Supercritical fluids for high power switching Citation for published version (APA): Zhang, J. (2015). Supercritical fluids for high power switching. [Phd Thesis 1 (Research TU/e / Graduation TU/e), Electrical Engineering]. Technische Universiteit Eindhoven. Document status and date: Published: 01/01/2015 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected]providing details and we will investigate your claim. Download date: 18. Oct. 2022
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Supercritical fluids for high power switching
Citation for published version (APA):Zhang, J. (2015). Supercritical fluids for high power switching. [Phd Thesis 1 (Research TU/e / GraduationTU/e), Electrical Engineering]. Technische Universiteit Eindhoven.
Document status and date:Published: 01/01/2015
Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.
Figure 4.9 – Current through the SC switch (C) with different settings of inductance L2 in
the arc interruption testing circuit (figure 3.18), under N2 pressure of 50 bar and gap width of
0.986 mm. The value of C2 is 1.3 nF. The measured current rate-of-rise and current frequency
is di/dt = 0.14−3 A/μs, f = 100 kHz for L2 = 9.8 mH; di/dt = 0.3−4.2 A/μs, f = 83 kHz for
L2 = 3.8 mH; di/dt = 1.4−29 A/μs, f = 285 kHz for L2 = 800 μH, respectively.
0 50 100 150 200 250−100
−50
0
50
100
Cur
rent
[A]
0 50 100 150 200 250−100
−50
0
50
100
Time [μs]
Cur
rent
[A]
C2=2 nF
C2=1.3 nF
Figure 4.10 – Current through the SC switch (C) with different settings of capacitance C2in
the arc interruption testing circuit (figure 3.18), under N2 pressure of 50 bar and gap width of
0.986 mm. The value of L2 is 800 μH.
the results we find that under the current waveform shown in figure 4.9(b)-(c) (correspond-
ing to L2 ≤ 3.8 mH), the SC switch was not able to interrupt the current at a gap width of
d ≤ 1 mm. However, at a gap width of d ≥ 1.2 mm, we observed temporary current inter-
ruptions at the current zero crossing point, at t = 30 μs or later after the breakdown of the
56 4. EXPERIMENTAL INVESTIGATION OF BREAKDOWN AND RECOVERY IN SCFS
switch.
An example current waveform at p = 60 bar and d = 1.29 mm is illustrated in figure
4.11. The circuit parameters are L2 = 3.8 mH and C2 = 1 nF. No forced N2 flushing was
supplied during the experiment. From the enlarged view of the selected time region in
figure 4.11, we see that from 29.4 μs after the breakdown onward, at every current zero
crossing point, the current is interrupted temporarily, both in positive and negative slopes.
The arc current measured by the current probe indicated in figure 3.18 is influenced by the
current flowing in C2, and also by the current induced by the stray capacitance of the SC
gap. This explains the slightly negative current during the charging process of capacitor C2
(in the time between −10 μs and 15 μs in figure 4.11), and the displacement of the current
interruption from zero crossing in the measured current waveforms.
0 50 100 150 200 250 300 350 400
-40
-20
0
20
40
60
Time [ s]
Curre
nt [A
]
29.3 29.4 29.5 29.6-2
0
2
53 53.5 54-10
-5
190 192 194 196 198
-2
0
2
Figure 4.11 – Measured current through SC switch (C) under the arc interruption testing circuit
given in figure 3.18. Pressure 60 bar, gap width 1.29 mm, L2 = 3.8 mH, and C2 = 1 nF. No
forced N2 flushing was supplied during the experiment.
In case of the arc current shown in figure 4.9(a), a successful interruption was observed
at 2 ms after the breakdown in a gap of d > 1.7 mm, with forced flushing estimated to be
50 Liter/h (corresponding to 2.73 m3/h at STP), i.e. flow velocity approximately 0.05 m/s
in a 1.7 mm gap. Examples of the arc voltage and arc current under the situation of success-
ful arc interruption are illustrated in figure 4.12 and figure 4.13, in the scenario of forced N2
flushing with volume of 50 Liter/h and no flushing, respectively. In the arc voltage meas-
urements the voltage induced in the measurement loop by the oscillating current, though
not shown here, is measured to be � 10 % of the signal.
Figure 4.12(b) gives the enlarged view of the selected two time regions for the tempor-
ary current interruptions: t = 901.8 μs and t = 906.4 μs. From the figure we can see that
the current was temporarily interrupted, which can be identified by a short duration rise of
4.4. CURRENT INTERRUPTION ANALYSIS 57
0 1000 2000 3000 4000 5000 6000-500
0
500Vo
ltage
[V]
6
Temporary interruption
Successful interruption
500 V
0 1000 2000 3000 4000 5000 6000-4
-2
0
2
4
Time [ s]
Cur
rent
[A]
900 950 1000 1050 1100-4
-2
0
2
4
6
Time [ s]
Curre
nt [A
]
(a) Voltage and current waveforms in SC switch (C). An enlarged view of a half cycle of the current
waveform, in the range of t = 900−1100 μs, is plotted as well.
902 904 906 908 910-400
-200
0
200
400
Time [ s]
Volta
ge [V
]
902 904 906 908 910-0.5
0
0.5
Time [ s]
Curre
nt [A
]
(b) Enlarged view of three selected time regions where temporary current interruptions are observed:
t = 901.8 μs and t = 906.4 μs. The other current interruption moments are not enlarged here.
Figure 4.12 – Voltage and current waveforms measured in SC switch (C) in the arc interruption
circuit. Pressure 50 bar, gap width 1.814 mm, L2 = 9.8 mH, and C2 = 1 nF. Forced N2 flushing
of about 50 Liter/h (2.73 m3/h at STP), i.e. flow velocity approximately 0.05 m/s in the gap,
was supplied during the experiment.
transient recovery voltage. After a temporary interruption of 0.2−0.3 μs , the SC switch
undergoes a re-ignition, which is represented by the sudden voltage collapse and continu-
58 4. EXPERIMENTAL INVESTIGATION OF BREAKDOWN AND RECOVERY IN SCFS
0 1000 2000 3000 4000 5000 6000-500
0
500Vo
ltage
[V]
0 1000 2000 3000 4000 5000 6000-4
-2
0
2
4
6
Time [ s]
Curre
nt [A
]
2.35 ms
Successful interruption
-250 V
Figure 4.13 – Voltage and current waveforms measured in SC switch (C) in the arc interrup-
tion circuit. Pressure 50 bar, gap width 1.814 mm, L2 = 9.8 mH, and C2 = 1 nF. No forced N2
flushing was supplied during the experiment.
ation of (arc) current. The sudden collapse of the voltage indicates that the switch undergoes
a dielectric breakdown, i.e., the arc was interrupted thermally before the re-ignition of the
switch. After about 2 ms the current is successfully interrupted, and the voltage on the an-
ode of the switch rises up to 500 V and remains almost constant. By comparing figure 4.12
and figure 4.13 we observed that the current was interrupted 0.15 ms later without flushing
than with forced N2 flushing through the gap. Without forced flushing, the arc voltage in
SC N2 switch has a value of ≥ 100 V from 100 μs onward after the breakdown. Under
situation of forced flushing the value of arc voltage is higher than that without flushing, and
increases after each temporary interruption in the scenario of forced flushing. This increase
might reflect the negative current-voltage characteristics of arcs in gaseous media.
The dependence of the interruption capability on the pressure of the medium and on the
flushing situation was investigated next. The current and voltage slopes at the moment of
successful arc interruption di/dt and du/dt at the pressure of 10−40 bar are illustrated in
figure 4.14. The value of the current rise slope di/dt first slightly increases with pressure,
then for p > 20 bar, decreases slightly with pressure, while in conventional gas media the
behavior is a monotone increase with pressure. This abnormality needs more investiga-
tion. The rate-of-rise of transient recovery voltage du/dt increases with pressure, which is
consistent with the observations in gas media. The results under forced flushing condition
presented in figure 4.14 suggest that forced flushing results in faster recovery of the former
arc channel. However, from the rate-of-rise of the dielectric recovery voltage corresponding
to the temporary arc interruption shown in figure 4.15, we observe that the value of du/dt,
at 0.9 ms and 0.43 ms after the current initiation, decreases with the media pressure. This
phenomenon needs further investigation as well.
4.4. CURRENT INTERRUPTION ANALYSIS 59
0 5 10 15 20 25 30 35 40 450
1000
2000
3000
4000
5000
6000
Pressure [bar]
di/d
t [A/
s]
Forced flushing
Non-flushing
d=2.139 mm d=2.112 mm
d=2.151 mm
d=2.195 mm
0 5 10 15 20 25 30 35 40 450
1
2
3
4
5
6 x 106
Pressure [bar]
du/d
t [V/
s]
Forced flushing
Non-flushing
d=2.139 mm
d=2.112 mm
d=2.151 mm d=2.195 mm
Figure 4.14 – The rate-of-rise of current di/dt and rate-of-rise of recovery voltage du/dt at the
moment of successful arc interruption in SC switch (C), under situation of forced flushing and
no flushing situation, at various pressures.
