# Linear voltage recovery after a breakdown in a pulsed dc system

Anton Saressalo<EMAIL_ADDRESS>Helsinki Institute of Physics and
Department of Physics, University of Helsinki, PO Box 43 (Pietari Kalmin katu
2), 00014 Helsingin yliopisto, Finland  Dan Wang  Helsinki Institute of
Physics and Department of Physics, University of Helsinki, PO Box 43 (Pietari
Kalmin katu 2), 00014 Helsingin yliopisto, Finland  State Key Laboratory of
Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an,
China  Flyura Djurabekova  Helsinki Institute of Physics and Department of
Physics, University of Helsinki, PO Box 43 (Pietari Kalmin katu 2), 00014
Helsingin yliopisto, Finland

###### Abstract

Breakdowns may occur in high-voltage applications even in ultrahigh vacuum
conditions. Previously, we showed that it is important to pay attention to the
post-breakdown voltage recovery in order to limit the appearance of secondary
breakdowns associated with the primary ones. This can improve the overall
efficiency of the high-voltage device. In this study, we focus on the
optimization of the linear post-breakdown voltage recovery, with the principle
aim of alleviating the problem of the secondary breakdowns. We investigate
voltage recovery scenarios with different starting voltages and slopes of
linear voltage increase by using a pulsed dc system. We find that a higher
number of pulses during the voltage recovery produces fewer secondary BDs and
a lower overall breakdown rate. Lowering the number of pulses led to more
dramatic voltage recovery resulting in higher breakdown rates. A steeper
voltage increase rate lead to a more localized occurrence of the secondary
breakdowns near the end of the voltage recovery period. It was also found that
the peak BD probability is regularly observed around $1\text{\,}\mathrm{s}$
after the end of the ramping period and that its value decreases exponentially
with the amount of energy put into the system during the ramping. The value
also decays exponentially with a half-life of $1.4\pm 0.3\text{\,}\mathrm{ms}$
if the voltage only increased between the voltage recovery steps.

##  I Introduction

Vacuum arc breakdown (BD) is an event where a conductive plasma channel forms
between two metal surfaces separated by a vacuum gap Boxman  _et al._ (1995).
Such events are common in any application utilizing high electric fields,
including particle accelerators Aicheler  _et al._ (2018); Pesce  _et al._
(2011), radio-frequency power sources Catalan-Lasheras  _et al._ (2014);
Cahill  _et al._ (2018) and vacuum interrupters Schade and Shmelev (2003).

The frequency of BD events affects the efficiency of these machines and limits
their performance. Therefore, pivotal in minimizing the BD frequency is not
only the understanding of the basic mechanisms driving the system towards the
formation of a BD spot, but also the BD characteristics and their possible
correlation with macroscopically adjustable parameters. Such understanding
contributes to the development of safe and cost-efficient high voltage
applications. Although the BD phenomenon has been studied for more than a
hundred years Hull and Burger (1928); Snoddy (1931); Hoffman  _et al._ (1997);
Aicheler  _et al._ (2012), the details of the process and its different stages
are still extensively debated Antoine  _et al._ (2011); Latham (1995);
Engelberg  _et al._ (2018); Anders (2009)

One of the plausible scenarios of development of a BD spot was associated with
the activation of dislocations in the presurface region under applied high
electric fields Pohjonen  _et al._ (2013); Nordlund and Djurabekova (2012);
Engelberg  _et al._ (2018); Jacewicz  _et al._ (2020). In this approach, the
dislocations are proposed to induce irregularities on the flat surface. These
irregularities can be further sharpened by the locally enhanced electric
fields Jansson  _et al._ (2016), transforming the irregularities into field
emitters. These may produce dark current bursts Engelberg  _et al._ (2020);
Lachmann  _et al._ (2021) that could eventually lead to BDs Kyritsakis  _et
al._ (2018); Gao  _et al._ (2020).

According to the statistical behavior of BD events, the latter were suggested
to be divided into two major groups: primary (pBD) and secondary (sBD) BD
events Wuensch  _et al._ (2017); Saressalo  _et al._ (2020a, b). The pBDs were
both spatially and temporally independent of the preceding events, whereas the
sBDs always follow the pBDs, typically correlating also spatially with the
location of the previous BD spot. This suggests that it is very likely that
there are different mechanisms contributing to generation of a BD spot of each
type.

We showed recently Saressalo  _et al._ (2020b) that the post-breakdown voltage
recovery must be performed slowly by a gradual increase in the voltage from
some initial value ($V_{0}$) to the target one ($V_{t}$), which was crucial
for the mitigation of sBD. A sBD was almost guaranteed to appeared if the
surface was exposed to the target electric field immediately after the
breakdown. We also showed that it was essentially equally optimal to recover
from a BD by the voltage increase (voltage ramp) either linearly or in twenty
voltage increment steps. Although both scenarios were found to be the best out
of the tested ones, they still showed a significant number of sBDs and it is
desirable to explore the ways of further improvement of the post-breakdown
voltage recovery.

