6. Transmission Line Models

This detailed technical content discusses various aspects of transmission line models, including the properties of conductors with line charges, capacitance and electric field relations, inductance calculations, and symmetric bundles with equivalent radii for double, triple, and quadruple configurations. The material also covers representation of three conductors, equivalent line charges, and symmetric properties for analysis. Prof. Mack Grady's insights from the TAMU Relay Conference Tutorial in 2015 provide valuable knowledge in this field.

• Transmission Line Models
• Conductor Properties
• Symmetric Bundles
• Capacitance
• Inductance

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1. 6. Transmission Line Models Conductor with radius r, modeled electrically as a line charge ql at the center Wire over Earth capacitance Electric Field b q l B = a E Volts per meter r h 2 a r o A Surface of Earth bi ai h Image conductor, at an equal distance below the Earth, and with negative line charge -ql = = r b r bi q q b bi b ai l l = = = + = = a a ln ln ln V E a E ai ab r i r 2 2 a ai a bi o o r r q 2 h l ln V for h >> r. rg 2 r o 2 8.854 pF per meter length = 55.6 pF / meter 2 q = = l o C ln(10000) = 9.2, C = 6.0 pf/m ln(1000) = 6.9, C = 8.1 pf/m ln(100) = 4.6, C = 12.1 pf/m 2 h V Reasonable estimate is 10 pF/m ln rg r Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 1

2. 6. Transmission Line Models, cont. Conductor of radius r, carrying current I Wire over Earth inductance Magnetic field intensity H I h = H Amperes per meter . 2 r Surface of Earth Note, the image flux exists only above the Earth h Image conductor, at an equal distance below the Earth ( x h 2 ) ( ) 2 h h r r 2 2 I I I dx dx h h r h r r = + = = ln ln o o o 2 2 x rh 2 I h ln o for h >> r. 2 r (4 10-7) / 2 Henries per meter length = 0.2 H/m 1 LC 1 I = ln(10000) = 9.2, L = 1.8 H/m ln(1000) = 6.9, L = 1.4 H/m ln(100) = 4.6, L = 0.92 H/m 2 N h ( 0 . 1 )( ) o = = ln L 6 12 10 10 10 ext 2 r 9 9 1 10 10 = = = Reasonable estimate is 1 H/m 10 10 18 10 10 8 3 10 Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 2

3. 6. Transmission Line Models, cont. Symmetric bundles have an equivalent radius Double Bundle, Each Conductor Has Radius r N 1 A = 1 N 2 req rA = eq r NrA Triple Bundle, Each Conductor Has Radius r 3 2 A req= 3rA Quadruple Bundle, Each Conductor Has Radius r 4 3 req= 4rA A Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 3

4. 6. Transmission Line Models, cont. Three phases Three Conductors Represented by Their Equivalent Line Charges b Conductor radii ra, rb, rc Dab a Dac c Daai Surface of Earth Daci ci Dabi ai Images bi D D D 1 aai r abi aci = + + ln ln ln V q q q ag a b c 2 D D o a ab ac V p p p q ag aa ab ac a 1 = V p p p q bg ba bb bc b 2 o V p p p q cg ca cb cc c If the transmission line is symmetric, then the P matrix has the equal diagonal, equal off-diagonal property that permits 0-1-2 analysis rather than a-b-c analysis Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 4

5. 6. Transmission Line Models, cont. D D D aai r abi aci ln ln ln D D I a ab ac a a D D D o bai bbi r bci = ln ln ln I b b 2 D D ba b bc I c c D D D cai cbi cci r ln ln ln D D ca cb c If the transmission line is symmetric, then the L matrix has the equal diagonal, equal off-diagonal property that permits 0-1-2 analysis rather than a-b-c analysis + 2 0 0 L L S M avg L 012 = 0 0 0 L L S M 0 L L S M Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 5

6. 6. Transmission Line Models, cont. Summary of Positive/Negative Sequence Capacitance and Inductance Calculations Computation of positive/negative sequence capacitance 2 o GMD = farads per meter, C + / + / ln GMR + / C where 3 = meters, GMD D D D + / ab ac bc , , where are D D D ab ac bc distances between phase conductors if the line has one conductor per phase, or distances between phase bundle centers if the line has symmetric phase bundles, and where is the actual conductor radius r (in meters) if the line has one conductor per GMR phase, or +/ C N 1 N = if the line has symmetric phase bundles. GMR N r A + / C Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 6

