Physics 202, Lecture 18Today’s Topics
Reminder RL Circuits RC Circuits LC (RLC) Circuits and Electromagnetic Oscillations
AC Circuits with AC Source Phasors
Phaser
Turn on RL Circuit (reminder)
Note: the time constant is τ=L/R
�
I = V0R(1− e
− tL /R )
Turn off RL Circuit (reminder)
Note: the time constant is τ=L/R
�
I = I0e− tL /R
Charging a Capacitor in RC Circuit
Charging
)1()( /RCteCtq −−= ε RCteR
tI /)( −= ε
Note: τ≡RC is called time constant
0)(/)( =−−dttdqRCtqε
Discharging a Capacitor in RC Circuit
discharging
RCtQetq /)( −=
�
I(t) = − QRC
e−t /RC
Note the time constant τ=RC
�
q(t) /C + R dq(t)dt
= 0
Demo: RC Circuit
Light Bulb
r
LC Circuit and Oscillation Exercise: Find the oscillation frequency of a LC circuit
0)(/)( =−−dttdILctq
)sin()cos(
1
0)(1)(
max
max
2
2
2
φωωφω
ω
ω
+−=+=
=
=+
tQItQq
LC
dttqdtq
0)(/)( 2
2
=+dttqdLCtq
eq. of Harmonic Oscillation
Total Energy is conserved
I
AC Power Source ΔV = ΔVmax Sin(ωt+φ0) = ΔVmax Sin(ωt)
Initial phase, usually set φ0=0
T=2π/ω
ω: angular frequencyω=2πf
t=0
AC Circuit Find out current i and voltage difference ΔVR, ΔVL, ΔVC.
Notes: • Kirchhoff’s rules still apply !• A technique called phasor analysis is convenient.
i
ΔVmax Sin(ωt)
Phasor A sinusoidal function y= Asinφ can be represented
graphically as a phasor vector with length A andangle φ (w.r.t. to horizontal)
φ
A
Asi
nφ
Resistors in an AC Circuit Ohm’s Law: ΔV=IR at any time
i
iR=ΔV/R=Imax sinωt, Imax=ΔVmax/R
The current through an resistor is in phase with the voltage across it Phasor view
Function view
Inductors in an AC Circuit ΔV - Ldi/dt =0
iL=Imax sin(ωt-π/2)Imax=ΔVmax/XL, XL= ωL inductive reactanceThe current through an inductor is 90o behind the voltage across it.
Phasor view
Function viewi
Capacitors in an AC Circuit
ΔV - q/C=0, dq/dt =i
i
iL=Imax sin(ωt+π/2)Imax=ΔVmax/XC, XC= 1/(ωC) capacitive reactanceThe current through a capacitor is 90o ahead of the voltage across it. Phasor view
Function view
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