Section 4_11 and 4_12 rearangement REVIWED

Chapter 4 Rearrangement PhL 9.23.14 Here I renamed old Section 4.11 to be 4.12 and took part this new section 4.12 to be Section 4.11. This seemed to clear up a lot of mysteries I was having. The text below starts with a section of notes followed by the actual text that was installed today 9/23/14 into lines doc. The old 4.12 on μ is now 4.13. I archived off the old Section 4.11 first. All done with this I think. Reviewed 9/23/14. Here are the major things being done below 1) Renumber equations from existing Section 4.11 to be Section 4.12 DONE 2) Insert a new Section 4.11 from a separate doc "tying in G". DONE 3) Edit the new section 4.12. 4) update external references from lines doc such that (4.11 becomes (4.12 . // A big effort, some stuff has been left in red to deal with soon. 5) further check on these references. 6) update Appendix S with this same change I will take notes above the line. // At 2 PM 9/23 I have a first cut of the entire Section 4.11 rewritten below. I then redid Section 4.12 and changed equation so I will have to go through all of lines doc and update equations numbers searching for (4.11 !!! I am going to archive off the old Section 14.11 from my backup save of yesterday which is before I started messing with it. DONE. Now install what is below on top of existing Section 12. DONE. The Figure numbers are messed up. Go edit in lines doc. DONE. OK, we are installed, but there are many loose ends. Equation numbers in italics many are wrong, especially in big boxes! –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 4.11 Relations involving C and G and the charges and currents in a transmission line Before continuing our development of the transmission line equations, we need to establish the connections between parameters C and G and the charges and currents in a transmission line Then we resume the development in Section 4.12. Consider this picture showing a section of a transmission line of length dz : Fig 4.1 We focus on the upper conductor C1. Over the distance dz, the current i(z) in this conductor is reduced by amount - di(z) = i(z) - i(z+dz) > 0 by the fact that current flows transversely to feed the surface charge qs and to feed the conductance G. Since the blue Gaussian box embedded just inside the upper conductor contains no free charge, we know from (1.1.35) that div J = 0 and ∫S J dS = 0. The latter means that the sum of all currents crossing the box boundary is 0. Thus, - di(z) = iG(z) + is(z) . " current loss feeds capacitance and conductance" (4.11.1) The two currents may be written iG(z) = [Gdz] V(z) // iG flows through the dielectric (4.11.2) is(z) = jω[qs(z)dz] // is feeds the true surface charge per length qs (4.11.3) The section of dielectric has some transverse resistance Rt = 1/[Gdz] and then iG(z) =V(z)/Rt. Quantity G is the conductance of the dielectric per unit length of the transmission line. The true total surface charge per length is qs and is feeds this charge according to is(z) = jω[qs(z)dz] ( is = ∂t[qsdz] in the time domain). But we know for the dz-length capacitor that qs(z) = CV(z) where C is the capacitance per length. Then (4.11.1) can be written - di(z) = iG(z) + is(z) = [Gdz] V(z) + jω[CV(z)dz] = [ G + jωC ] V(z)dz . (4.11.4) Dividing by dz and taking dz→0 then gives, ∂zi(z) = - [ G + jωC ] V(z) (4.11.5) This is in fact one of the "two transmission line equations" we shall be deriving below. Quantity qs(z) is the true surface charge (per length) at location z. Corresponding to this charge is the so-called