15 20 25 30 35 40 45 50 55
4
6
8
10
12x 108
Pressure [bar]
du/d
t [V/
s]
0.89 ms from current intiation0.43 ms from current intiation
d=1.814 mm
d=1.79 mm
d=1.763 mm
d=1.744 mm
Figure 4.15 – The rate-of-rise of recovery voltage du/dt as the function of the pressure at differ-
ent moment after the initiation of the arc in the switch gap, under the situation of forced flushing
in the gap.
60 4. EXPERIMENTAL INVESTIGATION OF BREAKDOWN AND RECOVERY IN SCFS
Tabl
e4.
1–
Ex
per
imen
tal
resu
lts
of
the
arc
inte
rru
pti
on
test
ing
of
SC
swit
ch(C
).’×
’:fa
ilu
reo
fcu
rren
tin
terr
up
tio
n;
’T’:
cap
able
of
tem
po
rary
curr
ent
inte
rru
pti
on
;’�
’:su
cces
sfu
lcu
rren
tin
terr
up
tio
n.
L2=
3.8
mH
,C
2=
1.2
nF
L2=
6.8
mH
,
C2=
1.0
nF
SC
swit
chd≤
1m
md>
1.2
mm
d>
1.4
mm
d>
1.4
mm
(C)
p[b
ar]
20
30
40
50
20
30
40
50
60
50
50
60
Cu
rren
td
[mm
]
Inte
rru
pti
on
0.9
17
0.9
41
0.9
56
0.9
86
1.2
11
.24
1.2
51
.26
51
.29
1.4
96
1.4
96
1.5
13
No
flu
shin
g×
××
×T
TT
TT
TT
T
Fo
rced
flu
shin
g×
××
×T
TT
T-
TT
T
L2=
9.8
mH
,C
2=
1.0
nF
d>
1.4
mm
d>
1.7
mm
d>
2.0
mm
p[b
ar]
30
40
50
60
20
30
40
50
10
20
30
40
d[m
m]
1.4
58
1.4
73
1.4
96
1.5
13
1.7
44
1.7
63
1.7
90
1.8
14
2.1
12
2.1
36
2.1
51
2.1
95
No
flu
shin
gT
TT
T�
�T
��
��
�F
orc
edfl
ush
ing
TT
TT
��
��
��
��
4.5. ICCD IMAGE OF DISCHARGE IN SC N2 61
4.5 ICCD image of discharge in SC N2
Discharge imaging inside the SC switch provides valuable information for the theoretical
analysis of the breakdown and recovery in SCFs. In this section the photographs of the
discharge channels in SC N2 switch are taken with an intensified CCD camera mounted
with a microscope lens. The ICCD camera is a 4 Picos from Stanford Computer Optics
with 780×580 pixels, 8.3×8.3 μm pixel size.
Figure 4.16 – Schematic of the electric circuit for triggering of ICCD camera.
The camera is synchronized with the pulsed voltage supplied to the trigger electrode
of the SC switch, using a triggering circuit sketched in figure 4.16. Simultaneously with
the charging of capacitor Ch in the main circuit (capacitor embedded in SC switch (B)),
the capacitor C2 in the triggering circuit, via an inductance L2 = 21 mH, is charged with a
relatively slower voltage increasing slope, but to higher amplitude than that on Ch. During
the charging process of Ch, the voltage on the trigger pin also increases due to the LCR
system explained in section 3.3.2. In order to prevent the voltage increases on the right
side of S1 (2-stage TLTL side), an isolating capacitor Ci = 200 pF is applied. From the
impedance equation Z = 1/(ω ·Ci) we can see that impedance of Ci is high at low frequency,
hence it can block the energy flow from the trigger pin to S1 during the slow charging
process of Ch.
At a time moment after the voltage on Ch reaches the plateau, the voltage on C2 reaches
the threshold voltage of the air switch S1. Once S1 breaks down, a voltage impulse is
generated and transmitted to the trigger electrode of SC switch (B) through a 2-stage TLT.
The 2-stage TLT has two functions: 1) to amplify the peak-voltage of the impulse by two
62 4. EXPERIMENTAL INVESTIGATION OF BREAKDOWN AND RECOVERY IN SCFS
times, and 2) to introduce a delay of 100 ns for the impulse to reach the trigger electrode of
SC switch (B). Ci, under this high frequency pulse, has low impedance, hence the voltage
pulse can transmit through Ci, and be applied to the trigger pin.
-2 -1 0 1 2 3x 104
-1
0
1
2
3
4
x 104
Time [ns]
Volta
ge [V
]
-400 -200 0 200 400 -1
0
1
2
3
4 x 10 4
Time [ns]
Volta
ge [V
]
trigger pin spark gap S1
camera opening
signal sent to camera
SC switch anode
Figure 4.17 – The typical voltage waveforms measured on the main capacitor Ch, trigger capa-
citor C2, isolating capacitor Ci, and the trigger pin of the SC switch.
-100 0 100 200 300 400 500 600
0
100
200
300
400
500
600
700
800
Time [ns]
Curre
nt [A
]
Current through SC switchCamera monitoring signal
Camera opening
Gap width 0.32 mm
Discharge channel at camera opening
Figure 4.18 – Imaging of discharge in the SC switch (B) with an ICCD camera. The N2 pressure
is 70 bar and the gap width 0.32 mm. The exposure time of the camera is 0.3 ns; the ND filter
intensity is 100 X. The camera opening signal illustrated in figure 4.18 represents the feed back
signal from the ICCD camera to the oscilloscope. The time delay for the waveforms caused by
the cables has already been deducted.
At the moment of S1 firing, a noise pick up coil placed nearby S1 induces a voltage sig-
4.5. ICCD IMAGE OF DISCHARGE IN SC N2 63
nal. Via an integrator circuit shown in appendix A2 and a buffer circuit, this signal serves
as the triggering signal for the ICCD camera. The trigger signal is sent to the camera about
100 ns earlier than the firing of the main gap. This 100 ns compensates the reaction time of
the camera, in practice ≈ 97 ns (including the delay of signal generator and internal delay
of ICCD camera). Once the voltage pulse reaches the trigger electrode and superimposes
on the originally induced voltage, the trigger gap (toroidal gap width ≈ 0.1 mm) fires and
initiates the breakdown of the main gap. The opening of the ICCD camera is synchronized
with the breakdown of the SC switch, hence to capture the discharge channel in SCF. The
time sequence of the typical voltage waveforms measured on the main capacitor Ch, the trig-
gering capacitor C2, the isolating capacitor Ci, and on the trigger electrode of the SC switch
during the operation is illustrated in figure 4.17. The time delay of the signal transmission
caused by the cable has been deducted from the waveforms shown in figure 4.17.
0 ns
5 ns
10 ns
Td:
60 ns
80 ns
Pressure: 70 bar Gap width: 0.38 mm Breakdown voltage: 25 kV Exposure time: 1 ns ND filter intensity: 100 X
Pressure: 40 bar Gap width: 0.35 mm Breakdown voltage: 25 kV Exposure time: 3.5 ns ND filter intensity: 200 X
Figure 4.19 – Images of the discharge channels in the SC N2 switch (B) taken by an ICCD
camera. Td: time moment after the current appearing in the gap.
The light emission from the SC N2 breakdown is very strong. In order to prevent the
overexposure of the camera, a neutral density (ND) filter (filtering intensity 2−400 X) is
placed between the optical window of the SC switch and the microscope lens of the ICCD
camera. At the moment of current detected in the SC N2 gap, the discharge diameter meas-
ured as full width at half maximum (FWHM) of the peak intensity is estimated to be 70 μm,
as can be seen in figure 4.18. This value is larger than the predicted streamer diameter by
the similarity law (∼ 3 μm at 70 bar, as calculated from the parameters in table 5.2 given in
chapter 5), indicating that the images we captured is somewhat after the streamer has bridges
the gap and expanded. Here the possible explanation is given. The streamer propagation
velocity in N2 at a pressure of 1 bar was measured to be 7×104 m/s under the applied elec-
64 4. EXPERIMENTAL INVESTIGATION OF BREAKDOWN AND RECOVERY IN SCFS
tric field of 0.31 kV/mm [189], and the velocity increases with the applied electric field.
