By David Archer

The purpose of this article is to develop an uncertainty
analysis for the noise measurement system described in the course MEAS403
– Basic to RF and Microwave Noise Measurement. In that course a
hypothetical noise measurement system was described that used the Y-factor
method to measure the noise temperature of an unknown noise source. This system is based on a system that
was actually implemented, with some of the details omitted. We will go into a
fair amount of detail in the analysis, but there still will be some questions,
such as details concerning the receiver, that will be left for a future
article.

Also, after presenting most of the uncertainty analysis, we
will leave one part of the analysis unfinished as an exercise so that
experience with uncertainty analysis can be gained by the reader. We may provide the solution as a future
article if there is some interest.

Before starting in on the uncertainty analysis, we will
review some of the contributing factors to the noise measurement.

In the course we learned that the definition of noise
temperature is the power-per-unit-bandwidth divided by BoltzmannÕs constant,
i.e. the relationship of the power received by a receiver with bandwidth B and
the noise temperature is

where k=BoltzmannÕs constant=1.38044 x 10^{-23}
joules per Kelvin.

We also pointed out that noise temperature is not quite the
same as the physical temperature of a load that would deliver the same power
and that the relationship between noise temperature and physical temperature
for a load is

where

and h=PlanckÕs constant=6.62517 x 10^{-34} joule
second.

If the uncertainty that is desired for the measurement
system is such that this correction is necessary, it should be included in the
analysis of the system.

Many noise sources are not calibrated in noise temperature,
but in what is called Excess Noise Ratio or ENR. The definition of ENR is

where 290K is a reference temperature that is used in the
definition, ENR is typically expressed in dB

.

The following diagram schematically describes the
hypothetical noise measurement system.

in the course we derived the following measurement equation
for the noise temperature of the device under test:

where Tr, the noise temperature of the receiver is

for completeness, the Ònoise efficienciesÓ , i=DUT,h,c are

and the mismatch terms N_{i} are

where

and is the
reflection coefficient of the receiver.

These equations describe how to determine the noise
temperature of the device under test. Of course all the reflection coefficients
and S-parameters of the system must be measured in some manner before hand, with
something like a network analyzer, and the noise temperature of the ÒhotÓ and
ÒcoldÓ standards, as well as the ambient temperature, must be known.

Obviously this would be an automated system, nobody would
want to calculate the results by hand, but the advantage of such a system is
that a complete uncertainty analysis could also be automated into the
measurement. We will no turn to the subject of the uncertainty analysis.

In general when one has a measurement equation

where M is the measureand and the x_{i} are the
component measurements that are required to determine its value, and each of
the x_{i} has an associated uncertainty , the RSS uncertainty in the measureand can be written

This the technique that will be used to determine the
various terms in the uncertainty analysis. We will not go into a great detail
on how the individual partial derivatives are performed, but just state the
results.

Performing this type of analysis on the measurement
equation, the following expression for the uncertainty in the DUT noise
temperature is obtained.

where

This then is the expression for the uncertainty of the
device under testÕs noise temperature, unfortunately we are not finished. There
are still eight terms in the above equations that we do not know. Each of these
eight terms will require a separate analysis, these eight terms are

Of these terms, the uncertainty in the cold standard noise
temperature is probably available as some sort of calibration certificate, or
the result of some other uncertainty analysis, and the uncertainty in the
ambient temperature is a function of the device used to measure the ambient
temperature.

The uncertainty of the Y-factor measurement is a function of
the linearity of the receiver, and so can be easily determined as the power
ratio measurement uncertainty of the receiver.

The remaining five terms each require an uncertainty
analysis of their own. That will be the task of the remaining sections of this
document.

Using the same analysis technique, the uncertainty
associated with the receiver noise is

where

Again the uncertainty in the Y-factor is the receiver power
ratio measurement accuracy, and
the uncertainty in the noise temperature of the ÒhotÓ noise standard either
comes from a calibration certificate or a separate uncertainty analysis. There
are four uncertainties associated with the hot standard we have not seen yet.

At this point it is worth noting that in the course there
was a tradeoff between the time it takes to make the measurement and the amount
of drift in the receiver noise temperature. If the receiver noise drifts say a
factor of 10 less than the calculated uncertainty over the course of a
measurement, it probably can be ignored. Otherwise it is a tradeoff of
measurement uncertainty due to the drift in the noise temperature vs the
uncertainty associated with not having enough samples to get a good average.
This tradeoff can only be determined empirically.

At this stage we have all but six uncertainty terms, the
three Õs and the three Õs. These are actually only two additional analysis, but
there is some difficulty. Since these parameters are measured typically with
some sort of network analyzer, and network analyzer uncertainties are usually
expressed in terms of magnitude and phase uncertainties, we need to express
these in terms of magnitude an phase. We will start with the mismatch terms:

The numerator is already expressed in terms of magnitudes, to express this in terms of magnitude and phase, we will let

and

and defining

results in

We can now express the uncertainty in this quantity as follows:

where

It should be noted here that in the above equations all angles and angle uncertainties are expressed in radians. The uncertainty terms that we do know above can be determined from network analyzer specifications. These terms are

We are almost done with the mismatch term, the only quantities we donÕt know at this point are the two uncertainties

These can be determined from network analyzer specifications if the reflection coefficient is measured directly through the switch. If not, the magnitude and phase must be written in terms of the four S-parameter magnitudes and phases, and the magnitude and phase of the noise standard reflection coefficient, a total of 10 terms.

Starting with the expression for the magnitude one obtains

where

It is convenient to write the uncertainty in the magnitude
in the following form

To facilitate the writing of the 10 coefficients we make the
following definitions

it then follows that

This completes the derivation of the uncertainty in the
magnitude of , what remains is to determine the uncertainty in the phase
to complete the uncertainty analysis of the mismatch terms. Again the phase is
a function of 10 terms.

One can write for the phase

where

and

The uncertainty in the phase can no be written

where

and

This completes the analysis for the mismatch terms, all that
remains is the uncertainty analysis of the noise efficiency.

We start with
the expression for the noise efficiency

This expression has to be rewritten in terms of magnitude
and phase before the appropriate partial derivatives can be taken to complete
the uncertainty analysis.

We are going to define

then

so that

Now we know

and that solves the problem of the uncertainty analysis of
this hypothetical noise system provided all the above partial derivatives are
calculated. That will be left as an exercise, after all this is a training
site. There should be enough examples here to allow one to do this. The
procedure is to write the quantity G as a function of the magnitude and phase
of all its components, and then take all the partial derivatives. It should
only take a few pages of complex algebra.

Except for the last part that is left as an exercise, the
complete uncertainty analysis of the hypothetical noise system has been
presented. One could implement a real system with the information provided
here.