# Measuring Noise Parameters Using an Open, Short, Load, and
$\lambda$/8-length Cable as Source Impedances

D. C. Price, C. Y. E. Tong, A. T. Sutinjo, L. J. Greenhill, N.Patra D. C.
Price, A. Sutinjo and N. Patra are with the International Centre for Radio
Astronomy Research, Curtin University, Bentley WA 6102, Australia.L. J.
Greenhill and E. Tong are with Center for Astrophysics, Harvard & Smithsonian,
60 Garden Street, Cambridge, MA 02143, USAManuscript received Oct xx 2021;
revised mmm dd 20yy.

###### Abstract

Noise parameters are a set of four measurable quantities which determine the
noise performance of a radio-frequency device under test. The noise parameters
of a 2-port device can be extracted by connecting a set of 4 or more source
impedances at the device’s input, measuring the noise power of the device with
each source connected, and then solving a matrix equation. However, sources
with high reflection coefficients ($\left|\Gamma\right|\approx 1$) cannot be
used due to a singularity that arises in entries of the matrix. Here, we
detail a new method of noise parameter extraction using a singularity-free
matrix that is compatible with high-reflection sources. We show that open,
short, load and an open cable (“OSLC”) can be used to extract noise
parameters, and detail a practical measurement approach. The OSLC approach is
particularly well-suited for low-noise amplifiers at frequencies below 1 GHz,
where alternative methods require physically large apparatus.

###### Index Terms:

noise measurements, noise parameters, low-noise amplifiers

##  I Introduction

The noise performance of a radio-frequency amplifier, or other device under
test (DUT), is commonly characterized in terms of its noise parameters: a set
of four real-valued terms from which noise characteristics can be derived for
all input and output impedances [1]. Alternatively—but equivalently—a “noise
wave” representation may be used, which defines noise in terms of incoming and
outcoming waves [2, 3]. Regardless of representation, the measurement of noise
parameters is an important task when determining and optimizing the signal-to-
noise performance of a radio receiver.

This article, we present two main results. Firstly, we reintroduce and expand
on a matrix formulation for determining noise parameters, which allows for
sources with $\left|\Gamma_{s}\right|\approx 1$ to be used. Central to this
approach is a singularity-free matrix formed from the reflection coefficients
of four sources. We show the relationship between the singularity-free matrix
and the traditional admittance-based matrix formulation [4], then show that
the singularity-free formulation yields smaller measurement errors. Compared
to standard techniques, no change in measurement apparatus is required; as
such our approach can be used a substitute for the admittance-based matrix
formulation.

Secondly, we detail a cold-source technique for measurement of noise
parameters based upon the singularity-free matrix formulation. Our measurement
technique requires only a 1/8-wavelength coaxial cable and open, short and
load termination. A key feature of this technique is the use of open and short
source impedances, for which well-characterized commercial offerings are
readily available as part of precision vector network analyzer (VNA)
calibration kits. The technique can be used with any unconditionally stable
DUT, and is ideally suited to low-frequency application ($<$1 GHz).

This paper is organized as follows. We first give an overview of noise
parameters (Sec. II) and matrix-based approaches (Sec. III), then introduce a
singularity-free matrix formulation (Sec. IV). We then outline how noise
parameters can be measured using an open, short, load and 1/8-wavelength
coaxial cable as reference source impedances (Sec. VI and VII). In Sec. VIII,
we use our approach to measure the noise parameters of a Minicircuits
ZX60-3018G-S+ amplifier across 50–300 MHz. The paper finishes with a
discussion and concluding remarks (Sec. IX).

##  II Noise parameters

In terms of source reflection coefficient $\Gamma_{s}$, or source admittance
$Y_{s}=G_{s}+jB_{s},$ the noise temperature $T$ of a 2-port DUT can be
expressed as :

$\displaystyle T(\Gamma_{s})$ $\displaystyle=T_{{\rm
min}}+T_{0}\frac{4R_{N}}{Z_{0}}\frac{\left|\Gamma_{s}-\Gamma_{{\rm
opt}}\right|^{2}}{(1-\left|\Gamma_{s}\right|^{2})\left|1+\Gamma_{{\rm
opt}}\right|^{2}}$ (1) $\displaystyle T(Y_{s})$ $\displaystyle=T_{{\rm
min}}+T_{0}\frac{R_{N}}{G_{s}}\left|Y_{s}-Y_{{\rm opt}}\right|^{2}$ (2)
$\displaystyle T(G_{s},B_{s})$ $\displaystyle=T_{{\rm
min}}+T_{0}\frac{R_{N}}{G_{s}}\left[(G_{s}-G_{{\rm opt}})^{2}+(B_{s}-B_{{\rm
opt}})^{2}\right]$ (3)

where $T_{0}=290$ K and $Z_{0}=1/Y_{0}$ is the characteristic impedance. $T$
is comprised of the following noise parameters:

  * •

$T_{{\rm min}}$ is the minimum noise temperature, also commonly expressed as
the minimum noise factor, $F_{{\rm min}}=(1+T_{{\rm min}}/T_{0})$

  * •

$Y_{{\rm opt}}=G_{{\rm opt}}+jB_{{\rm opt}}$ is the optimum admittance, or
equivalently, $\Gamma_{{\rm opt}}=\gamma_{{\rm opt}}\,exp\left(j\theta_{{\rm
opt}}\right)$ is the optimum reflection coefficient.

  * •

$R_{N}$ is the equivalent noise resistance. Alternatively, the unitless
quantity $N=R_{N}G_{{\rm opt}}$ may be used, which is invariant under
reciprocal lossless transformations.

There are several approaches to extract noise parameters from measurements of
the noise temperature $T(\Gamma_{s})$. In all approaches, as there are four
unknown (real-valued) noise parameters, at least $n$$\geq$4 independent
measurements of $T(\Gamma_{s})$ must be made. The noise parameters are then
found by casting the problem as a matrix equation (see Sec. III) or by
equivalent least-squares methods. The loci of the $n$ reference
$\Gamma_{s_{i}}$ on the Smith chart will form an “impedance pattern”, and it
has long been recognized that loci should be “well spread” across the Smith
chart [5, 6, 7, 8]. In general, a vector network analyzer (VNA) is used to
accurately measure $\Gamma_{s}$, and a noise receiver is used to measure
$T(\Gamma_{s})$, see Fig. 1.

Figure 1: Diagram showing setup of a device-under-test (DUT) for noise
parameter measurements using the cold-source method. A set of four (or more)
reference source impedances $\Gamma_{s}$ are connected to the DUT and
$T(\Gamma_{s})$ is measured with a noise receiver (RX). The noise figure of
the DUT must first be measured for a single source impedance, by using a
calibrated noise source.

While any four (or more) source impedances may be used, impedance tuners are
commonly employed as they offer a convenient way to generate well-spaced
impedances. We will refer to the case in which four source impedances are
used, with a load ($\Gamma=0$) selected as one source impedance, as the “four-
point” method. An alternative approach is to use a source impedance which
exhibits rapid phase wraps across frequency, such as a long coaxial cable [9,
10]. Under the assumption that the noise perfomance of the DUT does not change
appreciably across a small frequency range, the phase wrapping may be used to
effectively sample a range of impedances across the Smith chart.

Regardless of approach, a well-characterized noise source is required to
calibrate receiver noise power measurements (using the Y-factor method [11]).
By inserting a two-port impedance tuner between the noise source and the DUT,
the noise figure can be measured for different source impedances, as required
for noise parameter extraction. An alternative “cold-source” technique can
also be used, in which the noise source is directly connected to the DUT and
the noise figure is measured, after which a set of passive source impedances
is connected to provide measurements at different source impedances [12]. In
the cold-source method, the noise figure of the DUT is measured for only one
source impedance—that of the noise source—and one-port devices are used as
source impedances.

All variations of these approaches require that reflection coefficients are
not too large, due to a singularity (division by zero) caused by the
$(1-\left|\Gamma_{s}\right|^{2})$ term in Equation 1. Himmelfarb & Belostotski
(henceforth HB16) [8] provides a mathematical basis to show that one source
impedance should be a 50$\Omega$ load, and three (or more) reflection
coefficients should satisfy $0.4<\left|\Gamma_{s_{i}}\right|<0.9$. By doing
so, the well-spaced requirement is met, and the
$(1-\left|\Gamma_{s}\right|^{2})$ singularity is not encountered.

At low frequencies ($<$100 MHz), the long cable approach is troublesome, as
the cable can become prohibitively long. For example, an application to
measure noise parameters at 50–200 MHz for radio astronomy suggests the use of
a 25 m cable [13]. Low-frequency impedance tuners are also physically large,
and can be prohibitively expensive.

