# Propagation Constant Measurement Based on a Single Transmission Line
Standard Using a Two-port VNA

Ziad Hatab, Arezoo Abdi, Gregor Steinbauer, Michael Ernst Gadringer, and
Wolfgang Bösch
Christian Doppler Laboratory for Technology Guided Electronic Component Design
and Characterization
Institute of Microwave and Photonic Engineering
Graz University of Technology, Austria
{z.hatab, arezoo.abdi, michael.gadringer<EMAIL_ADDRESS>Software
implementation and measurements available online:
https://github.com/ZiadHatab/two-port-single-line-propagation-constant

###### Abstract

This study presents a new method for measuring the propagation constant of
transmission lines using a single line standard and without prior calibration
of a two-port vector network analyzer (VNA). The method provides accurate
results by emulating multiple line standards of the multiline calibration
method. Each line standard is realized by sweeping an unknown network along a
transmission line. The network need not be symmetric or reciprocal, but must
exhibit both transmission and reflection. We performed measurements using a
slab coaxial airline and repeated the measurements on three different VNAs.
The measured propagation constant of the slab coaxial airline from all VNAs is
nearly identical. By avoiding disconnecting or moving the cables, the proposed
method eliminates errors related to repeatability of connectors, resulting in
improved broadband traceability to SI units.

_Keywords_ microwave measurement $\cdot$ network analyzers $\cdot$ propagation
constant $\cdot$ traceability

##  1 Introduction

The propagation constant is a critical parameter in transmission line
analysis, providing valuable information about the electrical properties of
materials at different frequencies. The need for accurate measurement of the
propagation constant arises in various applications, such as material
characterization [1, 2, 3, 4], or estimation of the characteristic impedance
of transmission lines, which allows impedance renormalization in various
vector network analyzer (VNA) calibration methods [5]. Furthermore, knowledge
of the propagation constant allows the analysis of losses along a transmission
line, which is a critical aspect in signal integrity applications [6, 7]. In
general, there are many reasons to measure the propagation constant of guided
wave structures such as transmission lines.

There are several methods to measure the propagation constant using a two-port
VNA, but the most versatile method because of its broadband applicability is
the multiline technique [8]. In this method, multiple lines of different
lengths are measured to sample the traveling wave along the line standards in
a broadband scheme. However, this approach has several drawbacks, including
the need for multiple lines, the possibility of uncertainties in their
geometry, and the requirement for accurate repeated connection or probing, all
of which contribute to measurement uncertainties [9, 10, 11].

To address some of the problems of the multiline method, some techniques have
been introduced, such as the multireflect method [12, 13] and the line-
network-network method [14, 15, 16, 17]. The multireflect method uses multiple
identical reflect standards with different offsets to provide broadband
measurement of the propagation constant. However, because it requires multiple
identical independent standards, it is susceptible to repeatability errors due
to repeated connection or probing. In addition, the propagation constant must
be solved using optimization techniques that could diverge if not well
conditioned. The line-network-network method involves moving an unknown
symmetric and reciprocal network along a transmission line and solving for the
propagation constant using the derived similarity equations. This method has
limitations, such as the restriction to three offsets, which limits the
frequency range, and the requirement to use symmetric and reciprocal offset
networks. Results of relative effective permittivity measurements using this
method were presented in [18], highlighting the sensitivity and limitations of
this solution.

It is noteworthy that there is a significant amount of literature discussing
the broadband measurement of the propagation constant using only two line
standards of varying lengths, commonly known as the line-line method [19, 20,
21, 22]. Despite the different mathematical formulations used, all these
methods are based on solving the characteristic polynomial of the eigenvalue
problem associated with the thru-reflect-line (TRL) calibration [23]. However,
because of the use of only two lines, which often have a significant length
difference to cover lower frequencies, the result of the propagation constant
exhibits multiple resonance peaks, caused by integer multiples of half-
wavelength occurrences in the electrical length of the transmission lines. To
mitigate this issue, some authors have proposed post-processing techniques to
filter the resonance peaks [24, 25].

There are several indirect techniques for determining the propagation constant
of transmission lines, which involve evaluating the permittivity of materials
separately. These methods can be broadly classified into two categories: the
resonant method and the transmission/reflection method. The resonant method,
described in [26, 27], estimates the permittivity from S-parameters at
resonant frequencies, resulting in measurements only at specific frequencies.
In contrast, the transmission/reflection method estimates the permittivity of
a sample placed between two waveguides from the measured transmission and
reflection coefficients. This method can be implemented using various
configurations such as free space, rectangular waveguide, and coaxial line, as
discussed in [28, 29, 30].

In this paper, we present a different approach for measuring the propagation
constant using a single transmission line standard, without the need for prior
calibration of a two-port VNA. Our approach builds on the general idea
introduced in [14] by shifting an unknown network along a transmission line.
Unlike the approach in [14], our method is not constrained by the number of
offsets one can use, nor does the unknown network need to be symmetric or
reciprocal. To combine all offset measurements, we propose a weighted $4\times
4$ eigenvalue problem, inspired by the modified multiline method introduced in
[31]. One of the key advantages of our proposed method is that it only
requires a single transmission line to generate equations similar to those of
the multiline method. Furthermore, highly repeatable measurements are possible
because cable reconnection is not required. The remaining uncertainty is
mainly due to the dimensional motion of the unknown network and the intrinsic
noise of the VNA. We demonstrate the effectiveness of our method on a
commercial slab coaxial airline tuner, where the offset network is the sliding
tuning element. We performed measurements with three different VNA brands. The
measured propagation constant obtained by the different VNAs show overlapping
agreement. Our proposed approach offers a promising alternative to existing
methods for measuring the propagation constant.

The remainder of this paper is structured as follows. In Section 2, we provide
a detailed explanation of the mathematical derivation of the eigenvalue
problem formulation that allows for the adaptation of the multiline method.
Subsequently, in Section 3, we discuss the use of normalized eigenvectors to
extract the complex exponential terms, which contain the propagation constant,
and the utilization of least squares to derive an accurate estimate of the
propagation constant. In Section 4, we describe the experimental setup, where
we perform measurements using various VNAs and present the measured
propagation constant of the slab coaxial airline, as well as a comparison with
EM simulation. Finally, a conclusion is given in Section 5.

