EE101: Resonance in RLC circuits
M. B. Patil
mbpatil@ee.iitb.ac.in
www.ee.iitb.ac.in/~sequel
Department of Electrical Engineering
Indian Institute of Technology Bombay
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
I
VR
Vm 6 0
VL
VC
Vm ∠ 0
Vm
=
≡ Im ∠ θ , where
R + jωL + 1/jωC
R + j(ωL − 1/ωC )
–
»
Vm
ωL − 1/ωC
.
= p
, θ = − tan−1
R
R 2 + (ωL − 1/ωC )2
I=
Im
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
I
VR
Vm 6 0
VL
VC
Vm ∠ 0
Vm
=
≡ Im ∠ θ , where
R + jωL + 1/jωC
R + j(ωL − 1/ωC )
–
»
Vm
ωL − 1/ωC
.
= p
, θ = − tan−1
R
R 2 + (ωL − 1/ωC )2
I=
Im
* As ω is varied, both Im and θ change.
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
I
VR
Vm 6 0
VL
VC
Vm ∠ 0
Vm
=
≡ Im ∠ θ , where
R + jωL + 1/jωC
R + j(ωL − 1/ωC )
–
»
Vm
ωL − 1/ωC
.
= p
, θ = − tan−1
R
R 2 + (ωL − 1/ωC )2
I=
Im
* As ω is varied, both Im and θ change.
* When ωL = 1/ωC , Im reaches its maximum value, Immax = Vm /R, and
θ becomes 0, i.e., the current I is in phase with the applied voltage.
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
I
VR
Vm 6 0
VL
VC
Vm ∠ 0
Vm
=
≡ Im ∠ θ , where
R + jωL + 1/jωC
R + j(ωL − 1/ωC )
–
»
Vm
ωL − 1/ωC
.
= p
, θ = − tan−1
R
R 2 + (ωL − 1/ωC )2
I=
Im
* As ω is varied, both Im and θ change.
* When ωL = 1/ωC , Im reaches its maximum value, Immax = Vm /R, and
θ becomes 0, i.e., the current I is in phase with the applied voltage.
* The above condition is called “resonance,” and the corresponding frequency is
called the “resonance frequency” (ω0 ).
√
ω0 = 1/ LC
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
VR
VL
I
Vm 6 0
VC
»
–
Vm
ωL − 1/ωC
Im = p
, θ = − tan−1
.
R
R 2 + (ωL − 1/ωC )2
Resonance in series RLC circuits
f0
VR
VL
Vm 6 0
VC
Im (A)
0.1
I
0
102
R = 10 Ω
L = 1 mH
C = 1 µF
103
104
Frequency (Hz)
105
»
–
Vm
ωL − 1/ωC
Im = p
, θ = − tan−1
.
R
R 2 + (ωL − 1/ωC )2
* As ω deviates from ω0 , Im decreases.
Resonance in series RLC circuits
f0
VR
VL
Vm 6 0
VC
Im (A)
0.1
I
0
102
R = 10 Ω
L = 1 mH
C = 1 µF
103
104
Frequency (Hz)
105
»
–
Vm
ωL − 1/ωC
Im = p
, θ = − tan−1
.
R
R 2 + (ωL − 1/ωC )2
* As ω deviates from ω0 , Im decreases.
* As ω → 0, the term 1/ωC dominates, and θ → π/2.
Resonance in series RLC circuits
f0
90
VC
Im (A)
0.1
I
Vm 6 0
f0
VL
0
102
R = 10 Ω
L = 1 mH
C = 1 µF
103
104
Frequency (Hz)
θ (degrees)
VR
105
0
−90
102
103
104
Frequency (Hz)
105
»
–
Vm
ωL − 1/ωC
Im = p
, θ = − tan−1
.
R
R 2 + (ωL − 1/ωC )2
* As ω deviates from ω0 , Im decreases.
* As ω → 0, the term 1/ωC dominates, and θ → π/2.
* As ω → ∞, the term ωL dominates, and θ → −π/2.
