Transmission Lines And Power Flow Analysis

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 fluxΦ B d𝑠 𝐼 ,𝑂𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛SΦL 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 (𝜇) can beobtained as follows:The total magnetic field Bx at x via Amperes law:𝜇𝑜 𝑥𝐵𝑥 𝐼 (𝑊𝑏/𝑚)22𝜋𝑟The flux d through slice dx is given by:𝑑Φ 𝐵𝑥 𝑑𝑎 𝐵𝑥 𝑙 𝑑𝑥 (𝑊𝑏)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:𝑟Φ 𝑟𝑑Φ 00𝑥𝑟2𝜇𝑜 𝑥 𝐼𝑙𝑑𝑥2𝜋𝑟 2Integrating with subsequent algebra to obtain inductance-per-m yields:𝐿𝑤𝑖𝑟𝑒 𝑠𝑒𝑙𝑓 0.5 10 7 (𝐻/𝑚)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 7lnDr′(H/m)where 𝑟 ′ 𝑟𝑥′ 𝑟𝑦′ 𝑒1 4𝑟𝑥41

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

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

School of Engineering𝐿𝑥 2 10 7𝐷𝑥𝑦ln𝐷𝑥𝑥where𝐷𝑥𝑦 𝑀𝑁𝑁𝑀𝐷𝑘𝑚𝑘 1𝑚 1′𝑎𝑛𝑑𝐷𝑥𝑥 𝑁2𝑁𝑀𝐷𝑘𝑚𝑘 1𝑚 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 ‘transposition’ The inductance (H/m) of a completely transposed three phase line may also becalculated46

School of Engineering𝐿𝑎 2 𝐷𝑒𝑞 10 73𝐷𝑒𝑞ln𝐷𝑠𝐷12 𝐷13 𝐷23La has units of (H/m)Ds is the conductor GMR for stranded conductors or r’ for solid conductors47

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

School of Engineering𝐿𝑎 2 10 7𝐷𝑒𝑞ln𝐷𝑆𝐿La 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: 𝐷𝑆𝐿 3-conductor bundle: 𝐷𝑆𝐿 4-conductor bundle: 𝐷𝑆𝐿 1694𝐷𝑆 𝑑2 𝐷𝑆 𝑑𝑑3 𝐷𝑆 𝑑𝑑𝑑 24𝐷𝑆 𝑑3𝐷𝑆 𝑑 24 1.091 𝐷𝑆 𝑑 349

School of EngineeringTL Parameters - Capacitance50

School of EngineeringQ CVC 𝑆 𝜀 𝐸 𝑑𝑠𝑙𝐸 𝑑𝑙𝜀RC 𝜎

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 – which actsas an insulating medium – 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 EngineeringGauss’s Law may be used to calculate the electric field E and thepotential V for this geometry:𝑞𝐸𝑥 (𝑉/𝑚)2𝜋𝜀𝑥The potential (voltage) of the wire is found using this electric field:𝑉12𝑞 2𝜋𝜀𝐷2𝐷1𝑑𝑥𝑞𝐷2 𝑙𝑛𝑥2𝜋𝜀𝐷1𝑉𝑜𝑙𝑡𝑠The 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:𝐶𝑛 2𝜋𝜀𝑙𝑛𝐷𝑟(𝐹/𝑚)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 yields𝐶𝑎𝑛2𝜋𝜀 �� 3𝐷𝑎𝑏 𝐷𝑎𝑐 𝐷𝑏𝑐59

School of EngineeringThe Capacitance of a Transposed Line60

School of EngineeringThe Capacitance of a Transposed Line𝐶𝑎𝑛 2𝜋𝜀𝐷𝑒𝑞𝑙𝑛 𝐷𝑆𝐶(𝐹/𝑚)where𝐷𝑒𝑞 3𝐷𝑎𝑏 𝐷𝑎𝑐 𝐷𝑏𝑐2-conductor bundle: 𝐷𝑆𝐶 3-conductor bundle: 𝐷𝑆𝐶 𝑟𝑑3𝑟 𝑑244-conductor bundle: 𝐷𝑆𝐶 1.091 𝑟 𝑑 361

School of EngineeringTLs ContinuedDr. Greg MowryAnnie SebastianMarian Mohamed1

School of EngineeringMaxwell’s Eqns. & theTelegraph Eqns.2

School of Engineering3

School of EngineeringMaxwell’s Equations & Telegraph Equations The ‘Telegrapher's Equations’ (or) ‘Telegraph Equations’ 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 1880’s. The telegraph equations lead to the conclusion that voltages andcurrents propagate on TLs as waves.4

