LR Circuit: Time Constant
Physics 202, Lecture 17
turning on
Today’s Topics
„
„
Review LR Circuits
RC Circuits (Ch. 28.4)
„
LC (LRC) Circuits and Electromagnetic Oscillations
„
I=
AC Circuits with AC Source
turning off
t
−
V0
(1− e L / R )
R
I = I 0e
−
t
L /R
Time Constant of the LR circuit: τ=L/R
Charging a Capacitor in RC Circuit
Discharging a Capacitor in RC Circuit
q(t ) = Qe − t / RC
ε − q(t ) / C − R dq(t ) = 0
q(t ) = εC (1 − e − t / RC )
ε
I (t ) = e −t / RC
dt
Q −t / RC
I(t) = −
e
RC
q(t) /C + R
dq(t)
=0
dt
R
time constant τ
= RC
Charging
Again the time constant τ=RC
discharging
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Demo: RC Circuit
LC Circuit and Oscillation
‰ Exercise: Find the oscillation frequency of a LC circuit
dI (t )
Æ − q(t ) / C − L dt
r
Æ q(t ) / C + L
=0
d 2 q( t )
=0
dt 2
1 d 2 q(t )
=0
ω 2 dt 2
Î
1
ω≡
LC
q = Qmax cos(ωt + φ )
q( t ) +
Light Bulb
I
eq. of Harmonic Oscillation
I = −ωQmax sin(ωt + φ )
LC Circuit and Oscillation: Potential View
‰ Exercise: Find the potential across C , L
ΔVC
Æ Across C:
ΔVC= -Q/C=-(Qmax/C) cos(ωt+φ)
ΔVL
Æ Across L:
ΔVL =-LdI/dt = Lω2Qmax cos(ωt+φ)
=+(Qmax/C) cos(ωt+φ) =-(Qmax/C) cos(ωt+φ+180ο)
Î | ΔVC| = |ΔVC| but with an 180o phase difference.
Next: What about the total energy stored?
LC Circuit and Oscillation: Energy View
‰ Exercise: Find the energies stored in C, L
Æ in C: UC= ½ Q2/C = ½ (Qmax2/C) cos2(ωt+φ)
Æ in L: UL= ½ LI2 = ½ Lω2 Qmax2 sin2(ωt+φ)
= ½ (Qmax2/C) sin2(ωt+φ)
ÎTotal energy: U
U= UL+U
UC = ½ (Qmax2/C) = constant!
No energy is lost in a pure LC oscilator.
U=UC+UL
Next: What about adding an R?
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LC Oscillation with Resistor: LRC Circuit
LRC Circuit: Damped Oscillation
‰ C: overdamped “oscillation”
‰ Exercise: Find the oscillation frequency of a LC circuit
Æ − q(t ) / c − RI (t ) − L
dI (t )
=0
dt
2
Æ q( t ) / C + R dq(t ) L d q(t ) = 0
2
dt
dt
dq(t ) 1 d 2 q(t )
+ 2
=0
dt
ω dt 2
R
1
, ω '2 = ω02 − ( ) 2 < ω 02
2L
LC
q(t ) + RC
Î
ω0 =
ω’: effective frequency with damping
‰ B: critically damped oscillation
‰ A: underdamped oscillation
eq. of Harmonic Oscillation
with damping
AC Power Source
‰ Find out current i and voltage difference ΔVR, ΔVL, ΔVC.
‰ ΔV = ΔVmax Sin(ωt+φ0) = ΔVmax Sin(ωt)
Initial phase,
usually set φ0=0
AC Circuit
ω: angular frequency
ω=2πf
T=2π/ω
i
ΔVmax Sin(ωt)
t=0
recall: T=1/f
• Kirchhoff’s rules still apply !
• A technique called phasor analysis is convenient.
Î Solution will be given in next lecture
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Phasor
Asinφ
‰ A sinusoidal function x= Asinφ can be represented
graphically as a phasor vector with length A and
angle φ (w.r.t. to horizontal)
φ
Please preview Ch 33 before next lecture
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