7. 6. Transmission Line Models, cont. Summary of Positive/Negative Sequence Capacitance and Inductance Calculations Computation of positive/negative sequence inductance GMD L where +/ GMD is the same as for capacitance, and for the single conductor case, + / o = henrys per meter, ln + / 2 GMR + / L is the conductor gmr 4 adjustment for internal inductance. If gmr (in meters), which takes GMR r +/ e L / 1 into account both stranding and the is r / 1 4 = re not given, then assume , and rgmr 1 N N for bundled conductors, = if the line has symmetric phase GMR N gmr r A + / L bundles. Computation of positive/negative sequence resistance R is the 60Hz resistance of one conductor if the line has one conductor per phase. If the line has symmetric phase bundles, then divide the one-conductor resistance by N. Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 7

8. 6. Transmission Line Models, cont. Summary of Zero Sequence Capacitance and Inductance Calculations Io Io 3Io Io 3Io Io + Io + Io 3Io Vo Vo Cbundle Lbundle Io Io Lo 3Io Io 3Io Io Lo + Io + Io 3Io Lo Co Co Co Vo Vo 2 GMD 1 o = farads per meter, C 0 3 0 C ln GMR 0 C where Earth. is the average height (with sag factored in) of the a-b-c bundle above perfect is computed using GMD GMD 0 C 0 C 2 ab 2 ac 2 bc = meters, GMD D D D D D D 9 0 C i i i i i i aa bb cc 3 C 2 ab 2 ac 2 bc = GMR GMR D D D meters 9 0 C + / Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 8

9. 6. Transmission Line Models, cont. Summary of Zero Sequence Capacitance and Inductance Calculations Computation of zero sequence inductance 2 The geometric mean bundle radius is computed using 2 3 / 0 ab L L + o = Henrys per meter, 3 ln L 0 2 GMR 0 L = where skin depth meters. f o 2 ac 2 bc = meters, GMR GMR D D D 9 where , , , and were shown previously. GMR D D D +/ L bc ab ac Computation of zero sequence resistance There are two components of zero sequence line resistance. First, the equivalent conductor resistance is the 60Hz resistance of one conductor if the line has one conductor per phase. If the line has symmetric phase bundles with N conductors per bundle, then divide the one-conductor resistance by N. Second, the effect of resistive Earth is included by adding the following term to the conductor resistance: f 10 869 . 9 3 ohms per meter (see Bergen), 7 Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 9

10. 6. Transmission Line Models, cont. Summary of Zero Sequence Capacitance and Inductance Calculations As a general rule, C +/ L C is usually about 6 picoF per meter. 0 L is usually about 2 microH per meter if the line has ground wires and typical Earth resistivity, or about 3 microH per meter for lines without ground wires or poor Earth resistivity. 1 , is approximately the speed of light (3 x 108 m/s) for positive usually works out to be about 12 picoF per meter, works out to be about 1 microH per meter (including internal inductance). +/ 0 The velocity of propagation, LC and negative sequences, and about 0.8 times that for zero sequence. Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 10

11. 6. Transmission Line Models, cont. Ready for Use! QL absorbed P1+ jQ1 I1 P2+ jQ2 I2 R j L + 1 1 + VR/ R - 200kVrms j C/2 j C/2 - QC1 QC2 produced produced One circuit of the 345kV line geometry, 100km long Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 11

12. 6. Transmission Line Models, cont. 345kV Double-Circuit Transmission Line 345kV Double-Circuit Transmission Line Scale: 1 cm = 2 m 5.7 m 7.8 m 8.5 m 7.6 m 22.9 m at tower, sags down 10 m at mid-span to 12.9 m. 7.6 m 4.4 m 22.9 m at tower, and sags down 10 m at mid-span to 12.9 m. Double conductor phase bundles, bundle radius = 22.9 cm, conductor radius = 1.41 cm, conductor resistance = 0.0728 /km Single-conductor ground wires, conductor radius = 0.56 cm, conductor resistance = 2.87 /km Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 12 Tower Base Double conductor phase bundles, bundle radius = 22.9 cm, conductor radius = 1.41 cm, conductor resistance = 0.0728 /km Single-conductor ground wires, conductor radius = 0.56 cm, conductor resistance = 2.87 /km