high freq limit of fm gm hm REVIEWED

Appendix D new piece PhL 3.26.05 I presume this is superseded by version 2. D.9 Symmetry and high frequency limit of the round wire E fields Recall from earlier in this appendix that, Second summary of the E field solutions : Rdc = β'2 = β2 - βd2 (D.2.33) Ez(r,m) = (1/4) ηm I Rdc (aβ') fm where fm = [ - ] x = β'r xa = β'a Er(r,m) = (j/4) ηm I Rdc (aβd) gm where gm = [ + - ] Eθ(r,m) = (1/4) ηm I Rdc (aβd) hm where hm = [ - + + ] (a) Symmetry of fm, gm and hm. From NIST (10.4.1) we know that for integer m, J-m(x) = (-1)mJm(x) For fm we find that f-m = [ - ] = - [ - ] = fm Next, gm = [ + ] g-m = [ + ] = [ + ] = gm hm = [ - ] h-m = [ - ] = [ - ] = - [ - ] = hm 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.9.1) 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.9.2) 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.9.3) 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.9.4) 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.9.5) 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) ejπm/2 ejπ/4 [ 1 -j cm (1/x) - dm (1/x2) + ... ] . (D.9.6) Using this asymptotic formula, we start examining the ratios that appear in our E field coefficients. From (D.9.6) we see by inspection that = e(1+j)(r-a)/δ . (D.9.7) Next , keeping only leading terms in 1/x and 1/xa, = e(1+j)(r-a)/δ ej2π/2 ≈ - e(1+j)(r-a)/δ [ 1 - jcm+1(1/x) + jcm-1(1/xa) ] . (D.9.8) Our third object of interest for gm and hm is = (2m/x) e(1+j)(r-a)/δ ejπ/2 + j (2m/x) e(1+j)(r-a)/δ [ 1 - jcm (1/x) + j cm-1(1/xa)] . (D.9.9) For fm we have = e(1+j)(r-a)/δ ej(-1)π/2 ≈ -j e(1+j)(r-a)/δ [ 1 - jcm (1/x) + j cm+1(1/xa)] . (D.9.10) = e(1+j)(r-a)/δ ej(+1)π/2 ≈ +j e(1+j)(r-a)/δ [ 1 - jcm (1/x) + j cm-1(1/xa)] . (D.9.11) Keeping only the leading terms, we find then that fm = [ - ] = -2j e(1+j)(r-a)/δ . (D.9.12) Next, again keep terms through O(1/x), gm = [ + - ] = + j (2m/x) e(1+j)(r-a)/δ + e(1+j)(r-a)/δ + e(1+j)(r-a)/δ = e(1+j)(r-a)/δ [ 2 + 2jm/x ] ≈ 2 e(1+j)(r-a)/δ . (D.9.13) Finally, hm = [ - + + ] = - j (2m/x) e(1+j)(r-a)/δ + e(1+j)(r-a)/δ - e(1+j)(r-a)/δ [ 1 - jcm+1 (1/x) + j cm-1(1/xa)] = e(1+j)(r-a)/δ { -2mj/x + 1 - 1 + jcm+1 (1/x) - j cm-1(1/xa) } = j e(1+j)(r-a)/δ { (cm+1 - 2m) (1/x) - cm-1(1/xa) } ≈ 0 . (D.9.14) The results are then Large ω limits of the E field solutions : Rdc = (D.9.15) Ez(r,m) = (1/4) ηm I Rdc (aβ) fm where fm = -2j e(1+j)(r-a)/δ x = β'r xa = β'a Er(r,m) = (j/4) ηm I Rdc (aβd) gm where gm = 2 e(1+j)(r-a)/δ Eθ(r,m) = (1/4) ηm I Rdc (aβd) hm where 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)