Transmission Lines And Power Flow Analysis

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School of Engineering (SOE)University of St Thomas (UST)Transmission Lines andPower Flow AnalysisDr. Greg MowryAnnie SebastianMarian Mohamed1

School of EngineeringThe UST MissionInspired by Catholic intellectual tradition, theUniversity of St. Thomas educates studentsto be morally responsible leaders who thinkcritically, act wisely, and work skillfully toadvance the common good. civilization depends on power, power is for theโ€˜common goodโ€™!!2

School of EngineeringOutlineI. Background materialII. Transmission Lines (TLs)III. Power Flow Analysis (PFA)3

School of EngineeringI. Background Material4

School of EngineeringNetworks & Power Systems In a network (power system) there are 6 basicelectrical quantities of interest:1. Current๐‘– ๐‘ก ๐‘‘๐‘ž/๐‘‘๐‘ก2. Voltage๐‘ฃ ๐‘ก ๐‘‘๐œ‘/๐‘‘๐‘ก3. Power๐‘ ๐‘ก ๐‘ฃ ๐‘ก ๐‘– ๐‘ก ๐‘‘๐‘ค๐‘‘๐‘ก(FL) ๐‘‘๐ธ๐‘‘๐‘ก5

School of EngineeringNetworks & Power Systems 6 basic electrical quantities continued:4. Energy (work)w t ๐‘ก๐‘ 5. Charge (q or Q)๐‘ž ๐‘ก ๐‘ก๐‘– ๐œ‘ ๐‘ก ๐‘ก๐‘ฃ 6. Flux๐œ ๐‘‘๐œ ๐‘ก๐‘ฃ ๐œ ๐‘– ๐œ ๐‘‘๐œ๐œ ๐‘‘๐œ๐œ ๐‘‘๐œ6

School of EngineeringNetworks & Power Systems A few comments: Voltage (potential) has potential-energy characteristicsand therefore needs a reference to be meaningful; e.g.ground 0 volts I will assume that pretty much everything we discuss& analyze today is linear until we get to PFA7

School of EngineeringNetworks & Power Systems Linear Time Invariant (LTI) systems: Linearity:If๐‘ฆ1 ๐‘“ ๐‘ฅ1&๐‘ฆ2 ๐‘“(๐‘ฅ2 )Then๐›ผ ๐‘ฆ1 ๐›ฝ ๐‘ฆ2 ๐‘“(๐›ผ ๐‘ฅ1 ๐›ฝ ๐‘ฅ2 )8

School of EngineeringNetworks & Power Systems Implications of LTI systems: Scalability:if ๐‘ฆ ๐‘“(๐‘ฅ), then ๐›ผ๐‘ฆ ๐‘“(๐›ผ๐‘ฅ) Superposition & scalability of inputs holds Frequency invariance:๐œ”๐‘–๐‘›๐‘๐‘ข๐‘ก ๐œ”๐‘œ๐‘ข๐‘ก๐‘๐‘ข๐‘ก9

School of EngineeringNetworks & Power Systems ACSS and LTI systems: Suppose: ๐‘ฃ๐‘–๐‘› ๐‘ก ๐‘‰๐‘–๐‘› cos ๐œ”๐‘ก ๐œƒ Since input output, the only thing a linear system cando to an input is: Change the input amplitude: Vin VoutChange the input phase: in out10

School of EngineeringNetworks & Power Systems ACSS and LTI systems:These LTI characteristics along with Eulerโ€™s theorem๐‘’ ๐‘—๐œ”๐‘ก cos ๐œ”๐‘ก ๐‘— sin ๐œ”๐‘ก ,allow ๐‘ฃ๐‘–๐‘› ๐‘ก ๐‘‰๐‘– cos ๐œ”๐‘ก ๐œƒ to be represented as๐‘ฃ๐‘–๐‘› ๐‘ก ๐‘‰๐‘– cos ๐œ”๐‘ก ๐œƒ ๐‘‰๐‘– ๐‘’ ๐‘—๐œƒ ๐‘‰๐‘– ๐œƒ11

School of EngineeringNetworks & Power Systems In honor of Star Trek,๐‘‰๐‘– ๐‘’ ๐‘—๐œƒ ๐‘‰๐‘– ๐œƒis called a โ€˜Phasorโ€™ 12

School of Engineering๐‘‰๐‘– ๐‘’ ๐‘—๐œƒ ๐‘‰๐‘– ๐œƒ13

School of EngineeringNetworks & Power Systems ACSS and LTI systems: Since time does not explicitly appear in a phasor, anLTI system in ACSS can be treated like a DC systemon steroids with j ACSS also requires impedances; i.e. AC resistance14

School of EngineeringNetworks & Power Systems Comments: Little, if anything, in ACSS analysis is gained bythinking of j 1 (that is algebra talk) Rather, since Euler Theorem looks similar to a 2Dvector๐‘’ ๐‘—๐œ”๐‘ก cos ๐œ”๐‘ก ๐‘— sin ๐œ”๐‘ก๐‘ข ๐‘Ž๐‘ฅ ๐‘๐‘ฆ15

School of Engineering๐‘’ ๐‘—90 cos 90 ๐‘— sin 90 ๐‘—Hence in ACSS analysis,think of j as a 90 rotation16

School of EngineeringOperatorsv(t )V V dvdtj V vdtVj 17

School of EngineeringOhmโ€™s Approximation (Law)Summary of voltage-current relationshipElementTime domainFrequency domainRv RiV RILdiv LdtV j LICdvi CdtV Ij C18

School of EngineeringImpedance & AdmittanceImpedances and admittances of passive elementsElementRImpedanceZ RLZ j LC1Z j CAdmittance1Y RY 1j LY j C18

School of EngineeringThe Impedance TriangleZ R jX18

School of EngineeringThe Power Triangle (ind. reactance)S P jQ18

School of EngineeringTriangles Comments: Power Triangle Impedance Triangle The cos( ) defines the Power Factor (pf)22

