1. EE 369 POWER SYSTEM ANALYSIS
Lecture 5
Development of Transmission Line Models
Tom Overbye and Ross Baldick

2. Reading For lectures 5 through 7 read Chapter 4 Read Section 1.5,
we will not be covering sections 4.7, 4.11, and 4.12 in detail
Read Section 1.5,
HW 4 is chapter 4 case study questions A through D, and Problems 2.31, 2.41, 2.43, 2.48, 4.1, 4.3, 4.6, due Thursday 9/26.
HW 5 is Problems 4.9, 4.11, 4.13, 4.18, 4.21, 4.22, 4.24, 4.25 (assume Cardinal conductor and look up GMR in Table A.4); due Thursday 10/2.

3. Substation Bus

4. Inductance Example
Calculate the inductance of an N turn coil wound tightly on a toroidal iron core that has a radius of R and a cross-sectional area of A. Assume
1) all flux is within the coil
2) all flux links each turn
3) Radius of each turn is negligible compared to R
Circular path Γ of radius R within the iron core encloses all N turns of the coil and hence links total enclosed current of Ie = NI.
Since the radius of each turn is negligible compared to R, all circular paths within the iron core have radius approximately equal to R.

5. Inductance Example, cont’d

6. Inductance of a Single Wire
To develop models of transmission lines, we first need to determine the inductance of a single, infinitely long wire. To do this we need to determine the wire’s total flux linkage, including:
1. flux linkages outside of the wire
2. flux linkages within the wire
We’ll assume that the current density within the wire is uniform and that the wire is solid with a radius of r.
In fact, current density is non-uniform, and conductor is stranded, so our calculations will be approximate.

7. Flux Linkages outside of the wire

8. Flux Linkages outside, cont’d

9. Flux linkages inside of wire

10. Flux linkages inside, cont’d
Wire cross section x r

11. Line Total Flux & Inductance

12. Inductance Simplification

13. Two Conductor Line Inductance
Key problem with the previous derivation is we assumed no return path for the current. Now consider the case of two wires, each carrying the same current I, but in opposite directions; assume the wires are separated by distance D. D
To determine the
inductance of each
conductor we integrate
as before. However
now we get some
field cancellation.
Creates counter-
clockwise field
Creates a
clockwise field

14. Two Conductor Case, cont’d
Direction of integration D R
Key Point: Flux linkage due to currents in each conductor tend
to cancel out. Use superposition to get total flux linkage.
Left Current
Right Current

15. Two Conductor Inductance

16. Many-Conductor Case Now assume we now have n conductors,
with the k-th conductor having current ik, and
arranged in some specified geometry.
We’d like to find flux linkages of each conductor.
Each conductor’s flux
linkage, lk, depends upon
its own current and the
current in all the other
conductors.
For example, to derive the flux linkage for conductor 1, l1, we’ll be integrating from
conductor 1 (at origin) to the right along the x-axis.

17. Many-Conductor Case, cont’d
Rk is the
distance
from con-
ductor k
to point c.
We’d like to integrate the flux crossing between b to c. But the flux crossing between a and c is easier to calculate and provides a very good approximation of l1k. Point a is at distance d1k from conductor k.
At point b the net
contribution to l1
from ik , l1k, is zero.

18. Many-Conductor Case, cont’d

19. Many-Conductor Case, cont’d

20. Symmetric Line Spacing – 69 kV

21. Line Inductance Example
Calculate the reactance for a balanced 3f, 60Hz
transmission line with a conductor geometry of an
equilateral triangle with D = 5m, r = 1.24cm (Rook conductor) and a length of 5 miles.

22. Line Inductance Example, cont’d

23. Line Inductance Example, cont’d

24. Conductor Bundling
To increase the capacity of high voltage transmission
lines it is very common to use a number of
conductors per phase. This is known as conductor
bundling. Typical values are two conductors for
345 kV lines, three for 500 kV and four for 765 kV.

25. Bundled Conductor Flux Linkages
For the line shown on the left,
define dij as the distance between
conductors i and j.
We can then determine lk for conductor k.
Assuming ¼ of the phase current flows
in each of the four conductors in
a given phase bundle, then for conductor 1:

26. Bundled Conductors, cont’d

27. Bundled Conductors, cont’d

28. Inductance of Bundle

29. Inductance of Bundle, cont’d

30. Bundle Inductance Example
Consider the previous example of the three phases
symmetrically spaced 5 meters apart using wire
with a radius of r = 1.24 cm. Except now assume
each phase has 4 conductors in a square bundle,
spaced 0.25 meters apart. What is the new inductance
per meter?
0.25 M

31. Transmission Tower Configurations
The problem with the line analysis we’ve done so far is we have assumed a symmetrical tower configuration.
Such a tower configuration is seldom practical.
Therefore in
general Dab 
Dac  Dbc
Unless something
was done this would
result in unbalanced
Phases.
Typical Transmission Tower
Configuration

32. Transposition
To keep system balanced, over the length of a transmission line the conductors are “rotated” so each phase occupies each position on tower for an equal distance.
This is known as transposition.
Aerial or side view of conductor positions over the length
of the transmission line.

33. Line Transposition Example

34. Line Transposition Example

35. Transposition Impact on Flux Linkages
“a” phase in
position “1”
“a” phase in
position “3”
“a” phase in
position “2”

36. Transposition Impact, cont’d

37. Inductance of Transposed Line

38. Inductance with Bundling

39. Inductance Example
Calculate the per phase inductance and reactance of a balanced 3, 60 Hz, line with:
horizontal phase spacing of 10m
using three conductor bundling with a spacing between conductors in the bundle of 0.3m.
Assume the line is uniformly transposed and the conductors have a 1cm radius.

40. Inductance Example