##  III Matrix-based noise parameter approaches

The extraction of noise parameters from a DUT requires connecting
${\color[rgb]{0,0,0}{n}}$$\geq$4 reference sources and measuring the noise
output power spectra of the DUT using a receiver. The reflection coefficients
$\Gamma_{s}$, or equivalent admittances $Y_{s}$, must be known or measured for
each source. In matrix notation, the noise receiver measurements form a
($n\times$1) vector t, the $\Gamma_{s}$ measurements form a ($n$$\times$4)
matrix A, and we wish to find the (4$\times$1) noise parameter vector t, which
is related by:

${\textbf{{A}}}{\textbf{{x}}}={\textbf{{t}}}.$ (4)

To solve this (i.e. find t) requires inverting the matrix
${\textbf{{A}}}^{-1}$ (if $n$=4) or forming the pseudoinverse
${\textbf{{A}}}^{+}=({\textbf{{A}}}^{T}{\textbf{{A}}})^{-1}{\textbf{{A}}}^{T}$
if $n>$4:

${\textbf{{x}}}^{+}={\textbf{{A}}}^{+}{\textbf{{t}}}.$ (5)

The entries of the matrix A depend upon the formulation used, of which there
are a several. In Lane’s technique [4], the $i$th row of A is formed from
admittances:

$\displaystyle{{\textbf{{A}}}}_{i}^{G}$
$\displaystyle=\left[1,\frac{\left|Y_{s_{i}}\right|^{2}}{G_{s_{i}}},\frac{1}{G_{s_{i}}},\frac{B_{s_{i}}}{G_{s_{i}}}\right],$
(6)

and the vector ${\textbf{{x}}}=[a,b,c,d]^{T}$, whose entries can be converted
into the four noise parameters by:

$\displaystyle T_{{\rm min}}$ $\displaystyle=a+\sqrt{4bc-d^{2}}$ (7)
$\displaystyle R_{N}$ $\displaystyle=b$ (8) $\displaystyle G_{{\rm opt}}$
$\displaystyle=\sqrt{4bc-d^{2}}/2b$ (9) $\displaystyle B_{{\rm opt}}$
$\displaystyle=-d/2b.$ (10)

An alternative formulation is found in [9], which defines a matrix in terms of
the magnitude $\gamma_{s}$ and phase $\theta_{s}$ of
$\Gamma_{s}=\gamma_{s}exp(j\theta_{s})$

$\displaystyle{{\textbf{{A}}}}_{i}^{\gamma}$
$\displaystyle=\left[1,\frac{1}{1-\gamma_{i}^{2}},\frac{\gamma_{i}cos\theta_{i}}{1-\gamma_{i}^{2}},\frac{\gamma_{i}sin\theta_{i}}{1-\gamma_{i}^{2}}\right],$
(11)

and noise parameters111Note the minus signs in Equation 15 – omitted in [9] –
are important to ensure the correct quadrant is returned when using
$tan^{-1}$.  are obtained as

$\displaystyle T_{{\rm min}}$ $\displaystyle=a+\frac{b+\Delta}{2}$ (12)
$\displaystyle R_{N}$ $\displaystyle=\frac{\Delta}{4Y_{0}}$ (13)
$\displaystyle\gamma_{{\rm opt}}$
$\displaystyle=\sqrt{\frac{b-\Delta}{b+\Delta}}$ (14)
$\displaystyle\theta_{{\rm opt}}$
$\displaystyle=tan^{-1}\left(\frac{-d}{-c}\right),$ (15)

where $\Delta=\sqrt{b^{2}-c^{2}-d^{2}}$.

##  IV A reflection coefficient based source matrix

When measuring noise parameters, the choice of source impedances is crucial to
minimize measurement error. In this section, we introduce the matrix
${\textbf{{A}}}^{\Gamma}$, and show that by using ${\textbf{{A}}}^{\Gamma}$ in
lieu of ${\textbf{{A}}}^{\gamma}$ or ${\textbf{{A}}}^{G}$ yields lower
measurement errors.

The invertibility of matrices ${\textbf{{A}}}^{\gamma}$ and
${\textbf{{A}}}^{G}$ depends upon the characteristics of the reference
sources. Of the $n$ reference loci, one is almost always chosen to be the
$\Gamma=0$ reference impedance. A singularity is encountered in the entries
${\textbf{{A}}}^{\gamma}$ and ${{\textbf{{A}}}}^{G}$ if
$\left|\Gamma_{s}\right|^{2}\rightarrow 1$, so open and short references
cannot be used.

In Sutinjo et. al (henceforth SUT20) [14], it is shown that the singularities
in ${\textbf{{A}}}^{\gamma}$ and ${\textbf{{A}}}^{G}$ can be removed after
multiplication by $(1-\left|\Gamma_{s}\right|^{2})$ [15, 14]. From Equation 1,
as $\left|\Gamma_{s}\right|\rightarrow 1$ we see that
$T(\Gamma_{s})\rightarrow\infty$ due to the $(1-\left|\Gamma_{s}\right|^{2})$
term in the denominator. However, in the limit
$\left|\Gamma_{s}\right|\rightarrow 1$, we have

$\lim_{|\Gamma_{s}|\to
1}\left((1-\left|\Gamma_{s}\right|^{2})T(\Gamma_{s})\right)=T_{0}\frac{4R_{N}}{Z_{0}}\frac{\left|\Gamma_{s}-\Gamma_{{\rm
opt}}\right|^{2}}{\left|1+\Gamma_{{\rm opt}}\right|^{2}};$

that is, the quantity $(1-\left|\Gamma_{s}\right|^{2})T(\Gamma_{s})$ is non-
zero. The two matrices thus become:

$\displaystyle{{\textbf{{A}}}}_{i}^{G^{\prime}}$
$\displaystyle=(1-\left|\Gamma_{s_{i}}\right|^{2})\left[1,\frac{\left|Y_{s_{i}}\right|^{2}}{G_{s_{i}}},\frac{1}{G_{s_{i}}},\frac{B_{s_{i}}}{G_{s_{i}}}\right]$
(16) $\displaystyle{{\textbf{{A}}}}_{i}^{\gamma^{\prime}}$
$\displaystyle=\left[1-\gamma_{i}^{2},1,\gamma_{i}cos\theta_{i},\gamma_{i}sin\theta_{i}\right].$
(17)

By removing the singularity, SUT20 provided a physical and mathematical basis
for why loci in the impedance pattern should be “well spread”. For $n=4$
measurements, the matrix ${\textbf{{A}}}^{\prime}$ is $4\times 4$ and the
maximum spread on the Smith chart corresponds to maximizing the magnitude of
the matrix determinant $|$det$|$. SUT20 also show that the condition number of
the matrix ${\textbf{{A}}}^{\prime}$ is strongly anti-correlated, and is
minimized for maximum $|$det$|$; in contrast, for un-regularized matrices,
i.e. ${\textbf{{A}}}^{G}$ and ${\textbf{{A}}}^{\gamma}$, the condition number
is an unreliable predictor [6].

Here, we highlight that ${\textbf{{A}}}^{G^{\prime}}$can be rewritten as

$\displaystyle{{\textbf{{A}}}}_{i}^{G^{\prime}}$
$\displaystyle=\left[1-\left|\Gamma_{s_{i}}\right|^{2},\,\left|1-\Gamma_{s_{i}}\right|^{2},\,\left|1+\Gamma_{s_{i}}\right|^{2},-2{\rm
Im}(\Gamma_{s_{i}})\right]\equiv{{\textbf{{A}}}}_{i}^{\Gamma}.$ (18)

A derivation of this result is provided in Appendix A. The matrix
${\textbf{{A}}}^{\Gamma}$, is numerically equivalent to
${\textbf{{A}}}^{G^{\prime}}$, but its entries are simple expressions of
$\Gamma_{s}$. As such, there is no need to convert source reflection
coefficients into admittances. Using ${\textbf{{A}}}^{\Gamma}$, the matrix
relation becomes

$\displaystyle{\textbf{{A}}}^{\Gamma}{\textbf{{x}}}$
$\displaystyle={\textbf{{t}}}^{\prime}$ (19) x $\displaystyle=[a,b,c,d]^{T}$
(20) $\displaystyle{\textbf{{t}}}^{\prime}$
$\displaystyle=(1-\left|\Gamma_{s_{i}}\right|^{2}){\textbf{{t}}}$ (21)

where the noise parameters are related to the ${\textbf{{x}}}=[a,b,c,d]^{T}$
vector by:

$\displaystyle T_{{\rm min}}$ $\displaystyle=a+\sqrt{4bc-d^{2}}$ (22)
$\displaystyle R_{N}$ $\displaystyle=b/(Y_{0}T_{0})$ (23) $\displaystyle
G_{{\rm opt}}$ $\displaystyle=Y_{0}\sqrt{4bc-d^{2}}/2b$ (24) $\displaystyle
B_{{\rm opt}}$ $\displaystyle=-Y_{0}d/2b.$ (25)

The reverse relations are provided in Appendix B.

The source impedance matrix ${\textbf{{A}}}^{\Gamma}$ (Eq. 18) is a central
result of this paper. It can be employed in any noise parameter extraction
technique based upon ${\textbf{{A}}}^{G}$ by minor modification to the matrix
relation (Eqs.19–21). Similarly, the matrix
${\textbf{{A}}}^{\gamma}{{}^{\prime}}$ can be used in lieu of
${\textbf{{A}}}^{\gamma}$. We will now show that these substitutions are well
motivated as they minimize errors arising from matrix inversion.

##  V Comparison of measurement errors

The propagation of errors in matrix inversion problems is non-trivial,
particularly if row entries are covariant [16]. Within each row of A, entries
are indeed highly covariant as they are formed from the same source
admittance/reflection measurement. Nevertheless, simple arguments about worst-
case scaling errors (i.e. propagation of relative errors due to matrix
inversion) can be made based upon the matrix determinant.

