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09/16/2010
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Series RLC
When a circuit contains an inductor and capacitor in
series, the reactance of each tend to cancel. The total
reactance is given by X tot  X L  X C
2
The total impedance is given by Ztot  R2  X tot
X 
The phase angle is given by   tan 1  tot 
 R 
R
VS
L
C
Variation of XL and XC
At the frequency where XC=XL,
the circuit is at series resonance.
Below the resonant
frequency, the circuit is
predominantly capacitive.
Above the resonant
frequency, the circuit is
predominantly inductive.
Reactance
In a series RLC circuit, the circuit can be capacitive or inductive,
depending on the frequency.
XC>XL
XL>XC
XC
XL
XC=XL
f
Series
resonance
Impedance of Series RLC
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?
The total reactance is X tot  X L  X C  2.0 kW  5.0 kW  3.0 kW
2
2
2
2
The total impedance is Ztot  R  X tot  1.0 kW +3.0 kW  3.16 kW
The phase angle is   tan 1 
X tot
 R

1  3.0 kW 
o

tan

  71.6

 1.0 kW 

The circuit is capacitive,
so I leads V by 71.6o.
R
VS
L
C
1.0 kW XL = XC =
2.0 kW 5.0 kW
Impedance of Series RLC
What is the magnitude of the impedance for the circuit?
X L  2 fL  2 100 kHz  330 m H   207 W
1
1
XC 

 796 W
2 fC 2 100 kHz  2000 pF 
X tot  X L  X C  207 W  796 W  589 W
Z=
 470 W   589 W
2
2
753 W
R

VS
470 W
L
C
330 mH 2000 pF
f = 100 kHz
Impedance of Series RLC
Depending on the frequency, the circuit can appear to be
capacitive or inductive. The circuit in the Example-2 was
capacitive because XC>XL.
X
XL
XC
XL
XC
f
Impedance of Series RLC
What is the total impedance for the circuit when the
frequency is increased to 400 Hz?
X L  2 fL  2  400 kHz  330 m H   829 W
1
1
XC 

 199 W
2 fC 2  400 kHz  2000 pF 
X tot  X L  X C  829 W  199 W  630 W
Z=
 470 W   630 W  786 W
2
2
R
The circuit is
now inductive.
VS
L
C
470 W 330 mH 2000 pF
f = 400 kHz
Impedance of Series RLC
By changing the frequency, the circuit in Example-3 is
now inductive because XL>XC
X
XL
XL
XC
XC
f
Voltages in Series RLC
Voltages
inacross
a series
RLCcomponents
circuits must add to
The voltages
the RLC
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.
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.
Series Resonance
The formula for resonance can be found by setting
XC = XL. The result is
fr 
1
2 LC
What is the resonant frequency for the circuit?
fr 

1
R
2 LC
470 W
1
2
 330 μH  2000 pF 
 196 kHz
VS
L
C
330 mH 2000 pF
Series Resonance
Ideally, at resonance the sum of VL and VC is zero.
VS
5.0
Vrms
V=0
R
What is VR at
resonance?
470 W
VS
5.0 Vrms
By KVL,
VR = VS
L
C
330 mH 2000 pF
5.0 Vrms
Impedance of Series RLC
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
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
fr 
1
2 LC
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.
Circuit response:
Vout
Resonant circuit
L
C
Vin
Vout
R
f
Series
resonance
Series Resonant Filters
The response has a peak because at the series resonant frequency,
the current is maximum at resonance and falls off before and after
resonance. This develops the maximum voltage across the resistor
at resonance.
I or Vout
Passband
The bandwidth (BW) of the
1.0
filter is the range of frequencies for
which the output is equal to or
0.707
greater than 70.7% of the
maximum value. f1 and f2 are
commonly referred to as the
critical frequencies, cutoff
f1 fr f2
frequencies or half-power
frequencies.
BW
f
Series Resonant Filter
Decibels
Filter responses are often given in terms of decibels, which is
defined as
 Pout 
dB  10log 

P
 in 
Because it is a ratio, the decibel is dimensionless. One of the most
important decibel ratios occurs when the power ratio is 1:2. This is
called the 3 dB frequency, because
1
dB  10 log    3 dB
2
Another useful definition for the decibel, when measuring voltages
across the same impedance is
 Vout 
dB  20log 

