Coaxial Line - USPAS
Lecture 5June 17, 2003A. NassiriTransmission Line ReviewEquations,Definitions,ProcessesReflection and Transmissions CoefficientsCoaxial LineArbitrary ImpedanceArbitrary ImpedanceShort CircuitOpen CircuitMatched ImpedanceComplex Plane (Conformal Mapping) and Smith ChartMicrowave Physics and TechniquesUCSB –June 2003
Transmission Line EquationsApply Kirchhoff’s voltage and current laws: i (z ,t )ν(z ,t ) R zi (z ,t ) L z ν(z z ,t ) t ν(z z ,t )i (z ,t ) G zν(z z ,t ) C z i (z z ,t ) tDivide by z and taking the limit z0: i (z ,t ) ν(z ,t ) Ri (z ,t ) L z t i (z ,t ) ν(z ,t ) Gν(z ,t ) C z tPhasorMicrowave Physics and Techniques V (z ) (R jωL )I (z ) z I (z ) (G jωC )V (z ) zUCSB –June 2003
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 zMicrowave Physics and TechniquesUCSB –June 2003
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 β z φ V 0 γz V 0 γzI (z ) e eZ0Z0R j ωLZ0 G j ωCMicrowave Physics and Techniques)2πλ βων p λfβUCSB –June 2003
The Lossless LineIn 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 V 0 e γz V 0 eV 0 γz V 0 γzI (z ) e eZ0Z0ω1νp βLCMicrowave Physics and TechniquesUCSB –June 2003
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.Field Theory :W m µ0*H Hds 4 SCircuit Theory :W m Self Inductance : L LI02ε0*E Eds 4S24µ′I0We We * HHds H m SMicrowave Physics and TechniquesC V0C 24ε′V02* EEds F m SUCSB –June 2003
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.Field Theory : P c Rs2Circuit Theory : P c RGper Meter R * H H dlC 1 C 22R I0RsI0Pd 2Pd 2*2ε′′ H H dl Ω mC 1 C 2Microwave Physics and Techniques*E Eds SGV02G 2ωε′′V02* EEds S m SUCSB –June 2003
Example 1yρµ,εbV0ˆE e γz ρρ ln(b a )θaRSxH I02πρe 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.εMicrowave Physics and TechniquesUCSB –June 2003
Example 1 – Calculation of Lyρµ,εbH θaRSxL µI0I02πρ*2e γz ˆφµ2π b H H ds (2π)2 S01 ρ 2 ρdρdφaµ b ln 2π a Microwave Physics and TechniquesUCSB –June 2003
Example 1 – Calculation of Ryρµ,εbH θaR I02C12πρe γz ˆφxRSRSI02π2π R11S * φ φadbdH Hdl 2 22 ab()π2φ 0φ 0 C2RS 1 1 2π a b Microwave Physics and TechniquesUCSB –June 2003
Parameters for Some Common TLswµ,εbaaRSµbL ln 2π a dDµπL cosh 1 2πε′C ln (b a )RS 1 1 R 2π a b 2πωε′′G ln(b a )aD 2a πε′C cosh 1 ((D 2 a ))RSR πaG coshL 1((D 2a ))Microwave Physics and TechniqueswC R πωε′′µdG ε ′wd2R Swωε′′wUCSB –June 2003d
Problem 1: Find characteristic impedance of Coax, with a 0.4cm, b 1.14cmand εr 1.5Zo L' C'µbln2π a η ln b2πε2π ablnaµoε oε r 1 .14 ln2π0 .4120 π ln 2 .85 50 Ω.2π 1 .5Microwave Physics and Techniques12UCSB –June 2003
Average Power transmitted by CoaxSnIP S abb ar S d A S ab1Re{ E H2*r} dA1 Vo VoV o2b2 π rdr πln.2 r ηrηaMicrowave Physics and Techniques13UCSB –June 2003
Problem 2: Find max transmitted power for Coax from Problem 1EmaxVo 2 MV / maV o a 2 10 62Vo212ba 4 10 1.5 bP π ln πln 2.6MWη aa120πMicrowave Physics and Techniques14UCSB –June 2003
Lossless Coaxial LinePropagation ConstantWave ImpedanceCharacteristic ImpedancePower Flowβ ω µε ω LCE ρ ωµµZw ηHφβεV 0 E ρ ln(b a ) Z0 I02πH φP 11**E Hds VI0 02 S 2The 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.This is good to know!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.Microwave Physics and TechniquesUCSB –June 2003
Coaxial cable as a transmission line with TEM modeWhen coaxial cable is terminated by characteristic impedance, the line is perfectly matched and the voltage is constant along theline. The VSWR 1.Coaxial line terminated by characteristic impedance.E-field in coaxial line. Orientation depends on phase - position along the line.Microwave Physics and TechniquesUCSB –June 2003
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 lineMicrowave Physics and TechniquesUCSB –June 2003
