old version of D.11 just before install of new ARCHIVE

D.11 Low frequency limit of the round wire E fields This has been completely replaced by a new Section D.11. Our assumed Appendix D wave z dependence is e-jkz from (D.1.1). (a) An Inaccurate Model for Low Frequency Recall from Chapter 4 that for high frequencies we derived the transmission line equations, = - z i(z) = - y V(z) (4.11.14b) and from those we derived the second order uncoupled transmission line equations, - zy V(z) = 0 - zy i(z) = 0 . (4.11.15) These last equations imply the following z-dependence for V(z) and i(z) and then presumably for all other transmission line non-power quantities (like the fields), e-jkz where k = -j= -j Summary of the E field solutions : Rdc = β'2 = β2 - k'2 (D.2.33) Ez(r,m) = (1/4) ηm I Rdc (aβ') fm fm = [ - ] x = β'r Er(r,m) = (j/4) ηm I Rdc (ak) gm gm = [ + ] xa = β'a Eθ(r,m) = (1/4) ηm I Rdc (ak) hm hm = [ - ] For large ω, we show in Appendix Q that For large ω: Re(β'd) = ω Im(β'd) = - ω >> (R/L) and ω >> (G/C) Since the wave z dependence is e-jk'z = e-jRe(k')z e-[-Im(k')]z we can interpret Re(β'd) as the "wavenumber" of the wave, and we know that this wavenumber = ω/vd where vd is the phase velocity of the wave. Thus, we identify vd = 1/ in terms of which Re(β'd) = ω /vd = k -Im(β'd) = (RC+GL)(vd/2) . This last quantity is the attenuation parameter sometimes called α, so α = (RC+GL)(vd/2) . Thus we have a reasonable model for high frequency waves on our transmission line, including the attenuation due to the losses caused by R and G. We can go on to evaluate β'2 and take its large ω limit, β'2 = β2 - k'2 = -jμσω - [ω /vd - jα ]2 → -jμσω - (ω/vd)2 ≈ -jμσω . In the last equality, although it seems the ω2 term should win out at large ω, we have shown elsewhere that for any ω of transmission line interest, σ is large and vd is large so - (ω/vd)2 can be ignored. For example : f = 1000 GHz ω = 2π x 1012 Hz μ0σω = 4π x 5.81 x 2π x 1012 = .46 x 1015 (ω/vd)2 ≈ (ω/c)2 = [2π x 1012/ 3x108]2 = [(2π/3) x 104]2 = .44 x 109 In Section D.10 we went on to take the large β' limit of the fm, gm and hm functions. This led to an interesting asymmetric Jz distribution which we associated with the proximity effect. For small ω things are much more hazy. The main issue is that the transmission line equations quoted above are only justified from Maxwell's equations at high frequencies where the skin effect is at least starting to set in. We show in Appendix K that for the network model of a transmission line, the transmission line equations are valid at all ω. One must keep in mind that this network model is just a model, and we have shown in Chapter 4 that it does not correspond to the real world at low frequencies. Despite this caveat, we can pretend that the transmission line equations are accurate at low ω and see what happens. We quote again from Appendix Q for a transmission line with G = 0 (vacuum dielectric) since this is assumed in our boundary condition (D.2.24), For small ω: Re(β'd) = = α // a new α - Im(β'd) = = α G = 0 and ω << (R/L) This is an extremely lossy wave, since it damps out in a half wave cycle. We then have e-jk'z = e-jRe(k')z e-[-Im(k')]z = e-jαz e-σz = e-(j+1)αz . This has the exact form shown in (2.1.8) so what we have here is in effect a longitudinal skin effect where the wave is trying to travel down the transmission line but it can't get very far. The longitudinal skin depth is 1/α . We can blindly proceed and compute β'2 = β2 - k'2 = -jμσω - ωRC/2 → 0 as ω→0 In this case, we want to take the small argument