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Frequency Response Prof. Seungchul Lee Industrial AI Lab. Most - - PowerPoint PPT Presentation

Jul 05, 2023 789 likes •1.66k views

Frequency Response Prof. Seungchul Lee Industrial AI Lab. Most slides from Signals and Systems (MIT 6.003) by Prof. Denny Freeman Time Response Previously, we have determined the time response of linear systems to arbitrary inputs and

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SLIDE 1

Frequency Response

  • Prof. Seungchul Lee

Industrial AI Lab.

Most slides from Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 2

Time Response

  • Previously, we have determined the time response of linear systems to arbitrary inputs and initial

conditions

  • We have also studied the character of certain standard systems to certain simple inputs

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SLIDE 3

Frequency Response

  • Only focus on steady-state solution
  • Transient solution is not our interest any more
  • Input sine waves of different frequencies and look at the output in steady state
  • If 𝐻(𝑡) is linear and stable, a sinusoidal input will generate in steady state a scaled and shifted

sinusoidal output of the same frequency

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SLIDE 4

Response to a Sinusoidal Input

  • When the input 𝑦 𝑢 = 𝑓𝑘𝜕𝑢 to an LTI system
  • Output is also a sinusoid

– same frequency – possibly different amplitude, and – possibly different phase angle

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SLIDE 5

Response to a Sinusoidal Input

  • When the input 𝑦 𝑢 = 𝑓𝑘𝜕𝑢 to an LTI system
  • Output is also a sinusoid

– same frequency – possibly different amplitude, and – possibly different phase angle

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SLIDE 6

Fourier Transform

  • Definition: Fourier transform
  • 𝐼 𝑘𝜕 𝑓𝑘𝜕𝑢 rotates with the same angular velocity 𝜕

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SLIDE 7

Response to a Sinusoidal Input: MATLAB

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SLIDE 8

Response to a Sinusoidal Input: MATLAB

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transient

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SLIDE 9

Frequency Response to a Sinusoidal Input

  • Two primary quantities of interest that have implications for system performance are:

– The scaling = magnitude of 𝐼(𝑘𝜕) – The phase shift = angle of 𝐼(𝑘𝜕)

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SLIDE 10

Frequency Response to a Sinusoidal Input: MATLAB

  • Given input 𝑓𝑘𝜕𝑢
  • 𝑧 = 𝐵𝑓𝑘(𝜕𝑢+𝜚)

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SLIDE 11

From Laplace Transform to Fourier Transform

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Eigenfunctions and Eigenvalues

  • Eigenfunctions

– If the output signal is a scalar multiple of the input signal, we refer to the signal as an eigenfunction and the multiplier as the eigenvalue

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SLIDE 13

Eigenfunctions and Eigenvalues

  • Fact: Complex exponentials are eigenfunctions of LTI systems.
  • If 𝑦 𝑢 = 𝑓𝑡𝑢 and ℎ(𝑢) is the impulse response then
  • The eigenvalue associated with eigenfunction 𝑓𝑡𝑢 is 𝐼(𝑡)

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SLIDE 14

Rational Transfer Functions

  • Eigenvalues are particularly easy to evaluate for systems represented by linear differential equations

with constant coefficients.

  • Then the transfer function is a ratio of polynomials in 𝑡
  • Example

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SLIDE 15

Vector Diagrams

  • The value of 𝐼(𝑡) at a point 𝑡 = 𝑡0 can be determined graphically using vectorial analysis.
  • Factor the numerator and denominator of the system function to make poles and zeros explicit.
  • Each factor in the numerator/denominator corresponds to a vector from a zero/pole to 𝑡0, the point
  • f interest in the s-plane

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SLIDE 16

Vector Diagrams

  • The value of 𝐼(𝑡) at a point 𝑡 = 𝑡0 can be determined by combining the contributors of the vectors

associated with each of the poles and zeros

– The magnitude is determined by the product of the magnitudes – The angle is determined by the sum of the angles

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SLIDE 17

Frequency Response

  • Given the system described by
  • Find the response to the input 𝑦 𝑢 = 𝑓2𝑘𝑢

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SLIDE 18

Vector Diagrams for Frequency Response

  • The magnitude and phase of the response of an LTI system to 𝑓𝑘𝜕𝑢 is the magnitude and phase of

𝐼 𝑡 at s = 𝑘𝜕

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SLIDE 19

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 20

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 21

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 22

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 23

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 24

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 25

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 26

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 27

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 28

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 29

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 30

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 31

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 32

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 33

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 34

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 35

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 36

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 37

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 38

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 39

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 40

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 41

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 42

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 43

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 44

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 45

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 46

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 47

Vector Diagrams at 𝒕 = 𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 48

System Design in S-plane

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 49

Frequency Response (Frequency Sweep): MATLAB

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Frequency Response and Bode Plots

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SLIDE 51

Frequency Response: ȁ 𝑰(𝒕) 𝒕←𝒌𝝏

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 52

Poles and Zeros

  • Frequency response
  • Thinking about systems as collections of poles and zeros is an important design concept.