School of EngineeringReminder of Useful Network Theorems KVL โ€“ Energy conservation๐‘ฃ๐ฟ๐‘œ๐‘œ๐‘ 0 KCL โ€“ Charge conservation๐‘–๐‘๐‘œ๐‘‘๐‘’ ๐‘œ๐‘Ÿ ๐‘๐‘ข๐‘  0 Ohmโ€™s Approximation (Law) Passive sign Convention (PSC)23

School of EngineeringPassive Sign Convection โ€“ Power System24

School of EngineeringPSCSourceLoad or Circuit ElementPSource viPLoad viPSource PLoad25

School of EngineeringReminder of Useful Network Theorems Pointing Theorem๐‘† ๐‘‰ ๐ผ Thevenin Equivalent โ€“ Any linear circuit may berepresented by a voltage source and series Theveninimpedance Norton Equivalent โ€“ Any linear circuit my berepresented by a current source and a shunt Theveninimpedance26๐‘‰๐‘‡๐ป ๐ผ๐‘ ๐‘๐‘‡๐ป

School of Engineering (SOE)University of St Thomas (UST)Transmission Lines andPower Flow AnalysisDr. Greg MowryAnnie SebastianMarian Mohamed1

School of EngineeringII. Transmission Lines2

School of EngineeringOutline General Aspects of TLs TL Parameters: R, L, C, G 2-Port Analysis (a brief interlude) Maxwellโ€™s Equations & the Telegraph equations Solutions to the TL wave equations3

School of Engineering4

School of Engineering5

School of EngineeringTransmission Lines (TLs) A TL is a major component of an electricalpower system. The function of a TL is to transport powerfrom sources to loads with minimal loss.6

School of EngineeringTransmission Lines In an AC power system, a TL will operateat some frequency f; e.g. 60 Hz. Consequently, ๐œ† ๐‘/๐‘“ The characteristics of a TL manifestthemselves when ๐€ ๐‘ณ of the system.7

School of EngineeringUSA example The USA is 2700 mi. wide & 1600 mi. fromnorth-to-south (4.35e6 x 2.57e6 m). For 60 Hz, ๐œ† 3๐‘’860 5๐‘’6 ๐‘š Thus ๐€๐Ÿ”๐ŸŽ ๐‘ฏ๐’› ๐‘ณ๐‘ผ๐‘บ๐‘จ and consequently thecontinental USA transmission system will exhibitTL characteristics8

School of EngineeringOverhead TL Components9

School of EngineeringOverhead ACSR TL Cables10

School of EngineeringSome Types of Overhead ACSR TL 038152156-801813127/7 Wires AAC Wasp Conductor Aluminium Electric Conductor.html

School of EngineeringTL Parameters12

School of EngineeringModel of an Infinitesimal TL Section13

School of EngineeringGeneral TL Parametersa. Series Resistance โ€“ accounts for Ohmic (I2R losses)b. Series Impedance โ€“ accounts for series voltage drops Resistive Inductive reactancec. Shunt Capacitance โ€“ accounts for Line-Charging Currentsd. Shunt Conductance โ€“ accounts for V2G losses due to leakagecurrents between conductors or between conductors and14ground.

School of EngineeringPower TL Parameters Series Resistance: related to the physical structure ofthe TL conductor over some temperature range. Series Inductance & Shunt Capacitance: producedby magnetic and electric fields around the conductorand affected by their geometrical arrangement. Shunt Conductance: typically very small soneglected.15

School of EngineeringTL Design ConsiderationsDesign ConsiderationsResponsibilitiesElectrical Factors1. Dictates the size, type and number of bundle conductors per phase.2. Responsible for number of insulator discs, vertical or v-shaped arrangement,phase to phase clearance and phase to tower clearance to be used3. Number, Type and location of shield wires to intercept lightning strokes.4. Conductor Spacing's, Types and sizesMechanical FactorsFocuses on Strength of the conductors, insulator strings and support structuresEnvironmental Factors Include Land usage, and visual impactEconomic FactorsTechnical Design criteria at lowest overall cost16

School of EngineeringTransmission Line ComponentsComponentsConductorsInsulatorsShield wiresSupport StructuresMade ofTypesACSR - Aluminum Conductor Steel ReinforcedAAC - All Aluminum ConductorAAAC - All Aluminum Alloy ConductorACAR - Aluminum Conductor Alloy ReinforcedAluminum replacedAlumoweld - Aluminum clad Steel ConductorcopperACSS - Aluminum Conductor Steel SupportedGTZTACSR - Gap-Type ZT Aluminum ConductorACFR - Aluminum Conductor Carbon ReinforcedACCR - Aluminum Conductor Composite ReinforcedPin Type InsulatorPorcelain, ToughenedSuspension Type Insulatorglass and polymerStrain Type InsulatorSteel, AlumoweldSmaller Cross section compared to 3 phase conductorsand ACSRWooden PolesLattice steel Tower, Reinforced Concrete PolesWood FrameSteel PolesLattice Steel towers.17

School of EngineeringACSR Conductor gWolfLynxPantherZebraDearMooseBersimisOverall Dia DC Resistance Current Capacity(mm)(ohms/km)(Amp) 75 .0559683635.040.0424299818

School of EngineeringTL Parameters - Resistance19

School of Engineering TL conductor resistance depends on factors suchas: Conductor geometry The frequency of the AC current Conductor proximity to other current-carrying conductors Temperature20

School of EngineeringResistanceThe DC resistance of a conductor at a temperature T is given by: ๐‘น(๐‘ป) ๐†๐‘ป๐’๐‘จwhere T conductor resistivity at temperature Tl the length of the conductorA the current-carry cross-sectional area of the conductorThe AC resistance of a conductor is given by: ๐‘น๐’‚๐’„ ๐‘ท๐’๐’๐’”๐’”๐‘ฐ๐ŸwherePloss โ€“ real power dissipated in the conductor in wattsI โ€“ rms conductor current21

