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DC/AC Fundamentals: A Systems
Approach
Thomas L. Floyd
David M. Buchla
RLC Circuits and Resonance
Chapter 13
Ch.13 Summary
Series RLC Circuits
When a circuit contains an inductor and capacitor in series,
the reactance of each opposes (i.e., cancels) the other.
Total series LC reactance is found using:
Xtot  XL  XC
The total impedance is found using:
2
Ztot  R 2  X tot
The phase angle is found using:
R
L
C
VS
X 
  tan 1 tot 
 R 
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
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Ch.13 Summary
XL and XC Vs. Frequency
A series RLC circuit can be capacitive, inductive, or resistive,
depending on the frequency.
The frequency where XC=XL is
called the resonant frequency.
Below the resonant
frequency, the circuit is
predominantly capacitive.
Above the resonant
frequency, the circuit is
predominantly inductive.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
Reactance
XC > XL
XL > XC
XC
XL
XC = XL
f
Series
resonance
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Ch.13 Summary
Series RLC Circuit Impedance
What is the total impedance and phase angle of the series
RLC circuit below?
The total reactance is X tot  XL  XC  2 kΩ  5 kΩ  3 kΩ
2
2
2
2
The total impedance is Ztot  R  X tot  1kΩ  3 kΩ  3.16 kW
 X tot
 R
The phase angle is θ  tan 1
The circuit is capacitive,
and I leads V by 71.6o.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd

o
1 3 kΩ 

tan


  71.6
 1 kΩ 

VS
R
XL
1 kW
2 kW
XC
5 kW
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Ch.13 Summary
Series RLC Circuit Impedance
What is the magnitude of the impedance for the circuit below?
XL  2fL  2(100 kHz)(330 mH)  207 Ω
XC 
1
1

 796 Ω
2fC 2(100 kHz)(2000 pF)
X tot  XL  XC  207 Ω  796 Ω  589 Ω
2
Z  R 2  X tot
470 W
 (470 Ω)2  (589 Ω)2

753 W
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
L
R
VS
C
330 mH 2000 pF
f = 100 kHz
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Ch.13 Summary
Series RLC Circuit Impedance
Depending on the frequency, the circuit can appear to be
capacitive or inductive. The circuit in the previous slide was
capacitive because XC > XL.
X
XL
XC
XC
XL
f
100 kHz
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Series RLC Circuit Impedance
What is the total impedance for the circuit when the frequency is
increased to 400 kHz?
XL  2fL  2(400 kHz)(330 mH)  829 Ω
1
1
XC 

 199 Ω
2fC 2(400 kHz)(2000 pF)
X tot  XL  XC  829 Ω  199 Ω  630 Ω
2
Z  R 2  X tot
 (470 Ω)2  (630 Ω)2
 786 W
The circuit is now inductive.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
L
R
470 W
VS
C
330 mH 2000 pF
f = 400 kHz
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Ch.13 Summary
Impedance of Series RLC Circuits
By changing the frequency, the circuit in the previous slide
inductive (because XL > XC).
X
XL
XL
XC
XC
400 kHz
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
f
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Ch.13 Summary
Series RLC Circuit Voltages
The voltages across the RLC components must add to the
source voltage in accordance with KVL. Because of the
opposite phase shift due to L and C, VL and VC effectively
subtract.
Notice that VC is out of
phase with VL. When
they are algebraically
added, the result is….
VL
0
VC
This example is inductive.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
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Ch.13 Summary
Series Resonance
At series resonance, XC and XL cancel. VC and VL also
cancel because they are equal in magnitude and opposite
in polarity. The circuit is purely resistive at resonance.
0
Algebraic
sum is zero.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
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Ch.13 Summary
Series Resonance
A formula for resonance can be found by assuming XC =
XL and solving. The result is:
1
fr 
2 LC
What is the resonant frequency for the circuit?
fr 
1
2 LC
1

2 (330 mH)(2000 pF)
R
470 W
VS
L
C
330 mH 2000 pF
 196 kHz
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Series Resonance
Ideally, at resonance the sum of VL and VC is zero.
VS
0V
What is VR at
resonance? 5.0 Vrms
R
470 W
VS
5 Vrms
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
By KVL,
VR = VS
L
C
330 mH 2000 pF
5.0 Vrms
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Ch.13 Summary
Series RLC Circuit Impedance
The general shape of the
impedance versus frequency
for a series RLC circuit is
superimposed on the curves
for XL and XC. Notice that at
the resonant frequency, the
circuit is resistive, and Z = R.
X
XL
Z
XC
Z=R
f
Series
resonance
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Series Resonance
Summary of important concepts for series resonance:
• Capacitive and inductive reactances are equal.
• Total impedance is a minimum and is resistive.
• The current is maximum.
• The phase angle between VS and IS is zero.
• fr is calculated using:
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
1
fr 
2 LC
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Ch.13 Summary
Series Resonant Filters
Series resonant circuits can be used as filters. A
band-pass filter allows signals within a range of
frequencies to pass.
Vout
Resonant circuit
L
Circuit response
C
Vin
Vout
R
f
Series resonance
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Series Resonant Filters
The response curve has a peak; meaning the current is
maximum at resonance and falls off at frequencies below and
above resonance. The maximum current (at resonance) develops
maximum voltage across the series resistor(s).
The bandwidth (BW) of the filter
is the range of frequencies over
which the output is equal to or
greater than 70.7% of its
maximum value. f1 and f2 are
commonly referred to as the
critical frequencies, cutoff
frequencies or half-power
frequencies.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
I or Vout
Passband
1.0
0.707
f
f1 fr f2
BW
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Ch.13 Summary
Decibels
Filter responses are often described in terms of decibels
(dB). The decibel is defined as:
 Pout
dB  10 log
 Pin



