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)
ÎSolution
is same as mass on spring ⇒ oscillations
q = qmax cos (ω t + θ )
ω=
1
LC
‹ qmax
is the maximum charge on capacitor
‹ θ is an unknown phase (depends on initial conditions)
ÎCharge
oscillation
‹ Angular
frequency ω, frequency f = ω/2π
‹ Period: T
ÎExample:
= 1/ f = 2π / ω = 2π LC
L = 2.0 H, C = 0.5 μF
(
)
ω = 1/ 2.0 × 0.5 × 10−6 = 1/ 10−6 = 103 rad/s
f = ω / 2π = 1000 / 6.283 = 159 Hz
T = 1/ f = 0.0063 sec
PHY2049: Chapter 31
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LC Oscillations (3)
ÎCalculate
current: i = dq/dt q = qmax cos (ω t + θ )
i = −ω qmax sin (ωt + θ ) ≡ −imax sin (ωt + θ )
ÎThus
both charge and current oscillate
angular frequency ω, frequency f = ω/2π
‹ Period: T = 2π / ω = 2π LC
‹ Same
ÎCurrent
and charge differ in phase by 90°
i = −imax sin (ωt + θ ) = imax cos (ωt + θ + π / 2 )
‹ More
on that later!
PHY2049: Chapter 31
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Plot Charge and Current vs t
q = qmax cos (ωt )
i = −imax sin (ωt )
ωt
PHY2049: Chapter 31
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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⎠
UL
1 dx 2
dx
Use
=x
dt
2 dt
2
q
1 Li + 1
= const
2
2 C
2
UL + UC = const
UC
PHY2049: Chapter 31
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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
9
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 )
ÎEnergies
Utot
dq
i=
= −0.4sin ( 500t )
dt
= q2max/2C = 0.016J
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 )
ÎSo
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
‹ (4) All are same
C
C
A
ω = 1/ LCtot
C
B
C
C
C
C has smallest capacitance, therefore highest
frequency, therefore shortest period
PHY2049: Chapter 31
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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
14
RLC Circuit Example
ÎCircuit
‹L
parameters
= 12mH, C = 1.6μF, R = 1.5Ω
ω, ω’, f and T
‹ ω = 7220 rad/s
ω = 1/
ÎCalculate
‹ ω’
= 7220 rad/s
‹ f = ω/2π = 1150 Hz
‹ T = 1/f = 0.00087 sec
ÎTime
( 0.012 ) (1.6 ×10−6 ) = 7220
ω ′ = 72202 − (1.5 / 0.024 ) = 7219.7 ω
2
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
‹ So every 13 periods the charge amplitude falls by a factor of 2
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
Energy time constant = L/R
PHY2049: Chapter 31
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AC Circuits with RLC Components
ÎEnormous
impact of AC circuits
‹ Power
delivery
‹ Radio transmitters and receivers
‹ Tuners
‹ Filters
‹ Transformers
ÎBasic
components
‹R
‹L
‹C
‹ Driving
ÎNow
emf
we will study the basic principles
PHY2049: Chapter 31
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AC Circuits and Forced Oscillations
+ “driving” EMF with angular frequency ωd
ε = ε m sin ω d t
ÎRLC
L
di
q
+ Ri + = ε m sin ω d t
dt
C
ÎGeneral
solution for current is sum of two terms
“Transient”: Falls
exponentially & disappears
“Steady state”:
Constant amplitude
Ignore
i ∼ e−tR / 2 L cos ω ′t
PHY2049: Chapter 31
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