This document discusses power system operation and transmission line modeling. It provides an overview of the goals of developing simple transmission line models and gaining intuition about how line geometry affects the model parameters. It also reviews relevant magnetic concepts like magnetomotive force, magnetic field intensity, flux density, flux, inductance, and Faraday's law. Homework assignments and exam dates are provided.
2. 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
3. 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
4. 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
5. 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
6. 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.
,φ
,λ
6
7. 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
Γ
Γ
= =
Γ
∫
∫
H l
H
l
l
gÑ
Ñ
= Algebraic sum of current linked by Γ 7
8. 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
9. 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 frees
r
µ µ
µ µ µ
µ
H
B
B H
-7
pace = 4 10 H m
= relative permeability 1 for airr
π
µ
×
≈ 9
10. 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
φ
φ
=
∫
B
B a
a
g
density B is uniform and perpendicular to an
area A then
= BAφ
10
11. 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
12. 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 given
by the “Right-hand” Rule
13. Single Line Example, cont’d
4
0 0
5 0
4
7
0
5
2
2
2 10 2
T Gauss
2
(1 meter)
2
5 5
ln 2 10 ln
2 4 4
4.46 10 Wb
A
I
d xH I H
x
I
B H
x x x
I
dA dx
x
I
I
π
π
µ µ
π
µ
φ
π
φ µ
π
φ
Γ
−
−
−
= = ⇒ =
×
= = = =
= = ×
= = ×
= ×
∫
∫ ∫
H l
B
g
g
Ñ
For reference,
the earth’s
magnetic field is
about 0.6 Gauss
(Central US)
13
H is perpendicular
to surface of square
14. Flux linkages and Faraday’s law
i=1
Flux linkages are defined from Faraday's law
d
= , where = voltage, = flux linkages
d
The flux linkages tell how much flux is linking an
turn coil:
=
If flux links every coil then
N
i
V V
t
N
N
λ
λ
λ φ
φ λ φ=
∑
14
15. Inductance
• For a linear magnetic system; that is, one
where B = µ H,
• we can define the inductance, L, to be the
constant of proportionality relating the
current and the flux linkage: λ = L I,
• where L has units of Henrys (H).
15
16. 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
µ
φ
λ φ
Γ
= =
=
=
=
∫
∫
H l
B H
B
g
g
Ñ
n coil)
/ (inductance)L Iλ=