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
f0
90
VC
Im (A)
0.1
I
Vm 6 0
f0
VL
0
102
R = 10 Ω
L = 1 mH
C = 1 µF
103
104
Frequency (Hz)
θ (degrees)
VR
105
0
−90
102
103
104
Frequency (Hz)
105
»
–
Vm
ωL − 1/ωC
Im = p
, θ = − tan−1
.
R
R 2 + (ωL − 1/ωC )2
* As ω deviates from ω0 , Im decreases.
* As ω → 0, the term 1/ωC dominates, and θ → π/2.
* As ω → ∞, the term ωL dominates, and θ → −π/2.
(SEQUEL file: ee101 reso rlc 1.sqproj)
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
VR
VL
I
Vm 6 0
VC
Imax
m
√
Imax
m / 2
0
ω1
ω0
ω2
ω
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
VR
VL
I
Vm 6 0
VC
Imax
m
√
Imax
m / 2
0
ω1
ω0
ω2
ω
* The maximum power that can be absorbed by the resistor is
1
1
Pmax = (Immax )2 R = Vm2 /R.
2
2
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
VR
VL
I
Vm 6 0
VC
Imax
m
√
Imax
m / 2
0
ω1
ω0
ω2
ω
* The maximum power that can be absorbed by the resistor is
1
1
Pmax = (Immax )2 R = Vm2 /R.
2
2
√
* Define ω1 and ω2 (see figure) as frequencies at which Im = Immax / 2, i.e., the
power absorbed by R is Pmax /2.
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
VR
VL
I
Vm 6 0
VC
Imax
m
√
Imax
m / 2
0
ω1
ω0
ω2
ω
* The maximum power that can be absorbed by the resistor is
1
1
Pmax = (Immax )2 R = Vm2 /R.
2
2
√
* Define ω1 and ω2 (see figure) as frequencies at which Im = Immax / 2, i.e., the
power absorbed by R is Pmax /2.
* The bandwidth of a resonant circuit is defined as B = ω2 − ω1 , and the quality
factor as Q = ω0 /B. Quality is a measure of the sharpness of the Im versus
frequency relationship.
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
Vm
Im = p
.
R 2 + (ωL − 1/ωC )2
For ω = ω , Im = I max = Vm /R .
0
Imax
m
√
2
Imax
/
m
m
√
For ω = ω1 or ω = ω2 , Im = Immax / 2 .
0
ω1
ω0
ω2
ω
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
Vm
Im = p
.
R 2 + (ωL − 1/ωC )2
For ω = ω , Im = I max = Vm /R .
0
m
√
For ω = ω1 or ω = ω2 , Im = Immax / 2 .
1
⇒ √
2
Imax
m
√
2
Imax
/
m
0
ω1
„
Vm
R
«
Vm
= p
2
R + (ωL − 1/ωC )2
ω0
ω2
ω
for ω = ω1,2 .
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
Vm
Im = p
.
R 2 + (ωL − 1/ωC )2
For ω = ω , Im = I max = Vm /R .
0
m
√
For ω = ω1 or ω = ω2 , Im = Immax / 2 .
1
⇒ √
2
Imax
m
√
2
Imax
/
m
0
ω1
„
Vm
R
«
Vm
= p
2
R + (ωL − 1/ωC )2
ω0
ω2
ω
for ω = ω1,2 .
2 R 2 = R 2 + (ωL − 1/ωC )2 → R = ±(ωL − 1/ωC ) .
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
Vm
Im = p
.
R 2 + (ωL − 1/ωC )2
For ω = ω , Im = I max = Vm /R .
0
m
√
For ω = ω1 or ω = ω2 , Im = Immax / 2 .
1
⇒ √
2
Imax
m
√
2
Imax
/
m
0
ω1
„
Vm
R
«
Vm
= p
2
R + (ωL − 1/ωC )2
ω0
ω2
ω
for ω = ω1,2 .
2 R 2 = R 2 + (ωL − 1/ωC )2 → R = ±(ωL − 1/ωC ) .
Solving for ω (and discarding negative solutions), we get
s„ «
R 2
R
1
ω1,2 = ∓
+
+
.