Results in [190] indicate that the streamer velocity is inversely proportional to the medium
pressure. If we assume a linear increase of the streamer velocity versus the electric field,
the streamer velocity at the pressure of 70 bar under the electric field of 80 kV/mm is es-
timated to be 2.58×105 m/s, i.e., the streamer transition time across a 0.32 mm SC N2 gap
is 1.2 ns. Fast expansion of the streamer channel is assumed to happen, from the streamer
bridging the gap till the current is detected by our current sensor. We observe typically
only one spark channel during the breakdown of the SC switch, at varying positions of the
electrode ring.
Figure 4.19 shows the images of the discharge at Td = 0−80 ns (Td means time after
the current appeared in the gap) at p = 70 bar, d = 0.38 mm; as well as at Td = 100−600 ns
at p = 40 bar, d = 0.35 mm. From the images we observe that the bright channel expands
in the time span of Td = 0−80 ns. From Td = 100 ns onward, the diameter of the bright
channel keeps almost constant, while only the intensity of light emission becomes weaker.
It is reasonable to conclude that for the discharge channel in SC N2, fast expansion happens
at time Td ≤ 100 ns. It agrees with the observation in [191, 192] that fast expansion of the
spark radius happens within the first few hundred nanoseconds after the spark onset.
4.6 Conclusions
In this chapter we investigated the dielectric strength, dielectric recovery, and current inter-
ruption capability of SC N2 switches. The electric field across the SC gap and the discharge
radius were observed, providing important data for the theoretical simulations which will be
introduced later in chapter 5. The following conclusions are drawn from the experimental
results.
Dielectric strength
The dielectric strength of SC N2 have been recorded under different voltage pulses: slow
(1.66×103 kV/s), moderate (2.5×106 kV/s), and fast (2.0×109 kV/s) pulses.
Figure 4.20 summarizes the breakdown field of SC N2 under theses pulses at a regime
of gap width for N2 pressure of 70 bar. Data for other pressure values are not available.
From the dashed line in the figure we can clearly see that at fixed gap width and pressure
e.g., p = 70 bar and d = 0.3 mm, the breakdown field in SC N2 increases under increased
voltage slope of the impulses. Under the same voltage pulse, the breakdown field is higher
at smaller gap width.
Dielectric recovery
The dielectric recovery characteristics of a SC N2 are derived from the experimental results:
• The recovery breakdown voltage increases with N2 pressure and gap width;
4.6. CONCLUSIONS 65
40
60
80
100
120
140
160
180
200
Brea
kdow
n fie
ld [k
V/m
m]
kV/s kV/s kV/s
Figure 4.20 – Breakdown field of SC N2 under slow (1.66×103 kV/s), moderate
(2.5×106 kV/s), and fast (2.0×109 kV/s) charging slopes, at the pressure of 70 bar. Each
bar represents a regime of gap width. The dashed line represents the breakdown field at gap
width of 0.3 mm under the three voltage rising rates.
• The recovery breakdown voltage decrease at higher repetition rate at gap width of
0.25 mm;
• The recovery breakdown voltage at smaller gap width (0.15 mm) show less significant
relationship to the repetition rate compared to that at larger gap width (0.25 mm).
Current interruption capability
The current interruption capability of high-frequency (≥ 7 kHz) and low (< 500 A) cur-
rent of SC N2 is investigated experimentally. From the experimental results, we draw the
following conclusions:
I. Under the charging voltage supplied by our testing circuit (50−70 kV), the cur-
rent with rate-of-rise in the range of 0.14−3 A/μs and oscillation frequency up to
100 kHz can be successfully interrupted in a SC switch with fixed electrodes, at ap-
proximately 2 ms after the current initiation;
II. Higher pressure results in a slight decrease of di/dt at the moment of successful arc
interruption, and an increase of the rate-of-rise of transient recovery voltage du/dt;
III. Forced flushing results in faster recovery of the former arc channel: increased arc
voltage, earlier successful current interruption, and increased rate-of-rise of transient
recovery voltage du/dt.
At the current interruption the returning voltage was between 200−500 V at 1.5 mm
gap width. The current slope di/dt at the interruption moment was between 2000 and
5000 A/s, in a 40 bar, 2.2 mm gap.
66 4. EXPERIMENTAL INVESTIGATION OF BREAKDOWN AND RECOVERY IN SCFS
Higher arc voltage is beneficial to limitation of di/dt and thus arc current crosses current
zero earlier. The increased arc voltage observed in SC switch under forced N2 flushing
situation indicates the advantage of SC N2 in arc interruption. Successful arc interruption
within 2 ms in non-moving contacts and small inter-electrode distance in present work also
provides evidence of high arc interruption capability of SC N2.
CHAPTER 5
THEORETICAL MODELING OF
DISCHARGE AND RECOVERY IN SCFS
The content of this chapter is based in parts on two recent publications by the author andco-authors of this work, see publications [188] and [7]. The writer of this thesis is firstauthor and acknowledges the contributions from the co-authors of these papers.
5.1 Introduction
The dielectric strength and subsequent dielectric recovery in SC N2 have been studied by
experimental investigations (chapter 4). The dielectric strength of SC N2 is in the range of
60−180 kV/mm, which is as high as most solid dielectrics. The dielectric strength of a SC
N2 switch recovers to 80 % of the cold breakdown value within 200 μs after the breakdown.
In order to verify the superiority of SCFs in high power switching, theoretical analysis is
also an important approach to understand the discharge characteristics of SCFs.
In this chapter we introduce two models for the simulation of the discharge and recovery
processes in SCFs. In section 5.2 we introduce a simple analytic model. The purpose of this
simple model is to get a rough idea about the recovery time of a SCF switch, thus to provide
design data for the SC switches introduced earlier in chapter 3. The electric field across the
gap during the discharge of SC N2 is analyzed with simplified circuits in section 5.3. In
section 5.4 an extended physical model is generated, aiming on deeply understanding the
discharge and recovery characteristics of a SCF.
67
68 5. THEORETICAL MODELING OF DISCHARGE AND RECOVERY IN SCFS
5.2 Simple analytic model
5.2.1 Model description
The physical model of a SC switch in this model is introduced in figure 5.1. The switch is
composed of an anode (a), a hollow-cylindrical trigger electrode mounted inside the anode
(b), and a plane cathode (c). The anode is a 40.0 mm cup with a hole of 21.0 mm in dia-
meter. A cylindrical trigger electrode with inner diameter of 9.0 mm and outer diameter of
18.0 mm is mounted in the hole of the anode, forming a gap distance of 1.5 mm between
the trigger electrode and the anode. The axial thickness of the plane cathode is 50 mm, and
distance between anode and cathode is in the range of 0−10 mm. The medium is blown
into the spark gap from both the trigger gap and the center of the trigger electrode, then
flows out via six symmetrical exists, with the arrow indicating the gas flow direction.
Figure 5.1 – Schematic of the switch for the simple analytical model. (a) anode; (b) hollow
cylindrical trigger pin; (c) cathode; (d) example of spark channel generated in the switch.
When the switch breaks down, we assume that all the energy from the external source is
deposited into the spark channel (d) generated in the inter-electrode gap shown in figure 5.1.
In order to analyze the recovery performance of the switch, we calculate the temperature
decay of the spark channel by assuming two separate processes: adiabatic expansion and
heat transfer (the latter being adapted from a model with spherical electrodes in [193]).
The process of volume expansion is much faster than that of the heat transfer, so these two
processes are assumed to happen in succession. The criterion of recovery of the switch is
defined as the moment at which the temperature inside the spark channel drops to a value
of 550 K, corresponding to a breakdown voltage equal to ≈ 80 % of the static dielectric
strength [194].
This simple thermodynamic model is assumed to be suitable for an estimate of the re-
covery time in a SCF, because:
I) In our range of temperature T and pressure p the behavior is near the behavior of an
ideal gas. Although SCF has density ρ similar to liquids, its compressibility is high
5.2. SIMPLE ANALYTIC MODEL 69
as in gases. This makes the adiabatic expansion law applicable for SCF.
II) The heat transfer model is general and applies to both gases and liquids.
5.2.2 Model Formulation
Once the switch breaks down, a finite amount of energy E (in our simulation we take the
value as 0.7 J) suddenly releases into a cylindrical spark channel developed in the inter-
electrode gap [195]. The initial condition of our simulation follows Plooster’s assumption:
a very rapid heating of a column of gas in very short time, before it starts to expand [196].
This means we assume a cylinder with homogeneous temperature and density, following
the state equation of a real gas. The external gas pressure, temperature and flow velocity
are kept constant in time and space. Gas inside the cylindrical spark channel undergoes a
rapid temperature rise while the gas density remains constant and equals to the background
density, following the equation:
E = cvρ0V0(T0 −Tg,0), (5.1)
in which cv [J/(kg ·K)] is the isochoric specific heat capacity of gas, Tg,0 = 300 K the tem-
perature of the gas before energy deposition, T0 [K] the temperature of the channel after
heating, ρ0 [kg/m3] the gas density after heating (equaling to the background density), V0
[m3] the initial volume of spark channel.
