Chapter 31: RLC Circuits
PHY2049: Chapter 31
1
Topics
ÎLC
Oscillations
‹ Conservation
ÎDamped
‹ Energy
ÎAC
of energy
oscillations in RLC circuits
loss
current
‹ RMS
ÎForced
quantities
oscillations
‹ Resistance,
reactance, impedance
‹ Phase
shift
‹ Resonant frequency
‹ Power
ÎTransformers
‹ Impedance
matching
PHY2049: Chapter 31
2
LC Oscillations
ÎWork
out equation for LC circuit (loop rule)
q
di
− −L =0
C
dt
ÎRewrite
C
using i = dq/dt
d 2q
q
d 2q
2
+
ω
L 2 + =0 ⇒
q=0
2
C
dt
dt
‹ω
L
ω=
1
LC
(angular frequency) has dimensions of 1/t
ÎIdentical
m
d 2x
dt
2
to equation of mass on spring
+ kx = 0 ⇒
d 2x
dt
2
+ω x = 0
2
PHY2049: Chapter 31
k
ω=
m
3
LC Oscillations (2)
is same as mass on spring ⇒ oscillations
k
q = qmax cos (ω t + θ )
ω=
m
ÎSolution
‹ qmax
is the maximum charge on capacitor
‹ θ is an unknown phase (depends on initial conditions)
ÎCalculate
current: i = dq/dt
i = −ω qmax sin (ω t + θ ) = −imax sin (ω t + θ )
ÎThus
both charge and current oscillate
frequency ω, frequency f = ω/2π
‹ Period: T = 2π/ω
‹ Current and charge differ in phase by 90°
‹ Angular
PHY2049: Chapter 31
4
Plot Charge and Current vs t
q = qmax cos (ωt )
i = −imax sin (ωt )
ωt
PHY2049: Chapter 31
5
Energy Oscillations in LC Circuits
ÎTotal
energy in circuit is conserved. Let’s see why
di q
L + =0
Equation of LC circuit
dt C
di
q dq
L i+
=0
dt C dt
( )
Multiply by i = dq/dt
( )
Ld 2
1 d 2
i +
q =0
2 dt
2C dt
d ⎛ 1 2 1 q2 ⎞
⎜⎜ 2 Li + 2 ⎟⎟ = 0
dt ⎝
C⎠
dx 2
dx
Use
= 2x
dt
dt
2
q
1 Li 2 + 1
= const
2
2 C
UL + UC = const
PHY2049: Chapter 31
6
Oscillation of Energies
ÎEnergies
can be written as (using ω2 = 1/LC)
2
q 2 qmax
UC =
=
cos 2 (ω t + θ )
2C
2C
2
q
2
U L = 12 Li 2 = 12 Lω 2 qmax
sin 2 (ω t + θ ) = max sin 2 (ω t + θ )
2C
2
qmax
ÎConservation of energy: U C + U L =
= const
2C
ÎEnergy
oscillates between capacitor and inductor
‹ Endless
oscillation between electrical and magnetic energy
‹ Just like oscillation between potential energy and kinetic energy
for mass on spring
PHY2049: Chapter 31
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UC (t )
Plot Energies vs t
U L (t )
PHY2049: Chapter 31
Sum
8
LC Circuit Example
ÎParameters
‹C
= 20μF
‹ L = 200 mH
‹ Capacitor initially charged to 40V, no current initially
ÎCalculate
‹ω
ω, f and T
= 500 rad/s
‹ f = ω/2π = 79.6 Hz
‹ T = 1/f = 0.0126 sec
ÎCalculate
ω = 1/ LC = 1/
( 2 ×10−5 ) ( 0.2 ) = 500
qmax and imax
= CV = 800 μC = 8 × 10-4 C
= ωqmax = 500 × 8 × 10-4 = 0.4 A
‹ qmax
‹ imax
ÎCalculate
‹ UC
maximum energies
= q2max/2C = 0.016J UL = Li2max/2 = 0.016J
PHY2049: Chapter 31
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LC Circuit Example (2)
ÎCharge
and current
q = 0.0008cos ( 500t )
dq
i=
= −0.4sin ( 500t )
dt
ÎEnergies
U C = 0.016cos 2 ( 500t )
U L = 0.016sin 2 ( 500t )
ÎVoltages
VC = q / C = 40cos ( 500t )
VL = Ldi / dt = − Lω imax cos ( 500t ) = −40cos ( 500t )
ÎNote
how voltages sum to zero, as they must!
PHY2049: Chapter 31
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Quiz
ÎBelow
are shown 3 LC circuits. Which one takes the least
time to fully discharge the capacitors during the
oscillations?
‹ (1)
A
‹ (2) B
‹ (3) C
C
A
ω = 1/ LC
C
C
C
B
C
C
C has smallest capacitance, therefore highest
frequency, therefore shortest period
PHY2049: Chapter 31
11
RLC Circuit
ÎThe
loop rule tells us
di
q
L + Ri + = 0
dt
C
ÎUse
i = dq/dt, divide by L
d 2q
R dq q
+
+
=0
2
L dt LC
dt
ÎSolution
q = qmax e
ÎThis
slightly more complicated than LC case
−tR / 2 L
cos (ω ′t + θ ) ω ′ = 1/ LC − ( R / 2 L )
2
is a damped oscillator (similar to mechanical case)
‹ Amplitude
of oscillations falls exponentially
PHY2049: Chapter 31
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Charge and Current vs t in RLC Circuit
q (t )
i (t )
e
PHY2049: Chapter 31
−tR / 2 L
13
RLC Circuit Example
ÎCircuit
‹L
parameters
= 12mL, C = 1.6μF, R = 1.5Ω
ÎCalculate
‹ω
ω, ω’, f and T
= 7220 rad/s
‹ ω’ = 7220 rad/s
‹ f = ω/2π = 1150 Hz
‹ T = 1/f = 0.00087 sec
ÎTime
ω = 1/
( 0.012 ) (1.6 ×10−6 ) = 7220
ω ′ = 72202 − (1.5/ 0.024 )
2
ω
for qmax to fall to ½ its initial value e−tR / 2 L = 1/ 2
‹t
= (2L/R) * ln2 = 0.0111s = 11.1 ms
‹ # periods = 0.0111/.00087 ≈ 13
PHY2049: Chapter 31
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RLC Circuit (Energy)
di
q
L + Ri + = 0
dt
C
Basic RLC equation
di
q dq
2
L i + Ri +
=0
dt
C dt
Multiply by i = dq/dt
d ⎛ 1 2 1 q2 ⎞
2
Li
+
=
−
i
R
⎜⎜ 2
⎟⎟
2
dt ⎝
C⎠
Collect terms
(similar to LC circuit)
d
(U L + U C ) = −i 2 R
dt
Total energy in circuit
decreases at rate of i2R
(dissipation of energy)
U tot ∼ e −tR / L
PHY2049: Chapter 31
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Energy in RLC Circuit
UC (t )
U L (t )
Sum
e
−tR / L
PHY2049: Chapter 31
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