V
in


Selectivity
Greatest
Selectivity
Medium Selectivity
Selectivity describes the basic
frequency response of a
resonant circuit. (The 3 dB
frequencies are marked.)
The bandwidth is inversely
proportional to Q in
accordance with the formula,
f
0
BW  r
Q
Least Selectivity
f
BW1
BW2
BW3
Which curve represents the
highest Q? The one with the greatest selectivity.
Bandstop
Series Resonant Filters
By taking the output across the resonant circuit, a bandstop (or notch) filter is produced.
Circuit response:
Vin
Vout
R
Resonant
circuit
Vout
L
Stopband
1
0.707
C
f
f1 fr f2
BW
f2
Conductance, Susceptance, and Admittance
Conductance,
susceptance,
andand
admittance
Recall that conductance,
susceptance,
admittance
were defined in Chapter 10 as the reciprocals of
resistance, reactance and impedance.
Conductance is the reciprocal of resistance. G 
1
R
Susceptance is the reciprocal of reactance. B 
1
X
Y
1
Z
Admittance is the reciprocal of impedance.
Impedance of Parallel RLC
The admittance can be used to find the impedance.
Start by calculating the total susceptance: Btot  BL  BC
2
The admittance is given by Y  G 2  Btot
The impedance is the reciprocal of the admittance: Z tot 
 Btot 

 G 
1
Y
The phase angle is   tan 1 
VS
R
L
C
Impedance of Parallel RLC
What is the total impedance of the parallel RLC circuit
if R= 1.0 kW, XL = 2.0 kW, and XC = 5.0 kW?
First determine the conductance
The total admittance is:
2
and total susceptance as follows:
Ytot  G 2  Btot
1
1
2
2
G 
 1.0 mS

1.0
mS
+
0.3
mS
 1.13 mS
R 1.0 kW
1
1
1
1
Z 
 881 W
BL 

 0.5 mS
Y 1.13 mS
X L 2.0 kW
1
1
BC 

 0.2 mS VS
R
XL =
XC =
X C 5.0 kW
1.0 kW
Btot  BL  BC  0.3 mS
2.0 kW
5.0 kW
Sinusoidal Response of Parallel RLC
A typical current phasor diagram for a parallel RLC circuit is
IC
The total current is given by:
Itot  I R2   IC  I L 
+90o
2
The phase angle is given by:
IR
 I CL 

I
 R 
  tan 1 
90o
IL
What is Itot and  if IR = 10 mA, IC = 15 mA and IL = 5 mA?
Itot  10 mA2 + 15 mA  5.0 mA   14.1 mA
2
  45 mA
Currents in a Parallel RLC
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.
IC
Notice that IC is out of
phase with IL. When
they are algebraically
added, the result is….
0
IL
Currents in a Parallel RLC
IC
Draw a diagram of the phasors if IR = 12 mA,
IC = 22 mA and IL = 15 mA?
20 mA
10 mA
• Set up a grid with a scale that will allow
all of the data– say 2 mA/div.
• 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
0 mA
10 mA
20 mA
IL
IR
Parallel
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
they are algebraically
added, the result is….
IC
0
IL
Parallel
The formula for the resonant frequency in both
parallel and series circuits is the same, namely
fr 
fr 

1
2 LC
(ideal case)
What is the resonant frequency for the circuit?
1
2 LC
1
2
 680 μH 15 nF 
 49.8 kHz
VS
R
1.0 kW
C
L
680 mH 15 nF
Parallel Resonance in Nonideal
In practical circuits, when the coil resistance is
considered, there is a small current at resonance and
the resonant frequency is not exactly given by the
ideal equation. The Q of the coil affects the equation
for resonance:
fr 
1
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.
Bandwidth of Resonant Circuit
At the parallel resonant frequency, impedance is maximum, so
current is a minimum at resonance. 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
BW
f
Parallel
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 f r 
1
2 LC
Parallel Resonant
Parallel resonant circuits can also be used for band-pass
or band-stop filters. A basic band-pass filter is shown.
Circuit response:
Vout
R
Vin
Passband
Vout
L
C
1.0
0.707
Resonant
circuit
f
Parallel resonant
band-pass filter
f1 fr f2
BW
Bandpass
Parallel Resonant
For the band-stop filter, the resonant circuit and
resistance are reversed as shown here.
Circuit response:
Vout
C
Vin
Vout
L
R
Stopband
1
0.707
Resonan
t circuit
Parallel resonant
band-stop filter
f
f1 fr f2
BW
Parallel Resonant Filters: Band-stop
Key Ideas for Resonant
•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.
Admittance of Parallel RLC
Selected Key Terms
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.
A condition in a parallel RLC circuit in which the
reactances ideally are equal and the impedance is a
maximum.
A parallel resonant circuit.
Selected Key Terms
Half-power The frequency at which the output power of a
frequency 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.
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
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
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
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
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
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
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.
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.
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
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
Quiz
Answers:
1. c
6. d
2. c
7. b
3. a
8. d
4. a
9. d
5. b
10. a