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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.http://www.stthomas.edu/mission/Since civilization depends on power, power is for theโcommon 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 ๐ก ๐๐/๐๐ก2. Voltage๐ฃ ๐ก ๐๐/๐๐ก3. Power๐ ๐ก ๐ฃ ๐ก ๐ ๐ก ๐๐ค๐๐ก(FL) ๐๐ธ๐๐ก5

School of EngineeringNetworks & Power Systems 6 basic electrical quantities continued:4. Energy (work)w t ๐ก๐ 5. Charge (q or Q)๐ ๐ก ๐ก๐ ๐ ๐ก ๐ก๐ฃ 6. Flux๐ ๐๐ ๐ก๐ฃ ๐ ๐ ๐ ๐๐๐ ๐๐๐ ๐๐6

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:If๐ฆ1 ๐ ๐ฅ1&๐ฆ2 ๐(๐ฅ2 )Then๐ผ ๐ฆ1 ๐ฝ ๐ฆ2 ๐(๐ผ ๐ฅ1 ๐ฝ ๐ฅ2 )8

School of EngineeringNetworks & Power Systems Implications of LTI systems: Scalability:if ๐ฆ ๐(๐ฅ), then ๐ผ๐ฆ ๐(๐ผ๐ฅ) Superposition & scalability of inputs holds Frequency invariance:๐๐๐๐๐ข๐ก ๐๐๐ข๐ก๐๐ข๐ก9

School of EngineeringNetworks & Power Systems ACSS and LTI systems: Suppose: ๐ฃ๐๐ ๐ก ๐๐๐ cos ๐๐ก ๐ 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 Eulerโs theorem๐ ๐๐๐ก cos ๐๐ก ๐ sin ๐๐ก ,allow ๐ฃ๐๐ ๐ก ๐๐ cos ๐๐ก ๐ to be represented as๐ฃ๐๐ ๐ก ๐๐ cos ๐๐ก ๐ ๐๐ ๐ ๐๐ ๐๐ ๐11

School of EngineeringNetworks & Power Systems In honor of Star Trek,๐๐ ๐ ๐๐ ๐๐ ๐is called a โPhasorโ 12

School of Engineering๐๐ ๐ ๐๐ ๐๐ ๐13

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 ๐๐๐ก cos ๐๐ก ๐ sin ๐๐ก๐ข ๐๐ฅ ๐๐ฆ15

School of Engineering๐ ๐90 cos 90 ๐ sin 90 ๐Hence in ACSS analysis,think of j as a 90 rotation16

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

School of EngineeringOhmโs 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 conservation๐ฃ๐ฟ๐๐๐ 0 KCL โ Charge conservation๐๐๐๐๐ ๐๐ ๐๐ข๐ 0 Ohmโs 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 ๐ผ 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 Theveninimpedance26๐๐๐ป ๐ผ๐ ๐๐๐ป

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) Maxwellโs 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 ๐/๐ The characteristics of a TL manifestthemselves when ๐ ๐ณ 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 3๐860 5๐6 ๐ Thus ๐๐๐ ๐ฏ๐ ๐ณ๐ผ๐บ๐จ 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: ๐น(๐ป) ๐๐ป๐๐จwhere 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: ๐น๐๐ ๐ท๐๐๐๐๐ฐ๐wherePloss 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:๐น๐๐๐๐ ๐๐จ๐ (๐ ๐)๐ โฆ/m๐๐๐๐๐ where๐ ๐๐๐๐ - resistance of the wound conductor (โฆ)1 (๐p๐๐๐๐ 1๐๐๐๐๐)2 - Length of the wound conductor (m)๐ ๐ก๐ข๐๐2๐๐๐๐ฆ๐๐- relative pitch of the wound conductor (m)๐๐ก๐ข๐๐ - length of one turn of the spiral (m)2๐๐๐๐ฆ๐๐ - 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 โeffectiveโ 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 โProximity 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 1.02๐น๐จ๐ช ๐ ๐น๐ซ๐ช25