A prominent feature in the step-wise recovery was seen to be an increased BD
probability at the beginning of each ramping step. We have previously
identified two main factors responsible for this increase Saressalo  _et al._
(2020b): the $20\text{\,}\mathrm{s}$ idle time between the voltage steps and
the increase in voltage before the next ramping step. The former was
attributed to the redeposition of vacuum residual gases on the surface during
the idle time, since the reactivity of the surface increases due to the
surface cleaning during conditioning. In our previous experiments, it was not
possible to avoid pauses during the voltage recovery procedures as it was
required for adjusting to new values.

In this work, we developed our setup to exclude the necessity of pauses and
focus on linear voltage ramp scenarios only, since these ramping scenarios
were the most promising due to time-efficiency and technical accessibility.
Moreover, this new approach allows the separation of the effect of the vacuum
residual gases from the surface effects, as it allows studying of BD
generation at different voltage increase rates while keeping the vacuum level
constant. This allows us to focus on the processes which are triggered by the
increased applied electric field. In this study, we focus on the short-term
conditioning of the electrode surfaces, which is necessary for reducing the
amount of sBDs.

##  II Experimental setup

The voltage recovery characteristics were explored using a Pulsed DC system
intended for BD studies. The system is described in detail in Refs. Saressalo
_et al._ (2020b, a); Profatilova  _et al._ (2020). It includes a power supply
coupled with a Marx generator which magnifies the input DC voltage and
converts it into short square voltage pulses. Repetition rate of
$2000\text{\,}\mathrm{Hz}$ and pulse length of
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ are used in typical BD experiments.
In the system, two cylindrical Cu electrodes are enclosed in a vacuum chamber,
separated by a gap of $d=$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The top
electrode (anode) has a contact area diameter of $40\text{\,}\mathrm{mm}$ and
is connected to the Marx generator via a high-voltage (HV) cable, while the
bottom electrode (cathode) has a contact area diameter of
$60\text{\,}\mathrm{mm}$ and is grounded. Together, the electrodes form a
parallel plate capacitor with a capacitance of around
$700\text{\,}\mathrm{pF}$ (including the capacitance of the HV cable).
Voltages of up to $V=$6000\text{\,}\mathrm{V}$$
($E=$150\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$$, assuming $E=V/d$
as argued by Ref. Korsbäck  _et al._ (2020)) can be used in the system.

Different voltage recovery scenarios were studied in three sets of
measurements, all described in Table 1. Each measurement consists of 1000 BDs
produced using a constant target voltage. The pulsing voltage remained
constant at $V_{t}$, throughout each measurement set, except for the voltage
recovery after each BD. Before each measurement, a short re-conditioning was
performed to ensure that the electrode surfaces are in an identical state in
the beginning of each of the measurements. However, the electrode surface
state evolved in between the measurement sets, so the results are only
comparable within each measurement set, but not across the sets.

Table 1:  The voltage recovery scenarios applied in the experiments with the linear voltage ramp (one slope). The first column shows symbol each measurement is denoted as. The total number of ramping pulses $N_{\text{pulses}}$ and the fraction of the starting voltage $V_{0}$ to the target voltage $V_{t}$ are shown in the second and third column. The final two columns contain the corresponding overall ramping slopes and the ramp rates (voltage increase per pulse), respectively. The last measurement set included scenarios with pulsing at a constant voltage before the start of the voltage ramp. For these runs $N_{\text{pulses}}$ in each phase are denoted in the second column. # | $N_{\text{pulses}}$ | $V_{0}/V_{t}$ | Voltage slope | Increase per pulse
---|---|---|---|---
1 | 500 | $20\text{\,}\mathrm{\char 37\relax}$ | $15\,360\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $8\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
2 | 1000 | $20\text{\,}\mathrm{\char 37\relax}$ | $7680\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $4\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
3 | 2000 | $20\text{\,}\mathrm{\char 37\relax}$ | $3840\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $2\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
4 | 4000 | $20\text{\,}\mathrm{\char 37\relax}$ | $1920\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $1\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
5 | 8000 | $20\text{\,}\mathrm{\char 37\relax}$ | $960\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $0.5\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
I | 2000 | $80\text{\,}\mathrm{\char 37\relax}$ | $880\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $0.4\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
II | 4000 | $60\text{\,}\mathrm{\char 37\relax}$ | $880\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $0.4\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
III | 6000 | $40\text{\,}\mathrm{\char 37\relax}$ | $880\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $0.4\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
IV | 8000 | $20\text{\,}\mathrm{\char 37\relax}$ | $880\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $0.4\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
V | 10 000 | $0\text{\,}\mathrm{\char 37\relax}$ | $880\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $0.4\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
A | 6000+4000 | $60\text{\,}\mathrm{\char 37\relax}$ | $920\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $0.5\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
B | 4000+6000 | $40\text{\,}\mathrm{\char 37\relax}$ | $920\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $0.5\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
C | 6000+4000 | $40\text{\,}\mathrm{\char 37\relax}$ | $1380\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $0.7\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
D | 8000+2000 | $40\text{\,}\mathrm{\char 37\relax}$ | $2760\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ | $1.4\text{\,}\mathrm{V}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$

In the first set of measurements (scenarios 1–5), ramping scenarios with
different voltage slopes were studied. In each measurement, $V_{t}$ was
$4800\text{\,}\mathrm{V}$
($120\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$) and the ramping
started from $20\text{\,}\mathrm{\char 37\relax}$ of the target voltage.
Voltage slopes between $1\text{\,}\mathrm{kV}$ and $15\text{\,}\mathrm{kV}$
were used. The theoretical voltage increase curves are presented in Fig. 1.

