high freq limit of fm gm hm v2 REVIEWED

Appendix D large ω limits of fm etc PhL 5.18.14 Earliest work on this subject, when it was Section D.9 of lines doc. That is now Section D.10. This is version 2 of this document, so I guess that supersedes the earlier version. Some of the stuff below is verbatim now in Section D.10. (10.8.14) D.9 Symmetry and the high frequency limit of the round wire E fields 1 (a) Symmetry of fm, gm and hm and expansions for Ei(r,θ) 1 (c) High frequency evaluation of fm, gm and hm and the E fields 2 D.9 Symmetry and the high frequency limit of the round wire E fields Recall from (D2.2.33) that, Summary of the E field solutions : Rdc = β'2 = β2 - βd2 (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 (aβd) gm gm = [ + ] xa = β'a Eθ(r,m) = (1/4) ηm I Rdc (aβd) hm hm = [ - ] (a) Symmetry of fm, gm and hm and expansions for Ei(r,θ) From NIST (10.4.1) we know that for integer m, J-m(x) = (-1)mJm(x) (D.9.1) For fm we find that f-m = [ - ] = - [ - ] = fm Next, gm = [ + ] g-m = [ + ] = [ + ] = gm hm = [ - ] h-m = [ - ] = [ - ] = - [ - ] = hm Thus we have shown that f-m = fm g-m = gm h-m = hm . (D.9.2) If the surface charge n(θ) happens to be even in θ, we know from (D.1.7) that ηm = Nm/N0 = η-m. In this case, the E field components in (r,θ) can be written as in (D.1.7), Ei(r,θ) = Ei(r,m=0) + 2!Syntax Error, IEi(r,m) cos(mθ) (D.9.3) Then for even n(θ) the E fields are Ez(r,θ) = (1/4) I Rdc (aβ') [ f0 + 2 Σm=1∞ fm ηm cos(mθ) ] Er(r,θ) = (j/4) I Rdc (aβd) [ g0 + 2 Σm=1∞ gm ηm cos(mθ) ] Eθ(r,θ) = (1/4) I Rdc (aβd) [ h0 + 2 Σm=1∞ hm ηm cos(mθ) ] . (D.9.4) (c) High frequency evaluation of fm, gm and hm and the E fields For large ω we assume β' = β as discussed below (D.2.2). From NIST 10.17.2, keeping a few leading terms in each inverse power expansion, we have this large x behavior for Jm(x) , Jm(x) = (2/πx)1/2 { cos(w) [a0(m) - a2(m)/x2 + O(1/x4)] - sin(w) [a1(m)/x + O(1/x3)] ] } w = x - mπ/2 -π/4 => e-jw = e-j(x-mπ/2-π/4) = e-jx ejπm/2 ejπ/4 a0(m) = 1 a1(m) = ≡ cm a2(m) = ≡ dm . (D.9.5) The expansion is in fact valid for all real and complex values of the parameter m, but we shall only use the expansion for m = -1,0,1,2....∞. Using abbreviations cm and dm one gets, Jm(x) = (2/πx)1/2[ cos(w) (1-dm/x2) - sin(w) (cm/x ) ] . (D.9.6) Recall that inside the round wire, δ ≡ = skin depth // ωμσ = 2/δ2 (2.2.20) β = ej3π/4 (/δ) = (j-1)/δ (2.2.21) so x = βr = ej3π/4 (/δ) r = (j-1) (r/δ) xa = βa = ej3π/4 (/δ) a = (j-1) (a/δ) . (D.9.7) Since x has a large positive imaginary part for small δ, so does w. Then cos(w) = [ ejw + e-jw]/2 ≈ (1/2) e-jw sin(w) = [ ejw - e-jw]/2j ≈ -(1/2j) e-jw = (j/2)e-jw . (D.9.8) Then our large-x expansion above becomes