It has been previously noted that higher $|$det$|$ for A corresponds to lower
absolute uncertainties [7]. SUT20 show that the maximum possible $|$det$|$ for
${\textbf{{A}}}^{G^{\prime}}$ (and thus ${\textbf{{A}}}^{\Gamma}$) is 41.57,
when points are maximally spread; for the $|\Gamma_{s}|<0.9$ case (required
for ${\textbf{{A}}}^{G}$) the maximum $|$det$|$ $<27.7$. This suggests that
${\textbf{{A}}}^{\Gamma}$ has the potential to yield lower worst-case scaling
errors by allowing the use of reference sources with $|\Gamma_{s}|>0.9$.

Seemingly counter to this, HB16 argues that high-reflection sources
intrinsically introduce measurement uncertainties separate to scaling errors
due to matrix inversion. To illustrate, they use concentric noise circles of
0.1-dB around $\Gamma_{{\rm opt}},$which when plotted on the Smith chart
become denser toward the edge. However, by scaling measurements by
$(1-|\Gamma_{s}|)^{2}$, the space between concentric rings is constant,
negating this effect.

In four-point methods, several authors have noted that measurement error is
minimized when three of the source impedances are purely real (i.e. the phase
of $\Gamma$ is $0^{\circ}$, or $180^{\circ}$), and one impedance is located at
$90^{\circ}$ or $270^{\circ}$ on the Smith chart [6, 7, 17]. HB16 explains
that this corresponds to forming a matrix of source impedances that is
diagonally dominant.

A previous study determined that errors do not increase meaningfully as long
as a minimum spacing of 30 degrees between points in the impedance pattern is
maintained [7]. This requirement is equivalent to setting a minimum acceptable
$|$det$|$; for ${\textbf{{A}}}^{\Gamma}$, the requirement corresponds to
$|$det$|\apprge$ 15.5. Using a similar argument, SUT20 suggests $|$det$|$
$\apprge$ 10.

To summarize, the selection of reference sources affects the errors we expect
due to matrix inversion. Errors are minimized by maximizing $|$det$|$, and by
choosing sources that correspond to a diagonally dominant matrix. We now
demonstrate that for a given four-point pattern, measurement errors can be
improved just by choosing to use ${\textbf{{A}}}^{\Gamma}$ in lieu of
${\textbf{{A}}}^{G}$.

To qualitatively compare how the choice of matrix A affects errors, we ran a
Monte Carlo simulation of a “toy” DUT with $T_{{\rm min}}=200$ K,
$\left|\Gamma_{{\rm opt}}\right|=0.3$, $\theta_{{\rm opt}}=90^{\circ}$ and
$N$=0.25. These values were chosen to be similar to those measured in Sec.
VIII for a Minicircuits ZX60-3018G-S+ amplifier. Using the relations Appendix
B, we converted noise parameters into a vector x, and then computed the
measurement vector ${\textbf{{t}}}={\textbf{{A}}}{\textbf{{x}}}$ for both
matrices ${\textbf{{A}}}^{G}$ and ${\textbf{{A}}}^{\Gamma}$. The A matrices
were formed using an four-point pattern $\Gamma_{s_{i}}=(0,0.9,-0.9,0.9\angle
90^{\circ})$, which has maximum spacing between loci on the Smith chart.

Figure 2: Scatterplot matrix showing noise parameter uncertainties for a toy
model DUT. Each scatter plot within the matrix visualizes the covariance
relationship between a pair of parameters. Points computed using the
traditional admittance matrix ${\textbf{{A}}}^{G}$ are shown in black, and the
reflection-coefficient matrix ${\textbf{{A}}}^{\Gamma}$ in red.

To simulate errors in $\Gamma_{s}$, we generated normally-distributed 0.1 dB
magnitude and $1^{\circ}$ phase offsets, then computed error-corrupted noise
vectors $\tilde{{\textbf{{x}}}}$ via:

$\tilde{{\textbf{{x}}}}=\left(\tilde{{\textbf{{A}}}}\right)^{-1}{\textbf{{t}}},$
(26)

where $\tilde{{\textbf{{A}}}}$ is the source matrix with errors. This approach
is similar to that used by National Institute of Standards and Technology
(NIST) for noise parameter uncertainty analysis [18]. We ran this procedure
1024 times, then formed scatterplot matrices of noise parameter estimates
(Figure 2); the scatterplot shows the covariance between parameters. We find
that the traditional approach with ${\textbf{{A}}}^{G}$ exhibits order-of-
magnitude larger uncertainties in the retrieved values for $N$, $|\Gamma_{{\rm
opt}}|$ and $\theta_{{\rm opt}}$, but smaller uncertainties on $T_{{\rm min}}$
(i.e. if covariance is ignored). We conclude that the removal of the
singularity as $\left|\Gamma_{s}\right|\rightarrow 1$ does indeed lead to
smaller errors due to matrix inversion.

##  VI Using open and short source references

In the previous section, we showed that for a given four-point pattern, using
${\textbf{{A}}}^{\Gamma}$ in lieu of ${\textbf{{A}}}^{G}$ leads to smaller
errors. Here, we discuss the use of four-point patterns comprised of highly-
reflective sources, which cannot be used with ${\textbf{{A}}}^{G}$.

The use of open and short impedances ($\Gamma_{{\rm op}}=1$ and $\Gamma_{{\rm
sh}}=-1$ ) is well motivated by previous four-point studies, which found that
measurement errors are minimized when the phase of $\Gamma_{s}$ is
$0^{\circ}$, or $180^{\circ}$ for three of the impedances [6, 7, 8]. In
particular, HB16 identifies four regions on the Smith chart in which
measurements should be made: regions “B” and “C” contain open and short loci,
but the $|\Gamma|<0.9$ requirement precluded their use.

While using open and short maximizes $|$det$|$ of A, leading to smaller matrix
inversion errors, we must also consider the measurement accuracy for
$\Gamma_{s}$. In general, fractional $S_{11}$ measurement uncertainties on a
given VNA are greater for highly-reflective sources; this is a separate error
term that runs counter to maximizing $|$det$|$. Fortunately, physical models
of short and open are provided by calibration kit manufacturers, and methods
for precise and accurate measurement of these standards are well understood
[19, 20]. As such, physical models of open and short sources can be used to
characterize $\Gamma_{\rm{open}}$ and $\Gamma_{\rm{short}}$, to precision
exceeding that possible by VNA measurement alone.

Let us consider the case where ideal open, short and load ($\Gamma_{{\rm
ld}}=0$) are used as reference sources, along with an open 1/8-wavelength
cable. A lossless open (or shorted) 1/8-wavelength cable introduces a
$90^{\circ}$ phase shift (or $-90^{\circ}$), such that $\Gamma_{{\rm cbl}}=\pm
1j$.

For these four references, ${\textbf{{A}}}^{\Gamma}$ is

${\textbf{{A}}}^{\Gamma}=\left(\begin{array}[]{c}A_{1}^{\Gamma_{{\rm ld}}}\\\
A_{2}^{\Gamma_{{\rm op}}}\\\ A_{3}^{\Gamma_{{\rm sht}}}\\\ A_{4}^{\Gamma_{{\rm
cbl}}}\end{array}\right)=\left(\begin{array}[]{cccc}1&1&1&0\\\ 0&0&4&0\\\
0&4&0&0\\\ 0&2&2&\mp 2\end{array}\right).$ (27)

Note the entry $A_{44}=-2$ if an open cable is used, or $+2$ if a shorted
cable is used. ${\textbf{{A}}}^{\Gamma}$ is invertible, with $|$det$|$=32 and
condition number $c_{A}=5.62$. The inverse
$\text{(${\textbf{{A}}}^{\Gamma}$})^{-1}$

$\text{(${\textbf{{A}}}^{\Gamma}$})^{-1}=\frac{1}{4}\left(\begin{array}[]{cccc}4&-1&-1&0\\\
0&0&1&0\\\ 0&1&0&0\\\ 0&1&1&\mp 2\end{array}\right).$ (28)

From this, the solutions to
${\textbf{{x}}}={\textbf{{A}}}{\textbf{{t}}}^{\prime}$ are

$\displaystyle a$ $\displaystyle=t^{\prime}_{{\rm ld}}-(t^{\prime}_{{\rm
op}}+^{\prime}t_{{\rm sh}})/4,$ (29) $\displaystyle b$
$\displaystyle=t^{\prime}_{{\rm sh}}/4,$ (30) $\displaystyle c$
$\displaystyle=t^{\prime}_{{\rm op}}/4,$ (31) $\displaystyle d$
$\displaystyle=(t^{\prime}_{{\rm op}}+t^{\prime}_{{\rm sh}})/4\mp
t^{\prime}_{{\rm cbl}}/2,$ (32)

from which we note that 1) $b$ and $c$ are directly given by the measurement
of short and open standards, respectively; and 2) the load measurement is only
required to compute $T_{{\rm min}}$, as the A term only appears in Equation
22.

