Modeling and measuring the non-ideal characteristics of transmission lines
aa r X i v : . [ phy s i c s . e d - ph ] A ug Modeling and measuring the non-ideal characteristics of transmission lines
J. S. Bobowski ∗ Department of Physics, University of British Columbia, Kelowna, British Columbia, Canada V1V 1V7 (Dated: August 11, 2020)We describe a simple method to experimentally determine the frequency dependencies of theper-unit-length resistance and conductance of transmission lines. The experiment is intended as asupplement to the classic measurement of the transient response of a transmission line to a voltagestep or pulse. In the transient experiment, an ideal (lossless) model of the transmission line is usedto determine the characteristic impedance and signal propagation speed. In our experiment, theinsertion losses of various coaxial cables are measured as a function of frequency from 1 to 2000 MHz.A full distributed circuit model of the transmission line that includes both conductor and dielectriclosses is needed to fit the frequency dependence of the measured insertion losses. Our model assumesphysically-sensible frequency dependencies for the per-unit-length resistance and conductance thatare determined by the geometry of the coaxial transmission lines used in the measurements.
I. INTRODUCTION
Lumped-element circuit analysis fails when the wave-lengths of the signals of interest approach the size of thecircuit elements and/or connecting wires. In this limit,the voltage and current along, for example, the lengthof a pair of wires are not uniform and a distributed cir-cuit model of the wires must be used to properly analyzethe circuit behavior.
These so-called transmission lineeffects are rich in physics and, in many cases, can defycommon intuition. Furthermore, due to the ever decreas-ing size of circuits and increasing data rates, transmissionline effects are more prevalent than ever. From a pedagogical standpoint, transmission lines arecommonly used when deriving the expression for the ther-mal noise radiated by a resistor. In the derivation atransmission line, with both ends terminated by matchedload resistors, is treated as a 1-D blackbody. The thermalpower radiated by one resistor is completely absorbedby the other and the emitted radiation satisfies stand-ing wave conditions (normal modes) set by the lengthof the transmission line. Transmission lines models havealso been used to analyze problems in thermodynamicsand mechanics. An RC transmission line circuit modelhas been used to understand diffusion of heat along thelength of a conducting bar, and an analogy has beenmade between a system of coupled pendula and coupledtransmission lines. It is also worth pointing out that there have been veryclever uses of discrete transmission lines. In one exam-ple, reverse-biased variable capacitance diodes (varactordiodes) were used to construct a nonlinear transmissionline that supports solitons. In a second example, the ca-pacitance in some sections of the discrete transmissionline was changed to mimic a change in refractive index.These structures were used to experimentally demon-strate the principles behind highly-reflective dielectricmirrors and confinement of electromagnetic (EM) wavesby Bragg reflection. Many of the undergraduate transmission line experi-ments described in the literature focus on the propaga-tion of short pulses along the length of a line that is as- sumed to be lossless. These measurements allow studentsto determine the propagation speed of the pulse and ob-serve phase changes resulting from reflections at variousload terminations. By tuning the load resistance to elim-inate the reflections, students can also estimate the char-acteristic impedance of the transmission line.
An-other experiment in which dissipative effects are typicallyneglected is the transient response of a transmission lineto an applied pulse that is many times longer than thetime required to travel the length of the line.
This mea-surement is particularly interesting because a wide vari-ety of intricate voltage transients are possible dependingon the values of the source and load impedances used.In this paper, we demonstrate one of the simpler casesusing an open-circuit load and a large source resistance.After using the measured transient response to deter-mine the characteristics of a lossless line, we turn to ourmain focus which is to quantitatively measure and ana-lyze transmission line dissipation due to conductor anddielectric losses. There are examples of undergraduatelaboratories in which the dissipation due to a length oftransmission line has been measured.
