# Bode plot of 3rd order polynomial

Cuthbert Nyack
Applet shows the Bode plot of a system whose open loop transfer function has a 3rd order polynomial s3 + as2 + bs + c as denominator. The magnitude starts with slope zero and ends with slope -60dB, phase starts at zero and ends at -1.5p.

eg parameters
(K, 3.0, 3.0, 1.0, 0.0, 1.0) oscillates with K = 8 at angular frequency ~1.73rad/s.
(K, 3.0, 3.0, 3.0, 0.0, 1.0) oscillates with K = 6 at angular frequency ~1.73rad/s.

The parameter r can be used to find the dominant closed loop poles.
eg (3.0, 3.0, 4.0, 3.0, r, 1.0) with r = 0, the poles are to the left of the imaginary axis. Changing r to -0.31(equivalent to moving the imaginary axis to -0.31) shows that the system becomes oscillatory. This means that the dominant closed loop poles are at x ± jy = ~-0.31 ± j1.57(from period of oscillation) and the real pole is at -(c + K)/(x2 + y2) = ~-2.34.

The applet below show the closed loop poles and the phase and gain margins for a system whose characteristic eqn is a third order polynomial.
eg parameters
(K, 3.0, 3.0, 1.0, 2.0) for K = 8, the closed loop poles are at ~-3.0 and ~±j1.732 and both the phase and gain margins are ~ 0.0 at 1.732rad/s.
For (3.0, 3.0, 4.0, 3.0, 2.0) the poles are at ~ -2.378 and ~ -0.31 ± ~ j1.558 in aagreement with the approximate values above.

Gif image below shows how applet should appear. 