School of EngineeringMaxwell’s Equations & Telegraph Equations The Transmission Line equation can be derived using Maxwell’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 EngineeringMaxwell’s Equations – 4 LawsGauss’ 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.𝝆 𝑫 𝝐The ‘no name’ ME: The net magnetic flux through any closedsurface is zero. 𝑩 𝟎7

School of EngineeringFaraday’s law: Faraday's ‘law of induction’ 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. 𝑩 𝑿𝑬 𝒕Ampere’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. 𝑫 𝑿𝑯 𝑱 𝒕8

School of EngineeringThe Constitutive Equations Maxwell’s 4 equations are further augmented by the‘constitutive equations’ 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:𝑩 𝝁𝑯𝝁 𝝁𝒓 𝝁𝒐𝑫 𝜺𝑬𝝐 𝜺𝒓 𝜺𝒐9

School of EngineeringThe Telegraph Eqns.10

School of EngineeringTL Equations TL equations can be derived by two methods: Maxwell’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 R’ Conductor resistance per unit length, Ω/m L’ Conductor inductance per unit length, H/m C’ Capacitance per unit length between TL conductors, F/m G’ 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 ‘abcd’ loopKCL at node bVL L i tOLL for an inductorIC C v tOLC for a capacitorSet 𝑅′ 𝐺 ′ 0 for the Lossless TL case15

School of Engineering Apply KVL around the “abcd” loop: 𝑉(z) V(z z) 𝑅′ z 𝐼(𝑧)′I(z) 𝐿 z 𝑡 Divide all the terms by z : 𝑉 𝑧 𝑧 𝑉 𝑧 𝑧 𝑧 𝑅′ 𝐼 𝑧 𝐿′ 𝐼(𝑧) 𝑡 As z 0 in the limit and R’ 0 (lossless case): 𝑉(𝑧) 𝑧′ – 𝑅𝐼 𝑧 𝑽(𝒛) 𝒛 ′ 𝐼(𝑧) 𝐿 𝑡′ 𝑰(𝒛)–𝑳 𝒕(Eq.1)

School of Engineering Apply KCL at node b: 𝐼 (z) I(z z) 𝐺 ′ z 𝑉(𝑧 𝑧)′V(z z) 𝐶 z 𝑡 Divide all the terms by z: 𝐼 𝑧 𝑧 𝐼 𝑧 𝑧 𝑧 𝐺 ′ 𝑉 𝑧 𝑧 𝐶 ′ 𝑉(𝑧 𝑧) 𝑡 As z 0 in the limit and G’ 0 (lossless case): 𝐼(𝑧) 𝑧 – 𝐺 ′𝑉 𝑧 𝐶′ 𝑰(𝒛) 𝒛 –′ 𝑽(𝒛)𝑪 𝒕 𝑉(𝑧) 𝑡(Eq.2)

School of EngineeringSummarizing(Eqn. 1) 𝑉 𝑧 ,𝑡 𝑧(Eqn. 2) 𝐼 𝑧 ,𝑡 𝑧 ′ 𝐼 𝑧 ,𝑡𝐿 𝑡′ 𝑉 𝑧 ,𝑡𝐶 𝑡Equations 1 & 2 are called the “Coupled First Order Telegraph Equations”18

School of Engineering Differentiating (Eq.1) wrt ‘z’ yields: 2 𝑉(𝑧) 2 𝑧 –2′ 𝐼(𝑧)𝐿 𝑧 𝑡(Eq.3) Differentiating (2) wrt ‘t’ yields: 2 𝐼(𝑧) 𝑧 𝑡 –2′ 𝑉(𝑧)𝐶 𝑡 2(Eq.4) Plugging equation 4 into 3: 𝟐 𝑽(𝒛) 𝒛𝟐 𝟐 𝑽(𝒛) 𝑳′ 𝑪′ 𝒕𝟐(Eq.5)19

School of EngineeringReversing the differentiation sequence Differentiating (Eq.1) wrt ‘t’ yields: 2 𝑉(𝑧) 𝑧 𝑡 – 2 𝐼(𝑧)′𝐿 𝑡 2(Eq.6) Differentiating (2) wrt ‘z’ yields: 2 𝐼(𝑧) 2 𝑧 –2′ 𝑉(𝑧)𝐶 𝑧 𝑡(Eq.7) Plugging equation 6 into 7: 𝟐 𝑰(𝒛) 𝒛𝟐 𝟐 𝑰(𝒛) 𝑳′ 𝑪′ 𝟐 𝒕(Eq.8)20

