Poles and Zeros - Electrical Engineering (EE) PDF Download

 Poles and Zeros
In general the transfer function of a process is the ratio of two polynomials
                110

The roots of the denominator polynomial, P(s), are called poles and roots of the numerator polynomial, Z(s), are called zeros . For an example, poles and zeros of a few processes are given below:

Qualitative analysis of process response
Suppose the transfer function of a process is factorized in the following manner
             101

Where Pi are the roots (or poles) of the process. P₁,P₂ are real and distinct poles, P₃ is a real and multiple pole which is repeated m times,    are the pair of complex conjugate poles, P₅ is a pole which is located at the origin and hence its numerical value is zero. Fig III.9 shows the location of these poles in a complex plane

he time domain solution of the model equation (eq.111) is given as follows:

     112

As P₅ = 0, the final term will be a constant term C₅. Let us assume the complex conjugate poles are expressed as   . Then,

                       113

Hence, the revised form of eq. 112 would be   
         114

By analyzing the above equation, we find three types of time functions parts, viz . polynomial, exponential and sinusoidal. The exponential term is most important among the three. This term remains as a coefficient term with all other functional parts. If the pole of the system is negative    or the real part of complex pole is negative    then at    these exponential time functions would vanish to zero. The term containing sinusoidal term will have an amplitude that will decrease with time before vanishing to zero. This is a perfect condition of stability for the system. However, if one of them is positive    the corresponding time function will exponentially lead to ∞ as   , the sinusoidal term would possess an ever increasing amplitude which eventually lead to instability of the system. If a = 0, the sinusoidal term would possess a constant amplitude throughout, which eventually lead to sustained oscillatory behavior.

The Figure III.10 shows plots that indicate the stability conditions for various poles. Hence, a dynamic system, having all its poles at the left hand side of the imaginary axis of the complex plane, is a stable system. If the transfer function of the dynamic system has even one pole that has positive real part, the system is unstable