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ECE4991 Chapter 2

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Chapter 6 Frequency Response & Systems Concepts AC circuit analysis methods to study the frequency response of electrical circuits Understanding of frequency response ... – PowerPoint PPT presentation

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Title: ECE4991 Chapter 2


1
Chapter 6 Frequency Response Systems Concepts
  • AC circuit analysis methods to study the
    frequency response of electrical circuits
  • Understanding of frequency response aided by the
    concepts of phasors and impedance.
  • Filtering a new concept will be explored

2
Objectives
  • Understand significance of frequency domain
    analysis
  • Introduction of Fourier series as a tool for
    computation of Fourier spectrum.
  • Analyze first and second-order electrical filters
    by determining their filtering properties.
  • Computation of frequency response and its
    graphical representation as Bode plot.

3
Sinusoidal Frequency Response
  • Provides a circuit response to a sinusoidal input
    of arbitrary frequency.
  • The frequency response of a circuit is a measure
    of voltage or current (magnitude and phase) as a
    function of the frequency of excitation (source)
    signal.

4
Methods to Compute Frequency Response
  • Thevenin equivalent source circuit


Zs
Z1
ZT
VL


VT
VT
Z2
-
5
Load Voltage VT

ZT
ZL

VT
6
Frequency Response
  • From definition
  • VL(j?) is a phase-shifted and amplitude-scaled
    version of VS(j?) ?

7
Frequency Response (cont)
  • Phasor form of the load voltage

8
Example 6.1
  • Compute the frequency response Hv(j?) of the
    circuit for R1 1k?, C10?F and RL 10k?.

9
Magnitude Phaze
10
Fourier Analysis
  • Let x(t) be a periodic signal with period T.
  • x(t) x(tnT) for n1,2,3,

11
Fourier Series
  • A signal x(t) can be expressed as an infinite
    summation of sinusoidal components know as
    Fourier Series
  • Sine-cosine (quadrature) representation
  • Magnitude and Phase form
  • Fundamental Frequency and Period T

12
Fourier Series
  • It can be shown
  • Or similarly

13
Fourier Series Aproximation
  • Infinite summation practically not possible
  • Replaced by finite summation that leads to
    approximation.
  • Higher order coefficients n, are associated with
    higher frequencies (2?/T)n. ?
  • Better approximations require larger bandwidths.

14
Fourier Series
  • Odd and Even Functions

15
Frequency Spectrum
16
Computation of Fourier Series Coefficients

17
Example of Fourier Series Approximation
  • Square wave and its representation by a Fourier
    series. (a) Square wave (even function) (b)
    first three terms (c) sum of first three terms

18
Example 6.3 Computation of Fourier Series
Coefficients
  • Problem Compute the complete Fourier Spectrum of
    the sawtooth function shown in the Figure below
    for T1 and A1

19
Solution
  • x(t) is an odd function.
  • Evaluate the integral in equation

20
Solution (cont)
  • Spectrum computation

21
Matlab Simulation
  • Components of the sawtooth wave function

22
Matlab Simulation
  • Fourier Series approximation of sawtooth wave
    function

23
Example 6.4
  • Problem Compute the complete Fourier series
    expansion of the pulse waveform shown in the
    Figure for ?/T0.2
  • Plot the spectrum of the signal

24
Solution
  • Expression for x(t)
  • Evaluate Integral Equations

25
Solution (cont)
26
Spectrum Computation
  • Magnitude
  • Phase

27
Graphical Representation
28
Matlab Simulation
29
Matlab Simulation
30
Linear Systems Response to Periodic Inputs
  • Any periodic signal x(t) can be represented as a
    sum of finite number of pure periodic terms

31
General Input-Output Representation of a System
32
Linear Systems
  • For Linear Systems - by definition Principle of
    superposition applies

Tax1(t) bx2(t) aTx1(t) bTx2(t)
a
x1
ax1n bx2n
T
y Tax1(t) bx2(t)
x2
b
a
aTx1n
T
x1
y aTx1(n)bTx2(n)
T
x2
bTx2n
b
33
Linear System View of a Circuit
  • Output of a circuit y(t) as a function of the
    input x(t)

34
Example 6.6 Response of Linear System to Periodic
Input
  • Problem
  • Linear system
  • Input sawtooth waveform approximated with only
    first two Fourier components of the input
    waveform.

35
Solution
  • Approximation of the sawtooth function with first
    two terms of Fourier Series
  • Spectrum Computation

36
Frequency Response
  • Magnitude and Phase
  • Computation of Frequency Response for two
    frequency values of ?1 8? and ?2 16?

37
Frequency Response (cont)
  • Computation of steady-state periodic output of
    the system

38
Matlab Simulation
39
Matlab Simulation
40
Filters
  • Low-Pass Filters

Simple RC Filter
41
Low-Pass Filter

42
Low-Pass Filter
  • ?0
  • H(j?)1 ? Vo(j?)Vi(j?)
  • ?gt0

43
Low-Pass Filter
Cutoff Frequency
44
Example 6.7
  • Compute the response of the RC filter to
    sinusoidal inputs at the frequencies of 60 and
    10,000 Hz.
  • R1k?, C0.47?F, vi(t)5cos(?t) V
  • ?01/RC 2,128 rad/sec
  • ? 120? rad/sec ? ?/?0 0.177
  • ? 20,000? rad/sec ? ?/?0 29.5

45
Solution

46
High-Pass Filters

47
High-Pass Filter
  • The expression in previous slide can be written
    in magnitude-and-phase form

48
High-Pass Filter Response
49
Band-Pass Filters
50
Frequency Response of Band-Pass Filter
51
Analysis of the Second Order Circuit
  • ?1, ?2 and are the two frequencies that determine
    the pass-band (bandwidth) of the filter.
  • A gain of the filter.

52
Magnitude and Phase form

53
Magnitude and Phase form
54
Frequency Response and Bandwidth
55
Frequency Response and Bandwidth
56
Normalized Frequency Response Bandwidth
57
Frequency Response of the Filter with R1 k?
C10 ?F L5 mH
58
Frequency Response of the Filter with R10 ?
C10 ?F L5 mH
59
Bode Plots
  • Logarithmic Plots of Systems Frequency Response
  • Both plots are function of frequency also
    represented in log scale.

60
Bode Plot
  • Example of Low-Pass Filter

61
Bode Plot of Low-Pass Filter
62
Bode Plot of Low-Pass Filter
?n? Cut-off Frequency
3dB
-20 dB Slope
63
Bode Plot of Low-Pass Filter
64
Bode Plot of Low-Pass Filter
-450 dB Slope
65
Correction Factors
?/?0 Magnitude Response Error in dB Phase Response Error in dB
0.1 0 -5.7
0.5 -1 4.9
1 -3 0
2 -1 -4.9
10 0 5/7
66
Bode Plots of Higher-Order Filters
  • Bode Plots Higher-Order Filters may be obtained
    by combining Bode-Plots of lower-order functions
  • H(j?) H1(j?) H2(j?) H3(j?) Hn(j?)
  • H(j?)dB H1(j?)dB H2(j?)dB H3(j?)dB
    Hn(j?)dB
  • ?H(j?) ? H1(j?) ?H2(j?) ?H3(j?)
    ?Hn(j?)

67
Example of High-Order Filter
68
Magnitude and Phase in Bode Plots
69
Composite Bode Plot
70
Bode Plot Approximation Example
71
Straight Line Aproximation of Bode Plots
72
Actual Magnitude and Phase
73
Bode Plot Approximation Example
74
Bode Plots
75
Bode Plots
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