# 4 Complex Conjugate Poles Step Response

Cuthbert Nyack
The step response of a 4 pole type 0 system is shown in the 3 applets below.
The open and closed loop transfer function for the system is
G(s) = K/((s + a + jb)(s + a - jb)(s + c + jd)(s + c - jd)) C(s)/R(s) = K/((s + a + jb)(s + a - jb)(s + c + jd)(s + c - jd) + K)

The step response depends on the relative locations of the complex poles relative to the imaginary axis.
The values (0.2, 0.2, 1.0, 0.1, 3.0, 3.0) show the transient response is a combination of the responses of both pairs of poles.
The values (0.2, 0.2, 1.0, 0.4, 3.0, 3.0) show the transient response is predominantly determined by the pole pair a ± jb.
The values (0.2, 3.0, 1.0, 0.1, 3.0, 3.0) show the transient response is predominantly determined by the pole pair c ± jd.

Since this is a 4 pole system then instability is possible.
The parameter values (8.0, 1.0, 1.0, 1.0, 1.0, 3.0),
(32.4, 4.0, 1.0, 0.2, 3.0, 3.0),
(40.0, 4.0, 1.0, 0.1, 5.0, 1.0),
(25.7, 0.1, 1.0, 4.0, 4.0, 2.0),
shows the step response when the system is unstable. In each case the oscillation frequency is different.

Gif image below shows how applet should appear.