Turn on RL Circuit (reminder)
Physics 202, Lecture 18
Today’s Topics
I=

Reminder
 RL Circuits
 RC Circuits
 LC (RLC) Circuits and Electromagnetic Oscillations

AC Circuits with AC Source
 Phasors
t
−
V0
(1− e L / R )
R
Phaser
Note: the time constant is τ=L/R
Turn off RL Circuit (reminder)
I = I0e
−
Charging a Capacitor in RC Circuit
t
L /R
ε − q(t ) / C − R dq(t ) = 0
q(t ) = εC (1 − e − t / RC )
ε
I (t ) = e −t / RC
R
Note: τ≡RC is called time constant
Note: the time constant is τ=L/R
Discharging a Capacitor in RC Circuit
Demo: RC Circuit
r
q(t ) = Qe − t / RC
Q −t / RC
I(t) = −
e
RC
q(t) /C + R
dq(t)
=0
dt
Note the time constant τ= RC
Light Bulb
discharging
dt
Charging
LC Circuit and Oscillation
 Exercise: Find the oscillation frequency of a LC circuit
dI (t )
 − q(t ) / c − L dt = 0
AC Power Source
 ΔV = ΔVmax Sin(ωt+φ0) = ΔVmax Sin(ωt)
Initial phase,
usually set φ0=0
I
d 2 q( t )
 q( t ) / C + L
=0
dt 2
1 d 2 q( t )
=0
ω 2 dt 2

1
ω=
LC
q = Qmax cos(ωt + φ )
ω: angular frequency
ω=2πf
T=2π/ω
q( t ) +
eq. of Harmonic Oscillation
t=0
Total Energy is conserved
I = −ωQmax sin(ωt + φ )
AC Circuit
Phasor
 A sinusoidal function y= Asinφ can be represented
graphically as a phasor vector with length A and
angle φ (w.r.t. to horizontal)
Asinφ
 Find out current i and voltage difference ΔVR, ΔVL, ΔV C.
i
A
φ
ΔVmax Sin(ωt)
Notes:
• Kirchhoff’s rules still apply !
• A technique called phasor analysis is convenient.
Resistors in an AC Circuit
Inductors in an AC Circuit
 ΔV - Ldi/dt =0
 Ohm’s Law: ΔV=IR at any time
i
i
Function view
iR=ΔV/R=Imax sinωt, Imax =ΔV max /R
The current through an resistor is
in phase with the voltage across it
Phasor view
Function view
 i L=Imax sin(ωt-π/2)
Imax =ΔV max /XL,
XL= ωL  inductive reactance
The current through an inductor is
90 o behind the voltage across it.
Phasor view
Capacitors in an AC Circuit
 ΔV - q/C=0, dq/dt =i
i
Function view
 i L=Imax sin(ωt+π/2)
Imax =ΔV max /XC,
X C= 1/(ωC)  capacitive reactance
The current through a capacitor is
90 o ahead of the voltage across it.
Phasor view