Section D.9 mateiral REVIEWED

This is the Title PhL 3.26.05 This material is more or less now installed as lines doc Section D.9. D.9 About the boundary condition Er(a,θ) = (jω/σ) n(θ) . 1 (a) The notion of Debye Surface Currents 1 (b) The role of Debye Surface Currents in the Charge Pumping Boundary Condition 2 (c) Where does surface charge n(θ) come from? 4 D.9 About the boundary condition Er(a,θ) = (jω/σ) n(θ) . The "charge pumping boundary condition" appears in (D.2.23) and here we want to examine it more closely. Our concern is that the derivation of (D.2.23) ignores surface currents that we know exist on the surface of a transmission line conductor as the surface charge moves around in response to tangential E fields. The first issue then is to define and quantify the nature of these surface currents. (a) The notion of Debye Surface Currents We continue in the context of our classical treatment of the conductor surface. In Appendix E it is pointed out that the surface charge on a transmission line conductor exists in an incredibly thin surface layer we shall call the Debye layer for want of a better name. For copper the thickness λD of this layer is on the order of one atomic radius. In addition to the normal conduction electrons, this thin layer contains extra free electrons that are piled up just below the surface (negative surface charge) or are depleted from this thin region (positive surface charge), as shown by the red curve in Fig E.1. We want first so show : Fact 1: In a good conductor, the volume density of free electron carriers piled up at a surface (to make up the surface charge) is negligible compared to the volume density of conduction electrons. (D.9.1) Proof: From (E.7) the free charge density in the Debye layer (assume x is the inward surface normal direction) is given by ρ(x) = ρ(0) e-x/λ . The effective free surface charge n is then given by n = !Syntax Error, Idx ρ(x) = ρ(0) !Syntax Error, Idx e-x/λ = ρ(0)λD . The free electron density ne is then ne = ρ(0)/e = n / (eλD) . As a typical example, consider a parallel plate capacitor with close plate spacing s. The E field in the gap is E = V/s and the surface charge density from (1.1.47) is n = εE = εV/s. For V = 10 volts and s = 1 mm we find n = ε0V/s = 8.85 x 10-12 * 10 / 10-3 ≈ 101-12+1+3 = 10-7 Coul/m2 . Then the free electron density is ne = n / (eλD) ≈ 10-7 Cou/m2 / [ 1.6 x 10-19 Coul * 10-10m] ≈ 0.6 * 10-7+19+10 ≈ 1022 electrons/m3 As noted in (N.1.2), in copper the conduction electron density (one electron per atom) is 1029 /m3, QED. Corollary: The conductivity σD inside the Debye later is basically the same as σ outside that layer. (D.9.2) Proof: From (N.1.10) conductivity is σ = (nq2τ/m) where n is the electron density. The Fact above shows that this density is the same in the Debye layer as in the bulk conductor, so σD = σ. (We ignore the possibility that the collision time τ could differ in the Debye layer vs. in the bulk volume. ) QED Consider now this drawing which shows a tiny slice of width dx of a piece of a transmission line conductor cross section at its surface. The yellow Debye surface charge layer is greatly exaggerated in thickness. Recall from the comment below Fig E.2 that at 100 GHz one has δ ≈ 4000 λD so δ >> λD at all frequencies of transmission line interest. Fig D.6 The Debye layer holds the surface charge, and when this surface charge moves, one has a Debye surface current. We now show : Fact 2: The total current in the Debye layer is negligible compared to that in the skin effect layer. (D.9.3) Proof: The field Ez is transverse to the conductor surface, so we know from (1.1.41) that it is continuous through the (gradual) boundary at the bottom of the Debye layer. Then the ratio of the currents in the two layers is, = = = ≈ 1 * 1 * = << 1 . QED (b) The role of Debye Surface Currents in the Charge Pumping Boundary Condition Now referring to the Debye surface currents as KzD and KθD we reconsider the derivation of the charge pumping boundary