##  2 Formulating the eigenvalue problem

The general idea of the measurement setup is to move an unknown network along
a transmission line. For each movement of the network, either to the left or
to the right, we create two offset elements that are complementary to each
other. When the offset length is zero, the offset elements are reduced to a
thru connection, which we refer to as the reference plane. An illustration of
this concept is shown in Figure 1.

Figure 1: Illustration of network offset on a transmission line.

Before proceeding with the mathematical derivation, we need to define the sign
convention for the offset shift. In our analysis, we define that moving the
network to the right results in a positive offset, while moving the network to
the left results in a negative offset. This convention is shown in Figure 2 as
modeled by the error box model of a two-port VNA [32].

Figure 2: VNA two-port error box model of a network offsetted by a length
$l_{i}$ (positive or negative). Each offset results in two complementary
offset boxes. All blocks are given by their T-parameters.

With the definition of the offsets in Figure 2, the measured T-parameters of
the offsetted network by the length $l_{i}$ are given as follows:

$\boldsymbol{M}_{i}=\underbrace{k_{a}k_{b}}_{k}\underbrace{\left[\begin{matrix}a_{11}&a_{12}\\\
a_{21}&1\end{matrix}\right]}_{\boldsymbol{A}}\boldsymbol{L}_{i}\boldsymbol{N}\boldsymbol{L}_{i}^{-1}\underbrace{\left[\begin{matrix}b_{11}&b_{12}\\\
b_{21}&1\end{matrix}\right]}_{\boldsymbol{B}},$ (1)

where $k$, $\boldsymbol{A}$, and $\boldsymbol{B}$ are the error terms of an
uncalibrated two-port VNA. The matrices $\boldsymbol{L}_{i}$ and
$\boldsymbol{N}$ are given as follows:

$\boldsymbol{L}_{i}=\begin{bmatrix}e^{-\gamma l_{i}}&0\\\\[5.0pt] 0&e^{\gamma
l_{i}}\end{bmatrix},\quad\boldsymbol{N}=\begin{bmatrix}\frac{-S_{11}S_{22}+S_{12}S_{21}}{S_{21}}&\frac{S_{11}}{S_{21}}\\\\[5.0pt]
\frac{-S_{22}}{S_{21}}&\frac{1}{S_{21}}\end{bmatrix}.$ (2)

Here, $\gamma$ represents the propagation constant of the transmission line
and $\\{S_{11},S_{12},S_{21},S_{22}\\}$ are the S-parameters of the network
$\boldsymbol{N}$. The S-parameters of the offset network are generally
unknown, and the network can be asymmetric or non-reciprocal. However, the
network must satisfy some basic criteria, which are listed below:

  1. 1.

All S-parameters must be non-zero within the considered frequency range
($|S_{ij}|>0$).

  2. 2.

The S-parameters of the network should not change as the network is moved.

  3. 3.

The network should not lead to the generation of additional modes along the
transmission line.

Although the first condition is unique to our method’s formulation, the
remaining two conditions are also similar to the multiline method [8, 31],
which requires single-mode propagation and repeated error boxes. Fortunately,
it is not difficult to design a system that satisfies these requirements. We
will show this later in a Section 4, where we used a commercial sliding tuner
that was not designed for our application, but met our conditions.

We now define the T-parameters of a new network by taking the difference in
T-parameters of two offset networks of different lengths $l_{i}$ and $l_{j}$,
which is given by

$\displaystyle\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern 1.5mu_{i,j}$
$\displaystyle=\boldsymbol{L}_{i}\boldsymbol{N}\boldsymbol{L}_{i}^{-1}-\boldsymbol{L}_{j}\boldsymbol{N}\boldsymbol{L}_{j}^{-1}$
(3)
$\displaystyle=\nu_{i,j}\begin{bmatrix}0&\frac{S_{11}}{S_{21}}e^{-\gamma(l_{i}+l_{j})}\\\\[5.0pt]
\frac{S_{22}}{S_{21}}e^{\gamma(l_{i}+l_{j})}&0\end{bmatrix},$

where

$\nu_{i,j}=e^{-\gamma(l_{i}-l_{j})}-e^{\gamma(l_{i}-l_{j})}.$ (4)

The expression in (3) is very similar to a line standard in multiline
calibration, but now the line standard is described by an antidiagonal matrix
and with additional multiplication factors. We define an equivalent
measurement of a line standard by

$\mkern 1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu_{i,j}=\boldsymbol{M}_{i}-\boldsymbol{M}_{j}=k\boldsymbol{A}\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu_{i,j}\boldsymbol{B}.$ (5)

Similar to the multiline calibration, we also need an equation that describes
the inverse of the measurements. This is given by

$\widehat{\mkern 1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu}_{i,j}=\boldsymbol{M}_{i}^{-1}-\boldsymbol{M}_{j}^{-1}=\frac{1}{k}\boldsymbol{B}^{-1}\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu}_{i,j}\boldsymbol{A}^{-1},$ (6)

where the matrix $\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern 1.5mu}_{i,j}$ is
given by

$\displaystyle\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern 1.5mu}_{i,j}$
$\displaystyle=\boldsymbol{L}_{i}^{-1}\boldsymbol{N}^{-1}\boldsymbol{L}_{i}-\boldsymbol{L}_{j}^{-1}\boldsymbol{N}^{-1}\boldsymbol{L}_{j}$
(7)
$\displaystyle=-\nu_{i,j}\begin{bmatrix}0&\frac{S_{11}}{S_{12}}e^{-\gamma(l_{i}+l_{j})}\\\\[5.0pt]
\frac{S_{22}}{S_{12}}e^{\gamma(l_{i}+l_{j})}&0\end{bmatrix}.$

Given the expressions in (5) and (6), we can construct an eigenvalue problem
in terms of $\boldsymbol{A}$ as follows:

$\mkern 1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu_{i,j}\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu}_{n,m}=\boldsymbol{A}\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu_{i,j}\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu}_{n,m}\boldsymbol{A}^{-1},$ (8)

where the matrix product $\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu_{i,j}\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern 1.5mu}_{n,m}$ is
given by

$\mkern 1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu_{i,j}\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu}_{n,m}=-\kappa\nu_{i,j}\nu_{n,m}\begin{bmatrix}e^{-\gamma(l_{i,j}^{+}-l_{n,m}^{+})}&0\\\
0&e^{\gamma(l_{i,j}^{+}-l_{n,m}^{+})}\end{bmatrix},$ (9)

with

$\kappa=\frac{S_{11}S_{22}}{S_{21}S_{12}},\qquad
l_{i,j}^{+}=l_{i}+l_{j},\qquad l_{n,m}^{+}=l_{n}+l_{m}.$ (10)