Figure 1: Theoretical voltage increase in each of the ramping scenario in the
first set of measurements as a function of time and the number of pulses. With
the repetition rate of $2000\text{\,}\mathrm{Hz}$, one second equals 2000
pulses. The figure also contains a 20 step ramping scenario used in Saressalo
_et al._ (2020a); Profatilova  _et al._ (2019) for comparison.

In the scenarios, the voltage was set to the low value $V_{0}$ after a BD. For
the linear voltage recovery with 2000 pulses, this means that the voltage
increases from $V_{0}=$960\text{\,}\mathrm{V}$$ ($20\text{\,}\mathrm{\char
37\relax}$) to $V_{t}=$4800\text{\,}\mathrm{V}$$ ($100\text{\,}\mathrm{\char
37\relax}$) within one second (the repetition rate is
$2000\text{\,}\mathrm{Hz}$). The voltage slope is thus
$3840\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$. This is rather fast rate and
the system can not adjust to the new value instantaneously, causing delays in
the voltage change due to the RC coupling between the power supply and the
rest of the circuit. At this rate, the voltage changes from pulse to pulse by
$\Delta V=$2\text{\,}\mathrm{V}$$. Although the voltage changes linearly
including the time within the $1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ pulse,
this change is insignificant, and is only $4\text{\,}\mathrm{mV}$. Hence, the
change from pulse to pulse is more drastic, which causes the deflection in the
shape of the voltage ramp from the linear one.

The second set of measurements (I–V in Table 1) focused on the effect of the
starting voltage. The slope of the voltage increase remained the same, but the
starting voltages ranged from $0\text{\,}\mathrm{\char 37\relax}$ to
$80\text{\,}\mathrm{\char 37\relax}$ of $V_{t}$ ($4400\text{\,}\mathrm{V}$).
Note that scenario IV ($N_{\text{pulses}}=8000$ and
$V_{0}/V_{t}=$80\text{\,}\mathrm{\char 37\relax}$$) has identical parameters
to scenario 5, except for the slightly different $V_{t}$. In the last set of
measurements (denoted A–D), the voltage ramp was combined with pulsing at a
constant non-zero voltage prior to the voltage increase. Each measurement had
the same amount of pulses. $V_{t}$ of $4600\text{\,}\mathrm{V}$ was used in
this set of measurements.

Additionally, the shape of the actual voltage increase was studied and
compared to the theoretical one via bypassing the Marx generator and measuring
directly the output voltage of the power supply across a high-power capacitor
of similar capacitance compared to that of the electrode pair. Using a
capacitor instead of the Cu electrode pair ensures that no BDs occur during
the measurement. Bypassing the Marx generator assumes that the shape of the
voltage increase curve is fully determined by the limitations of the power
supply and its RC coupling, as the generator is only used to magnify the
incoming voltage by a constant factor.

##  III Results

The measured voltage recovery curves for each of the ramping scenarios are
shown in Fig. 2. In each measurement, the target voltage was
$1000\text{\,}\mathrm{V}$ and the capacitance $650\text{\,}\mathrm{pF}$.

Figure 2: Actual measured voltage increase in each of the ramping scenario as
a function of time and the number of pulses. The measurement was performed by
using the power supply to increase the voltage to $1000\text{\,}\mathrm{V}$
over a circuit with a capacitance of $650\text{\,}\mathrm{pF}$.

In the figure, we see that the actual voltage ramping follows the curves that
are slightly different from the theoretical ones of 1 as it was explained in
Sec. II. The steeper the ramping, the more dramatic the delay caused by the RC
coupling is. It also affects the time, when the voltage reaches $V_{t}$. For
instance, in the scenario with 500 pulses, the actual voltage in the system is
only at $60\text{\,}\mathrm{\char 37\relax}$ of $V_{t}$ at the time when the
ramping should theoretically end ($0.25\text{\,}\mathrm{s}\mathrm{)}$. For the
other ramping scenarios of the first set, with 1000, 2000, 4000, and 8000
pulses, these values are $80\text{\,}\mathrm{\char 37\relax}$,
$91\text{\,}\mathrm{\char 37\relax}$, $97\text{\,}\mathrm{\char 37\relax}$ and
$99\text{\,}\mathrm{\char 37\relax}$ of Vt, respectively. For the step-wise
voltage ramping scenario with 20 steps, the theoretical and measured curves
coincide, since a short pause was used in these experiments in between the
voltage increase and the start of pulsing to properly adjust the power supply
to the new voltage.