However, inthese cases, after making an attenuation measurement,there is little to no discussion about the origins of thedissipation and the relative importance of the varioussources of loss. One of the objectives of this paper isto make a simple, yet reasonably precise, measurementof the power dissipation due to a length of transmissionline over a wide frequency range. Another, more signifi-cant, objective is to use the data and physical insights toquantitatively determine the relative magnitudes of con-ductor and dielectric losses as a function of frequency.The pedagogical benefits of our work are as follows:(1) We describe two experiments that yield precise re-sults, yet are relatively simple and can be done withinthe usual three-hour period of a university teaching lab-oratory. (2) Some of the theory needed to interpret theexperimental results is typically not explicitly covered inan undergraduate physics program. Therefore, studentsare required to research and develop some of the relevanttheory themselves. (3) The main focus is characterizingdissipation in transmission lines which is of great practi- (a)(b)FIG. 1. (a) A transmission line of length ℓ connected to a signal source with output impedance Z at x = − ℓ and a loadimpedance Z L at x = 0. (b) The distributed circuit model of a transmission line. cal importance, but not often emphasized in a classroomsetting. (4) In order to fit the attenuation data, studentsmust develop models for conductor and dielectric lossesin a coaxial cable. These exercises require students to ac-tivate prior knowledge which gets integrated it into theirmodels and leads to new physical insights.The outline of the paper is as follows: In Sec. IIthe distributed circuit model of a transmission line andsome of its important features are reviewed. Section IIIpresents the transient response of transmission lines to along-duration pulse. The experimental results are usedto determine the per-unit-length capacitance and induc-tance of ideal (lossless) transmission lines. Dissipativeeffects are treated in Sec. IV. First, the expected fre-quency response of the transmission line insertion lossis calculated and then the experimental measurementsare presented. In Sec. IV A, physically-motivated mod-els for the frequency dependencies of the per-unit-lengthresistance and conductance of coaxial transmission linesare developed. These models are then used to fit theexperimentally-measured insertion loss in Sec. IV B. Fi-nally, the main results are summarized in Sec. V. II. DISTRIBUTED CIRCUIT MODEL OF ATRANSMISSION LINE
Figure 1(a) shows a transmission line of length ℓ . Oneend, at x = − ℓ , is connected to a signal source that hasoutput impedance Z and the opposite end, at x = 0, isterminated by load impedance Z L . The equivalent dis-tributed circuit model of the transmission line is shownin Fig. 1(b). It is made up of n = ℓ/ ∆ x daisy-chainedsegments, where ∆ x is the length of each segment. Thesegments have series inductance and resistance L ∆ x and R ∆ x , respectively and shunt capacitance and conduc-tance C ∆ x and G ∆ x , respectively. In this model, R represents conductor losses and G accounts for dielec- tric losses. The distributed circuit model describes thebehavior of real transmission lines in the limit that thenumber of segments n → ∞ or, equivalently, ∆ x → We also referthe reader to alternative treatments given in Refs. 1–3with additional insights provided in Ref. 19. In the limit∆ x →
0, an analysis of the Kirchhoff loop and junctionrules for a single segment of the transmission line circuitmodel leads to a pair of coupled first-order partial dif-ferential equations in voltage and current known as thetelegrapher’s equations ∂v∂x = − L ∂i∂t − Ri (1) ∂i∂x = − C ∂v∂t − Gv, (2)where v = v ( x, t ) and i = i ( x, t ). Taking the Laplacetransform, defined as F ( x, s ) = R ∞ o f ( x, t ) e − st dt , leadsto the following representation of the of the telegrapher’sequations in the complex frequency, or s , domain ∂V∂x = − ( R + sL ) I (3) ∂I∂x = − ( G + sC ) V, (4)where, for simplicity, we have assumed that the ini-tial transmission line voltage and current are everywherezero. Equations (3) and (4) can be re-expressed as a pairof independent second-order partial differential equations ∂ V∂x = γ V (5) ∂ I∂x = γ I, (6)where γ ≡ p ( R + sL ) ( G + sC ). The solutions toEqs. (5) and (6) are traveling waves V ( x, s ) = V + e − γx + V − e γx (7) I ( x, s ) = 1 Z c (cid:2) V + e − γx − V − e γx (cid:3) , (8)where V + and V − terms represent signals propagating inthe + x and − x directions, respectively, and Z c = r R + sLG + sC , (9)is the characteristic impedance of the transmission line.The V + and V − coefficients are related via V − = Γ V + where Γ = Z L − Z c Z L + Z c , (10)is the reflection coefficient determined by the mismatchbetween Z c and the load termination. Equation (10) isobtained by requiring V (0) /I (0) = Z L at the load termi-nation.The steady state solutions for harmonic voltages andcurrents of angular frequency ω are given by Eqs. (7) and(8) with s → jω . In this case, V and I represent the volt-age and current amplitudes along the length of the trans-mission line and the propagation constant γ = α + jβ iscomplex, where α and β are the attenuation and phaseconstants, respectively. The time-dependent transientsolutions, on the other hand, can be obtained by takingthe inverse Laplace transform of V ( x, s ) and I ( x, s ). A. Lossloss transmission lines
