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Resonant Circuits
SEE 1023 Circuit Theory
Frequency Response
1
Series RLC Circuit
L
R
I
+
+
Vs
w
(varied)
VR
-
VL
+
VC
C
-
When w varies, the impedance of the circuit will vary.
Then, the current and the real power will also vary.
We would like to study the frequency response of
these quantities.
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Series RLC Circuit
Impedance as a function of frequency:
1
Z (w )  R  j (wL 
)
wC
Reactance as a function of frequency: X w  w  L 
Current as a function of frequency: I w 
1
wC
Vs
2
1 

R  w  L 

w  C

2
Power as a function of frequency: P w 
Vs
2
 2 
1  
R

w

L




w

C



2
R
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Series RLC Circuit
Series RLC
Circuit
Excitation
(Input)
Constant input voltage: Vs
Variable Source angular
frequency: w
Response
(Output)
Main response: current
Other responses:
Power, Impedance,
reactance, etc.
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Series RLC Circuit in PSpice
R
I
L
2
1
+
+
VR
-
VL
3
+
Vs
VC
w
C
-
(varied)
0
It is too hard to study the frequency response of
these quantities manually.
It is too easy to study the frequency response of
these quantities PSpicely.
5
Series RLC Circuit in PSpice
Series resonant Circuit
Vs 1 0 AC 10V
R1 1 2 10
L1 2 3 100mH
C1 3 0 10uF
.AC LIN 1001 100Hz 220Hz
.Probe
.end
To Display
graph
Total PTS.
End FREQ.
Start FREQ.
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In the Probe windows
Trace Expression
M(V(1)/I(R1))
Response
Magnitude of Z
P(V(1)/I(R1))
Phase of Z
R(V(1)/I(R1))
Real part of Z
IMG(V(1)/I(R1))
Imaginary part of Z
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In the Probe windows
Trace Expression
Response
M(I(R1))
Magnitude of I
P(I(R1))
Phase of I
R(I(R1))
Real part of I
IMG(I(R1))
Imaginary part of I
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In the Probe windows
Trace Expression
Response
V(1,2)
Magnitude of VR
V(2,3)
Magnitude of VL
V(3)
Magnitude of VC
I(R1)*I(R1)*10
Real power, P
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Frequency Response of The Current
Run Pspice File
10
Frequency Response of The Current
(Variation of the current with frequency)
1.2
1
0.81
I( w )
0.61
0.41
0.22
800
840
880
920
960 1000 1040 1080 1120 1160 1200
w
At Resonance, the current is maximum
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Basic Questions
What is the minimum value of Z?
What is the maximum value of I?
What is the maximum value of P?
Z= R
I  Io 
Vs
R
Vs2
P  Po  I R 
R
2
o
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Basic Questions
When the power P = Po/2, what is
The angular frequency?
w1 lower half power frequency
w2 higher half power frequency
The magnitude of I?
I1 
Vs
2R
The magnitude of Z?
Z  2R
The magnitude of X?
X  R
X  R
at w1
at w2
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Resonant Condition
By definition the resonant angular frequency, wo, for
the RLC series circuit occurs at the peak of the
current response. Under this condition:
 The real power is maximum
 The magnitude of impedance is minimum
 The circuit is purely resistive
 The imaginary part of the impedance is zero
 The pf = 1
 The current is in phase with the voltage source
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Lower half-power angular frequency, w1, condition
By definition lower half-power angular frequency,
w1, occurs when the power is Po/2 and the angular
frequency is below the resonant angular frequency.
 The real power is Po/2
 The current is Io /2
 The magnitude of impedance is 2R
 X = -R
 The circuit is predominantly capacitive
 The pf = cos(45) leading
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Lower half-power angular frequency, w2, condition
By definition lower half-power angular frequency,
w2, occurs when the power is Po/2 and the angular
frequency is above the resonant angular frequency.
 The real power is Po/2
 The current is Io /2
 The magnitude of impedance is 2R
 X = +R
 The circuit is predominantly inductive
 The pf = cos(45) lagging
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The Voltage Phasor Diagram at wo
For R:
I is in phase with VR
For L:
I lags VL by 90
For C:
I leads VC by 90
VL
at wo
For series circuit, use I as the
reference.
I
VR = VS
The circuit is purely
resistive.
VC
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The Voltage Phasor Diagram at w1
For R:
I is in phase with VR
For L:
I lags VL by 90
For C:
I leads VC by 90
The circuit is predominantly capacitive.
VL
For series circuit, use I as a
reference.
VR
I
at w1
VL+VC
VS
VC
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The Voltage Phasor Diagram at w2
For R:
I is in phase with VR
For L:
I lags VL by 90
For C:
I leads VC by 90
VL
For series circuit, use I as the
reference.
VS
VL+VC
at w2
I
VR
VC
The circuit is predominantly inductive.
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Learning Sheet 3
Five Resonant Parameters:
1. Resonant Angular frequency,
wo
2. Lower cut-off angular frequency,
w1
3. Upper cut-off angular frequency,
w2
4. Bandwidth of the resonant circuit,
BW
5. Quality factor of the resonant circuit,
Q
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Learning Sheet 3
Five Resonant Parameters:
1. Resonant Angular frequency, wo 
1
LC
2
2. Lower cut-off angular frequency,
R
1
 R 
w1  
   
