The skin-eﬀect in ferromagnetic electrodes for wire-EDM

B. Schacht

∗

, J.-P. Kruth

∗

, B. Lauwers

∗

, P. Vanherck

∗

Abstract

High frequency current pulses in wire-EDM lead to excellent machining performance, in

terms of work piece roughness, material integrity of the cut and material removal rate. To

reach the highest frequencies the wire-EDM generator mostly consists of a voltage source

with an as low as possible internal inductance. The working current delivered to the spark

and hence the material removal rate of the process depends on the total impedance of the

electrical circuit. In this article the importance of the wire’s impedance will be shown. Due

to the skin-eﬀect this impedance depends on the frequency of the current signal, especially

for ferromagnetic wires, such as steel wire. Coatings will prove to be primordial to prevent

machining speed to drop signiﬁcantly.

1 The high current needle-impulse generator

All modern wire EDmachines are equipped with a high current needle-impulse generator(ﬁgure 1)

[10]. It is able to deliver very short (0.2µs) and high current pulses (400A). The generator consists

of two independent voltage sources U

L

and U

H

. One is dedicated to deliver the ignition voltage

(U

H

) and current, the other must deliver a higher current (the working current) at lower voltage

(U

L

) necessary for material removal. The current of the ﬁrst voltage source U

H

is limited to a few

(e.g. 16) Amperes, so that in fact it acts as a current source. In the spark’s ignition phase (1) S

2

and D

H

are engaged, after a while (2) the working current is imposed by engaging S

1

and S

2

. For

interrupting the spark (3), it is commuted to D

1

and D

2

. In this stage S

1

and S

2

are open. U

L

pulls the voltage down to enable a fast drop of the current. This excess of energy is stored in the

internal inductance of U

L

and used in the following spark.

A voltage source is chosen because it is more easy to produce steep current slopes (needle

impulse) with the latter source than with a current source. A needle-impulse generator minimizes

thermal inﬂuences on the work piece [1, 10]. On the other hand there is more electrode wear but

this is not so important in wire-EDM since the wire is continuously renewed. A voltage source

delivers a current that depends on the load: impedance of the machine, the wire, the gap and

the work piece, which is a disadvantage while compared to current sources, who are supposed to

deliver a ﬁxed current independently of the process’s impedance. Two independent voltage sources

minimize energy loses in internal resistances. The current can reach 400A.

2 The skin-eﬀect in ferromagnetic wire-electrodes

The skin-eﬀect results from eddy currents within the wire that counteract the current in the core,

and forces the current to the surface of the wire. Figure 2 shows the resulting current density

distribution along the wire’s radius for a steel and a copper wire as it is simulated at 100kHz [7].

Generally speaking the current ﬂows over a smaller area of the cross section of the wire due to the

skin-eﬀect. It hence experiences higher resistance and lower inductance. For a steel wire of 250

µm diameter the current density J at the surface of the wire is almost 500 times as big as on the

axis. For a copper wire it is only 1% bigger. The following subsections will discuss the frequency

and material dependence of both.

∗

Katholieke Universiteit Leuven, Division PMA, Celestijnenlaan 300B, 3001 Leuven, Bel-

gium

1

U

L

U

H

R

H

R

W

L

S

D

H

D

2

D

1

S

1

S

2

i

I(t)

t

t

U(t)

td tdte to te

Figure 1: Principle scheme of the high current needle-impulse generator

Steel wire

Wire radius r (µm)

Current density ratio j(r)/j(0)

Wire radius r (µm)

Copper wire

1

Figure 2: Eﬀect of skin-eﬀect on current density distribution at 100kHz

2

Steel 0.05% C Steel 0.7% C Copper Brass CuZn

37

ν(Hz) 100000 100000 100000 100000

σ

¡

S

m

¢

7.7e

6

5.6e

6

5.7e

7

1.5e

7

µ

r

5000 4000 1 1

diameter (µm) 100 250 100 250 100 250 100 250

ξ 3.08 7.70 2.35 5.87 0.12 0.29 0.06 0.15

R

0

¡

Ω

m

¢

16.53 2.64 22.73 3.63 2.23 0.36 8.48 1.35

R

¡

Ω

m

¢

55.21 21.01 59.55 22.25 2.23 0.36 8.48 1.35

R

R

0

3.34 7.96 2.62 6.13 1 1.003 1 1.0002

L

0

µH

m

200 200 250 250 0.05 0.05 0.05 0.05

L

µH

m

80 32 84 34 0.05 0.05 0.05 0.05

L

L

0

0.16 0.13 0.59 0.24 1 1 1 1

Table 1: mathematical values related to the skin-eﬀect in plain wires

2.1 The frequency dependence of a wire’s resistance

2.1.1 Plain wires

The frequency dependence of a wire’s resistance is approximately given in [4] by

R

R

0

≈ 1 +

ξ

4

3

if ξ << 1

R

R

0

≈

1

4

+ ξ +

3

64ξ

if ξ >> 1 (1)

The value ξ (given by equation 2) indicates whether the skin-eﬀect is important for the wire

under consideration. This depends on its radius r

0

, its electrical conductivity σ, its magnetic

permeability µ and the frequency of the current signal ν.

ξ =

r

0

2

√

πνσµ (2)

R

0

is the D.C. resistance per unit length of the wire and is given by Pouillet’s formula as

R

0

=

1

πσr

2

0

(3)

The value of equations 1, 2 and 3 is calculated for steel and copper wires of diameter 100 and

250µm at 100kHz. This is the ground frequency of the current signal. The results are given

in table 2.1.1. It shows that the skin-eﬀect only plays an important role for steel wires, and

especially for steel wires with big diameter. The reason for the presence of this eﬀect in steel wires

is their high magnetic permeability. The result is an increased resistance. For wires of 250µm the

resistance goes up by a factor 6 to 8, depending on the carbon content of the wire. Interesting

to see is that the lower carbon content steel wire has a lower resistance at D.C., but at high

frequencies it rises to the same level as the higher resistant high carbon content wire. This can

be fully explained by equation 2: the importance of the skin-eﬀect rises with conductivity and

permeability. The resulting resistance for steel wires at 100kHz is 60 times as high as for copper.

For the steel wire of diameter 100µm the inﬂuence of the skin-eﬀect is less, it yields an increase

of the resistance by a factor 2 to 3, resulting in a value which is 30 times as high as a copper wire.

Since the resistance of a steel wire at the frequencies under consideration seems to be independent

of carbon content, the 0.7%C wire can be chosen to proﬁt from its higher tensile strength in

precision applications [6, 8].

2.1.2 Coated wires

The inﬂuence of the skin-eﬀect on the overall resistance of a steel wire can be minimized by

introducing non-magnetic-permeable materials in the coating that are good conductors. Of course

3

Name Core Subcoating Coating R

0

R R/R

0

thickness (µm) thickness (µm) thickness (µm) (Ω/m) (Ω/m)

Macrocut Steel 0.64% C CuZn42 Ag 2.364 3.485 1.47

102.9 22 0.1

Compeed Steel 0.075% C Cu CuZn43 0.977 1.105 1.13

97 20 8

E

0/1

∗

Steel 0.7% C None Zn 3.523 17.322 4.91

124 0 1

E

0/10

Steel 0.7% C None Zn 2.771 5.908 2.13

115 0 10

E

10/1

Steel 0.7% C Cu Zn 1.496 2.082 1.39

114 10 1

E

10/10

Steel 0.7% C Cu Zn 1.396 1.787 1.28

105 10 10

Table 2: mathematical values related to the skin-eﬀect in coated wires

in this case the D.C. resistance will be lowered, but furthermore at high frequencies the raising of

the resistance will be damped. The current that is forced out of the core will chose the path of

least resistance and will be conducted in the coating.

Since all known coatings (Cu, CuZn

x

, Zn,...) are a-magnetic, the skin-eﬀect can be neglected

in the calculation of the resistance of the coating. For a coating with outer radius r

o

and inner

radius r

i

it is given by

R

coating

=

1

σ

coating

π(r

2

o

− r

2

i

)

(4)

The total value of the wire’s resistance is then found by assuming that the core of the wire is

electrically parallel to the coating. For a steel wire the resistance of the core and hence of the

whole wire is frequency dependent. It is calculated according to section 2.1.1 For coated brass or

copper wires there is no dependency at the frequency under investigation (ν = 100kHz). Table

2.1.2 summarizes the obtained values for some commercial and some experimental diameter 250µm

steel wires.

The table shows ﬁrst of all that the D.C-resistance of the coated wires is smaller because of the

good conductive coating that is electrically in parallel, but secondly the rising of the resistance

is less compared to plain wires (table 2.1.1). For thick coatings the D.C. as well as the high

frequency resistances are comparable to those of plain brass wires, while the tensile strength of

their cores stays considerably higher. For wires of smaller diameter, e.g. 100 µm, this will be valid

to a lesser extent, because the increase in resistance due to the skin-eﬀect is smaller in absolute

numbers (table 2.1.1). Relatively speaking it is even smaller compared to the D.C. resistance.

Because of the smaller diameter this is already one order of magnitude higher compared to 250

µm wires (equation 3). Moreover the coating’s cross sectional area is relatively smaller too. In

this sense the resistance electrically parallel to the steel core is smaller and less able to lower the

total resistance of the wire. When calculated for a steel wire (0.7%C) with a 4 µm Zn-coating

the D.C. resistance is 17.32 Ω/m, compared to 22.73 Ω/m without coating. It increases to 24.27

Ω/m at 100kHz. This is a factor 1.4 compared to 2.62 without coating. For even thinner wires

the inﬂuence of the skin-eﬀect and the conductive properties of the coating drops further.

∗

E

x/y

= Steel 0.7%C core with x µm Cu and y µm Zn.

4

2.2 The frequency dependence of a wire’s inductance

2.2.1 Plain wires

The frequency dependence of a wire’s internal inductance is approximately given by equation 5,

with ξ given by equation 2.

2πνL

R

0

≈ ξ

2

−

ξ

6

6

if ξ << 1

2πνL

R

0

≈ ξ −

3

64ξ

+

3

128ξ

3

if ξ >> 1 (5)

The wire’s inductance L

0

at low frequencies equals

µ

8π

[7]. The calculated values are given in

table 2.1.1. Again there is no noticeable eﬀect of frequency on the inductance of a copper or brass

wire, there is however an eﬀect on the inductance of a steel wire. With increasing frequency the

inductance decreases by 40% for a 100µm steel wire and 75% for a 250µm steel wire. This eﬀect is

hence advantageous for the use of the high current needle-impulse generator (discussed in section

1 and [10]), but still the inductance of steel wire remains three orders of magnitude higher than

e.g. the inductance of a brass wire.

It should be noted that in this section only the internal inductance of the wire was calculated.

The magnetic energy stored in the ﬁeld outside the wire was disregarded. The total inductance is

very diﬃcult to calculate, since the geometry of the work piece and the work piece material must

be taken into account. The total inductance of the circuit excluding the wire is estimated to be

in the order of magnitude of 400nH. However, since this paper only aims at comparing diﬀerent

wires, the testing conditions are chosen in such a manner that only the internal inductance is

variable. This is achieved by ﬁxing the work piece geometry and material (in this case B¨ohler

K107 steel (DIN X210CrW12) of 30 mm height).

2.2.2 Coated wires

The internal inductance of a coated a-magnetic wire can be calculated by calculating the energy

stored in the magnetic ﬁeld inside the wire. It is possible to do this for coated a-magnetic wires

but it is rather diﬃcult to calculate inductance values for coated steel wires at high frequencies.

A simple decomposition into parallel inductances, analogous to the calculation of the resistance,

is not valid. Mutual induction is here to be taken into account. An attempt is made in [7]

using energetic considerations. Appropriate conclusions can however also be made from physical

insights.

The dropping of the inductance due to the skin-eﬀect in steel wires, as explained in section

2.2.1, will be counteracted by using a thin conductive coating which is non-permeable (copper,

brass, zinc, . . . ) in the same way the rising of the resistance in counteracted. After all the steel

core is thinner, if a coating is used, and hence less subjected to the skin-eﬀect.

When the coating becomes thick on the other hand the inductance will again be lowered. Most

of the current will ﬂow in this non-permeable high conductive coating, which is electrically parallel

to the steel core. It will hence see a low inductance. The thicker the coating becomes, the closer

the inductance will come to that of a plain wire made out of the coating material. As shown

in table 2.1.1, the inductance of plain wires of these materials is much lower. The use of thick

coatings on steel wires is hence good for lowering the inductive load on the generator.

3 Inﬂuence of wire impedance on the process’s performance

The wire impedance is immediately reﬂected in the working current and the attained surface

roughness of the work piece. A less correlation is found to the material removal rate.

5