To have a valid eigenvalue problem, we need at least three unique offsets,
where one of the offsets $l_{n}$ or $l_{m}$ can be equal to $l_{i}$ or
$l_{j}$, but $l_{i}\neq l_{j}$, or vice versa. However, with three offsets, we
have three possible pairs of eigenvalue problems. In fact, for $N\geq 3$
offsets, we have $N(N-2)(N^{2}-1)/8$ possible pairs of eigenvalue problems.
This is because for a set of $N$ offsets, we have $N(N-1)/2$ pairs, and when
we create pairs from $N(N-1)/2$ pairs, we substitute the equation into itself,
resulting in $N(N-2)(N^{2}-1)/8$ pairs of pairs.

To address the issue of multiple eigenvalue problems, we refer to our previous
work in [31, 33], where a similar problem was presented in the context of
multiline calibration. This problem was solved by combining all measurements
using a weighting matrix, reducing the problem to solving a single $4\times 4$
eigenvalue problem, regardless of the number of lines. This method not only
reduced the size of the problem, but also allowed us to express both error
boxes $\boldsymbol{A}$ and $\boldsymbol{B}$ simultaneously in a single matrix
using Kronecker product notation. By applying the techniques described in [31,
33], we obtain the following set of equations:

$\displaystyle\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern 1.5mu$
$\displaystyle=k\boldsymbol{X}\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern 1.5mu,$ (11a)
$\displaystyle\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu}^{T}\boldsymbol{P}$ $\displaystyle=\frac{1}{k}\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu}^{T}\boldsymbol{P}\boldsymbol{X}^{-1},$ (11b)

with,

$\displaystyle\boldsymbol{X}$
$\displaystyle=\boldsymbol{B}^{T}\otimes\boldsymbol{A},$ (12a)
$\displaystyle\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern 1.5mu$
$\displaystyle=\begin{bmatrix}\mathrm{vec}\left({\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu_{1,2}}\right)&\cdots&\mathrm{vec}\left({\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu_{i,j}}\right)\end{bmatrix},$ (12b) $\displaystyle\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern 1.5mu$
$\displaystyle=\begin{bmatrix}\mathrm{vec}\left({\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu_{1,2}}\right)&\cdots&\mathrm{vec}\left({\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu_{i,j}}\right)\end{bmatrix},$ (12c) $\displaystyle\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern 1.5mu}$
$\displaystyle=\begin{bmatrix}\mathrm{vec}\left({\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu}_{1,2}}\right)&\cdots&\mathrm{vec}\left({\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu}_{i,j}}\right)\end{bmatrix},$ (12d) $\displaystyle\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern 1.5mu}$
$\displaystyle=\begin{bmatrix}\mathrm{vec}\left({\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu}_{1,2}}\right)&\cdots&\mathrm{vec}\left({\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu}_{i,j}}\right)\end{bmatrix},$ (12e) $\displaystyle\boldsymbol{P}$
$\displaystyle=\begin{bmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{bmatrix},\quad\text{where,
}\boldsymbol{P}=\boldsymbol{P}^{-1}=\boldsymbol{P}^{T}.$ (12f)

Details on the definition and properties of the Kronecker product ($\otimes$)
and the matrix vectorization ($\mathrm{vec}\left({}\right)$) can be found in
the reference [34].

We now formulate the main eigenvalue problem by defining a new matrix
$\boldsymbol{W}$, which we multiply on the right side of (11a). We call this
matrix the weighting matrix. In the next step, we construct the weighted
eigenvalue problem by multiplying the new equation on the left side of (11b).
This results in

$\underbrace{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu\boldsymbol{W}\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu}^{T}\boldsymbol{P}}_{\boldsymbol{F}}=\boldsymbol{X}\underbrace{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu\boldsymbol{W}\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu}^{T}\boldsymbol{P}}_{\boldsymbol{H}}\boldsymbol{X}^{-1}.$ (13)

The expression presented in (13) represents a similarity problem between the
matrices $\boldsymbol{F}$ and $\boldsymbol{H}$, with $\boldsymbol{X}$ as the
transformation matrix. The purpose of introducing the weighting matrix
$\boldsymbol{W}$ is to transform this similarity problem into an eigenvalue
problem by forcing $\boldsymbol{H}$ into a diagonal form. It turns out that if
$\boldsymbol{W}$ is any non-zero skew-symmetric matrix, then $\boldsymbol{H}$
takes a diagonal form [31]. However, we do not only want to diagonalize
$\boldsymbol{H}$, but also want to maximize the distance between the
eigenvalues, which in turn minimizes the sensitivity in the eigenvectors [35].
For multiline calibration, the optimal form of $\boldsymbol{W}$ was derived in
[31], and since the formulation in (13) is similar to that discussed in [31],
we use the same choice of $\boldsymbol{W}$ with some scaling modifications.
The optimal weighting matrix $\boldsymbol{W}$ can be written as follows,
taking into account the scaling factors:

$\boldsymbol{W}^{H}=-\kappa(\boldsymbol{z}\boldsymbol{y}^{T}-\boldsymbol{y}\boldsymbol{z}^{T}),$
(14)

where

$\displaystyle\boldsymbol{y}^{T}$
$\displaystyle=\begin{bmatrix}\nu_{1,2}e^{\gamma
l_{1,2}^{+}}&\ldots&\nu_{i,j}e^{\gamma l_{i,j}^{+}}\end{bmatrix},$ (15a)
$\displaystyle\boldsymbol{z}^{T}$
$\displaystyle=\begin{bmatrix}\nu_{1,2}e^{-\gamma
l_{1,2}^{+}}&\ldots&\nu_{i,j}e^{-\gamma l_{i,j}^{+}}\end{bmatrix}.$ (15b)

As a result of choosing $\boldsymbol{W}$ as defined in (14), the expression in
(13) takes an eigendecomposition form, as given below.