condition of (D.2.24) where we had this figure Fig D.2 If we include the Debye surface currents in the θ and z direction in our application of continuity, div J = - jωρ -jω[∫V ρ dV] = ∫S J dS (D.2.22) the result is -jω n(θ,z) = - Jr(r=a-ε, θ, z) + ∂zKz(θ,z) + a ∂θKθ(θ,z) . (D.9.4) where we assume that the dielectric outside the round wire is vacuum with σd = 0. The gaussian box selected here is that shown in red in Fig ***. The bottom face lies below the Debye layer so Jr(a-ε, θ, z) is the value of Jr in the normal skin effect region close to the surface. The Debye surface currents may be written approximately as KzD = JzD λD = σDEzD λD = σ Ez(r=a,θ) λD // dim(K) = amp/m KθD = JθD λD = σDEθD λD = σ Eθ(r=a,θ) λD = 0 // (D.9.2) and (3.7.0) (D.9.5) where we use the Corollary above that σD = σ. From (3.7.0) we have Eθ = 0 at the surface so KθD = 0 and we have only the Debye current KzD to worry about. Recall that Eθ(a,θ) = 0 is the second boundary condition (D.2.27) used in Section D.2 to evaluate the am and Km coefficients, and that this condition is itself a topic of interest in Section D.8, and we assume it is valid. We then have, -jω n(θ,z) = - Jr(a,θ,z) + ∂zKzD(θ,z) = - Jr(a,θ,z) + ∂z [σ Ez(a,θ,z) λD] (D.9.6) Assuming everything has z dependence ej(ωt-βz) as in (D.1.4), we replace ∂z → -jβd and then suppress the z arguments to get -jω n(θ) = - Jr(a,θ) -jβd [σ Ez(a,θ) λD] = - σEr(a,θ) -jβd [σ Ez(a,θ) λD] = - σEr(a,θ)[ 1 - jβdλD ] . (D.9.7) If we assume that the second term in (D.9.7) can be ignored, we get the charge pumping boundary condition Er(r=a-ε,θ) = (jω/σ) n(θ) (D.2.24) (D.9.8) which in return yields the E fields as stated in (D.2.33) where we see that roughly ~ | | . (D.9.9) Thus, our self-consistent condition for ignoring the second term in (D.9.7) is βd λD * << 1 βd λD * | | << 1 λD |β| << 1 λD | ej3π/4 (/δ)| << 1 (λD/δ) << 1 (δ/λd) >> 1 // ignore But we know from above that (δ/λd) >> 1 for any f < 100GHz, so for such f the second term in (D.9.7) can in fact be ignored. We have just proven: Fact 3: For f < 100 GHz, the Debye surface currents can be ignored in the derivation of the boundary condition Er(a-ε,θ) = (jω/σ) n(θ) . (D.9.10) (c) Where does surface charge n(θ) come from? According to our traveling-wave ansatz (D.1.1), all E field related quantities move down a transmission line at vd as ej(ωt-βz) . For a low-loss line and a vacuum dielectric, vd ≈ c, the speed of light. Therefore, n(θ,z,t) = n(θ,0,0) ej(ωt-βz) βd = (ω/vd) . (D.9.11) One can ponder and then discard a list of hypotheses concerning where n(θ) "comes from" as it increases and decreases over time at some location z on one of the conductors. The first hypothesis might be that the individual electrons which make up n(θ) simply travel at vd in the z direction down the conductor surface, and n(θ) is not fed by any radial currents inside the conductor. In this case one would have KzD(θ) = vd n(θ). But we know this is not what happens. Apart from the massive energy required to achieve relativistic electron velocities, we know from Appendix N.1 that the electrons in the Debye layer in fact drift along at something like ~ 1 mm/sec, just as do the regular conduction electrons in the conductor bulk. The second hypothesis is a variation of the first, where we now allow that the Debye surface current works like any other conduction current, and when one electron moves "to the right" at some point z, a distant electron at z+L moves to the right at nearly the same time, all electrons in a long string moving to the right one position, giving the illusion that a particular electron moved very fast. This does in fact happen, and if it were all that happened, again we would have KzD(θ) = vd n(θ). Comment : Assume some skin depth δ ≤ a/10 so the bulk current is flowing in a sheath of thickness δ just under the conductor surface. The total sheath current is then roughly 2πaδJz(a,θ). We can regard this current flow as due to an effective "full surface current" Kz = Jz δ. Notice