The matrix in Equation 28 is only correct at a central frequency
$f_{0}=v_{c}/\lambda_{0}$. For a 1/8-wavelength cable at a central frequency
$f_{0}=v_{c}/\lambda_{0}$, we can enforce a minimum $|$det$|$ for the
${\textbf{{A}}}^{\Gamma}$ matrix, and find a fractional bandwidth over which
our $|$det$|$ requirement is satisfied. The phase of $\Gamma_{{\rm cbl}}$ is
given by $\theta(f)=4\pi L/v_{c}f$, where $v_{c}$ is the velocity factor of
the cable. Setting $|$det$|$min=10, based on the recommendation of SUT20, we
compute a corresponding frequency range $f_{{\rm low}}$ to $f_{{\rm high}}$ :

$\displaystyle f_{{\rm low}}$ $\displaystyle=0.2f_{0}$ (33) $\displaystyle
f_{{\rm high}}$ $\displaystyle=1.8f_{0}.$ (34)

Or put another way, $f_{{\rm high}}=9f_{{\rm low}}$, covering a 9:1 band. For
example, if $f_{0}=1$ GHz, then the range over which $|$det$|$ $>$10 is
0.2–1.8 GHz.

##  VII An Open-Short-Load-Cable Measurement method

In this section, we present a practical method for noise parameter extraction
via the use of a load, open, short, and an open 1/8-wavelength cable. In this
method, which we call “OSLC”, we form a measurement vector
$\text{{{{t}}}'},$which includes a $(1-\left|\Gamma_{s}\right|^{2})$ term that
naturally arises when measuring the output power of a 2-port DUT. This term
cancels out the singularities inherent in previous methods that use matrices
${\textbf{{A}}}^{\Gamma}$ and ${\textbf{{A}}}^{G}$.

The OSLC method relies on the matrix ${\textbf{{A}}}^{\Gamma}$ (introduced in
Section 3), and requires the following:

  * •

A calibrated noise source to generate “hot” and “cold” temperature references,
$T_{{\rm hot}}$ and $T_{{\rm cold}}$. The reflection coefficients,
$\Gamma_{{\rm hot}}$ and $\Gamma_{{\rm cold}}$ must be known or measured, and
should satisfy $\Gamma_{{\rm hot}}\approx\Gamma_{{\rm cold}}$.

  * •

A broadband load, open, and short, with known or measured reflection
coefficients $\Gamma_{{\rm ld}}$, $\Gamma_{{\rm op}}$, and $\Gamma_{{\rm
sht}}$. Additionally, an open cable (or other mismatch device) with
$\Gamma_{{\rm cbl}}\approx\pm 1j$. To minimize pickup of radio interference,
we recommend that the cable is placed inside an RF-shielded box with an SMA
feedthru connection.

  * •

A radio receiver to measure power spectral density (PSD) with linear response.
The input reflection coefficient $\Gamma_{{\rm rx}}$ should be known or
measured, and the receiver must have high reverse isolation
($S_{12}S_{21}\approx 0$) for the analog component before the digitizer.

  * •

A 2-port VNA to measure the S-parameters of the DUT, and any unknown
reflection coefficients. However, If the DUT is highly directional and well-
matched to the receiver – such that
$\left|S_{12}S_{21}\Gamma_{rx}\right|\approx 0$ – only $S_{11}$ is required.

To show how the OSLC approach may be used, let us start by considering a power
measurement made by a radio receiver. The power $P_{s}$ measured at the output
of a 2-port network with scattering matrix [$S$], connected to a source
impedance $Z_{s}$ and load impedance $Z_{{\rm rx}}$ (see Figure 1), is given
by

$P_{s}=\mathcal{D}_{{\rm rx}}k_{B}\Delta f\mathcal{G}_{{\rm
rx}}\mathcal{G}_{{\rm DUT}}\left(T_{s}+T_{{\rm n}}+\frac{T_{{\rm
rx}}}{\mathcal{G}_{{\rm DUT}}(Z_{s})}\right)$ (35)

where $k_{B}$ is the Boltzmann constant, $\Delta f$ is the noise equivalent
bandwidth, $T_{n}$ is the noise temperature of the DUT (when connected to
$Z_{s}$ and $Z_{{\rm rx}}$), $T_{s}$ is the noise temperature of the source
($T_{s}=T_{{\rm amb}}$ for passive networks at ambient temperature), and
$\mathcal{G}_{{\rm DUT}}=\mathcal{G}_{{\rm DUT}}(Z_{s})$ is the available gain
of the DUT. Here, $\mathcal{D}_{{\rm rx}}$ encompasses all (unknown) digital
conversion and gain factors within the receiver, assumed to be linear.

The available gain [21], denoted here with $\mathcal{G}$, of the DUT is given
by

$\mathcal{G}_{{\rm
DUT}}(Z_{s})=\frac{\left|S_{21}\right|^{2}\left(1-\left|\Gamma_{s}\right|^{2}\right)}{\left|1-S_{11}\Gamma_{s}\right|^{2}\left(1-\left|\Gamma_{{\rm
out}}(Z_{s})\right|^{2}\right)}.$ (36)

When connected to the source impedance $Z_{s}$, the two-port network will be
mismatched with a reflection coefficient, $\Gamma_{{\rm out}}(Z_{s})$:

$\Gamma_{{\rm
out}}(Z_{s})=S_{22}+\frac{S_{12}S_{21}\Gamma_{s}}{1-S_{11}\Gamma_{s}}.$ (37)

For a calibrated noise source with low reflection coefficient, and requiring
$\Gamma_{{\rm hot}}\approx\Gamma_{{\rm cold}}$, we define $\Gamma_{{\rm
ns}}=(\Gamma_{{\rm hot}}+\Gamma_{{\rm cold}})/2$. It follows that the ratio

$\frac{P_{{\rm hot}}-P_{{\rm cold}}}{T_{{\rm hot}}-T_{{\rm
cold}}}=\mathcal{G}_{{\rm rx}}\mathcal{G}_{{\rm DUT}}k_{B}\Delta\nu.$ (38)

Where $\mathcal{G}_{\rm{rx}}$ is the available gain of the receiver’s analog
components. Referring to Figure 1, the receiver sees an input impedance
$\Gamma_{{\rm DUT}}=\Gamma_{{\rm out}}(Z_{s})$, which depends upon the source
impedance $Z_{s}$. So, the total cascaded gain (as seen at the receiver
output) is given by $\mathcal{G}_{{\rm casc}}(Z)=\mathcal{G}_{{\rm
DUT}}(Z)\mathcal{G}_{{\rm rx}}(Z_{{\rm out}})$. For a receiver (with high
reverse isolation), the ratio

$\frac{\mathcal{G}_{{\rm casc}}(Z_{s})}{\mathcal{G}_{{\rm casc}}(Z_{{\rm
ns}})}=\frac{\left(1-\left|\Gamma_{s}\right|^{2}|\right)}{\left(1-\left|\Gamma_{{\rm
ns}}\right|^{2}\right)}\frac{\left|1-S_{11}\Gamma_{{\rm
ns}}\right|^{2}}{\left|1-S_{11}\Gamma_{s}\right|^{2}}\frac{\left|1-\Gamma_{{\rm
rx}}\Gamma_{{\rm out}}(Z_{{\rm ns}})\right|^{2}}{\left|1-\Gamma_{{\rm
rx}}\Gamma_{{\rm out}}(Z_{{\rm s}})\right|^{2}}.$ (39)

We now define a scale factor $\alpha$, which converts from power to
temperature units, and a mismatch factor $M_{s}$ (similar to Eq. 30 of [14]):

$\displaystyle\alpha$ $\displaystyle=\frac{T_{{\rm hot}}-T_{{\rm
cold}}}{P_{{\rm hot}}-P_{{\rm cold}}}$ (40) $\displaystyle M_{s}$
$\displaystyle=\left(1-\left|\Gamma_{{\rm
ns}}\right|^{2}\right)\frac{\left|1-S_{11}\Gamma_{s}\right|^{2}}{\left|1-S_{11}\Gamma_{{\rm
ns}}\right|^{2}}\frac{\left|1-\Gamma_{{\rm rx}}\Gamma_{{\rm
out}}(Z_{s})\right|^{2}}{\left|1-\Gamma_{{\rm rx}}\Gamma_{{\rm out}}(Z_{{\rm
ns}})\right|^{2}},$ (41)

and from Equation 35 we retrieve

$\displaystyle\alpha
P_{s}M_{s}-\left(1-\left|\Gamma_{s}\right|^{2}\right)\left(T_{s}+\frac{T_{{\rm
rx}}}{\mathcal{G}_{{\rm DUT}}(Z_{s})}\right)$
$\displaystyle=\left(1-\left|\Gamma_{s}\right|^{2}\right)T_{n}.$ (42)

Note that the terms $\mathcal{G}_{{\rm DUT}}^{-1}$ and $T_{{\rm rx}}$ are
dependent upon the source impedance ($Z_{s}$); however, for sources where
$\left|\Gamma_{s}\right|\approx 1$ and/or $\mathcal{G}_{{\rm DUT}}$ is large,
the factor can be discarded, simplifying to:

$\alpha
P_{s}M_{s}-\left(1-\left|\Gamma_{s}\right|^{2}\right)\left(T_{s}\right)\approx\left(1-\left|\Gamma_{s}\right|^{2}\right)T_{n}.$
(43)