School of EngineeringTL Conductor Resistance depends on: Spiraling:The purpose of introducing a steel core inside the strandedaluminum conductors is to obtain a high strength-to-weight ratio. Astranded conductor offers more flexibility and easier tomanufacture than a solid large conductor. However, the totalresistance is increased because the outside strands are larger thanthe inside strands on account of the spiraling.22

School of EngineeringThe layer resistance-per-length of each spirally woundconductor depends on its total length as follows:๐‘น๐’„๐’๐’๐’… ๐†๐‘จ๐Ÿ (๐…๐Ÿ)๐Ÿ โ„ฆ/m๐’‘๐’„๐’๐’๐’…where๐‘…๐‘๐‘œ๐‘›๐‘‘ - resistance of the wound conductor (โ„ฆ)1 (๐œ‹p๐‘๐‘œ๐‘›๐‘‘ 1๐‘ƒ๐‘๐‘œ๐‘›๐‘‘)2 - Length of the wound conductor (m)๐‘™ ๐‘ก๐‘ข๐‘Ÿ๐‘›2๐‘Ÿ๐‘™๐‘Ž๐‘ฆ๐‘’๐‘Ÿ- relative pitch of the wound conductor (m)๐‘™๐‘ก๐‘ข๐‘Ÿ๐‘› - length of one turn of the spiral (m)2๐‘Ÿ๐‘™๐‘Ž๐‘ฆ๐‘’๐‘Ÿ - Diameter of the layer (m)23

School of Engineering Frequency: When voltages and currents change in time, current flow (i.e. currentdensity) is not uniform across the diameter of a conductor. Thus the โ€˜effectiveโ€™ current-carrying cross-section of a conductor withAC is less than that for DC. This phenomenon is known as skin effect. As frequency increases, the current density decreases from that at thesurface of the conductor to that at the center of the conductor.24

School of Engineering Frequency cont. The spatial distribution of the current density in a particular conductor isadditionally altered (similar to the skin effect) due to currents in adjacentcurrent-carrying conductors. This is called the โ€˜Proximity Effectโ€™ and is smaller than the skin-effect. Using the DC resistance as a starting point, the effect of AC on theoverall cable resistance may be accounted for by a correction factor k. k is determined by E&M analysis. For 60 Hz, k is estimated around 1.02๐‘น๐‘จ๐‘ช ๐’Œ ๐‘น๐‘ซ๐‘ช25

School of EngineeringSkin Effect26

School of Engineering Temperature: The resistivity of conductors is a function of temperature. For common conductors (Al & Cu), the conductorresistance increases linearly with temperature๐œŒ ๐‘‡ ๐œŒ0 1 ๐›ผ ๐‘‡ ๐‘‡๐‘œ the temperature coefficient of resistivity27

School of EngineeringTL Parameters - Conductance28

School of EngineeringConductance Conductance is associated with power losses between theconductors or between the conductors and ground. Such power losses occur through leakage currents on insulatorsand via a corona Leakage currents are affected by: Contaminants such dirt and accumulated salt accumulated on insulators. Meteorological factors such as moisture29

School of EngineeringConductance Corona loss occurs when the electric field at the surface of aconductor causes the air to ionize and thereby conduct. Corona loss depends on: Conductor surface irregularities Meteorological conditions such as humidity, fog, and rain Losses due to leakage currents and corona loss are often smallcompared to direct I2R losses on TLs and are typically neglectedin power flow studies.30

School of EngineeringCorona loss31

School of EngineeringCorona loss32

School of EngineeringTL Parameters - Inductance33

School of EngineeringInductance is defined by the ratio of the total magnetic flux flowingthrough (passing through) an area divided by the current producingthat fluxฮฆ B d๐‘  ๐ผ ,๐‘‚๐‘๐‘ ๐‘’๐‘Ÿ๐‘ฃ๐‘Ž๐‘ก๐‘–๐‘œ๐‘›SฮฆL tance is solely dependent on the geometry of the arrangementand the magnetic properties (permeability ) of the medium.Also, B H where H is the mag field due to current flow only.34

School of EngineeringMagnetic Flux35

School of EngineeringInductance When a current flows through a conductor, a magnetic flux is setup which links the conductor. The current also establishes a magnetic field proportional to thecurrent in the wire. Due to the distributed nature of a TL we are interested in theinductance per unit length (H/m).36

School of EngineeringInductance - Examples37

School of EngineeringInductance of a Single WireThe inductance of a magnetic circuit that has a constant permeability (๐œ‡) can beobtained as follows:The total magnetic field Bx at x via Amperes law:๐œ‡๐‘œ ๐‘ฅ๐ต๐‘ฅ ๐ผ (๐‘Š๐‘/๐‘š)22๐œ‹๐‘ŸThe flux d through slice dx is given by:๐‘‘ฮฆ ๐ต๐‘ฅ ๐‘‘๐‘Ž ๐ต๐‘ฅ ๐‘™ ๐‘‘๐‘ฅ (๐‘Š๐‘)38

School of EngineeringThe fraction of the current I in the wire that links the area da dx l is:xNf r2The total flux through the wire is consequently given by:๐‘Ÿฮฆ ๐‘Ÿ๐‘‘ฮฆ 00๐‘ฅ๐‘Ÿ2๐œ‡๐‘œ ๐‘ฅ ๐ผ๐‘™๐‘‘๐‘ฅ2๐œ‹๐‘Ÿ 2Integrating with subsequent algebra to obtain inductance-per-m yields:๐ฟ๐‘ค๐‘–๐‘Ÿ๐‘’ ๐‘ ๐‘’๐‘™๐‘“ 0.5 10 7 (๐ป/๐‘š)39

School of EngineeringInductance per unit length (H/m) for twoWires (Cables)To calculate the inductance per unit length (H/m) for two-parallelwires the flux through area S must be calculated.40

School of EngineeringThe inductance of a single phase two-wire line is given by:L 4 10 7lnDrโ€ฒ(H/m)where ๐‘Ÿ โ€ฒ ๐‘Ÿ๐‘ฅโ€ฒ ๐‘Ÿ๐‘ฆโ€ฒ ๐‘’1 4๐‘Ÿ๐‘ฅ41

