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matick section 4.5 REVIEWED

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Phil's review of Matick's Section 4.5 (pp 108-116) on a lossy stripline with width much greater than gap and metal thicker than the skin depth. It follows Matick's Helmholtz equation, the transcendental eigenvalue equation (4-66) for γ, and the skin depth and inductance results. It converts his small-loss solution to Phil's wavenumber k, giving an attenuation factor and the "drag effect" on wavelength. It also corrects an erratum in Matick's equation (4-50).

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Matick Section 4.5 PhL 1.9.14 Since I mention this section in my original lines doc, I might as well read it (probably for the first time). [p 108] The stripline is drawn, width W and separation S with W >> S. What I usually call the z direction (wave direction) he has as his x direction. [ p 109] This is all statement of Maxwell's equations. [ p 110] He obtains the vector Helmholtz equation for the E field, same as mine. His E points only across the gap, while B is in the W direction. His equation (4.50) is strange since the LHS is a vector which is 2E whereas the rest of the equation is scalar. I have fixed this up. [p 111-113] He assumes the form (4-52) for his longitudinal E field where γ and Γ1 are TBD. He then just goes on and on for several pages. He is matching Ez values at the conductor surface. He then gets a transcendental equation for constant γ in (4-66). [p 114-116] He discovers the skin effect δ value. He computes inductance L internal and external. The following is now in lines doc Chapter 5: In the 12-page Section 4.5 of his book Matick studies a lossy-transmission line in the simplest possible case which is a stripline geometry whose gap S is small compared to the width and whose metal strips are much thicker than the skin depth δ. His parameter γ is related to our parameter k by γ = jk, and his longitudinal direction is x instead of our z. He ends up with a transcendental "eigenvalue equation" (4-66) for γ, but if loss is very small, he can approximately solve for γ with these results Im(γ) = β(1+δ/2S) Re(γ) = β (δ/2S) // Matick (4-75,76,77) which with γ = jk we translate to Im(k) = - Re(γ) = - β (δ/2S) Re(k) = Im(γ) = β(1+δ/2S) k = β(1+δ/2S) -j β (δ/2S) = β - [-(δ/2S) + j(δ/2S)] so Δ = -(δ/2S) + j(δ/2S) . Thus, for such a stripline transmission line, the longitudinal dependence of all functions has this form, q(z,t) = ej(ωt-[β-Re(Δ)]z) e–Im(Δ)z = ej(ωt-[β+δ/2S]z) e–(δ/2S)z which shows the exponential loss factor and an increased wavenumber β+δ/2S which corresponds to a decreased wavelength λ and a decreased wave velocity v = ω/k = ωλ/2π = fλ, the "drag effect". Matick has an errata in this section which is a bit confusing, so we repair it right here. His equation (4-50) should read 2E = ( + ) + ( + ) = (jωμσ - ω2με)(Ex + Ez) Matick (4-50)