1043

CHAPTER 31 SUMMARY

Phasors and alternating current: An alternator or acsource produces an emf that varies sinusoidally withtime. A sinusoidal voltage or current can be representedby a phasor, a vector that rotates counterclockwise withconstant angular velocity equal to the angular fre-quency of the sinusoidal quantity. Its projection on thehorizontal axis at any instant represents the instanta-neous value of the quantity.

For a sinusoidal current, the rectified average andrms (root-mean-square) currents are proportional to thecurrent amplitude Similarly, the rms value of a sinu-soidal voltage is proportional to the voltage amplitude

(See Example 31.1.)V.

I.

v

(31.3)

(31.4)

(31.5)Vrms =V

12

Irms =I

12

Irav =2p

I = 0.637I

Voltage, current, and phase angle: In general, the instan-taneous voltage between two points in an ac circuit isnot in phase with the instantaneous current passingthrough those points. The quantity is called the phaseangle of the voltage relative to the current.

f

(31.2)

v = Vcos1vt + f2i = Icosvt

Resistance and reactance: The voltage across a resistoris in phase with the current. The voltage across an

inductor leads the current by whilethe voltage across a capacitor lags the current by

The voltage amplitude across eachtype of device is proportional to the current amplitude An inductor has inductive reactance and acapacitor has capacitive reactance (SeeExamples 31.2 and 31.3.)

XC = 1>vC.XL = vL,

I.90° 1f = -90°2.

C1f = +90°2,90°L

R

(31.7)

(31.13)

(31.19)VC = IXC

VL = IXL

VR = IR

Impedance and the L-R-C series circuit: In a general accircuit, the voltage and current amplitudes are related bythe circuit impedance In an L-R-C series circuit, thevalues of L, R, C, and the angular frequency determinethe impedance and the phase angle of the voltage rela-tive to the current. (See Examples 31.4 and 31.5.)

f

v

Z.

(31.22)

(31.23)

(31.24) tanf =vL - 1>vC

R

= 2R2 + 3vL - 11>vC242Z = 2R2 + 1XL - XC22V = IZ

Power in ac circuits: The average power input to anac circuit depends on the voltage and current amplitudes(or, equivalently, their rms values) and the phase angle of the voltage relative to the current. The quantity iscalled the power factor. (See Examples 31.6 and 31.7.)

cosff

Pav (31.31)

= VrmsIrms cosf

Pav = 12 VI cosf

Resonance in ac circuits: In an L-R-C series circuit, thecurrent becomes maximum and the impedance becomesminimum at an angular frequency called the resonanceangular frequency. This phenomenon is called reso-nance. At resonance the voltage and current are inphase, and the impedance is equal to the resistance (See Example 31.8.)

R.Z

(31.32)v0 =1

2LC

O

I

i 5 I cos vt

vt

v

V

I

V cos f

O

f

vt

aC

bi

q �q

i

a R b

i

a L b

i

V 5 IZ

I

VR 5 IR

VC 5 IXC

VL 5 IXL

VL 2 VC

O

fvt

I (A)

2000 V

500 V

200 V0.50.40.30.20.1

v (rad/s)1000 2000O

v, i, pPav 5 VI cos f

t

v

i

p

fv

12