Alternating Current (AC) Circuits

AC Voltages and PhasorsResistors, Inductors and Capacitors in AC CircuitsRLC CircuitsPower and ResonanceTransformers

AC SourcesAC sources have voltages and currents that vary sinusoidally.The current in a circuit driven by an AC source also oscillates at the same frequency.The standard AC frequency in the US is 60 Hz (337 rad/s).

)sin()( max tVtv ωΔ=Δ

ωπ21

==f

T

t

ΔVmax

−ΔVmax

Δv(t)

PhasorsConsider a sinusoidal voltage source: )sin()( max tVtv ωΔ=Δ

This source can be represented graphically as a vector called a phasor:

ωtT

ΔVmaxΔv(t)b

c

b

c

d

e

d

aa e

Resistors in AC Circuits

R

R

vtVvv

Δ=Δ=Δ−Δωsin

0

max

tItRV

Rvi R

R ωω sinsin maxmax =

Δ=

Δ=

RVI max

maxΔ

=

Since both have the same arguments in the sine term, iR and ΔvR are in line, they are in phase.

tRIvR ωsinmax=Δ

tIiR ωsinmax=

tRIRit R ω22max

2 sin)( ==P

tRIRit R ω22max

2 sin)( ==P

maxmax

22max

707.02

21

III

RIRI

rms

rmsav

==

=⎟⎠⎞

⎜⎝⎛=P

maxmax 707.02

VVVrms Δ=Δ

=Δ

Root Mean Square (RMS) Current and Voltage

Inductors in AC Circuits

dtdiLtV

vv L

=Δ

=Δ−Δ

ωsin

0

max

tdtL

Vdi ωsinmaxΔ=

tL

VdiiL ωω

cosmaxΔ−== ∫

⎟⎠⎞

⎜⎝⎛ −

Δ=

2sinmax πω

ωt

LViL

LXV

LVI maxmax

maxΔ

=Δ

=ω

LX L ω=

tXIv LL ωsinmax−=Δ

Inductive Reactance

For a sinusoidal voltage, the current in the inductor always lags the voltage across it by 90°.

Capacitors in AC Circuits

tVvv C ωsinmaxΔ=Δ=Δ

tVCq ωsinmaxΔ=

⎟⎠⎞

⎜⎝⎛ +Δ=

Δ==

2sin

cos

max

max

πωω

ωω

tVCi

tVCdtdqi

C

C

CXVVCI max

maxmaxΔ

=Δ=ω

CXC ω

1= Capacitive

Reactance

tXIv CC ωsinmax=Δ

For a sinusoidal voltage, the current in the capacitor always leads the voltage across it by 90°.

RLC Series CircuittVv ωsinmaxΔ=Δ

( )φω −= tIi sinmax

tVtRIv RR ωω sinsinmax Δ==Δ

tVtXIv LLL ωπω cos2

sinmax Δ=⎟⎠⎞

⎜⎝⎛ +=Δ

tVtXIv CCC ωπω cos2

sinmax Δ−=⎟⎠⎞

⎜⎝⎛ −=Δ

CLR vvvv Δ+Δ+Δ=Δ

Using Phasors in a RLC Circuit( )

( ) ( )( )22

maxmax

2maxmax

2maxmax

22max

CL

CL

CLR

XXRIV

XIXIRIV

VVVV

−+=Δ

−+=Δ

Δ−Δ+Δ=Δ

( )22

maxmax

CL XXR

VI−+

Δ=

( )22CL XXRZ −+≡ Impedance

ZIV maxmax =Δ

⎟⎠⎞

⎜⎝⎛ −

= −

RXX CL1tanφ

R = 425 ΩL = 1.25 HC = 3.5 μFω

= 377 s-1

ΔVmax = 150 V

( )( ) Ω=== − 7585.3377

111 FsC

XC μω

( )( ) Ω=== − 47125.1377 1 HsLX L ω

( ) ( ) ( ) Ω=Ω−Ω+Ω=−+= 513758471425 2222CL XXRZ

AVZVI 292.0

513150max

max =Ω

=Δ

=

°−=⎟⎠⎞

⎜⎝⎛

ΩΩ−Ω

=⎟⎠⎞

⎜⎝⎛ −

= −− 34425

758471tantan 11

RXX CLφ

( )( ) VARIVR 124425292.0max =Ω==Δ

( )( ) VAXIV LL 138471292.0max =Ω==Δ

( )( ) VAXIV CC 221758292.0max =Ω==Δ

( ) tVvC 377cos221−=Δ

( ) tVvL 377cos138=Δ

( ) tVvR 377sin124=Δ

Example 33.4

Power in an AC Circuit( ) tVtIvi ωφω sinsin maxmax Δ−=Δ=P

( )φωωφω

ωφωφω

sincossincossin

sinsincoscossin

maxmax2

maxmax

maxmax

ttVItVI

tttVI

Δ−Δ=

−Δ=

P

P

φcos21

maxmax VIav Δ=P

φcosrmsrmsav VI Δ=P

RIVvR maxmax cos =Δ=Δ φ

22max

max

maxmax RIIV

RIVI rmsrmsav =Δ

⎟⎠⎞

⎜⎝⎛ Δ=P

RIrmsav2=P

For a purely resistive load, φ=0

rmsrmsav VI Δ=P

No power loss occurs in an ideal capacitor or inductor.

Resonance in Series RLC Circuits

ZVI rms

rmsΔ

=( )22

CL

rmsrms

XXR

VI−+

Δ=

A circuit is in resonance when its current is maximum.

Irms =max when XL -XC =0

CL

XX CL

00

1ω

ω =

=

LC1

0 =ω Resonance Frequency

Power in Series RLC Circuits ( ) ( )

( )22

2

2

22

CL

rmsrmsrmsav XXR

RVRZVRI

−+Δ

=Δ

==P

( ) ( )220

22

222 1 ωω

ωωω −=⎟

⎠⎞

⎜⎝⎛ −=−

LC

LXX CL

( )( )22

02222

22

ωωω

ω

−+

Δ=

LR

RVrmsavP

( )R

Vrmsav

2

max,Δ

=P ω=ω0

RL00 ω

ωω

=Δ

=Q Quality Factor

Transformersdt

dNV BΦ−=Δ 11

dtdNV BΦ

−=Δ 22

11

22 V

NNV Δ=Δ

2211 VIVI Δ=Δ

LRVI 2

2Δ

=eqRVI 1

1Δ

=

Leq RNNR

2

2

1⎟⎟⎠

⎞⎜⎜⎝

⎛=

Power TransmissionPower is transmitted in high voltage over long distances and then gradually stepped down to 240V by the time it gets to a house.

22 kV 230 kV 240 V

P

g = 20 MWCost = 10¢/kWh

AV

WV

I g 8710230

10203

6

=××

=Δ

=P

( ) ( ) kWARI 15287 22 =Ω==P

( )( )( ) 36$1.0$2415 == hkWTotalCost

For 22kV transmission:

Total Cost = $4100

For Next ClassMidterm 2 Review on MondayMidterm 2 on TuesdayReading Assignment for Wednesday

Chapter 34 – Electromagnetic Waves

WebAssign: Assignment 11 (due Monday, 5 PM)