School of EngineeringSummarizing(Eqn. 5) 2 V(z) z2(Eqn. 8) 2 I(z) z2 2 V(z) L′ C′ t2 2 I(z) L′ C′ 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 ‘wave’ propagates with Phase Velocity 𝑢u ωβ ωω L ′ C′ 1L ′ C′22

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 𝑢𝑡 determines the direction of wave propagation: 𝑓 𝑧 𝑢𝑡 Wave traveling in the z direction 𝑓 𝑧 𝑢𝑡 Wave traveling in the –z direction23

School of EngineeringThe ‘ z’ Traveling WE Solution 𝑓 𝑧, 𝑡 𝑓(𝑧 𝑢𝑡) 𝑓 𝑧 𝑢𝑡) 𝑓(0 𝑧 𝑢𝑡 0 𝑧 𝑢𝑡24

School of Engineering The argument of the wave equation solution is called thephase. Hence the culmination of all of this is that, “Wavespropagate by phase change” 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) L′ C′ t2 2 I(z) L′ C′ 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:𝒗𝒔𝒔 𝒕 𝟏𝟎 cos 107 𝑡 30 𝒗𝒐𝒍𝒕𝒔 From the time function we form the source phasor & know thefrequency that the TL operates at (in ACSS operation):𝛚 𝟏𝟎𝟕 𝒔 𝟏𝑽𝒔𝒔 𝟏𝟎 𝒋𝟑𝟎𝒆𝒗𝒐𝒍𝒕𝒔30

School of EngineeringACSS TL Analysis Under ACSS conditions the voltage and current solutions to the WEmay be expressed as: 𝑉 𝑧, 𝑡 𝑅𝑒 𝑉0 𝑒 𝑗(𝜔𝑡 γ𝑧) 𝑉0 𝑒 𝑗(𝜔𝑡 γ𝑧) 𝑉 𝑧 𝑉0 𝑒 𝑗γ𝑧 𝑉0 𝑒 𝑗γ𝑧Time Harmonic Phasor Form I 𝑧, 𝑡 𝑅𝑒 𝐼0 𝑒 𝑗(𝜔𝑡 γ𝑧) 𝐼0 𝑒 𝑗(𝜔𝑡 γ𝑧) I 𝑧 𝐼0 𝑒 𝑗γ𝑧 𝐼0 𝑒 𝑗γ𝑧 𝑉 𝑗γ𝑧𝑒𝑍0 𝑉 𝑗γ𝑧𝑒𝑍0Time Harmonic Phasor Form

School of EngineeringACSS TL Analysis Recipe Step 1: Assemble 𝑍0 , 𝑍𝐿 , 𝑙, 𝑎𝑛𝑑 γ Step 2: Determine Г𝐿 Step 3: Determine 𝑍𝑖𝑛 Step 4: Determine 𝑉𝑖𝑛 Step 5: Determine 𝑉0 Step 6: Determine 𝑉𝐿 Step 7: Calculate 𝐼𝐿 and 𝑃𝐿32

School of EngineeringStep 1: Assemble 𝒁𝟎 , 𝒁𝑳 , 𝒍, , and 𝜸 Characteristic Impedance:𝒁𝒐 𝑳′𝑽 𝑽𝒐𝒐 𝑪′𝑰𝒐 𝑰𝒐 Load Impedance: 𝑽𝑳𝑽 𝑽𝟎𝟎𝒁𝑳 𝒁𝟎 𝑰𝑳𝑽𝟎 𝑽𝟎 Complex Propagation Constant: ( 0 for lossless TL)𝜸 𝜶 𝒋𝜷 𝒋𝝎 𝑳′ 𝑪′33

School of EngineeringAdditional Initial Results Phase velocity uP: Identity:𝛚𝐮𝐏 𝛃𝟏𝐋′ 𝐂 ′𝐋′ 𝐂 ′ 𝛍 𝛆34

School of EngineeringStep 2: Determine Г𝑳 Reflection Coefficient: Г𝐿 𝑉0 𝑉0 Г𝐿 𝑍𝐿 𝑍0𝑍𝐿 𝑍0 𝑧𝐿 𝑍𝐿𝑍0 𝐼0 𝐼0𝒛𝑳 𝟏Г𝑳 𝒛𝑳 𝟏 Reflection Coefficient at distance z (from load to source -l):Г 𝒛 𝜸𝒛𝑽 𝒐 𝒆 𝜸𝒛𝑽 𝒐 𝒆 Г𝑳 𝒆𝒋𝟐𝜷𝒛35