The volume of the cylindrical spark channel has to be given as an initial parameter. It
was reported that limiting temperature exist for specific gases, which is reached in a suffi-
ciently strong discharge [197]. As the limiting temperature is reached (in air 43000 K and in
N2 41000 K), the discharge emission spectrum is close to that of the blackbody, and further
released energy does not lead to an increase in the discharge plasma temperature. Instead,
the channel diameter increases, causing a larger activated gas volume [198]. So the initial
radius of the spark channel R0 can be calculated from equation (5.1) with T0 = 41000 K (for
N2). If the calculated radius R0 is less than 50 μm, then R0 is set to be R0 = 50×10−6 m.
V0 is adapted to the new value according to R0, and the value of initial pressure p0 [Pa]
before expansion can be calculated from the gas state equation. It is clear that with constant
density, increasing temperature causes large increase of gas pressure. The theory of cyl-
indrical strong shock waves gives the asymptotic solution for a strong shock radius versus
time [196]:
t0 =R2
0
Rc · a , (5.2)
in which Rc =√
E/(lgapBγpg,0) [m] is a characteristic value determined by the initial con-
ditions; B is a dimensionless constant specific to the characteristics of gases: 3.94 for air
and 3.37 for N2; γ = cp/cv is the ratio of specific heats; a [m/s] is the velocity of sound in
the background gas; lgap [m] is the inter-electrode gap width; pg,0 [Pa] is the gas pressure
before the energy deposition. A time of t0 [s] is needed for the spark to expand from the
characteristic value Rc [m] to the initial radius R0 [m] in our situation.
70 5. THEORETICAL MODELING OF DISCHARGE AND RECOVERY IN SCFS
Following this initially very rapid heating, the spark channel starts to expand due to the
pressure difference between the in- and outside of the channel, in which process, conserva-
tion of mass is applicable. During the volume expansion process, no heat transfer between
the spark channel and the environmental gas is considered (adiabatic expansion). The pres-
sure and temperature change due to the adiabatic expansion is described by equation (5.3):
p1
p0=
(V0
V1
)γ; (5.3a)
T1
T0=
(V0
V1
)γ−1
, (5.3b)
in which notation 0, 1 stands for before and after expansion respectively; γ = cp/cv is the
ratio of specific heats. The adiabatic expansion stops when the pressures in and outside the
channel are equal p1 = pg,0, i.e., the left side of equation (5.3a) equals pg,0/p0. The radius
of the spark channel after adiabatic expansion R1 [m] can be calculated with the equation:
R1 =
√V1
π · lgap. (5.4)
This adiabatic expansion is assumed to follow the the theory of moderate shock waves, so
the time duration of the expansion, denoted as t1 [s], can be calculated as equation [199]:
t1 =
√E
pg,0 · lgapγ ·B · a2+
R21
a2. (5.5)
After the adiabatic expansion, the model continues with the start of the heat transfer
phase. Now, heat transfer from the channel to the surroundings is seen as the only contribu-
tion to the temperature decay of the spark channel. In the heat transfer process the governing
equation is:
Q = (Tg(t2)−Tg,0)S =−cp ·m dT
dt2, (5.6)
where Tg(t2) [K] is the gas temperature in the spark channel; S [W/K] is sum of the products
of heat transfer coefficients and surface areas; m [kg] is the mass of gas inside the spark
channel. The term S is composed of the heat convection part (to the environmental gas)
with surface area Aconv [m2], and heat conduction part (to the electrodes) with surface area
Acond [m2], and the details are given by equation:
S =Nu ·kf
LAconv+
kaver
xAcond , (5.7)
where kf [W/(m ·K)] is the coefficient of thermal conduction at film temperature Tf [K]
(arithmetic mean of the spark channel wall temperature and the far-end gas temperature);
5.2. SIMPLE ANALYTIC MODEL 71
kaver [W/(m ·K)] is the averaged thermal conductivity coefficient of the hot gas over the
temperature decay range; x = 5×10−2 m is the length of the electrodes; Nu (Nusselt num-
ber ,dimensionless) is calculated with physics of forced heat convection across a circular
cylinder [200]:
Nu = C ·RemPr1/3; (5.8a)
Pr = υ/D; (5.8b)
Re = uL/υ , (5.8c)
in which Pr is Prandtl number (dimensionless); Re is Reynolds number (dimensionless); u
[m/s] is the velocity of gas flowing through the gap; υ [m2/s] is the kinetic viscosity of the
gas; D [m2/s] is the thermal diffusion coefficient of the gas. The constants C and m for
equation (5.8) can be found in table 5.1.
Table 5.1 – Constants for equation (5.8) for circular cylinders in cross flow [200].
Re 0.4−4 4−40 40−4000 4000−4×104 4×104 −4×105
C 0.989 0.911 0.683 0.193 0.027
m 0.330 0.385 0.466 0.618 0.805
The analytic solution for the gas temperature Tg(t2) in equation (5.6) is expressed as:
Tg(t2)−Tg,0
T1 −Tg,0= exp
( −t2
cpρV/S
), (5.9)
in which τ = cvρV/S is the time constant of the exponential decay of the gas temperature in
the heat transfer stage; T1 is the gas temperature after adiabatic expansion.
5.2.3 Results and discussions
The recovery time of a SC N2 switch after breakdown is calculated with this simple model.
Figure 5.2 gives the prediction of the recovery time of a SC switch insulated with 300 K,
150 bar SC N2, at a gap width of 0.4 mm and various flow rates after breakdown accompan-
ied by 0.7 J energy deposition. The recovery time shown in figure 5.2 consists of three parts:
1) t0: the time needed for the spark channel to expand from the characteristic radius Rc to
the initial radius R0; 2) t1: the time needed to expand from R0 to the radius after the adia-
batic expansion R1; 3) t2: the time needed for the cooling of the gas temperature to 550 K.
The value of t0 and t1 are found be in nanosecond and microsecond range, respectively.
Hence the time of heat transfer is the major part of the total recovery time.
With flow rates up to 0.675 m3/h (flow velocity equals to 6 m/s in a 0.4 mm gap) at
actual pressure and temperature (corresponding to 98 m3/h at STP) we see the following
phenomenon: a larger flow rate results in faster recovery of the SC switch. The recovery
time in a SC N2 switch is predicted to be about 1.5 ms after the energy deposition. The
72 5. THEORETICAL MODELING OF DISCHARGE AND RECOVERY IN SCFS
0 20 40 60 80 1000
1
2
3
4
5
6
7
8
9
Flow rate at STP [m3/h]
Estim
ated
reco
very
tim
e [m
s]
Figure 5.2 – Estimated recovery time of a SC N2 switch after the energy deposition of 0.7 J. Gap
width 0.4 mm, flow rate at STP 8−98 m3/h, corresponding to 0.05−0.675 m3/h at working
pressure of 150 bar (flow velocity equals to 0.4−6 m/s at 0.4 mm gap width).
discontinuity of the curve at 55 m3/h STP volume flow corresponds to the critical Reynolds
number Re (part of the Nusselt number Nu that appears in equation (5.7)) transition from
laminar to turbulent flow [200]. The recovery time in an air plasma switch with the same
amount of energy deposition and flushing volume velocity at STP was also calculated by
this simple model (later in chapter 6.3.1). The simulation results reveals that the recovery
time inside SC N2 of 150 bar is about 5 times shorter than that in a 2.5 bar air switch.
This simple model provides important design data for our SC switches. However, the
model is too simple to produce accurate data on the recovery process. Therefore the calcu-
lated recovery time is an order of magnitude estimate, which for example can be seen from
a comparison with the measured data given later in chapter 6.3.1. Hence we have to develop
an extended physical model for the discharge and recovery in SCFs.
5.3 Electric field across the gap
For the extended physical modeling which will be described later in section 5.4, the electric
field E across the gap during the discharge of SC N2 is an important input parameter. From
the measured current i(t) through the gap, we can calculate the time evolution of E by using
the simplified discharge circuit of the switch [201]. The simplified discharge circuit (marked
with dotted box in figure 3.10) is given in figure 5.3.