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 ๐ ๐0 1 ๐ผ ๐ ๐๐ 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 ๐ผ ,๐๐๐ ๐๐๐ฃ๐๐ก๐๐๐SฮฆL 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) can beobtained as follows:The total magnetic field Bx at x via Amperes law:๐๐ ๐ฅ๐ต๐ฅ ๐ผ (๐๐/๐)22๐๐The flux d through slice dx is given by:๐ฮฆ ๐ต๐ฅ ๐๐ ๐ต๐ฅ ๐ ๐๐ฅ (๐๐)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:๐ฮฆ ๐๐ฮฆ 00๐ฅ๐2๐๐ ๐ฅ ๐ผ๐๐๐ฅ2๐๐ 2Integrating with subsequent algebra to obtain inductance-per-m yields:๐ฟ๐ค๐๐๐ ๐ ๐๐๐ 0.5 10 7 (๐ป/๐)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 7lnDrโฒ(H/m)where ๐ โฒ ๐๐ฅโฒ ๐๐ฆโฒ ๐1 4๐๐ฅ41

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

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

School of Engineering๐ฟ๐ฅ 2 10 7๐ท๐ฅ๐ฆln๐ท๐ฅ๐ฅwhere๐ท๐ฅ๐ฆ ๐๐๐๐๐ท๐๐๐ 1๐ 1โฒ๐๐๐๐ท๐ฅ๐ฅ ๐2๐๐๐ท๐๐๐ 1๐ 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 โtranspositionโ The inductance (H/m) of a completely transposed three phase line may also becalculated46

School of Engineering๐ฟ๐ 2 ๐ท๐๐ 10 73๐ท๐๐ln๐ท๐ ๐ท12 ๐ท13 ๐ท23La 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 Engineering๐ฟ๐ 2 10 7๐ท๐๐ln๐ท๐๐ฟLa 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: ๐ท๐๐ฟ 3-conductor bundle: ๐ท๐๐ฟ 4-conductor bundle: ๐ท๐๐ฟ 1694๐ท๐ ๐2 ๐ท๐ ๐๐3 ๐ท๐ ๐๐๐ 24๐ท๐ ๐3๐ท๐ ๐ 24 1.091 ๐ท๐ ๐ 349

School of EngineeringTL Parameters - Capacitance50

School of EngineeringQ CVC ๐ ๐ ๐ธ ๐๐ ๐๐ธ ๐๐๐RC ๐

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 EngineeringGaussโs Law may be used to calculate the electric field E and thepotential V for this geometry:๐๐ธ๐ฅ (๐/๐)2๐๐๐ฅThe potential (voltage) of the wire is found using this electric field:๐12๐ 2๐๐๐ท2๐ท1๐๐ฅ๐๐ท2 ๐๐๐ฅ2๐๐๐ท1๐๐๐๐ก๐ 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:๐ถ๐ 2๐๐๐๐๐ท๐(๐น/๐)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 yields๐ถ๐๐2๐๐ ๏ฟฝ๏ฟฝ 3๐ท๐๐ ๐ท๐๐ ๐ท๐๐59

School of EngineeringThe Capacitance of a Transposed Line60

School of EngineeringThe Capacitance of a Transposed Line๐ถ๐๐ 2๐๐๐ท๐๐๐๐ ๐ท๐๐ถ(๐น/๐)where๐ท๐๐ 3๐ท๐๐ ๐ท๐๐ ๐ท๐๐2-conductor bundle: ๐ท๐๐ถ 3-conductor bundle: ๐ท๐๐ถ ๐๐3๐ ๐244-conductor bundle: ๐ท๐๐ถ 1.091 ๐ ๐ 361

School of EngineeringTLs ContinuedDr. Greg MowryAnnie SebastianMarian Mohamed1

School of EngineeringMaxwellโs Eqns. & theTelegraph Eqns.2

School of Engineering3

School of EngineeringMaxwellโs Equations & Telegraph Equations The โTelegrapher's Equationsโ (or) โTelegraph 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 1880โs. The telegraph equations lead to the conclusion that voltages andcurrents propagate on TLs as waves.4

School of EngineeringMaxwellโs Equations & Telegraph Equations The Transmission Line equation can be derived using MaxwellโsEquation. 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 EngineeringMaxwellโs 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 ๐ซ ๐The โno nameโ ME: The net magnetic flux through any closedsurface is zero. ๐ฉ ๐7

