rlc circuits and resonance chapter 13 thomas l. floyd david m. buchla dc/ac fundamentals: a systems...
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RLC RLC Circuits and ResonanceCircuits and Resonance
Chapter 13
Thomas L. Floyd
David M. Buchla
DC/AC Fundamentals: A Systems DC/AC Fundamentals: A Systems ApproachApproach
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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:
The total impedance is found using:
The phase angle is found using:
Ch.13 Summary
Series RLC Circuits
R L C
VS
XX X CLtot
22tottot XRZ
R
X tot1tan
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Rea
ctan
ce
f
XC XL
A series RLC circuit can be capacitive, inductive, or resistive, depending on the frequency.
The frequency where XC=XL is called the resonant frequency.
Series resonance
XC = XL
Below the resonant frequency, the circuit is predominantly capacitive.
Above the resonant frequency, the circuit is predominantly inductive.
XC > XL XL > XC
Ch.13 Summary
XL and XC Vs. Frequency
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
What is the total impedance and phase angle of the series RLC circuit below?
The total reactance is
The total impedance is 3.16 k
The circuit is capacitive, and I leads V by 71.6o.
The phase angle is 71.6o
Ch.13 Summary
Series RLC Circuit Impedance
1 k 2 k 5 k
R
VS
XCXL
kΩ 3 kΩ 5 kΩ 2 X L Ctot XX
2222 kΩ 3kΩ 1tottot XRZ
kΩ 1
kΩ 3tantan 11
R
X totθ
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
What is the magnitude of the impedance for the circuit below?
753
Ch.13 Summary
Series RLC Circuit Impedance
f = 100 kHz
470 330 mH 2000 pF
R
VS
CL
Ω 207H) kHz)(330 (10022 m fLXL
Ω 796pF) kHz)(2000 (1002
1
2
1
fCXC
Ω 589 Ω 796Ω 207 XX X CLtot
22
22
Ω) (589Ω) (470
totXRZ
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
XL
f
XC
X
Depending on the frequency, the circuit can appear to be capacitive or inductive. The circuit in the previous slide was capacitive because XC > XL.
XL
XC
Ch.13 Summary
Series RLC Circuit Impedance
100 kHz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
What is the total impedance for the circuit when the frequency is increased to 400 kHz?
The circuit is now inductive.
786
Ch.13 Summary
Series RLC Circuit Impedance
Ω 298H) kHz)(330 (40022 m fLXL
Ω 991pF) kHz)(2000 (4002
1
2
1
fCXC
Ω 630 Ω 991Ω 829 XX X CLtot
22
22
Ω) (630Ω) (470
totXRZ
f = 400 kHz
470 330 mH 2000 pF
R
VS
CL
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
XL
f
XC
X
By changing the frequency, the circuit in the previous slide inductive (because XL > XC).
XL
XC
Ch.13 Summary
Impedance of Series RLC Circuits
400 kHz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
0
Notice that VC is out of phase with VL. When they are algebraically added, the result is…. VC
VL
This example is inductive.
Ch.13 Summary
Series RLC Circuit Voltages
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.13 Summary
Series Resonance
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
A formula for resonance can be found by assuming XC = XL and solving. The result is:
What is the resonant frequency for the circuit?
196 kHz
Ch.13 Summary
Series Resonance
LCfr
2
1
m
pF) H)(2000 (3302
12
1
LCfr
470 330 mH 2000 pF
R
VS
CL
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
What is VR at resonance?
Ideally, at resonance the sum of VL and VC is zero.
5.0 Vrms
0 VVS
By KVL, VR = VS
Ch.13 Summary
Series Resonance
470 330 mH 2000 pF
CLR
VS
5 Vrms
5.0 Vrms
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
XL
f
XC
XThe 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.
Z
Series resonance
Z = R
Ch.13 Summary
Series RLC Circuit Impedance
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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:
Ch.13 Summary
Series Resonance
LCfr
2
1
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Series resonant circuits can be used as filters. A band-pass filter allows signals within a range of frequencies to pass.
R
VoutVin
Resonant circuit
f
Vout
Series resonance
Circuit response
Ch.13 Summary
Series Resonant Filters
L C
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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).
0.707
f1 fr f2
BW
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.
f
I or VoutPassband
1.0
Ch.13 Summary
Series Resonant Filters
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Filter responses are often described in terms of decibels (dB). The decibel is defined as:
When circuit input and output voltages are known, the filter response can be calculated using:
Example: When output power is half the input power, the ratio of Pout/Pin = ½, and
Ch.13 Summary
Decibels
dB 32
110logdB
in
out
P
Plog10dB
in
out
V
VdB log20
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Selectivity describes the basic frequency response of a resonant circuit. (The -3 dB frequencies are marked by the dots.)
f0
BW3
Least Selectivity
BW2
Medium Selectivity
BW1
Greatest Selectivity
The greater the Q of a filter at a given resonant frequency, the higher it’s selectivity.
Which curve represents the highest Q?
The one with the greatest selectivity.
Ch.13 Summary
Selectivity
Q
fBW r
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
By taking the output across the resonant circuit, a band-stop (or notch) filter is produced.