In the simplified circuit, the SC switch and the spark channel generated in the gap is
replaced by the series connection of an inductance La and a resistance Ra. The electric field
5.3. ELECTRIC FIELD ACROSS THE GAP 73
Figure 5.3 – Simplified circuit of the dotted box in figure 3.10. i(t): measured current in the cir-
cuit; Vb: measured breakdown voltage; La: arc inductance; Ra: arc resistance; Ch: high voltage
capacitance; Ls: stray inductance in the circuit; R0: total resistance in the circuit (including stray
resistance and load resistance).
across the switching gap can be calculated with the following formulas:
Ra(t) = 1i(t)
×{
Vb − 1Ch
∫ t0 i(t)dt− [Ls+La(t)] di(t)
dt −[R0+
dLa(t)dt
]i(t)
} ; (5.10)
E(t) =1
d
[i(t) ·Ra(t)+La(t)
di(t)
dt
]. (5.11)
In the equations above the variables are explained here:
− Ra(t): the resistance of the arc generated in the gap of SC switch;
− i(t): measured current through the gap, with a typical waveform shown in figure 3.11;
− Vb: the measured breakdown voltage of the SC switch;
− Ls: the stray inductance in the circuit. SC switch (B) in this case can be seen as a
set of coaxial cables with different diameters of the inner conductors, outer metal shield,
and insulators. So the value of Ls can be taken as the equivalent inductance of the co-axial
cables in series. The value of Ls is calculated to be Ls = 104 nH.
− R0: the total resistance of the circuit, which consists of the input impedance of the
4-stage TLT: RTLT = 12.5 Ω, the resistance of the electrode heads (W/Cu 75/25) Rcopper, the
resistance of the electrode bodies (stainless steel) Rss1, and the resistance of the stainless
steel plate denoted as ’6’ in figure 3.7 Rss2. The resistance of the grounded return path
(aluminum housing) of the switch is negligible, because the surface area is very large. Hence
the total resistance of the circuit can be calculated as R0 = RTLT+Rcopper+Rss1+Rss2.
− La: the arc inductance which can be calculated from equation [202, 203]:
La = lμ0
2πln
rc
rs(t), (5.12)
in which rc is the radius of the return path of the current to ground, rs = 35 μm is the radius
of the discharge channel generated in the gap. The value of rs is estimated by imaging with
an intensified CCD camera, as described in section 4.5. La(t) is found to be much smaller
than the value of Ls.
74 5. THEORETICAL MODELING OF DISCHARGE AND RECOVERY IN SCFS
0 100 200 300 400 500−1
0
1
2
3
4
5
6
7
8 x 107
Time [ns]
Elec
tric
field
[V/m
](a)
0 100 200 300 400 500−1
0
1
2
3
4
5
6
7
8 x 107
Time [ns]
Elec
tric
field
[V/m
]
(b)
Figure 5.4 – (a): Estimated average electric field E from the measured arc current i(t) in figure
3.11, with applied voltage of 25 kV and gap width of 0.3 mm; (b): Smoothed electric field E in
(a) with a span of 10 %.
With the current given in figure 3.11, the calculated electric field E across the gap up
to 500 ns after the start of the current is shown in figure 5.4(a). The oscillation in the tail
of the curve is due to the resonance oscillation between the inductance and capacitance in
the circuit. We use the smoothed curve of the calculated E with moving average of 10 % of
the total number of data points, shown in figure 5.4(b), as the electric field profile that will
be applied as the input parameter in the extended physical model for discharge in SCFs in
chapter 5.
5.4 Extended physical model for discharge in SCFs
5.4.1 General model description
The goal of this model is to study the complete breakdown and subsequent recovery pro-
cesses in SC N2. The physical model is of SC switch (B), which has been introduced in
chapter 3. If the switch undergoes a breakdown, the discharge is assumed to occur in the re-
gion (6) in figure 3.8, i.e., in the region of a rather uniform background field. We use results
of previous work in streamer propagation as input parameters, and simulate the streamer-to-
5.4. EXTENDED PHYSICAL MODEL FOR DISCHARGE IN SCFS 75
spark transition phase and discharge and post-discharge phase of the discharge. The rough
time scale for physics during these processes and the estimated temperature of neutral N2 is
given in figure 5.5.
Conservation of mass, momentum, total energy ns
μs
0
Time defined in the model
300 K 5000 K Temperature on axis 20000 K
Streamer stage
Streamer-spark transition stage
Spark-decay stage
Figure 5.5 – Stages in our extended physical model and the estimated temperature on the axis of
the spark channel in SC N2.
The modeled discharge phases in our work are briefly introduced here, while the detailed
explanation is given in section 5.4.3-5.4.4. In the streamer-to-spark transition phase the
streamer forms and propagates, and electric energy from the external source is deposited
into the gap. The discharge energy is transferred to different levels: translational motion,
rotational levels, vibrational levels, and electronically excited levels as well as dissociation
and ionization of N2 molecules. Energy in some excited levels is relaxed to gas heating
instantaneously, while the energy in the other excited levels takes time to relax fully. All
energy going into gas heating is denoted as Qin. During the relaxation process there is
energy output due to heat conduction and radiation, denoted as Qout. The total energy ε is a
result of the energy input, energy output and the gas dynamics.
We assume the streamer-to-spark transition phase ends when the gas temperature in the
discharge center is larger than 5000 K, then the discharge and post-discharge phase begins.
In this stage the remaining energy in excited levels continues relaxing with a certain time
constant; the total energy of the spark channel changes under combined mechanisms; the
thermodynamic properties of the N2 in the spark channel recover and finally the dielectric
strength of SC N2 can recover. During the discharge and post-discharge phase, the forced
flushing of the N2 might push the spark channel to the outer-edge of the inter-electrode gap,
where turbulent flow cools down the spark channel more fast. For the simplicity, we neglect
this effect during our simulation.
76 5. THEORETICAL MODELING OF DISCHARGE AND RECOVERY IN SCFS
Stor
ed in
ele
ctro
nic
leve
ls
+ sp
ent o
n io
niza
tion
Pres
umed
ele
ctric
fiel
d
Elec
tric
cond
uctiv
ity
Kin
etic
mod
elin
g
Elec
tric
ener
gy
Neg
lect
Stor
ed in
vib
ratio
n le
vel
Goe
s to
gas k
inet
ics
ion
Qq
nμ
σe
ee
iμsm
all V
ibra
tiona
l ene
rgy
VεR
elax
ed to
gas
ki
netic
s
recomb
ion
eF
F/dt
dn
out
QG
as d
ynam
ics
Ener
gy ε
E
Goe
s to
exci
ted
leve
l +
spen
t on
ioni
zatio
n
Rel
axed
to g
as k
inet
ics
Imm
edia
tely
rela
xed
Via
VT
Ener
gy lo
ss
Elec
troni
c ex
cite
d +
ioni
zatio
n en
ergy
Eε
Rel
axed
to g
as
kine
tics
Via
2e
eR
Eσ
QQ
RtransQ
ηR
ion
Vele
rot
Q)
ηη
η(η
Rele
rot
Q)
η3.0(η
Rion
ele
Q)
η(0.7η
ET
VT
Q
ET
Q
RVQ
η
Ener
gy in
put
inQ
Goe
s to
gas
heat
ing
TQ
p·dV
Figu
re5.
6–
Flo
wch
art
of
the
ener
gy
tran
siti
on
inth
en
um
eric
alm
od
elo
fd
isch
arg
ean
dre
cover
yp
roce
ssin
sid
ea
SC
F.
E:
elec
tric
fiel
d
esti
mat
edfr
om
equ
atio
n(5
.11
);n
e:
nu
mb
erd
ensi
tyo
fel
ectr
on
s;F
ion
and
Fre
com
b:
the
ion
izat
ion
and
reco
mb
inat
ion
coef
fici
ents
of
N2;
QR
:th
eel
ectr
icp
ow
erd
ensi
ty;
Qe
and
Qio
n:
elec
tro
nan
dio
nco
mp
on
ent
of
the
dis
char
ge
pow
erin
pu
td
ensi
ty;
σ e:
elec
tric
alco
nd
uct
ivit
y
of
the
elec
tro
ns;
μ e:
elec
tro
nm
ob
ilit
y;
qe:
elec
tro
nch
arg
e;η t
rans,
η rot,
η V,
η ele
,an
dη i
on:
frac
tio
ns
of
ener
gy
that
go
into
tran
slat
ion
al
level
,ro
tati
on
alle
vel
,v
ibra
tio
nal
level
,el
ectr
on
icex
cite
dle
vel
,an
dio
niz
atio
no
fN
2;
QT
:el
ectr
icp
ow
erd
ensi
tyg
oin
gd
irec
tly
into
the
gas
hea
tin
g;
QV
T:
the
pow
erd
ensi
tyre
laxed
fro
mv
ibra
tio
nal
totr
ansl
atio
nen
erg
yle
vel
of
N2;
QE
T:
the
pow
erd
ensi
tyre
laxed
fro
m
elec
tro
nic
ally
exci
ted
level
sas
wel
las
fro
md
isso
ciat
ion
and
ion
izat
ion
of
N2
mo
lecu
les;
τ VT
and
τ ET
:ti
me
scal
eso
fth
ev
ibra
tio
nal
rela
xat
ion
and
the
elec
tro
nic
ally
rela
xat
ion
inN
2;
Qin
and
Qout:
the
loca
lp
ow
erin
pu
tan
do
utp
ut
den
sity
;ε:
the
tota
len
erg
yd
ensi
ty;
ε V:
the
vib
rati
on
alen
erg
yd
ensi
ty;
ε E:
the
sum
of
the
elec
tro
nic
exci
ted
ener
gy
and
ener
gy
spen
to
nio
niz
atio
no
fN
2;
p·d
V:
gas
dy
nam
ics
con
trib
uti
on
toth
eto
tal
ener
gy
den
sity
.