We may now form the measurement vector ${\textbf{{t}}}^{\prime}$ by applying
calibration Equation 42 to our measured power $P_{s}$:

${\textbf{{t}}}^{\prime}_{i}=\left(\alpha
P_{s_{i}}M_{s_{i}}-\left(1-\left|\Gamma_{s_{i}}\right|^{2}\right)(T_{s_{i}}+\mathcal{G}_{{\rm
DUT}}^{-1}(Z_{s})T_{{\rm rx}})\right).$ (44)

Specifically, if we connect a load, open, short, and a (lossless) open
1/8-wavelength cable to a DUT, the noise parameters are retrieved via

x
$\displaystyle=\text{(${\textbf{{A}}}^{\Gamma}$})^{-1}\left(\begin{array}[]{c}\alpha
P_{{\rm ld}}M_{{\rm ld}}-\left(1-\left|\Gamma_{{\rm
ld}}\right|^{2}\right)(T_{{\rm amb}}+\frac{T_{{\rm rx}}}{\mathcal{G}_{{\rm
DUT}}})\\\ \alpha P_{{\rm op}}M_{{\rm op}}-\left(1-\left|\Gamma_{{\rm
op}}\right|^{2}\right)(T_{{\rm amb}}+\frac{T_{{\rm rx}}}{\mathcal{G}_{{\rm
DUT}}})\\\ \alpha P_{{\rm sh}}M_{{\rm sh}}-\left(1-\left|\Gamma_{{\rm
sh}}\right|^{2}\right)(T_{{\rm amb}}+\frac{T_{{\rm rx}}}{\mathcal{G}_{{\rm
DUT}}})\\\ \alpha P_{{\rm cbl}}M_{{\rm cbl}}-\left(1-\left|\Gamma_{{\rm
cbl}}\right|^{2}\right)(T_{{\rm amb}}+\frac{T_{{\rm rx}}}{\mathcal{G}_{{\rm
DUT}}})\end{array}\right)$ (49)
$\displaystyle=\frac{1}{4}\left(\begin{array}[]{cccc}4&-1&-1&0\\\ 0&0&1&0\\\
0&1&0&0\\\ 0&1&1&-2\end{array}\right)\left(\begin{array}[]{c}\alpha P_{{\rm
ld}}M_{{\rm ld}}-(T_{{\rm amb}}+\frac{T_{{\rm rx}}}{\mathcal{G}_{{\rm
DUT}}})\\\ \alpha P_{{\rm op}}M_{{\rm op}}\\\ \alpha P_{{\rm sh}}M_{{\rm
sh}}\\\ \alpha P_{{\rm cbl}}M_{{\rm cbl}}\end{array}\right)$ (58)

where $\text{(${\textbf{{A}}}^{\Gamma}$})^{-1}$ is given by Equation 28; note
that $M_{{\rm ld}}\approx 1$.

##  VIII Measurement example

Figure 3: Example measurement setup as used to extract noise parameters. A DUT
(Minicircuits ZX60-3018G-S+) is connected to a simple noise receiver,
consisting of commercially-available amplifiers, filters, and cables connected
to an ADC. The receiver starts with a semi-rigid cable, connected to a 3 dB
attenuator to improve impedance matching. A pair of Minicircuits ZFL-500HLN+
amplifiers provide $\sim 38$ dB of gain, and a K&L Microwave lowpass filter
and SHP-50+ highpass filter isolate the 35–310 MHz band. In the photo, an
Agilent 346A noise source connected to the DUT.  Figure 4: $|$det$|$ for the
$4\times 4$ ${\textbf{{A}}}^{\Gamma}$ matrix formed from $\Gamma_{s}$
measurements of the reference sources, using an open 15-cm coaxial cable. As
the relative phase of $\Gamma_{{\rm cbl}}$ changes with frequency, the
$|$det$|$ is maximized at $\sim 140$ MHz, close to the ideal value of 32.
Figure 5: Measured power (uncalibrated) for the DUT (ZX60-3018G-S+) connected
to the load, open, short and cable references (top). Mismatch factors $M_{s}$
applied during calibration (middle). Calibrated entries
$(1-|\Gamma_{s_{i}}|^{2})T_{n}$ in the ${\textbf{{t}}}^{\prime}$ vector
(bottom). Figure 6: Noise parameters $T_{{\rm min}}$, $N$ , $|\Gamma_{{\rm
opt}}|$ and $\theta_{{\rm opt}}$ for the DUT (Minicircuits ZX60-3018G-S+)
extracted using the method detailed in Sec VII (black); errorbars were
determined using the Monte Carlo approach in V. For comparison, the SUT20
noise parameter measurement for the same model amplifier (using a different
technique and apparatus) are also plotted (red) [14].

We applied our technique to measure the noise parameters of a Minicircuits
ZX60-3018G-S+ amplifier, across 50–250 MHz. The amplifier has a manufacturer-
supplied noise figure of 2.7–2.9 dB across the 50–250 MHz band, and a gain of
$\sim 25.5$ dB. Here, we used an SMA-terminated 15-cm RG-400 coaxial cable,
with $v_{c}\approx 0.69.$ Based on Equations 33 and 34, the nominal frequency
range for this cable is $\sim$30–300 MHz. The $|$det$|$ for the
${\textbf{{A}}}^{\Gamma}$ matrix using an open, short, load, and 15-cm cable,
is shown as a function of frequency in Fig 4.

To generate hot and cold reference loads, we used a Keysight HP346A noise
source with an ENR of 5.56–5.49 dB across 10 MHz to 1 GHz. As ENR is only
quoted at intervals, we use a 2-order polynomial fit to generate values across
50–250 MHz.

Measurements of power spectra were generated using a custom receiver, based on
a 14-bit Signatek PX1500-2 analog to digital converter (ADC) running at 650
Msamples/s; the noise performance of the receiver was characterized prior to
measurement. Power spectra were generated from ADC samples via an
autocorrelation spectrometer, which applies a 4096-channel Fast Fourier
Transform (FFT), signal detection, and time averaging. The receiver has a pair
of Minicircuits ZFL-500HLN+ amplifiers to provide an extra $\sim 38$ dB of
gain before digitization, and a 50-MHz highpass filter (Minicircuits SHP-50+)
and K&L Microwave 300-MHz lowpass filter were added at the ADC input as an
anti-aliasing filter. The receiver temperature, $T_{\rm{rx}}$, is 1350–1450 K
across the band. We captured 60 s of data per measurement.

To measure reflection coefficients of the DUT, load, cable, and the receiver,
we used a Fieldfox N9915A VNA, calibrated with an Agilent 85052D calibration
kit. Passive components were measured using a high (0 dBm) port power, whereas
lower power (-30 dBm) was used to measure the [$S$] matrix of the DUT and
$\Gamma_{{\rm rx}}$. Measurements were saved in S2P (Touchstone) format, and
read using the Python package scikit-rf222www.scikit-rf.org. To determine
$\Gamma_{{\rm sh}}$ and $\Gamma_{{\rm op}}$, we used a physical model based on
the Agilent 85052D calibration standard definitions (see Appendix C).

Figure 5 shows the measure power $P_{s}$ for the DUT with the load, open short
and cable reference sources (top panel). The computed mismatch factors $M_{s}$
(Equation 41) which are applied during calibration (Equation 42) are shown in
the middle panel, and calibrated entries to the vector
${\textbf{{t}}}^{\prime}$ are shown in the bottom panel.

Based on the measurement vector ${\textbf{{t}}}^{\prime}$ and $4\times 4$
matrix ${\textbf{{A}}}^{\Gamma}$, we solve for the noise parameter vector x
(Equation 20) via matrix inversion
${\textbf{{x}}}=\text{(${\textbf{{A}}}^{\Gamma}$})^{-1}{\textbf{{t}}}^{\prime}$
. We then solve for the standard noise parameters via Equations 22–25. The
derived noise parameters $T_{{\rm min}}$, $N$, and $\Gamma_{{\rm opt}}$ are
shown in Figure 6. Errors are derived using the Monte Carlo approach detailed
in Section V, assuming VNA measurements of the coaxial cable are accurate to
$\pm 0.1$ dB in magnitude and $0.5^{\circ}$ in phase. Errors on the open,
short and load impedance are modelled as gaussian random variables, with
uncertainties based upon manufacturer-supplied electrical specifications [22].

Also plotted in Figure 6 are measurements from SUT20 [14, Fig 12] of the same
amplifier model. The authors followed a different measurement methodology,
involving a Focus Microwaves CCMT-101 single-probe slide screw tuner and
Keysight PXA N9030A receiver. Note that 1) while the model is identical, we
did not use the same DUT and 2) uncertainties in SUT20 were derived using a
different approach, so cannot be directly compared. Nevertheless, our results
are in close agreement with those presented in SUT20. A $\approx 15^{\circ}$
offset between $\theta_{\rm{opt}}$ is apparent, which could be due to
variation between amplifiers or differences in operating conditions.

