Lecture 11

Transmission Line Parameters

Reading: 4.1 – 4.6 ; 4.8 – 4.10

Homework 3 is due on Feb. 20th.

Dr. Lei Wu

Department of Electrical and Computer Engineering

EE 333

POWER SYSTEMS ENGINEERING

Outline

� Develop simple model for transmission lines

� Line resistance (R)

� Line conductance (G)

� Line inductance (L)

� Line capacitance (C)

� Analyze how the geometry of the transmission lines will affect the model parameters

2

R L

CG

,T

dc T

lR

A

ρ=

i0 ln

2

GMDL

GMR

µπ

=

( )( ) ( )2

11 12 1 21 22 2 1 2n

n n n n nnGMR D D D D D D D D D= ⋯ ⋯ ⋯ ⋯

( )( ) ( )11' 12 ' 1 21' 22 ' 2 1' 2 'nm

xy m m n n nmGMD D D D D D D D D D= ⋯ ⋯ ⋯ ⋯

Line inductance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing in a

bundle is 45cm. Find La.

3

Line inductance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing is 45cm.

Find La.

� GMR of one stranded conductor

( ) ( )67*711 12 13 14 15 16 17 71 72 73 74 75 76 77

2.1767 3.9181

sGMR D D D D D D D D D D D D D D

r cm

=

= =

12 16 17 2D D D r= = =13 15 2 3D D r= = 14 4D r=

71 72 73 74 75 76 2D D D D D D r= = = = = =

11 77 ' 0.7788D D r r= = =

4

Line inductance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing is 45cm.

Find La.

� GMR of one bundle

� GMR of each phase

44

34

2

1.091 26.6567

b s

s

GMR GMR GMR dd d

GMR d cm

= =

= =

( )2 21 1,1' 0.266567 * 18 20 2.678bGMR GMR D m= = + =

2 2,2 ' 0.266567 * 24 2.5293bGMR GMR D m= = =

( )2 23 3,3 ' 0.266567 * 18 20 2.678bGMR GMR D m= = + =

5

Line inductance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing is 45cm.

Find La.

� GMR of the three-phase double circuit

31 2 3 2.6275GMR GMR GMR GMR m= =

6

Line inductance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing is 45cm.

Find La.

� GMD of the three-phase double circuit

312 23 13 16.5572GMD GMD GMD GMD= =

412 12 12 ' 1'2 1'2 '

4 2 2 2 2 2 2 2 210 4 10 20 10 22 10 2 15.4718

GMD D D D D

m

=

= + + + + =

423 23 23 ' 2 '3 2 '3 ' 15.4718GMD D D D D m= =

413 13 13 ' 1'3 1'3 '

4 2 2 2 220 2 16 20 2 20 18.9619

GMD D D D D

m

=

= + + =7

Line inductance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing is 45cm.

Find La.

70 ln 3.6814 *10 /2a

GMDL H m

GMR

µπ

−= =

2X fLπ=

8

Line capacitance

� Transmission line conductors exhibit capacitance with respect

to each other due to the potential difference between them.

� The relationship between charge q and potential V is

represented by the capacitance

212

0 1

-120

ln2

where ε =8.85*10 Farad/m,

permittivity of free space

DqV

Dπε=

V

qC =

9

Line capacitance

� Transmission line conductors exhibit capacitance with respect

to each other due to the potential difference between them.

� The relationship between charge q and potential V is

represented by the capacitance

� Assuming that

V

qC =

10

10

ii

1ln

2

where D

nkj

ij kk ki

i

DV q

D

r

πε =

=

=

∑

0n

ii

q =∑

Line capacitance

� For a single-phase two-wire line

11

1 2 0q q+ =

12 2212 1 2

0 11 21

0 0

1ln ln

2

1 1ln ln 2 ln

2 2

D DV q q

D D

D r Dq q q

r D r

πε

πε πε

= +

= − =

012

2

2 lnC

D

r

πε=

Line capacitance

� General formula for calculating capacitance

0

c

2

ln

where GMR is similar to GMR except that r is used instead of r'

c

CGMD

GMR

πε=

12

Line capacitance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing in a

bundle is 45cm. Find Ca and admittance.

13

Line capacitance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing is 45cm.

Find Ca and admittance.

� GMD of the three-phase double circuit

312 23 13 16.5572GMD GMD GMD GMD= =

14

Line capacitance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing is 45cm.

Find Ca and admittance.

� GMR of one stranded conductor

( ) ( )67*711 12 13 14 15 16 17 71 72 73 74 75 76 77

2.2558 4.0605

sGMR D D D D D D D D D D D D D D

r cm

=

= =

12 16 17 2D D D r= = =13 15 2 3D D r= = 14 4D r=

71 72 73 74 75 76 2D D D D D D r= = = = = =

11 77D D r= =

15

Line capacitance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing is 45cm.

Find Ca and admittance.

� GMR of one bundle

� GMR of each phase

44

34

2

1.091 26.9079

b s

s

GMR GMR GMR dd d

GMR d cm

= =

= =14 4D r=

( )2 21 1,1' 0.269079 * 18 20 2.6908bGMR GMR D m= = + =

2 2,2 ' 0.269079 * 24 2.5412bGMR GMR D m= = =

( )2 23 3,3 ' 0.269079 * 18 20 2.6908bGMR GMR D m= = + =

16

Line capacitance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing is 45cm.

Find Ca and admittance.

� GMR of the three-phase double circuit

� Capacitance

31 2 3 2.6400GMR GMR GMR GMR m= =

17

11023.03*10 /

lna

c

C F mGMD

GMR

πε −= =

11 82 2 * 60 *3.03*10 1.14 *10 /aY j fC j j S mπ π − −= = =

Line inductance & capacitance

� Example: three-phase double circuit, bundled conductors. Each

conductor is stranded, r=1.8cm and conductor spacing is 45cm.

18

Single circuit Double circuit

Inductance (H/m) 7.75*10-7 3.68*10-7

Capacitance (F/m) 1.44*10-11 3.03*10-11

Additional Transmission Topics

� Multi-circuit lines: Multiple lines often share a common transmission right-of-way. This DOES cause mutual inductance

and capacitance, but is often ignored in system analysis.

� Cables: There are about 3000 miles of underground ac cables in U.S. Cables are primarily used in urban areas. In a cable the

conductors are tightly spaced, (< 1ft) with oil impregnated paper

commonly used to provide insulation

� inductance is lower

� capacitance is higher, limiting cable length

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Additional Transmission topics

� DC Transmission: Because of the large fixed cost necessary to convert ac to dc and then back to ac, dc transmission is only

practical for several specialized applications

� long distance overhead power transfer (> 400 miles)

� long cable power transfer such as underwater

� providing an asynchronous means of joining different power

systems (such as the Eastern and Western grids).

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