Lecture 16 Bode Analysis, stability, Gain and Phase Margins North China Electric Power University Sun Hairong

Topics of this lecture  Gain and phase margins.  System type and steady-state error from bode diagram. (Reading Module 16)  Sample problems

1. Gain and phase margins(GM & PM) in the Nyquist diagram To know some parameters The frequency ----at which the phase is The frequency ----at which the M db is 0db. Application If GM>1,the system is stable, otherwise it ’ s not. If PM>0,the system is stable, otherwise it ’ s not.

2. Gain and phase margins (GM & PM) in Bode diagram The frequency ----at which the phase is The frequency ----at which the M db is 0db. If GM>0db,the system is stable, otherwise it ’ s not. If PM>0,the system is stable, otherwise it ’ s not.

Example: The open-loop transfer function is given by PM Plot the Bode diagram and point out the gain and phase margins

System type When The general transfer function may be wrote as The above equation is called system’s low-frequency asymptotes. And the system is called “Type v” system. 3. System type and steady-state error from Bode diagrams

Type 0 system The general transfer function may be wrote as The system’s low-frequency asymptotes is Plot the bode diagram of type 0 system. The ‘K’ in the plot is also the position error constant.

Type 1 system The general transfer function may be wrote as The system’s low-frequency asymptotes is Plot the bode diagram of type 1 system. The ‘K’ in the plot is also the velocity error constant. 20lgK

Type 2 system The general transfer function may be wrote as The system’s low-frequency asymptotes is Plot the bode diagram of type 2 system. The ‘K’ in the plot is also the acceleration error constant. 20lgK

[sample problem 1] Plot the Bode diagram for K=45, and determine the gain and phase margins. Calculate the maximum value of K consistent with stability, and check the answer using Routh’s array. SP16.1 page 331

[sample problem 2] For the following system, sketch the Bode diagram, and from the straight-line approximations to the gain and phase plots, estimate the maximum value of K for which the system is stable. P16.1 page 338

[sample problem 3] FigP16.4 (Page 339) shows a unity-gain feedback control system, calculate the value of K such that the system has a 20 degree phase margin. And the open-loop transfer function is P16.4 page 339

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