School of EngineeringFaradayโs law: Faraday's โlaw 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. ๐ฉ ๐ฟ๐ฌ ๐Ampereโs 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. ๐ซ ๐ฟ๐ฏ ๐ฑ ๐8

School of EngineeringThe Constitutive Equations Maxwellโs 4 equations are further augmented by theโconstitutive 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:๐ฉ ๐๐ฏ๐ ๐๐ ๐๐๐ซ ๐บ๐ฌ๐ ๐บ๐ ๐บ๐9

School of EngineeringThe Telegraph Eqns.10

School of EngineeringTL Equations TL equations can be derived by two methods: Maxwellโs 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, โฆ/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 โabcdโ loopKCL at node bVL L i tOLL for an inductorIC C v tOLC for a capacitorSet ๐ โฒ ๐บ โฒ 0 for the Lossless TL case15

School of Engineering Apply KVL around the โabcdโ loop: ๐(z) V(z z) ๐ โฒ z ๐ผ(๐ง)โฒI(z) ๐ฟ z ๐ก Divide all the terms by z : ๐ ๐ง ๐ง ๐ ๐ง ๐ง ๐ง ๐ โฒ ๐ผ ๐ง ๐ฟโฒ ๐ผ(๐ง) ๐ก As z 0 in the limit and Rโ 0 (lossless case): ๐(๐ง) ๐งโฒ โ ๐ ๐ผ ๐ง ๐ฝ(๐) ๐ โฒ ๐ผ(๐ง) ๐ฟ ๐กโฒ ๐ฐ(๐)โ๐ณ ๐(Eq.1)

School of Engineering Apply KCL at node b: ๐ผ (z) I(z z) ๐บ โฒ z ๐(๐ง ๐ง)โฒV(z z) ๐ถ z ๐ก Divide all the terms by z: ๐ผ ๐ง ๐ง ๐ผ ๐ง ๐ง ๐ง ๐บ โฒ ๐ ๐ง ๐ง ๐ถ โฒ ๐(๐ง ๐ง) ๐ก As z 0 in the limit and Gโ 0 (lossless case): ๐ผ(๐ง) ๐ง โ ๐บ โฒ๐ ๐ง ๐ถโฒ ๐ฐ(๐) ๐ โโฒ ๐ฝ(๐)๐ช ๐ ๐(๐ง) ๐ก(Eq.2)

School of EngineeringSummarizing(Eqn. 1) ๐ ๐ง ,๐ก ๐ง(Eqn. 2) ๐ผ ๐ง ,๐ก ๐ง โฒ ๐ผ ๐ง ,๐ก๐ฟ ๐กโฒ ๐ ๐ง ,๐ก๐ถ ๐กEquations 1 & 2 are called the โCoupled First Order Telegraph Equationsโ18

School of Engineering Differentiating (Eq.1) wrt โzโ yields: 2 ๐(๐ง) 2 ๐ง โ2โฒ ๐ผ(๐ง)๐ฟ ๐ง ๐ก(Eq.3) Differentiating (2) wrt โtโ yields: 2 ๐ผ(๐ง) ๐ง ๐ก โ2โฒ ๐(๐ง)๐ถ ๐ก 2(Eq.4) Plugging equation 4 into 3: ๐ ๐ฝ(๐) ๐๐ ๐ ๐ฝ(๐) ๐ณโฒ ๐ชโฒ ๐๐(Eq.5)19

School of EngineeringReversing the differentiation sequence Differentiating (Eq.1) wrt โtโ yields: 2 ๐(๐ง) ๐ง ๐ก โ 2 ๐ผ(๐ง)โฒ๐ฟ ๐ก 2(Eq.6) Differentiating (2) wrt โzโ yields: 2 ๐ผ(๐ง) 2 ๐ง โ2โฒ ๐(๐ง)๐ถ ๐ง ๐ก(Eq.7) Plugging equation 6 into 7: ๐ ๐ฐ(๐) ๐๐ ๐ ๐ฐ(๐) ๐ณโฒ ๐ชโฒ ๐ ๐(Eq.8)20

