Measuring the Microwave Magnetic Permeability of Small Samples Using the Short-Circuit Transmission Line Method

We present a method for estimating the bandwidth and the measurement errors of the shorted transmission line method used to obtain the permeability and the permittivity of magnetic materials. The permeability and the permittivity of a toroidal test sample are determined by measuring the reflection coefficients of a shorted coaxial transmission line and by using Newton's minimization algorithm. The permeabilities of two samples with different concentrations of cobalt nanoparticles are measured, and the validity of the shorted transmission line method is confirmed by comparing the results to a two-port transmission line approach. The sensitivity of the shorted line method to different measurement parameters is investigated in detail, and the connection between the spatial dimensions of the sample and the measurement uncertainty is studied. The bandwidth of the method is limited by the cavity resonance at the upper end of the frequency range and by the reduction of the sensitivity at the lower end. This is important because in magnetic materials research the sample sizes are generally small, which sets a limit to the lowest measurement frequency using transmission line methods. The reduction in sensitivity is demonstrated by measuring the permeability of three Teflon samples (each having different lengths), and by developing a theoretical model for the measurement errors.