First and Second Order Systems
In this chapter, the properties of the transfer function and frequency response of first and second order systems are studied on some examples from electrical circuit laws. We show that their properties are governed by the poles (i.e., the zeros of the denominator) of the transfer function which is a rational fraction. A geometric argument based on the location of the poles of the transfer function in the complex plane allows a qualitative interpretation of the behavior of the frequency response with varying frequency. This geometric interpretation is easily generalized to situations with any number of zeros and poles. It proves useful for the understanding of the general behavior of filters. The study begins here with the simplest system, the first order system. Then the second order circuit system is presented thoroughly. The logarithmic Bode representation of the frequency gain is introduced and its advantages demonstrated. The quality factor Q of a resonant circuit is defined.