Transmission Line A. Nassiri
Lecture 2Transmission LineA. NassiriMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 2010
Equations Definitions Processes Reflection and Transmissions Coefficients- Coaxial Line Arbitrary Impedance Arbitrary Impedance Short Circuit Open Circuit Matched Impedance Complex Plane (Conformal Mapping) and Smith ChartMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 20102
Transmission Line Equations Apply Kirchhoff’s voltage and current laws: i (z ,t ) ν(z z ,t )ν(z ,t ) R zi (z ,t ) L z t ν(z z ,t ) i (z z ,t )i (z ,t ) G zν(z z ,t ) C z t Divide by z and taking the limit z0: i (z ,t ) ν(z ,t ) Ri (z ,t ) L t z ν(z ,t ) i (z ,t ) Gν(z ,t ) C t zMassachusetts Institute of Technology V (z ) (R jωL )I (z ) z I (z ) (G jωC )V (z ) zRF Cavities and Components for AcceleratorsUSPAS 20103
Traveling Wave Solutions 2V (z ) I (z ) (R jωL )2 z z I (z ) (G jωC )V (z ) z 2V (z ) (R jωL )[ (G jωC )V (z )]2 z 2V (z ) (R jωL )(G jωC )V (z )2 zMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 20104
Traveling Wave SolutionsV (z ) V 0 e γz V 0 e γzγ α jβ (R jωL )(G jωC )()(ν(z ,t ) V 0 cos ωt β z φ V 0 cos ωt βz φ V 0 γz V 0 γzI (z ) e eZ0Z0R j ωLZ0 G j ωCMassachusetts Institute of Technology)2πλ βων p λfβRF Cavities and Components for AcceleratorsUSPAS 20105
The Lossless Line In many practical cases, the loss of the line is very small and socan be ignored.γ jβ jω LCL Y0CZ0 12π2π λ β ω LCV (z ) γz V0 e γz V 0 eV 0 γz V 0 γzeeI (z ) Z0Z01ωνp βLCMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 20106
Field Analysis of T.L.EBC2C1C1and C2 are line integration contours,S is the cross-sectional surface.SField lines on an arbitrary TEM transmission line.µField Theory : Wm 0 H H * ds4 SLICircuit Theory : Wm 04Self Inductance : L µ′I0Massachusetts Institute of Technology22 εWe 0 E E * ds4 SC V0We 4H H * ds H mC SRF Cavities and Components for Accelerators2ε′V02 E E*ds FmSUSPAS 20107
Field Analysis of T.L.EBC2C1C1and C2 are line integration contours,S is the cross-sectional surface.SField lines on an arbitrary TEM transmission lineRsField Theory : Pc 2Circuit Theory : Pc RGper meter R Massachusetts Institute of TechnologyRsI02 *H H dlC1 C 22R I02 H H * dl Ω mPd ε′′2 E E * dsSG V0Pd 2G C1 C 2RF Cavities and Components for Acceleratorsωε′′V02 2E E * ds SmSUSPAS 20108
Exampleyρµ,εbV0ˆE e γz ρρ ln(b a )θaxH RSI02πρe γz ˆφGeometry of a coaxial line with surface resistance RS on the inner andouter conductors.γ is the propagation constant of the line.RS is the surface resistivity. ε′ jε′′µ µ 0µ r.εMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 20109
Example – Calculation of Lyρµ,εbH θaxRSL Massachusetts Institute of TechnologyµI0I02πρµ2π b H H ds (2π)2 S0*2e γz ˆφ1 ρ 2 ρdρdφaµ b ln 2π a RF Cavities and Components for AcceleratorsUSPAS 201010
Example – Calculation of Ryρµ,εbH θaI02πρe γz ˆφxRSR RSI02C12π2π R11S * adbd φ φH Hdl 2 22 ab()2πφ 0φ 0 C2RS 1 1 2π a b Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201011
Parameters for Some Common TLswµ,εbRSaaaµbL ln 2π a 2πε′C ln(b a )RS 1 1 R 2π a b 2πωε′′G ln(b a )Massachusetts Institute of TechnologydDµπD 2a πε′C cosh 1 ((D 2a ))L cosh 1 RSR πaG coshL 1((D 2a ))RF Cavities and Components for AcceleratorswC R πωε′′µdG USPAS 2010ε′wd2R Swωε′′wd12
Problem : Find characteristic impedance of Coax, with a 0.4cm, b 1.14cmand εr 1.5L'Zo C'µ bln2π a η ln b2πε2π ablnaµoε oε r 1.14ln 2π0.4120πln 2.85 50Ω. 2π 1.5Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201013
Average Power transmitted by CoaxSnI P S dA S ab 1* 2 Re{E H } dASabbVo2b1 Vo Vo2πrdr πln . aη2 r ηraMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201014
Problem : Find max transmitted power for Coax from Problem 1EmaxVo 2 MV / maVo a 2 1062212Vo ba 4 101.5 bP πln πln 2.6 MWηMassachusetts Institute of Technologya120πRF Cavities and Components for AcceleratorsaUSPAS 201015