5.4. EXTENDED PHYSICAL MODEL FOR DISCHARGE IN SCFS 77
The numerical modeling of the discharge process follows the flow chart of the energy
transition shown in figure 5.6. From the detailed streamer simulations [204–206], it is
known that in the interior of streamer channel the electron density can be assumed to be
uniform in the axial direction far from the electrodes. Since we do not take into account
the effect of the electrodes, we assume a 1D configuration (in the radial direction) to be
sufficient for the present work.
5.4.2 Model Formulation
The Euler equations [30, 207, 208] which cover the equations of conservation of mass, mo-
mentum, and energy are applied in the model. The balance equations of the vibrational
energy and electronic excited energy of N2 are included in the following set of equations:
∂∂ t
⎡⎢⎢⎢⎢⎣
ρρu
εεV
εE
⎤⎥⎥⎥⎥⎦+
1
r
∂∂ r
r
⎡⎢⎢⎢⎢⎣
ρu
ρu2
u(ε +p)
εVu
εEu
⎤⎥⎥⎥⎥⎦+
∂∂ r
⎡⎢⎢⎢⎢⎣
0
p
0
0
0
⎤⎥⎥⎥⎥⎦ =
⎡⎢⎢⎢⎢⎣
0
0
Qin −Qout
ηVQR −QVT
ηEQR −QET
⎤⎥⎥⎥⎥⎦ , (5.13)
where ρ [kg ·m−3] is the mass density, u [m ·s−1] the velocity, p [Pa] the pressure, ε [J ·m−3]the total energy density, εV [J ·m−3] the vibrational energy density, εE [J ·m−3] the electronic-
ally excited energy density, Qin [J ·s−1 ·m−3] the local power input density, Qout [J ·s−1 ·m−3]the local power output density, QR [J · s−1 ·m−3] the external discharge power input density,
QVT [J · s−1 ·m−3] the power density relaxed from vibrational to translational energy level
of N2, and QET [J · s−1 ·m−3] the power density relaxed from electronically excited levels
as well as dissociation and ionization of N2 molecules, ηV the fraction of the energy which
goes into vibrational excited level, ηE the energy used for ionization together with the part
of electronic excited energy which will not be relaxed immediately. The detailed derivation
for the Euler equations in cylindrical coordinate can be found in appendix A4. All the units
listed in this work are SI unit, unless otherwise specified, and the terms and units in the
equations below will not be repeatedly described.
The total energy per unit volume ε is defined as ε = ρu2/2+p/(γ −1), where γ is the
specific heat ratio depending on gas temperature and pressure [209]. The input power dens-
ity Qin is expressed as
Qin = QT+QVT+QET , (5.14)
where QT is the electric power density going directly into the gas heating; QVT and QET
have been introduced before. The output power density Qout is described by
Qout = Qrad+Qcond+Qelectrode . (5.15)
where Qrad is the output power density due to radiation heat transfer, Qcond the output due
to heat conduction in the radial direction of the spark channel, and Qelectrode the output due
78 5. THEORETICAL MODELING OF DISCHARGE AND RECOVERY IN SCFS
to heat conduction to the metal electrodes. Energy dissipation due to heat convection is
neglected, since the temperature in the outer domain of the spark channel is almost equal to
the background temperature.
5.4.3 Streamer-to-spark transition phase
The critical electric field that can initiate a breakdown in N2 is normally Eb = 20N/N0 kV·cm−1 [210], in which N0 is the density under standard temperature and pressure (STP) and
N is the density at working pressure. At voltage below Eb, streamer model is able to explain
the breakdown mechanism, while at electric fields much higher than Eb, runaway electrons
from the main avalanche lead to a broad channel breakdown [211]. Since the electric field
applied in this work (seen figure 5.4) is lower than Eb, the streamer mechanism is applicable
in our model. Streamer properties like radius, maximum field on tip, propagation velocity,
electron density distribution etc., are very well studied [204, 212–214]. Results of previous
work [215, 216] in streamer propagation are taken as input parameters for our modeling.
In [217, 218] scaling (or similarity) laws have been derived. We use these laws to obtain
streamer properties in SC N2 by scaling of literature values at STP. However, as corrections
to scaling law at pressure substantially above one bar have been suggested in [213], and
as the radius of the discharge channel briefly after the streamer phase has been measured
to be 35 μm in chapter 4.5, we use 35 μm as the initial radius applied in our model. The
streamer develops into a conducting channel and electric energy from the external source,
with power density denoted as QR, deposits into the channel. When the N2 temperature
in the channel heats up to Tg = 5000 K [30, 34], we move to the next modeling domain:
discharge & post-discharge phase.
QR is composed of the ion and electron contributions, denoted as Qion and Qe respect-
ively. The ion component of the energy is neglected, as for the time scales considered in
the present work ion mobility is negligible compared to the electron mobility. So the elec-
tric energy input is considered to be equal to the electron component of the electric energy
QR = Qe. After the streamer bridges the gap, the electric field E is assumed as uniform in
the gap [219]. For a given E, the electron energy deposition rate can be calculated with
the equation Qe = σeE2 = qeμeneE2 , where σe is the electrical conductivity due to electron
mobility; qe, μe and ne are the electron charge, mobility, and number density of electrons,
respectively.
The electron mobility μe is dependent on the reduced electric field E/nN2and was ob-
tained from [220]. The balance of the electron density ne can be analyzed through the
kinetic processes governing the species reactions such as direct ionization, step-wise and
associative ionization, attachment, detachment, etc.. In present work we consider only ion-
ization and recombination mechanisms. We write the kinetic equation in the following form
dne(r)
dt= nN2
(r)ne(r)Fion(r)−ne(r)n+(r)Frec(r) , (5.16)
where Fion(r) [m3 · s−1] and Frec(r) [m6 · s−1] are the ionization and recombination coeffi-
5.4. EXTENDED PHYSICAL MODEL FOR DISCHARGE IN SCFS 79
cients of N2, obtained from [220, 221]. In appendix A5 we use a zero-dimensional model
to analyze detailed temporal dynamics of species in a streamer channel. The results reveal
that during the discharge N2+ is converting very rapidly to N4
+, and the dominant electron
loss mechanism in the streamer channel is electron-ion dissociative recombination between
N4+ and electrons [222]. In equation (5.16) the drift and diffusion of the electrons and ions
are justified to be neglected in the nanosecond time range in our situation. 1
As described before, part of the discharge power density QR is transferred into the ex-
cited levels and needs time to relax. Figure 5.7 gives the fractions of energy that go into
Table A-2 – Parameters used in state equation of nitrogen.
k Nk ik jk lk
1 0.924803575275 1.0 0.25 0
2 −0.492448489428 1.0 0.875 0
3 0.661883336938 2.0 0.5 0
4 −0.192902649201×101 2.0 0.875 0
5 −0.622469309629×10−1 3.0 0.375 0
6 0.349943957581 3.0 0.75 0
7 0.564857472498 1.0 0.5 1
8 −0.161720005987×101 1.0 0.75 1
9 −0.481395031883 1.0 2.0 1
10 0.421150636384 3.0 1.25 1
11 −0.161962230825×10−1 3.0 3.5 1
12 0.172100994165 4.0 1.0 1
13 0.735448924933×10−2 6.0 0.5 1
14 0.168077305479×10−1 6.0 3.0 1
15 −0.107626664179×10−2 7.0 0.0 1
16 −0.137318088513×10−1 7.0 2.75 1
17 0.635466899859×10−3 8.0 0.75 1
18 0.304432279419×10−2 8.0 2.5 1
19 −0.435762336045×10−1 1.0 4.0 2
20 −0.723174889316×10−1 2.0 6.0 2
21 0.389644315272×10−1 3.0 6.0 2
22 −0.212201363910×10−1 4.0 3.0 2
23 0.408822981509×10−2 5.0 3.0 2
24 −0.551990017984×10−4 8.0 6.0 2
25 −0.462016716479×10−1 4.0 16.0 3
26 −0.300311716011×10−2 5.0 11.0 3
27 0.368825891208×10−1 5.0 15.0 3
28 −0.255856846220×10−2 8.0 12.0 3
29 0.896915264558×10−2 3.0 12.0 4
30 −0.441513370350×10−2 5.0 7.0 4
31 0.133722924858×10−2 6.0 4.0 4
32 0.264832491957×10−3 9.0 16.0 4
33 0.196688194015×102 1.0 0.0 2
34 −0.209115600730×102 1.0 1.0 2
35 0.167788306989×10−1 3.0 2.0 2
36 0.262767566274×104 2.0 3.0 2
110 APPENDIX
δ 2
(∂ 2αr
∂τ2
)τ=
6
∑k=1
ik(ik −1)Nkβ ikτ jk
+32
∑k=7
jkNkβ ikτ jk × exp(−β lk )[
(ik − lkδ lk )(ik −1− lkδ lk )− l2kδ lk]
+36
∑k=33
Nkβ ikτ jkexp(−φk(β −1)2 −βk(τ − γk)2
)×{[ik −2δϕk(δ −1)]2 − ik −2δ 2ϕk
}.