transport charge (per unit length) qc(z), where qc(z) = (ξd/εd)qs(z) (4.11.6) as shown in (1.5.17). Here qs and qc are the integrals of the corresponding surface charge densities ns and qs over the surface of the C1 conductor section shown in Fig **. The transport charge density qc is larger than qs because it accounts for both the surface charge and the charge lost due to leakage into the dielectric, as described in Section 1.5. The two charges are related to the real and complex capacitances in this manner qs(z) = C V(z) C = real capacitance (per length) (4.11.7) qc(z) = C'V(z) C' = complex capacitance including effect of G (per length) (4.11.8) where the second line really defines the complex capacitance C' . Therefore using (4.11.6) and (1.5.1c) for ξd, C'/C = qc(z)/qs(z) = (ξd/εd) = [εd - jσd/ω]/ εd = 1 + (1/jω) (σd/εd) or C' = C + (C/jω) (σd/εd) or jωC' = C(σd/εd) + jωC . (4.11.9) The current feeding the transport charge in the transmission line section of length dz is given by ic(z) = is(z) + iG(z) (4.11.10) where ic(z) = jω [qc(z)dz] . // ic = ∂t[qcdz] in the time domain (4.11.11) Using (4.11.8) this says ic(z) = jω [C'V(z)dz] . (4.11.12) Then (4.11.10) and (4.11.4) imply jω [C'V(z)dz] = [ G + jωC ] V(z)dz (4.11.13) so that jωC' = G + jωC . (4.11.14) Comparing this with (4.11.9) shows that G = C(σd/εd) (4.11.15) which is an interesting relationship between G and C for a transmission line with arbitrary conductor shapes. We saw this relationship just below (1.5.20) for the special case of a parallel plate capacitor. Finally, were we to assume that transmission line quantities all have the simple z dependence e-jkz (implying a traveling wave ej(ωt-kz) ), then starting with (4.11.5), ∂zi(z) = - [ G + jωC ] V(z) (4.11.5) -jk i(z) = - jωC' V(z) // e-jkz dependence and (4.11.14) or i(z) = (ω/k) C' V(z) or i(z) = qc(z) (ω/k) // using (4.11.8) or i(z) = qc(z) v . // v ≡ (ω/k) , e-jkz assumed (4.11.16) In the last line we use the fact that v ≡ (ω/k) is the complex phase velocity of the wave ej(ωt-kz). This last equation can be interpreted as saying that the total current i(z) acts as if the transport charge qc(z) were traveling at speed v down the transmission line. If G = 0, then ξd = εd and qc = qs . Furthermore, if ω is large, then k = βd0 = ω = ω/vd [ see (1.5.1b) ] where vd which is the speed of light in the dielectric. In this case one finds that i(z) = qs(z) vd . G = 0 and large ω (4.11.17) Here one has the illusion that the current i(z) consists of the surface charge qs(z) moving at vd. Since vd is something near the speed of light, we know that the surface charge electrons are not really flowing down the line at such a speed. This interesting issue is addressed in Appendix D.9 (c). 4.12 The Classical Transmission Line Equations The results of the Sections 4.1 through 4.10 of this chapter may be succinctly summarized as: (4.12.1) = = K (4.4.7) Le = = KL (4.10.8) K ≡ !Syntax Error, Idx1' dy1' α1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' α2(x2',y2') ln(s222/s122) (4.4.8) KL ≡ !Syntax Error, Idx1' dy1' b1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' b2(x2',y2') ln(s222/s122) (4.10.9) Notice that we have made no assumptions whatsoever about the cross-sectional shape of the transmission line. We have only assumed that the transverse dimensions are small compared to the wavelength λ that corresponds to β -- this was the transmission line limit. (a) Initial Processing There are several equations from Chapter 1 we shall now press into service: E = - grad φ - ∂tA (1.3.1) div A = - μdεd ∂tφ - μσφ . // the King gauge (1.3.18) In the frequency domain these become, E = - grad φ - jωA div A = - j (βd2/ωφ . // the King gauge, see (1.5.1a) and (1.5.5) re βd2 (4.12.2) According to the Fact stated in (4.7.1), potential A has only component Az, so these equations become Ez(x) = - ∂zφ(x) - jωAz(x) ∂zAz(x) = - j (βd2/ωφ(x) . (4.12.3) However, as was shown at the end of Step 1 below (3.7.8), the second line of (4.12.3) can only be justified in the strong or extreme skin effect regimes, and we continue then to assume our transmission line is operating at sufficiently high ω to be in the small δ regime. The potentials in the above equations are those due to both conductors and were denoted as φ12 and Az12 in the previous sections. We then rewrite the above as Ez(x) = - ∂zφ12(x) - jωAz12(x) (4.12.4a) ∂zAz12(x) = - j (βd2/ωφ12(x) . (4.12.4b) Recall now the conductor-surface-located points x1 and x2 as shown for example in Fig 4.10. If we evaluate each of the above equations at x = x1 and then x = x2 and then subtract, we get Ez(x1) - Ez(x2) = -∂z[φ12(x1) - φ12(x2)] - jω[Az12(x1) - Az12(x2)] (4.12.5a) ∂z[Az12(x1) - Az12(x2)] = - j (β2/ω[φ12(x1) - φ12(x2)] . (4.12.5b) Then using these definitions (again, we are assuming the strong or extreme skin depth regime) V(z) ≡ φ12(x1) - φ12(x2) (4.4.1) W(z) ≡ Az12(x1) - Az12(x2) (4.10.1) we may rewrite (4.12.5) in this simple manner, Ez(x1) - Ez(x2) = - ∂zV - jωW (4.12.6a) ∂zW = - j (βd2/ωV . (4.12.6b) The quantity Ez(x1) is the longitudinal electric field at point x1 on the surface of conductor C1. It is related to the conductor's at-the-surface current density by Jz(x1) = σEz(x1). If the conductor were "perfect", we would have σ = ∞ and Ez(x1) = 0, but real conductors are not perfect. However, since we are assuming the strong or extreme skin effect all along here in our analysis, we do know that Ez(x1) and Ez(x2) are very small. (b) Averaging Repair and the Transmission Line Equations Our theory now has an inconsistency which needs to be fixed. We know that for a general transmission line operating at ω > 0, the current density Jz inside the conductors will not be uniformly distributed. It will be larger in the conductor region closest to the other conductor. This "proximity effect" is discussed in Appendix P from an eddy current point of view, see Fig P.13 for an example. The Jz current non-uniformity can be very dramatic as for example in a transmission line having this cross section, where Jz will be large near the gap and small far from the gap: Fig 4.12 Since Jz is non-uniform in each conductor, so is Ez, and so we expect Ez(x1) to be a strong function of the point x1 on the perimeter of C1, certainly for the above cross section example. This means that the left side of (4.12.6a) is a function of x1 = (x1,y1,z) and x2 = (x2,y2,z) whereas the right side in our theory is a function only of z. To remedy this inconsistency, we now have to think of V and W as having very slight dependence on x1 and x2 which we generally ignore, but which we must face up to in (4.12.6a). In reality we have V(x1,x2) and W(x1,x2). This is a manifestation of the fact that in reality φ ≈ constant and Az ≈ constant on the boundaries (with ≈ and not = ). In the extreme skin effect regime (think a very good conductor), the left side of (4.12.6a) can be a violent function of x1 and x2 as in the case of the above figure, but the left side is always very small, even where it is largest, and its variation can be accommodated by the right side of (4.12.6a) which is the difference of large-valued