In a lossless transmission line R and G are assumed tobe negligible. In this limit, the propagation constant andcharacteristic impedance become γ = s √ LC = s/v (11) Z c = p L/C, (12)where v = 1 / √ LC is the signal propagation speed. Notethat, if v and Z c are measured, then the inductanceper unit length and capacitance per unit length of thetransmission line can be determined via L = Z c /v and C = ( v Z c ) − . III. TRANSIENT RESPONSE
We now consider a length of lossless transmission linewith one end open ( Z L → ∞ ) and the opposite end con-nected to a resistance R g ≫ Z c . A square pulse of height V is applied to the free end of R g . The width of pulseis chosen to be very long compared to the time ℓ/v thatit takes signals to travel the length of the transmission (a)(b) . . . . . . . . . . . time ( ➭ s) V g / V FIG. 2. (a) Schematic diagram of the experimental setup usedto measure the transient response of a transmission line. (b)The transient response of a long semi-rigid UT-141 coaxialcable. The voltage V g ( t ) has been scaled by the maximumvoltage reached a long time after the pulse is applied. Al-though data was collected up to 10 µ s, only the data up to2 µ s are shown in order to highlight the step-like features in V g ( t ). line. After the pulse is applied, the time evolution of thevoltage V g at the junction between R g and the transmis-sion line is measured. The experimental setup is shownschematically in Fig. 2(a). To ensure a well-defined pulseshape, R g is chosen to be much larger than the outputimpedance of the pulse generator. In our experiments,we used R g = 1 kΩ and an HP 8011A pulse generatorwith a 50-Ω output impedance. A list of the equipmentneeded to perform all of the measurements described inthis paper along with possible vendors and cost estimatesis given in the appendix.Although we do not provide a full analysis of this prob-lem here, we refer the reader to the treatment given inSec. 14.4 of Ref. 1. Figure 2(b) shows a measurement of V g as a function to time using a 8 . V g ( t ) (a) . . . . . N t N ( ➭ s ) (b) − . − . − . − . − . N l n ( ∆ V g / V ) N FIG. 3. Analysis of the transient response of the semi-rigid coaxial cable. (a) Step time versus step number. The error barsare smaller than the point size. The red line is a linear fit to the data. (b) Plot of ln (∆ V g /V ) N as a function of step number.The red line is a linear fit to the data. is due to repeated reflections at the two ends of thetransmission line. The reflection coefficient at the openend is Γ = 1 and at the source end it is given byΓ g = ( R g − Z c ) / ( R g + Z c ) ≈ . t = 2 ℓ/v whichcorresponds to the time required for signals to traveltwice the length of the line. A plot of the time of the N th step versus N results in a straight line with slope m = ∆ t which can then be used to determine the signalpropagation speed as has been done in Fig. 3(a).For steps N ≥
2, the change in V g is given by (cid:18) ∆ V g V (cid:19) N = 1 − Γ g Γ N g , (13)such that, because 0 < Γ g <
1, the voltage steps decreasein size as N increases. A plot of ln (∆ V g /V ) N versus N is linear with slope m = ln Γ g which allows for adetermination of the characteristic impedance Z c of thetransmission line.Figures 3(a) and (b) show results of the analysis ofthe transient voltage steps for the semi-rigid coaxial ca-ble. The slopes of the linear fits and the correspondingvalues of v and Z c are given in Table I. The table alsoincludes determinations of C , L , and the dielectric con-stant ε ′ = ( c/v ) of the insulator separating the innerand outer conductors of the coaxial cables. Althoughnot shown, the transient responses of an RG-58 BNCcoaxial cable and a high-quality (HQ) sma coaxial cablewere also measured. As will be shown in Sec. IV B, theconductor and dielectric losses of the HQ cable are lowcompared to those of the other cables measured. Thiscable, of unknown make and model, was donated to theundergraduate lab and has 9-mm diameter, a cloth-woven TABLE I. Parameters extracted from an analysis of the tran-sient response of various coaxial cables.