that this "surface current" is different from the "Debye surface current". Based on Fact 2 above, we certainly expect Kz >> KzD . A third hypothesis is that somehow n(θ,z,t) is fed by azimuthal Debye surface currents, or some combination of these along with the z-directed KzD(θ). Our condition (3.7.0) that Eθ = 0 puts a stop to the possibility of feeding by azimuthal Debye surface currents. What we have learned from Fact 3 is that none of the above hypotheses explains where n(θ) comes from. The analysis above shows that, although the surface motions of the Debye surface charges do create Debye surface currents, these currents are so small that they play no role in div J = -∂tρ for the Gaussian box shown in red in Fig **. The charge n(θ) "comes from" inside the wire and is fed by the radial current density Jr just below the surface according to (D.2.24), Jr(r=a-ε,θ) = jω n(θ) n(θ) = (1/jω) Jr(a-ε,θ) (D.9.12) Here is a suggestive picture, Fig D.7 where the white boxes are little "radial charge pumps" delivering the required Jr needed to feed the changing surface charge n(θ). Apart from the miniscule KzD , charges in n(θ) don't move in the z direction in this picture, they just appear to be doing that due to the choreographed radial pumping in and out at the wire surface. A wave front of the n(θ) wave travels at vd, and this is just a phase velocity. In an analogous situation, in a deep ocean wave the individual particles of water travel in small ellipses and do not travel along with the wave, though there are small scale longitudinal motions due to those ellipses. Comment: In our transmission line theory, the exterior problem in the dielectric is solved using the capacitor method, from which one learns n(θ). The boundary condition Jr(r=a-ε,θ) = jω n(θ) couples this exterior information into the wire interior, allowing one to solve for the fields and currents inside. In Section 6.5 we show how div E = 0 inside the conductor (or div J = 0) forces a relationship between Jr and Jz just below the conductor surface. When that relationship (6.5.18) is combined with the charge pumping boundary condition (D.9.8), one finds that Ez(a,θ) = (-jω/σ) (β/βd) n(θ) (6.5.19) or Jz(a,θ) = (-jω) (β/βd) n(θ) . (D.9.13) This same result is obtained in a different manner as (6.5.13). The "full surface current" Kz was defined in a Comment above as Kz(θ) = δ Jz(a,θ). Thus, Kz(θ) = δ Jz(a,θ) = [δ (-jω) (β/βd)] n(θ) (D.9.14) But = so [δ (-jω) (β/βd)] = δ (-jω) = -j ej3π/4 vd = -j (j-1)/ * vd = (1+j) vd and we end up with Kz(θ) = (1+j) vd n(θ) Re(Kz(θ)) = Im(Kz(θ)) = vd n(θ) (D.9.15) Once again, this last equation gives the illusion that the surface charge density n(θ) moves "to the right" at speed vd to create the real or imaginary part of the full δ-thick surface current Kz. This is the equation that replaces the incorrect equation KzD(θ) = vd n(θ) which assumes there is no radial charge pumping. (d) What happens if the dielectric conducts? Ignoring the Debye surface currents as per section (b) above, if the dielectric has some conductivity σd, the charge pumping boundary condition (D.2.23) becomes Jr(a-α,θ) - Jr(a+α,θ) = jω n(θ) or σ Er(a-α,θ) - σdEr(a+α,θ) = jω n(θ) // this is div J = -jωρ where α > 0 is a tiny distance (ε is already used for dielectric constant). Another boundary condition at the surface is provided by (1.1.47) which says ( points into medium 1 which is the dielectric) [ε1En1 - ε2En2] = nfree // this is continuity of Dn at the surface or [εdEr1 - ε0Er2] = n(θ) or [εd Er(a+α,θ) - ε0 Er(a-α,θ)] = n(θ) A seeming third boundary condition is (1.1.48), ξ1En1 = ξ2En2 or (εd + σd/jω) Erd = (ε0 + σ/jω) Er ≈ (σ/jω) Er or (εd + σd/jω) Er(a+α,θ) ≈ (σ/jω) Er(a-α,θ) . There seem to be these three boundary conditions at the round wire surface, σ Er(a-α,θ) - σdEr(a+α,θ) = jω n(θ) // modified cpbc from div J = -jωρ (D.9.16) εd Er(a+α,θ) - ε0 Er(a-α,θ) = n(θ) // div D = ρ (straddle) (D.9.17) (εd + σd/jω) Er(a+α,θ) ≈ (σ/jω) Er(a-α,θ) // ξ1En1 = ξ2En2 (D.9.18) but only