Direct-Current Circuit Analysis of Time Domain Reflectometry with Variable Coaxial Cable Length for Electrical Conductivity Measurements

The improved formula of time domain reflectometry (TDR) with variable coaxial cable length for electrical conductivity (EC) measurements was derived based on the theory of multi-section transmission line when the frequency was approaching zero. The derivation procedure was quite complicated. The goal of this study was to derive the formula when the multi-section transmission line was at zero frequency. At zero frequency, the multi-section transmission line became a direct-current circuit which consisted of the resistance of the soil sample, the series resistance and shunt conductance of the coaxial cable, and the characteristic impedence of the cable tester. Kirchhoff's voltage law was used to formulate the ratio of the voltage at the interface between the coaxial cable and the cable tester to the voltage of the cable tester. The formula for soil EC measurement derived from zero frequency was identical to that derived from the procedure when the frequency was approaching zero.