Dispersion in Coaxial Cables

Transcription

1 Dispersion in Coxil Cbles Steve Ellingson June 1, 2008 Contents 1 Summry 2 2 Theory 2 3 Comprison to Welch s Result 4 4 Findings for RG58 t LWA Frequencies 5 Brdley Dept. of Electricl & Computer Engineering, 302 Whittemore Hll, Virgini Polytechnic Institute & Stte University, Blcksburg VA USA. E-mil: ellingson@vt.edu 1

2 1 Summry At the LWA Pre-SRR nd Kickoff Meeting in Albuquerque in September 2008, TAC Member Jck Welch suggested tht we consider the possibility tht coxil feedlines would introduce significnt dispersion in our frequency rnge. He provided memo written in 1990 describing nlysis nd mesurement of dispersion on 1/2-in cox t frequencies in the rnge 90 MHz to bout 1 GHz. I hve generlized his nlysis nd pplied it to the specil cse of RG58-type cbles in our frequency rnge. I find tht the excess dely due to dispersion is ) ) 1/2 l f τ d = 4.78 ns) 1) 100 m 10 MHz This is significnt; for comprison consider tht the free-spce light trvel time over 4 m roughly the element spcing in the sttion rry) is bout 13 ns nd frequency independent. The dispersion cn be improved slightly but probbly only t gret expense) by using coxil cbles hving lrger crosssection with inner nd outer conductors hving higher conductivity. This dispersion will probbly need to be tken into ccount in clibrtion nd observtions of ny resonble bndwidth. Formuls re provided for computing the dispersion dely nd ttenution of ny coxil cble given the relevnt mteril nd geometricl prmeters. 2 Theory In this section I will use essentilly the sme generl pproch s Jck but with different strting point nd retining ll the relevnt prmeters so tht results cn be generted for vriety of cble types. A reference which contins ll the formuls nd constnts used here is [1] my personl fvorite), but the sme informtion cn be found in vrious textbooks on engineering electromgnetics. The field inside ny trnsmission line propgtes ccording to the fctor e γz where γ is the complex-vlued propgtion constnt nd z is distnce long the line. The propgtion constnt is given by γ = R + jωl)g + jωc) 2) where R, L, G, nd C re the series resistnce, series inductnce, shunt conductnce, nd shunt cpcitnce, respectively; ω = 2πf where f is frequency; nd j = 1. The shunt conductnce describes the lekge of current through the medium seprting the inner nd outer conductors nd is negligible for well-designed trnsmission lines. For exmple, for RG59 which is similr to RG58 nd for which I just hppen to hve these prmeters worked out from pst project) C is bout 100 pf/m nd G is bout 200 µs/m, so t 10 MHz the cpcitive rectnce is ωc 6000 µs/m; i.e., lredy more thn 30 times greter thn G nd incresing with frequency. Thus we pproximte: γ R + jωl)jωc = ω 2 LC + jωrc 3) At this point is useful to identify nd mke explicit the frequency dependence. C is given by C = 2πǫ ln b/) 4) where ǫ is the permittivity of the medium between the inner nd outer conductors of the cox, nd nd b re the rdii of the inner conductor nd the fcing surfce of the outer conductor, respectively. ǫ = ǫ r ǫ 0 where ǫ 0 = F/m nd ǫ r is the reltive permittivity, which is property of the mteril which is essentilly frequency-independent over the rnge of frequencies of interest. Thus, C is frequency independent. L ccounts for two sources of inductnce. The idel inductnce is tht 2

3 ssocited with the mgnetic component of the propgting field between the conductors nd is given by L 0 = µ 2π ln b 5) where µ = 4π 10 7 H/m is the permebility of the medium between the inner nd outer conductors of the cox; essentilly the sme s tht of free spce for ny sne choice of spcing mteril. Note tht L 0, like C, is frequency-independent. The other source of series inductnce is tht ssocited with the mgnetic component of the propgting fields interior to the inner nd outer conductors, which is possible becuse they re not perfectly conductive. This inductnce is for the inner nd outer conductors respectively: L = µδ 4π nd 6) L b = µδ b 7) 4πb where δ nd δ b re the skin depths for the inner nd outer conductors respectively. Skin depth is given by δ = πµσf) 1/2 8) where σ is conductivity. It should be noted tht Equtions 6 nd 7 re vlid only for the cse tht δ mteril thickness; however this is the cse in the present problem over the rnge of frequencies of interest. The totl series inductnce L is the sum of L 0, L, nd L b. After substitutions nd some lgebr, we obtin: L = L 0 + f 1/2, where 9) = µ1/2 4π 3/2 σ 1/2 + σ 1/2 b b The resistnce R rises from the exct sme current ssocited with L nd L b. For good conductors it is known tht rel nd imginry prts of wve impednce re equl; thus R = ωl + L b ) which cn be written: R = 2π f 1/2 11) It is noted tht ny frequency dependence rises from non-zero, which mnifests s non-zero R nd frequency-dependent L. After substituting Equtions 9 nd 11 into Eqution 3 nd some lgebr, we find: γ = jβ j) L 0 f 1/2, where 12) ) 10) β 0 = ω L 0 C 13) Note tht β 0 is the wvenumber for n idel trnsmission line; tht is, one with perfectly-conducting mterils such tht R = = 0 nd thus is frequency-independent. Note tht ) = 1 L 0 2 σ 1/2 πµ + σ 1/2 b ln b ) 1 14) b The second term under the rdicl in Eqution 12 is smll compred to 1. For exmple, using the prmeters determined for RG58 in the next section, we find tht for f = 10 MHz: L 0 f 1/2 = 0.04 Estimted for RG58 t 10 MHz) 15) Thus it is resonble to pply the smll x pproximtion 1 + x x to Eqution 12. We obtin: 1 γ = β 0 f 1/2 + jβ ) f 1/2 16) 2 L 0 2 L 0 3