Lossless Coaxial Lineβ ω µε ω LC Propagation Constant Wave Impedance Characteristic Impedance Power FlowE ρ ωµµZw η HφεβV 0 E ρ ln(b a ) Z0 I02πH φP 11**E Hds VI0 02 S 2 The flow of power in a transmission line takes place entirely via the electric andmagnetic fields between the two conductors; power is not transmitted through theconductors themselves. For the case of finite conductivity, power may enter the conductors, but thispower is then lost as heat and is not delivered to the load.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201016
Coaxial cable as a transmission line with TEM modeWhen coaxial cable is terminated by characteristic impedance, the line is perfectly matched and the voltage isconstant along the line. The VSWR 1Coaxial line terminated by characteristic impedance.E-field in coaxial line. Orientation depends on phase - position along the line.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201017
Surface charge density induced in coaxial line. The sign depends on phase - position alongthe line.H-field in coaxial line. Orientation depends on phase - position along the lineMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201018
Surface current in coaxial line. Orientation depends on phase - position along the line.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201019
Voltage on coaxial line does not depend on time and position, because the load is matched.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201020
Voltage on coaxial line does not depend on time and position, because the load is matched.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201021
Coaxial line terminated by an impedance different than the characteristic impedance.E-field in coaxial line. Orientation depends on phase - position along the line. The VSWR depends onreflection coefficient.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201022
Voltage on coaxial line depends on time and position.Voltage on coaxial line depends on time and position, but does not go to zero.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201023
Coaxial line terminated by short circuit. The VSWR indefinitely large.Coaxial line terminated by open circuit. The VSWR indefinitely large.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201024
E-field in coaxial line. Orientation depends on phase - position along the line.Voltage on coaxial line depends on time and position. The VSWR indefinitely large.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201025
Voltage on coaxial line depends on time and position, and at nodes, goes to zero .Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201026
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Smith ChartSmith Chart was developed in 1939 by P. Smith at the BellTelephone Laboratory.One can develop intuition about transmission line andimpedance-matching problems by learning to think in terms ofthe Smith Chart.Smith Chart is essentially a plot of the voltage reflectioncoefficient, Γ, inthe complex plane.It can be used to convert from voltage reflection coefficient ( Γ )to normalized impedances (z Z/Z0) and admittances (y Y/Y0),and vice versa.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201039
Smith ChartMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201040
Complex Γ Plane with z CirclesThe Smith Chart is a plot of the voltage reflection coefficient, Γ, onthe complex plane superimposed with impedance circles.The complex Γ plane.The impedance circles.Zz r jxZ Yy g jbY z 1yMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201041
Conformal Mapping - Γand ZIf a lossless line of characteristic impedance Z0 is terminated with a loadimpedance ZL, the reflection coefficient at the load can be written as:z L 1Γ Γ e jθz L 1ZLwhere z L Z0This relation can be solved for zL in terms of Γ to give:1 Γ e jθ 1 Γr jΓizL rL jx Ljθ1 Γr jΓi1 Γe(())The real and imaginary part of the above equation can be found bymultiplying the numerator and denominator by the complex conjugate of thedenominator to give:rL Massachusetts Institute of Technology1 Γr2 Γi2(1 Γr ) Γi22xL RF Cavities and Components for Accelerators2Γi(1 Γr )2 Γi2USPAS 201042
