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A novel method of measuring the relative complex permittivity (/spl epsi/=/spl epsi/'-j/spl epsi/'') and relative complex permeability (/spl mu/=/spl mu/'-j/spl mu/'') of a material at K/sub /spl alpha// Band (26.5-40 GHz) using a partially filled waveguide (PFW) (rectangular) and a vector network analyzer (VNA) is presented. The method is based on 1) placing a material sample of length L/spl tilde/, width /spl alpha//spl tilde/ (waveguide width), and height d/spl tilde//spl les/b/spl tilde/ (b/spl tilde/ is the waveguide height) in a rectangular waveguide, 2) measuring the S-parameters of the sample using the VNA, and 3) inferring the /spl epsi/', /spl epsi/'', /spl mu/', and /spl mu/'' parameters by comparing the experimental S-parameters with numerically generated S-parameters. The paper presents a method of moments analysis and also a variational formulation of the scattering that occurs from a finite length sample that partially fills a waveguide. Formulas to calculate the complex Poynting power and energy in the waveguide are derived to check the degree to which the numerical solutions obey the conservation of complex power. Numerical methods to extract the material parameters from the S-parameter data are proposed. The experimental PFW S-parameters of a radar absorbing material are measured and its dielectric material parameters are inferred.