Example: When output power is half the input power, the
ratio of Pout/Pin = ½, and
 1
dB  10log   3 dB
2
When circuit input and output voltages are known, the filter
response can be calculated using:
V
dB  20 log out
 Vin
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd



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Ch.13 Summary
Selectivity
Selectivity describes the basic
frequency response of a resonant
circuit. (The -3 dB frequencies are
marked by the dots.)
Greatest Selectivity
Medium Selectivity
Least Selectivity
The greater the Q of a filter at a
given resonant frequency, the
higher it’s selectivity.
fr
BW 
Q
0
f
BW1
Which curve represents the highest Q?
The one with the greatest selectivity.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
BW2
BW3
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Ch.13 Summary
Series Resonant Filters
By taking the output across the resonant circuit, a
band-stop (or notch) filter is produced.
Circuit response
Vin
Vout
R
Resonant
circuit
Vout
L
Stop band
1
0.707
C
f
f1 fr f2
BW
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Conductance, Susceptance, and
Admittance
Conductance, susceptance, and admittance were defined in Chapter
10 as the reciprocals of resistance, reactance and impedance. As a
review:
Conductance is the reciprocal of
resistance.
1
G
R
Susceptance is the reciprocal of
reactance.
1
B
X
Admittance is the reciprocal of
impedance.
1
Y 
Z
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Parallel RLC Circuit Impedance
The admittance can be used to find
the impedance. Start by calculating
the total susceptance:
Btot  BL  BC
The admittance is given by:
2
Y  G2  Btot
The impedance is the reciprocal of the
admittance:
The phase angle is:
 Btot 
  tan 

 G 
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
Z tot
1

Y
1
VS
R
L
C
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Ch.13 Summary
Parallel RLC Circuit Impedance
What is the total impedance of the parallel RLC circuit below?
First, determine the conductance
and total susceptance as follows:
1
1
G 
 1 mS
R 1 kΩ
BL 
The total admittance is:
Ztot 

1
1

 0.5 ms
X L 2 kΩ
1
Y
1

1.13 mS
881 W
Btot  BL  BC  0.3 mS
2
Ytot  G 2  Btot
 1mS 2  0.3 mS 2  1.13 mS
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
VS
R
1 kW
XL
2 kW
XC
5 kW
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Ch.13 Summary
AC Response of Parallel RLC Circuits
The total current is given by:
IC
A typical
current phasor
diagram for a
parallel RLC
circuit is shown.
Itot  IR2  (IC  IL )2
+90o IR
The phase angle is given by:
IL
-90o
 ICL
θ  tan 
 IR
1
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd



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Ch.13 Summary
Parallel RLC Circuit Currents
The currents in the RLC components must add to the source
current in accordance with KCL. Because of the opposite phase
shifts of IL and IC (relative to VS) they effectively subtract.
Notice that IC is out of
phase with IL. When
they are algebraically
added, the result is….
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
IC
0
IL
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Ch.13 Summary
Parallel RLC Circuit Currents
Draw a diagram of the phasors having
values of IR = 12 mA, IC = 22 mA and IL =
15 mA.
IC
20 mA
10 mA
•Set up a grid.
•Plot the currents on the appropriate axes.
0 mA
•Combine the reactive currents.
10 mA
•Use the total reactive current and IR to find
the total current.
20 mA
IR
IL
In this case, Itot = 16.6 mA
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Parallel Resonance
Ideally, IC and IL cancel at parallel resonance because they
are equal and opposite. Thus, the circuit is purely resistive at
resonance.
Notice that IC is out of
phase with IL. When
they are algebraically
added, the result is….
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
IC
The algebraic
sum is zero.
0
IL
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Ch.13 Summary
Parallel Resonance
The formula for the resonant frequency in both
parallel and series circuits is the same:
fr 
1
2 LC
(ideal case)
What is the resonant frequency for the circuit?
fr 
1
2 LC
1

2π (680 mH)(15 nF)