The rL CirclesrL 1 Γr2 Γi2(1 Γr )2 Γi2rL 2 rL Γr rL Γr2 rL Γi2 1 Γr2 Γi22 rL Γr Γr2 Γi2 11 Γr 1 ΓrrL rL 1 Γr 1 ΓrrL rL Γr 1 Γr 22 rL 1 Γr2 2 rL Γr Γr2 Γi2 1 1 Γr 1 2 Γi 1 Γr Massachusetts Institute of Technology 2RF Cavities and Components for AcceleratorsUSPAS 201043
The rL CirclesVx x00.5130r0.513U-xConstant resistance lines inthe z r jxMassachusetts Institute of TechnologyΓ planeRF Cavities and Components for AcceleratorsUSPAS 201044
The xL CirclesxL 2Γi(1 Γr ) Γi22(1 Γr ) Γi 2(1 Γr )2(1 Γr )222ΓixL 1 Γi xL xL2222 1 xL 1 1 Γi xL xL Massachusetts Institute of Technology22RF Cavities and Components for AcceleratorsUSPAS 201045
The xL CirclesxV310.5310.5-0.5-1rU-0.5-3Constant reactance lines in (for r 0)in the z r jxMassachusetts Institute of Technology-1-3Γ planeRF Cavities and Components for AcceleratorsUSPAS 201046
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The rL xL CirclesVmapping of z r j 1(r 0)xΓ 1z 1 j1 z j1z 0Γ 1 90 z -j11Γ 0.447 63.4 z r j1Γ 0 rΓ 1U z 1-j1mapping of z 1 jxz 0 jxmapping of z 0 jxz 1 jxz plane Z R jX z r jx Z Z Massachusetts Institute of TechnologyΓ 0.447 63.4 Γ plane(Γ U jVRF Cavities and Components for Accelerators)USPAS 201060
Features of the Smith ChartAll resistance circles have centers on the horizontal Γi 0 axis and allpass through the point Γ 1.The reactance circles have centers on the vertical Γr 1 line, and passthrough the point Γ 1.The resistance and reactance circles are orthogonal.Smith Chart can be used to compute the normalized input impedance ata distance l away from the load. Plot the reflection coefficient ( Γ )at the load Rotate the point cw an amount 2βl around the center of the chart (toward thegenerator)One full rotation around the center of S.C. corresponding to a phaseshift of 0.5λ . A 180 rotation corresponding to λ/4 transformation; thisfacilitate the impedance circles to be used as admittance circles.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201061
Features of the Smith ChartΓ plotted directly in magnitude and phaseOutside of chart (r 0 circle) is Γ 1Center pf chart is Γ 0 (i.e.: Z Z0 )Top half – inductive x or capacitive b circlesBottom half – capacitive x or inductive b circlesLeft side is short – right side is openPosition on TL? – Rotation, λ toward generator or loadMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201062
NomographVSWRVoltage Reflection CoefficientPower Reflected (%)Return Loss (dB)Power Transmitted (%)Transmission Loss (dB)Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201063
The Z Smith ChartOnly impedance circles are plottedon the Γ-plane.The chart looks quite “clean” butconversion between Y and Z mustbe done by a λ/4 rotation.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201064
The ZY Smith ChartImpedance and admittancecircles are both plottedon the Γ-plane.The chart looks confusingbut conversion betweenY and z is automatic.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201065
standing waves result when a voltage generator of outputvoltage (VG 1sinωt) and source impedance ZG drive a loadimpedance ZL through a transmission line having characteristicimpedance Z0, where ZG Z0/ZL and where angular frequencyω corresponds to wavelength l (b). The values shown in Figurea result from a reflection coefficient of 0.5.Here, probe A is located at a point at which peak voltagemagnitude is greatest—the peak equals the 1-V peak of thegenerator output, or incident voltage, plus the in-phase peakreflected voltage of 0.5 V, so on your oscilloscope you wouldsee a time-varying sine wave of 1.5-V peak amplitude (tracec). At point C, however, which is located one-quarter of awavelength (l/4) closer to the load, the reflected voltage is180 out of phase with the incident voltage and subtracts fromthe incident voltage, so peak magnitude is the 1-V incidentvoltage minus the 0.5-V reflected voltage, or 0.5 V, and youwould see the red trace. At intermediate points, you’ll seepeak values between 0.5 and 1.5 V; at B (offset l/8 from thefirst peak) in c, for example, you’ll find a peak magnitude of1 V. Note that the standing wave repeats every halfwavelength (l/2) along the transmission line. The ratio of themaximum to minimum values of peak voltage amplitudemeasured along a standing wave is the standing waveratio,SWR, SWR 3.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsFigure 1USPAS 201066