##  IX Discussion

Here, we have shown that the commonly-used admittance-based matrix
${\textbf{{A}}}^{G}$, after removal of its singularity, can be rewritten as
simple expression of reflection coefficients (${\textbf{{A}}}^{\Gamma}$,
Equation 18). We also show that uncertainties in noise parameter estimates due
to errors in the source impedance matrix A are significantly lowered by using
${\textbf{{A}}}^{\Gamma}$ instead of ${\textbf{{A}}}^{G}$. Combined, these
suggest that singularity-free formulations of A should be used where possible.

We have presented a straightforward method to measure noise parameters using a
broadband load, open, short, and 1/8-wavelength cable as reference sources
(“OSLC”). Our method leverages a singularity-free matrix, to allow the use of
highly reflective reference loads. Specifically, our method allows for open
and short calibration standards to be used as reference sources.

The ability to use highly reflective reference loads allows the spread of loci
in the Smith chart to be maximized, which also maximises $|$det$|$, the
magnitude of the matrix determinant. It follows that highly-reflective
references will yield lower worst-case scaling errors due to matrix inversion.
However, the use of highly-reflective sources requires that the DUT is
unconditionally stable. Also, VNA measurements of highly-reflective sources
are prone to larger fractional errors (the magnitude of which depend upon VNA
specifications and calibration approach). Physical models of open and short
circuits, as provided in VNA calibration kits, can be used in lieu of VNA
measurements and are more accurate; methods for precise characterization of
high-impedance references can also be employed [19, 20].

The OSLC four-point method can be used across a range 0.2–1.8$\lambda_{0}$,
for a cable of length $\lambda_{0}/8$. If a larger range is required, one
could use a set of cables with varying lengths and repeat the process at
different central frequencies. Alternatively, rows may be added to
${\textbf{{A}}}^{\Gamma}$ and the pseudoinverse may be used when solving
(Equation 5).

The OSLC method is well-suited to low-frequency application ($<$1 GHz), as the
only source impedances are a 1/8-wavelength cable and a VNA calibration kit:
both are accessible and affordable. Approaches that leverage rapid phase
wrapping [9, 10, 13] may require lengths of cable that are unwieldy or
prohibitively long at low frequencies. Admittance tuners, which are rated down
to a half-wavelength, are comparatively expensive and physically bulky.

The OSLC method is more practical than previous approaches for _in-situ_
measurement in the field using a portable VNA and spectrum analyzer. On the
flip side, at higher frequencies, a 1/8-wavelength cable may become
prohibitively small. Precision airlines offer very low loss and may be a
suitable alternative; nevertheless, further research is needed to validate
comparable methodologies at millimeter frequencies. Modifications to the
approach may also be needed for single-transistor DUTs, which can be poorly
matched at low frequencies.

## Acknowledgments

The authors would like to thank Budi Juswardy, Dave Kenney and Daniel Ung for
help with laboratory measurements. This research made use of Scikit-RF, an
open-source Python package for RF and Microwave applications, and Corner.py
[23]. All calculations were performed in Python 3.7, using the Numpy package
[24].

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##  Appendix A Derivation of ${\textbf{{A}}}^{\Gamma}$

To show that ${{\textbf{{A}}}}_{i}^{G^{\prime}}={\textbf{{A}}}^{\Gamma}$, we
note that

$\displaystyle\left|\Gamma_{s}\right|^{2}$
$\displaystyle=\left(\frac{Y_{0}-Y_{s}}{Y_{0}+Y_{s}}\right)\left(\frac{Y_{0}-Y_{s}}{Y_{0}+Y_{s}}\right)^{*}$
(59)

where * denotes the complex conjugate, such that $Y_{s}^{*}=G_{s}-jB_{s}$.
Equation 59 simplifies to:

$\left|\Gamma_{s}\right|^{2}=\frac{Y_{0}^{2}-2G_{s}+(G_{s}^{2}+B_{s}^{2})}{Y_{0}^{2}+2G_{s}+(G_{s}^{2}+B_{s}^{2})}.$
(60)

Similarly, we may rewrite $\Gamma_{s_{I}}$ with the same denominator by
multiplying through by $1=(Y_{0}+Y_{s}^{*})/(Y_{0}+Y_{s}^{*})$:

$\Gamma_{s}=\left(\frac{Y_{0}-Y_{s}}{Y_{0}+Y_{s}}\right)\left(\frac{Y_{0}+Y_{s}^{*}}{Y_{0}+Y_{s}^{*}}\right)=\frac{Y_{0}^{2}-j2Y_{0}B_{s}+(G_{s}^{2}+B_{s}^{2})}{Y_{0}^{2}+2G_{s}+(G_{s}^{2}+B_{s}^{2})}.$

from this, the following four quantities can be rewritten into terms that
share the denominator of Equation 60:

$\displaystyle 1-\left|\Gamma_{s}\right|^{2}$
$\displaystyle=\frac{4G_{s}}{Y_{0}^{2}+2G_{s}+(G_{s}^{2}+B_{s}^{2})}.$ (61)
$\displaystyle\left|1-\Gamma_{s}\right|^{2}$
$\displaystyle=\frac{4(G_{s}^{2}+B_{s}^{2})}{Y_{0}^{2}+2G_{s}+(G_{s}^{2}+B_{s}^{2})}$
(62) $\displaystyle\left|1+\Gamma_{s}\right|^{2}$
$\displaystyle=\frac{4Y_{0}^{2}}{Y_{0}^{2}+2G_{s}+(G_{s}^{2}+B_{s}^{2})}$ (63)
$\displaystyle{\rm Im}(\Gamma_{s})$
$\displaystyle=\frac{-2Y_{0}B_{s}}{Y_{0}^{2}+2G_{s}+(G_{s}^{2}+B_{s}^{2})}.$
(64)

Using these three identities

$\displaystyle\frac{\left(1-\left|\Gamma_{s}\right|^{2}\right)}{G_{s}}$
$\displaystyle=\frac{\left|1+\Gamma_{s}\right|^{2}}{G_{0}}$ (65)
$\displaystyle\left(1-\left|\Gamma_{s}\right|^{2}\right)\frac{\left|Y_{s}\right|^{2}}{G_{s}}$
$\displaystyle=G_{0}\left|1+\Gamma_{s}\right|^{2}$ (66)
$\displaystyle\left(1-\left|\Gamma_{s}\right|^{2}\right)\frac{B_{s_{i}}}{G_{s_{i}}}$
$\displaystyle=-2{\rm Im}(\Gamma_{s})$ (67)

We thus have that ${\textbf{{A}}}^{G^{\prime}}={\textbf{{A}}}^{\Gamma}$
(setting $Y_{0}=G_{0}=1$).

##  Appendix B Reverse relations

The reverse relations for Equations 22–25 are:

$\displaystyle a$ $\displaystyle=T_{{\rm min}}-2R_{N}T_{0}G_{{\rm opt}}$ (68)
$\displaystyle b$ $\displaystyle=R_{N}T_{0}Y_{0}$ (69) $\displaystyle c$
$\displaystyle=\frac{R_{N}T_{0}}{Y_{0}}(G_{{\rm opt}}^{2}+B_{{\rm opt}}^{2})$
(70) $\displaystyle d$ $\displaystyle=-2R_{N}T_{0}B_{{\rm opt}}.$ (71)

These reverse relations are used in Section IV to simulate noise parameter
measurements.

##  Appendix C Open and short circuit models

Open and short source impedances are physically modelled by a line terminated
with an inductance (short) or capacitance (load) (see [19, 25]). Following
[26], the reflection coefficient of a terminated line is given by

$\Gamma_{i}=\frac{\Gamma_{1}\left(1-e^{-2\gamma\ell}-\Gamma_{1}\Gamma_{T}\right)+e^{-2\gamma\ell}}{1-\Gamma_{1}\left[e^{-2\gamma\ell}\Gamma_{1}+\Gamma_{T}\left(1-e^{-2\gamma\ell}\right)\right]}$
(72)

where $\Gamma_{1}$ is the transmission line reflection coefficient,
$\Gamma_{T}$ is the impedance of the termination, $\ell$ is the transmission
line length, $\gamma$ is the propagation constant along the transmission line.
$\Gamma_{1}$ is related to $Z_{c}$, the characteristic impedance of the line,
by

$\Gamma_{1}=\frac{Z_{c}-Z_{r}}{Z_{c}+Z_{r}}.$ (73)

Calibration kits provide a table of calibration standard definitions, from
which $Z_{c}$ can be determined for a given frequency. Specifically, the
manufacturer provides an offset delay $\tau_{\rm{ofs}}$ (in ps), offset loss
at 1 GHz $\l_{{\rm ofs}}$ (G$\Omega$/s), and offset impedance $Z_{{\rm ofs}}$,
from which $Z_{c}$ and $\gamma\ell$ are found:

$\displaystyle Z_{C}$ $\displaystyle=\left(Z_{{\rm ofs}}+\frac{Z_{{\rm
ofs}}}{2\omega}\sqrt{f_{{\rm GHz}}}\right)-j\left(\frac{l_{{\rm
ofs}}}{2\omega}\sqrt{f_{{\rm GHz}}}\right)$ (74) $\displaystyle\gamma\ell$
$\displaystyle=\left(\frac{l_{{\rm ofs}}\tau_{{\rm ofs}}}{2Z_{{\rm
ofs}}}\sqrt{f_{{\rm GHz}}}\right)+j\left(\omega\tau_{{\rm ofs}}+\frac{l_{{\rm
ofs}}\tau_{{\rm ofs}}}{2Z_{{\rm ofs}}}\sqrt{f_{{\rm GHz}}}\right)$ (75)

In the Agilent 85052D calibration kit [22], the inductance model for a short
circuit is a third-order polynomial in frequency

$\displaystyle L_{\rm{sh}}(f)$
$\displaystyle=L_{0}+L_{1}f+L_{2}f^{2}+L_{3}f^{3}$ (76) $\displaystyle
Z_{\rm{sh}}$ $\displaystyle=j2\pi fL_{\rm{sh}}(f).$ (77)

Similarly, the capacitance model for an open circuit is

$\displaystyle C_{\rm{op}}(f)$
$\displaystyle=C_{0}+C_{1}f+C_{2}f^{2}+C_{3}f^{3}$ (78) $\displaystyle
Z_{\rm{op}}$ $\displaystyle=-j(2\pi f)^{-1}.$ (79)

From these equations, a model for $\Gamma_{\rm{op}}$ and $\Gamma_{\rm{sh}}$
may be formed. Calibration standard definitions for the open and short are
summarized in Tab. I.

