The purpose of this appendix is to describe the transient response of a typical feedback control system which is referred to many times in this book. We consider a very common configuration in which a two-phase ac servomotor is enclosed by a simple unity feedback loop. Figure 4.1 illustrates the block diagram of this second-order system. For purposes of simplicity, the gain of the amplifier driving the motor is assumed to be unity.
The closed-loop transfer function of this system is given by
By defining the undamped natural frequency ωn and the dimensionless damping ratio ζ as
Eq. (B.1) can be rewritten as
The parameters ωn and ζ are very important for characterizing a system's response. Note from Eq. (B.3) that ωn turns out to be the radian frequency of oscillation when ζ = 0. As ζ increases from 0, the oscillation decays exponentially and becomes more damped. When ζ 1, an oscillation does not occur.
We assume that the initial conditions ...