School of EngineeringSummarizing(Eqn. 5) 2 V(z) z2(Eqn. 8) 2 I(z) z2 2 V(z) Lโฒ Cโฒ t2 2 I(z) Lโฒ Cโฒ 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 โwaveโ propagates with Phase Velocity ๐ขu ฯฮฒ ฯฯ L โฒ Cโฒ 1L โฒ Cโฒ22

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 ๐ข๐ก determines the direction of wave propagation: ๐ ๐ง ๐ข๐ก Wave traveling in the z direction ๐ ๐ง ๐ข๐ก Wave traveling in the โz direction23

School of EngineeringThe โ zโ Traveling WE Solution ๐ ๐ง, ๐ก ๐(๐ง ๐ข๐ก) ๐ ๐ง ๐ข๐ก) ๐(0 ๐ง ๐ข๐ก 0 ๐ง ๐ข๐ก24

School of Engineering The argument of the wave equation solution is called thephase. Hence the culmination of all of this is that, โWavespropagate 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) Lโฒ Cโฒ t2 2 I(z) Lโฒ Cโฒ 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:๐๐๐ ๐ ๐๐ cos 107 ๐ก 30 ๐๐๐๐๐ From the time function we form the source phasor & know thefrequency that the TL operates at (in ACSS operation):๐ ๐๐๐ ๐ ๐๐ฝ๐๐ ๐๐ ๐๐๐๐๐๐๐๐๐30

School of EngineeringACSS TL Analysis Under ACSS conditions the voltage and current solutions to the WEmay be expressed as: ๐ ๐ง, ๐ก ๐ ๐ ๐0 ๐ ๐(๐๐ก ฮณ๐ง) ๐0 ๐ ๐(๐๐ก ฮณ๐ง) ๐ ๐ง ๐0 ๐ ๐ฮณ๐ง ๐0 ๐ ๐ฮณ๐งTime Harmonic Phasor Form I ๐ง, ๐ก ๐ ๐ ๐ผ0 ๐ ๐(๐๐ก ฮณ๐ง) ๐ผ0 ๐ ๐(๐๐ก ฮณ๐ง) I ๐ง ๐ผ0 ๐ ๐ฮณ๐ง ๐ผ0 ๐ ๐ฮณ๐ง ๐ ๐ฮณ๐ง๐๐0 ๐ ๐ฮณ๐ง๐๐0Time Harmonic Phasor Form

School of EngineeringACSS TL Analysis Recipe Step 1: Assemble ๐0 , ๐๐ฟ , ๐, ๐๐๐ ฮณ Step 2: Determine ะ๐ฟ Step 3: Determine ๐๐๐ Step 4: Determine ๐๐๐ Step 5: Determine ๐0 Step 6: Determine ๐๐ฟ Step 7: Calculate ๐ผ๐ฟ and ๐๐ฟ32

School of EngineeringStep 1: Assemble ๐๐ , ๐๐ณ , ๐, , and ๐ธ Characteristic Impedance:๐๐ ๐ณโฒ๐ฝ ๐ฝ๐๐ ๐ชโฒ๐ฐ๐ ๐ฐ๐ Load Impedance: ๐ฝ๐ณ๐ฝ ๐ฝ๐๐๐๐ณ ๐๐ ๐ฐ๐ณ๐ฝ๐ ๐ฝ๐ Complex Propagation Constant: ( 0 for lossless TL)๐ธ ๐ถ ๐๐ท ๐๐ ๐ณโฒ ๐ชโฒ33

School of EngineeringAdditional Initial Results Phase velocity uP: Identity:๐๐ฎ๐ ๐๐๐โฒ ๐ โฒ๐โฒ ๐ โฒ ๐ ๐34

School of EngineeringStep 2: Determine ะ๐ณ Reflection Coefficient: ะ๐ฟ ๐0 ๐0 ะ๐ฟ ๐๐ฟ ๐0๐๐ฟ ๐0 ๐ง๐ฟ ๐๐ฟ๐0 ๐ผ0 ๐ผ0๐๐ณ ๐ะ๐ณ ๐๐ณ ๐ Reflection Coefficient at distance z (from load to source -l):ะ ๐ ๐ธ๐๐ฝ ๐ ๐ ๐ธ๐๐ฝ ๐ ๐ ะ๐ณ ๐๐๐๐ท๐35