To determine the best voltage recovery scenario, we analyze the BD rate
measured as the number of BDs per pulse (BDR), the percentage of sBDs out of
all BDs ($N_{\text{sBD}}$), the mean number of consecutive BDs between two
pBDs (including the pBD) ($\mu_{\text{sBD}}$) and $\mu^{\prime}_{\text{sBD}}$,
which is defined as the fraction $N_{\text{sBD}}/N_{\text{pBD}}$, excluding
those pBDs that were followed by no sBDs. The results are shown in Table 2.
The uncertainties were estimated by the standard error of the mean. The sBDs
were determined by fitting a two-term exponential curve to the probability
density function (PDF) of the number of the pulses between two consecutive BD
events (Fig. 3). As introduced in Refs. Wuensch  _et al._ (2017); Saressalo
_et al._ (2020b), the cross-point $N_{\text{cross}}$ of the two exponential
curves was used to classify the events so that the BDs that occurred with the
smaller number of pulses in-between ($N<N_{\text{cross}}$) were determined as
sBDs, while the pBDs happen with longer waiting time, i.e., more pulses, in
between the two BD events. The exponential coefficients $\alpha$ and $\beta$
correspond to the breakdown rates of pBDs and sBDs, respectively, and were
found from the fit to the PDFs in Fig. 3 as

$\text{PDF}=A\exp{(-\alpha N)}+B\exp{(-\beta N)}$ (1)

Wuensch  _et al._ (2017), where $N$ is the number of pulses between
consecutive BDs. It must be noted that the results are fully comparable only
within each set of measurements, as the surface state of the electrodes may
have changed in between the measurement sets.

Table 2:  Key figures for the BD experiments with different ramping scenarios. Values of breakdown rate (BDR), two-exponential PDF fit parameters ($\alpha$ and $\beta$), cross-point of the two exponentials ($N_{\text{cross}}$), percentage of sBDs ($N_{\text{sBD}}$), the average number of BDs between two pBDs (including the previous pBD) ($\mu_{\text{sBD}}$) and the fraction of the number of sBDs to pBDs for those series with at least one sBD ($\mu^{\prime}_{\text{sBD}}$) for each scenario are listed in the table. The two-exponential fit could not be performed for scenario I, so the values denoted with † are not fully comparable to the other measurements. # | Scenario |  $V_{0}/V_{t}$ [%] | BDR [bpp] | $\alpha$ | $\beta$ | N${}_{\text{cross}}$ | N${}_{\text{sBD}}$ [%] | $\mu_{\text{sBD}}$ | $\mu^{\prime}_{\text{sBD}}$
---|---|---|---|---|---|---|---|---|---
1 | 500 pulses | 20 | $6.46\text{\times}{10}^{-6}$ | $9.4\text{\times}{10}^{-5}$ | $0.003$ | $2067$ | $30.2$ | $2.06\pm 0.10$ | $3.6\pm 0.7$
2 | 1000 pulses | 20 | $3.59\text{\times}{10}^{-6}$ | $4.2\text{\times}{10}^{-5}$ | $0.003$ | $2607$ | $26.0$ | $1.91\pm 0.10$ | $2.7\pm 0.7$
3 | 2000 pulses | 20 | $4.19\text{\times}{10}^{-6}$ | $7.4\text{\times}{10}^{-5}$ | $0.002$ | $3686$ | $23.1$ | $1.76\pm 0.10$ | $3.1\pm 1.0$
4 | 4000 pulses | 20 | $3.48\text{\times}{10}^{-6}$ | $1.8\text{\times}{10}^{-5}$ | $0.003$ | $5588$ | $22.6$ | $1.53\pm 0.05$ | $2.2\pm 0.5$
5 | 8000 pulses | 20 | $2.02\text{\times}{10}^{-6}$ | $1.7\text{\times}{10}^{-5}$ | $0.002$ | $10\,489$ | $21.4$ | $1.35\pm 0.03$ | $1.8\pm 0.4$
I† | 2000 pulses | $80$ | $7.34\text{\times}{10}^{-5}$ | - | - | - |  $77.4$† |  $8.0\pm 0.7$† |  $12\pm 4$†
II | 4000 pulses | $60$ | $3.60\text{\times}{10}^{-5}$ | $5.5\text{\times}{10}^{-5}$ | $0.002$ | $5378$ | $58.7$ | $4.3\pm 0.4$ | $7\pm 3$
III | 6000 pulses | $40$ | $3.75\text{\times}{10}^{-5}$ | $4.2\text{\times}{10}^{-5}$ | $0.002$ | $7284$ | $39.9$ | $2.5\pm 0.2$ | $4.1\pm 1.4$
IV | 8000 pulses | $20$ | $2.81\text{\times}{10}^{-5}$ | $3.7\text{\times}{10}^{-5}$ | $0.001$ | $10\,352$ | $62.3$ | $3.7\pm 0.2$ | $4.7\pm 0.6$
V | 10 000 pulses | $0$ | $3.30\text{\times}{10}^{-5}$ | $4.8\text{\times}{10}^{-5}$ | $0.001$ | $11\,943$ | $45.4$ | $2.5\pm 0.1$ | $3.4\pm 0.7$
A | 6000 pulses flat + 4000 pulses ramp | $60$ | $5.48\text{\times}{10}^{-5}$ | $6.2\text{\times}{10}^{-5}$ | $0.001$ | $12\,755$ | $70.2$ | $7.5\pm 1.0$ | $13\pm 7$
B | 4000 pulses flat + 6000 pulses ramp | $40$ | $3.31\text{\times}{10}^{-5}$ | $4.1\text{\times}{10}^{-5}$ | $0.001$ | $12\,228$ | $42.8$ | $2.6\pm 0.3$ | $5\pm 2$
C | 6000 pulses flat + 4000 pulses ramp | $40$ | $1.39\text{\times}{10}^{-5}$ | $2.8\text{\times}{10}^{-5}$ | $0.001$ | $13\,404$ | $39.6$ | $1.84\pm 0.08$ | $2.9\pm 0.6$
D | 8000 pulses flat + 2000 pulses ramp | $40$ | $8.84\text{\times}{10}^{-6}$ | $1.0\text{\times}{10}^{-5}$ | $0.001$ | $14\,739$ | $38.4$ | $1.96\pm 0.13$ | $2.8\pm 0.9$
Figure 3: Probability density function for a BD to occur at a number of pulses
after the previous BD event for the different voltage recovery scenarios. The
vertical dashed lines indicate the cross-point between the two fitted
exponential curves (not shown) which correspond to the breakdown rates of the
primary and secondary events, respectively.

Fig. 3 shows the PDF as a function of the number of pulses between two
consecutive BDs. We see that the location of the BD peak probability
correlates with the number of the ramping pulses and is typically registered
during the early pulses at the target voltage. We notice not only a shift in
the peak location towards the greater number of pulses between consecutive
BDs, but also the change of the shape of the peak probability. The slower the
ramping is performed, the lesser the probability of a BD to happen at the end
of the ramping process and the BDs become more evenly distributed over
different number of pulses between BDs.

Comparing the results for the scenarios 1–5 presented in Table 2 and Fig. 3,
we note that the lowest values in BDR, $N_{\text{sBD}}$ and $\mu_{\text{sBD}}$
were obtained for the scenarios with the most pulses in the voltage recovery.
However, the exponential coefficients $\alpha$ and $\beta$ were relatively
similar for all scenarios. This consistency indicates the same nature of the
processes responsible for the sBD and the pBD events. The striking consistency
in the values of the $\beta$ coefficient, (the rate of sBDs) shows that this
process does not depend on the particular ramping scenario due to the
correlation existing between the subsequent events. Slight gradual decrease of
the $\alpha$ coefficient with increase of the number of ramping pulses
indicates a quieter conditioning process via the pBDs only. We also note that
despite the similarities in $\alpha$ and $\beta$ coefficients for all the
measurements in this series, both values are consistently lower for the
scenario with 8000 voltage recovery pulses.

The similar $\alpha$ and $\beta$ values in all the measurements lead to the
consistent shift of about 2000 pulses ($1\text{\,}\mathrm{s}$) in the
$N_{\text{cross}}$ towards a greater number of pulses after the voltage
recovery is finished . The combination of the smallest $\alpha$ and $\beta$
values in the recovery scenarios with the most pulses ($\geq 8000$) resulted
also in the largest shift in $N_{\text{cross}}$ towards the greater number of
pulses between the consecutive BDs. Despite this, we conclude that in our
measurements almost all BDs, separated by more than an extra
$\sim$$1\text{\,}\mathrm{s}$ after the voltage recovery period, are pBDs.

Since the results of scenarios 1–5 clearly indicate that the slope of the
voltage ramp affects the BD activity during the short-term conditioning, we
now turn to the analysis of the results obtained in the measurements performed
with the purpose of reducing the total ramping time while keeping the lowest
slope. In the next measurement set (scenarios I–V), we start the voltage
recovery from different fractions of $V_{t}$ while keeping the slope of the
voltage increase unchanged.

Unfortunately, in Table 2 we see that the shortest ramping time in this set of
measurements, i.e., scenario I with $V_{0}=$80\text{\,}\mathrm{\char
37\relax}$$, performed by far the worst, as more than
$50\text{\,}\mathrm{\char 37\relax}$ of the scenarios occurred on the first
pulse after the previous BD. This made it impossible to fit the two-
exponential curve to the PDF of scenario I. The scenarios with other starting
voltages were nearly equal in terms of minimizing the amount of sBDs.
Subsequently, the best result was obtained with the scenario starting from
zero voltage, which also had the greatest number of ramping pulses. Hence, we
conclude that not only the slope of ramping, but also the total number of
ramping pulses is important for a stable conditioning procedure with a minimal
breakdown rate.

Subsequently, we now verify this observation by fixing the total number of
pulses, but using a combination of a fixed slope with varied initial voltage
value or the fixed initial value with a varied slope. In these scenarios A–D,
we applied the voltage ramping with a delay: the pulses for the first
$2\text{\,}\mathrm{s}$ or $3\text{\,}\mathrm{s}$ had a flat, non-zero pulsing
voltage, with the voltage ramp that started immediately after. Although the
results from these measurements look very similar because of the same amount
of the recovery pulses, we still observe that the scenario A (starting from
the $60\text{\,}\mathrm{\char 37\relax}$ of $V_{t}$) performed the worst. In
this case, the long pulsing at a relatively high voltage value, although
without ramping, increased the unstable conditioning performance of the PDF.
Namely, we see that the distribution has the broadest peak, which means that
the BDs start before the voltage is ramped to the target value and continue
long after $V_{t}$ is reached.

In the scenarios B–D, the conditioning performance is nearly identical,
although scenario D stands out by a small margin. It is a surprising result,
which is rather counter-intuitive taking into account our previous observation
of dependence of the conditioning process on the voltage slope. In Fig. 2, we
clearly see that the scenario D had the steepest voltage increase slope,
although it resulted in BDs occurring at the largest number of pulses within
the ramping period. It also had the narrowest PDF peak (the appearance of BDs
is more localized around its peak value) and a fast decay towards the higher
number of the pulses between subsequent BDs.

Fig. 4 shows the peak BD probabilities from Fig. 3 versus the total energy $E$
that is entered into the system via pulsing during the ramping. Scenario I is
excluded from the figure since its peak BD probability is several orders of
magnitude higher compared to the other measurements. The energy is calculated
as the integral of the measured voltage increase curves in Fig. 2. It is
interesting to observe that all points obtained in different measurements
follow closely the exponential dependence fitted as

$\text{PDF}_{\text{peak,slopes}}=A\exp(BE)$ (2)

with the fitting coefficients $A=$(3\pm 0.4)\text{\times}{10}^{-3}$$ and
$B=$(-1\pm 0.2)\text{\times}{10}^{-2}$$ and the coefficient of determination
$R^{2}=0.97$.

Figure 4: Peak BD probability during linear voltage recovery as a function of
measured average voltage increase per pulse, $\Delta V$. Also a logarithmic
fit to the points is shown.

Finally, we revisit the step-wise voltage recovery measurement ($V_{t}$ is
reached using 2000 pulses in 20 steps), which is commonly used in metal
surface BD conditioning experiments as reported in Saressalo  _et al._
(2020b); Profatilova  _et al._ (2020). In Fig. 5, we analyze the data within
each step (between two subsequent changes in the voltage values) to determine
the decay rate of the BD probability within a ramping step. In the scenario,
the step length is 100 pulses, during which the voltage stays constant. In the
figure, we plot the PDF produced by a step-wise voltage recovery scenario as a
function of the number of pulses between two consecutive BDs. We noticed that,
during the ramping process, the PDF shows a saw-tooth behavior, as was
observed in Refs. Saressalo  _et al._ (2020a, b). The behaviour shows a strong
increase in the PDF at the first pulses after the voltage change and a rapid
decay until the end of the step. We now analyze the decay rate of the BD
probability $P$ by fitting it to an exponential function

$P(t)=\exp\left(-\frac{\ln{(2)}}{\tau_{1/2}}t\right),$ (3)

where $t$ is the time (or the number of pulses between the BDs). The half-life
$\tau_{1/2}$ was found to be $2.9\pm
0.6\text{\,}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{s}$,
which equals $1.4\pm 0.3\text{\,}\mathrm{ms}$ (coefficient of determination
$R^{2}=0.95)$. This shows a very sharp decay in the PDF after the change of
the voltage, confirming the strong effect of the idle time during the voltage
change. Furthermore, we analyze the relationship between the peak BD
probability during a ramping step and the voltage increase after the previous
step in the inset of Fig. 5. This dependence can be fitted to the logarithmic
function of the voltage increase between the ramping steps

$\text{PDF}_{\text{peak,steps}}=a_{\text{steps}}\log(\Delta
V)+b_{\text{steps}}$ (4)

are $a_{\text{steps}}=$(3\pm 2)\text{\times}{10}^{-3}$$ and
$b_{\text{steps}}=$(-2\pm 1)\text{\times}{10}^{-2}$$ (coefficient of
determination $R^{2}=0.80$). The non-ideal fitting could be explained by the
fact that in this case, not only the voltage increase, but also the idle time
between the ramping steps affects the BD probability Saressalo  _et al._
(2020b).

Figure 5: Decay of BD probability as a function of pulses (or time) since the
beginning of the ramping step. The red line shows an exponential fit on the
data. The inset shows the relationship between the peak BD probability during
a ramping step and the increase of voltage since the previous step.

##  IV Discussion

The results shown in Table 2 confirm that the overall behavior of most linear
ramping scenarios tested in this work are very similar. Since there were no
pauses in these experiments, neither during nor after the voltage ramp, the
PDF decay slopes seen in Fig. 3 after the peak values are very similar and did
not strongly depend on whether a short or long ramping time was used.
Quantitatively the shape similarity is confirmed by the similar values of
$\alpha$ and $\beta$, i.e., the respective pBD and sBD rates for all the
scenarios. However, we noticed that the most quiet and, hence, least
detrimental conditioning process is achieved in the longest voltage recovery
process, i.e., the one with the lowest ramping slope and the greatest number
of recovery pulses. Hence, we separately analyzed which of the two parameters
is the most important for optimization of the conditioning process.

Fixing the ramping slope, but aiming to reduce the time of the recovery
process, we performed the measurements with different starting voltages
(scenarios I–IV). The results emphasize the importance of the slope, showing
very comparable results for almost all the scenarios except for the shortest
one. The scenario with $V_{0}=$80\text{\,}\mathrm{\char 37\relax}$$ performed
significantly worse than the others. This can be explained by the fact that
starting the voltage recovery at such high voltage was very close to scenario
in which no recovery was used, which was found detrimental in Saressalo  _et
al._ (2020b). Yet, comparing the measurement data, we still found that, by a
small margin, the conditioning was performed the best, when the voltage was
recovered from zero values and, hence, had the greatest number of recovery
pulses ($10\,000$).

In the third set of measurements (scenarios A–D), we verified the effect of
the total number of pulses, alternating pulsing at the flat voltage value to
the voltage ramping mode. To our surprise, the scenario D with the steepest
ramping slope resulted in the least number of breakdowns. We believe that in
this experiment the combination of long pulsing at relatively high, but still
sufficiently low value, added the effect of a gentle cleaning of the surface,
which allowed to ramp the voltage even faster than usual without increasing
the number of breakdowns. On the contrary, the breakdown rate and the average
number of sBDs between the two subsequent pBD decreased dramatically (see
Table 2). However, the flat value in the voltage during the initial pulsing
plays role. Similar to scenario I, it was found that a high $V_{0}$ in
scenario A made it perform worse than the others. We also note that overall,
in Fig. 3 the scenarios A (long pulsing at $V_{0}$=$60\text{\,}\mathrm{\char
37\relax}$$V_{t}$) and B (short pulsing at $V_{0}$=$40\text{\,}\mathrm{\char
37\relax}$$V_{t}$) produced very similar PDF curves, while the curves of the
scenarios C and D (in both, the long pulsing at
$V_{0}$=$40\text{\,}\mathrm{\char 37\relax}$$V_{t}$) where slightly more
localized. This shows that the PDF shape is mostly determined by the slope of
the ramping curve, while the peak location correlates with the number of
ramping pulses.

Similarity in the PDF shapes of Fig. 3 confirms that the surface conditions
were similar during all the experiments, since other parameters of the
experiments, such as the gap length and the target voltage $V_{t}$, remained
unchanged. However, we noticed that the BD probability during the early pulses
after the voltage reached $V_{t}$, was increased as the ramping time was
shortened. Moreover, these BDs with the smallest number of pulses between
them, form their own distribution, which can be seen in Fig. 3 as short tails
on the left side of the graph. The shapes and the maxima of these
distributions are different for the different ramping slopes of the post-
breakdown voltage recovery.

Considering that the right-hand parts of the PDFs in Fig. 3 can be fitted to a
Poisson distribution using a two-exponential model, we can assume that both
the sBDs and pBDs are produced by distinct Poisson processes. This also
implies that each BD event must be independent of the preceding one, i.e., it
must appear with equal probability either immediately or after any number of
pulses since the previous BD. Existence of the tails in the region of the
smallest numbers of pulses between the consecutive BDs, with the peak values
PDF${}_{\text{peak}}$ indicates of enhancement of the BD probability for these
values compared to others, i.e., BDs are more probable to occur right after
the end of the ramping process compared to any other pulse number in the
region where the voltage is constant $V=V_{t}$. The difference in
PDF${}_{\text{peak}}$, consistent with the ramping slopes used during the
recovery process, reveals that the surface adjusts differently to the voltage
increase depending on the rate of the ramp.

In Fig. 4, we found out that the PDF${}_{\text{peak}}$, as a function of the
energy pumped into the system during the ramping pulses, follows an
exponential dependence. The more energy is put into the system during the
ramping, the lower the peak BD probability. This suggests that the work
performed by the ramping pulses affects the surface and broadens the PDF peak,
making the BD occurrence more delocalized in terms of pulses between
consecutive BDs. This is why a more localized probability is seen for the
shorter voltage recovery scenarios in Fig. 3. However, when the energy is
already significant, the further localization of sBDs during the first pulses
slows down.

Furthermore, we also note that the percentage of the sBDs and the average
length of the sBD series $\mu_{sBD}$ are also monotonically decreasing with a
decrease in the ramping slope. Surprisingly, the average length of the sBD
series ($\mu^{\prime}_{sBD}$), which actually occurred after a pBD, showed
much weaker dependence on the ramping slope, fluctuating between 2-3 sBD after
a pBD for the scenarios 1–5. It is clear that the shortening of $\mu_{sBD}$
with the decrease of the slope is explained by surface conditioning mainly via
less violent and smoother distributed surface pBDs.

As it was mentioned earlier, there is a consistent shift in the cross-point
$N_{\text{cross}}$ of the two exponential fits of the PDFs by approximately
2000 pulses ($1\text{\,}\mathrm{s}$) after the voltage ramp is finished.
However, this shift is stronger for the fast ramp and reduces with the
increase of the ramping time. We relate this trend to the delays in the
voltage ramping due to RC coupling. As it was shown in Fig. 2, the target
voltage is not reached immediately after the given number of pulses, but with
some delay. The delay is greater when the voltage increase is steeper. This
suggests that a vast majority of the BDs, that occur after the target voltage
is reached and stabilized, are pBDs. The apparent exception of this trend is
found in scenario 5, where we see that $N_{\text{cross}}$ in this case is
shifted deeper towards the greater number of pulses. However, we draw the
attention to the fact that we obtained the shortest series of consecutive sBDs
using this voltage recovery scenario, which reduces statistical accuracy of
the obtained results. In overall, we found the smaller sBD rates and much
lower localization of sBDs at the first pulses after the voltage ramp.
However, the $N_{sBD}$ is consistently lower than in any other experiments,
which confirms the conclusion about overall conditioning performed via pBDs
mainly.

From Fig. 3, it can be seen that the BDs start occurring closer to the end of
the ramping period when the voltage ramping slope is smaller. This indicates
that the metal surface can tolerate stronger electric fields if the voltage is
ramped slowly. The result shown in Fig. 4 is also in line with this argument.
Here we saw that the BD probability is lower if more energy is put into the
system during the early pulses.

As we saw previously in the experiments with the step-wise voltage recovery
scenarios, the increase in the voltage value before each following step
increased the BD probability drastically during the first few pulses. We
showed in Fig. 5 that this decay is very fast with a half-time of
$1.4\text{\,}\mathrm{ms}$. This suggests that any pulse, also in the linear
voltage ramp scenarios, with a voltage higher than that of the previous,
increases the BD probability for a short time and this increase depends on the
difference $\Delta V$ between the pulses.

The results above can be associated with the existence of short-lived field
emitters on the electrode surface. When the surface is exposed to an electric
pulse with an amplitude sufficient to initiate the formation of new field-
emitting features, fresh emitters can be born emitting electrons into vacuum.
Each emitter can either exhaust itself or enter into a runaway process, which
can eventually lead to a full BD event Kyritsakis  _et al._ (2018). One of the
known factors determining the outcome of the emitter is the electric field
strength, which has to be sufficiently high to launch the runaway process
Kyritsakis  _et al._ (2018). The hypothesis assumes that the amount of new
emitters born is proportional to the change in the electric field strength.
During the voltage change, there is a sudden increase in the external electric
field strength at the surface, which magnifies the local gradients, driving
the atomic migration on the surface. The relationship between the peak BD
probability and the ramping energy found in Fig. 4 also supports this
assumption. The work performed by the electric pulses at relatively low
voltages can be sufficient to bring out the field emitters, but not enough to
start the runaway process. Thus, when the system enters into pulsing with high
enough voltages that could cause BDs, many of the possible field emitters are
already exhausted, thus lowering the BD probability.

However, more studies would be required to confirm the hypothesis involving
the dynamics of the short-living field emitters.

Moreover, it is apparent that in the case of the longest ramping, the system
had experienced the largest number of pulses before the voltage reached the
target value. Electric fields exert tensile Maxwell stress on metal surfaces
Griffiths and Inglefield (2013), hence in the pulsing regime, this stress may
induce surface vibrations. These vibrations can contribute, e.g. in cleaning
of the surface by detachment of the loosely bound impurity atoms, which is
another possible way of reducing the BD probability.

The counter-intuitive results between the first and third measurement sets can
be explained by the fact that, once the surface is cleaned from the impurity
atoms due to the vibrations during the early ramping pulses, a higher strain
rate (produced by a higher voltage slope) may actually increase the yield
strength of the metal, thus resulting in fewer field emitters that could cause
BDs Liu  _et al._ (2008); Bandyopadhyay  _et al._ (2017). However, this means
that the voltage during the early ramping pulses needs to be high enough to
produce any cleaning effect, while at the same time the voltage cannot be so
high that it would cause BDs. It was found that $40\text{\,}\mathrm{\char
37\relax}$ of $V_{t}$ is a good compromise between these two.

##  V Conclusions

Different post-breakdown voltage recovery scenarios with linear ramping and no
pauses were investigated in this study. The main purpose was to determine the
best scenario resulting in the most efficient mitigation of the secondary BDs
and to understand the relationship between the rate of the voltage increase
and the BD probability.

We observed that the breakdown rate, the fraction of secondary BDs and the
average number of secondary BDs per a primary BD in a series were the lowest
for the scenarios with the most pulses to reach the target voltage. The slower
the voltage was recovered, the fewer BDs were produced during the recovery.

In all scenarios, the sBD probability was concentrated at the end of the
ramping period, regardless of the number of the ramping pulses. The
localization of the sBD probability was found to be dependent on the slope of
the voltage increase. It was also found that the peak BD probability is
dependent on the energy of the ramping pulses: the more energy is inserted
into the system during the ramping, the lower the peak BD probability.

The lowest breakdown rate was achieved with a combination voltage recovery
scenario with 8000 pulses of pulsing at a flat voltage,
$40\text{\,}\mathrm{\char 37\relax}$ of the target voltage, followed by a
rapid increase to the target voltage over
$2000\text{\,}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{s}$.

## Acknowledgements

The authors would like to thank the China Scholarship Council(CSC), which
supported parts of the work.

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