(A-12)
Table A-3 – Parameters used in state equation of nitrogen.
k φk βk γk
33 20 325 1.16
34 20 325 1.16
35 15 300 1.13
36 25 275 1.25
A2. Integrator of the triggering signal generator
Pick-up antenna
Figure A-1 – Integrator of the triggering signal generator for the ICCD camera.
111
A3. Calibration of current measured by Rogowski coil
From figure A-2 we can derive the relation between signal Vcoil, which comes out of the
Rowgowski coil and signal Vout collected by the oscilloscope
Vout =1
Ci
∫ (Vcoil −Vout
Ri− Vout
R1
)dt. (A-13)
in which Vout is the signal read from oscilloscope, and τ = Ri ·Ci with Ri and Ci respectively
1060 Ω and 879 pF. With known output voltage, the real voltage on the coil can be rewritten
with ∫Vcoildt = τVout+
(1+
Ri
R1
)∫Voutdt. (A-14)
Vcoil Vout
Ro
Ri
Ci R1
Figure A-2 – Circuit diagram of integrator of current measurement.
The arc current can be calculated from the voltage on the Rogowski coil
i(t) =1
M
∫Vcoildt, (A-15)
where M is the mutual inductance of the Rogowski coil. By interpreting equation (A-14)
into equation (A-16), we can get the expression of arc current:
i(t) =Voutτ
M+
(1+
Ri
R1
) ∫Vout
Mdt. (A-16)
112 APPENDIX
A4. Cylindrical coordinate in Euler system
The Euler equations which covers the equations of conservation of mass, momentum, and
energy are
∂ρ∂ t+� · (ρu) = 0; (A-17a)
∂ (ρu)
∂ t+� · (u
⊗(ρu))+�p = 0; (A-17b)
∂ε∂ t+� · (u(ε +p)) = 0. (A-17c)
In cylindrical coordinate, we have the expression:
� ·A = 1
r
∂ (A1r)
∂ r+
1
r
∂ (A2)
∂Φ+
∂ (A3)
∂z; (A-18a)
�A = e1∂ (A)
∂ r+ e2
1
r
∂ (A)
∂Φ+ e3
∂ (A)
∂z. (A-18b)
The total energy of the gas satisfies ∂ε∂ t+� · (u(ε +p)) = Qin −Qout, with Qin and Qout
representing the energy per unit time which comes in and goes out of the system. Also we
have energy stored in the vibration excited level εV and electronic excited level εE. Com-
bining equation (A-17) and (A-18), we can write the equation of conservation in cylindrical
coordinate as:
∂ρ∂ t+
1
r
∂ (ρur)
∂ r= 0; (A-19a)
∂ (ρu)
∂ t+
1
r
∂ (ρu2r)
∂ r+
∂p
∂ r= 0; (A-19b)
∂ε∂ t+
1
r
∂ (u(ε +p)r)
∂ r= Qin+Qout; (A-19c)
∂εV
∂ t+
1
r
∂ (uεVr)
∂ r) = ηVQR+QVT; (A-19d)
∂εE
∂ t+
1
r
∂ (uεEr)
∂ r) = ηEQR+QVT. (A-19e)
113
A5. Simulation of electron-ion recombination in N2 discharge
We use a zero-dimensional modeling platform ZDPlasKin [256] to model the dynamics of
species in a streamer channel. The description of dynamics is given by
d[ni]
dt= Si , (A-20)
where the source term Si represents the total rate of production and destruction of species i in
various processes. We use N2 kinetics from [227,256] which includes the following species
and states: e, N, N2, N+, N+2 , N+3 , N+4 , N2(A3 Σ+u , B3Πg, C3Πu, a′1Σ−u ), N2(X1, v = 1−8).
The rate constants for electron-neutral interactions are calculated using BOLSIG+ solver
[225].
For self-consistent coupling between the electric field and the equation (A-20) we use
the approach proposed in [257], where the electric field obeys the equation
dE
dt=
ε0
eμ (|E|) E ne . (A-21)
Essentially we apply the same initial conditions as in [257], but scaled up to 80 bar.
The maximum electric field is assumed to be 150 kV · cm−1 at standard temperature and
pressure (STP), which scales to E(0) = 12 MV · cm−1 at 80 bar according to the similarity
laws. Maximum electric field E(0) corresponds to the initial electron density from the table
5.2 and reads as ne(0) = ne,0 = 8.87×1020 m−3. The equations (A-20) and (A-21) are in-
tegrated together to obtain a decay of the electric field consistent with the conductivity. The
integration is continued up to the electric field reaches the value of 0.4 MVcm−1. After-
wards, we continue integration with a constant electric field of 0.4 MVcm−1 until end of
the pulse (total pulse duration is taken to be equal to 100 ns), and then in a vanishing field.
The results of ZDPlasKin run are imported and analyzed in an open source software
package PumpKin [222], which is freely available at www.pumpkin-tool.org. PumpKin
automates the process of finding all principal pathways, i.e. the dominant reaction se-
quences, in a chemical reaction system. We run PumpKin for the time interval of [0.01,1]ns and for the species of interest of electron. The result of analysis shows that the lifetime
of N+2 is very short about 12 fs, while N+4 has much longer lifetime of about 9.1 ps. Un-
der these conditions, the dominant electron loss mechanism is a electron-ion dissociative
recombination given by
e+N+4 → N2+N2 . (A-22)
According to PumpKin, this reaction is responsible for the 100 % of the destruction of
electron with a rate of 1.4 ·1022 cm−3 s−1. On the other hand, the electron impact ionization
e+N2 → N+2 +2e , (A-23)
is responsible for 98 % of the production of electrons with a rate of 2.0 ·1015 cm−3 s−1.
This is the reasoning why we, under the assumption that the similarity laws are applic-
able at 80 bar pressure, consider the kinetic equation in form of (5.16).
114 APPENDIX
A6. Ionization and dissociation mechanisms
Here we only consider the ionization up to double ionization stage, while the higher level
is neglected. We assume that all the molecules are assumed to be completely dissociated
before ionization begins, and all atoms are assumed to be singly ionized before the number
of doubly ionized particles becomes noticeable [38].
In this approach the state of equation and caloric equation are derived from refer-
ences [258]. The difference between the calculation here and the reference is that the con-
centration of molecules is also taken into account. The indexes used in the calculations are
given in table A-4 .
Table A-4 – The indexes used in the calculation.
Parameter Symbol Value
Number of total particle Ntotal NN2+N0 +Ne +N1 +N2
Number density of ntotal MN2NN2+M0N0+
total particle MeNe +M1N1 +M2N2
Electron mass Me 9.10×10−31 [kg]
Nitrogen atom mass M0 1.163×10−26 [kg]
Nitrogen ion mass Mi ≈ 1.163×10−26 [kg]
Nitrogen molecular mass MN22.326×10−26 [kg]
Concentration of molecule CN2NN2
/Ntotal
Concentration of atom C0 N0/Ntotal
Concentration of electron Ce Ne/Ntotal
Concentration of single ion C1 N1/Ntotal
Concentration of double ion C2 N2/Ntotal
Total Pressure P −Electron Pressure Pe P×Ce
Statistic weight gn g1 = 9; g2 = 6
of ion on stage n
Statistic weight of atom g0 g0 = 4
Boltzmann’s constant k 1.38×10−23 [J/K]
Plank’s constant h 6.626×1034 [J·s]
Dissociation energy I0 3.3484×104 [kJ·kg−1]
Single ionization energy I1 9.9806×104 [kJ·kg−1]
Double ionization energy I2 2.0321×105 [kJ·kg−1]
Gas constant R R = 2.8809×102 [J·K−1·kg−1]
Temperature T −
We know Saha’s equation for dissociation
n0n0
nN2=
g0 ·g0
gN2
[2πMN/2kT]3/2
h3e−I0RT , (A-24)
115
and Saha’s equation for ionization
ni+1ne
ni=
ge ·gi+1
gi
[2πMekT]3/2
h3e−Ii+1
RT . (A-25)
If we write the Saha’s equations in the format of concentration of different particles, we
have equation for dissociation
CN2
C0C0=
1
g0·g0gN2
[2πMN/2kT]3/2
h3 e−I0RT
·ntotal, (A-26)
in which C0C0CN2=
n0VntotalV
n0VntotalV
ntotalVnN2
V =n0n0nN2
1ntotal
; and Saha’s equation for ionization in format:
Ci
Ci+1Ce=
1
ge·gi+1gi
[2πMekT]3/2
h3 e−Ii+1
RT
·ntotal, (A-27)
in which Ci+1CeCi=
ni+1VntotalV
· neVntotalV
· ntotalVn0V =
ni+1nen0
1ntotal
. In the equation above if we write ntotal = P/kT,
then equation (A-27) actually is the same as the one given in [258].
