Resonance ppt.ppt

Chapter 13
© Copyright 2007 Prentice-Hall
Electric Circuits Fundamentals - Floyd
Chapter 13

Chapter 13
Summary
VS
R L C
Series RLC circuits
When a circuit contains an inductor and capacitor in
series, the reactance of each tend to cancel. The total
reactance is given by tot L C
X X X
 
The total impedance is given by 2 2
tot tot
Z R X
 
The phase angle is given by 1
tan tot
X
R
   
  
 

Chapter 13
Summary
Variation of XL and XC with frequency
Reactance
f
XC XL
In a series RLC circuit, the circuit can be capacitive or inductive,
depending on the frequency.
At the frequency where XC=XL,
the circuit is at series resonance.
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

Chapter 13
Summary
VS
R L C
What is the total impedance and phase angle of the series
RLC circuit if R= 1.0 kW, XL = 2.0 kW, and XC = 5.0 kW?
1.0 kW XC =
5.0 kW
The total reactance is 2.0 k 5.0 k 3.0 k
tot L C
X X X
   W W  W
The total impedance is
2 2 2 2
1.0 k +3.0 k
tot tot
Z R X
   W W  3.16 kW
XL =
2.0 kW
The circuit is capacitive,
so I leads V by 71.6o
.
The phase angle is 1 1 3.0 k
tan tan
1.0 k
tot
X
R
   W
   
  
 
 
W
 
 
71.6o
Impedance of series RLC circuits

Chapter 13
Summary
What is the magnitude of the impedance for the circuit?
R L C
VS
330 mH
f = 100 kHz
470 W 2000 pF
  
2 2 100 kHz 330 H 207
L
X fL
  m
   W
  
1 1
796
2 2 100 kHz 2000 pF
C
X
fC
 
   W
207 796 589
tot L C
X X X
   W W  W
   
2 2
= 470 589
Z W  W 
753 W

Chapter 13
Summary
XL
f
XC
X
Depending on the frequency, the circuit can appear to be
capacitive or inductive. The circuit in the Example-2 was
capacitive because XC>XL.
XL
XC

Chapter 13
Summary
R L C
VS
What is the total impedance for the circuit when the
frequency is increased to 400 Hz?
330 mH
f = 400 kHz
470 W 2000 pF
  
2 2 400 kHz 330 H 829
L
X fL
  m
   W
  
1 1
199
2 2 400 kHz 2000 pF
C
X
fC
 
   W
The circuit is
now inductive.
829 199 630
tot L C
X X X
   W W  W
   
2 2
= 470 630
Z W  W 
786 W

Chapter 13
Summary
XL
f
XC
X
By changing the frequency, the circuit in Example-3 is
now inductive because XL>XC
XL
XC

Chapter 13
Summary
Voltages in a series RLC circuits
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.

Chapter 13
Summary
Series resonance
At series resonance, XC and XL cancel. VC and VL also
cancel because the voltages are equal and opposite.
The circuit is purely resistive at resonance.
0
Algebraic sum
is zero.

Chapter 13
Summary
Series resonance
The formula for resonance can be found by setting
XC = XL. The result is
1
2
r
f
LC


What is the resonant frequency for the circuit?
R L C
VS
330 mH
470 W 2000 pF
  
1
2
1
2 330 μH 2000 pF
r
f
LC




 196 kHz

Chapter 13
Summary
Series resonance
What is VR at
resonance?
Ideally, at resonance the sum of VL and VC is zero.
R L C
VS
330 mH
470 W 2000 pF
5.0 Vrms
5.0 Vrms
V = 0
VS
By KVL,
VR = VS
5.0 Vrms

Chapter 13
Summary
XL
f
XC
X
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.
Z
Series
resonance
Z = R

Chapter 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 given by
1
2
r
f
LC



Chapter 13
Summary
Series resonant filters
An application of series resonant circuits is in filters. A
band-pass filter allows signals within a range of
frequencies to pass.
R
L C
Vout
Vin
Resonant circuit
f
Vout
Series
resonance
Circuit response:

Chapter 13
Summary
Series resonant filters
By taking the output across the resonant circuit, a band-
stop (or notch) filter is produced.
R
Vout
Vin
f
Vout
Circuit response:
L
C
Resonant
circuit
f1 fr
f2
BW
Stopband
0.707
1
f2

Chapter 13
Summary
Conductance, susceptance, and admittance
Recall that conductance, susceptance, and admittance
were defined in Chapter 10 as the reciprocals of
resistance, reactance and impedance.
Conductance is the reciprocal of resistance.
1
G
R

Admittance is the reciprocal of impedance.
Susceptance is the reciprocal of reactance.
1
B
X

