AC Voltages and Phasors Resistors, Inductors and Capacitors in

AC Circuits RLC Circuits Power and Resonance Transformers

AC 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

21

fT

t

Vmax

-Vmax

v(t)

Consider a sinusoidal voltage source: )sin()( max tVtv

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

t T

Vmax v(t)

b

c

b

c

d

e

d

a a e

R

R

vtV

vv

-

sin

0

max

tItR

V

R

vi RR sinsin max

max

R

VI max

max

Since both have the

same arguments in

the sine term, iR and

vR are in line, they

are in phase.

tRIvR sinmax

tIiR sinmax

tRIRit R 22

max

2sin)( P

tRIRit R 22

max

2sin)( P

maxmax

22

max

707.02

2

1

II

I

RIRI

rms

rmsav

P

maxmax 707.02

VV

Vrms

dt

diLtV

vv L

-

sin

0

max

tdtL

Vdi sinmax

tL

VdiiL

cosmax

-

-

2sinmax

tL

ViL

LX

V

L

VI maxmax

max

LX L

tXIv LL sinmax-

Inductive

Reactance

For a sinusoidal voltage,

the current in the inductor

always lags the voltage

across it by 90°.

tVvv C sinmax

tVCq sinmax

2sin

cos

max

max

tVCi

tVCdt

dqi

C

C

CX

VVCI max

maxmax

CXC

1

Capacitive

Reactance

tXIv CC sinmax

For a sinusoidal voltage,

the current in the

capacitor always leads

the voltage across it by

90°.

tVv sinmax

- tIi sinmax

tVtRIv RR sinsinmax

tVtXIv LLL

cos2

sinmax

tVtXIv CCC

cos2

sinmax -

-

CLR vvvv

22

maxmax

2

maxmax

2

maxmax

22

max

CL

CL

CLR

XXRIV

XIXIRIV

VVVV

-

-

-

22

maxmax

CL XXR

VI

-

22

CL XXRZ - Impedance

ZIV maxmax

- -

R

XX CL1tan

R = 425 W

L = 1.25 H

C = 3.5 mF

= 377 s-1

Vmax = 150 V

W

-758

5.3377

111 FsC

XCm

W - 47125.1377 1 HsLX L

WW-WW- 5137584714252222

CL XXRZ

AV

Z

VI 292.0

513

150maxmax

W

-

W

W-W

- -- 34

425

758471tantan 11

R

XX CL

VARIVR 124425292.0max W

VAXIV LL 138471292.0max W

VAXIV CC 221758292.0max W

tVvC 377cos221-

tVvL 377cos138

tVvR 377sin124

tVtIvi sinsin maxmax -P

sincossincossin

sinsincoscossin

maxmax

2

maxmax

maxmax

ttVItVI

tttVI

-

-

P

P

cos2

1maxmax VIav P

cosrmsrmsav VI P

RIVvR maxmax cos

22

max

max

maxmax RII

V

RIVI rmsrmsav

P

RIrmsav

2P

For a purely resistive load, =0

rmsrmsav VI P

No power loss occurs in

an ideal capacitor or

inductor.

Z

VI rms

rms

22

CL

rmsrms

XXR

VI

-

A circuit is in resonance when its current is maximum.

Irms=max when XL-XC=0

CL

XX CL

0

0

1

LC

10 Resonance Frequency

22

2

2

22

CL

rmsrmsrmsav

XXR

RVR

Z

VRI

-

P

22

0

2

2

22

2 1

-

--

L

CLXX CL

22

0

2222

22

-

LR

RVrmsavP

R

Vrmsav

2

max,

P =0

R

L00

Q Quality Factor

dt

dNV B

- 11

dt

dNV B

- 22

1

1

22 V

N

NV

2211 VIVI

LR

VI 2

2

eqR

VI 1

1

Leq RN

NR

2

2

1

Power 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 MW

Cost = 10¢/kWh

AV

W

VI

g87

10230

10203

6

P

kWARI 1528722 WP

36$1.0$2415 hkWTotalCost

For 22kV transmission:

Total Cost = $4100

Midterm 2 Review on Tuesday Midterm 2 on Wednesday Reading Assignment for Thursday

Chapter 34 – Electromagnetic Waves

WebAssign: Assignment 11 (due Tuesday, 5 PM)