RG-58 UT-141 HQ sma ℓ (m) 7 . ± .
02 8 . ± .
03 8 . ± . m (ns) 79 . ± . . ± . . ± . v /c . ± .
002 0 . ± .
003 0 . ± . ε ′ . ± .
02 2 . ± .
02 1 . ± . | m | . ± .
002 0 . ± .
001 0 . ± . Z c (Ω) 53 . ± . . ± . . ± . C (pF / m) 99 ± ± ± L (nH / m) 279 ± ± ± jacket, and is fitted with specialty sma connectors.The RG58C/U coaxial cable specifications fromPasternack give a velocity of propagation that is 0 . c and a capacitance of 101 .
05 pF / m, both of which arein reasonably good agreement with our results. Onthe other hand, Pasternack specifies a characteristicimpedance of 50 Ω which is about 3.5 standard devia-tions away from our measurement. We will discuss apossible reason for this difference in Sec. IV B. The spec-ifications for the UT-141-HA-M17 semi-rigid coaxial ca-ble from Micro-Coax are v = 0 . c , C = 98 . / m, and Z c = 50 ± Figure 2(b) shows a rounding of the steps that becomesmore pronounced as time evolves. When the transientresponses of the three cables are compared, there are noobvious differences in the observed rounding. This effectmay be due to the onset of higher-order modes in thecoaxial transmission lines. The TE mode has the lowestcutoff frequency given by f c ≈ . v / ( r + r ) where r is the radius of the center conductor and r is the innerradius of the outer conductor. For the semi-rigid coaxialcable, with r = 0 .
460 mm and r = 1 .
493 mm, the TE cutoff frequency is approximately 35 GHz. IV. DISSIPATION AND FREQUENCYRESPONSE
We now consider the simple scenario depicted inFig. 1(a). The signal source outputs a sinusoidal wave ofangular frequency ω and we wish to calculate the powerdelivered to the load impedance Z L . We assume thatboth R and G are small but not negligible and considerthe steady-state solutions given by Eqs. (7) and (8).Assuming that the second order cross term RG in γ is small and that, at all frequencies of interest, R/ ( ωL ) + G/ ( ωC ) ≪
1, the propagation constant canbe approximated as γ ≈ j ωv + 12 (cid:18) RZ + GZ (cid:19) ≡ j ωv + 12 α + , (14)where, as before, v = 1 / √ LC and we denote thecharacteristic impedance of a lossless transmission as Z = p L/C . As is usaully the case, the outputimpedance of the signal generator is assumed to also beequal to Z . Using the same approximations, the char-acteristic impedance of a lossy transmission line given byEq. (9) with s = jω can be written as Z c ≈ Z (cid:20) − jv ω (cid:18) RZ − GZ (cid:19)(cid:21) ≡ Z (cid:18) − jv ω α − (cid:19) . (15)In Eqs. (14) and (15), the quantities α ± ≡ ( R/Z ± GZ )have been defined. Next, using Eq. (7) and the fact that V − = Γ V + , V + can be expressed in terms of the voltageamplitude at x = − ℓ : V + = V − ℓ (cid:2) e γℓ + Γ e − γℓ (cid:3) − . Sub-stituting this result for V + back into Eq. (7) and usingEq. (10) for Γ allows one to solve for the voltage ampli-tude V L = V (0) at the load impedance Z L V L V − ℓ = 2 (cid:20)(cid:0) e γℓ + e − γℓ (cid:1) + Z c Z L (cid:0) e γℓ − e − γℓ (cid:1)(cid:21) − . (16)Substituting in the approximate forms of γ and Z c fromEqs. (14) and (15) yields (cid:18) V L V − ℓ (cid:19) − = (cid:20) cos ωℓv cosh α + ℓ Z Z L (cid:18) cos ωℓv sinh α + ℓ − v α − ω sin ωℓv cosh α + ℓ (cid:19)(cid:21) + j (cid:20) sin ωℓv sinh α + ℓ Z Z L (cid:18) sin ωℓv cosh α + ℓ v α − ω cos ωℓv sinh α + ℓ (cid:19)(cid:21) . (17)Identifying p and q as the real and imaginary partsof ( V L /V − ℓ ) − , respectively, allows one to calculate | V L /V − ℓ | = (cid:0) p + q (cid:1) − / and tan φ = − q/p , where φ isphase difference between the voltages at x = − ℓ and x = 0. In the lossless limit, α + = α − = 0, Eq. (17)reduces to V L V − ℓ = (cid:20) cos ωℓv + j Z Z L sin ωℓv (cid:21) − , (18)such that, as expected, | V L /V − ℓ | = 1 when Z L = Z .Our objective was to measure the ratio of the signalpower at x = 0 to the power at x = − ℓ as a function offrequency and then compare it to | V L /V − ℓ | calculatedfrom Eq. (17). This comparison requires models for thefrequency dependencies of the per-unit-length resistanceand conductance of the coaxial transmission lines usedin our measurements. These models are developed in thenext section. A. Models of resistance and conductance
First, we consider the resistance which is due to theusual Joule heating in conductors. Figure 4(a) shows aschematic diagram of a section of the center conductorfrom a coaxial cable. The current in the center conductoris restricted to a region that is within an EM skin depth d of the surface. For a good conductor d ≈ r ρµ ω , (19)where ρ is the resistivity of the conductor and µ is the permeability of free space. Equation (19) isvalid provided that d ≫ l e , where l e = m e v F / (cid:0) ne ρ (cid:1) is the conduction electron mean free path and m e isthe electron mass, v F ∼ m / s is the Fermi ve-locity, and n ∼ m − is the conduction electronnumber density. For copper, ρ ≈ . × − Ω msuch that l e ≈
20 nm and d ≈ (cid:0) µ m GHz / (cid:1) / √ f where f = ω/ (2 π ). These order-of-magnitude estimates showthat Eq. (19) is expected to be valid for f < ∼ (a)(b)FIG. 4. (a) Schematic drawing of the center conductor ofa coaxial cable of radius r . The shaded region depicts thecross-sectional area through which the current travels whichis determined by the frequency-dependent EM skin depth d .(b) Schematic drawing of a coaxial cable. The space betweenthe center and outer conductors is filled with a dielectricmaterial (shaded region) with complex relative permittivity ε r = ε ′ − jε ′′ . Referring again to Fig. 4(a), the cross-sectional areathrough which the current in the center conductor flowsis A ≈ πr d . A similar argument can be made for thecurrent in the outer conductor such that the total per-unit-length resistance of the coaxial cable can be esti-mated as R ≈ ρ A + ρ A = 12 π r µ ω (cid:20) √ ρ r + √ ρ r (cid:21) , (20)where we have allowed for the possibility that the innerand outer conductors have different resistivities given by ρ and ρ , respectively. Equation (20) can be expressedmore conveniently in terms of effective parameters R = 12 πr eff r µ ωρ eff , (21)where r − = r − + r − and √ ρ eff = r √ ρ + r √ ρ r + r . (22) Equation (22) has the desired property that the re-sistivity of the small-radius conductor is more heavilyweighted. The critical insight from Eqs. (20) and (21) isthat R ∝ ω / with a constant of proportionality that isdetermined by the conductor resistivities and geometricalfactors.Next, we turn our attention to G . A schematic diagramof the coaxial cable cross-section is shown in Fig. 4(b).The per-unit-length capacitance of a coaxial cable isgiven by C = 2 πε r ε ln ( r /r ) , (23)where ε r is the relative permittivity of the dielectric ma-terial filling the space between the center and outer con-ductors and ε is the permittivity of free space. For alossy dielectric, the relative permittivity is ε r = ε ′ − jε ′′ such that the capacitive admittance becomes Y C = jωC = 2 πωε ( jε ′ + ε ′′ )ln ( r /r ) . (24)The real term G = 2 πωε ε ′′ / ln ( r /r ) is identified asthe per-unit-length conductance.In general, both the real and imaginary parts of ε r can have their own nontrivial frequency dependen-cies. The dielectrics in our coaxial