2L
2L
LC
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
Vm
Im = p
.
R 2 + (ωL − 1/ωC )2
For ω = ω , Im = I max = Vm /R .
0
m
√
For ω = ω1 or ω = ω2 , Im = Immax / 2 .
1
⇒ √
2
Imax
m
√
2
Imax
/
m
0
ω1
„
Vm
R
«
Vm
= p
2
R + (ωL − 1/ωC )2
ω0
ω2
ω
for ω = ω1,2 .
2 R 2 = R 2 + (ωL − 1/ωC )2 → R = ±(ωL − 1/ωC ) .
Solving for ω (and discarding negative solutions), we get
s„ «
R 2
R
1
ω1,2 = ∓
+
+
.
2L
2L
LC
* Bandwidth B = ω2 − ω1 = R/L .
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
Vm
Im = p
.
R 2 + (ωL − 1/ωC )2
For ω = ω , Im = I max = Vm /R .
0
m
√
For ω = ω1 or ω = ω2 , Im = Immax / 2 .
1
⇒ √
2
Imax
m
√
2
Imax
/
m
0
ω1
„
Vm
R
«
Vm
= p
2
R + (ωL − 1/ωC )2
ω0
ω2
ω
for ω = ω1,2 .
2 R 2 = R 2 + (ωL − 1/ωC )2 → R = ±(ωL − 1/ωC ) .
Solving for ω (and discarding negative solutions), we get
s„ «
R 2
R
1
ω1,2 = ∓
+
+
.
2L
2L
LC
* Bandwidth B = ω2 − ω1 = R/L .
* Quality Q = ω0 /B = ω0 L/R .
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
Vm
Im = p
.
R 2 + (ωL − 1/ωC )2
For ω = ω , Im = I max = Vm /R .
0
m
√
For ω = ω1 or ω = ω2 , Im = Immax / 2 .
1
⇒ √
2
Imax
m
√
2
Imax
/
m
0
ω1
„
Vm
R
«
Vm
= p
2
R + (ωL − 1/ωC )2
ω0
ω2
ω
for ω = ω1,2 .
2 R 2 = R 2 + (ωL − 1/ωC )2 → R = ±(ωL − 1/ωC ) .
Solving for ω (and discarding negative solutions), we get
s„ «
R 2
R
1
ω1,2 = ∓
+
+
.
2L
2L
LC
* Bandwidth B = ω2 − ω1 = R/L .
* Quality Q = ω0 /B = ω0 L/R .
* Show that, at resonance (i.e., ω = ω0 ), |VL | = |VC | = Q Vm .
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
Vm
Im = p
.
R 2 + (ωL − 1/ωC )2
For ω = ω , Im = I max = Vm /R .
0
m
√
For ω = ω1 or ω = ω2 , Im = Immax / 2 .
1
⇒ √
2
Imax
m
√
2
Imax
/
m
0
ω1
„
Vm
R
«
Vm
= p
2
R + (ωL − 1/ωC )2
ω0
ω2
ω
for ω = ω1,2 .
2 R 2 = R 2 + (ωL − 1/ωC )2 → R = ±(ωL − 1/ωC ) .
Solving for ω (and discarding negative solutions), we get
s„ «
R 2
R
1
ω1,2 = ∓
+
+
.
2L
2L
LC
* Bandwidth B = ω2 − ω1 = R/L .
* Quality Q = ω0 /B = ω0 L/R .
* Show that, at resonance (i.e., ω = ω0 ), |VL | = |VC | = Q Vm .
√
* Show that ω0 = ω1 ω2 .
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
90
VL
0.1
Vm 6 0
VC
Im (A)
I = Im 6 θ
0
102
L = 1 mH
C = 1 µF
θ (degrees)
VR
R = 10 Ω
R = 20 Ω
103
104
Frequency (Hz)
105
R = 10 Ω
R = 20 Ω
0
−90
102
103
104
Frequency (Hz)
105
As R is increased,
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
90
VL
0.1
Vm 6 0
VC
Im (A)
I = Im 6 θ
0
102
L = 1 mH
C = 1 µF
θ (degrees)
VR
R = 10 Ω
R = 20 Ω
103
104
Frequency (Hz)
105
R = 10 Ω
R = 20 Ω
0
−90
102
103
104
Frequency (Hz)
105
As R is increased,
* The quality factor Q = ω0 L/R decreases, i.e., Im versus ω curve becomes
broader.
