is anything rea REVIEWEDl

Is anything Real ? PhL 9.22.14 I was having trouble seeing which things are real and which are not. The gist of this note is now installed in lines doc as a Technical Note at the end of Section 4.1. Basically everything floats against everything else. I added into lines doc some other comments on this subject. I think it is all resolved now. In particular, I show that even if something is complex, it can still be normalized to 1 in a certain sense explained in that Tech notes. That was the concern I guess. This little doc mainly relates to sections 4.1 and 4.7 so they will be stored in the Chapter 4 folder. This review done 9.23.14. Examine Chapter 2. We find there that E(r) = E(a) β = ej3π/4 (/δ) Even if E(a) were real for some reason, E(r) is in general very complex! Same for B(r). Examine Chapter 4. Opener is this φ1(x,ω) = ∫ ρ1(x',y',z',ω) dx'dy'dz' . R = |x - x'| (4.1.1) Here ρ1(x,y,z,ω) = ρ1(x,y,z) 2πδ(ω-ω1) ρ1(x,y,z,t) = ρ1(x,y,z) exp(jω1t) Can we say that ρ1(x,y,z) is real ? Maybe the answer is that we could insist this be real. Then ρ1(x,y,z,t)physical = Re[ρ1(x,y,z,t)] = ρ1(x,y,z) cos(ω1t) = real Well then what about Jz1(x,y,z) in (4.7.3) ? This is the current inside conductor C1 and I know from Chapter 2 that in general its phase varies with radius, so I cannot just say that it is real. In general, you have to say something like this for some general quantity Q Q(x,y,z,t) = Q(x,y,z) ejωt = |Q(x,y,z)| ejφ(x,y,z) ejωt and then Qphysical(x,y,z,t) = Re(Q) = |Q(x,y,z)| cos [ ω1t + φQ(x,y,z) ] So I guess in general it might be that ρ1(x,y,z,ω) on conductor C1 might have some complicated phase which is a function of position on the surface. Certainly it varies with z in Appendix D as ejkz. Conclusion: Assume that nothing is real! Now look at Chapter 4 in this light. φ1(x,ω) = ∫ ρ1(x',y',z',ω) dx'dy'dz' . R = |x - x'| (4.1.1) Both ρ1 and φ1 are assumed to have position-dependent phase. ρ1(x,y,z) = α1(x,y) q1(z) . (4.1.2) Here can we assume that α1(x,y) has the same phase at all points on a conductor perimeter? In the time domain, we would ask if the charge peaks at all points on the perimeter at the same instant. How would you answer such a question? This might be true at the driving end of a conductor. But maybe things change going down the line. For example, does n(θ) have a θ-dependent phase? I write elsewhere n(z,y,z,t) = e-jkz ejωt n(θ) where I always think of n(θ) as having the same phase at all θ. What can be said about the E field in some plane z = constant in a TL? This time-coincidence around a cross section of surface charge is a hidden assumption of my theory. It comes from the electrostatic capacitor problem idea. Go back to these equations in Ch 4: ρ1(x,y,z) = α1(x,y) q1(z) . (4.1.2) C/m3 1/m2 C/m !Syntax Error, Idx dy α1(x,y) = 1 . (4.1.3) I think I am really making an ansatz here that α1(x,y) is real, and the integral is real. If it had some weird phase around the perimeter, then perhaps the integral could not be made real. Big Problem. In (4.7.3) I write 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) I am pretty sure that I cannot assume b1(x,y) is real as I assumed α1(x,y) was real. I just showed that inside the round wire, Ez varies with r, so Ez varies with (x,y). How then can you impose the condition !Syntax Error, Idx dy b1(x,y) = 1 on a complex function? You are requiring that !Syntax Error, Idx dy b1(x,y) = !Syntax Error, Idx dy |b1(x,y,)| ejφ(x,y) = 1 Maybe I can bail this out. I think I could require that. Suppose we make our separation above and we find that !Syntax Error, Idx dy b1(x,y) = C = a complex number Define these new functions b'1(x,y) ≡ (1/C) b1(x,y) i'1(z) ≡ C i'(z) We then have Jz1(x,y,z) = b1(x,y) i1(z) = b'1(x,y) i'1(z) with !Syntax Error, Idx dy b'1(x,y) = 1 ρ1(x,y,z) = α1(x,y) q1(z) Then we have !Syntax Error, Idx dy b'1(x,y) = !Syntax Error, Idx dy b1(x,y)/C = C/C = 1 Then rewrite b1(x,y) i1(z) = b'1(x,y) i'1(z) !Syntax Error, Idx dy b1(x,y) = ejκ with κ = real. What this really says is that !Syntax Error, Idx dy | b1(x,y)| = 1 Now i1(z) has some unknown phase. Write !Syntax Error, Idx dy Jz1(x,y,z) = !Syntax Error, Idx dy b1(x,y) i1(z) = i1(z) !Syntax Error, Idx dy b1(x,y) = i1(z) ejκ Suppose we now make a new partition Jz1(x,y,z) = b'1(x,y) i'1(z) where b'1(x,y) = e-jk b1(x,y) i'1(z) = ejκ i1(z) Then for this new partition we will have !Syntax Error, Idx dy b'1(x,y) = !Syntax Error, Idx dy e-jk b1(x,y) = e-jk!Syntax Error, Idx dy b1(x,y) = 1 So yes, this does bail things out for normalization. So why not just assume that both α1(x,y) and b1(x,y) have phases which vary with (x,y). In each case, the partition allows the integral to be 1. I like it. I added a comment for α1(x,y) but I will now remove it and allow these things to be complex. Actions: I added a Tech Note at the end of Section 4.1 and a small paragraph after (4.7.4) to explain why the integral of a complex function like α1(x,y) or b1(x,y) can be set to the real number 1. I have not required that either of these functions be real! I scan down now through Chapter 4 looking for "phase and reality" issues. 