School of EngineeringAdditional Step-2 Results VSWR:𝟏 𝚪𝑳𝐕𝐒𝐖𝐑 𝟏 𝚪𝑳 Range of the VSWR:𝟎 𝑽𝑺𝑾𝑹 36

School of EngineeringStep 3: Determine 𝒁𝒊𝒏 Input Impedance:𝒁𝒊𝒏𝑽𝒊𝒏 𝒛 𝒍𝒁𝑳 𝒁𝟎 𝐭𝐚𝐧𝐡(𝜸𝒍) 𝒁𝟎𝑰𝒊𝒏 𝒛 𝒍𝒁𝟎 𝒁𝑳 𝒕𝒂𝒏𝒉(𝜸𝒍) For a lossless line tanh γl j tan(βl):𝒁𝒊𝒏𝒁𝑳 𝒋𝒁𝟎 𝒕𝒂𝒏(𝜷𝒍) 𝒁𝟎𝒁𝟎 𝒋𝒁𝑳 𝒕𝒂𝒏(𝜷𝒍) At any distance 𝑧1 from the load:𝒁𝑳 𝒋𝒁𝟎 𝐭𝐚𝐧(𝜷𝒛𝟏 )𝒁(𝒛𝟏 ) 𝒁𝟎𝒁𝟎 𝒋𝒁𝑳 𝒕𝒂𝒏(𝜷𝒛𝟏 )37

School of EngineeringStep 4: Determine 𝑽𝒊𝒏𝑽𝒊𝒏 𝑽𝑺𝑺𝒁𝒊𝒏 𝑽𝒊𝒏 𝒛 𝒍𝒁𝒊𝒏 𝒁𝑺38

School of EngineeringStep 5: Determine 𝑽𝟎 𝑉 𝑧 𝑉0 𝑒 𝑗𝛽𝑧 𝑉0 𝑒 𝑗𝛽𝑧 𝑉0 (𝑒 𝑗𝛽𝑧 Г𝐿 𝑒 𝑗𝛽𝑧 ) 𝑉𝑖𝑛 𝑉𝑇𝐿 𝑧 𝑙 𝑉0 (𝑒 𝑗𝛽𝑙 Г𝐿 𝑒 𝑗𝛽𝑙 )𝑽𝟎 𝑽𝒊𝒏 𝒋𝜷𝒍(𝒆 Г𝑳 𝒆 𝒋𝜷𝒍 )39

School of EngineeringStep 6: Determine 𝑽𝑳 Voltage at the load, VL:𝑽𝑳 𝑽𝑻𝑳 𝒛 𝟎 𝑽𝒐 𝟏 𝚪𝑳where:𝑽 𝟎𝑽𝒊𝒏 𝒋𝜷𝒍(𝒆 Г𝑳 𝒆 𝒋𝜷𝒍 )40

School of EngineeringStep 7: Calculate 𝑰𝑳 𝐚𝐧𝐝 𝑷𝑳𝑽𝑳𝑰𝑳 𝒁𝑳𝑷𝒂𝒗𝒈 𝑷𝑳 𝑰𝒏 𝒈𝒆𝒏𝒆𝒓𝒂𝒍,𝟏 𝟐𝑰𝑳 𝑹𝒆(𝒁𝑳 )𝟐𝑺𝑳 𝑽𝑳 𝑰 𝑳 𝑷𝑳 𝒋 𝑸𝑳41

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:𝑽𝑳 𝟓 𝒆 Then with 𝒋𝟔𝟎 𝒗𝒐𝒍𝒕𝒔𝛚 𝟏𝟎𝟕 𝒔 𝟏𝒗𝑳 𝒕 𝟓 cos 107 𝑡 60 𝒗𝒐𝒍𝒕𝒔42

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 ‘black box’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 “Black Box” 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:𝑦1𝑎11 𝑎21𝑦2𝑎12𝑎22𝑥1𝑥250

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:𝑦1𝑎11 𝑎21𝑦2𝑎12𝑎22𝑥1𝑥2 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 – Sending end voltage IS – Sending end current VR – Receiving end Voltage IR – Receiving end current54

School of EngineeringRepresentation of a TL by a 2-port network Relation between sending end and receiving end is given as:𝑉𝑆 𝐴 𝑉𝑅 𝐵 𝐼𝑅𝐼𝑆 𝐶 𝑉𝑅 𝐷 𝐼𝑅 In Matrix Form𝑉𝑆𝐴 𝐶𝐼𝑆𝐵𝐷𝑉𝑅𝐼𝑅𝐴𝐷 𝐵𝐶 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