The summary of all the particle concentrations should be
1 = CN2+C0+Ce+C1+C2+ ... (A-28)
The generation of electrons due to ionization satisfies the relation
Ce = C1+2C2+3C3... (A-29)
Combining equation (A-28) and (A-29), we have
1 = CN2+C0+2C1+2C2... (A-30)
All the concentrations can be expressed by the value of C0 and other parameters, seen as
CN2=
1
S0· P
kT·C2
0; (A-31a)
C1 =C0S1
Ce
kT
P; (A-31b)
C2 =C1S2
Ce
kT
P=
C0S1 ·S2
C2e
(kT)2
P2. (A-31c)
The values of S0, S1, and S2 are
S0 =g0 ·g0
gN2
[2πMN/2kT]3/2
h3e−I0RT ; (A-32a)
S1 =ge ·g1
g0
[2πMekT]3/2
h3e−I1RT ; (A-32b)
S2 =ge ·g2
g1
[2πMekT]3/2
h3e−I2RT . (A-32c)
116 APPENDIX
If we substitute equation (A-31a)-(A-32c) into equation (A-30), we can derive the solution,
though with an unknown value Ce inside. By assuming a value of Ce, we can calculate a
value of C0. The values of CN2, C1,C2 can be calculated with equation (A-31a)-(A-31c).
With equation (A-29) a new value of Cepre can be calculated. If the calculated Ce differs
larger than 0.1 % from the presumed value, a new value Ce−new = (Ce−pre+Ce)/2 is used
for the new round of calculation. The iterative process continues until the value of Ce
convergences to the required accuracy 0.1 %.
0 0.5 1 1.5 2 2.5x 104
0
20
40
60
80
100
120
Temperature [K]
Parti
cle
conc
entra
tions
[%]
CatomCion1Cion2CN2CNetotal concentration
Figure A-3 – Concentrations of particles in air for temperature up to 25000 K, pressure at 1 bar.
0 0.5 1 1.5 2 2.5x 104
0
20
40
60
80
100
120
Temperature [K]
Parti
cle
conc
entra
tions
[%]
CatomCion1Cion2CN2CNetotal concentration
Figure A-4 – Concentrations of particles in air for temperature up to 25000 K, pressure at 80 bar.
117
Figure A-3 and figure A-4 show the value of C0, C1, and C2 as function of temperature
at different pressures. From the figures we can see that thermal dissociation and ionization
happen at higher temperature when the pressure is higher. In both 1 bar and 80 bar situation,
the mechanisms become significant when the temperature is above 5000 K. However, from
literature survey, we learn that the establishing time scale of the ionization equilibrium for
nitrogen is quite long. The time scales of ionization equilibrium of several species against
the temperature can be found in [247]. According to the calculation, the time needed for
establishing ionization equilibrium in nitrogen in our case is between 1.5−60 μs. From
the time development of N2 dissociation under different temperatures given in [247], we
can see that for the temperature of 10000 K, the fraction of dissociation in nitrogen reaches
about 10−12 at the time of 1 μs. For temperature of 25000 K, the fraction reaches about
10−3 at 1 μs.
In our simulations given in chapter 5.4, the high temperature zone of the discharge chan-
nel (at 80 bar) vanishes with a time scale much shorter than the establishing time of disso-
ciation and ionization equilibrium, hence it is reasonable to neglect the impact of thermal
dissociation and ionization in our simulations.
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• J. Zhang, E.J.M. van Heesch, Takao Namihira, F.J.C.M. Beckers, and A.J.M. Pemen.
Breakdown strength and dielectric recovery investigation inside a supercritical switch.
Proceedings of 14th International Symposium on High Pressure Low Temperature
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141
142 LIST OF PUBLICATIONS
• E.J.M. van Heesch, J. Zhang, T. Namihira, A. Markosyan, A.J.M. Pemen, F.J.C.M.
Beckers, T. Huiskamp, and U. Ebert. Voltage recovery in supercritical switchingmedia. Proceedings of IEEE International Power Modulator and High Voltage Conference,
Santa Fe, June 1-5, 2014.
• E.J.M. van Heesch, W.F.L.M. Hoeben, J. Zhang, F.J.C.M. Beckers, T. Huiskamp and
A.J.M. Pemen. Pulsed Power Applications Research: From Super Critical Switchesto Renewable Fuels. Proceedings of IEEE International Power Modulator and High
Voltage Conference, Santa Fe, June 1-5, 2014.
2013
• J. Zhang, E.J.M. van Heesch. Recovery rate analysis of plasma switch and comparisonwith experimental results. Proceedings of XXth Symposium on Physics of Switching
Arc-FSO 2013, Czech Republic, Sep. 2013.
• A.H. Markosyan, J. Zhang, B. van Heesch, U. Ebert. Streamer to spark transition insupercritical N2 . Proceedings of XXth Symposium on Physics of Switching Arc-
FSO 2013, Czech Republic, Sep. 2013.
• J. Zhang, T. Furusato, F.J.C.M. Beckers, E.J.M. van Heesch, E.M. van Veldhuizen.
Study of breakdown inside a supercritical fluid plasma switch. Proceedings of IEEE
Pulsed Power & Plasma Science Conference, San Francisco, CA USA, June. 2013.
2012
• J. Zhang, F.J.C.M. Beckers, E.J.M. van Heesch. Breakdown voltage study of asupercritical medium switch based on experiment. Proceedings of EAPPC-BEAMS-
Conference, Karlsruhe, Germany, 2012.
ACKNOWLEDGEMENT
After the long hard time, I am so glad and proud about what I have managed to do. But I
know that I couldn’t have gone so far without the great help from my colleagues, my friends,
my family and everyone else who has given me help within the past five years.
First of all, I want to give my sincere appreciation to my first promoter prof.ir. Wil
Kling. When my husband Lei got a chance of interview for a Ph.D. candidate position in
EES-group in TU/e, Wil kindly invited me for a visit to the group. The encouragement and
care from Wil in our regular discussion during my Ph.D. study always gave me impetus to
work hard and cruise to the happy ending of the thesis.
Dr. Bert van Heesch is my daily supervisor and co-promoter. My highest thanks go
to Bert for his patient guidance. You gave me huge help throughout my five-year study,
from the collection of the initial materials for this project to the design and operation of the
supercritical switches. Thanks to the kind help from Bert, my contract of Ph.D. study was
successfully extended for extra six months.
I would like to give my acknowledgement to my second Promoter prof. Ute Ebert. She
gave me huge support in the theoretical modeling of my thesis. I am enlightened by her
intelligence and diligence.
My great thanks also go to Dr. Martin Seeger from ABB, Switzerland. You ceaseless
advertising is an indispensable part of my thesis. My appreciation to your patient guidance
and friendly attitude.
I would also like to appreciate prof.dr.ir. René Smeets. Thanks so much for your
kind patient guidance during the past years. During all the discussion and meetings, you
always gave me essential suggestions, and your kindness always made me feel relaxed and
confident.
I want to thank colleagues Tomohiro and Dr. Takao Namihira from Kumamoto University,
Japan. Thank you a lot for the useful discussion and sharing of information during your visit
to our group.
With this chance, I want to express my thankfulness to all the other committee members,
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144 ACKNOWLEDGEMENT
Prof.Dr.-Ing. A. Schnettle, dr.ing. A.J.M. Pemen, and dr. R.A.H. Engeln for their precious
time to review my thesis. Their useful comments help me improve this thesis so much.
My gratitude also goes to my cooperator Aram from CWI, Amsterdam. I enjoyed all
those hard working weekends when we built the theoretical models and prepared the papers.
Other colleagues from CWI, Anbang, Ashutosh, and Jannis also gave me kind support
during those days. My appreciations to all of you.
I would like to thank all the other colleagues from EES-group. I feel so lucky to
work with you during my Ph.D. student life. Thank you so much for creating the joyful
environment and your support whenever I need. I especially want to thank Anna, Annemarie,
