Bode Stability Criterion:
1. Definitions:
• The phase crossover frequency, wpc , is the frequency where  phase shift is equal to -180o.
• The gain crossover frequency, wgc, is the frequency where the amplittude ratio is 1, or when log modulus is equal to 0.
2. Stability Criterion:
If at the phase crossover frequency, the corresponding log modulus of G(iwpc) is less than 0 dB, then the feedback system is stable.
3. Stability Margins:

1. Gain Margin:   Let x = |G(iwpc)| then the gain margin is given by GM = 1/x.
1. Phase Margin:   Let q=arg(G(iwgc)) then the phase margin is given by PM = 180o + q.

Example 1:   Given the following Bode plots,

The phase crossover frequency is at 1.05 radians/sec, while the gain crossover frequency is at 1.6. Since at the phase crossover frequency the Log modulus is above 0dB, the system is closed loop unstable.

Example 2:  Given the following Bode plots,
The phase crossover frequency is at 1.43 rad/sec, while the gain crossover frequency is at 1.06 rad/sec. The system is closed loop stable with the following stability margins:

PM= 180o + (-142o) = 38o

20 log(1/GM) = -17     or     GM = 10(17/20) = 7.08

The Nyquist plot is shown below:

This page is maintained by Tomas B. Co (tbco@mtu.edu). Last revised 2/13/2000.
Tomas B. Co
Associate Professor
Department of Chemical Engineering
Michigan Technological University
1400 Townsend Avenue
Houghton, MI 49931-1295