– Simple: just a few numbers characterize entire system – Powerful: complete information about frequency response

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SLIDE 53

Bode Plots: Magnitude

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Asymptotic Behavior: Isolated Zero

  • The magnitude response is simple at low and high frequencies

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 55

Asymptotic Behavior: Isolated Zero

  • Two asymptotes provide a good approximation on log-log axes

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 56

Asymptotic Behavior: Isolated Pole

  • The magnitude response is simple at low and high frequencies

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 57

Asymptotic Behavior: Isolated Pole

  • Two asymptotes provide a good approximation on log-log axes

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 58

Check Yourself

  • Compare log-log plots of the frequency-response magnitudes of the following system functions

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 59

Asymptotic Behavior of More Complicated Systems

  • Constructing 𝐼(𝑡0)

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 60

Asymptotic Behavior of More Complicated Systems

  • The magnitude of a product is the product of the magnitudes
  • The log of the magnitude is a sum of logs

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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Bode Plot: Adding Instead of Multiplying

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 62

Bode Plot: Adding Instead of Multiplying

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 63

Bode Plot: Adding Instead of Multiplying

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 64

Bode Plot: Adding Instead of Multiplying

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 65

Bode Plots: Angle

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Asymptotic Behavior: Isolated Zero

  • The angle response is simple at low and high frequencies

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 67

Asymptotic Behavior: Isolated Zero

  • Three straight lines provide a good approximation versus log 𝜕

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 68

Asymptotic Behavior: Isolated Pole

  • The angle response is simple at low and high frequencies

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 69

Asymptotic Behavior: Isolated Pole

  • Three straight lines provide a good approximation versus log 𝜕

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 70

Bode Plot: Adding

  • The angle of a product is the sum of the angles
  • The angle of 𝐿 can be 0 or 𝜌 for systems described by linear differential equations with constant, real-

value coefficients

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 71

Bode Plot: Adding

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 72

Bode Plot: Adding

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 73

Bode Plot: Adding

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 74

Bode Plot: Adding

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 75

Summary: From Frequency Response to Bode Plot

  • The log of the magnitude is a sum of logs
  • The angle of 𝐼(𝑘𝜕) is a sum of angles
  • Bode plot (logarithmic plot) : separate plots for magnitude and phase

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Bode Plot: dB

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 77

Bode Plot: dB

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 78

Bode Plot: dB

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 79

Bode Plot: dB

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From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 80

Bode Plot: Accuracy

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  • The straight-line approximations are surprisingly accurate

From Signals and Systems (MIT 6.003) by Prof. Denny Freeman

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SLIDE 81

How to Draw Bode Plots by Hands

  • You should watch the following video clips

– https://www.youtube.com/watch?v=_eh1conN6YM&index=9&list=PLUMWjy5jgHK1NC52DXXrriwihVrYZKqjk – https://www.youtube.com/watch?v=CSAp9ooQRT0&index=10&list=PLUMWjy5jgHK1NC52DXXrriwihVrYZKqjk – https://www.youtube.com/watch?v=E6R2XUEyRy0&index=11&list=PLUMWjy5jgHK1NC52DXXrriwihVrYZKqjk – https://www.youtube.com/watch?v=O2Cw_4zd-aU&index=12&list=PLUMWjy5jgHK1NC52DXXrriwihVrYZKqjk – https://www.youtube.com/watch?v=4d4WJdU61Js&index=13&list=PLUMWjy5jgHK1NC52DXXrriwihVrYZKqjk – https://www.youtube.com/watch?v=GIlx9Yu__y8&index=14&list=PLUMWjy5jgHK1NC52DXXrriwihVrYZKqjk

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Frequency Response and Nyquist Plots

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Nyquist Plot

  • If we only want a single plot we can use 𝜕 as a parameter
  • A plot of 𝑆𝑓{𝐻(𝜕)} vs. 𝐽𝑛{𝐻(𝜕)} as a function of 𝜕

– Advantage: all information in a single plot

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Example: Nyquist Plot

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Example: Nyquist Plot

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SLIDE 86

Example: Nyquist Plot

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