Physics 202, Lecture 18Today’s Topics

Reminder RL Circuits RC Circuits LC (RLC) Circuits and Electromagnetic Oscillations

AC Circuits with AC Source Phasors

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