School of EngineeringAdditional Step-2 Results VSWR:๐ ๐ช๐ณ๐๐๐๐ ๐ ๐ช๐ณ Range of the VSWR:๐ ๐ฝ๐บ๐พ๐น 36

School of EngineeringStep 3: Determine ๐๐๐ Input Impedance:๐๐๐๐ฝ๐๐ ๐ ๐๐๐ณ ๐๐ ๐ญ๐๐ง๐ก(๐ธ๐) ๐๐๐ฐ๐๐ ๐ ๐๐๐ ๐๐ณ ๐๐๐๐(๐ธ๐) For a lossless line tanh ฮณl j tan(ฮฒl):๐๐๐๐๐ณ ๐๐๐ ๐๐๐(๐ท๐) ๐๐๐๐ ๐๐๐ณ ๐๐๐(๐ท๐) At any distance ๐ง1 from the load:๐๐ณ ๐๐๐ ๐ญ๐๐ง(๐ท๐๐ )๐(๐๐ ) ๐๐๐๐ ๐๐๐ณ ๐๐๐(๐ท๐๐ )37

School of EngineeringStep 4: Determine ๐ฝ๐๐๐ฝ๐๐ ๐ฝ๐บ๐บ๐๐๐ ๐ฝ๐๐ ๐ ๐๐๐๐ ๐๐บ38

School of EngineeringStep 5: Determine ๐ฝ๐ ๐ ๐ง ๐0 ๐ ๐๐ฝ๐ง ๐0 ๐ ๐๐ฝ๐ง ๐0 (๐ ๐๐ฝ๐ง ะ๐ฟ ๐ ๐๐ฝ๐ง ) ๐๐๐ ๐๐๐ฟ ๐ง ๐ ๐0 (๐ ๐๐ฝ๐ ะ๐ฟ ๐ ๐๐ฝ๐ )๐ฝ๐ ๐ฝ๐๐ ๐๐ท๐(๐ ะ๐ณ ๐ ๐๐ท๐ )39

School of EngineeringStep 6: Determine ๐ฝ๐ณ Voltage at the load, VL:๐ฝ๐ณ ๐ฝ๐ป๐ณ ๐ ๐ ๐ฝ๐ ๐ ๐ช๐ณwhere:๐ฝ ๐๐ฝ๐๐ ๐๐ท๐(๐ ะ๐ณ ๐ ๐๐ท๐ )40

School of EngineeringStep 7: Calculate ๐ฐ๐ณ ๐๐ง๐ ๐ท๐ณ๐ฝ๐ณ๐ฐ๐ณ ๐๐ณ๐ท๐๐๐ ๐ท๐ณ ๐ฐ๐ ๐๐๐๐๐๐๐,๐ ๐๐ฐ๐ณ ๐น๐(๐๐ณ )๐๐บ๐ณ ๐ฝ๐ณ ๐ฐ ๐ณ ๐ท๐ณ ๐ ๐ธ๐ณ41

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:๐ฝ๐ณ ๐ ๐ Then with ๐๐๐ ๐๐๐๐๐๐ ๐๐๐ ๐ ๐๐๐ณ ๐ ๐ cos 107 ๐ก 60 ๐๐๐๐๐42

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 โblack boxโwith 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 โBlack 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:๐ฆ1๐11 ๐21๐ฆ2๐12๐22๐ฅ1๐ฅ250

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:๐ฆ1๐11 ๐21๐ฆ2๐12๐22๐ฅ1๐ฅ2 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:๐๐ ๐ด ๐๐ ๐ต ๐ผ๐ ๐ผ๐ ๐ถ ๐๐ ๐ท ๐ผ๐ In Matrix Form๐๐๐ด ๐ถ๐ผ๐๐ต๐ท๐๐ ๐ผ๐ ๐ด๐ท ๐ต๐ถ 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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