![Page 1: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/1.jpg)
EE 372Fundamentals of Power SystemsTextbook:
John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994.
Objective: To teach the fundamental concepts of electric power system engineering.
![Page 2: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/2.jpg)
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
![Page 3: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/3.jpg)
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
![Page 4: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/4.jpg)
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)
![Page 5: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/5.jpg)
Sinusoidal Signals
THREE-PHASE SYNCHRONOUS GENERATOR
![Page 6: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/6.jpg)
How do you write the mathematical equation for this periodic function?
? ?
Sinusoidal Signals
![Page 7: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/7.jpg)
)1002cos(200)( tt Period : 0.01 s.Frequency : 100 Hz.
Sinusoidal Signals
![Page 8: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/8.jpg)
100)502sin(200)( tt Period : 0.02 s.Frequency : 50 Hz.
100)2
502cos(200)( tt
OR
Sinusoidal Signals
![Page 9: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/9.jpg)
)502sin(100)( tt )502cos(100)( tt
)2
502cos(100)( tt )
2502sin(100)(
tt
)2
cos()sin( )
2sin()cos(
Sinusoidal Signals
![Page 10: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/10.jpg)
?? -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
![Page 11: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/11.jpg)
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:
![Page 12: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/12.jpg)
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
![Page 13: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/13.jpg)
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.
![Page 14: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/14.jpg)
Derivative:
jdt
d
Phasors
tjtjjm
tjjm
tj ejeeVjeeVdt
de
dt
d VV
Consider the derivative of sinusoidal signal represented as a phasor
![Page 15: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/15.jpg)
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
![Page 16: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/16.jpg)
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.
![Page 17: EE 372 Fundamentals of Power Systems Textbook: John J. Grainger, William D. Stevenson, Jr., “Power System Analysis”, McGraw-Hill, Inc., 1994. Objective:](https://reader033.vdocuments.site/reader033/viewer/2022061504/56649e4d5503460f94b43066/html5/thumbnails/17.jpg)
0 .2 0 .1 0 .1 0 .2second
20
10
10
20
v(t)i(t)
Phasors