4 Recll tht the fields propgte long the trnsmission line ccording to the fctor e γz. Seprting γ into rel nd imginry prts γ = α + jβ, we hve e αx e jβx where the first nd second fctors describe ttenution nd phse, respectively, long the line. In our cse we hve β = Im {γ} = β ) f 1/2 17) 2 L 0 The phse φ t x ssuming φ = 0 t x = 0 is φ = βx. Let x = ct 0, where c is the speed of light in free spce. We then hve: φ = β 0 ct ) f 1/2 = 2πt 0 f + 1 ) f 1/2 18) 2 L 0 2 L 0 since β 0 c = 2πf. The ssocited dely τ is given by: τ = 1 dφ 2π df = t 0 The excess dely due to dispersion is therefore or, written in terms of the constitutive prmeters: 1 τ d = t 0 8 σ 1/2 πµ f 1/2 4 L 0 ) 19) 1 τ d = t 0 f 1/2 20) 4 L 0 + σ 1/2 b b ) ln b ) 1 f 1/2 21) This is the primry result of this nlysis. However, it is lso possible to obtin the ttenution constnt α, which is useful for checking the result. One finds: After substitutions nd some lgebr, we obtin: πǫ σ 1/2 α = 4 The ttenution is then given by ssuming α is expressed in m 1. 1 α = Re {γ} = β 0 f 1/2 22) 2 L 0 + σ 1/2 b b 3 Comprison to Welch s Result ) ln b ) 1 f 1/2 23) A = 20log 10 e α [db/m] 24) In his memo, Jck describes Cblewve 1/2-in cox hving copper inner conductor with 2 = in, n luminum outer conductor hving 2b = in, nd spcing mteril with ǫ r = He gives the index of refrction n s 1.228, nd n = ǫ r ). I used vlues of σ = S/m nd σ b = S/m from [1]. From Equtions 23 nd 24 I obtin db/m, wheres Jck obtins db/m; probbly OK given tht we my hve used slightly different conductivities Jck used skin impednce s opposed to conductivity). For τ d I obtin 5.28t 0 f 1/2 wheres Jck obtins 7.4t 0 f 1/2. Jck s result includes n empiricl correction bsed on mesurements of the cox which incresed the vlue of his constnt; I believe this ccounts for most of the difference between his result nd mine. 4

5 4 Findings for RG58 t LWA Frequencies To the best of my bility to determine, RG58 hs the following chrcteristics: Inner conductor twisted tinned copper strnds): = 0.42 mm, σ = S/m Outer conductor brided tinned copper strnds): b = 1.75 mm, σ b = S/m Spcing mteril polyethylene): ǫ r = The conductivities σ nd σ b re bit tricky to determine, becuse the inner nd outer conductors re not simple homogeneous medi. I used the following procedure: First, I ssumed σ = σ b = S/m, which is the vlue given for copper in [1]. From this, I obtined A = db/m t 100 MHz. The correct vlue should be more like db/m i.e., 5 db/100 ft)). I then guessed tht the use of brid in the outer conductor reduces the conductivity by n order of mgnitude, nd tht the use of strnds in the inner conductor reduces the conductivity by hlf, giving the vlues of listed bove. Using these new vlues I obtin very nerly the desired vlue of db/m, nd I therefore ssume these to be pproprite. For τ d I obtin Hz 1/2 )t 0 f 1/2. This cn be trnslted into more meningful expression by letting t 0 be l/0.66c), the trvel time long dispersion-free cble of length l, where c is the speed of light in free spce nd the fctor of 0.66 is the well-known velocity fctor for RG58. After substitutions nd some mth, we obtin: ) ) 1/2 l f τ d = 4.78 ns). 25) 100 m 10 MHz 5

6 References [1] W.H. Hyt, Jr., Engineering Electromgnetics, 4th Ed., McGrw-Hill,