CHAPTER 6 Frequency Response, Bode Plots, and Resonance

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CHAPTER 6Frequency Response, Bode Plots, and
Resonance

• Objectives
• State the fundamental concepts of Fourier
  analysis.
• Determine the output of a filter for a given
  input consisting of sinusoidal components using
  the filters transfer function.

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3. Use circuit analysis to determine the transfer
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. 6.
Draw the Bode plots for transfer functions of
first-order filters. 7. Calculate parameters for
series and parallel resonant circuits. 8. Select
and design simple filter circuits.
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Fourier Analysis
All real-world signals are sums of sinusoidal
components having various frequencies,
amplitudes, and phases.
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Filters
Filters process the sinusoid components of an
input signal 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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Transfer Functions
The transfer function H(f ) of the two-port
filter is defined to be the ratio of the phasor
output voltage (or current) to the phasor input
voltage (or current) as a function of frequency
Where X is an input quantity Y is an output
quantity
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Magnitude and Phase of the Transfer Function
The magnitude of the transfer function shows how
the amplitude of each frequency component is
affected by the filter. Similarly, the phase of
the transfer function shows how the phase of each
frequency component is affected by the filter.
Example
Q1 If the input signal is vin(t) 2cos(2000pt
40o) what is the output signal vout(t) ?
A1 vout(t) 6cos(2000pt 70o)
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Q2 If the input signal is vin(t) 5cos(4000pt
20o) what is the output signal vout(t) ?
A2 vout(t) 10cos(4000pt 80o)
Q3 If the input signal is vin(t) 4cos(2000pt
10o) 5cos(4000pt 20o) 3cos(6000pt 60o)
what is the output signal vout(t) ?
A3 vout(t) 12cos(2000pt 40o) 10cos(4000pt
80o)
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Determining the output of a filter for an input
with multiple components
1. Determine the frequency and phasor
representation for each input component. 2.
Determine the value of the transfer function for
each component. 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
various frequencies. Add these time functions to
produce the output.
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Linear circuits behave as if they

• 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 produce the
  output signal.

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FIRST-ORDER LOWPASS FILTERS
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DECIBELS
The magnitude (only!) of the transfer function
can be expressed in dB
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Example
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Cascaded Two-Port Networks
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Logarithmic Frequency Scales
A decade is a range of frequencies for which the
ratio of the highest frequency to the lowest is
10.
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BODE PLOTS
A Magnitude Bode Plot shows the magnitude of a
network function in decibels versus frequency
using a logarithmic scale for frequency.
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• A horizontal line at zero for f lt fB /10.
• 2. A sloping line from zero phase at fB /10 to
  90 at 10fB.
• 3. A horizontal line at 90 for f gt 10fB.

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Example
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FIRST-ORDER HIGHPASS FILTERS
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First order R-L High-Pass Filter
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SECOND ORDER FILTERS
Resonance is the state of the system when the
output signal is in phase with the input signal
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SERIES RESONANT CIRCUIT
Resonant frequency
Phasor diagram for the series RLC circuit at
resonance
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Series Resonant Circuit as a Bandpass Filter
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PARALLEL RESONANCE
Resonant frequency
Phasor Diagram for parallel RLC circuit at
resonance
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Ideal Filters
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