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
90
VL
0.1
Vm 6 0
VC
Im (A)
I = Im 6 θ
0
102
L = 1 mH
C = 1 µF
θ (degrees)
VR
R = 10 Ω
R = 20 Ω
103
104
Frequency (Hz)
105
R = 10 Ω
R = 20 Ω
0
−90
102
103
104
Frequency (Hz)
105
As R is increased,
* The quality factor Q = ω0 L/R decreases, i.e., Im versus ω curve becomes
broader.
* The maximum current (at ω = ω0 ) decreases (since Immax = Vm /R).
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
90
VL
0.1
Vm 6 0
VC
Im (A)
I = Im 6 θ
0
102
L = 1 mH
C = 1 µF
θ (degrees)
VR
R = 10 Ω
R = 20 Ω
103
104
Frequency (Hz)
105
R = 10 Ω
R = 20 Ω
0
−90
102
103
104
Frequency (Hz)
105
As R is increased,
* The quality factor Q = ω0 L/R decreases, i.e., Im versus ω curve becomes
broader.
* The maximum current (at ω = ω0 ) decreases (since Immax = Vm /R).
√
* The resonance frequency (ω0 = 1/ LC ) is not affected.
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
I
VR
Vm 6 0
VL
VC
Vm ∠ 0
Vm
=
≡ Im ∠ θ , where
R + jωL + 1/jωC
R + j(ωL − 1/ωC )
–
»
Vm
ωL − 1/ωC
.
= p
, θ = − tan−1
R
R 2 + (ωL − 1/ωC )2
I=
Im
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
I
VR
Vm 6 0
VL
VC
Vm ∠ 0
Vm
=
≡ Im ∠ θ , where
R + jωL + 1/jωC
R + j(ωL − 1/ωC )
–
»
Vm
ωL − 1/ωC
.
= p
, θ = − tan−1
R
R 2 + (ωL − 1/ωC )2
I=
Im
* For ω < ω0 , ωL < 1/ωC , the net impedance is capacitive, and the current leads
the applied voltage.
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
I
VR
Vm 6 0
VL
VC
Vm ∠ 0
Vm
=
≡ Im ∠ θ , where
R + jωL + 1/jωC
R + j(ωL − 1/ωC )
–
»
Vm
ωL − 1/ωC
.
= p
, θ = − tan−1
R
R 2 + (ωL − 1/ωC )2
I=
Im
* For ω < ω0 , ωL < 1/ωC , the net impedance is capacitive, and the current leads
the applied voltage.
* For ω = ω0 , ωL = 1/ωC , the net impedance is purely resistive, and the current
is in phase with the applied voltage.
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
I
VR
Vm 6 0
VL
VC
Vm ∠ 0
Vm
=
≡ Im ∠ θ , where
R + jωL + 1/jωC
R + j(ωL − 1/ωC )
–
»
Vm
ωL − 1/ωC
.
= p
, θ = − tan−1
R
R 2 + (ωL − 1/ωC )2
I=
Im
* For ω < ω0 , ωL < 1/ωC , the net impedance is capacitive, and the current leads
the applied voltage.
* For ω = ω0 , ωL = 1/ωC , the net impedance is purely resistive, and the current
is in phase with the applied voltage.
* For ω > ω0 , ωL > 1/ωC , the net impedance is inductive, and the current lags
the applied voltage.
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
I
VR
Vm 6 0
VL
VC
Vm ∠ 0
Vm
=
≡ Im ∠ θ , where
R + jωL + 1/jωC
R + j(ωL − 1/ωC )
–
»
Vm
ωL − 1/ωC
.
= p
, θ = − tan−1
R
R 2 + (ωL − 1/ωC )2
I=
Im
* For ω < ω0 , ωL < 1/ωC , the net impedance is capacitive, and the current leads
the applied voltage.