Surface current in coaxial line. Orientation depends on phase - position along the line.Microwave Physics and TechniquesUCSB –June 2003
Voltage on coaxial line does not depend on time and position, because the load is matched.Microwave Physics and TechniquesUCSB –June 2003
Voltage on coaxial line does not depend on time and position, because the load is matched.Microwave Physics and TechniquesUCSB –June 2003
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.Microwave Physics and Techniques21UCSB –June 2003
Voltage on coaxial line depends on time and position.Voltage on coaxial line depends on time and position, but does not go to zero.Microwave Physics and Techniques22UCSB –June 2003
Coaxial line terminated by short circuit. The VSWR indefinitely large.Coaxial line terminated by open circuit. The VSWR indefinitely large.Microwave Physics and Techniques23UCSB –June 2003
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.Microwave Physics and Techniques24UCSB –June 2003
Voltage on coaxial line depends on time and position, and at nodes, goes to zero .Microwave Physics and Techniques25UCSB –June 2003
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Line length LMicrowave Physics and Techniques29UCSB –June 2003
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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, in the 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.Microwave Physics and TechniquesUCSB –June 2003
Smith ChartsMicrowave Physics and Techniques39UCSB –June 2003
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 jxZoYy g jbYoz 1yMicrowave Physics and TechniquesUCSB –June 2003
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 1 Γr2 Γi2(1 Γr ) Γi22Microwave Physics and TechniquesxL 2Γi(1 Γr )2 Γi2UCSB –June 2003
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 ΓrrL2 rL 1 Γr 1 Γr rL 1 ΓrrL rL Γr 1 Γr 22 2 rL Γr Γr2 Γi2 1 1 Γr 1 2 Γi 1 Γr 2Microwave Physics and TechniquesUCSB –June 2003
The rL CirclesVx x00.5130 0.5 1 3rU-xConstant resistance lines inthe z r jxMicrowave Physics and TechniquesΓ planeUCSB –June 2003
The xL Circles2ΓixL 22(1 Γr ) Γi(1 Γr ) Γi 2(1 Γr )2(1 Γr )222ΓixL 1 Γi xL xL2222 1 xL 1 1 Γi xL xL 22Microwave Physics and TechniquesUCSB –June 2003
The xL CirclesxV310.5310.5-0.5-1rU-0.5-3Constant reactance lines in (for r 0)in the z r jx-1-3Γ planeMicrowave Physics and TechniquesUCSB –June 2003
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The rL xL CirclesVmapping of z r j 1(r 0)xΓ 1z 1 j1 z j1z 0Γ 1 90 o z -j11Γ 0.447 63.4 oz r j1Γ 0 rΓ 1U z 1-j1mapping of z 1 jxz 0 jxz 1 jxmapping of z 0 jxz plane Z R jX z r jx ZoZo Microwave Physics and TechniquesΓ 0.447 63.4 oΓ plane(Γ U jV )UCSB –June 2003
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 loadRotate 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.Microwave Physics and TechniquesUCSB –June 2003
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 loadMicrowave Physics and TechniquesUCSB –June 2003
NomographVSWRVoltage Reflection CoefficientPower Reflected (%)Return Loss (dB)Power Transmitted (%)Transmission Loss (dB)Microwave Physics and TechniquesUCSB –June 2003
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.Microwave Physics and TechniquesUCSB –June 2003
The ZY Smith ChartImpedance and admittancecircles are both plottedon the Γ-plane.The chart looks confusingbut conversion betweenY and z is automatic.Microwave Physics and TechniquesUCSB –June 2003
