Uncertainty in Material Characterization Using the Hessian and Filled Waveguide Method

This paper examines an error model that can predict the uncertainty in the electromagnetic material properties $(\epsilon_{r}=\epsilon_{r}^{\prime}-j\epsilon_{r}^{\prime\prime})$ obtained through a curve fitting algorithm. Using a model of the measurement method (filled waveguide, open-ended waveguide, etc.), the algorithm optimizes the material properties to match the scattering parameters obtained through measurements. Utilizing the Jacobian from the optimizer, we can calculate the Hessian matrix of the material properties $(\epsilon_{r})$. This matrix is then used to determine the expected standard deviation in the material properties for the given measurement method. The uncertainty model formulated in this paper is applied to the filled waveguide technique with measurements taken at X-band (8.2-12.4 GHz). Measurements of several materials with known properties are performed to evaluate the uncertainty model by comparing the predicted uncertainty to the error in the calculated materials over the frequency band. In addition, we explore several sources of uncertainty when performing measurements and their impact on the material properties that the algorithm calculates.