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Fundamentals of
Electric Circuits
Chapter 14
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Overview
• This chapter will introduce the idea of the
transfer function: a means of describing the
relationship between the input and output of
a circuit.
• Bode plots and their utility in describing the
frequency response of a circuit will also be
introduced.
• The concept of resonance as applied to LRC
circuits will be covered as well
• Finally, frequency filters will be discussed.
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Frequency Response
• Frequency response is the variation in a
circuit’s behavior with change in signal
frequency.
• This is significant for applications involving
filters.
• Filters play critical roles in blocking or
passing specific frequencies or ranges of
frequencies.
• Without them, it would be impossible to have
multiple channels of data in radio
communications.
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Transfer Function
• One useful way to analyze the
frequency response of a
circuit is the concept of the
transfer function H(ω).
• It is the frequency dependent
ratio of a forced function Y(ω)
to the forcing function X(ω).
H   
Y  
X  
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Transfer Function
• There are four possible input/output
combinations:
H    Voltage gain 
H    Current gain 
Vo  
Vi  
I o  
I i  
H    Transfer impedance 
H    Transfer admittance 
Vo  
I i  
I o  
Vi  
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Zeros and Poles
• To obtain H(ω), we first convert to frequency
domain equivalent components in the circuit.
• H(ω) can be expressed as the ratio of
numerator N(ω) and denominator D(ω)
polynomials.
N  
H   
D  
• Zeros are where the transfer function goes to
zero.
• Poles are where it goes to infinity.
• They can be related to the roots of N(ω) and
D(ω)
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Resonance
• The most prominent feature of the frequency
response of a circuit may be the sharp peak
in the amplitude characteristics.
• Resonance occurs in any system that has a
complex conjugate pair of poles.
• It enables energy storage in the firm of
oscillations
• It allows frequency discrimination.
• It requires at least one capacitor and
inductor.
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Series Resonance
• A series resonant circuit
consists of an inductor and
capacitor in series.
• Consider the circuit shown.
• Resonance occurs when the
imaginary part of Z is zero.
• The value of ω that satisfies
this is called the resonant
frequency
0 
1
rad/s
LC
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Series Resonance
• At resonance:
–
–
–
–
The impedance is purely resistive
The voltage Vs and the current I are in phase
The magnitude of the transfer function is minimum.
The inductor and capacitor voltages can be much more than
the source.
• There are two frequencies above and below
resonance where the dissipated power is half the
max:
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R
1
 R 
1  
 


2L
 2 L  LC
2
R
1
 R 
2 
 


2L
 2 L  LC
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Quality Factor
• The “sharpness” of the resonance is
measured quantitatively by the quality factor,
Q.
• It is a measure of the peak energy stored
divided by the energy dissipated in one
period at resonance.
Q
0 L
R

1
0CR
• It is also a measure of the ratio of the
resonant frequency to its bandwidth, B
B
R 0

L Q
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Parallel Resonance
• The parallel RLC circuit
shown here is the dual of the
series circuit shown
previously.
• Resonance here occurs when
the imaginary part of the
admittance is zero.
• This results in the same
resonant frequency as in the
series circuit.
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Parallel Resonance
• The relevant equations for the parallel
resonant circuit are:
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Table
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Passive Filters
• A filter is a circuit that is designed to pass
signals with desired frequencies and reject
or attenuate others.
• A filter is passive if it consists only of
passive elements, R, L, and C.
• They are very important circuits in that many
technological advances would not have been
possible without the development of filters.
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Passive Filters
• There are four types of filters:
– Lowpass passes only low
frequencies and blocks high
frequencies.
– Highpass does the opposite of
lowpass
– Bandpass only allows a range of
frequencies to pass through.
– Bandstop does the opposite of
bandpass
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Lowpass Filter
• A typical lowpass filter is formed
when the output of a RC circuit
is taken off the capacitor.
• The half power frequency is:
c 
1
RC
• This is also referred to as the
cutoff frequency.
• The filter is designed to pass
from DC up to ωc
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Highpass Filter
• A highpass filter is also
made of a RC circuit, with
the output taken off the
resistor.
• The cutoff frequency will be
the same as the lowpass
filter.
• The difference being that the
frequencies passed go from
ωc to infinity.
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Bandpass Filter
• The RLC series resonant
circuit provides a bandpass
filter when the output is taken
off the resistor.
• The center frequency is:
0 
1
LC
• The filter will pass frequencies
from ω1 to ω2.
• It can also be made by feeding
the output from a lowpass to a
highpass filter.
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Bandstop Filter
• A bandstop filter can be
created from a RLC circuit by
taking the output from the LC
series combination.
• The range of blocked
frequencies will be the same
as the range of passed
frequencies for the bandpass
filter.
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Active Filters
• Passive filters have a few drawbacks.
– They cannot create gain greater than 1.
– They do not work well for frequencies below the
audio range.
– They require inductors, which tend to be bulky
and more expensive than other components.
• It is possible, using op-amps, to create all the
common filters.
• Their ability to isolate input and output also
makes them very desirable.
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First Order Lowpass
• If the input and feedback
elements in an inverting
amplifier are selectively
replaced with capacitors, the
amplifier can act as a filter.
• If the feedback resistor is
replaced with a parallel RL
element, the amplifier
becomes a lowpass filter.
• The corner frequency will be:
c 
1
Rf C f
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First Order Highpass
• Placing a series RL
combination in place of the
input resistor yields a
highpass filter.
• The corner frequency of the
filter will be:
c 
1
Ri Ci
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Bandpass
• To avoid the use of an inductor, it is possible
to use a cascaded series of lowpass active
filter into a highpass active filter.
• To prevent unwanted signals passing, their
gains are set to unity, with a final stage for
amplification.
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Bandreject
• Creating a bandstop filter requires using a
lowpass and highpass filter in parallel.
• Both output are fed into a summing amplifier.
• It will function by amplifying the desired
signals compared to the signal to be
rejected.
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