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 each 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.
IntroductionIn numerous routine microwave attenuation measurement comparisons that have been carried out over several years in the UK under the auspices of the British Calibration Service, the differences between the RSRE and EQD results have consistently been lower than the conventionally calculated limits of mismatch uncertainty. This evidence motivated in 1979 some work [1] which showed that the calculated mismatch uncertainty can be reduced by measuring both the magnitudes and phases of the s-parameters of the attenuator under test. More recently, this evidence has also led to a reexamination of the nature of mismatch uncertainty in microwave measurements and the results of this study form the subject of the present paper. The new results confirm the original method of calculation for simple power measurement comparison [2, 3 ] ; but, for attenuation measurements, a more accurate method of calculation has been developed. This leads to more realistic values of mismatch uncertainty (for a given confidence level) which are in closer agreement with experience than the values given by the earlier expressions [2, 3].
Mismatch uncertainty for a simple power measurementIf the true power measured by an absorption wattmeter is defined as the power absorbed in a load of resistance equal to the real characteristic impedance of the guided wave system used, then, when the source has a voltage reflection coefficient F G and the absorption wattmeter (load) has a voltage reflection coefficient V L , the true power P o is found (by rearrangement of eqn 2.7 in Reference 3) to be related to the actual power absorbed P L by the following equation
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