$\boldsymbol{F}=\boldsymbol{X}\begin{bmatrix}0&0&0&0\\\ 0&\lambda&0&0\\\
0&0&-\lambda&0\\\ 0&0&0&0\end{bmatrix}\boldsymbol{X}^{-1},$ (16)

where $\lambda$ is real-valued and proportional to the square Frobenius norm
of the matrix $\boldsymbol{W}$, given by

$\lambda=\frac{1}{2}\left\|\boldsymbol{W}\right\|_{F}^{2}=\frac{1}{2}\sum_{i,j}|w_{i,j}|^{2}.$
(17)

There are two ways to compute $\boldsymbol{W}$, the first is the direct method
where we already know the propagation constant $\gamma$ and the factor
$\kappa$ which describes the unknown network. Naturally, the first option is
not practical since both $\gamma$ and $\kappa$ are unknown. The better option
is to apply a rank-2 Takagi decomposition to the left side of the following
equation, as described in [33] for multiline calibration.

$\underbrace{\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu}^{T}\boldsymbol{P}\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{M}\mkern-1.5mu}\mkern
1.5mu}_{\text{measurement}}=\underbrace{\widehat{\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu}^{T}\boldsymbol{P}\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{N}\mkern-1.5mu}\mkern
1.5mu}_{\text{model}}.$ (18)

Note that the left side of (18) contains only the measurement data, while the
right side describes the model. Also, the error boxes are not present in (18).
To determine $\boldsymbol{W}$, we need to calculate the rank-2 Takagi
decomposition. This is done in two steps. First, we compute the rank-2 of (18)
via signular value decomposition (SVD), and then we apply the Takagi
decomposition to decompose the matrix into its symmetric basis [36]. This
looks as follows:

$\widehat{\overline{\boldsymbol{N}}}^{T}\boldsymbol{P}\overline{\boldsymbol{N}}=\underbrace{s_{1}\boldsymbol{u}_{1}\boldsymbol{v}_{1}^{H}+s_{2}\boldsymbol{u}_{2}\boldsymbol{v}_{2}^{H}}_{\text{rank-2
SVD from
measurement}}=\underbrace{\boldsymbol{G}\boldsymbol{G}^{T}}_{\text{Takagi}}$
(19)

Then, the weighting matrix is given by

$\boldsymbol{W}^{H}=\pm\boldsymbol{G}\begin{bmatrix}0&j\\\
-j&0\end{bmatrix}\boldsymbol{G}^{T}$ (20)

The derivation process of the matrix $\boldsymbol{W}$ is described in more
detail in [33]. To resolve the sign ambiguity, one approach is to select the
answer that has the smallest Euclidean distance to a known estimate. Such as
an estimate can be obtained from an approximate knowledge of the material
properties of the transmission line.

The last step is the solution of the eigenvectors described by
$\boldsymbol{X}$ in (16). The solution of the eigenvectors has been discussed
in [31]. It is worth noting that we cannot solve the matrix $\boldsymbol{X}$
uniquely, but only up to a diagonal matrix multiplication. Therefore, to
define a unique solution for $\boldsymbol{X}$, we normalize its columns so
that the diagonal elements are equal to one. This is written as follows:

$\widetilde{\boldsymbol{X}}=\boldsymbol{X}\mathrm{diag}(a_{11}b_{11},b_{11},a_{11},1)^{-1},$
(21)

where $a_{11}$ and $b_{11}$ are part of the error boxes $\boldsymbol{A}$ and
$\boldsymbol{B}$ (see (1) and (12a)).

##  3 Least squares solution for the propagation constant

Knowing $\widetilde{\boldsymbol{X}}$ from the eigenvector solutions, we can
extract the complex exponential terms that contain the propagation constant.
To do this, we first multiply the inverse of the normalized error terms to all
vectorized measurements of the offset network. This is given by

$\boldsymbol{E}=\widetilde{\boldsymbol{X}}^{-1}\boldsymbol{M}=\mathrm{diag}(ka_{11}b_{11},kb_{11},ka_{11},1)\boldsymbol{N}^{\prime},$
(22)

where

$\displaystyle\boldsymbol{M}$
$\displaystyle=\begin{bmatrix}\mathrm{vec}\left({\boldsymbol{M}_{1}}\right)&\cdots&\mathrm{vec}\left({\boldsymbol{M}_{N}}\right)\end{bmatrix},$
(23a) $\displaystyle\boldsymbol{N}^{\prime}$
$\displaystyle=\begin{bmatrix}\mathrm{vec}\left({\boldsymbol{L}_{1}\boldsymbol{N}\boldsymbol{L}_{1}^{-1}}\right)&\cdots&\mathrm{vec}\left({\boldsymbol{L}_{N}\boldsymbol{N}\boldsymbol{L}_{N}^{-1}}\right)\end{bmatrix}.$
(23b)

Since we do not know the remaining error terms $k,a_{11},b_{11}$, as well as
the S-parameters of the network $\boldsymbol{N}$, we need to choose a
reference offset to eliminate these unknown factors. For simplicity, we choose
the first offset, which we define as zero, i.e., $l_{1}=0$ (any other choice
is also valid). As a result, the positive and negative complex exponential
terms are given as follows, using indexing notation based on Python.

$\displaystyle\boldsymbol{E}[1,1:]/\boldsymbol{E}[1,0]=$
$\displaystyle\begin{bmatrix}e^{2\gamma l_{2}}&e^{2\gamma
l_{3}}&\ldots&e^{2\gamma l_{N}}\end{bmatrix},$ (24a)
$\displaystyle\boldsymbol{E}[2,1:]/\boldsymbol{E}[2,0]=$
$\displaystyle\begin{bmatrix}e^{-2\gamma l_{2}}&e^{-2\gamma
l_{3}}&\ldots&e^{-2\gamma l_{N}}\end{bmatrix}.$ (24b)

Now that we have the complex exponential terms, we can extract the exponents
using the complex logarithm function and determine $\gamma$ using the least
squares method, while taking care of any phase unwrapping. First, since we
have both the positive and negative complex exponential terms, we can account
for both by averaging them. This is done by defining a new vector
$\boldsymbol{\tau}$:

$\boldsymbol{\tau}=\begin{bmatrix}\frac{e^{2\gamma l_{2}}+1/e^{-2\gamma
l_{2}}}{2}&\cdots&\frac{e^{2\gamma l_{N}}+1/e^{-2\gamma
l_{N}}}{2}\end{bmatrix}^{T}.$ (25)

The next step is to calculate the logarithm to extract the exponents, which is
given by

$\boldsymbol{\phi}=\log\left(\boldsymbol{\tau}\right)+j2\pi\boldsymbol{n},\quad\text{where,
}\boldsymbol{n}\in\mathbb{Z}^{N-1}.$ (26)

The phase unwrapping factor $\boldsymbol{n}$ can be estimated by rounding the
difference between $\boldsymbol{\phi}$ and an estimated value. This is given
by

$\boldsymbol{n}=\mathrm{round}\left(\frac{\operatorname{Im}\left({\boldsymbol{\phi}}\right)-2\gamma_{\mathrm{est}}\boldsymbol{l}}{2\pi}\right),$
(27)

where $\gamma_{\mathrm{est}}$ is a known approximation for $\gamma$, and
$\boldsymbol{l}$ is a vector containing all length offsets except the
reference zero offset. The initial estimate for $\gamma_{\mathrm{est}}$ can be
derived from the material properties of the transmission line.