School of EngineeringThe inductance per phase of a 3-phase 3 wire line with equal spacing is given by:La 2 10 7lnDrโ€ฒ(H/m) per phase42

School of EngineeringL of a Single Phase 2-conductor Line Comprisedof Composite Conductors43

School of Engineering๐ฟ๐‘ฅ 2 10 7๐ท๐‘ฅ๐‘ฆln๐ท๐‘ฅ๐‘ฅwhere๐ท๐‘ฅ๐‘ฆ ๐‘€๐‘๐‘๐‘€๐ท๐‘˜๐‘š๐‘˜ 1๐‘š 1โ€ฒ๐‘Ž๐‘›๐‘‘๐ท๐‘ฅ๐‘ฅ ๐‘2๐‘๐‘€๐ท๐‘˜๐‘š๐‘˜ 1๐‘š 1Dxy is referred to as GMD geometrical mean distance between conductors x & yDxx is referred to as GMR geometrical mean radius of conductor x44

School of EngineeringA similar Ly express exists for conductor y.LTotal Lx Ly(H/m)Great NEWs!!!!!Manufacturers often calculate these inductances for us forvarious cable arrangements 45

School of Engineering Long 3-phase TLs are sometimes transposed for positive sequence balancing. This is cleverly called โ€˜transpositionโ€™ The inductance (H/m) of a completely transposed three phase line may also becalculated46

School of Engineering๐ฟ๐‘Ž 2 ๐ท๐‘’๐‘ž 10 73๐ท๐‘’๐‘žln๐ท๐‘ ๐ท12 ๐ท13 ๐ท23La has units of (H/m)Ds is the conductor GMR for stranded conductors or rโ€™ for solid conductors47

School of EngineeringIn HV TLs conductors are often arranged in one of thefollowing 3 standard configurations48

School of Engineering๐ฟ๐‘Ž 2 10 7๐ท๐‘’๐‘žln๐ท๐‘†๐ฟLa again has units of (H/m)If bundle separation is large compared to the bundle size, then Deq the center-tocenter distance of the bundles2-conductor bundle: ๐ท๐‘†๐ฟ 3-conductor bundle: ๐ท๐‘†๐ฟ 4-conductor bundle: ๐ท๐‘†๐ฟ 1694๐ท๐‘† ๐‘‘2 ๐ท๐‘† ๐‘‘๐‘‘3 ๐ท๐‘† ๐‘‘๐‘‘๐‘‘ 24๐ท๐‘† ๐‘‘3๐ท๐‘† ๐‘‘ 24 1.091 ๐ท๐‘† ๐‘‘ 349

School of EngineeringTL Parameters - Capacitance50

School of EngineeringQ CVC ๐‘† ๐œ€ ๐ธ ๐‘‘๐‘ ๐‘™๐ธ ๐‘‘๐‘™๐œ€RC ๐œŽ

School of EngineeringCapacitance A capacitor results when any two conductors are separated by an insulatingmedium. Conductors of an overhead transmission line are separated by air โ€“ which actsas an insulating medium โ€“ therefore they have capacitance. Due to the distributed nature of a TL we are interested in the capacitance perunit length (F/m). Capacitance is solely dependent on the geometry of the arrangement and theelectric properties (permittivity ) of the medium52

School of EngineeringCalculating Capacitance To calculate the capacitance between conductors wemust calculate two things: The flux of the electric field E between the two conductors The voltage V (electric potential) between the conductors Then we take the ratio of these two quantities53

School of Engineering Since conductors in a power system are typically cylindrical, westart with this geometry for determining the flux of the electricfield and potential (reference to infinity)54

School of EngineeringGaussโ€™s Law may be used to calculate the electric field E and thepotential V for this geometry:๐‘ž๐ธ๐‘ฅ (๐‘‰/๐‘š)2๐œ‹๐œ€๐‘ฅThe potential (voltage) of the wire is found using this electric field:๐‘‰12๐‘ž 2๐œ‹๐œ€๐ท2๐ท1๐‘‘๐‘ฅ๐‘ž๐ท2 ๐‘™๐‘›๐‘ฅ2๐œ‹๐œ€๐ท1๐‘‰๐‘œ๐‘™๐‘ก๐‘ The capacitance of various line configurations may be found using55these two equations

School of EngineeringCapacitance - Examples56

School of EngineeringCapacitance of a single-phase 2-Wire LineThe line-to-neutral capacitance is found to be:๐ถ๐‘› 2๐œ‹๐œ€๐‘™๐‘›๐ท๐‘Ÿ(๐น/๐‘š)The same Cn is also found for a 3-phase line-to-neutral arrangement

School of EngineeringThe Capacitance of Stranded Conductorse.g. a 3-phase line with 2 conductors per phase58

School of EngineeringDirect analysis yields๐ถ๐‘Ž๐‘›2๐œ‹๐œ€ ๏ฟฝ๏ฟฝ 3๐ท๐‘Ž๐‘ ๐ท๐‘Ž๐‘ ๐ท๐‘๐‘59

School of EngineeringThe Capacitance of a Transposed Line60

School of EngineeringThe Capacitance of a Transposed Line๐ถ๐‘Ž๐‘› 2๐œ‹๐œ€๐ท๐‘’๐‘ž๐‘™๐‘› ๐ท๐‘†๐ถ(๐น/๐‘š)where๐ท๐‘’๐‘ž 3๐ท๐‘Ž๐‘ ๐ท๐‘Ž๐‘ ๐ท๐‘๐‘2-conductor bundle: ๐ท๐‘†๐ถ 3-conductor bundle: ๐ท๐‘†๐ถ ๐‘Ÿ๐‘‘3๐‘Ÿ ๐‘‘244-conductor bundle: ๐ท๐‘†๐ถ 1.091 ๐‘Ÿ ๐‘‘ 361

