Measurement Accuracy

The principal quantity characterizing measurement error of a six-port-based reflectometer is the uncertainty circle, drawn in the polar plot around the measured reflection coefficient G (see Fig. 2). Its radius u is such that the actual (true) result lies with a nearly 100% probability somewhere within this circle. From the radius of the uncertainty circle, all other uncertainty quantities can be derived. The actual radius of the uncertainty circle depends on operating conditions, such as frequency, measured reflection coefficient, input power, or ambient temperature. Therefore, usually a worst-case radius is specified for a defined range of operating conditions, or its typical value.

Fig. 2.  Reflection coefficient uncertainty circle.

If the radius of the uncertainty circle is u and the measured reflection coefficient is

where M is its modulus (absolute value, magnitude) and j its phase angle in degrees, then the uncertainty limits for various quantities of interest are as follows:

Reflection coefficient modulus

The formulas assume M  > u. If M < u then M_min = 0.

Reflection coefficient phase angle

where

is phase angle uncertainty in degrees. The formula derivation assumes M >> u. Obviously, given a value of u, the phase angle uncertainty grows when the reflection coefficient magnitude decreases. For M < u, the phase angle becomes meaningless.

Return loss R

Voltage standing wave ratio VSWR

Effective directivity and test port (source) match

Two independent quantities usually characterize heterodyne network analyzers. These are:

• Effective directivity E_D (critical for the measurement of small reflection coefficients).

• Effective test port (source) match E_S (affecting primarily the measurement of high reflection coefficients).

In the six-port reflectometer, effective directivity can be linked with the uncertainty circle radius u as

Effective test port match is a quantity that cannot be practically defined for this measurement principle.

See also Deembedded Reflection Coefficient Accuracy Considerations.

Measurement Uncertainty Example

An example of return loss uncertainty considerations is depicted in Fig. 3. If, for instance, the uncertainty radius is u = 0.02 (E_D = 34 dB) and the measured return loss is 20 dB (i.e., M = 0.1), then the true value lies somewhere between R_min = 18.42 dB and R_max = 21.94 dB.

If you wish to be sure that the true return loss is greater than 20 dB, the measured value must be so that R_min is at least 20 dB. The intersection of 20-dB horizontal graticule with the lower blue line (green circle) then shows that the measured return loss must be at least 22 dB.

Mathematically, R_min = 20 dB corresponds to M_max = 0.1. Because, as stated above,

M_max = M + u,

the measured reflection coefficient magnitude must not exceed the value

M_x = M_max – u = 0.1 – 0.02 = 0.08,

hence the measured return loss must be at least

R_x = ‑20 log(M_x) = ‑20 log(0.08) = 21.94 dB, 

which is approximately the 22 dB obtained from the graph.

Fig. 3.  Uncertainty circle radius and its effect on return loss measurement error.