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