An automatic regulator is used to control the field

Chapter 8, Problem 66

(choose chapter or problem)

An automatic regulator is used to control the field current of a three-phase synchronous machine with identical symmetrical armature windings (Stapleton, 1964). The purpose of the regulator is to maintain the system voltage constant within certain limits. The transfer function of the synchronous machine is

\(G_{s m}(s)\) = \(\frac{\Delta \delta(s)}{\Delta P_{m I}(s)}\) = \(\frac{M\left(s-z_1\right)\left(s-z_2\right)}{\left(s-p_1\right)\left(s-p_2\right)\left(s-p_3\right)}\)

which relates the variation of rotor angle, \(\Delta \delta(s)\), to the change in the synchronous machine's shaft power, \(\Delta P_m(s)\). The closed-loop system is shown in Figure P8.3, where G(s) = \(K G_C(s) G_{s m}(s)\) and K is a gain to be adjusted. The regulator's transfer function, \(G_C(s)\), is given by

\(G_C(s)\) = \(\frac{\mu / T_e}{s+\frac{1}{T_e}}\)

Assume the following parameter values:

\(\mu\) = 4, M = 0.117, \(T_e\) = 0.5, \(z_{1,2}\) = \(-0.071 \pm j 6.25\), \(p_1\) = -0.047, and \(p_{2,3}\) = \(-0.262 \pm j 5.1\),

and do the following:

Write a MATLAB M-file to plot the root locus for the system and to find the following:

a. The gain K at which the system becomes marginally stable

b. The closed-loop poles, p, and transfer function, T(S), corresponding to a 16% overshoot

c. The coordinates of the point selected on the root-locus corresponding to 16% overshoot

d. A simulation of the unit-step response of the closed-loop system corresponding to your 16% overshoot design. Note in your simulation the following values:

(1) actual percent overshoot,

(2) corresponding peak time, \(T_p\),

(3) rise time, \(T_r\),

(4) settling time, \(T_s\), and

(5) final steady-state value.