modification of Chapter 4 for mu REVIEWED

This is the Title PhL 3.26.05 This is all installed into lines doc, here is where I first wrote it up. Older notes at the bottom. 4.12 Modifications of Chapter 4 to account for μ ≠ μ1 ≠ μ2. First of all, the first part of Chapter 4 involving potentials φ and V(z) is completely unaffected. Thus, there is no modification to the first six sections of Chapter 4. If the equality μ = μ1 = μ2 assumed in Section 4.7 is broken, the result is that surface magnetization currents appear on one or both of the conductor surfaces and these cause an alteration of the theory. Thanks to the "Jm Theorem" proven in Appendix B, this alteration can be carried through with a very minimal impact, as we now show. In Appendix B conductor magnetization surface currents are studied in some detail. The reader interested in how the magnetic modification is carried out would do well to read Appendix B at this point. A reader less interested can accept the Appendix B results and then learn below that basically nothing changes! The first modification arises in Section 4.7 where now we no longer assume that μ1 of conductor C1 is the same as μ of the dielectric. As described in Appendix B.6, the modified version of (4.7.2) is this, Az1(x) = ∫ [ Jz1(x') + Jzm1(x') ] dx'dy'dz' . R = |x - x'| (4.7.2)' where Jzm1 includes only the surface component of the magnetization current on conductor C1. Appendix B.6 shows how this Jzm1 adder term in effect adds a certain homogeneous solution to the particular solution (first term above) of the Az Helmholtz equation such that the Az boundary conditions are duly satisfied at the magnetic conductor C1 boundary. We maintain the next two equations of Section 4.7 as is, Jz1(x,y,z) = b1(x,y) i1(z) A/m2 1/m2 A (4.7.3) where i1 is scaled such that !Syntax Error, Idx dy b1(x,y) = 1 . (4.7.4) This i1(z) is still the total conduction current in C1. But we now add two new equations, Jz1m(x,y,z) = b1m(x,y) i1m(z) A/m2 1/m2 A (4.12.1) where i1m is scaled such that !Syntax Error, Idx dy b1m(x,y) = 1 . (4.12.2) It is understood here that b1m(x,y) is a distribution which is restricted to the surface of C1, but we continue to write it as if it existed at all points in the cross section of C1. The integration in (4.12.2) is of course meant to include this surface distribution. We know from (B.1.11) and (B.1.12) that, for an arbitrarily shaped conductor C1, i1m(z) ≡ - ( - ) i(z) [ μ1 = conductor C1, μ = dielectric ], (4.12.3) and that the current ratio is therefore given by, f1m ≡ i1m(z)/ i(z) = - ( - ) . (4.12.4) With the above definitions, our modified (4.7.6) becomes Az1(x,y,z) = !Syntax Error, Idz' i(z') !Syntax Error, Idx' dy' [ b1(x',y') + f1m b1m(x',y') ] . = !Syntax Error, Idz' i(z') !Syntax Error, Idx' dy' [ b1(x',y') + f1m b1m(x',y') ] . (4.12.5) This leads us to define a new effective transverse current density, b'1(x,y) ≡ b1(x',y') + f1m b1m(x',y') = b1(x',y') + [1-] b1m(x',y') . (4.12.6) This new transverse density b'1 is still normalized to unity, !Syntax Error, Idx dy b'1(x,y) = !Syntax Error, Idx dy b1(x,y) + [1-]!Syntax Error, Idx dy b'1m(x,y) = * 1 + [1-] * 1 = 1 . (4.12.7) How does b'1 differ from b1? The difference is that b1 does not include a surface current and b'1 does. We can represent equation (4.12.6) in this symbolic graphic manner: (4.12.6) Thus, we have this new version of (4.7.6), Az1(x,y,z) = !Syntax Error, Idz' i(z') !Syntax Error, Idx' dy' b'1(x,y) . (4.7.6)' The differences are that the leading factor is μ instead of μ1, and b1 is replaced by b'1. Moving into Section 4.8 we find this new version of (4.8.1), Az12(x) = Az1(x) + Az2(x) = !Syntax Error, Idz' i(z') { !Syntax Error, Idx1' dy1' b'1(x1',y1') – !Syntax Error, Idx2' dy2' b'2(x2',y2') } (4.8.1)' which is identical to (4.8.1) except bi → b'i. Then in the transmission line limit, we get this new version of (4.9.1), Az12(x) = i(z) !Syntax Error, Idz'{ !Syntax Error, Idx1' dy1' b'1(x1',y1') – !Syntax Error, Idx2' dy2' b'2(x2',y2') } (4.9.1)' From this point onward, all equations are the same apart from bi → b'i. Here are some of those equations after modification: W(z) ≡ Az12(x1) - Az12(x2) (4.10.1)' = i(z) !Syntax Error, Idz' { !Syntax Error, Idx1' dy1' b'1(x1',y1') – !Syntax Error, Idx2' dy2' b'2(x2',y2') } – i(z) !Syntax