School of EngineeringTLs ContinuedDr. Greg MowryAnnie SebastianMarian Mohamed1

School of EngineeringMaxwellโ€™s Eqns. & theTelegraph Eqns.2

School of Engineering3

School of EngineeringMaxwellโ€™s Equations & Telegraph Equations The โ€˜Telegrapher's Equationsโ€™ (or) โ€˜Telegraph Equationsโ€™ are apair of coupled, linear differential equations that describe thevoltage and current on a TL as functions of distance and time. The original Telegraph Equations and modern TL model wasdeveloped by Oliver Heaviside in the 1880โ€™s. The telegraph equations lead to the conclusion that voltages andcurrents propagate on TLs as waves.4

School of EngineeringMaxwellโ€™s Equations & Telegraph Equations The Transmission Line equation can be derived using Maxwellโ€™sEquation. Maxwell's Equations (MEs) are a set of 4 equations that describeall we know about electromagnetics. MEs describe how electric and magnetic fields propagate, interact,and how they are influenced by other objects.5

School of EngineeringTime for History James Clerk Maxwell [1831-1879] was an Einstein/Newtonlevel genius who took the set of known experimental laws(Faraday's Law, Ampere's Law) and unified them into asymmetric coherent set of Equations known as Maxwell'sEquations. Maxwell was one of the first to determine the speed ofpropagation of electromagnetic (EM) waves was the same as thespeed of light - and hence concluded that EM waves and visiblelight were really descriptions of the same phenomena.6

School of EngineeringMaxwellโ€™s Equations โ€“ 4 LawsGaussโ€™ Law: The total of the electric flux through a closed surfaceis equal to the charge enclosed divided by the permittivity. Theelectric flux through an area is defined as the electric fieldmultiplied by the area of the surface projected in a planeperpendicular to the field.๐† ๐‘ซ ๐The โ€˜no nameโ€™ ME: The net magnetic flux through any closedsurface is zero. ๐‘ฉ ๐ŸŽ7

School of EngineeringFaradayโ€™s law: Faraday's โ€˜law of inductionโ€™ is a basic law ofelectromagnetism describing how a magnetic field interacts over aclosed path (often a circuit loop) to produce an electromotive force(EMF; i.e. a voltage) via electromagnetic induction. ๐‘ฉ ๐‘ฟ๐‘ฌ ๐’•Ampereโ€™s law: describes how a magnetic field is related to twosources; (1) the current density J, and (2) the time-rate-of-change ofthe displacement vector D. ๐‘ซ ๐‘ฟ๐‘ฏ ๐‘ฑ ๐’•8

School of EngineeringThe Constitutive Equations Maxwellโ€™s 4 equations are further augmented by theโ€˜constitutive equationsโ€™ that describe how materials interact withfields to relate (B & H) and (D & E). For simple linear, isotropic, and homogeneous materials theconstitutive equations are represented by:๐‘ฉ ๐๐‘ฏ๐ ๐๐’“ ๐๐’๐‘ซ ๐œบ๐‘ฌ๐ ๐œบ๐’“ ๐œบ๐’9

School of EngineeringThe Telegraph Eqns.10

School of EngineeringTL Equations TL equations can be derived by two methods: Maxwellโ€™s equations applied to waveguides Infinitesimal analysis of a section of a TL operating in ACSSwith LCRG parameters (i.e. use phasors) The TL circuit model is an infinite sequence of 2-portcomponents; each being an infinitesimally shortsegment of the TL.11

School of EngineeringTL Equations The LCRG parameters of a TL are expressed as perunit-length quantities to account for the experimentallyobserved distributed characteristics of the TL Rโ€™ Conductor resistance per unit length, โ„ฆ/m Lโ€™ Conductor inductance per unit length, H/m Cโ€™ Capacitance per unit length between TL conductors, F/m Gโ€™ Conductance per unit length through the insulating medium betweenTL conductors, S/m12

School of EngineeringThe Infinitesimal TL Circuit13

School of EngineeringThe Infinitesimal TL Circuit Model14

School of EngineeringTransmission Lines EquationsThe following circuit techniques and laws are used toderive the lossless TL equationsKVL around the โ€˜abcdโ€™ loopKCL at node bVL L i tOLL for an inductorIC C v tOLC for a capacitorSet ๐‘…โ€ฒ ๐บ โ€ฒ 0 for the Lossless TL case15

School of Engineering Apply KVL around the โ€œabcdโ€ loop: ๐‘‰(z) V(z z) ๐‘…โ€ฒ z ๐ผ(๐‘ง)โ€ฒI(z) ๐ฟ z ๐‘ก Divide all the terms by z : ๐‘‰ ๐‘ง ๐‘ง ๐‘‰ ๐‘ง ๐‘ง ๐‘ง ๐‘…โ€ฒ ๐ผ ๐‘ง ๐ฟโ€ฒ ๐ผ(๐‘ง) ๐‘ก As z 0 in the limit and Rโ€™ 0 (lossless case): ๐‘‰(๐‘ง) ๐‘งโ€ฒ โ€“ ๐‘…๐ผ ๐‘ง ๐‘ฝ(๐’›) ๐’› โ€ฒ ๐ผ(๐‘ง) ๐ฟ ๐‘กโ€ฒ ๐‘ฐ(๐’›)โ€“๐‘ณ ๐’•(Eq.1)

School of Engineering Apply KCL at node b: ๐ผ (z) I(z z) ๐บ โ€ฒ z ๐‘‰(๐‘ง ๐‘ง)โ€ฒV(z z) ๐ถ z ๐‘ก Divide all the terms by z: ๐ผ ๐‘ง ๐‘ง ๐ผ ๐‘ง ๐‘ง ๐‘ง ๐บ โ€ฒ ๐‘‰ ๐‘ง ๐‘ง ๐ถ โ€ฒ ๐‘‰(๐‘ง ๐‘ง) ๐‘ก As z 0 in the limit and Gโ€™ 0 (lossless case): ๐ผ(๐‘ง) ๐‘ง โ€“ ๐บ โ€ฒ๐‘‰ ๐‘ง ๐ถโ€ฒ ๐‘ฐ(๐’›) ๐’› โ€“โ€ฒ ๐‘ฝ(๐’›)๐‘ช ๐’• ๐‘‰(๐‘ง) ๐‘ก(Eq.2)