The 85052D open and short have electrical specifications of $\pm 0.65^{\circ}$
and $\pm 0.50^{\circ}$ deviation from nominal, respectively, at DC to 3 GHz.

TABLE I: Agilent 85052D open/short calibration definitions Open
---
Offset delay | 29.243 | ps
Offset loss | 2.2 |  G$\Omega$/s
$C_{0}$ | $49.433\times 10^{-15}$ | F
$C_{1}$ | $-310.13\times 10^{-27}$ | F/Hz
$C_{2}$ | $23.168\times 10^{-36}$ |  F/Hz2
$C_{3}$ | $-0.15966\times 10^{-45}$ |  F/Hz3
Short
Offset delay | 31.785 | ps
Offset loss | 2.36 |  G$\Omega$/s
$L_{0}$ | $2.0765\times 10^{-12}$ | H
$L_{1}$ | $-108.54\times 10^{-24}$ | H/Hz
$L_{2}$ | $2.1705\times 10^{-33}$ |  H/Hz2
$L_{3}$ | $-0.01\times 10^{-42}$ |  H/Hz3
Offset impedance $Z_{\rm{ofs}}$=50 $\Omega$

  *[DUT]: device under test
  *[VNA]: vector network analyzer
  *[NIST]: National Institute of Standards and Technology


# Design of simple stellarator using tilted toroidal field coils

Yasuhiro Suzuki<EMAIL_ADDRESS>National Institute for Fusion
Science, National Institutes of Natural Sciences, Toki 5095292, Japan  The
Graduate University for Advanced Studies, SOKENDAI, Toki 5095292, Japan  Jie
Huang  National Institute for Fusion Science, National Institutes of Natural
Sciences, Toki 5095292, Japan  Nengchao Wang  International Joint Research
Laboratory of Magnetic Confinement Fusion and Plasma Physics, State Key
Laboratory of Advanced Electromagnetic Engineering and Technology, School of
Electrical and Electronic Engineering, Huazhong University of Science and
Technology, Wuhan, 430074, China  Yonghua Ding  International Joint Research
Laboratory of Magnetic Confinement Fusion and Plasma Physics, State Key
Laboratory of Advanced Electromagnetic Engineering and Technology, School of
Electrical and Electronic Engineering, Huazhong University of Science and
Technology, Wuhan, 430074, China

###### Abstract

This paper deals with the design of the stellarator field with the simple coil
set. In order to realize the stellarator field by the simple coil set, the
tilted toroidal field coil uses for creating the rotational transform. Sixteen
tilted TF coils create the small radial field and large vertical field. With
reducing the vertical field of the tilted toroidal field coil by the
axisymmetric poloidal field coil, the stellarator field can be made. The
formation of clear and nested flux surfaces is confirmed, and the rotational
transform is proportional to the tilting angle of the toroidal field coil.
Main components of the magnetic field for the simple stellarator in this paper
are the large mirror ripple and small helical ripple. This is a similar
property to the quasi-isodynamic configuration like the Wendelstein 7-X
stellarator. The collisionless orbit for the proton is studied. For a moderate
tilting angle of the toroidal field coil, the confinement of the passing and
trapped particles improves.

##  I Introduction

The tokamak is the first candidate of the fusion reactor. However, although
the tokamak has the good confinement, the current driven instability
disrupting the plasma is a critical problem John Wesson (2011). Also, for the
high confinement mode like the H-mode, the MHD instability in the high
confinement region appears, for example, the edge localized mode (ELM) on the
pedestal Zohm (1996). The high-energy flux driven by the ELM damages the
material of the divertor, and difficulties appear for the realization of the
steady state fusion reactor Loarte  _et al._ (2003). On the other hand, the
stellarator Spitzer (1958) is an alternative. The stellarator has many
advantages for the steady state operation, stable detachment, and so on Boozer
(2008, 2020), but the transport property, which is comparing with the tokamak,
is not good due to the complex 3D magnetic field structure, in particular, for
the conventional stellarator Ho and Kulsrud (1987). For improving those
problems, optimized stellarators are designed, and the confirmation of the
optimization are examined experimentally and theoretically Grieger  _et al._
(1989); Anderson  _et al._ (1995); Klinger  _et al._ (2017). In many cases,
the optimized stellarator uses the poloidal modular coil with very
sophisticated coil shape Chu  _et al._ (1982); Merkel (1987). Therefore, a
simple question that those sophisticated coil shape can be extended and used
in the future reactor is very critical. In the history of stellarator
researches, the design using the simple coil set, the so-called simple
stellarator, had considered Funato  _et al._ (1983); Bykov  _et al._ (1989);
Todd (1990); Pedersen  _et al._ (2004); Clark  _et al._ (2014). The simple
stellarator is classified as a conventional stellarator, and the magnetic
field is not optimized. However, the construction is easy comparing with the
fully optimized stellarator, and it will be a good candidate to start up the
stellarator research.

This study deals with the design of a simple stellarator using tilted toroidal
field (TF) coils. The tilted TF coil produces not only the toroidal field but
also small radial field and large vertical fields. This small radial field is
a reason to make an error field, but that field can be used to make the
rotational transform. The several stellarator using the tilted TF coils were
proposed and constructed as the laboratory device. However, in previous study,
the generation of clear and nested flux surfaces were confirmed,but the
properties of the magnetic field was not studied in details yet. Also, as
pointed out in above, the simple stellarator has a great advantage to start up
the stellarator research. In particular, the facility of the tokamak device
can be used to the stellarator, and there is a further possibility to realize
the hybrid operation of the stellarator and tokamak. This leads the synergy by
the 3D physics. In the next section, we discuss a design of the simple
stellarator by the tilted TF coils. The device size and the number of TF coils
refer the J-TEXT tokamak Ge  _et al._ (2009). We then study the formation of
the flux surfaces, the rotational transform, and the spectrum of the magnetic
field on Boozer coordinate Boozer (1982). Also, the confinement of ions for
the bulk plasma energy is studied to understand the transport property
briefly. In the last section, we summarize this study.

##  II Realization of stellarator field by tilted toroidal field coils

###  II.1 Vacuum magnetic field by a tilted toroidal field coil

Here, the vacuum magnetic field produced by a tilted TF coil is considered.
Figure 1 shows a configuration of a tilted TF coil. The size of the TF coil is
0.5 m by 0.5 m and the shape is the square on the $R$-$Z$ plane. The center of
the squared TF coils is $R_{0}$ = 1 m. The size of the TF coil is decided from
the vessel design of the J-TEXT tokamak. The tilting angle, $\theta$, is 30
degree along the toroidal angle, $\phi$. Figure 2 shows profiles of $B_{R}$
and $B_{Z}$ along the toroidal angle, $\phi$ at $R_{0}$ = 1 m. The toroidal
magnetic field is $B_{0}$ = 1 T at $R_{0}$ = 1 m, in other words, the current
of the TF coil is estimated from the condition of $R_{0}B_{0}$ = 1 mT. The
tilting of the TF coil produces the both of $B_{R}$ and $B_{Z}$. In
particular, phases of $B_{R}$ and $B_{Z}$ are opposite, and this is a reason
why the tilting of the TF coil can be a source of the error field to make the
magnetic island in tokamak plasmas. The amplitude of $B_{Z}$ is much higher
than the amplitude of $B_{R}$. If the amplitude of $B_{Z}$ can be reduced by
the axisymmetric poloidal field (PF) coil up to the same order of $B_{R}$, the
vacuum magnetic field by th tilted TF coil and PF coils is equivalent to the
stellarator filed produced by the helical coil.

(a) square shaped TF coil | (b) tilting angle, $\theta$,
---|---
|
Figure 1: A schematic view of a tilted toroidal field (TF) coil for (a)
$R$-$Z$ plane and (b) $\phi$-$Z$ space. Figure 2: Profiles of the radial field
($B_{R}$) and vertical field ($B_{Z}$) produced by one tilted TF coil. The
toroidal field, $B_{0}$, is 1 T at the center of the TF coil, $R_{0}$ = 1 m.

