ee 458 electrical distribution systems books: sallam, a. a., “electric distribution systems (ieee...
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EE 458Electrical Distribution Systems
Books: Sallam, A. A., “Electric Distribution Systems (IEEE Press Series on Power Engineering)”, Wiley-IEEE Press, First Edition, 2011. Gönen, T., “Electric Power Distribution System Engineering”, CRC Press, Second Edition, 2011. Fehr, R. E., “Industrial Power Distribution”, Prentice-Hall, Inc., 2002. Beaty, W., “Electric Power Distribution Systems: A Nontechnical Guide”, Penn Well Pub. Company, 1998
Power: Instantaneous consumption of energy Power Units
Watts = voltage x current for dc (W)
kW – 1 x 103 Watt
MW – 1 x 106 Watt
GW – 1 x 109 Watt
Basics
Energy: Amount of Work Energy Units (for electrical power)
Wh -- 1 x 100 Watthour
kWh – 1 x 103 Watthour
MWh – 1 x 106 Watthour
GWh – 1 x 109 Watthour Relationship of power and energy
t
WP
Average Power
Energy Consumed
Duration
Basics
Sinusoidal Signals
Circular rotation of a magnetized rotor in Synchronous Generator produces sinusoidal voltage in stator windings due to FARADAY LAW. (Look at EE 471 Notes)
Sinusoidal Signals
THREE-PHASE SYNCHRONOUS GENERATOR
How do you write the mathematical equation for this periodic function?
? ?
Sinusoidal Signals
)1002cos(200)( tt Period : 0.01 s.Frequency : 100 Hz.
Sinusoidal Signals
100)502sin(200)( tt Period : 0.02 s.Frequency : 50 Hz.
100)2
502cos(200)( tt
OR
Sinusoidal Signals
)502sin(100)( tt )502cos(100)( tt
)2
502cos(100)( tt )
2502sin(100)(
tt
)2
cos()sin( )
2sin()cos(
Sinusoidal Signals
?? -400
-300
-200
-100
0
100
200
300
400
-0.0
200
-0.0
175
-0.0
150
-0.0
125
-0.0
100
-0.0
075
-0.0
050
-0.0
025
0.00
00
0.00
25
0.00
50
0.00
75
0.01
00
0.01
25
0.01
50
0.01
75
0.02
00
Time (seconds)
Vo
lts
, A
mp
ere
sCurrent
Voltage
)314sin(310)2sin()( ttfVt m Peak voltage : 310 V. Period : 0.02 s.Frequency : 50 Hz.
Peak current : 150 A. Period : 0.02 s.Frequency : 50 Hz.
)355.2314sin(150)2sin()( ttfIti m
0135
radian
Sinusoidal Signals
Complex Numbers
)sin()cos( RjReRyjxz j
Euler’s Formula : Relates exponential and sinusoidal functions
22 yxzR
x
yarctan
R
Re
ImRectangular Notation
Polar Notation R
jz 1
jz 1045
4)1arctan(
045
000 13518045
Attention:
Complex Numbers
Addition and subtraction of complex numbers are easier with the rectangular notation.
)()()()( dbjcadjcbja
)().()).(( BABA
Multiplication and division of complex numbers are easier with the polar notation.
)(
B
A
B
A
Attention:
Rectangular Polar
Phasor representation of a sinusoidal function:
Phasors
Phasor
)cos()( tVt m mj
m VeVV
If we multiply phasor V tje by and apply Euler’s formula
)sin()cos()( tVjtVeVeeVe mmtj
mtjj
mtjV
tjet Ve)(
Phasors are complex numbers used to represent sinusoids.
Derivative:
jdt
d
Phasors
tjtjjm
tjjm
tj ejeeVjeeVdt
de
dt
d VV
Consider the derivative of sinusoidal signal represented as a phasor
Phasors
Examples:
)()(
tdt
tidL )(
)(ti
dt
tdC
tjtj
edt
edL
VI
tjtj eeLj VI
tjtj
edt
edC
IV
tjtj eeCj IV
Inductor Capacitor
VI Lj IV Cj
Phasors
)(t
)(ti
)502cos(311)( tt )5
502cos(141)( tti
V2202
311rmsV A100
2
141rmsI
V00220V A36100 0I
Important: In power systems, RMS values are used for the magnitudes.
036
V
I
Ref.
0 .2 0 .1 0 .1 0 .2second
20
10
10
20
v(t)i(t)
Phasors