1. Definitions:

• The phase crossover frequency, w_pc , is the frequency where  phase shift is equal to -180^o.
• The gain crossover frequency, w_gc, 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(iw_pc) is less than 0 dB, then the feedback system is stable.
3. Stability Margins:
 
1. Gain Margin:   Let x = |G(iw_pc)| then the gain margin is given by GM = 1/x.
1. Phase Margin:   Let q=arg(G(iw_gc)) then the phase margin is given by PM = 180^o + q.

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:

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

Back to Homepage