EE 369 POWER SYSTEM ANALYSIS Lecture 18 Fault Analysis Tom Overbye and Ross Baldick 1.
EE369 POWER SYSTEM ANALYSIS Lecture 4 Power System Operation, Transmission Line Modeling Tom Overbye...
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Transcript of EE369 POWER SYSTEM ANALYSIS Lecture 4 Power System Operation, Transmission Line Modeling Tom Overbye...
EE369POWER SYSTEM ANALYSIS
Lecture 4Power System Operation, Transmission Line
ModelingTom Overbye and Ross Baldick
1
Reading and Homework• For lectures 4 through 6 read Chapter 4
– We will not be covering sections 4.7, 4.11, and 4.12 in detail,– We will return to chapter 3 later.
• HW 3 is Problems 2.43, 2.45, 2.46, 2.47, 2.49, 2.50, 2.51, 2.52, 4.2, 4.3, 4.5, 4.7 and Chapter 4 case study questions A through D; due Thursday 9/17.
• HW 4 is 2.31, 2.41, 2.48, 4.8, 4.10, 4.12, 4.13, 4.15, 4.19, 4.20, 4.22, due Thursday 9/24.
• Mid-term I is Thursday, October 1, covering up to and including material in HW 4.
2
Development of Line Models
• Goals of this section are:
1) develop a simple model for transmission lines, and
2) gain an intuitive feel for how the geometry of the transmission line affects the model parameters.
3
Primary Methods for Power Transfer
The most common methods for transfer of electric power are:
1) Overhead ac2) Underground ac3) Overhead dc4) Underground dcThe analysis will be developed for ac lines.
4
Magnetics Review
Magnetomotive force: symbol F, measured in ampere-turns, which is the current enclosed by a closed path,
Magnetic field intensity: symbol H, measured in ampere-turns/meter:– The existence of a current in a wire gives rise to an
associated magnetic field. – The stronger the current, the more intense is the
magnetic field H.Flux density: symbol B, measured in webers/m2
or teslas or gauss (1 Wb /m2 = 1T = 10,000G):– Magnetic field intensity is associated with a magnetic
flux density.5
Magnetics Review
Magnetic flux: symbol measured in webers, which is the integral of flux density over a surface.
Flux linkages measured in weber-turns.– If the magnetic flux is varying (due to a changing
current) then a voltage will be induced in a conductor that depends on how much magnetic flux is enclosed (“linked”) by the loops of the conductor, according to Faraday’s law.
Inductance: symbol L, measured in henrys:– The ratio of flux linkages to the current in a coil.
,
,
6
Magnetics Review• Ampere’s circuital law relates magnetomotive
force (the enclosed current in amps or amp-turns) and magnetic field intensity (in amp-turns/meter):
d
= mmf = magnetomotive force (amp-turns)
= magnetic field intensity (amp-turns/meter)
d = Vector differential path length (meters)
= Line integral about closed path (d is tangent to path)
e
e
F I
F
I
H l
H
l
l
= Algebraic sum of current linked by 7
Line Integrals•Line integrals are a generalization of “standard” integration along, for example, the x-axis.
Integration along thex-axis
Integration along ageneral path, whichmay be closed
Ampere’s law is most useful in cases of symmetry, such as a circular path of radius x around an infinitelylong wire, so that H and dl are parallel, |H|= H is constant,and |dl| integrates to equal the circumference 2πx.
8
Flux Density•Assuming no permanent magnetism, magnetic field intensity and flux density are related by the permeability of the medium.
0
0
= magnetic field intensity (amp-turns/meter)
= flux density (Tesla [T] or Gauss [G])(1T = 10,000G)
For a linear magnetic material:
= where is the called the permeability
=
= permeability of freesr
H
B
B H
-7pace = 4 10 H m
= relative permeability 1 for airr
9
Magnetic Flux
2
Magnetic flux and flux density
magnetic flux (webers)
= flux density (webers/m or tesla)
Definition of flux passing through a surface is
=
= vector with direction normal to the surface
If flux
A
A
d
d
B
B a
a
density B is uniform and perpendicular to an area A then
= BA10
Magnetic Fields from Single Wire
• Assume we have an infinitely long wire with current of I =1000A.
• Consider a square, located between 4 and 5 meters from the wire and such that the square and the wire are in the same plane.
• How much magnetic flux passes through the square?
11
Magnetic Fields from Single Wire• Magnetic flux passing through the square?
• Easiest way to solve the problem is to take advantage of symmetry.
• As an integration path, we’ll choose a circle with radius x, with x varying from 4 to 5 meters, with the wire at the center, so the path encloses the current I.
12
Direction of H is givenby the “Right-hand” Rule
Single Line Example, cont’d
4
0 0
5 04
70
5
22
2 10 2T Gauss
2
(1 meter)2
5 5ln 2 10 ln
2 4 4
4.46 10 Wb
A
Id xH I H
x
IB H
x x xI
dA dxx
II
H l
B
For reference, the earth’s
magnetic field is about 0.6 Gauss
(Central US)
13
H is perpendicularto surface of square
Flux linkages and Faraday’s law
i=1
Flux linkages are defined from Faraday's law
d= , where = voltage, = flux linkages
dThe flux linkages tell how much flux is linking an
turn coil:
=
If flux links every coil then
N
i
V Vt
N
N
14
Inductance
• For a linear magnetic system; that is, one where B = H,
• we can define the inductance, L, to be the constant of proportionality relating the current and the flux linkage: = L I,
• where L has units of Henrys (H).
15
Summary of magnetics.
16
d (enclosed current in multiple turns)
(permeability times magnetic field intensity)
(surface integral of flux density)
(total flux li
(c
nked by tur
urrent in a conductor)
e
A
F I
dA
I
N N
H l
B H
B
n coil)
/ (inductance)L I