R
1.0
1 kW
kW
C
L
680
680mH
mH 15 nF
49.8 kHz
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
VS
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Ch.13 Summary
Parallel Resonance in Non-ideal Circuits
In practical circuits, there is a small current through
the coil at resonance and the resonant frequency is
not exactly given by the ideal equation. The Q of the
coil affects the equation for resonance:
1
fr 
2 LC
Q2
Q2  1
(non-ideal)
For Q >10, the difference between the ideal and the non-ideal
formula is less than 1%, and generally can be ignored.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
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Ch.13 Summary
Bandwidth of Resonant Circuits
In a parallel resonant circuit, impedance is maximum and
current is minimum. The bandwidth (BW) can be defined in
terms of the impedance curve.
Ztot
A parallel resonant circuit is
commonly referred to as a
tank circuit because of its
ability to store energy like a
storage tank.
Zmax
0.707Zmax
f1 fr f2
f
BW
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Parallel Resonance
Summary of important concepts for parallel resonance:
• Capacitive and inductive susceptance are equal.
• Total impedance is a maximum (ideally infinite).
• The current is minimum.
• The phase angle between VS and IS is zero.
• The resonant frequency (fr) is given by
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
1
fr 
2 LC
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Ch.13 Summary
Parallel Resonant Filters
Parallel resonant circuits can also be used for band-pass
or band-stop filters. A basic band-pass filter is shown
below.
Vout
R
Vin
Vout
L
C
Passband
1.0
0.707
Resonant
circuit
f
Parallel resonant
band-pass filter
f1 fr f2
BW
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Parallel Resonant Filters
For the band-stop filter, the positions of the resonant circuit
and resistance are reversed as shown here.
C
Vin
Vout
Vout
L
R
Stop band
1
0.707
Resonant
circuit
Parallel resonant
band-stop filter
f
f1 fr f2
BW
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Key Ideas for Resonant Filters
A band-pass filter allows frequencies between two critical
frequencies and rejects all others.
•A band-stop filter rejects frequencies between two critical
frequencies and passes all others.
•Band-pass and band-stop filters can be made from both series
and parallel resonant circuits.
•The bandwidth of a resonant filter is determined by the Q and
the resonant frequency.
•The output voltage at a critical frequency is 70.7% of the
maximum.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Key Terms
Series
resonance
A condition in a series RLC circuit in which
the reactances ideally cancel and the
impedance is a minimum.
Resonant
frequency (fr)
The frequency at which resonance occurs;
also known as the center frequency.
Parallel
resonance
Tank circuit
A condition in a parallel RLC circuit in which
the reactances ideally are equal and the
impedance is a maximum.
A parallel resonant circuit.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
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Ch.13 Summary
Key Terms
Half-power
frequency
The frequency at which the output power of a
resonant circuit is 50% of the maximum value
(the output voltage is 70.7% of maximum);
another name for critical or cutoff frequency.
Decibel
Ten times the logarithmic ratio of two powers.
Selectivity
A measure of how effectively a resonant
circuit passes desired frequencies and rejects
all others. Generally, the narrower the
bandwidth, the greater the selectivity.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Quiz
1. In practical series and parallel resonant circuits,
the total impedance of the circuit at resonance will
be
a. capacitive
b. inductive
c. resistive
d. none of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
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Ch.13 Summary
Quiz
2. In a series resonant circuit, the current at the halfpower frequency is
a. maximum
b. minimum
c. 70.7% of the maximum value
d. 70.7% of the minimum value
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
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Ch.13 Summary
Quiz
3. The frequency represented by the red dashed line
is the
a. resonant frequency
X
b. half-power frequency
XL
c. critical frequency
d. all of the above
XC
f
f
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
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Ch.13 Summary
Quiz
4. In a series RLC circuit, if the frequency is below the
resonant frequency, the circuit will appear to be
a. capacitive
b. inductive
c. resistive
d. answer depends on the particular
components
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
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Ch.13 Summary
Quiz
5. In a series resonant circuit, the resonant frequency
can be found from the equation
b.
BW
Q
1
fr 
2 LC
c.
f r  0.707 I max
d.
fr 
a.
fr 
1
2 LC
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
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Ch.13 Summary
Quiz
6. In an ideal parallel resonant circuit, the total
impedance at resonance is
a. zero
b. equal to the resistance
c. equal to the reactance
d. infinite
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
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Ch.13 Summary
Quiz
7. In a parallel RLC circuit, the magnitude of the total
current is always the
a. same as the current in the resistor.
b. phasor sum of all of the branch currents.
c. same as the source current.
d. difference between resistive and reactive
currents.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
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Ch.13 Summary
Quiz
8. If you increase the frequency in a parallel RLC
circuit, the total current
a. will not change
b. will increase
c. will decrease
d. can increase or decrease depending on if it
is above or below resonance.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
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Ch.13 Summary
Quiz
9. The phase angle between the source voltage and
current in a parallel RLC circuit will be positive if
a. IL is larger than IC
b. IL is larger than IR
c. both a and b
d. none of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
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Ch.13 Summary
Quiz
10. A highly selectivity circuit will have a
a. small BW and high Q.
b. large BW and low Q.
c. large BW and high Q.
d. none of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
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Ch.13 Summary
Answers
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
1. c
6. d
2. c
7. b
3. a
8. d
4. a
9. d
5. b
10. a
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