functions which vary only slightly with x1 and x2. So first rewrite (4.12.6a) as Ez(x1) - Ez(x2) = - ∂z V(x1,x2) - jω W(x1,x2) . (4.12.6a)' Backing up another step, we write out of (4.12.4a) for the two perimeter points x1 and x2, (1/σ)Jz(x1) = Ez(x1) = - ∂zφ12(x1) - jωAz12(x1) x1 on perimeter of C1 (1/σ)Jz(x2) = Ez(x2) = - ∂zφ12(x2) - jωAz12(x2) x2 on perimeter of C2 (4.12.4a)' Calling the perimeter distances of the conductors P1 and P2, we then average each of these equations around its appropriate perimeter. Apply (1/P1) ∫C1 ds1 to the first equation and (1/P2) ∫C1 ds2 to the second to get [ ds1 is a distance element along the perimeter of C1 ] , (1/σ)<Jz(x1)>C1 = <Ez(x1) >C1 = - ∂z<φ12(x1) >C1 - jω<Az12(x1) >C1 (1/σ)<Jz(x2)>C2 = <Ez(x2) >C2 = - ∂z<φ12(x2) >C2 - jω<Az12(x2) >C2 . Subtract the second line from the first to get, [<Ez(x1) >C1 - <Ez(x2) >C2] = - ∂z[<φ12(x1) >C1 - <φ12(x2) >C2] - jω [<Az12(x1) >C1 - <Az12(x2) >C2 ] . We now redefine V and W to be the averages appearing in these equations, along with Ez1 and Ez2 : Ez1(z) ≡ <Ez(x1) >C1 = (1/P1) ∫C1 ds1 Ez(x1) Ez2(z) ≡ <Ez(x2) >C2 = (1/P2) ∫C2 ds2 Ez(x2) V(z) ≡ <φ12(x1) >C1 - <φ12(x2) >C2 = <V(x1,x2)>C1,C2 W(z) ≡ <Az12(x1) >C1 - <Az12(x2) >C2 = <W(x1,x2)>C1,C2 (4.12.7) with this result [Ez1(z) - Ez2(z)] = - ∂z V(z) - jω W(z) . (4.12.8) Meanwhile, the surface impedances on C1 and C2 are defined by (see C.2.1) , Ez1(x1) = Zs1(x1) i1(z) Ez2(x2) = Zs2(x2) i2(z) (C.2.1) which we average in the same way to obtain Ez1(z) = Zs1 i1(z) Zs1 ≡ (1/P1) ∫C1 ds1 Zs1(x1) Ez2(z) = Zs2 i2(z) Zs2 ≡ (1/P2) ∫C2 ds2 Zs2(x2) . (4.12.9) There will be some location on C1 where Ez1(x1) and thus Zs1(x1) will be maximal (for example on the walls of the gap in Fig 4.12). Referring to this value as Zs1,max we can define p1 ≡ (Zs1/Zs1,max)P1 p2 ≡ (Zs2/Zs2,max)P2 (4.12.10) where p1 is the effective length of the "active perimeter" of C1. This then provides a crude model for the symbol p which appears in (2.5.1) and Fig 2.16 which we replicate here, Fat twinlead Fig 2.16 Then using i(z) = i1(z) = -i2(z) and (4.12.9), rewrite (4.12.8) and (4.12.6b) as [Zs1 + Zs2] i(z) = - ∂z V(z) - jω W(z) ∂zW(z) = - j (βd2/ωV(z) (4.12.11) where the second equation above is the < >C1,C2 average of (4.12.6b). Continuing this repair effort, we back up to box (4.12.1) and write V(x1,x2) = q(z) / C'(x1,x2) = q(z) [ K(x1,x2) ] W(x1,x2) = i(z) Le(x1,x2) = i(z) [ KL(x1,x2)] (4.12.12) which we average in the same way to get V(z) = q(z) W(z) = i(z) Le ≡ (1/P1) ∫C1 ds1 (1/P2)∫C2 ds2 = < >C1,C2 Le ≡ (1/P1) ∫C1 ds1 (1/P2)∫C2 ds2 Le(x1,x2) = < Le(x1,x2)>C1,C2 . (4.12.13) The "constants" K and KL in (4.12.1) are similarly replaced with their <>C1,C2 averages. In the discussion below, we shall no longer mention the averaging process, but it should be understood that for closely spaced conductors the symbols Zs1, Zs1, Le, C', K, KL, V, W are the perimeter-averaged values discussed above. For widely spaced conductors, Jz is roughly uniform over the conductor cross sections and perimeters and the averaging process is not needed. The whole subject of averaging is reviewed in more detail in Appendix S. Inserting the equations on the first line of (4.12.13) into (4.12.11) we get (Zs1 + Zs2) i(z) = - ∂zV(z) - jω Le i(z) Le ∂z i(z) = - j (βd2/ωV(z) which we then rearrange as ∂zV(z) = - [ Zs1+ Zs1+ jωLe] i(z) ∂z i(z) = - [ jβd2/(ωLe)] V(z) . (4.12.14) These are the classical transmission line equations. They are usually written in this form: [ ∂/∂z = d/dz] = - z i(z) = - y V(z) (4.12.15) where z = Zs1+ Zs1+ jωLe = R + jωL // z and R are ohms/m y = jβd2/(ωLe) = G +jωC = jωC' . // y and G are mhos/m (4.12.16) Note: We have been using bold notation only for vectors, and we now break that guideline by bolding these complex quantities z and y. Our purpose for this bolding is to distinguish them from Cartesian coordinates z and y which typically appear in the same problem. In King's books, all complex parameters are put in bold font, but we do this only for z and y. The quantities z and y are called the transmission line impedance and admittance. On the right we have partitioned the expressions for z and y into their real and imaginary parts in terms of four real parameters R,L,G and C. Parameters R and L are defined to be the resistance and inductance of the transmission line (per unit length). When ω = 0, R = Rdc = the total DC resistance of both transmission line conductors, but when ω is large this is no longer true. One can associate this fact with the skin effect which displaces current away from the conductor's central region Comparing the right equation in (4.12.14b) to (4.11.5), we may immediately interpret the real parameters C and G as the capacitance and conductance of the transmission line (per unit length) as discussed in Section 4.11. The fact that G +jωC = jωC' has been shown in (4.11.14), where C' is the complex capacitance (per unit length) of the transmission line. If one applies ∂z to either of the above equations and then uses the other, one obtains these corresponding wave equations (but in the ω domain, so they are really Helmholtz equations), - zy V(z) = 0 - zy i(z) = 0 . (4.12.17) We refer to (4.12.14b) as the first order transmission line equations, and (4.12.15) as the second order transmission line equations. Jumping the gun a bit, if we assume now a traveling-wave z dependence ej(ωt-kz) for both V(z) and i(z), where k is the wave's (possibly complex) wavenumber, then ∂z → -jk and the transmission line equations become -jk V(z) = - z i(z) and -jk i(z) = - y V(z) or -jk = - z i(z)/V(z) and -jk = - y V(z)/i(z) . Equating these last two expressions gives - z i(z)/V(z) = - y V(z)/i(z) => z/y = [V(z)/i(z)]2 and we then have, Z0 ≡ V(z)/i(z) = = (4.12.18) where by definition Z0 is the characteristic impedance of the transmission line. Obviously Z0 is completely different from z even though both are referred to as an impedance. Looking at (4.12.14c) we see that jωC' = jβd2/(ωLe) or LeC' = (βd/ω)2 = μdξd // using (1.5.1a) => LeC = μdεd = 1/vd2 (4.12.19) Recall now from summary box (4.12.1) that = = K => C' = 4πξd/K (4.4.7) Le = = KL (4.10.8) (4.12.1) Inserting these expressions into (4.12.17) that [Le][C'] = μdξd gives [ KL ] [4πξd/K] = μdξd or KL = K (4.12.20) This is a remarkable connection between our two seemingly unrelated constants K and KL, K ≡ !Syntax Error, Idx1' dy1' α1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' α2(x2',y2') ln(s222/s122) (4.4.8) KL ≡ !Syntax Error, Idx1' dy1' b1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' b2(x2',y2') ln(s222/s122) . (4.10.9) Since K involves a peripheral line integral of surface charge densities αi whereas KL involves a full cross sectional area integral of the current densities bi, it seems unlikely these integrals would be equal, but they are equal. [Appendix S shows that this is not exactly true when the averaging is done for very closely spaced conductors, but it is still quite close and we ignore this small error. ] (d) An example of K = KL The equality even seems unlikely in a case with symmetric densities on round wires, so let's do a check