Analysis of Mode Chart for the Calculation of Hakki-Coleman Parameter, α in Characteristic Equation for Dielectric Permittivity Measurement

Variation of α values corresponding to β values in the characteristic equation for Hakki-Coleman measurement of dielectric permittivity is thoroughly studied in this paper. Since the characteristic equation for TE_0nl mode is a transcendental equation, graphical solution based on Mode Chart gives α_nversus β, but for β ≤ 12. The HakCol program quickly and accurately calculates the relative permittivity for the TE₀₁₁ mode from resonant frequency and resonator dimensions, but for β ≤ 10 Also restrictions are imposed in selecting the resonant frequency for a given set of resonator dimensions and vice versa. Such limitations of the Mode Chart are presented with proof. Also Mode Charts corresponding to β values for various combinations of resonant frequency, thickness and diameter of the resonator are plotted.