Re-examination of mismatch uncertainty when measuring microwave power and attenuation

The mismatch uncertainty associated with a simple microwave power measurement is precisely defined and shown, under certain assumptions, to have a U-shaped probability density distribution. Earlier expressions for the mismatch error when measuring microwave attenuation receive further mathematical development; and, when all the phase angles are unknown, it is shown that there are six independent contributions to the mismatch error, each with a U-shaped distribution. When the phase angles of the scattering parameters of the device under test are known, the mismatch error expression contains three independent contributions cach with a U-shaped distribution. The combination of U-shaped distributions is discussed in detail and new expressions are presented for the standard deviations of the mismatch uncertainties in the above two cases. The uncertainties yielded by these new expressions are in closer agreement with experience than those obtained by using methods based on maximum limits.