Stability and Frequency Compensation
Chapter 10
General Consideration
Unstable if
Alternatively,
(to add in phase)(to grow in magnitude)
Body Plots
(GX,Gain cross over frequency)
(PX, phase crossoverpoint)
Worst Case Scenario (=1)
( increases)Assumptions:1. does not depend on frequency.2. 1The magnitude plots are shifted down. The system becomesmore stable as is reduced.H() with =1 represents the worst case stability. H () is often used to analyze stability.
increases
Review Slides
Laplace Transform/Fourier Transform for RC LPF
p=1/(RC)(Fourier Transform)(Laplace Transform)
-p
Location of the pole in the left complexplaneComplex s plane
Rules of thumb: (applicable to a pole)Magnitude:20 dB drop after the cut-off frequency
3dB drop at the cut-off frequency
Phase:-45 deg at the cut-off frequency. Phase is more significantly affected by the pole than magnitude.
0 degree at one decade prior to the cut-frequency
90 degrees one decade after the cut-off frequency
Laplace Transform/Fourier Transform for RC HPF
p=1/(RC)Zero at DC.(Fourier Transform)(Laplace Transform)
-p
Location of the pole in the left complexplaneComplex s plane
Zero at the origin.Thus phase(f=0)=90 degrees.The high pass filter has a cut-off frequency of 100.
Time-Domain Response of a System Versus Position of Poles
(unstable)(constant magnitudeOscillation)(exponential decay)The location of the poles of a closedLoop system is shown.
One-Pole System
(one-pole feedforward amplifier)
Ione pole system isUnconditionally Stable.
Root Locus Plot for a One Pole System
As the loop gain increase (e.g. ), the pole moves away from the origin.
Two-Pole System
The system is stable since theloop gain is less than 1 at a frequencyFor which the angle(H())=-180.
When is reduced,the system becomesmore stable.
Assumption: does not dependon frequency.
Root Locus Plot for a Two-Pole System (1)
Root Locus Plot for a Two-Pole System (2)
Three-Pole System
Relative Location of GX and PX
Case 1:
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