cables are poly-tetrafluoroethylene (PTFE, Teflon) or polythene (PE).Attenuation in a dielectric is characterized by the losstangent tan δ = ( σ + ωε ′′ ) /ε ′ . For Teflon, the conduc-tivity σ ≪ ωε ′′ at all frequencies in the microwaverange such that tan δ ≈ ε ′′ /ε ′ . The loss tangent and ε ′ of Teflon have been measured precisely by a va-riety of techniques over a wide range of frequencies.From 100 MHz to 60 GHz, both ε ′ and tan δ have beenshown to be independent of frequency, with ε ′ ≈ . × − < tan δ < × − . In the analysispresented in Sec. IV B, which includes measurementsthat span 1 MHz to 2 GHz, we assume a frequency-independent ε ′′ such that G ∝ ω with the constant ofproportionality determined by ε ′′ and geometrical fac-tors. B. Power ratio measurements
The simple experimental setup shown schematically inFig. 1(a) was used to measure the rms power P L deliv-ered to a load impedance at the end of a transmissionline. The signal source used was a Rohde & SchwarzSMY 02 signal generator with a 50-Ω output impedanceand the load impedance was a Boonton 41-4E power sen-sor with a 50-Ω input impedance coupled with a Boon-ton Model (a) . . . . . . − − − − − − − Frequency (GHz) | V L / V − ℓ | ( d B ) high-quality smasemi-rigid smaRG-58 BNCRG-58 BNC R -only fit (b) − − − − − Frequency (Hz) R / Z , G Z ( m − ) RG-58 BNCsemi-rigid smahigh-quality sma GZ ∼ fR/Z ∼ f / FIG. 5. (a) The measured insertion loss of the three coaxial transmissions lines tested. The fits to the data (black lines), using | V L /V − ℓ | calculated from Eq. (17) and assuming R ∼ f / and G ∼ f , are excellent. The dashed line is a fit to the RG-58BNC cable data assuming R/Z ≫ GZ at all measurement frequencies. The fit returned a = (2 . ± . × − Ω s / m.(b) The R/Z (solid lines) and GZ (dashed lines) values extracted from the fits in (a) plotted as a function of frequency. frequency and the average of the values was recorded.Twenty averages were used in the measurements reportedin this section.The calibration of the Boonton power sensor is fre-quency dependent and there can be small variations inthe power output by the signal generator when sweepingover a wide frequency range. To remove both of these ef-fects from the measured data, we repeated the same fre-quency scan and measured P − ℓ with the power meter con-nected directly to the output of the signal generator. Theratio P L /P − ℓ is independent of both the calibration of thepower sensor and small variations in the output power.Note that this power ratio is equivalent to the scatteringparameter S which can measured quickly and preciselyusing a vector network analyzer (VNA). The results ofthe measurements for the RG-58 BNC, semi-rigid UT-141, and HQ sma cables are shown in Fig. 5(a). The dataare shown on a decibel scale and are a measure of the in-sertion loss resulting from the transmission lines. Therms power ratio P L /P − ℓ is also equivalent to | V L /V − ℓ | which can be calculated from Eq. (17). Table II showsthat the measured insertion losses are in good agreementwith manufacturer specifications. Fits to the insertion loss data, assuming R = af / and G = bf are also shown in Fig. 5(a). For all but theRG-58 cable, the v = 1 / √ LC and Z = p L/C valuesobtained from the transient response analysis were usedsuch that a and b were the only fit parameters. The fitsare are in good agreement with the developed theory butdo not capture the ≈ TABLE II. Measured insertion losses of the RG-58, UT-141,and HQ sma coaxial cables at various frequencies. The man-ufacturer specifications, when given, are shown in brackets.
