Chap 5 rewrite 1 REVIEWED

Rewrite of Chapter 5 PhL 1.8.14 Chapter 5: The Transverse Problem 5.1 Philosophy ?? maybe discard completely 5.2 The Helmholtz Equations and Separation of Variables Since Ja and ρa no longer exist, as described above, we set Ja = 0 in (1.5.1) and ρa= 0 in (1.5.2), so we have this pair of differential equations, ( 2 + β2 ) A = 0 (5.2.1) ( 2 + β2 ) φ = - (1/ε) ρ (5.2.2) where β ≡ ω (5.2.3) with ξ ≡ [ ε + σ/(jω) ] (5.2.4) Since φ and A are the potentials arising from the presence of both conductors C1 and C2, the φ and A appearing in (5.2.1) and (5.2.2) correspond to φ12and A12 of Chapter 4. The above equations apply at all points in space, both in the dielectric between the conductors, and inside the conductors as well. Note: A differential equation of the form ( 2 + β2 ) f = 0 is usually referred to as the Helmholtz equation, in honor of Hermann von Helmholtz (1821-1894), an early electromagnetic researcher. Let us first consider the equation (5.2.2) for φ. We shall do a separation of variables for both φ and ρ. For ρ, we perform a separation very similar to (4.1.2) through (4.1.4), ρ(x,y,z) = ρt(x,y) q(z) (5.2.5) where q(z) is the charge per unit length on conductor C1. This defines a transverse charge density ρt(x,y). The integral of ρt(x,y) over the surface of a C1 slice is +1, and over the surface of a C2 slice is -1. For the potential, we make the following separation: φ(x,y,z) = (1/2πε) q(z) φt(x,y) (5.2.6) We are of course free to set the scale factor arbitrarily, since changing the scale factor just changes the definition of φt(x,y). Our motivation for the form shown in (5.2.6) is the work of Chapter 4, in particular equation (4.2.5) In light of (4.1.13) and the related discussion, we realize that the separable form (5.2.6) is only possible if we are working in the "transmission line limit" where the wavelength λ along the transmission line is much larger than all transverse dimensions. When (5.2.5) and (5.2.6) are inserted into (5.2.2), the result is [ + 2π ] + = - β2 (5.2.7) which has the general form, [ h(x,y) ] + g(z) = - β2 (5.2.8) The only way this can be true for all x,y,z in a region is if g(z) = some constant. For reasons that will be clear later, we write this constant as kφ2. Then we get = kφ2 [ + 2π ] = - β2 - kφ2 (5.2.9) We can rewrite these as [ t2 + (β2 +kφ2)] φt(x,y) = -2πρt(x,y) (5.2.10) [ z2 - kφ2 ] q(z) = 0 (5.2.11) We now repeat the process for the vector potential component Az, which we know is the only significant component of concern. We make the following separation, similar to (5.2.6), which is motivated by (4.4.3), Az(x,y,z) = (μ/2π) i(z) Azt(x,y) (5.2.12) Putting (5.2.12) into (5.2.1) yields, [ ] + = - β2 (5.2.13) Again, this can only work if the second term is some constant, which we here call kA2. = kA2 [ ] = - β2 - kA2 (5.2.14) which leads to, [ t2 + (β2 +kA2 )] Azt(x,y) = 0 (5.2.15) [ z2 - kA2 ] i(z) = 0 (5.2.16) Consider now equations (5.2.11) and (5.2.16), which we now apply to the full potentials by making use of (5.2.6) and (5.2.12), - kA2 Az = 0 - kφ2 φ = 0 (5.2.17) We seek next to find the constants kA and kφ. Consider the differences that played such a major role in Chapter 4, V(z) = φ( x1, y1, z ) - φ( x2, y2, z ) W(z) = Az( x1, y1, z ) - Az( x2, y2, z ) (5.2.18) where x1 and x2 are points on the conductors at the same z. Applying (5.2.17) to these differences gives, - kA2 W = 0 - kφ2 V = 0 (5.2.19) Now, application of ∂/∂z to the transmission line equations (4.5.9) gives, - zy i = 0 - zy V = 0 (5.2.20) where constants z and y are the impedance and admittance of the transmission line. According to (4.5.1), W(z) = Le i(z), so we replace the first equation above with an identical one in W: - zy W = 0 - zy V = 0 (5.2.21) Comparison of (5.2.21) with (5.2.19) gives us the result we seek, kA2 = kφ2 = zy ≡ k2 (5.2.22) In retrospect, we are not surprised that the constants are equal in light of the Lorentz covariance discussion relating to equation (1.3.9). From (4.5.11) and (4.5.12) we know that z = Zi + jωLe, and y = jβd2/(ωLe ), where Zi is the total internal impedance of both conductors, and where βd is β of the dielectric. From (4.5.15) we can also say y = jωC'. Therefore k2 + βd2 = Zi(jω C') = Zi j βd2/ (ωLe) (5.2.23) For perfect conductors, Zi = 0. We might define a "low loss" transmission line as one where the right side of (5.2.23) is much smaller than βd2. Using β2 = ω2μξ this condition becomes, with μ=1, Zi << μ0f K = 4πK x 10-7 f ≈ K f(MHz) ohms/meter (5.2.24) where K is the dimensionless geometric integral defined in Chapter 4. Typically K is a number in the range 1-10, so the above statement is quite clear as a definition of "low loss". For a typical coax cable, Zi is about 1 ohm/m at 1 GHz, K ≈ 3, so the above condition is well met since 1 << 3000. Fact: For a "low loss" transmission line, as defined above, k2 ≈ - βd2 . There remains one important final connection to be made to our work of Chapter 4. The constants in the variable separations for φ and Az given in (5.2.6) and (5.2.12) were carefully selected to yield the following result, φt(x1) - φt(x2) = Azt(x1) - Azt(x2) = K (5.2.25) where x1 and x2 are any points lying on C1 and C2 in the same z plane. Recall that K determines the three transmission line parameters G, C and Le. The ingredients needed to show that (5.2.25) is true are (5.2.6) and (5.2.12) of this section, and (4.5.1) , (4.5.18) and (4.5.19). We now summarize the key results of this section: Separated Transmission Line Equations Variable separations: Eigenvalue relation: ρ(x,y,z) = ρt(x,y) q(z) k2 = zy φ(x,y,z) = (1/2πε) q(z) φt(x,y) (βd2 + k2) = jωZi C' Az(x,y,z) = (μ/2π) i(z) Azt(x,y) ≈ 0 "low loss" seems wrong Transverse equations: Transverse boundary conditions: [ t2 + (β2 + k2)] φt(x,y) = -2πρt(x,y) φt(x1) - φt(x2) = K [ t2 + (β2 + k2)] Azt(x,y) = 0 Azt(x1) - Azt(x2) = K Longitudinal equations: β2 = ω2 μξ [ z2 - k2 ] q(z) = 0 βd2 ≈ ω2 με = ω2/ v2 [ z2 - k2 ] i(z) = 0 βc2 ≈ j(2/δ2) , δ = (5.2.26)