# Calculating Mismatch Uncertainty

Mismatch uncertainty is a very common and often underestimated source of error in microwave power measurements. It arises from an incomplete knowledge of the phase of the reflection coefficients of the source and load impedances, plus their interconnection, and is usually a large component of the overall measurement uncertainty budget in a microwave power transfer. It is easy to estimate and should not be ignored.

#### Theory

From *Figure 1*

The power absorbed in the load is

The numerator of the fraction, 1–|Γ_{1}|^{2}, is known as the mismatch loss. If only the voltage reflection coefficient magnitudes of the source and load are given, and not their phases, the denominator term cannot be evaluated precisely. However, the maximum and minimum values can be determined.

To derive an easily remembered approximation for these formulae, using the binomial series

If this series is truncated to the first two terms, and z is replaced by product |Γ_{s}||Γ_{1}| and m by 2, then

The approximation is valid for |Γ_{s}||Γ_{1}| << 1, and the mismatch uncertainty is given approximately by M_{u} = ±200 |Γ_{s}||Γ_{1}|%. Some examples will clarify the sizes of the errors involved.

**Example 1**

A power sensor with a VSWR of 1.2:1 is connected to a signal generator with a VSWR of 1.5. What is the mismatch uncertainty?

*Solution:*

First the VSWR figures are converted to reflection coefficient magnitudes:

Then the mismatch uncertainty is: M_{u} = ±200 (0.091) (0.2%) = ±3.64% or ±10 x log10(103.64/100) = ±0.155 dB.

**Example 2**

A power sensor with a return loss of –23 dB is connected to a calibration source with a return loss of –20 dB. What is the mismatch uncertainty?

*Solution:*

First the return loss figures are converted to reflection coefficient magnitudes:

Then the mismatch uncertainty is: M_{u} = ±200 (0.071) (0.1%) = ±1.41% or ±10 x log10(101.41/100) = ±0.061 dB.

#### Conclusion

Mismatch uncertainty can explain some of the variation that is seen when repeating microwave power measurements when the test set-up is slightly different from the original. Perhaps an adaptor with a different electrical length was selected, or a connector might have been tightened with and without a torque wrench. A deeper analysis^{1} suggests that the more extreme values are actually quite likely. Calculating the mismatch limits is straightforward and serves to set expectations of repeatability.

#### Reference

1. I.A. Harris and F.L. Warner, “Re-examination of Mismatch Uncertainty When Measuring Microwave Power and Attenuation,” *IEE Proc.*, Vol. 128, Pt. H, No. 1, February 1981.

**Anthony Lymer *** received his BSc degree in electrical and electronics engineering from the University College of North Wales, Bangor, UK, in 1975. He then became a development engineer at the Marconi Research Laboratories in Great Baddow, UK, where he developed and field-tested a data-transmission scheme for VHF/UHF mobile radio systems. From 1977 to 1980, he was a research student at the University of Bath, UK, developing a single-sideband UHF mobile radio system. He joined Hewlett-Packard (now Agilent Technologies Inc.) in 1982, where he became a senior development engineer at the company’s Queensferry Telecommunications Division. He joined Satori-Technology Ltd. in February of 2006, and contributed to the design of the ST series power meters.*