limits of fm, gm and hm. We shall go ahead and do that below just for the sake of "completeness", but it should be realized that the results are not accurate. (b) Low frequency evaluation of fm, gm and hm and the E fields In the following we consider only m ≥ 0 since we know from (D.10.2) that f-m = fm , g-m = gm, h-m = hm. The small x limit for Jm(x) is given by NIST 10.7.3, Jn(x) = (x/2)n / n! . for n = 0,1,2,..... (D.11.14) Since Jm-1 appears in our coefficient expressions and since m = 0 is encountered, we have to deal with m = 0 as a special case since the above limit is not valid for n = -1. To this end we use NIST 10.2.2 which is valid for integer n, J-n(x) = (-1)nJn(x) ≈ (-1)n (x/2)n / n! (D.11.15) so that J-1(x) = - J1(x) ≈ - (x/2). Our small-x forms of interest are then Jn(x) = (x/2)n / n! for n = 0,1,2,..... J-1(x) = - (x/2) for n = -1 . (D.11.16) We now examine the small x limits of fm, gm, and hm . First fm for m > 0, and then for m = 0: fm = [ - ] = [ - ] = [ (m+1) (x/xa)m (2/xa) - (1/m) (x/xa)m(xa/2) ] = (x/xa)m [ (m+1) (2/xa) - (1/m) (xa/2) ] ≈ (x/xa)m (m+1) (2/xa) // as xa→ 0 f0 = [ - ] = [ + ] = 2 = 2 = 4/xa First gm for m > 0, and then for m = 0: gm = [ + ] = [ + ] = (x/xa)m+1 + (x/xa)m-1 g0 = [ + ] = [ + ] = 2 = 2 (x/xa) Results for hm are then obvious since there is only a sign change between the terms in gm, hm = (x/xa)m+1 - (x/xa)m-1 h0 = 0 The results are then ( we replace k → k' as discussed in section (a) above ) : Small ω limit of the E field solutions : Rdc = (D.11.17) Ez(r,m) = (1/4) ηm I Rdc (aβ') fm fm = (r/a)m (m+1) (2/β'a) f0 = 4/(aβ') Er(r,m) = (j/4) ηm I Rdc (ak') gm gm = (r/a)m+1 + (r/a)m-1 g0 = 2 (r/a) Eθ(r,m) = (1/4) ηm I Rdc (ak') hm hm = (r/a)m+1 - (r/a)m-1 h0 = 0 For m > 0: Ez(r,m) = (1/2) ηm I Rdc (r/a)m (m+1) Er(r,m) = (j/4) ηm I Rdc (ak') [(r/a)m+1 + (r/a)m-1] Eθ(r,m) = (1/4) ηm I Rdc (ak') [(r/a)m+1 - (r/a)m-1] For m= 0: Ez(r,0) = I Rdc Er(r,0) = (j/2) I Rdc (ak') (r/a) Eθ(r,0) = 0 Recall now these field expansions for the case that n(θ) is an even function of θ, Ez(r,θ) = (1/4) I Rdc (aβ') [ f0 + 2 Σm=1∞ fm ηm cos(mθ) ] Er(r,θ) = (j/4) I Rdc (ak') [ g0 + 2 Σm=1∞ gm ηm cos(mθ) ] Eθ(r,θ) = (1/4) I Rdc (ak') [ h0 + 2 Σm=1∞ hm ηm cos(mθ) ] . (D.10.4a) For low ω we insert the expressions above to get Ez(r,θ) = I Rdc [ 1+ Σm=1∞ (r/a)m (m+1) ηm cos(mθ) ] Er(r,θ) = (j/2) I Rdc (ak) [ (r/a) + Σm=1∞ [(r/a)m+1 + (r/a)m-1] ηm cos(mθ) ] Eθ(r,θ) = (1/2) I Rdc (ak) [ Σm=1∞ [(r/a)m+1 - (r/a)m-1] ηm cos(mθ) ] . (D.11.18) THIS SECTION ABOVE IS ON HOLD Observations on the E fields for small ω 2. The m = 0 term in Ez is just Ez = IRdc which says Jz = σ IRdc = σ I(1/σπa2) = I/(πa2). This is the current one would expect in a wire carrying DC current I. 3. Assuming a non-uniform n(θ) charge density on the wire surface, the ηm moments are non-zero and one concludes that Jz(r,θ) is asymmetric even as ω → 0. This conclusion is incorrect, and the reason is that we have improperly assumed STOP I = V/Z0 and according to (4.11.16), where G = 0 since we assumed a non-conducting dielectric, Z0 ≡ V(z)/i(z) = = . (4.11.16) Thus in the limit ω→0 we get Z0 → ∞. For a finite length transmission line, as one lowers ω, one must increase the size of the termination Z0 to maintain a properly terminated line. Thus, our situation here does not apply to taking the low frequency limit of a transmission line terminated by a fixed 75Ω or 8Ω resistor. See Section 6.5 (e) for more on this subject. The major point here is that the entire model is not valid for low ω and we cannot use it to study the limit ω → 0.