Lecture EE333 - lecture 11 PDF

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EE 333 POWER SYSTEMS ENGINEERING

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

Outline 

Develop simple model for transmission lines    

0

i

Line resistance (R) Line conductance (G) Line inductance (L) Line capacitance (C)

2

,

ln

G

2

11

11'



12

12 '

⋯ ⋯

1

1

21

21'

22

22 '

⋯

2

⋯

1

2

⋯

⋯

2

⋯

1'

2'

⋯

Analyze how the geometry of the transmission lines will affect the model parameters 2

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 12

16

17

71

72

73

11

77

'

2

13

74

75

2 3

15

76

14

4

2

0.7788 6

7*7 11

2.1767

12

13

14

15

16

17

71

72

73

74

75

76

77

3.9181 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 4

2

4 3

1.0914 

26.6567

GMR of each phase

1

1,1'

2

2, 2 '

3

3,3 '

0.266567 *

18 2

0.266567 * 24 0.266567 *

20 2

2.678

2.5293 18 2

20 2

2.678 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 3

1

2

3

2.6275

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 3

12 4

12 4

10 2

12

23

12 '

4 2 10 2 4

23

23

23 '

1' 2

16.5572

13

1' 2 '

20 2 10 2

2 '3

2'3'

22 2 10 2

22

15.4718

15.4718

4 13 4

20 2

13

13 '

1' 3

2 2 16 20 2

1' 3 '

2 2 20

18.9619 7

Line inductance 

Example: three-phase double circuit, bundled conductors. Each conductor is stranded, r=1.8cm and conductor spacing is 45cm. Find La. 0

2

ln

3.6814 *10

7

/

2

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

12

2

ln 0

2 1

where ε 0 =8.85*10 -12 Farad/m, permittivity of free space 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 1 2

∑ 0

∑

0

ln

1

where D ii 10

Line capacitance 

For a single-phase two-wire line 1

0

2

  1 ln 2 0 1  ln 2 0  1

12

2

12

2 ln

11

22 21

ln

  

2

   1

2 ln 0

0

12

2 ln 11

Line capacitance 

General formula for calculating capacitance

2

0

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

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 3

12

23

13

16.5572

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 12

16

17

71

72

73

11

77

2

13

74

75

2 3

15

76

14

4

2

6

7*7 11

2.2558

12

13

14

15

16

17

71

72

73

74

75

76

77

4.0605

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 4

2

4 3

1.0914 

14

4

26.9079

GMR of each phase

1

2

3

1,1'

2, 2 '

3,3 '

0.269079 *

18 2

0.269079 * 24 0.269079 *

18 2

20 2

2.6908

2.5412 20 2

2.6908 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 3



1

2

2.6400

3

Capacitance 2

0

3.03 *10

11

/

ln

2

2 * 60 * 3.03 *10

11

1.14 *10

8

/ 17

Line inductance & capacitance 

Example: three-phase double circuit, bundled conductors. Each conductor is stranded, r=1.8cm and conductor spacing is 45cm.

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

18

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

19

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).

20

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