two of these conditions are independent. For example, multiply (D.9.16) by (-1/jω) to get (σd/jω)Er(a+α,θ) - (σ/jω) Er(a-α,θ) = - n(θ) . Adding this to (D.9.17) then gives (εd + σd/jω) Er(a+α,θ) - (σ/jω) Er(a-α,θ) = 0 which is in fact the same as (D.9.18). When we solve the "capacitor problem" as in Section 6.5 a) to obtain n(θ) on the conductor surface, we are using (D.9.17) with the assumption that Er(a+α,θ) >> Er(a-α,θ). Typically one just says that in a good conductor Er(a-α,θ) = 0 and then n(θ) = εd Er(a+α,θ) . That is fine, but it is not clear what happens to (D.9.16) above. The first term is the product of a large quantity σ times a small quantity Er(a-α,θ) so can be the same size as the other terms in the equation. The resolution is provided by the discussion in Section 1.5 (c) where we encountered the equation (1.5.17) nc(x,ω) = (ξ1/ε1) ns(x,ω) . (1.5.17) which in our current context (1 = dielectric) becomes nc(θ) = (ξd/εd) n(θ) . (D.9.19) In that discussion it is noted that n(θ) is the actual free surface charge density, whereas nc(θ) is a related "transport charge density" having the same dimensions as n(θ). If we multiply (D.9.16) and (D.9.17) by (ξd/εd), our (redundant) triplet of boundary conditions becomes, 1 σ (ξd/εd) Er(a-α,θ) - σd (ξd/εd) Er(a+α,θ) = jω nc(θ) 2 ξd Er(a+α,θ) - (ξd/εd) ε0 Er(a-α,θ) = nc(θ) 3 ξdEr(a+α,θ) ≈ ξ Er(a-α,θ) . (D.9.20) We now use last of these three equations to eliminate Er(a+α,θ) in the first, which then becomes σ (ξd/εd) Er(a-α,θ) - σd (ξ/εd) Er(a-α,θ) = jω nc(θ) or [ σ ξd - σd ξ ]/εd Er(a-α,θ) = jω nc(θ) or [ σ (εd + σd/jω) - σd (ε0 + σ/jω) ]/εd Er(a-α,θ) = jω nc(θ) or [ (σ εd - σdε0)]/εd Er(a-α,θ) = jω nc(θ) // two large terms cancelled or [ (σ - σd(ε0/εd) ] Er(a-α,θ) = jω nc(θ) We now assume that ε0 (conductor) and εd (dielectric) are the same order of magnitude, and we assume that, even though the dielectric conducts, we still have σ >> σd . The last equation then reads Er(a-α,θ) = (jω/σ) nc(θ) = (jω/σ) (ξd/εd) n(θ) (D.9.21) Er(a-α,m) = (jω/σ) (ξd/εd) Nm . // partial waves (D.9.22) The above are the "modified" charge pumping boundary conditions which replaces (D.2.24) and (D.2.25) for a mildly conducting dielectric Er(r=a,θ) = (jω/σ) n(θ) . (D.2.24) Er(r=a,m) = (jω/σ) Nm . (D.2.25) How then does σd ≠ 0 alter the E field results summarized in box (D.2.33)? If we indicate σd = 0 quantities with no prime and σd > 0 quantities with a prime, then tracing through the development we find that n(θ) → (ξd/εd) n(θ) ≡ n'(θ) Nm → (ξd/εd) Nm ≡ N'm but ηm ≡ Nm /N0 remains the same Km → (ξd/εd) Km ≡ K'm am → (ξd/εd) am ≡ a'm βd = ω → ω ≡ βd' so βd' = βd (D.9.23) The normalization condition (D.2.31) I = 2πω (a/βd) N0 (D.2.31) becomes (where I' is the total current in the wire with σd > 0) I' = 2πω (a/β'd) N0' = 2πω (a/βd) (ξd/εd)N0 = 2πω (a/βd) N0 = I . (D.9.24) Then for example the Ez field becomes (see below (D.2.32)), E'z(r,m) = -j(β'/β'd) Jm(x) = -j(β'/β'd) (jω/2σ) N'm [ – ] Jm(x) = -j(β'/β'd) (jω/2σ) ηm N'0 [ – ] Jm(x) = -j(β'/β'd) [(j/4) (aβ'd) ηm I' Rdc] [ – ] Jm(x) = (1/4) ηm I' Rdc (aβ') [ - ] and for the Er field, E'r(r,m) = a'm x-1 Jm(x) + Jm+1(x) = [(jω/2σ) N'm] { 2m x-1 Jm(x) + [ – ] Jm+1(x) } = (j/4) (aβ'd) ηm I' Rdc { + - } = (j/4) (aβ'd) ηm I' Rdc { + }. The results (D.2.33) then become: Second summary of the E field solutions : Rdc = β'2 = β2 - β'd2 (D.9.25) βd' = βd where βd = ω I' = I E'z(r,m) = (1/4) ηm I' Rdc (aβ') fm fm = [ - ] x = β'r E'r(r,m) = (j/4) ηm I' Rdc (aβ'd) gm gm = [ + ] xa = β'a E'θ(r,m) = (1/4) ηm I' Rdc (aβ'd) hm hm = [ - ] Conclusion: Turning on σd causes the total current to "increase" from I to I' = I, which seems intuitive since the total current now has to feed the conductance between the conductors. However, the E field solutions have exactly the same form they had when σd = 0 when expressed in terms of the new larger current I' and the new β'd. This β'd differs from the βd for σd = 0, and this then gets mapped into a slight difference in β' since β'2 = β2 - β'd2 .