standing waves result when a voltage generator of output voltage(VG 1sinωt) and source impedance ZG drive a load impedance ZLthrough a transmission line having characteristic impedance Z0,where ZG Z0/ZL and where angular frequency ω corresponds towavelength l (b). The values shown in Figure a result from areflection 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 would see atime-varying sine wave of 1.5-V peak amplitude (trace c). At pointC, however, which is located one-quarter of a wavelength (l/4)closer to the load, the reflected voltage is 180 out of phase withthe incident voltage and subtracts from the incident voltage, sopeak magnitude is the 1-V incident voltage minus the 0.5-Vreflected voltage, or 0.5 V, and you would see the red trace. Atintermediate points, you’ll see peak values between 0.5 and 1.5 V;at B (offset l/8 from the first peak) in c, for example, you’ll find apeak magnitude of 1 V. Note that the standing wave repeats everyhalf wavelength (l/2) along the transmission line. The ratio of themaximum to minimum values of peak voltage amplitude measuredalong a standing wave is the standing wave ratio,SWR, SWR 3.Figure 1Microwave Physics and Techniques65UCSB –June 2003
Here, point L represents a normalizedload impedance zL 2.5 – j1 0.5/18 (Ichose that particular angle primarily toavoid the need for you to interpolatebetween resistance and reactance circlesto verify the results). The relationship ofreflection coefficient and SWR dependsonly on the reflection coefficientmagnitude and not on its phase. If pointL corresponds to G 0.5 and SWR 3, then any point in the complexreflection-coefficient plane equidistantfrom the origin must also correspond to G 0.5 and SWR 3, and a circlecentered at the origin and whose radiusis the length of line segment OLrepresents a locus of constant-SWRpoints. (Note that the SWR 3 circlehere shares a tangent line with the rL 3circle at the real axis; this relationshipbetween SWR and rL circles holds for allvalues of SWR.)Microwave Physics and Techniques66UCSB –June 2003
Example Problem:A 100-Ω transmission line with an air dielectric is terminated by a load 50-j80Ω. Determinethe values of the VSWR, the reflection coefficient, and the percentage of the reflected power.If the power incident at the load is 100 mW, calculate the power in the load. If the generatorfrequency is 3 GHz and the line is 73 cm long, find the input impedance at the generator.What are the values of the maximum and minimum impedances existing on the line?TLVG100 ΩZR 50-j80Ω73 cmMicrowave Physics and Techniques67UCSB –June 2003
Solution GStep 1: Normalize ZLNZ LN50 j 80 0.5 j 0.89100Step 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.ZGN ZLNLS 3.6Microwave Physics and Techniques68 P 0.561.68dBUCSB –June 2003
Solution cont. GNotice 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. ZGN M N Zc 3 1010 cm / secλ 10cm9f3 10 / secLNLTherefore, 73 cm is equivalent to7.3 λ.S 3.6Microwave Physics and Techniques69 P 0.561.68dBUCSB –June 2003
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 ZGN M N ZOn 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.LNLS 3.6Microwave Physics and Techniques70 P 0.561.68dBUCSB –June 2003
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 ZGN M N ZLNLS 3.6Microwave Physics and Techniques71 P 0.561.68dBUCSB –June 2003
Self Inductance : * . 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.
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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.
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Classical Mechanics and Electro-Magnetic Theory CM&EM. USPAS JAN 19-30, 2004, College of William & Mary, Williamsburg, VA 2 Literature Classical Mechanics, H. Goldstein, Addison-Wesley, 1950 The Classical Theory of Fields, L.D. Landau and E.
Andreas Werner The Mermin-Wagner Theorem. How symmetry breaking occurs in principle Actors Proof of the Mermin-Wagner Theorem Discussion The Bogoliubov inequality The Mermin-Wagner Theorem 2 The linearity follows directly from the linearity of the matrix element 3 It is also obvious that (A;A) 0 4 From A 0 it naturally follows that (A;A) 0. The converse is not necessarily true In .
