Characteristic impedance of unbalanced TDR probes

Time domain reflectometry (TDR) may be used to make simultaneous measurements of both dielectric constant and conductivity by means of a probe inserted into the medium. The air-spaced characteristic impedance of the probe is required in order to estimate the conductivity from the final value of the TDR waveform. Unbalanced probes derived from a coaxial line by replacing the outer shield with regularly spaced wires are favored for many applications because they eliminate the need for a balun. Until now, it has been necessary either to measure the probe characteristic impedance or to calibrate it in solutions of known conductivity. This paper shows how to deduce an expression for the probe characteristic impedance by means of a conformal transformation. For thin wires, a simpler formula has been derived which is suitable for hand calculation. The accuracy of these analytic expressions has been assessed by a method of moments numerical solution which exhibits very rapid convergence as extra basis functions are added. The characteristic impedance formula obtained by conformal transformation and its thin wire approximation are found to be accurate to 0.1% and 1%.