An Approximation for the Characteristic Impedance of Shielded-Slab Line

It is shown how the parametric expressions for t/b and w/b of shielded-slab line, given by means of elliptic integrals in terms of two independent real parameters a and k, can be inverted. First q´ = exp (-pi K/K´) is expressed as an odd power series in exp (-pi v/(b - t)) whose coefficients are irrational functions of t/b. Here, w is the width of the slab, t is its thickness, and b is the spacing between the infinite parallel plates. Then k is expressed in terms of q´ by a well-known formula, and an expression is derived which gives a in terms of q´ and w/(b - t). When the series expressions for a and k are substituted in the formulas which yield the total capacitance C₀, of the shielded-slab line, it is found that the fringing capacitance C_f0=C₀/4 - w/(b - t), is given by an even power series in exp (-pi w/(b - t)), at least to the eighth power. These coefficients, which are irrational functions of t/b, are given explicitly. Finally, comparisons are made with exact values of Z₀.