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Resonant Circuits. SEE 1023 Circuit Theory. Frequency Response. L. R. I. + V L -. + V R -. + V C -. V s. C. w (varied). Series RLC Circuit. When w varies, the impedance of the circuit will vary. Then, the current and the real power will also vary. - PowerPoint PPT Presentation
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Resonant CircuitsResonant Circuits
SEE 1023 Circuit TheorySEE 1023 Circuit Theory
Frequency ResponseFrequency Response
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Series RLC Circuit
When 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.
Vs
R L
C
I
+ VR -+ VL -
+
VC
-
(varied)
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Series RLC Circuit
Impedance as a function of frequency: )1
()(C
LjRZ
Current as a function of frequency: I Vs
R2 L1
C
2
Power as a function of frequency: P Vs
2
R2 L1
C
2
R
Reactance as a function of frequency: X L1
C
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Series RLC Circuit
Excitation(Input)
Response(Output)
Series RLC Circuit
Constant input voltage: VsVariable Source angular frequency:
Main response: current
Other responses: Power, Impedance, reactance, etc.
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Series RLC Circuit in PSpice
It is too hard to study the frequency response of these quantities manually.
Vs
R L
C
I
+ VR -+ VL -
+
VC
-
(varied)
It is too easy to study the frequency response of these quantities PSpicely.
0
1
2 3
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Series RLC Circuit in PSpice
Series resonant CircuitVs 1 0 AC 10VR1 1 2 10L1 2 3 100mHC1 3 0 10uF.AC LIN 1001 100Hz 220Hz.Probe.end
Start FREQ.
End FREQ.Total PTS.To Displaygraph
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In the Probe windows
Trace Expression
M(V(1)/I(R1)) Magnitude of Z
Response
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
M(I(R1)) Magnitude of I
Response
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
V(1,2) Magnitude of VR
Response
V(2,3) Magnitude of VL
V(3) Magnitude of VC
I(R1)*I(R1)*10 Real power, P
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Run Pspice File
Frequency Response of The Current
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800 840 880 920 960 1000 1040 1080 1120 1160 12000.22
0.41
0.61
0.81
1
1.2
I ( )
(Variation of the current with frequency)
Frequency Response of The Current
At Resonance, the current is maximumAt 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
R
VII so
R
VRIPP s
oo
22
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Basic Questions
The magnitude of I?
When the power P = Po/2, what is
The magnitude of Z?
The magnitude of X?
The angular frequency?
1 lower half power frequency
2 higher half power frequency
R
VI s
21
RZ 2
RX RX
at 1
at 2
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Resonant Condition
By definition the resonant angular frequency, o, 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, 1, condition
By definition lower half-power angular frequency, 1, 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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By definition lower half-power angular frequency, 2, 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
Lower half-power angular frequency, 2, condition
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The Voltage Phasor Diagram at o
For R: I is in phase with VR
For L: I lags VL by 90
For C: I leads VC by 90
For series circuit, use I as the
reference. VR = VSI
VL
VC
at o
The circuit is purely resistive.
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The Voltage Phasor Diagram at 1
For R: I is in phase with VR
For L: I lags VL by 90
For C: I leads VC by 90
For series circuit, use I as a
reference.
VS
I
VL
VL+VC
at 1
VR
VC
The circuit is predominantly capacitive.
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The Voltage Phasor Diagram at 2
For R: I is in phase with VR
For L: I lags VL by 90
For C: I leads VC by 90
For series circuit, use I as the
reference.
VS
I
VL
VL+VC
at 2
VR
VC
The circuit is predominantly inductive.
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Learning Sheet 3
Five Resonant Parameters:
1. Resonant Angular frequency,
2. Lower cut-off angular frequency,
4. Bandwidth of the resonant circuit,
3. Upper cut-off angular frequency,
5. Quality factor of the resonant circuit,
o
1
2
BW
Q
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Learning Sheet 3
Five Resonant Parameters:
1. Resonant Angular frequency,
2. Lower cut-off angular frequency,
4. Bandwidth of the resonant circuit,
3. Upper cut-off angular frequency,
5. Quality factor of the resonant circuit,
LCo
1
LCL
R
L
R 1
22
2
1
LCL
R
L
R 1
22
2
2
L
RBW
C
L
RR
LQ o 1
Note: Lower cut-off angular frequency is also popularly known aslower half-power angular frequency. The same is true for the upper.
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Learning Sheet 3
We know that,
Lower cut-off angular frequency,
Upper cut-off angular frequency,
LCL
R
L
R 1
22
2
1
LCL
R
L
R 1
22
2
2
Are the half-power frequencies symmetrical around o?
21 o
Generally No.
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 o . Then
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BWo
22
BWo and
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Example: Series RLC Resonant Circuit
Vs = 10 Vrms, R = 10 , L = 100 mH, C = 10 F
Vs
R L
C
I
+ VR -+ VL -
+
VC
-
(varied)
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Find:
(ii) The magnitude of the current at o
(iii) The real power P at o
(iv) The expression for i(t) at o
(v) The expression for vL(t) and vC(t) at o
(i) The impedance of the circuit at o
(vii) The current at 1 in polar form
(viii) The real power P at 1
(ix) The expression for i(t) at 1
(x) The expression for vC(t), vL(t) and vC(t)+vL(t) at 1
(vi) The impedance of the circuit at 1 in polar form
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(xii) The current at 2 in polar form
(xiii) The real power P at 2
(xiv) The expression for i(t) at 2
(xv) The expressions for vL(t), vC(t) and vL(t)+vC(t) at 2
(xi) The impedance of the circuit at 2 in polar form
(xvi) Draw the voltage phasor diagram at o
(xvii) Draw the voltage phasor diagram at 1
(xviii) Draw the voltage phasor diagram at 2
(ixx) Draw the waveforms of vC(t), vL(t) and vC(t)+vL(t) at o
(xx) Draw the waveforms of vC(t), vL(t) and vC(t)+vL(t) at 1
(xxi) Draw the waveforms of vL(t), vC(t) and vL(t)+vC(t) at 2
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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
