# Instrumentation & Measurement Magazine 24-2 - 80

```Fig. 2. Noise-parameter measurement system block diagram.

Fig. 1. ABCD noise model of a two-port LNA.

does not affect its accuracy [2]. While these assumptions allow
for relatively fast measurements, the obtained measurements
lack the precision required for applications, such as LNAs for

Noise Parameters
The simplifying assumptions associated with noise-factor
measurements cannot be used when measuring very-lownoise circuits that target radio-astronomy telescopes. Instead,
it must be obtained using the noise parameters, which can be
thought of as " vector " noise factors that account for the effects
of impedances at each node on available noise powers.
In general, all two-port devices, such as LNAs, can be modeled using two correlated noise sources, vn and in, as shown in
Fig. 1. The auto and cross correlations of these noise sources
can be described by noise correlation matrices as:

C ABCD 



1  vnvn
2 B  vn in


vn in 
 .	(1)
in in 


Fig. 1 shows an ABCD two-port model, where Ys and is
represent the LNA driving port admittance and input noise
current, respectively. An intuitive explanation of the terms in
the noise-correlation matrix is provided by the noise parameters, which are defined based on (1) as [3]:

C ABCD


Rn

 2 kbTn 
 Fmin  1
 2  RnYs ,opt



Fmin  1
 RnYs,opt 
2
	(2)
2

Rn Ys ,opt



The noise parameters consist of the device minimum noise
factor, Fmin, the complex optimum admittance (or impedance)
for minimum noise, Yopt ≡ Gopt + jBopt, and the equivalent noise
resistance, Rn, which is sometimes replaced with Lange invariant N. Unlike Rn, N is invariant under lossless transformations
and indicates the physicality of the measurement results [4].
The relationship between the noise factor and noise parameters is expressed by:

F Fmin 

2
2
Rn
N
Ys  Yopt Fmin 
Ys  Yopt 	(3)
Gs
GsGopt

where the complex signal-source admittance is Ys ≡ Gs + jBs.
The following section reviews noise-parameter measurement
methods and highlights some recent developments in this field.
80

Noise-Parameter Measurements
Although standard noise-parameter measurement methods
were developed by IRE as early as 1959 [5], researchers have
continued to search for faster and more accurate methods of
acquiring these measurements.
There are a few different approaches to measuring noise
parameters. Some approaches are based on representing noise
signals as power waves [6], while others involve performing a single noise-figure measurement and fitting the result
to a DUT noise model that is determined analytically or experimentally using other techniques [7]. Another approach to
measuring noise parameters is to measure the noise factors of
devices that are differently sized, but biased at the same current density [8].
Nonetheless, the most commonly used techniques of noise
parameter measurements employ source-impedance tuners
to generate a few different signal-source admittances, Ys, at
the LNA input, and receivers to measure the resultant noise
powers at the LNA output [5], [9]-[14]. A block diagram of
such a system is presented in Fig. 2, where " Matching network " represents an impedance tuner, impedance generator,
or any other method of modifying the signal-source impedance seen by the LNA. The noise parameters can be obtained
by using these measurements in conjunction with data-fitting
techniques.
All details that may contribute to minimizing measurement
uncertainties are critical when characterizing radio-astronomy very-low-noise amplifiers (vLNA) and receivers [10]. The
main sources of measurement uncertainty detailed in [10] are
described next.

Noise Source
One important conclusion from [10] is that the uncertainty of
noise-source Th could easily be much larger than the noise temperatures of the vLNAs. As such, it is necessary to calibrate
the noise sources beyond the standard calibration provided
by noise source manufacturers [1]. Such calibration services
are offered by various organizations, including NIST and METAS. The calibration process involves comparing the noise
power generated by the noise source to that of a known standard, which is typically created using a resistor that has been
cooled to a well-known cryogenic temperature [1]. For example, the method described in [1] entailed immersing a resistor
in liquid nitrogen in a pressurized dewar, as this cryogenic apparatus provides a well-known noise temperature that can

IEEE Instrumentation & Measurement Magazine

April 2021

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# Instrumentation & Measurement Magazine 24-2

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