###  II.2 A simple stellarator with tilted toroidal field and poloidal field
coils

As discussed in above, the tilted TF coil can make the stellarator field
without the toroidal current. Here, some reference designs are discussed.
Figure 3 shows a schematic view of a coil geometry. This configuration
consists of sixteen tilted TF coils (light green rectangles) and three pairs
of axisymmetric poloidal coils (light blue curves). For a reference, the shape
of the last closed flux surface of the plasma is also shown in the figure
(colors indicate the magnetic field strength). The major radius of the TF coil
is 1.0 m, and the size of the square is 0.5 m by 0.5 m. The $R_{0}B_{0}$ is
1.0 mT, that is, the toroidal magnetic field, $B_{0}$, at the center of the
helical coil, $R_{0}$ = 1 m, is 1 T along the counterclockwise direction. The
size and location of PF coils are also refereed from the J-TEXT tokamak.

Figure 3: A schematic view of a simple stellarator configuration. This
configuration consists of sixteen TF coils (light green rectangles) and three
pairs of axisymmetric PF coils (light blue curves). For a reference, the last
closed flux surface for $\theta$ = 20 degree is also shown. The major radius
of the TF coil is 1.0 m, and the toroidal field, $B_{0}$, is 1 T at $R_{0}$ =
1 m.

Figure 4 shows Poincaré plot of the vacuum magnetic field with different
tilting angle, $\theta$, to 10, 20, and 30 degree. A blue circle in each
figures indicates the first wall. As shown in figures, clear flux surfaces
appear, and the first wall limits the plasma volume, that is, this
configuration is the limiter configuration. For each cases, the magnetic axis
is helically deviates, and the deviation is proportional to the tilting angle,
$\theta$. In figure 5, profiles of the rotational transform, $\iota$, as the
function of the effective minor radius, $r_{\mathrm{eff}}$, are plotted as
configurations corresponding to figure 4. For a case of $\theta$ = 10 degree,
the rotational transform is very low ($\sim$ 0.2), and the magnetic shear is
very weak. With increased the tilting angle, $\theta$, the rotational
transform increases, because the $B_{R}$ and $B_{Z}$ increase. For another
case of $\theta$ = 30 degree, the rotational transform on the magnetic axis
increases up to the unity, and gradually decreases. In that case, the magnetic
shear becomes strong. However, as shown in the figure, the plasma volume
shrinks, because of the large helical deviation.

(a) $\theta$ = 10 degree
---
(b) $\theta$ = 20 degree
(c) $\theta$ = 30 degree
Figure 4: Poincaré plots of simple stellarators for different tilting angles,
$\theta$ = (a) 10, (b) 20, and (c) 30 degree. Figure 5: Profiles of the
rotational transform, $\iota$, for different tilting angles, $\theta$ = (a)
10, (b) 20, and (c) 30 degree.

###  II.3 Magnetic field spectrum on Boozer coordinate

In order to understand characteristics of the magnetic configuration, the
spectrum of the magnetic field is studied on Boozer coordinate Boozer (1982).
The magnetic field spectrum on Boozer coordinate is calculated from the zero-
beta 3D equilibrium by the VMEC code Hirshman (1983). Figure 6 shows the
spectrum of the magnetic field, $|B|$, on Boozer coordinate for $\theta$ = 10,
20, and 30 degree, respectively. The spectra omit the axisymmetric toroidal
field component, $(n,m)=(0,0)$. Since the magnetic field is mainly produced by
the TF coil, the first dominant component is the mirror ripple (TF ripple),
$(n,m)=(1,0)$. The second dominant component is the helical ripple,
$(n,m)=(1,1)$. This is a similar nature to the quasi-isodaynamic configuration
like W7-X. With increased the tilting angle, $\theta$, the mirror ripple
decreases, and the helical ripple increases. For a case of $\theta$ = 30
degree, the mirror and helical ripples are almost comparable in the plasma
edge. That might reflect that the orbit property of the ion confinement will
degrade because of the helical ripple loss. Other components are relatively
small comparing with the mirror and helical ripples. These spectra are studied
for only the vacuum field. For cases of $\theta$ = 10 and 20 degree, since the
rotational transform is small, the Shafranov shift due to the finite-$\beta$
effect might be large. An impact of the finite-$\beta$ effect on Boozer
spectrum must be studied.

(a) $\theta$ = 10 degree | (b) $\theta$ = 20 degree | (c) $\theta$ = 30 degree
---|---|---
|  |
Figure 6: Spectra of the magnetic field on Boozer coordinate for different
tilting angles, $\theta$ = (a) 10, (b) 20, and (c) 30 degree.

###  II.4 collisionless orbit property

According to the spectrum of the magnetic field on Boozer coordinate, these
configurations have a similar nature to the quasi-isodynamic configuration,
that is, the confinement of the trapped particle improves due to the strong
mirror ripple. On the other hand, for the small tilting angle, $\theta$ = 10
degree, the rotational transform is very small. In such a case, the poloidal
drift width is very large. Therefore, the orbit surfaces deviates from the
flux surface, and then, the prompt loss will increase. To confirm that
hypothesis and estimate the orbit confinement quantitatively, the
collisionless guiding-center orbit is studied for three cases. The guiding-
center of the 100 eV proton is followed by the GCR code Suzuki  _et al._
(2006) for the vacuum field. The starting points are set to the $Z$ = 0 plane
at the $\phi$ = 0. The pitch angle, $\tan^{-1}(v_{\perp}/v_{\parallel})$, is
scanned from 0 to $\pi$. Figure 7 shows the loss cone for three cased of
$\theta$ = 10, 20, and 30 degree, respectively. For a case of $\theta$ = 10
degree, due to the large poloidal drift width, the almost co-going orbit
losses. The trapped ion with $\tan^{-1}(v_{\perp}/v_{\parallel})\sim\pi/2$ is
confined in the inward of the torus, but that losses in the outward of the
torus. For another case of $\theta$ = 20 degree, the confinement of the co-
going ion improves, and small amount of the counter-going ion is confined.
Also, the almost trapped ion with
$\tan^{-1}(v_{\perp}/v_{\parallel})\sim\pi/2$ is confined. However, for the
last case of $\theta$ = 30 degree, the number of confined co-going ion
decreases, and all counter-going ions loss. The number of the trapped ion
increases due to the increased helical ripple, and the almost trapped ion is
confined. Since the orbit property is sensitive to the finite-$\beta$ effect,
the orbit following calculation for the finite-$\beta$ equilibrium will be
important.

(a) $\theta$ = 10 degree | (b) $\theta$ = 20 degree | (c) $\theta$ = 30 degree
---|---|---
|  |
Figure 7: Loss cone of 100 eV proton orbits for different tilting angles,
$\theta$ = (a) 10, (b) 20, and (c) 30 degree. The orbit following starts from
Z = 0 plane with the different pitch angle.

##  III Summary and outlook

This paper studied the simple design of the stellarator with tilted TF coils.
Coupling the tilted TF coil and axisymmetric PF coil, the stellarator field
produces. With tracing the magnetic field line for the vacuum field, the
formation of clear and nested flux surfaces are confirmed. The rotational
transform is proportional to the tilting angle of the TF coil, $\theta$, and
the magnetic shear is weak. The spectrum of the magnetic field on Boozer
coordinate is studied. The stellarator with the tilted TF coils has a similar
property to the quasi-isodynamic configuration, because the mirror ripple is a
dominant component of the magnetic field spectrum. With changing the tilting
angle of the TF coil, $\theta$, the configuration is scanned. For a case of a
low tilting angle, $\theta$ = 10 degree, the rotational transform is small and
the magnetic shear is very weak. In such a case, the drift width of the proton
orbit is large, and then the co-going ions cannot be confined. A small amount
of the counter-going ion in the inward of the torus can be confined. However,
for another case of $\theta$ = 20 degree, the rotational transform increases,
and the confinement of the co-going orbit improves. Also, the confinement of
the trapped ion for the pitch angle,
$\tan^{-1}(v_{\perp}/v_{\parallel})\sim\pi/2$, improves. However, for the last
case of $\theta$ = 30 degree, the confinement of the co-going ions degrades
again.

The design of the simple stellarator in this paper is studied for only the
vacuum field. Since the magnetic shear is weak, the magnetic field might be
sensitive to the finite-$\beta$ effect. Studies including the finite-$\beta$
effect is an important subject. Also, in this paper, we discussed only the
tilted TF coils according to a simple rule, that is, the TF coil tilts along
the toroidal angle, $\phi$, direction. From a study of CNT, it was confirmed
that the magnetic field is very sensitive to the small displacement of the
coil set Zhu  _et al._ (2018). Therefore, the optimization of the simple
stellarator will be possible. Those studies are future subjects, and those
will be reported in near future.

## Acknowledgments

This work is supported by the NINS (National Institute of Natural Sciences)
program of Strategic International Research Interaction Acceleration
Initiative (Grant No. UFEX404), and State Key Laboratory of Advanced
Electromagnetic Engineering and Technology (Grant No. AEET 2019KF002). Also,
this work was partially supported by ’PLADyS’, JSPS Core-to-Core Program, A.
Advanced Research Networks.

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