School of EngineeringSummarizing(Eqn. 1) ๐‘‰ ๐‘ง ,๐‘ก ๐‘ง(Eqn. 2) ๐ผ ๐‘ง ,๐‘ก ๐‘ง โ€ฒ ๐ผ ๐‘ง ,๐‘ก๐ฟ ๐‘กโ€ฒ ๐‘‰ ๐‘ง ,๐‘ก๐ถ ๐‘กEquations 1 & 2 are called the โ€œCoupled First Order Telegraph Equationsโ€18

School of Engineering Differentiating (Eq.1) wrt โ€˜zโ€™ yields: 2 ๐‘‰(๐‘ง) 2 ๐‘ง โ€“2โ€ฒ ๐ผ(๐‘ง)๐ฟ ๐‘ง ๐‘ก(Eq.3) Differentiating (2) wrt โ€˜tโ€™ yields: 2 ๐ผ(๐‘ง) ๐‘ง ๐‘ก โ€“2โ€ฒ ๐‘‰(๐‘ง)๐ถ ๐‘ก 2(Eq.4) Plugging equation 4 into 3: ๐Ÿ ๐‘ฝ(๐’›) ๐’›๐Ÿ ๐Ÿ ๐‘ฝ(๐’›) ๐‘ณโ€ฒ ๐‘ชโ€ฒ ๐’•๐Ÿ(Eq.5)19

School of EngineeringReversing the differentiation sequence Differentiating (Eq.1) wrt โ€˜tโ€™ yields: 2 ๐‘‰(๐‘ง) ๐‘ง ๐‘ก โ€“ 2 ๐ผ(๐‘ง)โ€ฒ๐ฟ ๐‘ก 2(Eq.6) Differentiating (2) wrt โ€˜zโ€™ yields: 2 ๐ผ(๐‘ง) 2 ๐‘ง โ€“2โ€ฒ ๐‘‰(๐‘ง)๐ถ ๐‘ง ๐‘ก(Eq.7) Plugging equation 6 into 7: ๐Ÿ ๐‘ฐ(๐’›) ๐’›๐Ÿ ๐Ÿ ๐‘ฐ(๐’›) ๐‘ณโ€ฒ ๐‘ชโ€ฒ ๐Ÿ ๐’•(Eq.8)20

School of EngineeringSummarizing(Eqn. 5) 2 V(z) z2(Eqn. 8) 2 I(z) z2 2 V(z) Lโ€ฒ Cโ€ฒ t2 2 I(z) Lโ€ฒ Cโ€ฒ 2 tEquations 5 & 8 are the v(z,t) and i(z,t) TL wave equations21

School of Engineering What is really cool about Equations 5 & 8 is that the solution to awave equation was already known; hence solving these equationswas straight forward. The general solution of the wave equation has the following form:f z, t f(z ut)where the โ€˜waveโ€™ propagates with Phase Velocity ๐‘ขu ฯ‰ฮฒ ฯ‰ฯ‰ L โ€ฒ Cโ€ฒ 1L โ€ฒ Cโ€ฒ22

School of Engineering The solution of the general wave equation, f(z,t), describes theshape of the wave and has the following form:f z, t f(z ut)orf z, t f(z ut) The sign of ๐‘ข๐‘ก determines the direction of wave propagation: ๐‘“ ๐‘ง ๐‘ข๐‘ก Wave traveling in the z direction ๐‘“ ๐‘ง ๐‘ข๐‘ก Wave traveling in the โ€“z direction23

School of EngineeringThe โ€˜ zโ€™ Traveling WE Solution ๐‘“ ๐‘ง, ๐‘ก ๐‘“(๐‘ง ๐‘ข๐‘ก) ๐‘“ ๐‘ง ๐‘ข๐‘ก) ๐‘“(0 ๐‘ง ๐‘ข๐‘ก 0 ๐‘ง ๐‘ข๐‘ก24

School of Engineering The argument of the wave equation solution is called thephase. Hence the culmination of all of this is that, โ€œWavespropagate by phase changeโ€ One other key point, the form of the voltage WE andcurrent WE are identical. Hence solutions to each mustbe of the same form and in phase.25

School of EngineeringWave Equation z increases as t increase for a constant phase26

School of EngineeringTL Solutions27

School of EngineeringTL Wave Equations - Recap(Eqn. 5) 2 V(z) z2(Eqn. 8) 2 I(z) z2 2 V(z) Lโ€ฒ Cโ€ฒ t2 2 I(z) Lโ€ฒ Cโ€ฒ 2 tEquations 5 & 8 are the v(z,t) and i(z,t) TL wave equations28

School of EngineeringACSS TL Analysis Recipe29

School of EngineeringInitial Steps Typically we will know or are given the source voltage. Forexample, it could be:๐’—๐’”๐’” ๐’• ๐Ÿ๐ŸŽ cos 107 ๐‘ก 30 ๐’—๐’๐’๐’•๐’” From the time function we form the source phasor & know thefrequency that the TL operates at (in ACSS operation):๐›š ๐Ÿ๐ŸŽ๐Ÿ• ๐’” ๐Ÿ๐‘ฝ๐’”๐’” ๐Ÿ๐ŸŽ ๐’‹๐Ÿ‘๐ŸŽ๐’†๐’—๐’๐’๐’•๐’”30

