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Parallel RLC Resonant Circuit
SEE 1023 Circuit Theory
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Parallel RLC Resonant Circuit
Parallel RLC
Circuit
Excitation
(Input)
Constant input current: Is
Variable Source angular
frequency: w
Response
(Output)
Main response: Voltage
Other responses:
Power, Admittance,
susceptance, etc.
The parallel Resonant is the dual of the series resonant.
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Parallel RLC Resonant Circuit
Duality between series and parallel
Series
Parallel
Voltage, V
Current, I
Impedance, Z
Admittance, Y
Reactance, X
Susceptance, B
Resistance, R
Conductance, G
Short-circuit
Open-circuit
Capacitor, C
Inductor, L
Inductor, L
Capacitor, C
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Parallel RLC Resonant Circuit
+
IS
w
(varied)
IR
R
IC
IL
L
C
V
-
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Frequency Response of The Voltage
(Variation of the voltage with frequency)
120
100
80
V( w )
60
40
20
0
800
880
1040
960
1120
1200
w
At Resonance, the voltage is maximum
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Frequency Response of The Admittance
(Variation of the Admittance with frequency)
0.005
0.004
0.003
Y( w )
0.002
0.001
0
800
900
1000
1100
1200
w
At Resonance, the admittance is minimum
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Frequency Response of The Susceptance
(Variation of the susceptance with frequency)
0.02
w C
0.01
1
w L
0
B( w )
0
0.01
0.02
800
900
1000
1100
1200
w
At Resonance, the susceptance is zero
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Frequency Response of The Power
(Variation of the power with frequency)
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At Resonance, the power is maximum
10
8
P( w )
6
4
2
0
800
840
880
920
960 1000 1040 1080 1120 1160 1200
w
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Frequency Response of The Power
Three main points on the power plot.
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(Po, wo)
10
8
P( w )
(0.5Po, w1)
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(0.5Po, w2)
4
2
0
800
840
880
920
960 1000 1040 1080 1120 1160 1200
w
At resonance, the power is maximum
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Analysis of Parallel RLC resonant circuit
Three main points on the power plot:
Point-0:
(Po, wo)
Point-1:
(0.5Po, w1)
Point-2:
(0.5Po, w2)
wo : angular resonant frequency
w1 and w2 : half-power angular frequencies
w1 : lower half-power angular frequency
w2: upper half-power angular frequency
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The Current Phasor Diagram at wo
For R:
IR is in phase with V
For L:
IL lags V by 90
For C:
IC leads V by 90
IC
at wo
For parallel circuit, use V as the
reference because V is common
For all three elements.
IR= Is
V
The circuit is purely
resistive.
IL
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The Current Phasor Diagram at w2
For R:
IR is in phase with V
For L:
IL lags V by 90
For C:
IC leads V by 90
IC
IL+IC+IR=IS
For parallel circuit, use V as the
IL+IC
at w2
reference because V is common
For all three elements.
V
IR
IL
The circuit is predominantly capacitive.
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The Current Phasor Diagram at w1
For R:
IR is in phase with V
For L:
IL lags V by 90
For C:
IC leads V by 90
The circuit is predominantly inductive.
For parallel circuit, use V as the
IC
reference because V is common
V
For all three elements.
IR
at w1
IL+IC
IR+IL+IC = IS
IL
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Learning Sheet 3a
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 3a
Five Resonant Parameters:
1. Resonant Angular frequency, wo 
1
LC
2
2. Lower cut-off angular frequency,
1
1
 1 
w1  
 
 
2 RC
 2 RC  LC
3. Upper cut-off angular frequency,
1
1
 1 
w2 
 
 
2 RC
 2 RC  LC
2
4. Bandwidth of the resonant circuit,
BW 
1
RC
5. Quality factor of the resonant circuit, Q  wo RC  R C
L
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Example: Parallel RLC Resonant Circuit
+
IS
w
(varied)
IR
R
IC
IL
L
C
V
-
Is = 0.1 Arms, R = 1k W, L = 100 mH, C = 10 mF
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Find:
(i) The admittance of the circuit at wo
(ii) The magnitude of the voltage at wo
(iii) The real power P at wo
(iv) The expression for i(t) at wo
(v) The expression for iL(t) and iC(t) at wo
(vi) The admittance of the circuit at w1 in polar form
(vii) The voltage at w1 in polar form
(viii) The real power P at w1
(ix) The expression for v(t) at w1
(x) The expression for iC(t), iL(t) and iC(t)+iL(t) at w1
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(xi) The admittance of the circuit at w2 in polar form
(xii) The voltage at w2 in polar form
(xiii) The real power P at w2
(xiv) The expression for v(t) at w2
(xv) The expressions for iL(t), iC(t) and iL(t)+iC(t) at w2
(xvi) Draw the current phasor diagram at wo
(xvii) Draw the current phasor diagram at w1
(xviii) Draw the current phasor diagram at w2
(ixx) Draw the waveforms of iC(t), iL(t) and iC(t)+iL(t) at wo
(xx) Draw the waveforms of iC(t), iL(t) and iC(t)+iL(t) at w1
(xxi) Draw the waveforms of iL(t), iC(t) and iL(t)+iC(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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