Jm(x) = (2/πx)1/2 (1/2) [e-jw (1- dm /x2) - j e-jw (cm /x) ] = (1/2πx)1/2 e-jw [ 1 -j cm (1/x) - dm (1/x2) + ... ] = (1/2πx)1/2 e-jx ejπm/2 ejπ/4 [ 1 -j cm (1/x) - dm (1/x2) + ... ] (D.9.9) It is not hard to show that this agrees with (2.3.5) through order 1/x. Notice that e-jx = e-j(j-1)(r/δ) = e(1+j)(r/δ) giving a convenient hybrid form Jm(x) = (1/2πx)1/2 e(1+j)(r/δ) (j)m ejπ/4 [ 1 -j cm (1/x) - dm (1/x2) + ... ] . (D.9.10) From (D.9.10) we see by inspection that, through O(1/x), = (j)m-n e(1+j)(r-a)/δ ≈ (j)m-n e(1+j)(r-a)/δ [ 1 - jcm/x + jcn/xa ] (D.9.11) and = e(1+j)(r-a)/δ // independent of m (D.9.12) Therefore for large ω, gm = [ + ] = 2 e(1+j)(r-a)/δ hm = [ - ] = 0 fm= [ - ] = (j)-1 e(1+j)(r-a)/δ [ 1 - jcm/x + jcm+1/xa ] - (j)+1 e(1+j)(r-a)/δ [ 1 - jcm/x + jcm-1/xa ] = - j e(1+j)(r-a)/δ [ 2 - 2jcm(1/x) + j(cm+1+cm-1) (1/xa) ≈ - 2j e(1+j)(r-a)/δ The results are then Large ω limits of the E field solutions : Rdc = (D.9.13) Ez(r,m) = (1/4) ηm I Rdc (aβ) fm fm = -2j e(1+j)(r-a)/δ x = β'r Er(r,m) = (j/4) ηm I Rdc (aβd) gm gm = 2 e(1+j)(r-a)/δ xa = β'a Eθ(r,m) = (1/4) ηm I Rdc (aβd) hm hm = 0 As observed earlier, the longitudinal current Jz is much larger than the radial current Jr by factor (β/βd). Notice the standard skin effect behavior both in amplitude and phase for all field components. We saw this earlier in several places: E(x,ω) = E(0,ω) e-x/δ e-jx/δ . x → (a-r) 1D example (2.1.8) = e(r-a)/δ r/δ > 3/= 2.1 . (2.3.7) The configuration space fields from ** are then, for even n(θ), Ez(r,θ) = (1/4) I Rdc (aβ') {-2j e(1+j)(r-a)/δ } [ 1+ 2 Σm=1∞ ηm cos(mθ) ] Er(r,θ) = (j/4) I Rdc (aβd) {2 e(1+j)(r-a)/δ } [ 1 + 2 Σm=1∞ 1 ηm cos(mθ) ] Eθ(r,θ) = 0 ηm = Nm/N0 (D.9.14) But for even n(θ) the [...] expansions shown here are just n(θ)/N0 from (D.1.7) where recall that N0 = (βd/2πωa) I . (D.2.31) Now since Rdc = (πa2)/σ we find, 2 (1/4) I Rdc(aβd)/N0 = (1/2) I Rdc (aβd) 2πωa/(βdI) = Rdc πωa2 = (ω/σ) so then for even n(θ) and large ω: Ez(r,θ) = - (jω/σ) e(1+j)(r-a)/δ n(θ) (β'/βd) Er(r,θ) = (jω/σ) e(1+j)(r-a)/δ n(θ) Eθ(r,θ) = 0 (D.9.15) Observations on the E fields for large ω In the extreme skin effect (small δ, large ω) regime, and for n(θ) an even function of θ : 1. There is no azimuthal field Eθ inside or on the surface of the round wire. 2. Both Ez and Er exhibit the standard skin effect form for amplitude and phase 3. The ratio Ez(r,θ)/Er(r,θ) = - (β'/βd) is very large and is constant in r and θ 4. Both Ez and Er track the surface charge density n(θ) for azimuthal dependence 5. If n(θ) ≠ constant, then Jz = σEz ≠ constant in θ and the longitudinal current density is asymmetric across the round wire cross section, which is known as the proximity effect.