### 4.2. CHARACTERISTIC RESPONSES OF SECOND-ORDER CONTROL SYSTEMS

The purpose of this section is to describe the transient response of a typical feedback control system. We consider a very common configuratuon in which a two-phase ac servomotor, whose transfer function is given by Eq. (3.101) 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 *ω*_{n} and the dimensionless damping ratio *ζ* as

Eq. (4.1) can be rewritten as

The parameters *ω*_{n} and *ζ* are very important for characterizing a system’s response. Note from Eq. (4.3) that *ω*_{n} 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 are zero and the input is a unit step. Therefore, *R*(*s*) = 1/*s*, and the Laplace transform ...