EE369
POWER SYSTEM ANALYSIS
Lecture 4
Power System Operation, Transmission Line
Modeling
Tom 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 ac
2) Underground ac
3) Overhead dc
4) Underground dc
The 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 ampturns) and magnetic field intensity (in ampturns/meter):
F
  H dl  I e
F = mmf = magnetomotive force (amp-turns)
H = magnetic field intensity (amp-turns/meter)
dl = Vector differential path length (meters)

= Line integral about closed path 
(dl is tangent to path)
I e = 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 the
x-axis
Integration along a
general path, which
may be closed
Ampere’s law is most useful in cases of symmetry,
such as a circular path of radius x around an infinitely
long 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.
H = magnetic field intensity (amp-turns/meter)
B = flux density (Tesla [T] or Gauss [G])
(1T = 10,000G)
For a linear magnetic material:
B =  H where  is the called the permeability
 = 0  r
0 = permeability of freespace = 4  10-7 H m
 r = relative permeability  1 for air
9
Magnetic Flux
Magnetic flux and flux density
  magnetic flux (webers)
B = flux density (webers/m 2 or tesla)
Definition of flux passing through a surface A is
 =
A B da
da = vector with direction normal to the surface
If flux density B is uniform and perpendicular to an
area A then
 = BA
10
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?
Direction of H is given
by the “Right-hand” Rule
• 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
Single Line Example, cont’d

H dl  2 xH  I  H 
I
2 x
H is perpendicular
to surface of square
2  104
2
B  0 H  0

T 
Gauss For reference,
2 x
x
x
the earth’s
5 0 I
magnetic field is
  A B dA  (1 meter)  4
dx
about 0.6 Gauss
2 x
(Central US)
I
5
5
7
  0 ln
 2  10 I ln
2 4
4
I
  4.46  105 Wb
13
Flux linkages and Faraday’s law
Flux linkages are defined from Faraday's law
d
V =
, where V = voltage,  = flux linkages
dt
The flux linkages tell how much flux is linking an
N turn coil:
 =
N
i
i=1
If flux  links every coil then   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.
I (current in a conductor)
F 
  H dl  I e (enclosed current in multiple turns)
B   H (permeability times magnetic field intensity)
 
A B dA (surface integral of flux density)
  N  (total flux linked by N turn coil)
L   / I (inductance)
16