10 MHz 0 . . RG-58 0.045 0.15 0.40 0.61(dB / m) (0.046) (0.16) (0.66)UT-141 0.040 0.11 0.25 0.37(dB / m) (0.26) (0.39)HQ sma 0.021 0.063 0.14 0.20(dB / m)TABLE III. Parameters extracted from fits to the insertionloss data shown in Fig. 5(a). For the RG-58 cable, an ad-ditional Z fit parameter was used. For the UT-141 cable,estimates of ρ eff and tan δ = ε ′′ /ε ′ were calculated from a and b , respectively. RG-58 UT-141 HQ sma a (10 − Ω s / m) 1 . ± .
003 1 . ± .
002 0 . ± . b (10 − s / Ω / m) 6 . ± .
03 1 . ± .
03 0 . ± . Z (Ω) 49 . ± . ρ eff (10 − Ω m) 1 . ε ′′ /ε ′ (10 − ) 2 . modulations has not been identified. The values of a and b extracted from the fits are given in Table III. Forthe semi-rigid UT-141, for which r and r are preciselyknown, we estimated the values of ρ eff and tan δ = ε ′′ /ε ′ using ρ eff = ( a r eff ) µ / (4 π ) (25)tan δ = b ln ( r /r )4 π ε ε ′ . (26)The results of these estimates using r = 0 .
460 mm, r = 1 .
493 mm, and r eff = 0 .
352 mm are given in Ta-ble III.The center conductor of the UT-141 cable is silver-plated copperweld (SPCW). Copperweld is a wire inwhich a copper cladding is bonded to a steel core. Theconductivity of the cladding can be anywhere from 30to 70 % IACS (International Annealed Copper Standard,58 . × Ω − m − ). The silver plating will reduce theoverall effective resistivity which, depending on the thick-ness of the plating, could be frequency dependent due tothe changes in the EM skin depth. Given these consider-ations, the extracted estimate of ρ eff , being about 15 %greater than that of copper, is very reasonable.The UT-141 coaxial cable has a Teflon dielectric. Theextracted value of tan δ falls directly within the rangeof values reported in the literature. This result isremarkable because typically precision resonator tech-niques are required to accurately measure dissipation inlow-loss dielectrics.When fitting the insertion loss data of the RG-58 cable,the characteristic impedance of 53 . V g measured in the transient analysis is reduced by0 . Z was included as an additional fit parameter. The best-fitvalue returned was Z = 49 .
71 Ω which results in oscilla-tion amplitudes that are much closer to those observedin the measured data. We note that it would also be pos-sible to deduce the characteristic impedance of the lineusing Eq. (10) and a separate measurement of the re-flection coefficient Γ as a function of the load impedance Z L .If Z is included as a parameter in the fits to the semi-rigid and HQ sma data, a fit value that is in experimentalagreement with results from the transient analysis arereturned. Finally, we note that the average period of theobserved oscillations highlighted in Fig. 6 agrees very wellwith the expected value of v / (2 ℓ ) = 12 . R and G for all of the transmission lines mea-sured using the values of a and b extracted from the
940 960 980 1000 1020 1040 1060 − . − . − . − . − . − . Frequency (MHz) | V L / V − ℓ | ( d B ) RG-58 BNC data Z = 53 . Z = 49 .
71 Ω fit
FIG. 6. Insertion loss frequency oscillations. The green lineshows the same RG-58 BNC data displayed in Fig. 5(a), butover a narrow frequency range. The purple line is a fit to thedata using a fixed value of Z = 53 . Z wasincluded as a fit parameter. fits (see Table III). At the lowest frequencies, R/Z isat least two orders of magnitude greater that GZ suchthat α + ≈ α − ≈ R/Z . However, because G increasesmore quickly with frequency than R , its contribution tothe overall insertion loss becomes more important as fre-quency is increased. In the case of the RG-58 cable, di-electric losses match conductor losses, and then surpassthem, by 2 GHz. This crossover occurs by 25 GHz for theUT-141 and HQ sma cables. The dashed line in Fig. 5(a)shows a fit to the insertion loss of the RG-58 cable assum-ing that G is negligible. Clearly, the dielectric losses needto be included to capture the full frequency dependenceof the measured insertion loss.Finally, we note that at high frequencies where dielec-tric losses dominate or in situations requiring very lowattenuation, air-dielectric coaxial cables are available. Inone example, a helical polyethylene spacer is used to keepthe inner conductor concentric with the outer conductor,both of which are made from corrugated copper. The in-sertion loss of the HELIFLEX air-dielectric coaxial cableis specified to be 0 .
