Transcript
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
CHAPTER 6
Frequency Response, BodePlots, and Resonance
1. State the fundamental concepts of Fourier
analysis.
2. Determine the output of a filter for a given
input consisting of sinusoidal componentsusing the filter’s transfer function.
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
3. Use circuit analysis to determine thetransfer functions of simple circuits.
4. Draw first-order lowpass or highpass filter
circuits and sketch their transfer functions.
5. Understand decibels, logarithmic frequency
scales, and Bode plots.
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
6. Draw the Bode plots for transfer functions of
first-order filters.
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
Fourier Analysis
All real-world signals are sums of sinusoidalcomponents having various frequencies,
amplitudes, and phases.
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
Filters
Filters process the sinusoid components of an inputsignal differently depending of the frequency of
each component. Often, the goal of the filter is to
retain the components in certain frequency ranges
and to reject components in other ranges.
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
Transfer Functions
The transfer function H(f ) of the two-port filter
is defined to be the ratio of the phasor output
voltage to the phasor input voltage as a function
of frequency:
( ) inout
V
V
= f H
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
The magnitude of the transfer function shows
how the amplitude of each frequency componentis affected by the filter. Similarly, the phase of
the transfer function shows how the phase of eachfrequency component is affected by the filter.
( )in
out
V
V= f H
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
Determining the output of a filterfor an input with multiple
components:
1. Determine the frequency and phasor
representation for each input component.
2. Determine the (complex) value of the transfer
function for each component.
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
3. Obtain the phasor for each output
component by multiplying the phasor for each
input component by the corresponding
transfer-function value.
4. Convert the phasors for the output
components into time functions of variousfrequencies. Add these time functions to
produce the output.
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
Linear circuits behave as if they:
1. Separate the input signal into components
having various frequencies.
2. Alter the amplitude and phase of each
component depending on its frequency.
3. Add the altered components to producethe output signal.
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
FIRST ORDER LOWPASS
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
FIRST-ORDER LOWPASS
FILTERS
RC f B π 2
1
=
( ) ( ) B f f j f H
+= 11
( )( )21
1
B f f f H
+=
( )
−=∠
B f f f H arctan
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
( ) ( )21
1
B f f f H +=
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ENGINEERINGPrinciples and
ApplicationsChapter 6
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ENGINEERINGPrinciples and
ApplicationsChapter 6
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
C S CASCA
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
DECIBELS, THE CASCADE
CONNECTION, AND LOGARITHMICFREQUENCY SCALES
( ) ( ) f H f H log20dB =
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ENGINEERINGPrinciples and
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
C d d T P t N t k
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ELECTRICAL
ENGINEERINGPrinciples and
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Frequency Response, Bode Plots, and Resonance
Cascaded Two-Port Networks
( ) ( ) ( ) f H f H f H 21 ×=
( ) ( ) ( )dB2dB1dB
f H f H f H +=
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Logarithmic Frequency ScalesOn a logarithmic scale, the variable is
multiplied by a given factor for equalincrements of length along the axis.
A decade is a range of frequencies for which
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
A decade is a range of frequencies for which
the ratio of the highest frequency to the
lowest is 10.
=
1
2logdecadesof number
f
f
An octave is a two-to-one change in frequency.
( )
( )
=
= 2loglog
logoctavesof number
12
1
2
2
f f
f
f
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
BODE PLOTS
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
BODE PLOTS
A Bode plot shows the magnitude of a network
function in decibels versus frequency using a
logarithmic scale for frequency.
( )( ) B f f j
f H
+=
1
1
( )
+−=
2
dB1log10
B f f f H
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
1. A horizontal line at zero for f < f B / 10.
2. A sloping line from zero phase at f B / 10 to –90° at 10 f B.
3. A horizontal line at –90° for f > 10 f B.
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ENGINEERINGPrinciples and
ApplicationsChapter 6
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
FIRST ORDER HIGHPASS
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
FIRST-ORDER HIGHPASS
FILTERS
( ) ( )( ) B
B
f f j
f f j f H
+==
1in
out
V
V
RC f B π 2
1
=
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ENGINEERINGPrinciples and
ApplicationsChapter 6
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
SERIES RESONANCE
Resonance is a phenomenon that can beobserved in mechanical systems and
electrical circuits.
f1
Q1
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
LC
f
π 2
0 =
R L f Q s02π =
CR f
Q s02π
=
( )
−+=
f f
f f jQ R f Z s s
0
0
1
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
Series Resonant Circuit as a
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
Bandpass Filter
( ) f f f f jQ s s R
001
1
−+=V
V
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ENGINEERINGPrinciples and
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Frequency Response, Bode Plots, and Resonance
L H f f B −=
sQ
f
B
0
=
20
B f f H +≅
20 B
f f L −≅
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
PARALLEL RESONANCE
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
( ) ( ) fL j fC j R Z p
π π 21211 −+=
LC f
π 2
10 = L f
RQ p02π
= CR f Q p 02π =
( ) f f f f jQ R Z
p
p
001 −+=
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ENGINEERINGPrinciples and
ApplicationsChapter 6
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ENGINEERINGPrinciples and
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Ideal Filters
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ENGINEERINGPrinciples and
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Ideal Filters
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ENGINEERINGPrinciples and
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Second-Order Lowpass Filter
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ENGINEERINGPrinciples and
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Second-Order Lowpass Filter
( ) ( )( ) f f f f jQ
f f jQ f H s
s
00
0
in
out
1 −+−==
VV
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ENGINEERINGPrinciples and
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ENGINEERINGPrinciples and
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DIGITAL SIGNAL
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
G S GN
PROCESSING
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
Conversion of Signals from
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ENGINEERINGPrinciples and
ApplicationsChapter 6
Frequency Response, Bode Plots, and Resonance
g
Analog to Digital Form
If a signal contains no components with
frequencies higher than f H
, the signal can be
exactly reconstructed from its samples, provided
that the sampling rate f s is selected to be more
than twice f H .
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ENGINEERINGPrinciples and
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Digital Lowpass Filter
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ENGINEERINGPrinciples and
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Frequency Response, Bode Plots, and Resonance
g p
( ) ( ) ( ) ( )n xanayn y −+−=11
T
T a
τ
τ
+= 1
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ENGINEERINGPrinciples and
ApplicationsChapter 6
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