Physics 202, Lecture 17
Component
Today’s Topics
Symbol
Behavior in circuit
Ideal battery, emf
ΔV=V+-V- =ε
Resistor (R)
  Review LR Circuits (last lecture)
  Review RC Circuits (Ch. 28.4)
Realistic Battery
  LC and LRC Circuits
  Electromagnetic Oscillations
  AC Circuits with AC Source

ΔV= -IR
ε r
(Ideal) wire
Capacitor (C)
ΔV=0 (R=0, L=0, C=0)
ΔV=V- - V+ = - q/C, dq/dt =I
Inductor (L)
(Ideal) Switch
ΔV= - LdI/dt
L=0, C=0, R=0 (on), R=∞ (off)
Transformer
Diodes,
Transistors,…
LR Circuit: Time Constant
switch on
Basic Circuit Components
Future Topics
Charging a Capacitor in RC Circuit
switch off
time constant τ
Time Constant of the LR circuit: τ=L/R
= RC
Charging
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Discharging a Capacitor in RC Circuit
Another look at the currents
In a capacitor:
charging: current I decreases
discharging: current in opposite direction, |I| decreases
For t=∞ , Either ways, I vanishes !
In an inductor:
switching on: current I increases
switching off: current I decreases, same direction
In a resistor:
switching on and off: current I instantaneous, steady.
Again the time constant τ=RC
discharging
LC Circuit and Oscillation
 Find the oscillation frequency of a LC circuit
LC Circuit and Oscillation: Potential View
 Find the potential across C , L
ΔVC

I

ΔVL
eq. of Harmonic Oscillation

 Across C:
ΔVC= -Q/C=-(Qmax/C) cos(ωt+φ)
 Across L:
ΔVL =-LdI/dt = Lω2Qmax cos(ωt+φ)
=+(Qmax/C) cos(ωt+φ) =-(Qmax/C) cos(ωt+φ+180ο)
 | ΔVC| = |ΔVC| but with an 180o phase difference.
This is an oscillation between L and C.
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LC Circuit and Oscillation: Energy View
 Find the energies stored in C, L
  in C: UC= ½ Q2/C = ½ (Qmax2/C) cos2(ωt+φ)
  in L: UL= ½ LI2 = ½ Lω2 Qmax2 sin2(ωt+φ)
= ½ (Qmax2/C) sin2(ωt+φ)
 Total energy: U= UL+UC = ½ (Qmax2/C) = constant!
No energy is lost in a pure LC oscillator.
LC Oscillation with Resistor: LRC Circuit
 Oscillation frequency of a LRC circuit

Solution: Q(t) = Qmax e –Rt/2L cosω t

U=UC+UL
Next: What about adding an R?
LRC Circuit: Damped Oscillation
 C: overdamped “oscillation”
ω’: effective frequency with damping
eq. of Harmonic Oscillation
with damping
AC Power Source
  ΔV = ΔVmax Sin(ωt+φ0) = ΔVmax Sin(ωt)
Initial phase,
usually set φ0=0
ω: angular frequency
ω=2πf
T=2π/ω
 B: critically damped oscillation
t=0
 A: underdamped oscillation
recall: T=1/f
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AC Circuit
i
ΔVmax Sin(ωt)
  A sinusoidal function x= 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, ΔVC.
Phasor
A
φ
•  Kirchhoff’s rules still apply  solving differential eqs!
•  A technique called phasor analysis is convenient.
 Solution will be given in next lecture
Please preview Ch 33 before next lecture
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