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: