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Experimental Validation of Sequence-Wise Predistorter for Evaluation of
Geometrically Shaped 128-QAM

Menno van den Hout, Sjoerd van der Heide, Sebastiaan Goossens, and Chigo
Okonkwo

High-Capacity Optical Transmission Laboratory, Eindhoven University of
Technology,

PO Box 513, 5600 MB, Eindhoven, the Netherlands.

<EMAIL_ADDRESS>
###### Abstract

A predistorter for transmitter nonlinearities is applied to the evaluation of
a geometrically shaped constellation, such that constellation points are
transmitted correctly during the evaluation of the geometrically shaped
constellation.

##  1 Introduction

As the need for higher per channel capacity continues to grow in optical
transmission systems, baud rate and constellation cardinality are increasing.
Hence more stringent requirements are placed on the transmitter in terms of
bandwidth and linearity of the driver amplifiers and Mach-Zehnder modulator
[1, 2]. Recently, constellation shaping via geometric shaping (GS) has been
demonstrated to increase the capacity per channel even further [3]. For GS ,
the precise transmission of constellation points is especially important,
since GS constellation points are chosen carefully such that they result in a
Gaussian-like profile. Various methods have been proposed to overcome
transmitter limitations. A precompensation technique for low electrical
bandwidth[4] and a Volterra series based nonlinear predistortion method [1]
have been presented, as well as a less complex memory polynomial based
predistorter[5]. In [6], a more easy to implement lookup table (LUT) based
method was proposed.

In this work, a sequence-wise (SW) predistortion method is demonstrated for
the evaluation of GS-128[]*128[1]/̄QAM and is compared to a LUT based method
proposed and demonstrated in [6]. For evaluation of GS constellations,
mitigation of the constellation point shift due to nonlinearities is
important, as otherwise the designed GS constellation is not optimally tested.
Hence, the maximum performance might not be achieved. Significant performance
increase over only linear precompensation is shown, and the SW method is shown
to outperform the LUT predistortion for practical table sizes. Due to the SW
nature of this method, the predistorter needs to be trained for every
transmitted sequence and hence, it is noted that technique is not suitable for
application to real-world transmission systems. However, for constellation
design and transmission evaluation, this remains a valuable tool, since
transmitter impairments will have less influence during experimental
validations.

(a)

(b)

(c)

Fig. 1: 1(a) Experimental setup 1(b) Original and received constellation
points. 1(c) Simulated NGMI vs. SNR

##  2 Sequence-wise predistortion

The SW predistorter is based on [6]. There, a LUT is created for each
dimension (i.e. XI, XQ, YI, YQ) where a pulse-amplitude modulation signal is
sent. The LUT contains the amplitude errors for the center symbol of each
possible symbol pattern with length $n$. The LUT is filled by transmitting and
receiving a pseudorandom bit sequence (PRBS) in an optical back-to-back (OBTB)
setup as described in Section 3, downsampling the resulting signal to 1 sample
per symbol, identifying the $n$-length patterns, and calculating the errors
for the center symbol. If a pattern occurs more than once, the error is
averaged. Increasing the LUT pattern length $n$, the amount of memory effects
the LUT can capture increases, at the cost larger table sizes. To apply the
symbol-wise LUT at the transmitter, the amplitude of each symbol is corrected
using the pattern-dependent error stored in the LUT .

For the sequence-wise (SW) predistorter, the pattern length $n$ is set equal
to the length of the transmit sequence, justifying the name of the
predistorter. The LUT will be able to capture memory effects as long as the
transmit sequence and will contain as many non-zero entries as there are
symbols in the transmit sequence. However, a certain LUT is only valid for the
same sequence as it was trained on. Using the same LUT structure as in [6]
would not be feasible, as the LUT size is given by the number of possible
symbol patterns $L^{n}$, where $L$ is the number of output amplitudes in a
single dimension. Note that as each pattern in the LUT occurs only once, the
errors for the SW method can be stored in a LUT with the same size as the
transmit sequence. Since each pattern occurs only once in the transmitted
sequence, the noise is not averaged out as for a LUT with a short pattern
length. Therefore, the LUT of the SW method is trained iteratively. The LUT is
filled and used to re-transmit the sequence, then, again errors are determined
and added to the values which are already in the LUT .

##  3 Experimental setup and results

The predistortion method is validated using the setup in Fig. 1(a). PRBSs
containing $2^{16}$ GS-128[]*128[1]/̄QAM symbols are generated at
$48.8\text{\,}\mathrm{GBd}\text{/}$, predistorted for nonlinear effects, root-
raised-cosine shaped with $1\text{\,}\%$ roll-off, precompensated for linear
transmitter bandwidth limitations and uploaded to a digital-to-analog
converter (DAC) . The $193.4\text{\,}\mathrm{THz}\text{/}$ tone produced by an
external cavity laser is modulated by a dual-polarization IQ-modulator and
transmitted over $75\text{\,}\mathrm{km}\text{/}$ of single-mode fiber (SMF) .
At the receiver, the signal is received by a coherent receiver, after which it
is digitized, followed by coherent receiver digital signal processing . For
OBTB measurements, the components in Fig. 1(a) denoted by _Optional for single
span_ are left out of the setup. At the transmitter, noise-loading is achieved
by filtering the amplified spontaneous emission of an erbium-doped fiber
amplifier .

(a)

(b)

(c)

Fig. 2: 2(a) NGMI vs. of DAC swing 2(b) Received constellations for OBTB
without (top) and with (bottom) predistortion 2(c) NGMI vs OSNR for
transmission over $75\text{\,}\mathrm{km}\text{/}$ of SMF

In Fig. 1(b), the GS-128[]*128[1]/̄QAM constellation, optimized as described
in [3], is shown. Fig. 2(a) depicts the normalized generalized mutual
information (NGMI) when transmitting this constellation using only linear
precomp for different output swings of the DAC . Higher swings are associated
with higher transmitter optical signal-to-noise ratios (OSNRs) , but with more
transmitter nonlinearities. As seen form Fig. 2(a), the maximum NGMI occurs at
$400\text{\,}\mathrm{mV}\text{/}$, for which the received constellation points
are given in Fig. 1(b). The points are shifted and compressed, resulting in
worse maximum performance. This is simulated in Fig. 1(c), where additive
white Gaussian noise is added to both the original GS-128[]*128[1]/̄QAM
constellation and the distorted one. A penalty of 0.02 in NGMI can be observed
between the original and the distorted constellation. indicating that without
predistortion, the tested GS constellation is not the same as the designed
one.

The SW method is compared with the LUT predistortion method for a short
pattern length. Due to the non-uniform placing of the constellation points for
the GS-128[]*128[1]/̄QAM constellation, the number of possible amplitudes
levels is not 12 as for 128[]*128[1]/̄QAM , but 36. Hence, increasing the
pattern length above 5 results in extremely large LUTs
($\approx$300\text{\times}{10}^{9}$$ entries for $n=7$) and therefore only
$n=3$ and $n=5$ are evaluated. In Fig. 2(c), the NGMI vs. OSNR is presented
for a DAC output swing of $400\text{\,}\mathrm{mV}\text{/}$. A NGMI forward
error correction (FEC) limit of 0.85 is also depicted in Fig. 2(c),
corresponding to a $540\text{\,}\mathrm{Gbit}\text{/}\mathrm{s}$ net data
rate[7, 8]. It can be seen that only the SW predistorter and the LUT method
with $n=5$ achieves the FEC limit. The received constellations with and
without the SW predistorter are shown in Fig. 2(b). The NGMI vs. DAC output
swing for the SW predistorter is given in Fig. 2(a). It can be seen that the
decrease in NGMI when increasing the drive voltage is lower when compared to
the only linear precompensation case, resulting in the ability to increase the
OSNR after the modulator.

##  4 Conclusions

A SW predistorter to compensate for the transmitter nonlinearities, aimed at
design and evaluation of advanced modulation formats, where the correct
transmission of constellation points is especially important, is presented.
The SW method is easier to implement and train, and is shown to outperform
existing LUT based methods for practical table sizes for the evaluation of a
geometrically shaped 128-QAM.
_Partial funding is from KPN-TU/e Smart Two and from the Dutch NWO Gravitation
Program (Grant Number 024.002.033)._

## References

  * [1] P. W. Berenguer _et al._ , J. LightwaveTechnology 34, 1739–1745 (2016). doi:10.1109/JLT.2015.2510962.
  * [2] A. Napoli _et al._ , OpticsCommunications 409, 52–65 (2018). doi:10.1016/j.optcom.2017.09.062.
  * [3] B. Chen _et al._ , in _ICTON,_ (2018), p. Mo.B3.1. doi:10.1109/ICTON.2018.8473932.
  * [4] A. Napoli _et al._ , J. LightwaveTechnology 34, 3053–3064 (2016). doi:10.1109/JLT.2016.2535487.
  * [5] G. Khanna _et al._ , in _OFC,_ (2017), p. M2C.4. doi:10.1364/OFC.2017.M2C.4.
  * [6] S. Zhalehpour _et al._ , in _OFC,_ (2019), p. Th1D.3. doi:10.1364/OFC.2019.Th1D.3.
  * [7] K. Sugihara _et al._ , in _OFC,_ (2013), p. OM2B.4. doi:10.1364/OFC.2013.OM2B.4.
  * [8] K. Kojima _et al._ , J. LightwaveTechnology 35, 1383–1391 (2017). doi:10.1109/JLT.2017.2662942.


# Robust approximation rules for critical electric field of dielectric gas
mixtures

N.A. Garland  Centre for Quantum Dynamics, Griffith University, Nathan 4111,
Australia  School of Environment and Science, Griffith University, Nathan
4111, Australia<EMAIL_ADDRESS>D.L. Muccignat  College of Science
and Engineering, James Cook University, Townsville 4810, Australia  G.J. Boyle
College of Science and Engineering, James Cook University, Townsville 4810,
Australia  R.D. White  College of Science and Engineering, James Cook
University, Townsville 4810, Australia

###### Abstract

A semi-analytic method for quickly approximating the density-reduced critical
electric field for arbitrary mixtures of gases is proposed and validated.
Determination of this critical electric field is crucial for designing and
testing alternatives to SF6 for insulating high voltage electrical equipment.
We outline the theoretical basis of the approximation formula from electron
fluid conservation equations, and demonstrate how for binary mixtures the
critical electric field can be computed from the transport data of electrons
in the pure gases. We demonstrate validity of the method in mixtures of N2 and
O2, and SF6 and O2. We conclude with an application of the method to
approximate the critical electric field for mixtures of SF6 and HFO1234ze(E),
which is a high interest mixture being actively studied for high voltage
insulation applications.

##  1 Introduction

Presently, most high voltage electrical infrastructure on Earth relies on
sulphur hexafluoride (SF6). This inert gas is utilised in equipment such as
circuit breakers, transformers, and transmission infrastructure due its
excellent electric insulation and arc-extinguishing capacity [1]. Identified
as a greenhouse gas in the Kyoto protocol [2], SF6 has since been labeled by
the Intergovernmental Panel on Climate Change as the most potent greenhouse
gas with a global warming potential (GWP) approximately 23,500 times that of
carbon dioxide (CO2), and with an anticipated lifetime of up to 3200 years.

The bulk of SF6 emissions come from the electric power industry, often during
maintenance of electrical equipment or simply due to aging or faulty storage
vessels and electrical equipment. Recently, some success has been reported in
legislatively constraining SF6 emissions [3, 4]. However, ultimately the
global SF6 gas content is still increasing year-on-year and growth in newly
installed gas-insulated equipment is overpowering industry regulation and
financial penalty policies in reducing SF6 emissions. [4].

To date, a number of replacements to SF6 have been deployed in the power
industry, such as dry air, N2, CO2, or various admixtures of natural and
proprietary synthetic gases [1]. Despite some adoption of replacements with
lower GWP, the widespread global use of SF6 prevails in many industrial
applications where it has not yet been bettered on an engineering basis for
insulation and/or current quenching requirements. In addition, the logistical
and/or financial burdens to replace SF6 based equipment are often significant.

Driving the effort towards replacing SF6 are many hours and millions of
dollars spent per year in academic and commercial research performing
dielectric breakdown experiments [5, 6] or swarm experiments [7, 8] to
characterise insulating gases. There are many complexities and challenges in
adequately performing and interpreting measurements of such experiments in
their varying geometries (e.g. needle-plane, parallel plate, coaxial) and
varying operating conditions (e.g. low or high pressure/temperature, gas
purity, pulsed Townsend or steady state Townsend in the case of swarm
experiments). As a result, theory and simulation are vital methods helping to
complement experiments.

The power and flexibility of theory and simulation applied to highly
unconstrained design problems can enable a quick and affordable scan of very
large parameter spaces of material choices or design configurations. This
computational sieve approach to rapidly explore, trial, and assess potential
scenarios then informs the design of targeted and efficient experiments and
solution prototypes. For example, insight through large simulations of
breakdown scenarios in often complex geometries [9] or detailed kinetic
calculations of electron energy distribution functions (EEDFs) in gas mixtures
over density or pressure windows [10] are now capable with advances in
computing. That being said, these non-trivial computational tasks can also
present their own challenges and barriers to entry, such as advanced
computational domain knowledge or access to sufficient computing power or
time.

In addition to brute force computational methods, analytic or simplified semi-
analytic methods can be employed to provide very simple and rapid estimations
of design parameters. In particular, simple design formulas can be accessed by
a wide user base, allowing experiment scenarios to be quickly explored. In
Section 2 we present a semi-analytic method to rapidly predict the critical
reduced electric field, $\left(E/n_{0}\right)_{c}$, of gas mixture
combinations based entirely on calculated electron transport properties of the
constituent pure gases only. We first demonstrate the validity of this model
in Section 3 by benchmarking the predictive capability with relatively simple
mixtures of well studied gases, such as N2-O2, and SF6-O2. Finally, in Section
4 we demonstrate the ability of the model to correctly predict the positive
synergy in $\left(E/n_{0}\right)_{c}$ for mixtures of SF6 and HFO-1234ze(E).
This molecule (1E)-1,3,3,3-Tetrafluoroprop-1-ene (C3H2F4), often termed
HFO-1234ze(E) or R-1234ze, is a hydrofluoroolefin (HFO) used as a refrigerant
and is studied as a potential SF6 replacement [11, 8].

##  2 Theory

The macroscopic properties of a gas, like breakdown voltage, are often used to
guide design of electrical equipment. Fundamentally, the macroscopic
behaviours are underpinned by the microscopic interactions between electrons
and gas molecules. Pulsed Townsend (PT) and steady-state Townsend (SST) swarm
experiments have been used to study electron transport properties in
insulating gases, such as drift velocity, diffusion coefficients, and
importantly ionisation and attachment rates [1]. One quantity of immediate
relevance to the community is the critical electric field, where the
ionisation rate equals the electron attachment rate. Beyond this field the
rate of ionisation will exceed electron attachment and so it marks the
transition of a gas from being a free electron sink to being a source,
precipitating the dielectric breakdown of the medium.

Primary methods of approximating swarm properties in gas mixtures have largely
been based on projecting functional dependence onto the reduced electric
field, $E/n_{0}$, which we will abbreviate in equations as $\check{E}=E/n_{0}$
for brevity. One then approximates quantities of a gas mixture as linear
combinations of pure gas quantities evaluated at a common value of $E/n_{0}$,
weighted by gas density fractions of the mixture. For example, for simple gas
mixtures, many have employed Blanc’s law [12] to approximate drift velocity

$\displaystyle\frac{1}{W_{\mathrm{m}}(\check{E})}=\frac{x}{W_{A}(\check{E})}+\frac{(1-x)}{W_{B}(\check{E})},$
(1)

where $W_{\mathrm{m}}$ is the electron drift velocity in the mixture, $W_{A}$
and $W_{B}$ are drift velocities in gas A and gas B, and

$\displaystyle x=\frac{n_{A}}{n_{A}+n_{B}},$ (2)

where $n_{A}$ and $n_{B}$ are present gas densities of constituent gases A and
B evaluated at a common value of $E/n_{0}$.

In a similar fashion, a so-called Wieland approximation has been used to
approximate ionisation, $k_{\mathrm{i,m}}$, or attachment, $k_{\mathrm{a,m}}$,
rates of gas mixtures [13]

$\displaystyle
k_{\mathrm{m}}(\check{E})=xk_{\mathrm{A}}(\check{E})+(1-x)k_{\mathrm{B}}(\check{E}),$
(3)

where $k_{\mathrm{m}}$ is a rate in the mixture and $k_{\mathrm{A}}$ and
$k_{\mathrm{B}}$ are the rates in pure gases A and B evaluated at a common
value of $E/n_{0}$.

Implicit in its derivation, Blanc’s Law [12] does not inherently include
inelastic collisions. Further assumptions inherent in the Blanc and Wieland
approximations that limit their validity are that (i) electron impact cross
sections can be added in simple linear combinations at a given value of
$E/n_{0}$, and (ii) the steady state EEDF is the same for each component gas,
and the combination mixture, at a given value of $E/n_{0}$. This can be a
reasonably good approximation for some certain conditions, however in general
this assumption of the EEDF can be overly restrictive.

Previous studies have highlighted shortcomings in the Wieland approximation
[14, 15], with one notable limitation being an inability to predict positive
synergism in gas mixtures - where a higher critical electric field of the
mixture, above that of either constituent gases, is obtained. To remedy
shortcomings, extensions on this style of approximation, allowing for a non-
linear perturbation around the basic linear combination, have been explored
[7, 8]. Indeed, van Brunt [15] proposed a novel alternative parameterisation
of swarm properties via consideration of electron temperatures, similar to
what is presented in this current study. However, while this parameterisation
identifies the crucial role of mean electron energy, it relies on multiple
free parameters that require specification for each gas being considered.

To better account for the dependence of energy transfer on inelastic
collisions as $E/n_{0}$ increases an alternative approach to Blanc’s law based
on electron mean energy was developed [16]. This alternative approach for
evaluating collision and transport data of electrons in gas mixtures assumes
equivalence of a shared common electron mean energy (CME),
$\langle\varepsilon\rangle$, instead of a common reduced electric field,
$E/n_{0}$ [16]. This assumption therefore allows for arbitrary EEDFs in either
constituent gas or the mixture itself, with the restriction that the
integrated moment of $mv^{2}/2$ over each EEDF must be equivalent. The CME
approach has been well adopted by many areas of the community, however use of
Blanc’s Law still persists in some studies.

Of particular note to this study is the prior demonstration [14] of using the
CME approximation for estimation of non-conservative collision rates,
$\nu^{*}$, in mixtures of molecular gases

$\displaystyle\nu^{*}_{\mathrm{m}}(\langle\epsilon\rangle)=x\nu^{*}_{\mathrm{A}}(\langle\epsilon\rangle)+(1-x)\nu^{*}_{\mathrm{B}}(\langle\epsilon\rangle),$
(4)

this assumption was shown to be a much better method of projecting complex,
non-linear electron kinetic effects onto a single macroscopic swarm variable.
In arriving at the semi-analytic model presented in this study, we will also
adopt the approximation for estimation of non-conservative collision rates in
equation (4), and leverage the CME approach to determine $E/n_{0}$ in the
mixture conditions; this will enable estimation of the critical reduced
electric field.

As a foundation for our approach, we refer to the momentum and energy
conservation equations for electron transport, that can be derived by taking
integral moments of the Boltzmann equation over trial functions for electron
momentum, $m\textbf{v}$, and kinetic energy, $mv^{2}/2$ [17, 18, 19, 20]. As
per the CME approximation idea [16], it has been previously demonstrated [16,
21] that in the steady state limit the electron momentum balance equations of
both pure gases A and B, and an arbitrary mixture of the two, yield the
following expression for the electron drift velocity in the mixture

$\displaystyle\frac{1}{W_{\mathrm{m}}{\left(\langle\epsilon\rangle\right)}}=x\frac{\check{E}_{\mathrm{A}}{\left(\langle\epsilon\rangle\right)}}{\check{E}_{\mathrm{m}}{\left(\langle\epsilon\rangle\right)}}\frac{1}{W_{\mathrm{A}}{\left(\langle\epsilon\rangle\right)}}+(1-x)\frac{\check{E}_{\mathrm{B}}{\left(\langle\epsilon\rangle\right)}}{\check{E}_{\mathrm{m}}{\left(\langle\epsilon\rangle\right)}}\frac{1}{W_{\mathrm{B}}{\left(\langle\epsilon\rangle\right)}}.$
(5)

While from the steady state energy balance equations of both pure gases A and
B, and an arbitrary mixture of the two, a second expression for the electron
drift velocity in the mixture can be found

$\displaystyle
W_{\mathrm{m}}{\left(\langle\epsilon\rangle\right)}=x\frac{\check{E}_{\mathrm{A}}{\left(\langle\epsilon\rangle\right)}}{\check{E}_{\mathrm{m}}{\left(\langle\epsilon\rangle\right)}}W_{\mathrm{A}}{\left(\langle\epsilon\rangle\right)}+(1-x)\frac{\check{E}_{\mathrm{B}}{\left(\langle\epsilon\rangle\right)}}{\check{E}_{\mathrm{m}}{\left(\langle\epsilon\rangle\right)}}W_{\mathrm{B}}{\left(\langle\epsilon\rangle\right)}.$
(6)

With (5) and (6) both explicitly referring to the value of $E/n_{0}$ in the
mixture, we can combine (5) and (6) to obtain either an expression for the
drift velocity if one desires

$\displaystyle
W^{2}_{\mathrm{m}}{\left(\langle\epsilon\rangle\right)}=\frac{x\check{E}_{\mathrm{A}}W_{\mathrm{A}}+(1-x)\check{E}_{\mathrm{B}}W_{\mathrm{B}}}{x\check{E}_{\mathrm{A}}W_{\mathrm{B}}+(1-x)\check{E}_{\mathrm{B}}W_{\mathrm{A}}}W_{\mathrm{A}}W_{\mathrm{B}},$
(7)

or an expression for the reduced electric field in the mixture

$\displaystyle\check{E}^{2}_{\mathrm{m}}{\left(\langle\epsilon\rangle\right)}=\frac{\left(x\check{E}_{\mathrm{A}}W_{\mathrm{A}}+(1-x)\check{E}_{\mathrm{B}}W_{\mathrm{B}}\right)\left(x\check{E}_{\mathrm{A}}W_{\mathrm{B}}+(1-x)\check{E}_{\mathrm{B}}W_{\mathrm{A}}\right)}{W_{\mathrm{A}}W_{\mathrm{B}}},$
(8)

where
$\check{E}_{\mathrm{A}}=\check{E}_{\mathrm{A}}{\left(\langle\epsilon\rangle\right)}$,
$\check{E}_{\mathrm{B}}=\check{E}_{\mathrm{B}}{\left(\langle\epsilon\rangle\right)}$,
$W_{\mathrm{A}}=W_{\mathrm{A}}{\left(\langle\epsilon\rangle\right)}$, and
$W_{\mathrm{B}}=W_{\mathrm{B}}{\left(\langle\epsilon\rangle\right)}$ are all
functions of the common mean electron energy being considered,
$\langle\epsilon\rangle$.

###  2.1 Approximation rule for critical reduced electric field

By using the earlier approximation for a mixture ionisation or attachment
collision frequency in equation (4), and the expression in equation (8) for
equivalent reduced electric field in the mixture, we can now state the
procedure to extract a value of critical electrical field as:

  1. 1.

Using a platform of choice (e.g. Boltzmann equation solver, Monte Carlo
simulation) compute steady state electron swarm parameters for pure gases A
and B.

  2. 2.

Extract necessary variables: reduced electric field ($E/n_{0}$), drift
velocity ($W$), mean energy ($\langle\epsilon\rangle$), ionisation coefficient
($k_{\mathrm{i}}$), attachment coefficient ($k_{\mathrm{a}}$).

  3. 3.

Using the available swarm data and desired density fraction given by (2),
evaluate ionisation and attachment coefficients of a mixture as a function of
mean energy

$\displaystyle
k_{\mathrm{i,m}}{\left(\langle\epsilon\rangle\right)}=xk_{\mathrm{i,A}}{\left(\langle\epsilon\rangle\right)}+(1-x)k_{\mathrm{i,B}}{\left(\langle\epsilon\rangle\right)},$
(9) $\displaystyle
k_{\mathrm{a,m}}{\left(\langle\epsilon\rangle\right)}=xk_{\mathrm{a,A}}{\left(\langle\epsilon\rangle\right)}+(1-x)k_{\mathrm{a,B}}{\left(\langle\epsilon\rangle\right)}.$
(10)

  4. 4.

Using the available data, evaluate (8) to obtain the reduced electric field,
$\check{E}_{\mathrm{m}}{\left(\langle\epsilon\rangle\right)}$, in the mixture
as a function of mean energy.

  5. 5.

Evaluate the net effective ionisation coefficient

$\displaystyle
k_{\mathrm{eff,m}}{\left(\langle\epsilon\rangle\right)}=k_{\mathrm{i,m}}{\left(\langle\epsilon\rangle\right)}-k_{\mathrm{a,m}}{\left(\langle\epsilon\rangle\right)},$
(11)

which can now be expressed, or plotted for convenience, using
$\check{E}_{\mathrm{m}}{\left(\langle\epsilon\rangle\right)}$ as a function
$k_{\mathrm{eff,m}}\left(E/n_{0}\right)$ to determine the zero-crossing and
thus the critical reduced electric field of the gas mixture.

So, with zero knowledge of the electron transport in the mixture, we have
detailed an approximation of the critical reduced electric field of a gas
mixture. This uses only the electron swarm data computed for pure gases, which
is a standard exercise being made evermore accessible by multiple freely
available software for download, or even codes evaluated online via a web
browser [22, 23, 24, 25]. To validate the use of this presented method, we now
present benchmarking results of approximating the critical reduced electric
field for mixtures of relatively well-studied molecular gases.

##  3 Benchmarking and model validation

To validate the proposed method to rapidly approximate the critical reduced
electric field, we compare to the calculated critical reduced electric field
determined for various mixture combinations. The calculated value is
determined by first solving the Boltzmann kinetic equation for the EEDF in a
mixture gas using a well-benchmarked multi-term solution technique and
software framework [19, 26, 27]. One can then compute the effective net
electron creation rate, $k_{\mathrm{eff,m}}$, as the difference of the
integrated ionisation and attachment rates
$k_{\mathrm{i,m}}-k_{\mathrm{a,m}}$. The critical reduced electrical field is
naturally determined when $k_{\mathrm{eff,m}}=0$.

We first present results for a simplified case of N2 and O2 mixtures, since
only the O2 target will contribute to electron attachment processes. We then
follow with benchmarking of SF6 and O2 mixtures, where the addition of SF6
provides a large attachment cross-section to rigorously test the proposed
approximation. For the calculations detailed in this study electron scattering
cross sections for SF6, O2, and N2 were obtained from the Biagi
database111Biagi-v7.1 database, www.lxcat.net, retrieved on November 30, 2023
of the LXCat online database [28].

###  3.1 Benchmark: N2 and O2 mixtures

Mixtures of molecular nitrogen and oxygen are often studied as a surrogate for
air. As a result, there has been a wealth of scientific data and understanding
produced for electron scattering cross-sections and subsequent transport in
pure N2, O2, and indeed mixtures of the two. Therefore, it is a good candidate
system for benchmarking and validation of theory and methods.

For validation in the case of N2 and O2 mixtures, in Figure 1 we present the
results of approximating both the drift velocity, $W$, and net effective
ionisation rate, $k_{\mathrm{eff}}$, as a function of $E/n_{0}$; for clarity
of plot presentation we show comparisons for mixture fractions at 25%, 50%,
and 75% ratios.

Figure 1: Comparison of (top) flux drift velocity, $W$, and (bottom) net
effective ionisation rate, $k_{\mathrm{eff}}$, for various mixtures of N2 and
O2 as a function of $E/n_{0}$. Exact values computed via multi-term solution
of the EEDF from the Boltzmann equation, approximations determined from
procedure proposed in this study. Horizontal axis in $k_{\mathrm{eff}}$ plot
drawn in thicker line to highlight where zero-crossing of curves denotes
critical reduced electric field.

The present study is chiefly concerned with the net effective ionisation rate,
however since drift velocity is also a common transport quantity used in
assessment of swarm analysis we include it in Figure 1 as an additional
demonstration of the applicability of the approximation method presented in
this work. For both transport quantities the approximation produces excellent
qualitative agreement compared to the exact calculations. As a quantitative
measure of the critical reduced electric field accuracy, Table 1 shows that,
at worst, the agreement is about 5-6%, which was representative of the general
case for other mixture ratios not shown in Figure 1.

Table 1: Errors in critical reduced electric field approximation for various mixtures of N2 and O2. | Critical $E/n_{0}$ [Td] |
---|---|---
| Actual | Approximate | Relative error
25% N2 | 117 | 122 | 4.3%
50% N2 | 116 | 123 | 6.0%
75% N2 | 109 | 112 | 2.8%

###  3.2 Benchmark: SF6 and O2 mixtures

Mixtures of sulphur hexafluoride, SF6, and molecular oxygen, O2, have been
well studied in the semiconductor etching industry. As it was for our previous
benchmark case, choosing a well studied pair of molecules is a wise choice for
benchmarking and validation of theory and methods. In particular, validation
in the presence of a highly attaching gas like SF6 accompanied by another
attaching gas is worth demonstrating before trialing the methods presented in
this study on more complex molecules such as HFO-1234ze(E).

For validation in the case of SF6 and O2 mixtures, in Figure 2 we present the
results of approximating both the drift velocity, $W$, and net effective
ionisation rate, $k_{\mathrm{eff}}$, as a function of $E/n_{0}$; for clarity
of presentation we show comparisons for mixture fractions at 25%, 50%, and 75%
ratios.

Figure 2: Comparison of (top) flux drift velocity, $W$, and (bottom) net
effective ionisation rate, $k_{\mathrm{eff}}$, for various mixtures of SF6 and
O2 as a function of $E/n_{0}$. Exact values computed via multi-term solution
of the EEDF from the Boltzmann equation, approximations determined from
procedure proposed in this study. Horizontal axis in $k_{\mathrm{eff}}$ plot
drawn in thicker line to highlight where zero-crossing of curves denotes
critical reduced electric field.

Once again, Figure 2 demonstrates excellent qualitative agreement produced by
the approximation, for both drift velocity and net creation rate. In these
cases the transition from pure SF6 to O2 extremes is somewhat monotonic. In
terms of the critical reduced electric field values, Table 2 values of
relative error summarise that there is generally improved accuracy compared
with the prior N2/O2 benchmark case.

Table 2: Errors in critical reduced electric field approximation for various mixtures of SF6 and O2. | Critical $E/n_{0}$ [Td] |
---|---|---
| Actual | Approximate | Relative error
25% SF6 | 206 | 217 | 5.3%
50% SF6 | 266 | 273 | 3.0%
75% SF6 | 313 | 317 | 1.3%

As an additional point of interest, Figure 3 shows the EEDFs at (a) the
critical reduced electric field point for pure SF6, O2, and a mixture of 50%
SF6 and 50% O2. Here, subfigure (a) shows that for the wide range of critical
$E/n_{0}$ values there is some variation in EEDFs. The mixture EEDF generally
resembles that for SF6, with some deviation around the distribution peak.
Notably the EEDF in O2 peaks at a lower energy. Subfigure (b) shows the same
EEDF for the mixture as in (a), however for pure SF6 and O2 the EEDFs are
shown corresponding to the same electron mean energy value at the mixture’s
critical $E/n_{0}$, $\langle\epsilon\rangle=7.0$ eV. This demonstrates that
for the bulk of the distribution population, the EEDFs are indeed quite
similar, particularly between approximately 0.1-20 eV. This demonstration
highlights that abstracting the dynamics of the bulk population of EEDFs onto
the common mean energy $\langle\epsilon\rangle$, invoked in the derivation of
our approximation method, is indeed a reasonable approach.

Figure 3: Comparison of EEDFs for SF6, O2, and a mixture of 50% SF6 and 50%
O2. (Left) EEDFs at each critical $E/n_{0}$ value for the three background gas
cases. (Right) EEDFs for each background gas case associated with a mean
electron energy $\langle\epsilon\rangle=7.0$ eV corresponding to the mixture
EEDF at the critical field threshold.

When discussing the nature of EEDFs it is often instructive to compare against
a Maxwellian distribution often used by the community. For this purpose,
Figure 3(b) also allows comparison against a Maxwellian EEDF of the same
$\langle\epsilon\rangle$ as the calculated distributions. This highlights that
while the bulk of these computed EEDFs reasonably follow the Maxwellian shape
about the peak, the sharp roll-off to a depleted tail of higher energy
electrons is not sufficiently captured by a Maxwellian.

##  4 Application to dielectric insulator design

In the search for alternatives to sulphur hexafluoride, SF6, industry and
scientific communities are considering a wide range of options. One such
option that has been heavily studied [29, 11, 5, 30] is
(1E)-1,3,3,3-Tetrafluoroprop-1-ene (C3H2F4), often abbreviated to
HFO-1234ze(E). This gas is one of an emerging generation of hydrofluoroolefins
(HFOs) originally targeted for use as refrigerants in place of traditional
hydrofluorocarbons. The industry use of HFOs is being spurred by their often
very low ozone depeletion and global warming potentials.

Mixtures of sulphur hexafluoride, SF6, and HFO-1234ze(E), are also being
studied to maximise insulating and arc quenching properties [8, 31] as part of
the large design and development research efforts to reduce SF6 use in the
high voltage electricity industry. For the HFO-1234ze(E) electron scattering
cross-section data, a recently obtained cross-section set from a forthcoming
study [32] applying neural network based swarm inversion techniques [33, 34]
is employed.

Here in Figure 4 we focus on probing mixtures of SF6 and HFO-1234ze(E) for
approximating the net effective ionisation rate, $k_{\mathrm{eff}}$, as a
function of $E/n_{0}$. For clarity of presentation we show comparisons for
mixture fractions of 25%, 50%, and 75%.

Figure 4: Comparison of net effective ionisation rate, $k_{\mathrm{eff}}$, for
various mixtures of SF6 and HFO-1234ze(E) as a function of $E/n_{0}$. Exact
values computed via multi-term solution of the EEDF from the Boltzmann
equation, approximations determined from procedure proposed in this study.
Horizontal axis in $k_{\mathrm{eff}}$ plot drawn in thicker line to highlight
where zero-crossing of curves denotes critical reduced electric field.

The results in Figure 4, and subsequent Figures 6 and 7, clearly demonstrate
the well established non-monotonic behaviour of the critical field increasing
with increasing SF6 fraction but then reducing as the fraction of SF6
approaches 100%. The ability of these mixtures to yield a critical reduced
field greater than either case of pure gas has been termed as a positive
synergy in the insulating gas literature. This positive synergy effect has
been well studied and discussed [8, 31, 35] for insulating gases, and so being
able to reproduce this important physics phenomenon from the present
approximation formula, despite only using known data for pure gas cases, is an
encouraging outcome of the present method.

Despite the added complication of the non-monotonic positive synergy
phenomenon our present method again provides a good approximation to the
critical reduced electric field for these mixtures in the regime of competing
ionising and attaching processes. We can see from Table 3 there is excellent
quantitative agreement between the approximated value for the critical reduced
electric field for mixtures, with relative errors on the order of 1-2%.

Table 3: Errors in critical reduced electric field approximation for various mixtures of SF6 and HFO-1234ze(E). | Critical $E/n_{0}$ [Td] |
---|---|---
| Actual | Approximate | Relative error
25% SF6 | 402 | 403 | 0.2%
50% SF6 | 426 | 434 | 0.7%
75% SF6 | 401 | 398 | 1.9%

Comparing the EEDFs, in Figure 5(a), we can see quite different distributions
for the case of pure SF6, HFO-1234ze(E), and a mixture of 50% SF6 and 50%
HFO-1234ze(E). At low energies the mixture EEDF resembles that of SF6, however
beyond about 1 eV the mixture EEDF adopts qualitative features of the pure
HFO-1234ze(E) distribution. To compare EEDFs corresponding to the same common
electron mean energy, we refer to subfigure (b). Here, we see that when
considering EEDFs yielding the same mean energy as the mixture critical point,
$\langle\epsilon\rangle=5.2$ eV, the distributions are once again very
similar. Once again, this outcome validates the use of the underlying common
mean energy assumption discussed in Section 2.

Figure 5: Comparison of EEDFs for SF6, HFO-1234ze(E), and a mixture of 50% SF6
and 50% HFO-1234ze(E). (Left) EEDFs at each critical $E/n_{0}$ value for the
three background gas cases. (Right) EEDFs for each background gas case
associated with a mean electron energy $\langle\epsilon\rangle=5.2$ eV
corresponding to the mixture EEDF at the critical field threshold.

We now demonstrate an example of high-throughput data generation and
understanding that the present semi-analytic method offers, compared to having
to calculate the exact EEDF via the Boltzmann equation for each system
configuration. Scanning over a range of mixture combinations $0\leq x\leq 1$
and a range of applied $E/n_{0}$ values allows a surface of the net effective
ionisation rate, $k_{\mathrm{eff}}$, to be constructed over the domain of
possible parameter space, as shown in Figure 6. Considering the colour
gradient changes projected to the base of the plot, one can clearly see the
non-linear and non-monotonic features of the levels of $k_{\mathrm{eff,m}}$.

Figure 6: Surface of net effective ionisation rate, $k_{\mathrm{eff}}$, for
mixtures of SF6 and HFO-1234ze(E) as a function of $E/n_{0}$ and % fraction of
SF6. Level curves projected on lower plane, with $k_{\mathrm{eff}}=0$ points
marked in black,  .

For the possible mixture combinations in the case of
$k_{\mathrm{eff,m}}\left(E/n_{0}\right)=0$, black marker points are placed to
highlight the critical field curve for the mixtures of SF6 and HFO-1234ze(E).
These points can be projected onto a more convenient 2D plot, like that
commonly presented in literature, in Figure 7. For comparison the pulsed
Townsend swarm measurements of Egüz et al. [7] have been plotted. Further, the
critical electron mean energy, $\langle\epsilon\rangle_{c}$, at a given
critical electric field is plotted against the right-hand axis of Figure 7.

Figure 7: (Left-hand axis) Approximate critical reduced electric field where
attachment transitions to ionisation for increasing fractions of SF6 in
mixture with HFO-1234ze(E). Present approximation result shown in black
markers,  , swarm measurements of Egüz et al.[8] shown in red markers,  .
(Right-hand axis) Mean electron energy, $\langle\epsilon\rangle$, at critical
electric field plotted in triangle markers,  , to demonstrate monotonic nature

In contrast to the non-monotonic nature of $\left(E/n_{0}\right)_{c}$, the
critical electron mean energy, $\langle\epsilon\rangle_{c}$, plotted in Figure
7 demonstrates a monotonic increase with increasing fraction of SF6. By
plotting and comparing $\left(E/n_{0}\right)_{c}$ and the equivalent critical
$\langle\epsilon\rangle_{c}$ we highlight why indeed a mixture approximation
of linear addition of constituent gas quantities with functional dependence on
$E/n_{0}$ may fail. In contrast, the monotonic nature in
$\langle\epsilon\rangle_{c}$ allows a linear combination of pure gas
quantities with mean energy functional dependence to be used effectively.

##  5 Conclusion

In summary, we have presented the formulation and validation of a semi-
analytic formula and method for approximating the critical reduced electric
field for insulating gas mixtures. These findings demonstrate a rapid and
accurate approximate approach to assessing the breakdown electric field
properties of arbitrary insulating gas mixtures. Importantly, our results show
immediate application and promise towards parameter space exploration and
design studies of potential insulating gases to replace SF6 in the high
voltage electricity industry. While this study provides valuable insights into
how industry standard electron transport data of individual pure gases can be
combined to estimate mixture properties, further investigation is needed, for
example, to develop improvements to the model to facilitate the use of
experimentally measured swarm quantities as a function of $E/n_{0}$. While
high-throughput approximation and prototyping can be done using the presented
method, the necessity of having calculated pure gas swarm data as a function
of mean energy is a limitation. Future studies will explore this by studying
functional dependencies of electron mean energy, characteristic energy, and
quantities measured directly in swarm experiments. By continuing to
investigate these aspects, we can expand the set of quality tools available to
scientists and engineers in the design and development of effective, safe, and
climate neutral insulating gas technologies for high voltage electricity
industries.

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