Here, point L represents a normalizedload impedance zL 2.5 – j1 0.5/18 (I chose that particular angle primarilyto avoid the need for you to interpolatebetween resistance and reactancecircles to verify the results). Therelationship of reflection coefficient andSWR depends only on the reflectioncoefficient magnitude and not on itsphase. If point L corresponds to G 0.5 and SWR 3, then any point in thecomplex reflection-coefficient planeequidistant from the origin must alsocorrespond to G 0.5 and SWR 3,and a circle centered at the origin andwhose radius is the length of linesegment OL represents a locus ofconstant-SWR points. (Note that theSWR 3 circle here shares a tangentline with the rL 3 circle at the realaxis; this relationship between SWRand rL circles holds for all values ofSWR.)Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201067
Using the standing-wave circle, you can determine inputimpedances looking into any portion of a transmission linesuch as Figure 1’s if you know the load impedance. Figure1, for instance, shows an input impedance Zin to bemeasured at a distance l0 from the load (toward thegenerator). Assume that the load impedance is as given bypoint L in Figure 2. Then, assume that l0 is 0.139wavelengths. (Again, I chose this value to avoidinterpolation.) One trip around the Smith chart isequivalent to traversing one-half wavelength along astanding wave, and Smith charts often include 0- to 0.5wavelength scales around their circumferences (usuallylying outside the reflection-coefficient angle scalepreviously discussed).Such a scale is show in yellow in Figure 2, whereclockwise movement corresponds to movement away fromthe load and toward the generator (some charts also includea counter-clockwise scale for movement toward the load).Using that scale, you can rotate the red vector intersectingpoint L clockwise for 0.139 wavelengths, ending up at theblue vector. That vector intersects the SWR 3 circle atpoint I, at which you can read Figure 1’s input impedanceZin. Point I lies at the intersection of the 0.45 resistancecircle and –0.5 reactance circle, so Zin 0.45 – j0.5.Massachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsFigure 2USPAS 201068
A 100-Ω transmission line with an air dielectric is terminated by a load 50-j80Ω. Determine thevalues of the VSWR, the reflection coefficient, and the percentage of the reflected power. If thepower incident at the load is 100 mW, calculate the power in the load. If the generator frequencyis 3 GHz and the line is 73 cm long, find the input impedance at the generator. What are thevalues of the maximum and minimum impedances existing on the line?TLVG100 ΩZR 50-j80Ω73 cmMassachusetts Institute of TechnologyRF Cavities and Components for AcceleratorsUSPAS 201069
SolutionStep 1: Normalize ZLNZ LN G50 j 80 0.5 j 0.89 100Step 2: Plot ZLN and draw VSWRcircle thru ZLN. M NStep 3: Mark off a length equal to thecircles’ radius on the H scale. Thereflection coefficient is P 0.56. Sameon the G scale for reflected power,P2 100 0.32 100 32%. The reflectedpower is (32/100) 100 32 mW andthe power in the load is 100-32 68mW. ZLNL S 3.6Massachusetts Institute of TechnologyZGNRF Cavities and Components for AcceleratorsP 0.561.68dBUSPAS 201070
Solution cont.Notice that the reflected power is 10log(32/100) -4.95dB w.r.t the incidentpower while the load power whencompared with the incident power is 10log(68/100) -1.68dBStep 4: From the center of the chartdraw a line thru ZLN and extend the lineto the peripheral scales. G M Nc 3 1010 cm / sec 10cmλ 9f3 10 / sec ZLNTherefore, 73 cm is equivalent to7.3 λ.L S 3.6Massachusetts Institute of TechnologyZGNRF Cavities and Components for AcceleratorsP 0.561.68dBUSPAS 201071