Finally, we can determine $\gamma$ through weighted least squares [37],

$\gamma=\frac{\boldsymbol{l}^{T}\boldsymbol{V}^{-1}\boldsymbol{\phi}}{\boldsymbol{l}^{T}\boldsymbol{V}^{-1}\boldsymbol{l}},$
(28)

where $\boldsymbol{V}^{-1}$ is given by

$\boldsymbol{V}^{-1}=\boldsymbol{I}_{(N-1)\times(N-1)}-\frac{1}{N}\boldsymbol{1}_{N-1}\boldsymbol{1}^{T}_{N-1},$
(29)

The matrix $\boldsymbol{I}$ is the identity matrix and $\boldsymbol{1}$ is a
vector of ones. The weighting matrix $\boldsymbol{V}^{-1}$ is necessary
because each measurement has a common reference, which is $l_{1}$. Therefore,
the correlation between the measurements had to be taken into account by the
matrix $\boldsymbol{V}^{-1}$ [37].

Figure 3 summarizes the mathematical derivation presented in this and the
previous sections, and provides a visual representation of the steps taken to
compute the propagation constant.

Figure 3: Block diagram summary of the proposed propagation constant
measurement method. The matrix $\boldsymbol{M}$ contains the T-parameter
measurements of all offsets. The vector $\boldsymbol{l}$ contains the relative
length of the offsets with respect to the reference offset (i.e., the zero
offset).

##  4 Experiment and Discussion

###  4.1 Measurement setup

For demonstration purposes, we used the slide screw tuner 8045P from Maury
Microwave as an implementation of the offset network, where the transmission
line is a slab coaxial airline that supports frequencies up to 18 GHz. The
tuner is depicted in Figure 4.

Figure 4: Maury Microwave 8045P tuner. The cross-section dimensions of the
airline are given as follows: $w=9.398\,\mathrm{mm}$, $h=40.691\,\mathrm{mm}$,
$d=3.040\,\mathrm{mm}$, and $p=2.778\,\mathrm{mm}$.

For our method to work, we require that the unknown network (i.e., the tuner
element) be both reflective and transmissive, as the factor $\kappa$ in (9)
can explode to infinity if the network is only reflective, and can be zero if
the network is only transmissive. Ideally, we want $\kappa=1$ to minimize its
effect on the eigenvalue problem. However, we also want to avoid scenarios
where the network causes the generation of additional modes or resonances.
Therefore, we adjusted the tuner with an already calibrated VNA to tune the
network to a desired response, as shown in Figure 5. It should be noted that
this step of tuning the tuner with an existing calibrated VNA is only
necessary because the tuner is a commercial product designed for circuit
matching applications and not for our purpose. If we were designing the
network ourselves, we would not need to measure it with a calibrated VNA
because we would have already designed it to meet our frequency
specifications. Also, the S-parameters of the network are never explicitly
used in the derivation of the propagation constant.

Figure 5: Calibrated measurement of the tuner after tuning. The highlighted
frequency range below 3 GHz is not usable due to small reflection and
resonance.

As shown in Figure 5, we set the lower frequency to 3 GHz to avoid very low
return loss and resonances. We then measured the airline using different
uncalibrated VNAs. This was done to demonstrate that even if we changed the
measurement system, we would still get consistent results because the error
boxes would not be affected by uncertainties caused by connector and cable
movement. For the offset lengths, we chose
$\\{0,21,66,81,84,93,117,123,171,192\\}\,\mathrm{mm}$, which ensures that the
eigenvalue $\lambda$ in (16) does not reach zero in the target frequency
range.

The VNAs used for the measurements are: Anritsu VectorStar, R&S ZNA and
Keysight ENA. The ENA is limited to 14 GHz. All VNAs were placed in the same
room to provide the same room conditions. The power level and IF bandwidth for
all VNAs were set to 0 dBm and 100 Hz, respectively. Due to the low loss of
the airline, an average measurement of 50 frequency sweeps was calculated to
reduce noise. Pictures of the three instruments are shown in Figure 6.

(a)

(b)

(c)

Figure 6: The VNAs used for the measurements. (a) Rohde & Schwarz ZNA, (b)
Anritsu VectorStar, and (c) Keysight ENA.

###  4.2 Results and discussion

All measurements of the different offsets were taken without prior calibration
of the VNAs. The collected data is then read in Python using the scikit-rf
package [38]. In Figure 6 we show the measured magnitude response of $S_{11}$
and $S_{21}$ from all three VNAs for the offset $123\,\mathrm{mm}$. From the
figure, we can see that all three VNAs give different responses because the
error boxes are different for each VNA.

Figure 7: Raw measurements from the three VNAs of the magnitude response of
$S_{11}$ and $S_{21}$ of the 8045P tuner, at an offset location of
$123\,\mathrm{mm}$.

After collecting all raw measurements for all the offsets and from all the
VNAs, the data were processed to extract the propagation constant according to
the discussion in Sections 3 and 4. For easier and better interpretation of
the extracted propagation constant, we have plotted in Figure 8 the real part
of the relative effective permittivity and the loss per unit length of the
slab coaxial airline from all three VNA measurements. The real part of the
relative effective permittivity and the loss per unit length are calculated
from the propagation constant as follows:

$\epsilon_{\mathrm{r,eff}}^{\prime}=-\operatorname{Re}\left({\left(\frac{c_{0}\gamma}{2\pi
f}\right)^{2}}\right)\quad(\text{Unitless}),\qquad\text{loss}=\frac{20\times
10^{-2}}{\ln 10}\operatorname{Re}\left({\gamma}\right)\quad(\text{dB/cm}),$
(30)

where $c_{0}$ is the speed of light in vacuum and $f$ is the frequency.

The relative effective permittivity and loss per unit length results presented
in Figure 8 show clear agreement between all VNA measurements, demonstrating
the high repeatability of the proposed method even when using different VNA
setups.

Figure 8: Extracted measurement of relative effective permittivity and loss
per unit length of the slab coaxial airline, as well as EM simulated results
for an anodic coating of $15\,\mu\mathrm{m}$ and inner conductor conductivity
of $35\%$ IACS.

We also performed an EM simulation with the dimensional parameters of the
airline given in Figure 4. Unfortunately, we did not have information on the
metal types of the inner and outer conductors. From the appearance of the
inner conductor, we think it is made of some kind of brass. For the ground
plates, we think they are made of aluminum because they have a black anodized
coating, which is typical for aluminum components. The anodized layer is often
based on aluminum oxide and typically has a relative permittivity of $8.3$
[39]. Since the thickness of the oxide layer and the exact conductivity of
brass are unknown, we ran some values for the thickness of the anodic layer
and the conductivity of brass. We found that a coating thickness of
$15\,\mu\mathrm{m}$ and a relative conductivity of $35\%$ IACS (International
Annealed Copper Standard) overlap with the measurement shown in Figure 8. The
value obtained for the thickness of the anodic layer is quite typical to
obtain a dark black coating [40]. The conductivity of the inner conductor of
$35\%$ IACS (=20.3 MS/m) is within the range of common brass types [41].

The purpose of the simulation is to show that the results obtained from the
proposed method of measuring the propagation constant do indeed translate into
realistic properties of the transmission line. In fact, with the proposed
method, one could characterize materials in reverse, as in our case, the
conductivity of the metal.

Another aspect that may be of interest is the quality of the extracted
propagation constant by varying the length and number of offsets. In the
results shown in Figure 8 we used 10 offsets ranging from 0 to 192 mm. Now we
consider different cases. These cases are listed in Table 1.

Table 1: Considered cases of different offset lengths. Cases | Offset lengths (mm)
---|---
Case 1 |  $0,21,81$
Case 2 |  $0,21,192$
Case 3 |  $0,21,66,117,192$
Case 4 |  $0,21,81,93,117,123,192$
All offsets |  $0,21,66,81,84,93,117,123,171,192$

In Figure 9 we show the results of the relative effective permittivity and the
loss per unit length of the slab coaxial airline from the VectorStar VNA
measurements for all the cases mentioned in Table 1. Cases 1 and 2 show the
results when only three offsets are considered. Case 2 differs from Case 1 in
that we have replaced the last offset with a much longer offset. The results
of both cases 1 and 2 are poor and show multiple resonances. For case 2 we see
more resonances than for case 1. This is the result of the eigenvalue crossing
zero at multiple frequencies (see Figure 10). In case 3 we spread the offsets
further to include 5 offsets. We can see a clear improvement over cases 1 and
2. We can further improve the accuracy of the extracted relative effective
permittivity and loss per unit length by further spreading the offsets as in
case 4, where we use 7 offsets. In case 4, we obtain results of similar
accuracy to the case of using all 10 offset lengths.

Figure 9: Extracted measurement of relative effective permittivity and loss
per unit length of the slab coaxial airline for various combinations of
offsets from the VectorStar VNA measurements.

The quality of the extracted propagation constant depends on the eigenvalue
$\lambda$ as defined in (16). As the eigenvalue approaches zero, the
eigenvectors become more sensitive, which in turn affects the calculation of
the extracted propagation constant. To visualize the differences between
different scenarios, we present a scaled representation of the eigenvalue
$\lambda$ for each case. This scaled representation excludes the influence of
the network through the common factor $\kappa$, which is invariant over all
offset lengths. Since $\kappa>0$ was established earlier, variations in the
eigenvalues can only be induced by the choice of offset lengths. Accordingly,
we define the normalized version of the eigenvalue by dividing it by the
absolute value of $\kappa$, as shown below:

$\lambda=\frac{1}{2}\left\|\boldsymbol{W}\right\|_{F}^{2}=\frac{|\kappa|^{2}}{2}\left\|\boldsymbol{z}\boldsymbol{y}^{T}-\boldsymbol{y}\boldsymbol{z}^{T}\right\|_{F}^{2}\quad\Longrightarrow\quad\lambda^{\prime}=\frac{\lambda}{|\kappa|^{2}}=\frac{1}{2}\left\|\boldsymbol{z}\boldsymbol{y}^{T}-\boldsymbol{y}\boldsymbol{z}^{T}\right\|_{F}^{2}$
(31)

In Figure 10, we present a plot of the scaled eigenvalue normalized to its
maximum value, which facilitates a consistent comparison as the number of
offsets varies. As illustrated in the figure, for cases 1 and 2, the
eigenvalue exhibits multiple zero crossings at various frequencies. Similarly,
in case 3, the eigenvalue approaches zero at several instances, although to a
lesser extent than in cases 1 and 2. In contrast, in case 4, the eigenvalue
never reaches zero, but attains values closer to zero at specific frequencies
than when all 10 offsets are utilized. Ideally, a flat eigenvalue over
frequency would be preferred, but this would necessitate employing even more
offsets. This is not different from the multiline calibration approach
proposed in [31], where a finer spacing between lines results in a flatter
eigenvalue over frequency. Therefore, utilizing a broader range of offset
lengths is highly advantageous for enhancing the accuracy of results across
frequency. It is also noteworthy that the eigenvalue possesses a bandpass
characteristic, whereby the lowest and highest frequency limits are bound by
the largest and smallest relative offset, respectively.

Figure 10: The scaled eigenvalue normalized to its maximum for the invested
offset length cases.

For comparison, it is worth noting that the multiline method necessitates the
measurement of multiple line standards of different lengths, a process that
can introduce errors due to connector repeatability. Achieving high
repeatability in this context poses a significant mechanical challenge,
especially concerning connectors, and automating this process represents an
even greater hurdle. In contrast, our proposed method eliminates the need for
physical contact between the sliding element and the transmission line.
Furthermore, although the sliding process is performed manually in the example
presented, it could be automated by employing a linear actuator, thus
eliminating the need for any user interaction with the measurement system.

##  5 Conclusion

We presented a new broadband method for measuring the propagation constant of
transmission lines that does not require the prior calibration of a two-port
VNA or the use of multiple line standards. This method provides accurate
results by emulating the use of multiple line standards through sweeping an
unknown network along a transmission line. The shifted network does not have
to be symmetric or reciprocal, but it must exhibit both transmission and
reflection properties and remain invariant when moved along the line. The
experimental results obtained using different VNAs on a slab coaxial airline
with a slider tuner showed consistent agreement with each other and with EM
simulation.

One of the significant advantages of the proposed method is that it uses the
same eigenvalue formulation as multiline calibration, but without the need for
disconnecting or moving the cables. As a result, it eliminates errors related
to connector repeatability and provides improved broadband traceability to the
SI units. Moreover, since the offsets are implemented by simply moving the
unknown network laterally, the process can be easily automated using an
automated linear actuator. Therefore, the proposed method can accurately
measure the propagation constant without requiring any physical interaction
from the user on the measurement system.

## Acknowledgment

The financial support by the Austrian Federal Ministry for Digital and
Economic Affairs and the National Foundation for Research, Technology, and
Development is gratefully acknowledged. The authors also thank ebsCENTER for
lending their Anritsu and Keysight VNAs, and Maury Microwave for their support
in providing the airline cross-section dimensions of 8045P tuner.

## References

  * [1] K. Orend, C. Baer, and T. Musch, “A compact measurement setup for material characterization in w-band based on dielectric waveguides,” _Sensors_ , vol. 22, no. 16, 2022, doi: 10.3390/s22165972.
  * [2] Z. Hatab, M. Gadringer, M. Habib, and W. Bösch, “mm-wave complex permittivity extraction of ltcc substrate under the influence of surface roughness,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 71, pp. 1–11, 2022, doi: 10.1109/TIM.2022.3152319.
  * [3] X. Liu and J. Yu, “Characterization of the dielectric properties of water and methanol in the d-band using a quasi-optical spectroscopy,” _Scientific Reports_ , vol. 9, no. 1, p. 18562, Dec 2019, doi: 10.1038/s41598-019-55126-6.
  * [4] J. Roelvink, S. Trabelsi, and S. O. Nelson, “Determining complex permittivity from propagation constant measurements with planar transmission lines,” _Measurement Science and Technology_ , vol. 24, no. 10, p. 105001, aug 2013, doi: 10.1088/0957-0233/24/10/105001.
  * [5] R. Marks and D. Williams, “Characteristic impedance determination using propagation constant measurement,” _IEEE Microwave and Guided Wave Letters_ , vol. 1, no. 6, pp. 141–143, 1991, doi: 10.1109/75.91092.
  * [6] E. Bogatin, _Bogatin’s Practical Guide to Transmission Line Design and Characterization for Signal Integrity Applications_. Artech House, 2020. ISBN 9781630816926
  * [7] J. Zhang, J. L. Drewniak, D. J. Pommerenke, M. Y. Koledintseva, R. E. DuBroff, W. Cheng, Z. Yang, Q. B. Chen, and A. Orlandi, “Causal rlgc($f$ ) models for transmission lines from measured $s$ -parameters,” _IEEE Transactions on Electromagnetic Compatibility_ , vol. 52, no. 1, pp. 189–198, 2010, doi: 10.1109/TEMC.2009.2035055.
  * [8] R. Marks, “A multiline method of network analyzer calibration,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 39, no. 7, pp. 1205–1215, 1991, doi: 10.1109/22.85388.
  * [9] D. DeGroot, D. Walker, and R. Marks, “Impedance mismatch effects on propagation constant measurements.” Proc., 5th Topical Mtg. Electrical Performance of Electronic Packaging, Napa, CA, 1996-10-01 1996, doi: 10.1109/EPEP.1996.564810.
  * [10] R. F. Kaiser and D. F. Williams, “Sources of error in coplanar-waveguide trl calibrations,” in _54th ARFTG Conference Digest_ , vol. 36, 2000, doi: 10.1109/ARFTG.1999.327367. pp. 1–6.
  * [11] K. Wong and J. Hoffmann, “Improving vna measurement accuracy by including connector effects in the models of calibration standards,” in _82nd ARFTG Microwave Measurement Conference_ , 2013, doi: 10.1109/ARFTG-2.2013.6737334. pp. 1–7.
  * [12] A. Lewandowski, W. Wiatr, D. Gu, N. D. Orloff, and J. Booth, “A multireflect-thru method of vector network analyzer calibration,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 65, no. 3, pp. 905–915, 2017, doi: 10.1109/TMTT.2016.2627036.
  * [13] N. B. Popovic, A. M. Hagerstrom, J. A. Drisko, J. C. Booth, E. J. Garboczi, C. J. Long, and N. D. Orloff, “Materials characterization with multiple offset reflects at frequencies to 110 ghz,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 68, no. 1, pp. 184–195, 2020, doi: 10.1109/TMTT.2019.2950023.
  * [14] H. Heuermann and B. Schiek, “Line network network (LNN): an alternative in-fixture calibration procedure,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 45, no. 3, pp. 408–413, 1997, doi: 10.1109/22.563340.
  * [15] I. Rolfes and B. Schiek, “Self-calibration procedures for vector network analyzers on the basis of reflection standards,” in _33rd European Microwave Conference Proceedings (IEEE Cat. No.03EX723C)_ , vol. 1, 2003, doi: 10.1109/EUMC.2003.1262226. pp. 93–96 Vol.1.
  * [16] T. Zhang, H. Gu, and W. Zhao, “A reformulation of lnn calibration,” in _2022 IEEE 9th International Symposium on Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications (MAPE)_ , 2022, doi: 10.1109/MAPE53743.2022.9935187. pp. 20–23.
  * [17] C. Orlob, T. Reinecke, E. Denicke, B. Geck, and I. Rolfes, “Compact unfocused antenna setup for x-band free-space dielectric measurements based on line-network-network calibration method,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 62, no. 7, pp. 1982–1989, 2013, doi: 10.1109/TIM.2013.2246905.
  * [18] P. M. Narayanan, “Microstrip transmission line method for broadband permittivity measurement of dielectric substrates,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 62, no. 11, pp. 2784–2790, 2014, doi: 10.1109/TMTT.2014.2354595.
  * [19] J. Reynoso-Hernandez, “Unified method for determining the complex propagation constant of reflecting and nonreflecting transmission lines,” _IEEE Microwave and Wireless Components Letters_ , vol. 13, no. 8, pp. 351–353, 2003, doi: 10.1109/LMWC.2003.815695.
  * [20] A. Koul, P. K. R. Anmula, M. Y. Koledintseva, J. L. Drewniak, and S. Hinaga, “Improved technique for extracting parameters of low-loss dielectrics on printed circuit boards,” in _2009 IEEE International Symposium on Electromagnetic Compatibility_ , 2009, doi: 10.1109/ISEMC.2009.5284646. pp. 191–196.
  * [21] K.-F. Fuh, “Broadband continuous extraction of complex propagation constants in methods using two-line measurements,” _IEEE Microwave and Wireless Components Letters_ , vol. 23, no. 12, pp. 671–673, 2013, doi: 10.1109/LMWC.2013.2283852.
  * [22] U. C. Hasar, “Propagation constant measurement of microwave networks with symmetric/asymmetric reflections,” _IEEE Sensors Journal_ , vol. 18, no. 12, pp. 4940–4946, 2018, doi: 10.1109/JSEN.2018.2829123.
  * [23] G. Engen and C. Hoer, “Thru-reflect-line: An improved technique for calibrating the dual six-port automatic network analyzer,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 27, no. 12, pp. 987–993, 1979, doi: 10.1109/TMTT.1979.1129778.
  * [24] K.-F. Fuh, B.-J. Tsai, A. Liu, and E. Liao, “Wideband extraction of dk and df of printed circuit boards using two transmission lines with sufficient length difference,” in _2012 7th International Microsystems, Packaging, Assembly and Circuits Technology Conference (IMPACT)_ , 2012, doi: 10.1109/IMPACT.2012.6420309. pp. 287–290.
  * [25] Y. Rodríguez-Velásquez, S. C. Sejas-García, and R. Torres-Torres, “A method of differences for determining the propagation constant from multiline measurements,” _IEEE Microwave and Wireless Components Letters_ , vol. 30, no. 3, pp. 300–303, 2020, doi: 10.1109/LMWC.2020.2968236.
  * [26] M. A. Stuchly and S. S. Stuchly, “Coaxial line reflection methods for measuring dielectric properties of biological substances at radio and microwave frequencies-a review,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 29, no. 3, pp. 176–183, Sep. 1980, doi: 10.1109/TIM.1980.4314902.
  * [27] Y. Kato and M. Horibe, “Broadband permittivity measurements up to 170-ghz using balanced-type circular-disk resonator excited by 0.8-mm coaxial line,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 68, no. 6, pp. 1796–1805, June 2019, doi: 10.1109/TIM.2018.2886864.
  * [28] A. Kazemipour, M. Hudlička, S.-K. Yee, M. A. Salhi, D. Allal, T. Kleine-Ostmann, and T. Schrader, “Design and calibration of a compact quasi-optical system for material characterization in millimeter/submillimeter wave domain,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 64, no. 6, pp. 1438–1445, June 2015, doi: 10.1109/TIM.2014.2376115.
  * [29] J. Baker-Jarvis, E. Vanzura, and W. Kissick, “Improved technique for determining complex permittivity with the transmission/reflection method,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 38, no. 8, pp. 1096–1103, Aug 1990, doi: 10.1109/22.57336.
  * [30] K. Dudeck and L. Buckley, “Dielectric material measurement of thin samples at millimeter wavelengths,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 41, no. 5, pp. 723–725, Oct 1992, doi: 10.1109/19.177352.
  * [31] Z. Hatab, M. Gadringer, and W. Bösch, “Improving the reliability of the multiline trl calibration algorithm,” in _2022 98th ARFTG Microwave Measurement Conference (ARFTG)_ , 2022, doi: 10.1109/ARFTG52954.2022.9844064. pp. 1–5.
  * [32] R. B. Marks, “Formulations of the basic vector network analyzer error model including switch-terms,” in _50th ARFTG Conference Digest_ , vol. 32, 1997, doi: 10.1109/ARFTG.1997.327265. pp. 115–126.
  * [33] Z. Hatab, M. E. Gadringer, and W. Bösch, “Propagation of linear uncertainties through multiline thru-reflect-line calibration,” 2023, doi: 10.48550/arXiv.2301.09126.
  * [34] J. Brewer, “Kronecker products and matrix calculus in system theory,” _IEEE Transactions on Circuits and Systems_ , vol. 25, no. 9, pp. 772–781, 1978, doi: 10.1109/TCS.1978.1084534.
  * [35] J. H. Wilkinson, _The Algebraic Eigenvalue Problem_. USA: Oxford University Press, Inc., 1988. ISBN 0198534183
  * [36] A. M. Chebotarev and A. E. Teretenkov, “Singular value decomposition for the takagi factorization of symmetric matrices,” _Applied Mathematics and Computation_ , vol. 234, pp. 380–384, 2014, doi: https://doi.org/10.1016/j.amc.2014.01.170.
  * [37] D. DeGroot, J. Jargon, and R. Marks, “Multiline trl revealed,” in _60th ARFTG Conference Digest, Fall 2002._ , 2002, doi: 10.1109/ARFTGF.2002.1218696. pp. 131–155.
  * [38] A. Arsenovic, J. Hillairet, J. Anderson, H. Forstén, V. Rieß, M. Eller, N. Sauber, R. Weikle, W. Barnhart, and F. Forstmayr, “scikit-rf: An open source python package for microwave network creation, analysis, and calibration [speaker’s corner],” _IEEE Microwave Magazine_ , vol. 23, no. 1, pp. 98–105, 2022, doi: 10.1109/MMM.2021.3117139.
  * [39] F. Argall and A. Jonscher, “Dielectric properties of thin films of aluminium oxide and silicon oxide,” _Thin Solid Films_ , vol. 2, no. 3, pp. 185–210, 1968, doi: 10.1016/0040-6090(68)90002-3.
  * [40] V. F. Henley, in _Anodic Oxidation of Aluminium and its Alloys_ , ser. The Pergamon Materials Engineering Practice Series, V. F. HENLEY, Ed. Pergamon, 1982.
  * [41] J. Davis, _Copper and Copper Alloys_ , ser. ASM Handbooks Series. ASM International, 2001. ISBN 9780871707260