School of EngineeringACSS TL Analysis Under ACSS conditions the voltage and current solutions to the WEmay be expressed as: ๐‘‰ ๐‘ง, ๐‘ก ๐‘…๐‘’ ๐‘‰0 ๐‘’ ๐‘—(๐œ”๐‘ก ฮณ๐‘ง) ๐‘‰0 ๐‘’ ๐‘—(๐œ”๐‘ก ฮณ๐‘ง) ๐‘‰ ๐‘ง ๐‘‰0 ๐‘’ ๐‘—ฮณ๐‘ง ๐‘‰0 ๐‘’ ๐‘—ฮณ๐‘งTime Harmonic Phasor Form I ๐‘ง, ๐‘ก ๐‘…๐‘’ ๐ผ0 ๐‘’ ๐‘—(๐œ”๐‘ก ฮณ๐‘ง) ๐ผ0 ๐‘’ ๐‘—(๐œ”๐‘ก ฮณ๐‘ง) I ๐‘ง ๐ผ0 ๐‘’ ๐‘—ฮณ๐‘ง ๐ผ0 ๐‘’ ๐‘—ฮณ๐‘ง ๐‘‰ ๐‘—ฮณ๐‘ง๐‘’๐‘0 ๐‘‰ ๐‘—ฮณ๐‘ง๐‘’๐‘0Time Harmonic Phasor Form

School of EngineeringACSS TL Analysis Recipe Step 1: Assemble ๐‘0 , ๐‘๐ฟ , ๐‘™, ๐‘Ž๐‘›๐‘‘ ฮณ Step 2: Determine ะ“๐ฟ Step 3: Determine ๐‘๐‘–๐‘› Step 4: Determine ๐‘‰๐‘–๐‘› Step 5: Determine ๐‘‰0 Step 6: Determine ๐‘‰๐ฟ Step 7: Calculate ๐ผ๐ฟ and ๐‘ƒ๐ฟ32

School of EngineeringStep 1: Assemble ๐’๐ŸŽ , ๐’๐‘ณ , ๐’, , and ๐œธ Characteristic Impedance:๐’๐’ ๐‘ณโ€ฒ๐‘ฝ ๐‘ฝ๐’๐’ ๐‘ชโ€ฒ๐‘ฐ๐’ ๐‘ฐ๐’ Load Impedance: ๐‘ฝ๐‘ณ๐‘ฝ ๐‘ฝ๐ŸŽ๐ŸŽ๐’๐‘ณ ๐’๐ŸŽ ๐‘ฐ๐‘ณ๐‘ฝ๐ŸŽ ๐‘ฝ๐ŸŽ Complex Propagation Constant: ( 0 for lossless TL)๐œธ ๐œถ ๐’‹๐œท ๐’‹๐Ž ๐‘ณโ€ฒ ๐‘ชโ€ฒ33

School of EngineeringAdditional Initial Results Phase velocity uP: Identity:๐›š๐ฎ๐ ๐›ƒ๐Ÿ๐‹โ€ฒ ๐‚ โ€ฒ๐‹โ€ฒ ๐‚ โ€ฒ ๐› ๐›†34

School of EngineeringStep 2: Determine ะ“๐‘ณ Reflection Coefficient: ะ“๐ฟ ๐‘‰0 ๐‘‰0 ะ“๐ฟ ๐‘๐ฟ ๐‘0๐‘๐ฟ ๐‘0 ๐‘ง๐ฟ ๐‘๐ฟ๐‘0 ๐ผ0 ๐ผ0๐’›๐‘ณ ๐Ÿะ“๐‘ณ ๐’›๐‘ณ ๐Ÿ Reflection Coefficient at distance z (from load to source -l):ะ“ ๐’› ๐œธ๐’›๐‘ฝ ๐’ ๐’† ๐œธ๐’›๐‘ฝ ๐’ ๐’† ะ“๐‘ณ ๐’†๐’‹๐Ÿ๐œท๐’›35

School of EngineeringAdditional Step-2 Results VSWR:๐Ÿ ๐šช๐‘ณ๐•๐’๐–๐‘ ๐Ÿ ๐šช๐‘ณ Range of the VSWR:๐ŸŽ ๐‘ฝ๐‘บ๐‘พ๐‘น 36

School of EngineeringStep 3: Determine ๐’๐’Š๐’ Input Impedance:๐’๐’Š๐’๐‘ฝ๐’Š๐’ ๐’› ๐’๐’๐‘ณ ๐’๐ŸŽ ๐ญ๐š๐ง๐ก(๐œธ๐’) ๐’๐ŸŽ๐‘ฐ๐’Š๐’ ๐’› ๐’๐’๐ŸŽ ๐’๐‘ณ ๐’•๐’‚๐’๐’‰(๐œธ๐’) For a lossless line tanh ฮณl j tan(ฮฒl):๐’๐’Š๐’๐’๐‘ณ ๐’‹๐’๐ŸŽ ๐’•๐’‚๐’(๐œท๐’) ๐’๐ŸŽ๐’๐ŸŽ ๐’‹๐’๐‘ณ ๐’•๐’‚๐’(๐œท๐’) At any distance ๐‘ง1 from the load:๐’๐‘ณ ๐’‹๐’๐ŸŽ ๐ญ๐š๐ง(๐œท๐’›๐Ÿ )๐’(๐’›๐Ÿ ) ๐’๐ŸŽ๐’๐ŸŽ ๐’‹๐’๐‘ณ ๐’•๐’‚๐’(๐œท๐’›๐Ÿ )37

School of EngineeringStep 4: Determine ๐‘ฝ๐’Š๐’๐‘ฝ๐’Š๐’ ๐‘ฝ๐‘บ๐‘บ๐’๐’Š๐’ ๐‘ฝ๐’Š๐’ ๐’› ๐’๐’๐’Š๐’ ๐’๐‘บ38

School of EngineeringStep 5: Determine ๐‘ฝ๐ŸŽ ๐‘‰ ๐‘ง ๐‘‰0 ๐‘’ ๐‘—๐›ฝ๐‘ง ๐‘‰0 ๐‘’ ๐‘—๐›ฝ๐‘ง ๐‘‰0 (๐‘’ ๐‘—๐›ฝ๐‘ง ะ“๐ฟ ๐‘’ ๐‘—๐›ฝ๐‘ง ) ๐‘‰๐‘–๐‘› ๐‘‰๐‘‡๐ฟ ๐‘ง ๐‘™ ๐‘‰0 (๐‘’ ๐‘—๐›ฝ๐‘™ ะ“๐ฟ ๐‘’ ๐‘—๐›ฝ๐‘™ )๐‘ฝ๐ŸŽ ๐‘ฝ๐’Š๐’ ๐’‹๐œท๐’(๐’† ะ“๐‘ณ ๐’† ๐’‹๐œท๐’ )39

