Lesson 5: “CAPACITANCE” OF TRANSMISSION LINE OUTLINE: Lesson 5: ON A TRANSMISSION LINE I. The Representation of Transmission Lines II. Two Port Network Analysis III.The Short Transmission Line (STL) IV.The Medium Transmission Line (MTL) V. The Long Transmission Line (LTL) “V-I RELATIONSHIP” OF TRANSMISSION LINE B. REPRESENTATION OF LINES BASED ON LENGTH TR ANS M IS S ION LINE (TL) TRANSMISSION LINE TRANSMISSION LINE RECEIVING END SENDING END A. INTRODUCTION KILOMETER (KM) MILES (MI) 80 below 50 below 80-240 50-150 Above 240 Above 150 ***What values we usually looked for? V&I S,P,Q, pf Line losses Efficiency of the line %VD & %VR NOTE: R,L,C of transmission lines are called the of TL and they are considered once installed. means NEGLECTED only “V-I RELATIONSHIP” OF TRANSMISSION LINE VSN f VRN , I R I S f VRN , I R TWO PORT NETWORK ANALYSIS used in analyzing transmission line. it includes: port and port. VSN A VRN B I R I S C VRN D I R O U T P U T I N P U T In a transmission line, I N VS /VSN P U IS T In a transmission line the GENERAL EQUATION is, VR /VRN TRANSMISSION LINES (RLC CONSTANTS) IR VSN O U T P U T IS V & I at “SENDING “ end A C B VRN D IR “ABCD CONSTANT” *dependent on the type of line V & I at “RECEIVING “ end “V-I RELATIONSHIP” OF TRANSMISSION LINE Constants Generalized circuit constants of transmission line using 2-port network. 1. Using Circuit Analysis CONSTANT 2. Using Two-Port Network Analysis I N VSN P U IS T VRN TRANSMISSION LINES (RLC CONSTANTS) VSN A VRN B I R I S C VRN D I R VSN IS A C B VRN D IR IR UNIT Unitless O U T P U T Ohm Mho Unitless TAKE NOTE: z= series impedance per unit length per phase y= shunt admittance per unit length per phase to neutral I = length of line Z= z(l) = total series impedance per phase Y= y(l) = total shunt admittance per phase to neutral “V-I RELATIONSHIP” OF TRANSMISSION LINE In Matrix Form, A. Single Phase, Line-to-Line Analysis VS IS 1 0 2 Z L VR IR 1 Constants CONSTANT FORMULA 1 2ZL VS 1 VR I S 0 VR 2 Z L I R 1 I R 0 1 “V-I RELATIONSHIP” OF TRANSMISSION LINE In Matrix Form, B. Single Phase, Line-to-Neutral Analysis VSN IS NOTE: 1 0 VS= 2VSN Z L VRN 1 IR VR= 2VRN Constants CONSTANT FORMULA 1 VSN 1 VRN I S 0 VRN Z L 1 ZL IR IR 0 1 “V-I RELATIONSHIP” OF TRANSMISSION LINE In Matrix Form, C. Three Phase, Line-to-Line Analysis VS IS 1 0 Z L 1 1-120 VR IR 1 Constants (line ab) CONSTANT FORMULA 1 (1-1cis120)*Z L VS 1 VR I S 0 VR 1 1-120Z L I R 1 I R 0 1 “V-I RELATIONSHIP” OF TRANSMISSION LINE In Matrix Form, D. Three Phase, Line-to-Neutral Analysis VSN IS 1 0 Z L VRN 1 IR Constants CONSTANT FORMULA 1 ZL VSN 1 VRN I S 0 VRN Z L 1 0 IR IR 1 “V-I RELATIONSHIP” OF TRANSMISSION LINE A. NOMINAL π MTL, Line-to-Neutral Analysis VSN Z LYC 1 VRN Z L I R 2 Z LYC2 I S YC 4 Z Y VRN 1 L C I R 2 In Matrix Form, Constants Z Y A 1 L C 2 Z LYC2 C YC 4 VSN B ZL D 1 IS Z LYC 2 Z LYC 1 2 Z LYC2 YC 4 ZL VRN Z LYC 1 2 IR “V-I