* For ω = ω0 , ωL = 1/ωC , the net impedance is purely resistive, and the current
is in phase with the applied voltage.
* For ω > ω0 , ωL > 1/ωC , the net impedance is inductive, and the current lags
the applied voltage.
* Let us look at an example (next slide).
M. B. Patil, IIT Bombay
Resonance in series RLC circuits
1
0.1
0
0
−1
i
Vs
R = 10 Ω
L = 1 mH
f =4.3 kHz
−0.1
1
0.1
0
0
−1
f =5 kHz ≃ f0
−0.1
1
0.1
0
0
f =5.9 kHz
C = 1 µF
Vs (V) (left axis)
−1
−0.1
0
100
200
Time (µsec)
300
i (A) (right axis)
400
M. B. Patil, IIT Bombay
Resonance in series RLC circuits: phasor diagrams
VR
VL
I
VC
Vs
R = 10 Ω
L = 1 mH
C = 1 µF
4
VL
3
2
VL
VL
1
Im(V)
VR
0
Vs
Vs
Vs , VR
VL
−1
VC
−2
VC
VR
VR
−3
−4
VR
VL
VC
f = 4.3 kHz
−1
0
1
Re(V)
2
f = f0 ≃ 5 kHz
−1
0
1
Re(V)
2
f = 5.9 kHz
−1
0
1
Re(V)
2
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits
IR
Im 6
0
IL
IC
V
Im ∠ 0 = Y V, where Y = G + jωC + 1/jωL (G = 1/R) .
Im ∠ 0
Im
V=
=
≡ Vm ∠ θ , where
G + jωC + 1/jωL
G + j(ωC − 1/ωL)
–
»
Im
ωC − 1/ωL
Vm = p
, θ = − tan−1
.
G
G 2 + (ωC − 1/ωL)2
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits
IR
Im 6
0
IL
IC
V
Im ∠ 0 = Y V, where Y = G + jωC + 1/jωL (G = 1/R) .
Im ∠ 0
Im
V=
=
≡ Vm ∠ θ , where
G + jωC + 1/jωL
G + j(ωC − 1/ωL)
–
»
Im
ωC − 1/ωL
Vm = p
, θ = − tan−1
.
G
G 2 + (ωC − 1/ωL)2
* As ω is varied, both Vm and θ change.
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits
IR
Im 6
0
IL
IC
V
Im ∠ 0 = Y V, where Y = G + jωC + 1/jωL (G = 1/R) .
Im ∠ 0
Im
V=
=
≡ Vm ∠ θ , where
G + jωC + 1/jωL
G + j(ωC − 1/ωL)
–
»
Im
ωC − 1/ωL
Vm = p
, θ = − tan−1
.
G
G 2 + (ωC − 1/ωL)2
* As ω is varied, both Vm and θ change.
* When ωC = 1/ωL, Vm reaches its maximum value, Vmmax = Im /G = Im R, and
θ becomes 0, i.e., the voltage V is in phase with the source current.
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits
IR
Im 6
0
IL
IC
V
Im ∠ 0 = Y V, where Y = G + jωC + 1/jωL (G = 1/R) .
Im ∠ 0
Im
V=
=
≡ Vm ∠ θ , where
G + jωC + 1/jωL
G + j(ωC − 1/ωL)
–
»
Im
ωC − 1/ωL
Vm = p
, θ = − tan−1
.
G
G 2 + (ωC − 1/ωL)2
* As ω is varied, both Vm and θ change.
* When ωC = 1/ωL, Vm reaches its maximum value, Vmmax = Im /G = Im R, and
θ becomes 0, i.e., the voltage V is in phase with the source current.
* The above condition is called “resonance,” and the corresponding frequency is
called the “resonance frequency” (ω0 ).
√
ω0 = 1/ LC
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits
–
ωL − 1/ωC
.
R
R 2 + (ωL − 1/ωC )2
»
–
ωC − 1/ωL
Im
, θ = − tan−1
.
= p
G
G 2 + (ωC − 1/ωL)2
Series RLC circuit: Im = p
Parallel RLC circuit: Vm
Vm
, θ = − tan−1
»
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits
–
ωL − 1/ωC
.
R
R 2 + (ωL − 1/ωC )2
»
–
ωC − 1/ωL
Im
, θ = − tan−1
.
= p
G
G 2 + (ωC − 1/ωL)2
Series RLC circuit: Im = p
Parallel RLC circuit: Vm
Vm
, θ = − tan−1
»
* The two situations are identical if we make the following substitutions:
I ↔ V,
R ↔ 1/R,
L ↔ C.
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits
–
ωL − 1/ωC
.
R
R 2 + (ωL − 1/ωC )2
»
–
ωC − 1/ωL
Im
, θ = − tan−1
.
= p
G
G 2 + (ωC − 1/ωL)2
Series RLC circuit: Im = p
Parallel RLC circuit: Vm
Vm
, θ = − tan−1
»
* The two situations are identical if we make the following substitutions:
I ↔ V,
R ↔ 1/R,
L ↔ C.
* Thus, our results for series RLC circuits can be easily extended to parallel RLC
circuits.
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits
–
ωL − 1/ωC
.
R
R 2 + (ωL − 1/ωC )2
»
–
ωC − 1/ωL
Im
, θ = − tan−1
.
= p
G
G 2 + (ωC − 1/ωL)2
Series RLC circuit: Im = p
Parallel RLC circuit: Vm
Vm
, θ = − tan−1
»
* The two situations are identical if we make the following substitutions:
I ↔ V,
R ↔ 1/R,
L ↔ C.
* Thus, our results for series RLC circuits can be easily extended to parallel RLC
circuits.
s„
«2
1
1
1
* Show that ω1,2 = ∓
+
+
2RC
2RC
LC
⇒ Bandwidth B = 1/RC .
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits
–
ωL − 1/ωC
.
R
R 2 + (ωL − 1/ωC )2
»
–
ωC − 1/ωL
Im
, θ = − tan−1
.
= p
G
G 2 + (ωC − 1/ωL)2
Series RLC circuit: Im = p
Parallel RLC circuit: Vm
Vm
, θ = − tan−1
»
* The two situations are identical if we make the following substitutions:
I ↔ V,
R ↔ 1/R,
L ↔ C.
* Thus, our results for series RLC circuits can be easily extended to parallel RLC
circuits.
s„
«2
1
1
1
* Show that ω1,2 = ∓
+
+
2RC
2RC
LC
⇒ Bandwidth B = 1/RC .
* Show that, at resonance (i.e., ω = ω0 ), |IL | = |IC | = Q Im .
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits
–
ωL − 1/ωC
.
R
R 2 + (ωL − 1/ωC )2
»
–
ωC − 1/ωL
Im
, θ = − tan−1
.
= p
G
G 2 + (ωC − 1/ωL)2
Series RLC circuit: Im = p
Parallel RLC circuit: Vm
Vm
, θ = − tan−1
»
* The two situations are identical if we make the following substitutions:
I ↔ V,
R ↔ 1/R,
L ↔ C.
* Thus, our results for series RLC circuits can be easily extended to parallel RLC
circuits.
s„
«2
1
1
1
* Show that ω1,2 = ∓
+
+
2RC
2RC
LC
⇒ Bandwidth B = 1/RC .
* Show that, at resonance (i.e., ω = ω0 ), |IL | = |IC | = Q Im .
√
ω1 ω2 .
* Show that ω0 =
M. B. Patil, IIT Bombay
Resonance in parallel RLC circuits: home work
IR
IL
Im 6 0
IC
Im = 50 mA
V
R = 2 kΩ
L = 40 mH
C = 0.25 µF
* Calculate ω0 , f0 , B, Q.
* Calculate IR , IL , IC at ω = ω0 , ω1 , ω2 .
* Verify graphically that IR + IL + IC = Is in each case.
* Plot the power absorbed by R as a function of frequency for f0 /10 < f < 10 f0 .
M. B. Patil, IIT Bombay