1
Y
Z


Chapter 13
Summary
Impedance of parallel RLC circuits
The admittance can be used to find the impedance.
Start by calculating the total susceptance: tot L C
B B B
 
The admittance is given by 2 2
tot
Y G B
 
The phase angle is 1
tan tot
B
G
   
  
 
The impedance is the reciprocal of the admittance:
1
tot
Z
Y

R L C
VS

Chapter 13
Summary
What is the total impedance of the parallel RLC circuit
if R= 1.0 kW, XL = 2.0 kW, and XC = 5.0 kW?
1.0 kW
XC =
5.0 kW
First determine the conductance
and total susceptance as follows:
The total admittance is:
2 2
2 2
1.0 mS + 0.3 mS 1.13 mS
tot tot
Y G B
 
 
881 W
XL =
2.0 kW
Impedance of parallel RLC circuits
R
VS
1 1
1.0 mS
1.0 k
G
R
  
W
1 1
0.5 mS
2.0 k
L
L
B
X
  
W
1 1
1.13 mS
Z
Y
  
1 1
0.2 mS
5.0 k
C
C
B
X
  
W
0.3 mS
tot L C
B B B
  

Chapter 13
Summary
Sinusoidal response of parallel RLC circuits
A typical current phasor diagram for a parallel RLC circuit is
+90o
90o
IC
IR
IL
The total current is given by:
 
2
2
tot R C L
I I I I
  
What is Itot and  if IR = 10 mA, IC = 15 mA and IL = 5 mA?
45 mA
   
1
tan CL
R
I
I
   
  
 
The phase angle is given by:
 
2
2
10 mA + 15 mA 5.0 mA
tot
I    14.1 mA

Chapter 13
Summary
Currents in a parallel RLC circuits
The currents in the RLC components must add to the
source current in accordance with KCL. Because of the
opposite phase shift due to L and C, IL and IC effectively
subtract.
Notice that IC is out of
phase with IL. When
added, the result is….
IC
IL
0

Chapter 13
Summary
Currents in a parallel RLC circuits
Draw a diagram of the phasors if IR = 12 mA,
IC = 22 mA and IL = 15 mA?
• Set up a grid with a scale that will allow
all of the data– say 2 mA/div.
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

Chapter 13
0
Summary
Parallel resonance
Ideally, at parallel resonance, IC and IL cancel because
the currents are equal and opposite. The circuit is
purely resistive at resonance.
The algebraic
sum is zero.
Notice that IC is out of
phase with IL. When
added, the result is….
IC
IL

Chapter 13
Summary
The formula for the resonant frequency in both
parallel and series circuits is the same, namely
1
2
r
f
LC


What is the resonant frequency for the circuit?
  
1
2
1
2 680 μH 15 nF
r
f
LC




 49.8 kHz
Parallel resonance
R L C
VS
680 mH
1.0 kW 15 nF
(ideal case)

Chapter 13
Summary
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.
• fr is given by
1
2
r
f
LC


Parallel resonance

Chapter 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.
R
Vin
Vout
Parallel resonant
band-pass filter
C
L
Resonant
circuit
0.707
f1 fr f2
BW
f
Vout
Passband
Circuit response:
1.0

Chapter 13
Summary
Parallel resonant filters
For the band-stop filter, the resonant circuit and
resistance are reversed as shown here.
f
Vout
f1 fr
BW
Stopband
0.707
1
R
Vin Vout
Parallel resonant
band-stop filter
C
L
Resonant
circuit
Circuit response:
f2

Chapter 13
Summary
Key ideas for resonant filters
• 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.

Chapter 13
Series
resonance
Resonant
frequency (fr)
Parallel
resonance
Tank circuit
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.
Key Terms
A condition in a parallel RLC circuit in which
the reactances ideally are equal and the
impedance is a maximum.
A parallel resonant circuit.

Chapter 13
Half-power
frequency
Decibel
Selectivity
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.
Key Terms
A measure of how effectively a resonant
circuit passes desired frequencies and rejects
all others. Generally, the narrower the
bandwidth, the greater the selectivity.

Chapter 13
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

Chapter 13
Quiz
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

Chapter 13
Quiz
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

Chapter 13
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

Chapter 13
Quiz
5. In a series resonant circuit, the resonant frequency can
be found from the equation
a.
b.
c.
d.
max
0.707
r
f I

1
2
r
f
LC


1
2
r
f
LC


r
BW
f
Q


Chapter 13
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

Chapter 13
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.

Chapter 13
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.

Chapter 13
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

Chapter 13
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.

Chapter 13
Quiz
Answers:
1. c
2. c
3. a
4. a
5. b
6. d
7. b
8. d
9. d
10. a

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