021 dB / m at 1 GHz which is an orderof magnitude less than the value found for the “high-quality” sma cable characterized in this work. Waveg-uides can also be used in low-loss applications. For ex-ample, Mega Industries offers WR950 waveguide that canbe used between 0 .
75 and 1 .
12 GHz and has an insertionloss less than 0 .
005 dB / m. V. SUMMARY
We have described two experiments that, together, canbe used to fully characterize the properties of transmis-sion lines. Both experiments are simple to set up andmake use of equipment that is either commonly avail-able in undergraduate labs or relatively inexpensive toacquire.First, the transient response to a voltage step wasused determine the transmission line signal propaga-tion speed and characteristic impedance or, equivalently,the per-unit-length capacitance and inductance. In thiswell-known experiment, the data analysis assumes thatthe transmission lines are lossless. We found that thisapproximation worked well for the relatively low-losssemi-rigid UT-141 and high-quality sma coaxial cables.However, we found evidence suggesting that losses inthe RG-58 coaxial cable were causing the characteristicimpedance to be systematically overestimated.Our main objective was to use insertion loss measure-ments to determine the per-unit-length resistance andconductance of the same coaxial cables used in the firstexperiment. With one end of the transmission line drivenby a sinusoidal voltage source, and assuming small butnon-negligible losses, an expression for the power deliv-ered to a load termination Z L was derived. Based on thegeometry of the coaxial cables, the frequency dependenceof the conductor losses was assumed to be determined bythe EM skin depth such that R ∝ f / . An analysis ofthe dielectric losses, due to ε ′′ , was used to deduce that G ∝ f .The insertion loss measurements, where possible, werecompared to manufacturer specifications and found to bein good agreement. Fits to the data using the theoret-ical model developed were excellent. It was shown thatboth the R and G contributions were required to cap-ture the full frequency dependence of the measured in-sertion losses. The parameters extracted from the semi-rigid UT-141 fit were used to make reasonable estimatesof the cable’s effective resistivity and the loss tangent ofthe Teflon dielectric.These measurements also serve to highlight the im-portance of the non-ideal characteristics of transmissionlines, which are often not emphasized in theoretical treat-ments at the undergraduate level. For example, despiteimpedance matching at the source and load, at 2 GHzonly 20 % of the incident power is delivered to the Z L termination at the end of a 7 . ℓ = 8 .
04 m), although the power transfer efficiency increased, at 58 % efficiency, substantial atten-uation was still observed.
Appendix: Parts and Suppliers
This appendix provides a list of the equipment, withpossible vendors and prices, required to reproduce allparts of the transient response and insertion loss experi-ments described in this paper.
RG-58 BNC Coaxial Cable – A 25-foot length (7 . ). Semi-Rigid Coaxial Cable – A 25-foot length (7 . ). Pusle Generator – The HP 8011A pulse generator usedfor the transient response experiment is obsolete. A pos-sible substitute is the Rigol DG812 10 MHz signal genera-tor which can be purchased for $299 from Digi-Key Elec-tronics. Note that it is also possible to create a suitablevoltage step using only a PP3 9-V battery and jumperwires.
Oscilloscope – We used a Tektronix TBS 1104 oscillo-scope to record the voltage transients. The two-channelversion of this oscilloscope (TBS 1102) can be purchasedfor $1190 from Newark ( ).Many college and university physics departments will al-ready have a suitable oscilloscope available.
Signal Generator – The signal for our insertion lossmeasurements was provided by a Rohde & SchwarzSMY02 9 kHz to 2080 MHz signal generator. A suit-able substitute would be the Rigol DSG815 9 kHzto 1 . ). Power Meter/Sensor – Our power measurements weremade a using Boonton 42BD power meter with Boon-ton 41-4E power sensors. The Mini-Circuits ZX47-50-S+power detector can be used from 10 MHz to 8 GHz tomeasure powers from −
45 to +15 dBm and costs $90( ). ACKNOWLEDGMENTS
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