School of EngineeringStep 6: Determine ๐‘ฝ๐‘ณ Voltage at the load, VL:๐‘ฝ๐‘ณ ๐‘ฝ๐‘ป๐‘ณ ๐’› ๐ŸŽ ๐‘ฝ๐’ ๐Ÿ ๐šช๐‘ณwhere:๐‘ฝ ๐ŸŽ๐‘ฝ๐’Š๐’ ๐’‹๐œท๐’(๐’† ะ“๐‘ณ ๐’† ๐’‹๐œท๐’ )40

School of EngineeringStep 7: Calculate ๐‘ฐ๐‘ณ ๐š๐ง๐ ๐‘ท๐‘ณ๐‘ฝ๐‘ณ๐‘ฐ๐‘ณ ๐’๐‘ณ๐‘ท๐’‚๐’—๐’ˆ ๐‘ท๐‘ณ ๐‘ฐ๐’ ๐’ˆ๐’†๐’๐’†๐’“๐’‚๐’,๐Ÿ ๐Ÿ๐‘ฐ๐‘ณ ๐‘น๐’†(๐’๐‘ณ )๐Ÿ๐‘บ๐‘ณ ๐‘ฝ๐‘ณ ๐‘ฐ ๐‘ณ ๐‘ท๐‘ณ ๐’‹ ๐‘ธ๐‘ณ41

School of EngineeringFinal Steps After we have determined VL in phasor form, we might beinterested in the actual time function associated with VL; i.e.vL(t). It is straight forward to convert the VL phasor back to thevL(t) time-domain form; e.g. suppose:๐‘ฝ๐‘ณ ๐Ÿ“ ๐’† Then with ๐’‹๐Ÿ”๐ŸŽ ๐’—๐’๐’๐’•๐’”๐›š ๐Ÿ๐ŸŽ๐Ÿ• ๐’” ๐Ÿ๐’—๐‘ณ ๐’• ๐Ÿ“ cos 107 ๐‘ก 60 ๐’—๐’๐’๐’•๐’”42

School of EngineeringSurge Impedance Loading (SIL)Lossless TL with VS fixed for line-lengths /443

School of Engineering2-Port Analysis44

School of EngineeringTwo Port Network45

School of Engineering A two-port network is an electrical โ€˜black boxโ€™with two sets of terminals. The 2-port is sometimes referred to as a fourterminal network or quadrupole network. In a 2-port network, port 1 is often considered asan input port while port 2 is considered to be anoutput port.46

School of Engineering A 2-port network model is used tomathematically analyze electrical circuits byisolating the larger circuits into smaller portions. The 2-port works as โ€œBlack Boxโ€ with itsproperties specified by a input/output matrix. Examples: Filters, Transmission Lines,Transformers, Matching Networks, Small Signal47Models

School of EngineeringCharacterization of 2-Port Network 2-port networks are considered to be linear circuitshence the principle of superposition applies. The internal circuits connecting the 2-ports are assumedto be in their zero state and free of independent sources. 2-Port Network consist of two sets of input & outputvariables selected from the overall V1, V2, I1, I2 set: 2 Independent (or) Excitation Variables 2 Dependent (or) Response Variables48

School of EngineeringCharacterization of 2-Port NetworkUses of 2-Port Network: Analysis & Synthesis of Circuits and Networks. Used in the field of communications, control system, power systems, andelectronics for the analysis of cascaded networks. 2-ports also show up in geometrical optics, wave-guides, and lasers By knowing the matric parameters describing the 2-port network, it can beconsidered as black-box when embedded within a larger network.49

School of Engineering2-Port Networks & Linearity Any linear system with 2 inputs and 2 outputs can always beexpressed as:๐‘ฆ1๐‘Ž11 ๐‘Ž21๐‘ฆ2๐‘Ž12๐‘Ž22๐‘ฅ1๐‘ฅ250

2-Port 48308/chapter-10-two-port-networks51

School of EngineeringReview A careful review of the preceding TL analysis reveals that all wereally determined in ACSS operation was Vin & Iin and VL & IL. This reminds us of the exact input and output form of any linearsystem with 2 inputs and 2 outputs which can always beexpressed as:๐‘ฆ1๐‘Ž11 ๐‘Ž21๐‘ฆ2๐‘Ž12๐‘Ž22๐‘ฅ1๐‘ฅ2 52

School of EngineeringTransmission Line as Two Port NetworkI1 IinZSI2 IL AVS -BV2 VLV1 VinCD-SourceZL-2-Port NetworkLoad

School of EngineeringRepresentation of a TL by a 2-port network Nomenclature Used: VS โ€“ Sending end voltage IS โ€“ Sending end current VR โ€“ Receiving end Voltage IR โ€“ Receiving end current54

School of EngineeringRepresentation of a TL by a 2-port network Relation between sending end and receiving end is given as:๐‘‰๐‘† ๐ด ๐‘‰๐‘… ๐ต ๐ผ๐‘…๐ผ๐‘† ๐ถ ๐‘‰๐‘… ๐ท ๐ผ๐‘… In Matrix Form๐‘‰๐‘†๐ด ๐ถ๐ผ๐‘†๐ต๐ท๐‘‰๐‘…๐ผ๐‘…๐ด๐ท ๐ต๐ถ 155

School of Engineering4 Cases of Interest Case 1: Short TL Case 2: Mediu

Transmission Line Components School of Engineering Components Made of Types Conductors Aluminum replaced copper ACSR - Aluminum Conductor Steel Reinforced AAC - All Aluminum Conductor AAAC - All Aluminum Alloy Conductor ACAR - Aluminum Conductor Alloy Reinforced Alumoweld - Aluminum clad

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