RVoutVin
f
Vout
Circuit response
Resonant circuit
f1 fr
BW
Stop band
0.707
1
f2
Ch.13 Summary
Series Resonant Filters
L
C
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Conductance, susceptance, and admittance were defined in Chapter 10 as the reciprocals of resistance, reactance and impedance. As a review:
Ch.13 Summary
Conductance, Susceptance, and Admittance
Conductance is the reciprocal of resistance. R
G1
Susceptance is the reciprocal of reactance. X
B1
Admittance is the reciprocal of impedance. Z
Y1
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.13 Summary
Parallel RLC Circuit ImpedanceThe admittance can be used to find the impedance. Start by calculating the total susceptance:
CLtot BBB
The admittance is given by: 22totBGY
The impedance is the reciprocal of the admittance: Y
Ztot
1
The phase angle is:
G
Btot1tan
VS R L C
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
What is the total impedance of the parallel RLC circuit below?
First, determine the conductance and total susceptance as follows:
The total admittance is:
881
Ch.13 Summary
Parallel RLC Circuit Impedance
mS 0.3 CLtot BBB
mS 1.13 mS 0.3mS 1 22
22
tottot BGY
mS
YZtot
13.1
1
1
mS1kΩ 1
11
RG
ms 0.5kΩ 2
11
LL X
B
VS R XL XC
1 k 2 k 5 k
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
A typical current phasor diagram for a parallel RLC circuit is shown.
+90o
-90o
IC
IR
IL
The total current is given by:
The phase angle is given by:
Ch.13 Summary
AC Response of Parallel RLC Circuits
22 )( LCRtot IIII
R
CL
I
I1tanθ
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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….
IC
IL
0
Ch.13 Summary
Parallel RLC Circuit Currents
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Draw a diagram of the phasors having values of IR = 12 mA, IC = 22 mA and IL = 15 mA.
•Set up a grid.
0 mA
10 mA
20 mA
10 mA
20 mA
IR
IC
IL
•Plot the currents on the appropriate axes.
•Combine the reactive currents.
•Use the total reactive current and IR to find
the total current.
In this case, Itot = 16.6 mA
Ch.13 Summary
Parallel RLC Circuit Currents
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
0
Ideally, IC and IL cancel at parallel resonance because they are equal and opposite. Thus, the circuit is purely resistive at resonance.
The algebraic sum is zero.
Notice that IC is out of phase with IL. When they are algebraically added, the result is….
IC
IL
Ch.13 Summary
Parallel Resonance
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
The formula for the resonant frequency in both parallel and series circuits is the same:
What is the resonant frequency for the circuit?
49.8 kHz
Ch.13 Summary
Parallel Resonance
(ideal case)LC
fr
2
1
m
nF) H)(15 (6802π
12
1
LCfr
R L CVS680 mH1.0 k 15 nF
R L CVS
1 k 15 nF680 mH
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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:
For Q >10, the difference between the ideal and the non-ideal formula is less than 1%, and generally can be ignored.
Ch.13 Summary
Parallel Resonance in Non-ideal Circuits
(non-ideal)12
12
2
Q
Q
LCfr
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
In a parallel resonant circuit, impedance is maximum and current is minimum. The bandwidth (BW) can be defined in terms of the impedance curve.
f
Zmax
0.707Zmax
f1 fr f2
BW
Ztot
A parallel resonant circuit is commonly referred to as a tank circuit because of its ability to store energy like a storage tank.
Ch.13 Summary
Bandwidth of Resonant Circuits
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.13 Summary
Parallel Resonance
LCfr
2
1
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Parallel resonant circuits can also be used for band-pass or band-stop filters. A basic band-pass filter is shown below.
R
VinVout
Parallel resonant band-pass filter
CLResonant circuit
0.707
f1 fr f2
BW
f
VoutPassband
1.0
Ch.13 Summary
Parallel Resonant Filters
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
For the band-stop filter, the positions of the resonant circuit and resistance are reversed as shown here.
f
Vout
f1 fr
BW
Stop band
0.707
1
R
Vin Vout
Parallel resonant band-stop filter
C
L
Resonant circuit
f2
Ch.13 Summary
Parallel Resonant Filters
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
•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.
A band-pass filter allows frequencies between two critical frequencies and rejects all others.
Ch.13 Summary
Key Ideas for Resonant Filters
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
A condition in a series RLC circuit in which the reactances ideally cancel and the impedance is a minimum.
The frequency at which resonance occurs; also known as the center frequency.
A condition in a parallel RLC circuit in which the reactances ideally are equal and the impedance is a maximum.
A parallel resonant circuit.
Ch.13 Summary
Key Terms
Series resonance
Resonant frequency (fr)
Parallel resonance
Tank circuit
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ten times the logarithmic ratio of two powers.
A measure of how effectively a resonant circuit passes desired frequencies and rejects all others. Generally, the narrower the bandwidth, the greater the selectivity.
Ch.13 Summary
Key Terms
Half-power frequency
Decibel
Selectivity
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.13 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
2. In a series resonant circuit, the current at the half-power frequency is
a. maximum
b. minimum
c. 70.7% of the maximum value
d. 70.7% of the minimum value
Ch.13 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
3. The frequency represented by the red dashed line is the
a. resonant frequency
b. half-power frequency
c. critical frequency
d. all of the above
XL
f
XC
X
f
Ch.13 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.13 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
5. In a series resonant circuit, the resonant frequency can be found from the equation
a.
b.
c.
d.
max0.707rf I
1
2rf
LC
1
2rf LC
r
BWf
Q
Ch.13 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.13 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.13 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.13 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.13 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.13 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
1. c
2. c
3. a
4. a
5. b
6. d
7. b
8. d
9. d
10. a
Ch.13 Summary
Answers