Error, Idz' { !Syntax Error, Idx1' dy1' b'1(x1',y1') – !Syntax Error, Idx2' dy2' b'2(x2',y2') } W(z) (4.10.3)' = i(z)!Syntax Error, Idz' {!Syntax Error, Idx1' dy1' b'1(x1',y1')( - ) -!Syntax Error, Idx2' dy2' b'2(x2',y2') (- ) } . W(z) = i(z) {!Syntax Error, Idx1' dy1' b'1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' b'2(x2',y2') ln(s222/s122) } (4.10.4)' KL ≡ !Syntax Error, Idx1' dy1' b'1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' b'2(x2',y2') ln(s222/s122) (4.10.9)' Le = = KL // no change (4.10.8) We then enter Section 4.11. The derivation of the transmission line equations (4.11.11) is unaffected by the above modifications; the only change is that the b'i appear in the integral KL in place of the bi. The derivation of the fact that K = KL ending in (4.11.26) is also unchanged! This at first seems strange since K has not changed, but we have apparently altered KL by the replacements bi → b'i. But KL is not an evaluation -- it is an integral equation relating KL to the b'i. In the self-consistent solution, the new functions (distributions) b'i adjust themselves so that KL does not change. KL cannot change because (4.11.26) says it must remain equal to K which is determined by the electrostatic side of the problem. It is perhaps helpful to look at (4.11.25) which says Le = KL . We know that if the dielectric μ value does not change, the external inductance Le of the transmission line cannot change so KL stays fixed. Changing μ1 and/or μ2 away from the value μ will of course change the internal inductances of the conductors, as noted below. Finally, if we look at the example associated with Fig 4.13, we still find explicitly that KL= K because the calculation leading to (4.11.29) is unchanged when bi are replaced with b'i, since the b'i are still normalized as shown in (4.12.7). The happy bottom line is that all of summary box (4.11.30) is unchanged except bi → b'i in the KL integral. The constant K can still be evaluated using the "capacitor problem" of Section 5.5 and it is unaffected. Having said this, let us now consider what happens to an operating transmission line which starts off with μ1 = μ2 = μ = μ0 and we then gradually turn a magic "permeability knob" so that μ1 gradually increases from μ0 to some value μ1 > μ0. That is to say, we gradually cause conductor C1 to become magnetic. The constant K (and therefore KL = K) does not change at all. This K determined by the potential φ part of the problem in Section 4.4 and does not even know about the magnetic modification. Thus, looking at (4.11.30), C', C, G and Le do not change. In particular, Le does not change because we have not altered μ of the dielectric. The following two items shown in box (4.11.30) do change : R = Re(Zs1+ Zs2) L = Le + (1/ω) Im(Zs1+ Zs2) . They change because Zs1 changes if we change μ1. This is so because Zs1 is always a function of the skin depth δ1, and δ1 ≡ from (2.2.20). In the special case that C1 is a round wire of radius a1 with an axially symmetric current distribution (such as the center wire of a coaxial cable), we showed in (2.4.11) that the surface impedance is given by Z1s(ω) = , (2.4.11) so certainly this Zs(ω) is a function of μ1 both due to the leading constant and through the five occurrences of δ1. Both the real and imaginary parts of Z1s(ω) will change as μ1 changes, so the transmission line parameters R and L both change. In the high frequency limit , Z1s(ω) ≈ (1+j) δ << 16a . (2.4.16) so now the variation with μ1 is through the single δ1 factor shown. Again, both real and imaginary parts of Z1s(ω) vary with μ1. Since R and L change as noted above, the transmission line characteristic impedance will also change, Z0 = = (K.11) This means, for example, if we drive a semi-infinite transmission line with some fixed voltage V(z), the driving current i(z) will vary in amplitude and phase as we turn our "permeability knob" for conductor C1. This is simply because i(z) = V(z)/Z0. So the good news is that the theory of Chapter 4 is easily extended to allow for magnetic conductors and or dielectric. Once again, the summary box (4.11.30) is unchanged when μ1 = μ2 = μ is broken except for the appearance of b'i in the KL integral, and except for the fact that Zs1 and Zs2 change as noted above, causing changes in R, L and Z0. At