Poles and zeros of the transfer function

In some cases (e.g. stability analysis) it is expedient to represent the rational transfer function according to Eq. (3.2) in the factorised form

For physical reasons only real coefficients occur. Therefore the poles and the zeros of , respectively, can be real or complex conjugate pairs. The terms zeros and poles are chosen, because the transfer function is zero at and infinite at . Zeros and poles can be graphically represented in the complex plane as shown in Figure 3.1. A linear time-invariant system without dead time is described completely by the distribution of its poles and zeros and the gain factor .

Figure 3.1: Example of the pole and zero distribution of a rational transfer function in the complex plane

Moreover, the poles and zeros of a transfer function have a further significance. Observing a system without input ( ) according to Eq. (3.1) and determining the time response for the given initial conditions, one has to solve the associated homogeneous differential equation

which corresponds exactly to Eq. (2.35). For the approach of Eq. (3.11) one obtains for the solution in s the characteristic equation

which was already mentioned in Eq. (2.37). This relation can be directly determined by setting the denominator of to zero (), as long as and have no common factor. The zeros of the characteristic equation are the poles of the transfer function. As already shown in section 2.5 the modes (i.e.  ) are described by the characteristic equation, so that the poles of a transfer function contain all of this information.

The zeros of a transfer function are those values for which . This means that the output signal does not contain any components which depend on . In order to explain this in more detail a stable system with a transfer function according to Eq. (3.10) is excited by the input signal

First for simplification the zero is assumed to be real. For this case the input signal is . Because one obtains from Eq. (3.6) the steady-state output signal as

In the case of complex conjugate pairs of zeros , both zeros have to be taken into consideration in the input signal

Eq. (3.6) leads also to the result . This shows that a zero of a system blocks the transmission of the input signal .

Example 3.5.1   The mass-spring-damper mechanical system in Figure 3.2 with the mechanical constants =1, =2, , =1 and =4 is excited by the force . The transfer function between the force and the position can be shown to be

which has the zeros . If this system is excited by the sinusoidal input signal

which is derived from this pair of zeros, the output signal decays to zero as shown in Figure 3.3b even though the input signal is a sinusoidal signal and the mass shows an undamped oscillation (see Figure 3.3c). The system does not pass this oscillation to the mass when the frequency matches the zeros .

Figure 3.2: Mass-spring-damper mechanical system used for the interpretation of zeros

Figure 3.3: Response to the sinusoidal input signal (a), (b) position of the mass and (c) position of the mass