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Title: Feedback: Principles


1
Feedback Principles Analysis
Dr. John Choma, Jr. Professor of Electrical
Engineering University of Southern
California Department of Electrical
Engineering-Electrophysics University Park
Mail Code 0271 Los Angeles, California
90089-0271 213-740-4692 OFF 626-915-7503
HOME 626-915-0944 FAX johnc_at_almaak.usc.edu
(E-MAIL)
EE 448 Feedback Principles Analysis
Fall 2001
2
Overview Of Lecture
  • Feedback
  • System Representation
  • System Analysis
  • High Frequency Dynamics
  • Open And Closed Loop Damping Factor
  • Open And Closed Loop Undamped Natural Frequency
  • Frequency Response
  • Phase Margin
  • High Speed Transient Dynamics
  • Step Response
  • Rise Time
  • Settling Time
  • Overshoot

65
3
Open Loop Model
  • Gain
  • Parameters
  • ? Zero Frequency Gain
  • ? Frequency Of Zero
  • ? Frequency Of Dominant Pole
  • ? Frequency Of NonDominant Pole
  • Frequency Of Zero Can Be Positive (RHP Zero) Or
  • Negative (LHP Zero)
  • Note That A Simple Dominant Pole Model Is Not
    Exploited
  • Input And Output Variables
  • Input Voltage Or Current Is X(s)
  • Output Voltage Or Current Is Y(s)

3
66
4
Open Loop Transfer Function
(0)
  • Damping Factor
  • Measure Of Relative Stability
  • Measure Of Step Response Overshoot And
    Settling Time
  • Undamped Natural Frequency
  • Measure Of Steady State Bandwidth
  • Measure Of "Ringing" Frequency And Settling
    Time
  • Poles
  • Dominant Pole Implies
  • Complex Poles Imply
  • Identical Poles Imply



gtgt

1

lt

1



1
4
67
5
Closed Loop Transfer Function
  • Loop Gain (Return Ratio w/r To Feedback
    Factor, f )
  • Closed Loop Gain


Obtained Through Substitution Of Open Loop Gain
Relationship Into Closed Loop Gain Expression

68
6
Closed Loop Parameters
(0)

  • Closed Loop Damping Factor
  • Closed Loop Undamped Frequency
  • "DC" Closed Loop Gain
  • T(0) Large For Intentional Feedback
  • T(0) Possibly Large For Parasitic Feedback


(0)
?
(0)


69
7
Closed Loop General Comments

  • Damping Factor
  • Potential Instability Increases With Diminishing
    Damping Factor
  • Potential Instability Strongly Aggravated By
    Large Loop Gain
  • Note Open Loop Damping Attenuation By Factor Of
  • Square Root Of One Plus "DC"
    Loop Gain
  • For Intentional Feedback Having Closed Loop Gain
    Of (1/f ),
  • Worst Case Is Unity Gain (f
    1), Corresponding To Maximal T(0)
  • Open Loop Zero
  • Closed Loop Damping Diminished, Thus Potential
    Instability
  • Aggravated, For Right Half
    Plane Open Loop Zero
  • Closed Loop Damping Increased, Thus Potential
    Instability
  • Diminished, For Left Half
    Plane Open Loop Zero
  • Undamped Frequency
  • Measure Of Closed Loop Bandwidth
  • Closed Loop Bandwidth Increases By Square Root
    Of One Plus
  • "DC" Loop Gain, In Contrast To
    Increase By One Plus "DC" Loop
  • Gain Predicted By Dominant Pole
    Analysis

70
8
Step Response Example Of Damping Factor
Effect
Transmission Zero Assumed To Lie At Infinitely
Large Frequency

t

71
9
Phase Margin
(
v
  • Unity Loop Gain Frequency
  • Assumes Frequencies Of Zero And Second Pole Are
    Larger Than
  • Substitutions
  • Phase Margin
  • Difference Between Actual Loop Gain Phase Angle
    And 180?
  • A Safety Margin For Closed Loop
    Stability
  • Approximate Phase Margin
  • Since Can Be Negative, k Can Be A Negative
    Number
  • Result Is Meaningful Only For


?
k


?

?
(k)

?
gt 1
72
10
Phase Margin Characteristic
Phase Margin (deg.)
120
100
T(0) 1
80
T(0) 5
60
40
20
T(0) 100
0
k
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
-20
-40
-60
73
11
Circuit Response Parameters


?


k
?






l Closed Loop Damping Factor l Closed Loop
Undamped Frequency l Phase Margin


?



?


?

?
(k)
74
12
Closed Loop Example Calculation
  • Given
  • Desire Maximally Flat Closed Loop Response,
    Which
  • Implies
  • Computations
  • Requisite Phase Margin
  • In Practical Electronics, Phase Margins In The
    60s Of Degrees
  • Are Usually Mandated, Which Requires
    That The Non-Dominant
  • Pole Frequency Be 2.5 -To- 4 Times
    Larger Than The Unity
  • Gain Frequency

75
13
Closed Loop Step Response Problem
Formulation
  • Problem Setup
  • (Damped Frequency Of Oscillation)
  • Normalized Variables
  • (Normalized Time Variable)
  • (Output Normalized
    To SteadyState Response)
  • (Error Between
    Steady State And Actual

  • Output Responses)


?

?

?


?
t
(t)
?

(t)
?

(s)
?

76
14
Closed Loop Step Response Solution

(t)
  • Solution
  • Assumptions
  • (Underdamped Closed Loop Response)
  • (Satisfied For Right
    Half Plane Zero)



?
t

?

?


?

lt 1
77
15
Closed Loop Step Response Example 1
M 1
78
16
Closed Loop Step Response Example 2
M 5
79
17
Closed Loop Settling Time

(x)
  • Observations
  • Magnitude Of Error Term Decreases Monotonically
    With x
  • Maxima Of Error Determined By Setting
    Derivative Of Error
  • Term With Respect To x To Zero
  • Maxima Are Periodic With Period ?
  • First Maximum Of Error Establishes Undershoot
    Point
  • Determine Second Maximum And Constrain To
    Desired Minimal Error
  • Procedure
  • Let Be The Normalized Time Corresponding
    To Second
  • Error Maximum
  • Let Be The Magnitude Of Error
    Corresponding To
  • If Is The Desired Settling Error,
    Represents The Settling Time
  • Of the Circuit

x
80
18
Closed Loop Settling Time Results

(x)
  • Results
  • For Large M (Far Right Half Plane Zero)




?

?


?

81
19
Closed Loop Settling Time Example
  • Requirements
  • Settling To Within One Percent In 1 nSEC
  • Assume Zero Is In Far Right Half Plane
    (Reasonable
  • Approximation For Common Gate And
    Compensated Source
  • Follower First Order Approximation For
    Common Source)
  • Assume Very Large "DC" Loop Gain
  • Computations
  • Second Pole Must Be At Least 2.7 Times
    Larger Than
  • Unity Gain Frequency


?

?
0.01
?

gt
2.73

?
5.575
?

?
(887.2 MHz)


?

?

?
(537 MHz)
?

?
) 69.9

82
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