very high frequency, one will have Z0 = and in this case Z0 is not altered, see (4.11.32). *************************************************************************** Paradox: The above stuff seems to say that since Le and W(z) don't change, then KL should not change when one "turns on" μ1 inside C1, say. But maybe the impedance changes, so if you keep the driving V(z) fixed, then i(z) will in fact change! Pause: What happens to "An example of K = KL" ???? Look at (4.11.29) and replace bi with b'i and we get down to KL = 2π [ln[(b-a1)2/a12] !Syntax Error, Ir1dr1 b'1(r1) - 2π [ln[(b-a2)2/a22] !Syntax Error, Ir2dr2 b'2(r2). But since we are using "widely spaced round wires", we know that Jz is uniform and thus we know from *** that !Syntax Error, Idx dy b'1(x,y) = 1 and so !Syntax Error, Ir1dr1 b'1(r1) = 2π just as before, and we end up with the same result KL = [ln[(b-a1)2/a12] - [ln[(b-a2)2/a22] = K so in this example, even with magnetic media, we still get KL= K !!!! Assume W(z) is still just a function of z but it has those b'i functions in there. This is a "new" W(z) by the way since it is a difference of "new" Az objects. 4.10.1 = OK with b'i 4.10.3 = OK with b'i 4.10.4 = OK with b'i and with μ on the outside as shown. What about the flux and Le stuff??? Our "modification" changes only the μi inside the conductors, the μ in the dielectric is unchanged. I think such μi changes only change B inside the conductors, and H is the same both inside and out. The B field in the dielectric is the same, so flux is same, so Le is same!! Note: From (2.2.30) the claim is that H(a) at the surface does not change (symmetric current case). 4.10.7. If Le is "the same", and W(z) is "new", and i(z) is unchanged, how can this equation be true? It suggests that somehow W(z) does not change! I think (4.10.8) is OK with the μ shown, but I think KL has been modified so Le is then not the same. This is all as we expect due to the b'i functions inside KL. 4.11.1 = OK as stated except for the b'i in KL. thru (4.11.3) = OK (4.11.4) = OK, but we have new Az12 objects now Now look at the second of (4.11.5). I guess we evaluate each Az12 potential at x1+ and x2+ just outside the magnetization current? We have really modified the Az potential because we have added in the mag current as if it were a real current!!!! But I think it is OK to evaluate at x1+ and x2+. These are not normal derivatives, so I think ∂zAz12(x1) does not have a jump at the surface. I think all is OK through (4.11.7) now. Surface impedance still exists as in (4.11.8). Here i1 is the conduction current in the conductor, something not affected by surface current. But is Ez1 affected by surface current? I think the answer is yes! This is a real current at the surface in our model here. Surface impedance is not geared up to handle having such a current I suspect. But only conduction current enters Ohm's law stuff! So Ez1 = Zs1 i1(z) Ez2 = Zs2 i2(z) . (4.11.8) Maybe these are still OK with i1 being the conduction current only. A sticky wicket. Let's assume these equations are OK and continue. Note: The Chapter 2 round wire shows that Zs1 is a complicated function of μ1 so changing μ1 away from μ will change Zs1. What about (4.11.9? It is the same as before, but W(z) is a new W(z) somehow. So these are now different pair of equations. Now in (4.11.1) we still have W(z) = Lei(z) and both Le and W are changed, but this equation is still true! So 4.11..10 = OK 4.11.11 = OK 4.11.12 = OK BUT the parameters are may be different! 4.11.13 = OK, but Le is different so L and z are both "new" as well. There has been a change. 4.11.14 = OK. 4.11.14 BUT: how can G and C be the same, but Le is different???? We have not changed μ in the dielectric, so we have not changed β2. Something is amiss here! What about the King gauge? Does it get modified?? I carefully took different μi into account doing that gauge development, I think it is exactly right. Whiteboard Theorem: A battery does not supply magnetization current along with conduction current! I convinced myself of this fact. Question. Go back to Az1(x) = ∫ [ Jz1(x') + Jm1(x') ] dx'dy'dz' . R = |x - x'| (4.7.2)' Does the addition of the Jm term change the value of Az1(x) ? I can answer this question for the round wire since I computed both terms in Appendix B