2L
 2 L  LC
3. Upper cut-off angular frequency,
R
1
 R 
w2 
   
2L
 2 L  LC
2
4. Bandwidth of the resonant circuit,
BW 
5. Quality factor of the resonant circuit,
Q
R
L
wo L
R

1 L
R C
Note: Lower cut-off angular frequency is also popularly known as
lower half-power angular frequency. The same is true for the upper.
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Learning Sheet 3
We know that,
2
Lower cut-off angular frequency,
R
1
 R 
w1  
   
2L
 2 L  LC
Upper cut-off angular frequency,
R
1
 R 
w2 
   
2L
 2 L  LC
2
Are the half-power frequencies symmetrical around wo?
Generally No.
w o  w1w 2
The resonant frequency is the geometric
mean of the half-power frequencies.
But, If Q  10, the half-power frequencies can be approximately
considered as symmetrical around wo . Then
w1  wo 
BW
2
and
BW
w2  wo 
2
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Example: Series RLC Resonant Circuit
L
R
I
+
+
Vs
w
(varied)
VR
-
VL
+
VC
C
-
Vs = 10 Vrms, R = 10 W, L = 100 mH, C = 10 mF
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Find:
(i) The impedance of the circuit at wo
(ii) The magnitude of the current at wo
(iii) The real power P at wo
(iv) The expression for i(t) at wo
(v) The expression for vL(t) and vC(t) at wo
(vi) The impedance of the circuit at w1 in polar form
(vii) The current at w1 in polar form
(viii) The real power P at w1
(ix) The expression for i(t) at w1
(x) The expression for vC(t), vL(t) and vC(t)+vL(t) at w1
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(xi) The impedance of the circuit at w2 in polar form
(xii) The current at w2 in polar form
(xiii) The real power P at w2
(xiv) The expression for i(t) at w2
(xv) The expressions for vL(t), vC(t) and vL(t)+vC(t) at w2
(xvi) Draw the voltage phasor diagram at wo
(xvii) Draw the voltage phasor diagram at w1
(xviii) Draw the voltage phasor diagram at w2
(ixx) Draw the waveforms of vC(t), vL(t) and vC(t)+vL(t) at wo
(xx) Draw the waveforms of vC(t), vL(t) and vC(t)+vL(t) at w1
(xxi) Draw the waveforms of vL(t), vC(t) and vL(t)+vC(t) at w2
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(xxii) The resonant frequency, fo
(xxiii) The lower cut-off frequency, f1
(xxiv) The upper cut-off frequency, f2
(xxv) The bandwidth, BW in Hertz
(xxvi) The Quality factor, Q
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