RELATIONSHIP” OF TRANSMISSION LINE B. NOMINAL T MTL, Line-to-Neutral Analysis L L VSN Z Y 1 L C 2 IS YC VRN Z L2YC I R VRN Z L 4 Z LYC 1 IR 2 In Matrix Form, Constants Z Y A 1 L C 2 C YC Z L2YC B ZL 4 Z Y D 1 L C 2 VSN IS Z Y 1 L C 2 YC Z L2YC ZL 4 VRN Z LYC IR 1 2 “V-I RELATIONSHIP” OF TRANSMISSION LINE SHORT TRANSMISSION LINE and MEDIUM TRANSMISSION LINE 1. A 100 mile, three phase transmission line delivers 55MVA at 0.8 pf lagging to the load at 132kV. The line is composed of “DRAKE” conductors with flat horizontal spacing of 11.9ft between adjacent conductors. Assume a wire temperature of 50°C. Determine the sending voltage, sending current, sending power and line losses using: a. “SHORT” line representation b. “MEDIUM (NOMINAL π)” line representation c. “MEDIUM (NOMINAL T)” line representation “V-I RELATIONSHIP” OF TRANSMISSION LINE where A. LONG TRANSMISSION LINE: SOLUTION TO DIFFERENTIAL EQUATIONS z= series impedance per unit length per phase yc= shunt admittance per unit length per phase to neutral x = length of line from “RECEIVING” end TAKE NOTE: x= Independent Variable I, V = Dependent Variables z, yc= constant parameters V C1e C yc z x yc z x yc z x yc z x C2 e C1e C2 e I z / yc z / yc “V-I RELATIONSHIP” OF TRANSMISSION LINE Thus, A. LONG TRANSMISSION LINE: SOLUTION TO DIFFERENTIAL EQUATIONS when x=0 V= VRN I=IR VRN C1 2 z / yc IR 2 VRN C2 2 z / yc IR 2 VRN V 2 z / yc I R e 2 VRN 2 z / yc I R e 2 VRN 1 I I R e 2 2 z / yc VRN 1 I R e 2 2 z / yc yc z x yc z x yc z x yc z x “V-I RELATIONSHIP” OF TRANSMISSION LINE Let: B. LONG TRANSMISSION LINE: INTERPRETATION OF THE EQUATION ZC z yc yc z j From Where: VRN V 2 V RN 2 z / yc 2 I R e z / yc I R e 2 VRN 1 I I R e 2 2 z / yc VRN 1 I R e 2 2 z / yc yc z x α= Attenuation Constant (nepers per unit length) β= Phase constant (radians per unit length) Thus yc z x yc z x yc z x ZC V V RN I R e x 2 2 ZC V RN I R e x 2 2 V 1 I RN I R e x 2 2Z C V 1 RN I R e 2 2Z C x “V-I RELATIONSHIP” OF TRANSMISSION LINE Constants A cosh x C. LONG TRANSMISSION LINE: HTPERBOLIC FORM OF THE EQUATIONS B Z c sinh x C V cosh x VRN Z c sinh x I R I sinh x VRN cosh x I R Zc cosh x sinh x I Zc D cosh x RECALL: Note in “RADIANS” cosh x cosh x j x In Matrix Form, V sinh x Zc Z c sinh x cosh x cosh x cosh x cos x j sinh x sin x VRN IR sinh x sinh x j x sinh x sinh x cos x j cosh x sin x “V-I RELATIONSHIP” OF TRANSMISSION LINE LONG TRANSMISSION LINE A three phase 60Hz transmission line is 250 mile long. The voltages at the sending end is 220kV. The parameters of the line are r= 0.2Ω/mi, xL=0.8Ω/mi and yC=5.3μmho/mi. a. Find the following at no load 1. 2. 3. IS and VR I and V at the middle of the line I and V at 50 miles from the sending end b. Find the following if the load on the line is 80MW at 220kV with unity power factor 1. 2. 3. IS , IR and VS I and V at the middle of the line I and V at 50 miles from the sending end Lesson 5: