j.cirpj.2017.09.005

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CIRPJ 439 No. of Pages 15
CIRP Journal of Manufacturing Science and Technology xxx (2017) xxx–xxx
Contents lists available at ScienceDirect
CIRP Journal of Manufacturing Science and Technology
journal homepage: www.elsevier.com/locate/cirpj
Analysis of micro-EDM electric characteristics employing plasma
property
A.B.M.A. Asad, M. Tanjilul Islam, Takeshi Masaki, M. Rahman* , Y.S. Wong
Department of Mechanical Engineering, National University of Singapore, 117575, Singapore
A R T I C L E I N F O
Article history:
Available online xxx
Keywords:
Micro-EDM
RLC network
RC power supply
Spark gap
Discharge energy
Plasma
Stray capacitance
A B S T R A C T
Micro-electro discharge machining (micro-EDM) involves simultaneous complex processes associated
with electrical characteristics and plasma properties, which lead to the removal of material from both
electrode and workpiece. Unfortunately, existing micro-EDM models are simple electro-thermal based
and not capable to connect the link between plasma and electrical circuit. This article presents a
theoretical study to model the electric properties of micro-EDM plasma discharge for RC power supply
with direct current (DC) source. Analysis presented in this study provides significant insight for realizing
changes in the current waveform due to changes in parameters like input voltage, capacitance,
inductance and the charging resistance. This research bridges a crucial gap in the present theoretical
understanding on the interaction of micro-EDM plasma with different circuit elements of micro-EDM
power supply. Furthermore, by employing this proposed model, a novel stray capacitance measurement
method has been presented to adjust pulse energy at minimum level for smallest machinable features.
The robustness of the analytical model is substantiated by validating it on multiple experimental results
of measured data as well as literature data.
© 2017 CIRP.
Introduction
Owing to recent trends in the miniaturization of products in
areas such as information technology, biotechnology, and the
environmental, medical and optics industries, micro-EDM process
is becoming a key supportive technology for micro machining.
Major benefits are the absence of cutting force, and flexibility for
machining irrespective of the material’s hardness. Material
removal mechanism in EDM process is associated with plasma
formation, which is generated by an electric breakdown in the
electrode and workpiece gap space filled with dielectric under a
condition when the electric field strength exceeds the dielectric
strength. Finally, this ignition process leads to a subsequent current
flow that generates an electric discharge, which lead to the
removal of material from both electrode and workpiece. Through
optimization of the process parameters, high-aspect-ratio micro
features can be achieved using micro-EDM process. A proper
analytical model is necessary to better interpret and predict results
for micro-EDM process. The analytical modelling of the EDM
process has been investigated by various researchers since 1971
[1]. Most of these existing models are based on electro-thermal
* Corresponding author.
E-mail address: mpemusta@nus.edu.sg (M. Rahman).
properties of EDM process. In electro-thermal model, researchers
considered various shapes of heat sources, e.g., disk heat source,
point heat source etc. to obtain temperature/plasma distribution
equations. Snoeys and Van Dijck [2] reported a study using a disk
heat source while DiBitonto et al. [3] proposed model using a point
heat source approximation. A comprehensive attempt to model the
plasma channel was reported by Lhiaubet et al. [4]. The properties
of diatomic plasma were taken as a constant and the fluid dynamic
equation was included in the model. Eubank et al. [5] reported
variable mass cylindrical plasma which expands with time. Recent
research efforts includes Gaussian distribution model as a heat
source to numerically investigate MRR in EDM process [6]. Unlike
the above mentioned models where the thermophysical properties
of the material are considered to be constant or average,
Weingärtner et al. [7] considered temperature varying thermophysical properties value over the whole temperature range from
solid to liquid melt. Researchers have also used calorimetric model
to investigate material removal rate (MRR) [8] and electromechanical model to estimate stress distribution inside the metal
due to the electrostatic force [9]). However, these models lacked
focus in linking between the plasma and electrical circuit. Due to
the complex simultaneous processes associated with micro-EDM
plasma, an analytical model based on micro-EDM plasma and
electric network is necessary for optimization of machining
https://doi.org/10.1016/j.cirpj.2017.09.005
1755-5817/© 2017 CIRP.
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parameters. Proper understanding of the electrical characteristics and
plasma properties in micro-EDM process can also help to minimize the
spark energy released from each spark to achieve smaller unit removal,
which will result in smaller machinable feature size and finer surface
roughness. Unfortunately, not much attempt has been made to
investigate on the discharge itself and on the plasma created during
the process. Descoeudres [10] has studied the EDM plasma with
electrical measurements. However, current literature lacks the
necessary modelling approach to bridge the gap between the power
supply/electrical circuit and plasma generation.
There are two major types of micro-EDM power supply, namely
resistance-capacitance (RC) or relaxation type and transistor type
power supply [11]. The relaxation type pulse generators are the
popular choice for micro EDM because it is difficult to obtain
significantly short pulse duration with constant pulse energy using
the transistor type pulse generator [12]. So, the RC based power
supply has found widespread applications in micro-EDM, and is
somewhat a rebirth after being replaced by transistor type power
supply for conventional EDM power supply. In an RC or relaxationtype circuit, discharge pulse duration is dominated by the
capacitance of the capacitor and the inductance of the wire
connecting the capacitor to the workpiece and the workpiece to
the tool [13] and the discharge energy is determined by the used
capacitance and applied voltage. In the case of typical RC type power
supply, the repetition of the charging and discharging occurs in
which capacitor C is charged through resistor R and discharged
between the electrode and workpiece produces an extremely short
width pulse discharge. For micro EDM in RC type power supply
alongside main capacitor there is stray capacitance contributed by
the electric feeder, the tool electrode holder and work table, and
between the tool electrode and workpiece. Stray capacitance
contributes in parallel to the installed capacitor and when the
installed capacitance is negligible the discharge energy per pulse is
determined by the stray capacitance. Thus, in micro-EDM in order to
reduce the pulse energy, it is important to minimize stray
capacitance between wire and workpiece. The existing method
for estimation of stray capacitance requires a voltage and a current
probe to be employed for simultaneous acquisition of voltage and
current waveform [14,15]. Actual discharge energy is then
computed as the product of the two waveforms obtained. While
the acquisition of the current waveform can be performed using a
contactless probe with negligible inductive loading on the EDM
circuit; however, generally a voltage probe is required to be directly
connected to the circuit which introduces additional stray
capacitance and therefore, may impact the accuracy of the
measurement.
To resolve the above mentioned issues, this paper aims to
provide an analytical model of the electric properties of microEDM plasma for a RC power supply circuit by incorporating the
electrical equivalent of plasma into an electrical network of RC
power supply circuit having DC source. Detailed analysis is
presented in this paper to provide significant insight for the
changes in the current waveform due to changes in parameters,
and to introduce the measurement of stray capacitance technique
based on non-linear least square fitting method.
Fig. 1. Parallel plate model of micro-EDM plasma.
During the discharge process the electrons will have thermal
energy of a few eV to bring atoms into excited states and from the
collision process dielectric molecules gets dissociated which
finally forms plasma to conduct current with very high current
density. Even though the usual RC based micro-EDM circuit is
powered by a constant DC power supply, the presence of the large
series resistor limits the current flowing into the channel. Most of
the power dissipated in the plasma is supplied by the capacitor
placed in parallel to the plasma and therefore micro-EDM
discharge can be compared to the commonly used capacitive
discharges in plasma engineering [17] as opposed to other type of
discharge known as inductively coupled plasma discharge where
the electric field is generated by a time-varying magnetic field of
transformer action.
In capacitive discharge, obviously, the electric field results from
surface charges on electrodes and charges in dielectric filling up
the gap space. Fig. 1(b) shows the homogeneous model of plasma
[16] where the plasma is divided into a central quasineutral bulk of
thickness b and space charge sheaths of width s1 and s2. A current i
flows across the discharge plates and the plates are separated by a
distance d = b+ s1 + s2 and having cross sectional area A. In response
to the current flow, a discharge plasma forms between the plates,
accompanied by a voltage v across the plates and a power discharge
P into the plasma. Assuming that the plasma is in a quasineutral
state, it can be considered that almost everywhere the electron
density and ion density are equal, i.e. ne n. In this given model it is
also considered that the sheath regions are much smaller
compared to the width of the bulk plasma; b (s1 + s2). According
to this model the plasma impedance is defined as the ratio of
voltage and current for current flow in a capacitor given by Eq. (1)
[18].
zb ¼
ub
1
¼
ivC b
Ib
ð1Þ
where, Ub is the voltage across plasma, Ib is the current through the
plasma, Cb is the capacitance of the plasma, Cb = ebC0; and C0 is the
vacuum capacitance of a capacitor given by C0 = e0A/d. e0 is the
dielectric constant of vacuum, eb is the plasma dielectric constant
which is given by Eq. (2) [16].
v2pe
vðv jvm Þ
Electrical equivalent network of micro-EDM plasma
eb ¼ 1 The two parallel electrode plates across which the EDM plasma
forms is shown in Fig. 1 [16].
Where due to a very high electric field strength initially a
weakly-ionized channel forms which then rapidly grows from one
electrode to the other and results in primary electron avalanche
starting from the cathode. Subsequently this forms a streamer as
the initiation of a discharge process as could be seen in Fig. 2.
In Eq. (2),v is the angular frequency of the oscillation of plasma,
vpe electron plasma frequency and nm is the effective electron–ion
collision frequency for momentum transfer. Using the value of Cb
and eb, finally the bulk plasma containing two parallel electrodes
can be represented by a combination of an inductor (representing
electron inertia) in series with a resistor (representing electron–
neutral collisions), and a parallel capacitor (representing the
ð2Þ
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3
Fig. 2. Breakdown mechanisms leading to a spark discharge. Propagation of: (a) the primary electron avalanche; (b) a positive streamer; (c) a negative streamer [29].
electric field in the bulk plasma) and could be expressed by Eq. (3)
[17].
1
1
¼ jvC 0 þ
Zb
Rb þ jvLb
ð3Þ
Where, C0 is the vacuum capacitance, Lb is the inductance of the
bulk plasma given by Eq. (4) and Rb is the resistance of the bulk
plasma given by Eq. (5).
Lb ¼
1
v2pe C 0
Rb ¼ vm Lb
ð4Þ
ð5Þ
In Eq. (4), electron plasma frequency is denoted as vpe, which can
be evaluated using Eq. (6); and vm is the effective electron–ion
collision frequency for momentum transfer.
sffiffiffiffiffiffiffiffiffiffiffi
ne e 2
ð6Þ
vpe ¼
e0 me
Where, me = mass of electron, ne = electron density in plasma,
e0 = vacuum permittivity, and e = charge of electron. Now from the
above discussion, the network shown in Fig. 3(a) can be considered
as the electric circuit equivalent of plasma and can be replaced as
the plasma in the electric network of power supply circuit shown
in Fig. 3(b). In Fig. 3(b), Rd is the resistance of the discharge line
including the resistance of the cable, L is the inductance of the
discharge path, C is the capacitance of the installed capacitor
including the stray capacitance and R is the resistance of the
resistor installed after the DC source.
Evaluation of electric components in plasma
Evaluation of C0
C0 is the vacuum capacitance of a parallel plate capacitor with
an Area of cross section A and a distance between the capacitor d,
as shown in Eq. (7).
C 0 ¼ e0
A
d
ð7Þ
Table 1 shows values of experimentally obtained electrode gap
at spark for Tungsten electrode and SUS-304 workpiece with a
simple implementation of RC setup without any gap control and
was performed on the universal multi-process machine tool
proposed by Rahman et al. [19].
Since, the theoretical analysis and the experimental works of
this research is focused on the context of micromachining in the
lower dimensional range of micromachining domain, practically
the maximum applied voltage can be 100V–110 V and capacitance
is 470 pF and the electrodes will be between 20 mm–40 mm
diameter. From Table 1 and above argument a fair assumption can
be made that the value of d will be between 1.2 mm–6 mm and the
range of cross section area A will be a theoretically bounded by the
surface area of 20 mm–200 mm diameter plate for the range of
voltage and capacitance settings.
Fig. 4(a) shows plot of C0 at different gap distance d and ranging
between surface area of 20 mm–200 mm electrode diameter. The
maximum obtained value of plasma capacitance is only 0.2782 pF
in the simulation range which is even 20–30 times smaller
compared to stray capacitance of the system and electrode holder
[14]. Practically, the value of C0 is even smaller as during the
discharge the plasma only forms over a small portion of the
electrode and not the common cross section on the electrodes
facing each other. Descoeudres et al. [20] reported that the contact
surface between the plasma and the electrodes can be estimated to
be equal from measurements of crater diameter, which was
10 mm.
Their experiments were conducted at a much higher power
settings compared to the usual power settings for micro-EDM
[14,15]. where the crater sizes ranges from 2.2 mm to 5 mm.
Practically, since the plasma temperature in micro-EDM is
extremely high and much higher than the boiling temperature
of SUS-304, electrode plasma interface diameter can never be
larger than the crater size as at the interface of plasma and
electrode definitely erosion of metal will occur rendering the
electrode plasma interface diameter to be slightly smaller or equal
to the crater diameter, r [20]. Therefore, it can be assumed that the
interface diameter is 0.1 mm–0.2 mm smaller than the crater
Table 1
Gap width at different voltage and capacitance.
Fig. 3. (a) Electrical equivalent circuit of plasma, (b) electrical network of RC power
supply with a DC source having micro-EDM plasma replaced by the equivalent circuit.
Voltage (V)
Capacitance (pF)
Electrode gap (mm)
60
60
100
100
Stray (11 pF)
470
Stray (11 pF)
470
1.20
3.30
3.30
5.90
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diameter and can be estimated to be in between 2 mm–5 mm for
this analysis, which was mentioned earlier.
Fig. 5(a) shows the plot of plasma inductance at varying
electrode diameter and electron density of plasma at 1.2 mm
electrode gap and Fig. 5(b) shows the same at 6 mm electrode gap.
The result shows that the plasma inductance value ranges from
0.65 pH–67.8 pH. Even with a power supply circuit configuration
with the minimum wire length of 10 cm, the inductance
contributed by the wire is 0.1090 mH for tin plated copper stranded
AWG20 wire [23] which is about 1600 times larger than the
inductance of this micro-sized plasma and therefore, the inductance of the plasma can also be neglected.
Evaluation of Rb
Eq. (5) states the value of plasma resistance Rb, and combining
Eqs. (4), (5) and (6), Eq. (9) can be obtained as follows
Rb ¼ vm Lb ¼
vm
v2pe C 0
¼
me vm d
ne e2 A
ð9Þ
Now, from the definition of resistivity,
d me vm d
R b ¼ rb ¼
A
ne e2 A
ð10Þ
Therefore, the plasma resistivity (rb) can be stated as the
following in Eq. (11)
rb ¼
m e vm
ne e2
ð11Þ
Fig. 4. (a) Plot of plasma capacitance vs electrode area at different electrode gap; (b)
the effective area contributing to the capacitance is taken as the electrode plasma
interface diameter of 2 mm–5 mm.
Where, vm is the effective electron–ion collision frequency in
quasineutral plasma. The electron–ion collision, vm can be
computed from Coulomb force and is given by Eq. (12) [17].
diameter. Fig. 4(b) shows the plot of C0 at different gap distance d
and ranging between surface area equalling to the electrode
plasma interface diameter of 2 mm–5 mm and this resulted in a
capacitance value of 1.7385 1004 pF making at approximately
35,000 times smaller than the stray capacitance. Therefore, it can
be concluded that C0 in micro-EDM plasma with the electrode
dimension and electrode gap at spark mentioned above can be
really small and in analyzing the electric interaction with the
capacitance C0 can be ignored.
vm ¼
Where, V is the characteristic velocity of a Maxwell–Botzmann
distribution and can be considered as equal to the mean thermal
velocity ðvth Þ given by Eq. (13) [30]
Evaluation of Lb
Where, T is the temperature of plasma in Kelvin and Botzmann’s
constant is denoted by kB. Now, replacing Eq. (13) in Eq. (12) the
following, Eq. (14), can be obtained.
From the definition of inductor it is known that an inductor will
try to resist any change in the flow of current. Under an electric
field the electrons in plasma gains directional kinetic energy and
any changes in the flow of current will be resisted by the inertia of
the particles; therefore, it can be said that the inductance of plasma
is the ensemble of the inertia of particles and is related to the
characteristic frequency at which the electrons oscillate among the
heavier immobile ions. This electron plasma frequency is related to
electron density ne and given by Eq. (6). From Eqs. (4), (6) and (7),
Eq. (8) can be obtained.
1
dme
Lb ¼ 2
¼
vpe C0 Ae2 ne
ð8Þ
Descoeudres et al. [21] reported that electron density at the
beginning of the discharge is 2 1018 cm3 and electron density
reported by Nagahanumaiaha [22] is 3.5 1018 cm3. For the
simulation of inductance value the electron density value can be
considered to be bounded in a range of 1 1018 cm3 to 3.5 1018
cm3. Concerning the surface area contributing to the inductance
of the plasma is also given by the electrode plasma interface
ne e4
ð12Þ
16Pe20 m2e V 3
V ¼ vth ¼
8kb T
Pme
12
ð13Þ
1
vm ¼
ne e4 P2
1
2
ð14Þ
3
ð4e0 Þ2 me ð8kb T Þ2
Now, replacing vm in Eq. (11) the following plasma resistivity
(rb) can be obtained:
1
rb ¼
1
m e vm
m2e e2 P2
¼
1
3
ne e2
ð4e0 Þ2 m2e ð8kb T Þ2
ð15Þ
This estimation of Coulombic collision was based on the
assumption that the deflection angle after collision is large; but a
more detailed treatment leads to Spitzer resistivity given by
Eq. (16) [17].
1
rb ¼
1
m e vm
m2e e2 P2
¼
ln A
3
2
ne e
ð4e0 Þ2 ð8kb T Þ2
ð16Þ
Where, the correction factor ln(L) ln(lD/bp/2) = ln(4pND), called
the Coulomb logarithm that is related to the number of particles ND
in a Debye sphere and from our assumption ND = 1, resulting in ln
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Fig. 5. (a) Plot of plasma inductance at various electron density and electrode plasma interface diameter at 1.2 mm electrode gap; (b) and at 6 mm electrode gap.
(L) 2.531, which also shows good agreement to the value of ln
(L) 2.2 obtained by Descoeudres et al. [20]. Descoeudres et al.
[20] reported from their observation that the plasma temperature
reaches 8110 K and remains constant throughout the discharge;
and Nagahanumaiaha [22], reported that plasma temperature
ranges from 5167 K to 7889 K with an average plasma temperature
of 6170 K. Therefore, it was estimated that the plasma temperature
remains in a range of 5167 K–8110 K to compute a possible range of
plasma resistivity which is shown in Fig. 6.
Now, again consider that the electrode plasma interface
diameter is 2 mm–5 mm for this analysis, as has been mentioned
earlier. Fig. 7(a) shows the computed resistance at different plasma
temperature at 1.2 mm electrode gap and at 6 mm electrode gap
shown in Fig. 7(b).
It can also be assumed that crater diameter ranging around
2.2 mm can only be obtained at very small energy settings and at
those settings the required electrode gap is in the range of 1.2–
1.5 mm. Therefore, in Fig. 7(a) the resistance value shown in higher
electrode plasma interface diameter has been ignored and can only
focus was between 2 mm–2.5 mm interface diameter based on the
earlier assumption that the interface diameter is 0.1 mm–0.2 mm
smaller than the observed crater diameter. In this range of
electrode plasma interface diameter the plasma resistance Rb
varies between 25 V–80 V (with mean around 50 V). Similar
arguments can be placed for the case of a 5 mm crater size which
can only be obtained at larger capacitance and voltage settings;
and that will yield a larger electrode gap, like 6 mm for spark. At
5 mm electrode plasma interface diameter and 6 mm electrode gap
it can be seen from Fig. 7(b) that the plasma resistance Rb also
varies in the range between 30 V–76 V (mean 50 V). Therefore,
it can be assumed that micro-EDM plasma resistance remains
between 25 V–80 V, with a mean around 50 V for electrode gap
within 1.2 mm–6 mm and for obtained crater diameter between
2.2 mm–5 mm.
Analysis of the RC power supply electric network involving
micro-EDM plasma
Fig. 6. Plasma resistivity at different temperature.
From the evaluation of the equivalent electric network of
plasma, the inductive and capacitive load of micro-EDM plasma
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Fig. 7. (a) Plot of plasma resistance at different temperature and electrode plasma interface diameter at 1.2 mm electrode gap; (b) and the same at 6 mm electrode gap.
can be essentially ignored and consider only the resistive network.
From the evaluation of plasma resistivity, rb it can be observed that
the plasma resistivity is only dependent on plasma temperature
and not on electron/ion density of the plasma. Observation from
Descoeudres et al. [20] and detailed theoretical analysis of microEDM plasma by Dhanik and Joshi [24] indicated that the plasma
temperature can be considered to remain constant throughout the
discharge process — and therefore, the resistivity can be
considered to remain constant. The electrode plasma interface
diameter was reported to grow slightly during the discharge by
Descoeudres et al. [20] but for very fine short pulse ranging
between 20 ns–100 ns — it can be considered that the electrode
plasma interface diameter remains constant; and from this
assumption the plasma can be simplified to a resistive network
with constant resistance during the discharge. This will allow
simplifying the electric network shown in Fig. 3(b) to the network
shown in Fig. 8.
Let’s, consider that the voltage across the capacitor is v, and the
current flowing to node marked A is i0 and the current flowing out
from node A to the plasma is i. Now, applying Kirchhoff’s voltage
law (KVL) in loop 2 Eq. (17) can be obtained, applying KVL in loop 1
Eq. (18) can be obtained and applying Kirchhoff’s current law (KCL)
in node A Eq. (19) can be obtained.
di
v ¼ Rd i þ L þ Rb i
dt
E v Ri0 ¼ 0
i ¼ i0 C
dv
dt
ð18Þ
ð19Þ
Now, combining Eqs. (18) and (19) to replace i0 Eq. (20) can be
obtained
i¼
Ev
dv
C
R
dt
ð20Þ
ð17Þ
Fig. 8. Simplified electrical network of plasma with the power supply network (RC
circuit with a DC source).
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Differentiating Eq. (20), Eq. (21) can be found
2
di
d v 1dv
¼ C 2 dt
R dt
dt
Now, replacing i and di/dt in Eq. (17)
!
2
Ev
dv
d v 1dv
Ev
dv
C
C
þL 2 þ Rb
v ¼ Rd
R
dt
R dt
R
dt
dt
ð21Þ
s1;2 ¼ Rb
L
1
þ RC
2
ffi
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u(
)
1u
Rb
1 2
1
Rb
t
þ
þ
4
2
RC
LC RLC
L
ð24Þ
ð26Þ
Now, the 3 cases of the solution exists, namely overdamped for
real distinct roots, critically damped for real repeated roots and
underdamped for complex conjugate roots. Investigating the
discriminant will allow to analyze the type of solution that is
required to be computed for some practical values of the circuit
element. Let, R = 1 kV, L = 0.1090 mH; and C = 53 pF for this
investigation which are some of the values reported by Masaki
and Kuriyagawa (2010a).
By taking the 1st and 2nd derivative of the discriminant with
respect to Rb given by Eq. (26), Eqs. (27) and (28) could be obtained
dD
2
2
¼ R dRb L2 b RLC
ð30Þ
Solution of underdamped case
The characteristic equation of the 2nd order differential
equation given by Eq. (24) is given by the equation in (25) and
the solutions are given by Eq. (26).
Rb
1
1
R
þ b ¼0
þ
ð25Þ
sþ
s2 þ
RC
LC RLC
L
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
R
þ b P2
Q¼
LC RLC
ð23Þ
Now, in Eq. (23) it can be observed that both Rd and Rb come as a
summed term. The resistance of AWG20 tin plated copper stranded
wire is about 34 mV/m and the resistance contributed by tungsten
electrode system is around another 20 mV. Therefore, even for a
1 m long wire with a 32 mm electrode in the discharge path will
make Rd = 54 mV; and thus it can be considered that Rd + Rb Rb to
obtain the following:
2
d v
Rb
1 dv
1
R
R
þ
þ b v¼ b E
þ
þ
2
RC dt
LC RLC
L
RLC
dt
underdamped, overdamped and critically damped cases of the
differential equation given by Eq. (26) the following equation for
voltage v and current i could be obtained (where, P and Q are given
by Eqs. (29) and (30)).
1 Rb
1
þ
ð29Þ
P¼
2 L
RC
ð22Þ
Rearranging Eq. (22) and dividing both the sides of the equation
by LC following equation can be obtained:
2
d v
Rd þ Rb
1 dv
1
R þ Rb
R þ Rb
þ
þ d
þ
E
þ
v¼ d
2
RC dt
LC
L
RLC
RLC
dt
7
ð27Þ
v¼E 1
Rb
P
Rb
E
ePt cosQt þ sinQt þ
Q
R þ Rb
R þ Rb
!
2
Rb
Ev
Pt P
þ Q sinQt þ
i ¼ CE 1 e
R
R þ Rb
Q
ð31Þ
ð32Þ
Solution of overdamped case
v¼
Rb E
1
1
ðP Q Þ þ 1 eðPþQ Þt þ
ðP Q ÞeðPQ Þt þ 1
R þ Rb
2Q
2Q
ð33Þ
2
3
P2 Q 2
Rb E 4 1
ðPþQ Þt
ðPQ Þt 5
e
ðP Q Þ þ 1 ðP Q Þe
i ¼ C
R þ Rb
2Q
2Q
þ
Ev
R
ð34Þ
Solution of critically damped case
v¼
Rb
E 1 ePt PePt
R þ Rb
i ¼ C
Rb E
Ev
PePt ð1 þ PÞ þ
R þ Rb
R
ð35Þ
ð36Þ
Model validation
Validation of the model by varying l
2
d D
dR2b
¼
2
L2
ð28Þ
By setting Eq. (27) equal to 0 and since Eq. (28) is always
positive, the minima of the discriminant of Eq. (26) is when
Rb = 2 V and at that value Eq. (26) results in complex conjugate
solution 1.8608 1007 j4.1605 1008 and demonstrates to be an
underdamped case. Similarly, for values of Rb computed in section
Evaluation of Rb to be between 25 V–80 V the solution yields a
complex conjugate root. On the other hand, when the value of Rb
changes to 93 V the solution turns to and overdamped having real
distinct root. Moreover, for sweeping the equation between wide
range of R and L values it is required to obtain all of the 3 solutions
of the differential equation given by Eq. (24). Solving the
Masaki and Kuriyagawa [14,15], added varying length of wire in
the discharge loop to experimentally evaluate the effect of varying
wire length (and essentially the effect of inductance, L) on
discharge current waveform. In their experiment, the authors
varied the wire length from 0.1 m (which is the minimum length
required to connect the spindle head (tool electrode) to the
capacitor) to 15 m and experiments were conducted at two
different capacitance values: stray capacitance and 53pF. In this
section, the proposed model of micro-EDM electric network
defined by the Eqs. (31)–(36) (which are the solutions of the
differential Eq. (24)) involving the plasma impedance will be
validated using experimental results reported by Masaki and
Kuriyagawa [14,15]. Table 2 shows the condition used for easy
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Table 2
Conditions and setting values for obtaining current waveform of varying L (adapted from Masaki and Kuriyagawa
[14,15], except the equivalent inductance value which was not computed in the original report).
Item
Conditions and value
Voltage [V]
Capacitor capacitance [pF]
Resistance [kV]
Cable length [m]
Equivalent inductance [mH]
Machined depth [mm]
Dielectric
Tool electrode
Workpiece material
60
6,47 + 6
1
0.1, 0.5, 1.1, 2, 3.9, 7.6, 15
0.1090, 0.7059, 1.7265, 3.3782, 7.1084, 14.8663, 31.3811
10
EDM oil (CASTY-LUBE EDS)
Tungsten 32 mm diameter
Stainless steel SUS304
referral. Masaki and Kuriyagawa [14,15] only reported the length of
the wire used in the experiment and the experimentally obtained
current waveform.
For the computation of the model defined by Eqs. (31)–(36), the
inductance value of the wire length is required to be computed
using Eq. (37) [23]. The computed inductance value is also shown
in Table 2 [23].
2l 3
ð37Þ
L ¼ 0:002l ln dw 4
Where, dw is radius of the wire in cm and l is the length of the wire
in cm and the obtained value of L is in mH.
In addition to the minimum wire length of 10 cm to configure
the discharge circuit, as reported by Masaki and Kuriyagawa
[14,15], there is also dead length of wire which has to be added in
the return path of the current after discharge which contributes to
additional inductance. Considering the structure of the equipment
used by Masaki and Kuriyagawa [14,15] (the distance between the
connections on the mandrel and XY table is approximately 20 cm)
and travel length of both X and Y axes are 5 cm it can be considered
that a minimum of additional 25 cm of wire is required to configure
the return path of the current (this required minimum length was
also confirmed from personal communication with Masaki and
Kuriyagawa [14,15]). Inductance is also contributed by the
electrode, feeding system and workpiece. Considering a 3 cm long
0.300 mm diameter tool electrode — the contributed inductance
should be around 0.0314 mH (this is about 35% of the contribution
by 10 cm wire). Therefore, it can be assumed that the system had
additional inductance equivalent to a total of 30 cm wire length
(0.3929 mH) in the return path of the discharge current contributed
by the wire in the return path, the tool electrode and other
component of the feeding system.
In order to validate the proposed model, computer simulation
was performed using Matlab using the Eqs. (31)–(36). The program
was written to compute the determinant equation based on the
input R, L and C value to identify the appropriate solution case
(underdamped, overdamped and critically damped). Simulation
was performed at 3 different plasma resistance values — taking
Rb = 40 V, 50 V and 60 V, keeping 50 V as the mean. The current
waveform obtained from the simulation and the experimental
waveforms of Masaki and Kuriyagawa [14,15] are compared in
Fig. 9. Two important parameters of current waveform are the
pulse width and peak current, which are compared between
experimental and simulation results (Fig. 9). Pulse width is defined
by the time between the beginning of a discharge to the first zero
crossing of the current waveform and peak current is defined by
the peak value of the current over the discharge duration. The
experimental current waveform for 0.1 m shows smaller peak
current (300 mA) compared to the value obtained from the model
(415 mA), but considering micro-EDM as a stochastic process and
the presence of a debris particle in gap space might change the
conductivity of plasma which may result in quite a significant
difference in peak current as well as in waveform.
Comparing Figs. 6 (a) and 10 (a) of Masaki and Kuriyagawa
[14,15], the same can be concluded as at the same setting one
profile shows a peak at 300 mA and the second one shows a peak at
400 mA. Moreover, usually the peak of the current profile shows a
well formed peak as could be seen in Fig. 10(a) of Masaki and
Kuriyagawa [14,15] as opposed to the valley shaped peak that could
be seen in Fig. 6(a) which indicates that the peak value of profile in
Fig. 10(a) is more appropriate.
Another factor that might have contributed to the little
difference is due to the fact that the current probe used for
performing these experiments was band limited between 1.2 kHz
to 200 MHz (Tektronix Current Probe — Model CT2) which signifies
that a very sharp movement happening in less than 5 ns will
display an averaged response in the oscilloscope. Therefore, some
variation with experimental data is expected but the congruency
between the overall waveform and the changes observed due to
varying L demonstrates excellent fit of the theoretical value to the
experimental data.
Masaki and Kuriyagawa [14,15] reported that the current
waveform generated at 15 m wire length was not showing clear
waveform to measure pulse width and peak current. This
phenomenon could be explained well using this model if it is
assumed that given the experimental settings and inter-electrode
gap if the steady state current drops below 30–40 mA the discharge
stops and Rb becomes an extremely large resistor until the next
breakdown, whereas if the steady state current remains above
50 mA then continuous arc discharge takes place — then. This
makes a fair assumption based on the fact that in plasmas energy is
dissipated in inelastic collisions, including the ionization events
which maintain the plasma, and excitation collisions which lead to
photon emission that makes the plasma visible; and electron
energy is also dissipated in elastic collisions with the background
neutral dielectric which causes the heat to get transferred to the
ambience formed by dielectric and electrodes, thus continuous
power to the plasma is needed to maintain it. Fig. 11 shows the
simulated waveform at 15 m wire length and it can be observed
that the slowly varying waveform did not cross below 30–40 mA at
steady state and this caused a continuous power flow to the plasma
to keep it at arcing condition thus the peak and the pulse width
were not identifiable.
Validation of the model from discharge energy
In the case of RC type power supply shown in Fig. 8, the
repetition of the charging and discharging occurs in which
capacitor C is charged through resistor R and discharged between
the electrode and workpiece and therefore, the energy deposited in
the gap from the spark is commonly computed by the energy that
can be stored in the capacitor which is given by 1/2CV2 [25]. But, this
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Fig. 9. Showing experimentally obtained (left column) [14,15] and simulated current waveform at 3 different Rb values (40 V, 50 V, 60 V). Observed peak current and pulse
width values are plotted in Fig. 10.
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Fig. 12. Comparison between discharge energy computation methods.
Fig. 10. Comparison between experimental and theoretical values for 53 pF and 6 pF
capacitance of (a) discharge peak current at varying wire length and (b) discharge
current pulse width at varying wire length.
The model shows excellent agreement to both the data
validating the model. The theoretical model deviates between
4%–7% from the 1/2CV2 method of computation, which is due to the
fact that the model accounts for the power imparted to the plasma
directly from the DC source during the discharge. Comparing to the
experimental values, both the values obtained from 1/2CV2 method
and values obtained using this proposed model fit nicely at smaller
discharge energy; but at higher energy both the values are
overestimated from the experimentally obtained result. This could
potentially indicate that the particular discharge captured was not
from a fully charged capacitor, or it could even deviate from the
assumption that the electrode plasma interface diameter remains
same throughout the discharge — which may not be the case for
larger pulse width; meaning that the plasma resistance was
reducing gradually due to the slow expansion of interface diameter
and thus less heat was deposited in the gap. But, this argument
requires further investigation.
assumption may not hold true always as the discharge can be from
a partially charged capacitor and as well as for a small power would
be supplied by the main DC source through the charging resistor R
to the plasma during the discharge. Masaki and Kuriyagawa [14,15]
proposed that discharge energy can be also measured by obtaining
both the voltage waveform and current waveform and then
multiplying the area under both the curves (since, power P(t) = V
(t) I(t)).
The proposed model given by Eqs. (31)–(36) has been used to
compute the discharge energy in a single discharge from the
theoretically computed voltage and current waveforms. Fig. 12
compares between the discharge energy obtained from the
theoretical computation, experimental data of Masaki and
Kuriyagawa [14,15] and the discharge energy obtained by
computing 1/2CV2.
Masaki and Kuriyagawa [14,15] performed experiments to
observe the effect of supply voltages at 40 V and below on peak
discharge current. The theoretically computed value by using this
proposed model at 60 V or below did not show good fit to the
experimental results for values of voltage below 60 V as could be
seen Fig. 13 (red line). This evaluation was done using Rb = 50 V
which has been computed with the assumption that the value of
electrode
plasma
interface
diameter
is
2 mm–2.5 mm
(Section Evaluation of Rb). But, practically for voltages lower
than 60 V the assumptions on diameter of electrode plasma
interface needs to be corrected as the crater size changes
Fig. 11. Shows that the theoretical pulse waveform remains above 50 mA which
may cause a continuous arc discharge.
Fig. 13. Plasma resistance for electrode plasma interface diameter 1 mm (For
interpretation of the references to colour in the text, the reader is referred to the
web version of this article.).
Validation of the model at lower DC supply voltage and at higher
discharge energy
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11
Fig. 14. Comparison between experimental and theoretical value of peak current at different voltage settings.
significantly at such low voltage settings. Masaki and Kuriyagawa
[14,15] observed that the crater size became 1.22 mm for 30 V input
voltage. Therefore, the electrode plasma interface diameter can be
considered as 1.0 mm (as it has been assumed that the electrode
plasma interface diameter is slightly smaller than the crater size).
Using plasma electrode interface diameter as 1.0 mm, plasma
resistance was recomputed to be 165 V–325 V (with mean around
250 V) for the model (Fig. 13). With this corrected assumption, the
model yielded good agreement with experimental findings at
lower voltages (Fig. 14). Using plasma resistance value 250 V
shows better agreement than the computation at 50 V which is
shown in Fig. 14 as well.
At the same time, this also needs to be observed that the model
predicts a linear relationship at different voltages and peak current
while the experiment indicates a sort of exponential relationship.
Furthermore, while the calculation based on corrected assumption
obtained better fit with the experimental observation for lower
voltages it deviated from the peak current that was observed at
60 V. This actually further establishes and validates the theoretical
analysis that the model is based on the plasma resistance. In the
mode value of plasma electrode interface diameter is required as
an input to the model and is estimated from the crater size.
Therefore, the value of inter-electrode gap and plasma electrode
interface diameter (given by the crater diameter) plays a very
important role in the understanding of micro-EDM energy
discharge characteristics and interaction with plasma.
The capability of the proposed model was further explored by
comparing the theoretical result to experimental results published
by Mahardika and Mitsui [26] which was done at higher voltage
and larger capacitance settings. Their experimental conditions
were: Supply voltage V = 110 V, C = 3300 pF, R = 1000 V and during
their experiments they used another additional voltage probe
(Tektronix P6109B) for taking voltage reading which added
additional inductance of approximately 0.288 mH (L = 25 cm, dw =
0.075 cm). From the experimental results of Han et al. [27] it can be
observed that the crater size obtained from this voltage and
capacitor settings is about 12 mm on tungsten and tungsten
carbide workpiece. Using this assumption the plasma resistance
was recomputed to be between 5 V–10 V. Fig. 15(a) shows the
experimentally obtained pulse profile and the simulated pulse
profile is shown in Fig. 15(b) where good agreement between the
pulse width (175 ns experimental value; simulated 165 ns–171 ns,
mean 168 ns) and the pulse peak (experimental value 4.9 A;
simulated 4.7 A–5.6 A, mean 5.1 A) can be observed.
Observation on the current waveform generated by the
proposed model
As the proposed model investigates into the interaction
between the micro-EDM power supply and micro-EDM plasma
it can be used for elucidating the theoretical relationship of
different circuit parameters and how that may change the energy
discharged on the workpiece by a spark. In this section the effect of
varying R on the pulse waveform will be analyzed which will lead
to the selection of an appropriate value of R for micro-EDM power
supply circuit to be developed. Effect of varying voltage V,
Fig. 15. Current waveform (left Figure: a) [26] and simulated value at C = 3300 pF, V = 110 V (right Figure: b). Excellent match of pulse width (175 ns experimental value;
simulated 165 ns–171 ns, mean 168 ns) and the pulse peak (experimental value 4.9 A; simulated 4.7 A–5.6 A, mean 5.1 A) can be observed.
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inductance L and capacitance C on current waveform will also be
observed from the model which will provide sufficient background
understanding for selection of these parameters during an
experiment for desired output. This model can also be used for
obtaining stray capacitance and inductance of the power supply
circuit contributed by obtaining a current waveform and then
using non-linear curve fitting of the current waveform to the
mathematical model.
Effect of varying resistor R and selection of R for micro-EDM power
supply
Resistor R, shown in Fig. 8, plays a very critical role in designing
the micro-EDM RC power supply. The main role of R is to limit the
supply of current to the capacitor as well as to the plasma
contained in the gap immediately after a discharge such that after
the spark the discharge stops due to the lack of necessary flow of
current required to maintain the plasma. This energy is needed to
maintain the plasma as there is loss from inelastic collisions,
ionization events which maintain the plasma, energy dissipated in
the form electromagnetic radiation loss ranging from visible light
to other frequencies of electromagnetic waves including X-rays,
and heat transferred to the background. Therefore, a large R is
preferred from this concern to minimize arcing current. On the
other hand after one discharge, as soon as the plasma is switched
off, the best condition is to have the capacitor fully charged for the
next discharge. But the charging time of the capacitor will be based
on the time constant formed by the capacitor and the resistor. For a
practical settings of micro-EDM experiment, C = 100 pF capacitor,
V = 60 V and R = 400 V, 1 kV, 2 kV will have time constant t = 10 ns,
100 ns and 200 ns respectively. Essentially this will result in
charging 63% of the capacitor and in most part of this period the
capacitor is not charged enough to make the spark happen. From
this perspective smaller settings of R will allow the capacitor to get
quickly charged. Therefore, due to this contradictory relationship
the value of R needs to be optimized such that it provides a balance
between minimizing charging time and reducing the chance of arc.
This is usually done by performing experiments as the fastest
machining time resembles the optimal setting. In arcing condition
the machine requires to perform back and forth motion repeatedly
to stop the short condition which extends the machining time
whereas, in slow charging case the machine takes longer time
simply because of the reduction in spark per second. Experiments
were conducted on the universal multi-process machine tool using
the machining conditions shown in Table 3 and the machining time
is shown in Fig. 16.
The machining time indicates 1 kV gave the best machining
time whereas 400 V took significantly longer machining time and
at 2 kV the machining time was slightly longer than for 1 kV, even
though statistically the difference was not significant (the same
experiment was performed 3 times to obtain experimental testretest repeatability).
Fig. 16. Machining time of 1 mm slot by WEDM using R = 400 V, 1 kV and 2 kV. The
error bars are one SD (n = 3).
The above experimental results could be similarly observed for
optimizing R for micro-EDM RC circuit by computing the waveform
profile at R = 400 V–2 kV using this proposed model, as could be
seen in Fig. 17.
The current waveform up to R = 600 V had no zero crossing and
remained above 100 mA at 60 V supply voltage beyond the first
90 ns which can be considered as large enough current to maintain
the plasma. From R = 700 V to R = 2 kV it had zero crossing within
the first 60 ns which may indicate that the power from the plasma
was cutoff within this time and thus arcing was minimized.
Therefore, practically the power supply circuit for micro-EDM
should be designed with resistance R > 700 V. At resistance values
above 900 V it can also be observed that the negative dip was
below 50 mA compared to those at 700 V and 800 V. Given that
micro-EDM is a stochastic process with significant amount of
process noise and presence of debris in the gap space changes the
waveform — it can be concluded that a value of above 900 V will
provide sufficient safety margin in avoiding continuous arcing in
the circuit.
Effect of inductance on the current waveform
In Section Validation of the model by varying l, as part of the
validation of the model, current waveforms at different wire length
in order to vary the inductance were generated using this model
and was compared with the experimental data shown in Fig. 9. In
this section, changes in current waveform with varying inductance
will be further discussed. It can be observed in Fig. 18 that the peak
current changes significantly from 660 mA to 530 mA and the pulse
becomes broader from 20 ns to 30 ns (first zero crossing of current
after the beginning of a discharge) for a change of inductance from
Table 3
Machining conditions for WEDM experiments varying R.
Item
Conditions and value
Setup used
Feedrate
Voltage [V]
Capacitor installed [pF]
Resistance [V]
Estimated inductance L [mH]
Machined slot size [mm]
Dielectric
Tool electrode
Workpiece material
WEDM
2.0 mm/s
60
100 pF
400, 1000, 2000
1.9043
1.0
Total EDM 3 oil
Tungsten wire (AgieCharmilles)
Stainless steel SUS304
Fig. 17. Current waveform computed theoretically at R = 400 V–2kV (C = 100 pF,
L = 1.9043 mH, V = 60 V, plasma resistance Rb = 50 V).
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Fig. 18. Current waveform at varying inductance value at 0.2457 mH, 0.5469 mH and
1.9043 mH (C = 100 pF, R = 1 kV, V = 60 V, plasma resistance Rb = 50 V) for equal
energy discharged in all 3 profiles around 190 nJ.
0.2457 mH (l = 0.20 m; AWG20 wire) to 0.5469 mH (l = 0.40 m;
AWG20 wire). At L = 1.9043 mH (l = 1.20m; AWG20 wire) the pulse
becomes much broader having a width of around 50 ns and the
pulse peak changes to 360 mA.
But the computed energy under all these three waveforms
remains the same at around 190 nJ. The similar effects of varying
inductance by varying the length of the wire can also be observed
in Fig. 9. Masaki and Kuriyagawa [14,15], observed that by
increasing the wire length, the craters on the surface became
larger, shallower and flat; and therefore, this can be applied to
improve the properties of the machined surface. One possible
explanation to this can be provided assuming that the disk heat
source based electro-thermal model is more appropriate at very
small energy spark compared to the point source head model [3]
and therefore, over longer spark duration but with the same
amount of energy delivered — there is sufficient time for the heat
to get conducted and material removal by melting action is higher.
On the other hand with very short pulse — there is rather less time
for conduction and due to higher energy density removal by
evaporation is increased where the heat is lost to the dielectric
rather quickly. However, this argument requires further investigation. But, this can be stressed from the experimental observation
that variation in L can be instrumental in changing surface
properties of machined workpiece. Larger L increases machining
time as the pulse gets larger but switching L can be an option where
the initial machining is done with smaller L value and the finishing
can be obtained by larger L value.
13
Fig. 19. Current waveform at varying capacitance (C = 50 pF, 100 pF, 200 pF)
(V = 60 V, R = 1 kV, L = 1.9043 mH, plasma resistance Rb = 50 V).
shown in Fig. 15 obtained at 110 V but a much higher capacitance
value which significantly changed the pulse width to 165 ns–171 ns
and the pulse peak to 4.7 A–5.6 A indicating a change in both pulse
width and peak.
On the contrary, changing the supply voltage V (60 V, 80 V and
100 V) did not change the value of pulse width (remained at 50 ns)
but significant change in the pulse peak was observed (345 mA,
460 mA, 575 mA respectively) as could be seen in Fig. 20.
This is also to be noted that the energy to be discharged in a
pulse changes proportionally to the square of the input voltage.
Therefore, it can be inferred from the earlier discussions in this
chapter that higher input voltage will create a deep crater and will
have increased proportion of removal by vaporization. Therefore,
for rough machining higher supply voltage value around 110V–
120 V is preferred and for final finishing cut the voltage needs to be
reduced to 60 V.
Non-linear curve fitting for estimation of stray capacitance and
inductance
Stray capacitance determines the amount of minimum energy
spark size that a circuit is able to provide and therefore it is desired
to have the equipment and power supply to be designed with
minimum stray capacitance. Depending on the length of the cable
stray capacitance will eventually not be negligible [28]. Existing
methods for estimation of stray capacitance depend on the fact
that the discharge current pulse width depends on the capacitance
which is a rather indirect and inaccurate inference for
Effect of capacitance C and supply voltage V on the waveform
Varying capacitance C and V increases the amount of energy
stored in the power supply (as Ed = 1/2CV2). The effect of C can be
observed from the simulation of pulse waveform shown in Fig. 19
with 3 capacitance values of C = 50 pF, C = 100 pF and C = 200 pF.
Pulse peaks were at 265 mA, 345 mA and 440 mA; and pulse width
was found to be 37 ns, 50 ns and 70 ns respectively (which are given
by the first zero crossing of current after the beginning of a
discharge).
It can be observed from the current waveform that both the
peak and pulse width changed due to the change in capacitance
which is similar to the change in inductance L. This is due to the fact
the C actually changes the impedance and thus modulates the
frequency of the waveform and at the same time there is higher
energy stored which forces the pulse peak to be higher.
Experimentally this was also observed in the pulse waveform
Fig. 20. Current waveform at varying supply voltage (V = 60 V, 80 V, 100 V)
(C = 100 pF, R = 1kV, L = 1.9043 mH, plasma resistance Rb = 50 V).
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measurement of stray capacitance. Masaki and Kuriyagawa [14,15]
proposed that the measurement of stray capacitance can be
performed from actual discharge energy by observing the current
and voltage waveform and computing their product. This requires
a voltage probe to be connected to the circuit for simultaneous
acquisition of a voltage and a current waveform which introduces
additional stray capacitance and therefore, incurring inaccuracy in
its measurement. During the computation of stray capacitance,
Masaki and Kuriyagawa [14,15], computed the total discharge
energy to be 31 pF and the system was installed with a 10 pF
capacitor. During the measurement they used a voltage probe with
13–17 pF capacitance and from this they concluded that the system
had a stray capacitance between 4–8 pF with a mean value of 6 pF.
This clearly indicates the inaccuracy in the measurement of stray
capacitance. Another alternative solution for stray capacitance
measurement is to obtain the current waveform alone (without
installing a voltage probe) from inductively coupled current probe
(inductively couple probe does not have any loading or minimal
loading effect on the power supply network) and then performing
non-linear curve fitting of the waveform with this model (given by
Eqs. (31) and (32)). The green profile in Fig. 21 shows one such plot
obtained from the system using only a current probe. As there is
sinusoidal oscillation in the current waveform, the underdamped
case was taken as the solution for fitting. Non-linear curve fitting
was done to obtain the value of C in the equation and the input
values were R = 1000 V; L = 0.5019 mH (total 10 cm wire for the
capacitor to tool electrode and 30 cm for workpiece to capacitor in
the return path; explained in Section Validation of the model by
varying l); Rb = 50 V, V = 60 V. The experimentally obtained current
waveform provided inputvalue for the current at stray capacitance.
The fitting was performed using custom written Matlab
program and the non-linear fitting was done using Matlab routine
lsqcurvefit (lsqcurvefit, R2012a). This routine performs non-linear
fitting to obtain least square error and finds coefficients x that best
fit the equation of the model (Eq. (32)) and the optimization is
done by computing the following equation:
m
1X
min 1
2
k F ðx; xdataÞ ydata k22 ¼
ðF ðx; xdatai Þ ydatai Þ
x 2
2
ð38Þ
1
Where, input data isxdata, and the observed output is ydata; and
xdata and ydata are vectors of length m and F (x, xdata) is a vectorvalued function (Eq. (31) in this case). Large scale optimization was
used as the fitting option with an initial guess of C = 50 pF. The
fitting program returned the value of stray capacitance based on
the best fitted curve.
Fig. 21. Non-linear curve fitting of the current waveform for computing stray
capacitance. For fitting input to the model was R = 1000 V; L = 0.5019 mH; Rb = 50 V,
V = 60 V and initial guess of C = 50 pF (For interpretation of the references to colour
in the text, the reader is referred to the web version of this article.).
Fig. 22. Non-linear curve fitting of the current waveform for computing the value of
inductance. R = 1000 V, C = 39 pF, Rb = 50 V, V = 60 V actual L = 0.5776 mH. With an
initial guess of L = 5 mH, the model computed the value of L = 0.58936 mH.
The red profile in Fig. 21 has been obtained from the model of
Eq. (32) computed using the value of stray capacitance, C, obtained
from curve fitting. After 18 iterations of computation the model
estimated the stray capacitance to be 6.58 pF which matches well
with the experimentally reported value of 6 pF by Masaki and
Kuriyagawa [14,15].
Similar fitting can also be performed for obtaining any other
parameters of the proposed model. Non-linear curve fitting was
performed with another current waveform (Fig. 22) for computing
the value of inductance. With an initial guess of L = 5 mH and input
values of R = 1000 V, C = 39 pF, Rb = 50 V and V = 60 V the model
computed the value of L = 0.58936 mH (actual L = 0.5776 mH).
Limitations
Although this work has generated many interesting and new
findings, it has also opened new questions where further
investigations are necessary to enhance our understanding of
micro-EDM plasma and process. Firstly, this proposed analytical
model is only applicable to micro-EDM process with electrode
diameter between 20 mm–40 mm and maximum voltage settings
are 100–110 V. Assumptions that plasma capacitance and inductance can be considered as negligible are limited within this scope
and for single pulse discharge. Beyond the condition of single pulse
discharge, this model is also not capable of explaining simultaneous multiple discharges, the impact from the presence of large
debris within the electrode plasma interface, arcing and short
circuit conditions, etc. Secondly, one distinct observation from this
proposed model employing micro-EDM plasma properties is that
the assumption of electrode plasma interface diameter is a critical
parameter and requires re-computation for various discharge
energy ranges.
These can be considered as major limitations of this model
leading to considerable simplification of a complex problem which
may then finally impact the outcome of the model rendering it to
be unrealistic, specially if applied beyond the micro-EDM ranges
defined. Further development of the proposed model integrating
the electrode plasma interface diameter will allow the model to
factor in the changes in electrode plasma interface diameter based
on discharge energy settings. Besides, the inclusion of plasma
capacitance and plasma inductance as parameters, instead of
considering them negligible within a specific range, will serve
towards developing a generalized model of EDM process employing plasma properties. Finally, the physical properties of plasma
are quite fascinating. Plasma physics is a complex phenomenon
that spans across various sub-disciplines of physics (e.g., material,
thermal, and electrical properties) and are topics of ongoing active
Please cite this article in press as: A.B.M.A. Asad, et al., Analysis of micro-EDM electric characteristics employing plasma property, NULL (2017),
https://doi.org/10.1016/j.cirpj.2017.09.005
G Model
CIRPJ 439 No. of Pages 15
A.B.M.A. Asad et al. / NULL xxx (2017) xxx–xxx
research. The advancement in our understanding of plasma physics
will provide us with better characterization of plasma properties
and can be employed in the context of micro-EDM to elucidate
properties and critical parameters for improved micromachining.
Conclusion
This paper presented a theoretical analysis of the electric
properties of micro-EDM plasma for an RC power supply circuit
with a DC source. The following conclusions, which summarize the
significance of this study, can be drawn from the analysis It is analytically established in this paper that for capacitive
plasma discharge the plasma can be resolved into capacitive,
resistive and inductive components. In case of micro-EDM
capacitive and inductive component of the plasma can be
ignored and the model is proposed using the resistive component of the plasma.
Presented analysis provides significant insight for realizing
changes in the current waveform due to changes in parameters
like input voltage, capacitance, inductance and the charging
resistance, which can be utilized for process parameter
optimization.
It has been demonstrated that the inductance and stray
capacitance in micro-EDM power supply can be estimated using
non-linear least square fitting of experimentally obtained
current waveform to the analytical model.
Presence of inductance in the power supply network has been
observed to minimize the peak height and increase pulse width
of the current waveform which results in shallower and flatter
craters and improves surface roughness.
The analysis and understanding developed through this study
will considerably leverage the design of a power supply fitting
the appropriate role of all of the three components Resistance
(R), Inductance (L) and Capacitance (C).
Acknowledgement
The authors would like to acknowledge the technical support
from Mikrotools Pte Ltd. (National University of Singapore Spin-off
Company).
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Please cite this article in press as: A.B.M.A. Asad, et al., Analysis of micro-EDM electric characteristics employing plasma property, NULL (2017),
https://doi.org/10.1016/j.cirpj.2017.09.005