flection coefficients. This article will
emphasize a model of Gaussian distribution in the real and imaginary parts
of the reflection coefficient, which results in a Rayleigh distribution of the
magnitude of the reflection coefficient.
The simplest statistical model,
sometimes called the “ring” model, assumes a constant (known) magnitude
reflection coefficient with uniformly
distributed phase. Classic (and now
updated with the information of this
article) Application Note 1449-3 from
Agilent Technologies3 calls this “case
b.” This can be visualized with Figure
1 showing a simulated distribution of
the reflection coefficient in the real-imaginary plane.
Because this model is obviously
too conservative in many cases, another model has been in use. The
“disk” model, shown in Figure 2, assumes that the maximum magnitude
is known but the complex reflection
coefficient will be equally likely to
have a value anywhere within a circle
in the complex plane. The effect of
modeling both the source and load
with the disk model is to give only half
as much amplitude uncertainty due to
mismatch as for the ring model.
The third model, newly-validated
by the research described here, is the
“Rayleigh” model, named because the
probability density of the magnitude
of the reflection coefficient is Rayleigh
distributed if the probability density of
both complex parts is Gaussian distributed. The simulation is shown in
Figure 3a. The simulation figure is annotated with the 95th percentile ring and
a “max” ring. These are also shown in
the probability density function plot in
Figure 3b on a different scale.
mismatch uncertainty which are three
to six times lower than the older methods.
POWER DELIVERY AND
The power delivered to an imperfect power sensor is given by this
(1) a gZ0
The term PgZ0 is the amount of
power delivered to an ideal matched
load. The numerator term, 1 - |Γι|2,
is accounted for in the calibration of
the power sensor. The denominator
term, |1 -ΓιΓg |2, represents the mismatch error. If we make the assumption that the cable between the DUT
and the power sensor is long, practically speaking we will often not know
the phase of the reflections, leading to
uncertainty that depends on Γι and Γg.
When we want to combine the uncertainty from the mismatches with
other measurement uncertainties, using the method of the GUM, we will
usually find the standard deviation of
the uncertainty and combine that (a
addition) with the standard deviation
of other uncertainties; we multiply
that by the coverage factor, usually
2, to get the 95th percentile measurement uncertainty. In order to do this,
we must have a model of the statistical distribution of the magnitudes of
Γι and Γg.
The older literature documents the
assumptions of known reflection magnitude and uniform-inside-a-circle re-
s Fig. 1 Simulation of reflection coefficient
in the complex plane for a constant magnitude .
s Fig. 2 Simulation with is equally likely
anywhere within a circle.