using our Section 4.5 example with widely-spaced round wires of unequal diameters. The first thing we need is a new picture to display the "kinematics" of the KL integral ( since densities are symmetric, one should regard this picture as having b much larger than shown relative to a1 and a2), Fig 4.14 As before, we read off the four distances of interest using the law of cosines. The new distances are all different than they were before since x1' and x2' are now each integrated over their respective disks instead of the bounding circles. s212 = r12 + (b-a1)2 - 2 r1(b-a1) cos(θ1) s112 = r12 + a12 - 2 r1 a1 cos(θ1) s222 = r22 + a22 + 2 r2 a2 cos(θ2) s122 = r22 + (b-a2)2 + 2 r2(b-a2) cos(θ2) . The integration rule is still !Syntax Error, Idθ ln (A ± Bcosθ) = 2π ln[(1/2)(A + )] . (4.5.5) The first integral is: !Syntax Error, Idθ1 ln(s212) = !Syntax Error, Idθ1ln([r12 + (b-a1)2 - 2 r1(b-a1) cos(θ1)] A = r12 + (b-a1)2 B = 2 r1(b-a1) A2-B2 = [r12 + (b-a1)2]2 - 4 r12(b-a1)2 = [r12 - (b-a1)2]2 => = (b-a1)2- r12 > 0 b >> a1 => !Syntax Error, Idθ1 ln(s212) = 2π ln[(1/2)( r12 + (b-a1)2 + (b-a1)2 - r12 ) = 2π ln[(b-a1)2] But this integral is the same as before! The s112 integral is obtained from the above with b-a1→a1 !Syntax Error, Idθ1 ln(s112) = 2π ln(a12) which is also the same as before. The other two integrals are found from 1→ 2. Our integral summary is then exactly the same as (4.5.6), !Syntax Error, Idθ1 ln(s212) = 2π ln[(b-a1)2] !Syntax Error, Idθ1 ln(s112) = 2π ln(a12) !Syntax Error, Idθ2 ln(s222) = 2π ln(a22) !Syntax Error, Idθ2 ln(s122) = 2π ln[(b-a2)2] . (4.5.6) We now assume that the current densities bi each have radial symmetry ("widely spaced wires") , b1(r1,θ1) = b1(r1) (4.12.21) where b1(r1) is a completely arbitrary function, with the following normalization of (4.7.4), !Syntax Error, Idθ1 !Syntax Error, Ir1dr1 b1(r1) = 1 => !Syntax Error, Ir1dr1 b1(r1) = 1/2π . (4.12.22) We now proceed to calculate the constant KL KL = !Syntax Error, Idθ1 !Syntax Error, Ir1dr1 b1(r1) ln(s212/s112) -!Syntax Error, Idθ2 r2dr2 b2(r2) ln(s222/s122) = !Syntax Error, Ir1dr1 b1(r1) !Syntax Error, Idθ1 ln(s212/s112) - !Syntax Error, Ir2dr2b2(r2) !Syntax Error, Idθ2 ln(s222/s122) = 2π !Syntax Error, Ir1dr1 b1(r1) [ln[(b-a1)2]- ln(a12)] - !Syntax Error, Ir2dr2b2(r2) [ ln[(b-a2)2] - ln(a22)] = 2π [ln[(b-a1)2/a12] !Syntax Error, Ir1dr1 b1(r1) - 2π [ln[(b-a2)2/a22] !Syntax Error, Ir2dr2 b2(r2) = [ln[(b-a1)2/a12] - [ln[(b-a2)2/a22] = ln [] = K as obtained in (4.5.7) (4.12.23) and we have then shown KL = K for this particular example. The key fact is that the dθ integrals appear to be functions of ri , but the ri2 terms cancel and so the dθ integrals are independent of ri. (e) Summary of Results Classical Transmission Line Equations and Parameters (ω domain) (4.12.24) K ≡ !Syntax Error, Idx1' dy1' α1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' α2(x2',y2') ln(s222/s122) (4.4.8) KL ≡ !Syntax Error, Idx1' dy1' b1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' b2(x2',y2') ln(s222/s122) , (4.10.9) K = KL real and dimensionless (4.12.30) = - z i(z) ( - zy) V(z) = 0 z = R + jωL transmission line equations = - yV(z) ( - zy) i(z) = 0 y = G +jωC (4.12.14), (4.12.15) z = Zs1 + Zs2 + jωLe (4.12.17) XL ≡ ωLe , XC ≡ 1/(ωC) y = jωC' = jωC + (σd/εd)C (4.12.24) G = (σd/εd)C (4.12.25) R = Re(Zs1+ Zs2 + jωLe) L = (1/ω) Im(Zs1+ Zs2 + jωLe) = Le + Li Le = K (4.12.29) and (4.12.30) C' = 4πξd/K (4.12.26) C' = (ξd/εd)C (4.12.23) C = 4πεd/K (4.12.26) G = 4πσd/K (4.12.26) + (4.12.25) => G/C = σd/εd LeC' = μdξd (4.12.27) LeC = μdεd = 1/vd2 (4.12.28) Z0 = = (4.12.16) Z0 (large ω) ≈ ≈ = (1/4π) K = (K/4π) Zm // See comments below λ >> D (4.3.6) assumed transmission line limit where βd = 2π/λ βd2 = μdεdω2 - jωμdσd = ω2μd ( εd - jσd/ω) = ω2μd ξd ξd ≡ εd - jσd/ω . (1.5.1a) F(z) = F(0) e-az e-jbz a ≡ Re() = Re[] b = Im() see (5.3.6) attenuation phase Comments: 1. In Chapter 2 we computed the surface impedance Zs for a round wire in the case of axially symmetric current and we found that, for large ω, Zs(ω) ≈ (1+j) (2.4.16) δ ≡ = skin depth (2.2.20) so that Zs(ω) ≈ (1+j) . (4.12.25) Presumably the result will be Zs(ω) ~ for any conductor cross section shape. Then L = Le + (1/ω) Im(Zs1+ Zs2) = Le + (stuff) 1/ → Le for large ω (4.12.26) For this reason, the high frequency characteristic impedance Z0 can be written as shown in (4.12.34). 2. Conductors have internal inductance Li as well as external inductance Le. In Appendix C.3 (a) we compute the low frequency internal inductance of a round wire to be Li = μ/8π = (μ/μ0) * 50 nH/m . Our Chapter 4 transmission line development makes no mention of Li. This can be traced to Figure 4.11 where only the external magnetic flux is involved. In fact, Li is accounted for in the imaginary part of the surface impedance Zs . For example, we found that for our round wire situation, Zs(ω) = + jω = Rs + jωLs // low frequency limit (2.4.12) and here one sees that Ls = = Li . 3. We have assumed that εd and μd are real. If not, the usual adjustments can be made in (4.12.34) for the interpretations of R,L,G and C. See for example (3.3.4) concerning σ being replaced by σeff if ε has an imaginary part. 4. Apart from the symmetric cases like the examples of Section 4.5 and 4.6, we do not yet have a way to compute K and the transmission line parameters since the charge and current distributions αi and bi are not known. This matter will be remedied in Chapter 5. 5. A strip transmission line of width w and separation s with s << w is the simplest example of the above summary: E = V/s n = εdE = εdV/s q = nw C = q/V = εdw/s => K = 4πεd/C = 4π(s/w) so C = 4πεd/K = εd (w/s) K = 4π (s/w) G = 4πσd/K = σd (w/s) Le = (μd/4π) K = μd (s/w) Z0 ≈ (K /) 30Ω = 4π (s/w) (1/) 30Ω = (s/w) (1/) 377Ω (4.12.27) (f) Time domain equations (telegraph equations) The results above are all stated in the frequency domain, but it is a simple matter to convert them to the time domain using jω ↔ ∂t. One then makes these replacements z = R+jωL → L∂t + R y = G+jωC → C∂t + G zy = (R+jωL)( G+jωC) → (R + L∂t)( G + C∂t) = LC∂t2 + (LG+RC)∂t + RG . (4.12.28) Here then are selected equations and their translations to the time domain: Transmission Line Equations (4.12.14b) : [ coupled first order PDE's] ∂zV = - z i ∂zV(z,t) = - L∂ti(z,t) - LR i(z,t) ∂z i = -y V ∂z i(z,t) = - C∂tV(z,t) - CGV(z,t) (4.12.29) Transmission Line Wave Equations (4.12.15) [ damped wave equations ] ( ∂z2 - zy) V(z) = 0 [ ∂z2 - LC ∂t2 - (LG+RC)∂t - RG] V(z,t) = 0 ( ∂z2 - zy) i(z) = 0 [ ∂z2 - LC ∂t2 - (LG+RC)∂t - RG] i(z,t) = 0 (4.12.30) If we set the loss parameters R and G both to 0 these equations become ∂zV(z,t) = -L∂ti(z,t) [ ∂z2 - LC ∂t2] V(z,t) = 0 [ undamped wave equations] ∂z i(z,t) = -C∂tV(z,t) [ ∂z2 - LC ∂t2] i(z,t) = 0 // lossless (4.12.31) At large ω one has L ≈ Le (note 1 above) and since (4.12.28) says LeC = μdεd = 1/vd2 we conclude that the factor LC appearing in the above wave equations is 1/vd2 where vd is the dielectric wave velocity. The various transmission line equations shown above in the time domain are often referred to as telegraph (telegrapher, telegrapher's) equations.