See online explanation at

http://www.physclips.unsw.edu.au/jw/AC.html

AC Circuits

Total Energy• Total energy of the system

is the sum of the electric and magnetic field energy.

tC

QU mC 2

2

cos2

tLIU mL 2

2

sin2

CL

I

B

CL+ + + +

E

22

22mm

LC

LI

C

QUUU

RLC Circuit• After the switch is closed

Kirchoff’s rules gives

• Solution:

• Critical Damping:

CLI + + + +

+

R

02

2

C

Q

dt

dQR

dt

QdL

teQQ dLRt

m cos2/

2/12

2

1

L

R

LCd

C

LRc

4

AC Source• The voltage of the power

supply follows that of a cosine function.

• Represent the source as a vector rotating in the complex plane.

• Real component of vector is the instantaneous voltage.

tVtv m cos)(

Im

Rev

Vm

t

Vm

-Vm

t

v

AC & Resistors

• Voltage and current are in phase, = 0.

v R

tR

Vti m cos)( tVtv m cos)(

vi

Im

Rev

Vm

ti

0 Riv

FromKirchoff

Power• Power used by a resistor is

• Average power is

• Effective voltage and current are

tR

VivtP m 2

2

cos)(

22

2mmm IV

R

VP

2m

rms

VV rmsrms IVP

2m

rms

II

AC & Inductors

• Voltage leads current by 90°, = /2.

2cos)(

tL

Vti m tVtv m cos)( Im

Rev

Vm

ti

v L

v

i

0dt

diLv

FromKirchoff

LX L Reactance

AC & Capacitors

• Voltage lags current by 90°, = -/2.

2cos)( tCVti m tVtv m cos)( Im

Rev

Vm

ti

v C

01

dtiC

v

FromKirchoff

CXC

1

Reactance

v

i

LRC Circuits & AC

C

L

R

v

CLR jvjvvv Re

Im

vL

vC vR

i

• In a series circuit current is the same everywhere, so add voltages as phasors.

• Phasors are rotating vectors in the complex plane.

• Assume we are at t = 0s.

Impedance

Re

Im

vSvL

vC

vR

22CL XXRZ

iZXXjRiv CL

R

XX CL1tan

CL iXjiXjiRv

• Express voltage as current times impedance.

ELI the ICE man• At resonance, 0, XL=XC and = 0.

• For <0, the circuit is capacitive and current leads voltage. Also < 0.

• For >0, the circuit is inductive and voltage leads current. Also > 0.

R

XX CL1tanLC

10

Quality Factor• As the AC frequency approaches resonance,

0, the current increases.

• Quality factor indicates how sharp the current peak is around resonance.

0

Im

Low Q

High Q

R

LQ 0

0

Transformers• A transformer consists of two

sets of coils which share the same magnetic flux.

• AC current through the primary generates a changing magnetic flux, which generates a changing voltage on the secondary.

RLv

Primary SecondaryP

P

SS V

N

NV

PPSS IVIV

Power Lines• Ex. How much power is lost if 120V is

delivered through a 10 transmission line to a 10 load?

Compare that to transmission at 12,000 V and then stepped down to 120V at the load.

RL120V12,000V

RT

RL120V

RT

PowerTransmission

Long Distance:230 kV

To Home:240 V

Through Town:20 kV

Less resistive lossat low current.