Theme 5

Alternating Current and Voltage

Alternating Voltage

• Acts in alternate directions periodically. • Alternating voltage generated usually by a

rotating machine called an alternator – in a sinusoidal manner.

• Otherwise Alternating voltages can be of rectangular type, triangular……

• Main interest is of a sinusoidal waveform- utility supply

• Can be easily stepped up or down using a transformer.

• Numerous devices in industry use alternatin voltage/current to operate. E.g. an induction motor.

)sin()( max tVtV

Vmax

-Vmax

time

2π0.5π π 1.5 π

Tf

1

T

fT

22

• Mathematically, a sinusoid that’s integrated or differentiated- results in a sinusoid of the same frequency:

• Summing different sinusoids of the same frequency and different amplitudes results in a sinusoid of the same frequency.

• A cosine wave is just a sine wave with shifted phase

• Phase shifted sinusoidal waveforms:• B Lags A by angle θ• Or, A Leads B by angle θ

θ

AB

)sin( tVA A

)sin( tVB B

• Phase shifted sinusoidal waveforms:• B Leads A by angle θ• Or, A Lags B by angle θ

θ

AB

)sin( tVA A

)sin( tVB B

Average Value of a Sinusoid Wave

t1=0 π=t2

tVtf m sin)(

• True Average value=0• Finite average value for half the sinusoid wave

can be found= average value of a sinusoid

Average Value of a Sinusoid Wave

mm

t

t

av

VtdtV

dttftt

F

2)(sin

1

)(1

0

12

2

1

t1=0 π=t2

tVtf m sin)(

RMS Value of a Sinusoidal Waveform

• If a resistor is connected across a sinusoidal voltage source, a sinusoidal current will flow in the resistor.

• The RMS value/effective Value is the current that produces the same heating effect as a direct current flowing…i.e.

RiP 2

• Average power dissipated by the resistor over a time T is;

T

T

av

dtiT

R

dtRiT

P

0

2

0

2

1

1

T

RMS

T

RMS

dttfT

F

OR

dtiT

I

0

2

0

2

)(1

1

• FRMS is the RMS of any function f(t)

Obtaining IRMS for a sinusoidal current waveform

t1=0 t2=2π

tItf m sin)( I

time

2

0

22

2

0

2

0

2

)(sin(2

1

)sin((2

1

)(1

;

2T Period

)sin()(

dttII

dttII

dttfT

I

Thus

tItf

mRMS

mRMS

T

RMS

m

t1=0 t2=2π

I

time

)sin()( tItf m

22

2

2sin

4

)()2cos1(4

2

2

0

2

2

0

2

mmRMS

mRMS

mRMS

III

tt

II

tdtI

I

• Similarly, VRMS is;

22

2mm

RMS

VVV

Crest/Peak Factor for a sinusoidal waveform

2

22

Factor max m

m

m

m

RMSVV

II

F

FCrest

Form Factor of a Sinusoid

11.122

222

2

Factor orm

m

m

m

m

RMS

av

V

VF

F

F

FF

OPERATOR j

• Alternating current or voltage is a vector quantity

• But instantaneous values are constantly changing with time

• Thus it can be represented by a ‘rotating’ phasor- which rotates are a constant angular velocity

t1=0 t2=2π

I

time

Phasor RotationAngular momentum=ώt

j

tjmeI

r

Im

Im

Im

t 0 deg

j-operator

901j

90 degrees

270 degrees

180 degrees 0 degrees 18012j

27013j

0136014j

• An operator which turns a phasor by 90 degrees

Polar and Rectangular form

tVeV mtj

m sin VeV j

sincos

sincos

VjV

VV

je j

Polar Form

Rectangular Form

Phasor Algebra

• Algebraic operations same as complex number manipulations

jdcA

jbaA

2

1

)()(21 bcadjbdacAA

22222

1 )()(

dc

adbcj

dc

bdac

A

A

Rectangular Form

Converting from rectangular to polar

a

b

baA

AjbaA

11

221

111

tan

)(

Polar Algebraic Manipulations

222

111

AA

AA

)(

)(

212

1

2

1

212121

A

A

A

A

AAAA

Assignment

• Example 7.7• Example 7.8• Problems 7.3, 7.4 ,• 7.7,• 7.8,• 7.19,• 7.20 (page 210-212)