Solution cont.Identical impedance values on amismatched line repeat every half-wavelength, which is the distance covered byone complete revolution on the Smithchart. A distance of 7.3 λ. will require14 complete revolution together withadditional rotation of 0.3 λ. in the CWdirection toward the generator. G M NOn the inner peripheral wavelengthscale, the distance from ZLN position tothe null position, N, is 0.118λ. We mustthen travel toward the generator afurther 0.3λ-0.118λ 0.182λon theoutmost peripheral scale and arrive atG. ZLNL S 3.6Massachusetts Institute of TechnologyZGNRF Cavities and Components for AcceleratorsP 0.561.68dBUSPAS 201072
Solution cont.Step 5: Draw a line from G to the centerof chart. Point ZGN of intersectionbetween this line and the VSWR circlerepresentthenormalizedinputimpedance at the generator; therefore,ZGN 1.18 j1.43 and the de-normalizedvalue is 118 j143Ω. The min. and max.impedance values on the line,respectively occur at N and M. At N thenormalized value is 0.28 j0 so min.impedance is 28 Ω (Z0/S) and the M is3.6 j0 so max. impedance is 360 Ω or(S Z0). G M N ZLNL S 3.6Massachusetts Institute of TechnologyZGNRF Cavities and Components for AcceleratorsP 0.561.68dBUSPAS 201073
Self Inductance : 2 0 2 0 . When coaxial cable is terminated by characteristic impedance, the line is perfectly matched and the voltage is constant along the line. The VSWR 1 Coaxial line terminated by characteristic impedance. E-field in coaxial line. Orientation depends on phase - position along the line.
2) Medium transmission line – the line length is between 80km to 160 km 3) Long transmission line – the line length is more than 160 km Whatever may be the category of transmission line, the main aim is to transmit power from one end to another. Like other electrical system, the transmission network also will have
transmission line model. A transmission line is always a transmission line. A Simpler Approximation At the lowest frequencies, there is another ideal electrical circuit element that can be used to approximate the impedance of a real, physical transmission line. The measured impedance of the real
going into the transmission line will change depending on the length of the transmission line. This can only happen of the input impedance to the transmission line is changing. Backward Wave Forward Wave 0 2 0 V e z V Transmission Line Behavior Slide 26 Derivation of Input Impedance, Zin(2 of 2)
5 What Are Lines And Stanzas? Line A line is pretty self-explanatory. Line A line of a poem is when it jumps Line To a new, well, line, Line Like this! Line Sometimes a line is a complete sentence. Line But it doesn’t Line Have to be! Line A stanza is kind of like a paragraph. Line Stanzas are made up of lines. Line This “stanza” has five lines.
SPICE has two transmission-line components: T, the lossless transmission-line component; and O, the lossy transmission-line component. The lossless line simulates only delay and characteristic impedance. For the lossy line; you must enter the primary line parameters, but at this time it is
BOLD- Transmission Line Design Considerations The responsibility to successfully implement the BOLD technology in real world transmission line projects ultimately falls on the transmission line engineer. Once a project has been identified as a candidate for BOLD, the transmission line process will be similar to a traditional line design project. As
The T-junction power divider Lossless divider, lossy divider The Wilkinson power divider Even-odd mode analysis, unequal power division divider, N-way Wilkinson divider The quadrature (90 ) hybrid branch-line coupler Coupled line directional couplers Even- and odd-mode Z 0, single-section and multisection coupled line
Department of Plant Biology, University of Newcastle upon Tyne, Newcastle upon Tyne, NEl 7RU, United Kingdom ABSTRACT: 200 taxa of algae were recovered from cultures of 24 "terres trial" and "hydro-terrestrialll soil and vegetation samples from Glerar dalur, northern Iceland. 22 of the samples were collected at heights of between 500 and 1300 m. The algae were divided between the classes .