4.1 as already noted, and surely φ1(x,y,z) has some phase which varies with x,y,z though I don't explicitly state that fact. 4.2 nothing need be done 4.3 nothing need be done 4.4 Here I assume that φ = constant on a slice. That suddenly says φ has a constant phase on that perimeter, but it does not say φ is real. We then get down to this: = = {!Syntax Error, Idx1' dy1' α1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' α2(x2',y2') ln(s222/s122) } = K and V(z) = q(z) K (4.4.7) Now something must be said! I kind of want K to be real here, but α1(x1',y1') is complex, so that is a problem Houston. Maybe I should just avoid stating that K is real! But I think I really have to say something here that α1(x1',y1') is taken to be real. Otherwise equations like C = 4πεd/K capacitance per unit length of the transmission line G = 4πσd/K conductance per unit length of the transmission line so C/G = εd/σd . (4.4.10) don't make any sense. I could say 4πξd/K = C' = C + G/jω and allow K to still be complex. Then for example C would be C = Re [4πξd/K ] and we are still OK. But I really want to see C = 4πεd/K because that is one of my main results! Eventually in Chapter 4 we assume that a z=constant perimeter has a constant φ and a constant Az. In this case, we can take α1(x1',y1') = real and also b1(x1',y1') = real. Then we get all the usual claims. Now how does φ,Az = constant on perimeter say something about α1(x1',y1') ? Well, these two conditions tell us that Eθ = 0 and then we cannot have ρ1(x,y,z) = α1(x,y) q1(z) peaking at different points on the perimeter at different times, because that would imply Eθ ≠ 0. Action: I added to the Tech note to say that φ ≈ constant allows us to take α1(x1',y1') = real. In this case K comes out real, and we get all our desired stuff above. Action: Cleaned up statement that K is real near equation (4.4.8). I now continue the scan through Ch 4. I have now to hold off on (4.4.11). We then skip through the two huge examples and we end up starting into Section 4.7. I have my little comment about b1(x,y) after (4.7.4), but I don't claim it is real!!! Since this b1(x,y) is a current inside the wire, I suspect it canNOT be taken real. I say nothing at this point about b1 being real. Nothing happens for the rest of Section 4.7 so We come to Section 4.8. It is OK as is. Section 4.9 also seems OK, not saying much. Now we come to Section 4.10. Here there will be issues. Equation (4.10.4) shows KL but I don't define it yet. I then come to my little red loop in Fig 4.11. I end up with W(z) = Le i(z) . (4.10.7) Do W(z) and i(z) have the same phase so Le = real ??? Don't know at this point. But then we come to Le = = KL (4.10.8) where KL is the following dimensionless real number, KL ≡ !Syntax Error, Idx1' dy1' b1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' b2(x2',y2') ln(s222/s122) . (4.10.9) I want to say KL is real, but the same argument does not work here! I know that b1(x1',y1') has a phase which varies especially with radius r, probably also with θ. Suppose we just assume Le is in general complex and KL is in general complex. I would then delete the word real appearing below (4.10.8) and that would take me to Section 4.11. opening text OK (a) OK (b) on repair. OK through (4.11.13). OK through (4.11.14a) . OK thru (4.11.16). OK through : z = R + jωL = Zs1 + Zs2 + jωLe (4.11.17) y = G + jωC = jβd2/(ωLe) (4.11.18) Comments: If Le were completely real, then since βd = ω/vd we would have y = completely imaginary. So I have already assumed that Le is complex!!! I did not realize that! OK through (4.11.18). I might delete text shown in red. I continue: I think after (4.11.18) things get muddled, something is just not right here. Way back in Section 4.4 I already say this For example, in analogy to what we did with a parallel plate capacitor in (1.5.19), we may define the complex capacitance C' per unit length of our transmission line using (4.4.6) as follows: = = {!Syntax Error, Idx1' dy1' α1(x1',y1') ln(s212/s112) -!Syntax Error, Idx2' dy2' α2(x2',y2') ln(s222/s122) } = K and V(z) = q(z) K (4.4.7) so I already have the symbol C' introduced. Maybe this is where my Digression on C' should go. I go on right away to say this Recall from (1.5.20) that (C', C and G are discussed further in Section 4.11 below) C' = C + 1/(jωR) = C + G/(jω) (4.4.9) So I already have C' involved and defined and active. No need to start it again. I am going to start a scratch Section 4.11 doc so I have a sandbox to play in without wrecking lines doc. ______________________________________________________________________ TBC below this red text post feeding. // Resuming. I now look in more detail into Fig 4.13 and surrounding discussion. I assume G = 0 in this little discussion, very similar to the CP BC. If we really have k = ω/v then I guess it is really true i(z) = q(z) vd (4.11.19a) Do I really believe this result? Why is there no capacitive current ? Because we are talking here free charge and real conduction current, I should clarify that both here and at CPBC. So no free charge can flow off the surface radially if G = 0 and even if there is C > 0. I guess this is OK as long as you remember that G = 0. I guess OK thru (4.11.19a). OK thru (4.11.20). But I am confused here. I already know that y = G + jωC = jβd2/(ωLe) = jωC' so (4.11.21) is already known?? Why not just say this: z = R + jωL = Zs1 + Zs2 + jωLe (4.11.17) y = G + jωC = jβd2/(ωLe) = jωC' (4.11.18) where Zs1, Zs2 and Le are the averages shown in (4.11.9) and (4.11.13). In the last line we define C' to be the complex capacitance C' = C + G/(jω), in analogy with