2, +5, …. i.e. real and without repeated values, they are called simple poles. A pole having same value twice or more than that is called repeated pole. A pair of poles with complex conjugate values is called pair of complex conjugate poles.
The poles are the roots of equation (s+4)2(s2+2s+2)(s+1) =0.
... Poles are s = -4, -4, -1±j1, -1
So T(s) has simple pole at s = -1,
Repeated pole at s= -4, (two poles)
Complex conjugate poles at s = -1±j1
Poles are indicated by 'X' (cross) in s-plane
2. characteristics Equation of a Transfer Function
Definition : The equation obtained by equating denominator of a T.F. to zero, where roots are the poles of that T.F. called characteristics equation of that system
F(s) = b0s2+ b1s2 + b2s2 + ...... + bn = 0
is called the characteristics equation.
3. Zeros of a Transfer Function
Similar to the poles, now if the values of 's' are substituted as sa, sb ……. + sm in the numerator of a T.F., its value become zero.
Definition : the values of 's' which make the T.F. zero after substituting in the numerator are called 'zeros' of that T.F.
Such zeros are the roots of the equation obtained by equating numerator of a T.F. to zero. Such zeros are indicated by a small circle 'o' in s-plane.
Poles and zeros may be real or complex-conjugates or combination of both the types.
Poles and zero may be located at the origin in s-plane.
Similar to the poles, the zeros also are called simple zeros, repeated zeros and complex conjugate zeros depending upon their nature.
This transfer function has zeros which are roots of the equation,
2(s+1)2 (s+2)(s2+2s+2) = 0
i.e. Simple zero at s = -2
Repeated zero at s = -1 (twice)
complex conjugate zeros at s = -1 ±j1.
The zeros are indicated by small circle or zero 'o' in the s-plane
4. Pole-Zero Plot
Definition : Plot obtained by locating all poles and zeros of a T.F. in s-plane is called pole-zero plot of a system.
5. Order of a Transfer Function
Definition : The highest power of 's' present in the characteristic equation i.e. in the denominator polynomial of a closed loop transfer function of a system is called 'Order' of a system.
6. D.C. Gian
The value of the transfer function obtained for s = 0 i.e. zero frequency is called the d.c. gain of the system.
Note : It is not possible to indicate the value of d.c. gain on pole-zero plot as it is a constant value. It is required to be separately specified, alongwith the pole-zero plot.
For example, consider example discussed earlier. The system T.F. is 1/(1+sRC).
So 1+sRC = 0 is its characteristics equation and system is first order system.
Then s = -1/RC is a pole of that system and T.F. has no zeros.
The corresponding pole-zero plot can be shown as in the Fig.1.
Similarly for example , the T.F. calculated is,
The characteristic equation is,
Now if values of R, L and C selected are such that both poles are real, unequal and negative, the corresponding pole-zero plot can be shown as in the Fig.2.
For a system having T.F. as,
The characteristic equation is,
s (s2+ 2s + 2) (s2+ 7s + 12) = 0 i.e. s(s2+ 2s + 2) (s+3) (s+4) = 0
s (s2+ 2s + 2) (s2+ 7s + 12) = 0 i.e. s(s2+ 2s + 2) (s+3) (s+4) = 0
i.e. system is 5th order and there are 5 poles. Poles are 0, -1±j, -3, -4 while zero is located at '-2'.
The corresponding pole-zero plot can be drawn as shown in the Fig.3.
After getting familiar with introductory remarks about control system, now it is necessary to see how overall systems are represented and the methods to represent the given system, based on the transfer function approach.
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