Method for Estimating Molten Volume of Current Collecting Materials at Contact Loss Point using ϕ-θ Theory

Abstract

In order to control electric wear of current collecting materials such as contact wires and contact strips in electric railways, it is necessary to understand the relationship between current and the molten volume at a contact loss point. In this paper, we propose a method for estimating the molten volume of the contact wire whose film resistance is taken into account, on the basis of the ϕ-θ theory. To verify the proposed method, we carried out wear tests under varying current conditions to measure the molten depth, molten radius, and molten volume. The wear test results showed that the experimental values of the molten volume of the contact wire are spread in the range where normalized potential at the contact boundary at which α was estimated to be 0.90 to 0.94.

1. Introduction

In electric railways, electric power is supplied to a vehicle through an overhead contact line and a pantograph. Current collecting materials such as the contact wire and the contact strip of pantographs are subject to wear due to the contact forces, sliding speeds and currents. Previous field survey [1] has reported that the wear due to current becomes locally significant at a contact loss point between the contact wire and the contact strip. Since the local wear of the contact wire at the contact loss point is a major issue for maintenance and replacement of the contact wire, it is required to determine an allowable value of current from the viewpoint of wear to suppress the local wear.

Iwase [2] discussed the allowable current of the pantograph from the viewpoint of the temperature rise of the contact wire. However, he pointed out that the local wear of the contact wire is not considered for determination of allowable current of the pantograph. Kono et al [3, 4] and Oda [5] carried out wear tests under current flowing condition and reported that the wear increased by an arc discharge at the time of contact loss. In these wear tests, the wear volume at the time of contact loss was not clarified because there was a mixture of wear in contact and in contact loss. Hayasaka et al [6] reported that a molten bridge is formed between the contact wire and the contact strip before the arc discharge occurs, and that the materials are worn by the scattering of the bridge at the time of the arc discharge. Sasamoto [7] and Kubono et al [8] theoretically discussed the formation of the molten bridge in an ideal condition where there is no film resistance. We think that it is necessary to consider the film resistance in the case of outdoor field conditions such as sliding contact between the contact wire and the contact strip.

Previously, we classified wear modes of current collecting materials due to Joule heat with wear tests under current flowing condition and discussed that the main factor of significant wear of the contact wire should be the molten bridge [9]. In addition, we analyzed a temperature distribution in the vicinity of the contact spot and clarified that the relationship curve between the electric potential and the temperature becomes parabolic [10]. We also formulated the melting condition of the contact wire and the contact strip taking into account the film resistance and proposed a wear mode map due to Joule heat [10].

In this paper, we proposed a method for estimating the molten volume at the contact loss point of the contact wire for which the wear volume can be measured in wear tests under current flowing condition. To verify the method, we carried out wear tests under current flowing condition in a material combination of a hard-drawn copper contact wire and an iron-based sintered alloy contact strip, and compared the estimated value with the experimental value.

2. Method for estimating molten volume

Figure 1 [10] shows an analysis model of the temperature distribution in two contact members under the condition of current flow I [A]. The relationship curve between the electric potential ϕ [V] and the temperature θ [K] along the z axis of the model under the condition of contact voltage V_c [V] becomes parabolic as shown in Fig. 2. We call this curve the electric potential-temperature parabolic curve, and propose a method for estimating the molten volume from this curve. Table 1 shows symbols used in this paper.

Fig. 1　Analysis model for contact temperature between contact wire and contact strip [10]

Fig. 2　Electric potential-temperature parabolic curve [10]

Table 1　Parameter list used for molten volume estimation [13]

I	Current	[A]	Rm1	Electric resistance of molten range of contact wire	[Ω]
Vc	Contact voltage	[V]	Rm2	Electric resistance of molten range of contact strip	[Ω]
ϕ	Electric potential	[V]	R	Contact resistance	[Ω]
ϕc	Electric potential at contact boundary	[V]	L	Lorentz number	[V2/K2]
α	Normalized potential at contact boundary		λ1	Heat conductivity of contact wire	[W/(m･K)]
β1	Normalized potential at meting point of contact wire		λ2	Heat conductivity of contact strip	[W/(m･K)]
β2	Normalized potential at meting point of contact strip		a	Radius of contact point	[m]
θ	Temperature	[K]	d1	Thickness of film resistance of contact wire	[m]
Tb	Boiling point	[K]	d2	Thickness of film resistance of contact strip	[m]
Tm1	Melting point of contact wire	[K]	h1	Molten depth of contact wire	[m]
Tm2	Melting point of contact strip	[K]	h2	Molten depth of contact strip	[m]
ρ1	Electric resistivity of contact wire	[Ω·m]	r1	Molten radius of contact wire	[m]
ρ2	Electric resistivity of contact strip	[Ω·m]	r2	Molten radius of contact strip	[m]
ρd1	Electric resistivity of film resistance of contact wire	[Ω·m]	V1	Molten volume of contact wire	[m3]
ρd2	Electric resistivity of film resistance of contact strip	[Ω·m]	V2	Molten volume of contact strip	[m3]

The electric potential-temperature parabolic curve shown in Fig. 2 is calculated with (1).

  

$$\theta ={\left[\frac{V_{\mathrm{c}}^2}{L}\left\{\left(\frac{\varphi }{V_{\mathrm{c}}}\right)-{\left(\frac{\varphi }{V_{\mathrm{c}}}\right)}^2\right\}+{300}^2\right]}^{1/2}$$（1）

where, L is the Lorenz number (=2.4×10⁻⁸ [V²/K²]). The normalized potential at the contact boundary between the contact wire and the contact strip in Fig. 2 is calculated with (2) [10].

  

$$\alpha =\frac{{\varphi }_{\mathrm{c}}}{V_{\mathrm{c}}}=\frac{\frac{{\rho }_2}{4a}+\frac{{\rho }_{\mathrm{d}2}{d}_2}{\pi a^2}}{\frac{{\rho }_1+{\rho }_2}{4a}+\frac{{\rho }_{\mathrm{d}1}{d}_1+{\rho }_{\mathrm{d}2}{d}_2}{\pi a^2}}$$（2）

where, ϕ_c [V] is the electric potential at the contact boundary, ρ₁, ρ₂ [Ω·m] are electric resistivities, ρ_d1, ρ_d2 [Ω·m] are electric resistivities of the film resistances, d₁, d₂ [m] are film thicknesses. Here, the subscripts 1 and 2 of each constant indicate the contact wire and the contact strip respectively. The radius of the contact spot a [m] is calculated with (3) defined in reference [11].

  

$$a=\frac{\sqrt{L}I\left({\lambda }_1+{\lambda }_2\right)}{8{\lambda }_1{\lambda }_2}$$（3）

where, I [A] is the current, λ₁, λ₂ [W/(m･K)] are the heat conductivities.

The molten bridge of the contact members boils and scatters at the time of arc discharge. At that time, the maximum temperature of the electric potential-temperature parabolic curve should be consistent with the boiling point of the member. Therefore, the contact voltage at the time arc discharge is calculated with (4) on the basis of the ϕ-θ theory [12].

  

$$V_{\mathrm{c}}={\left[4L\left(T_{\mathrm{b}}^2-{300}^2\right)\right]}^{1/2}$$（4）

where, T_b [K] is the boiling point of the member in which the maximum temperature occurs. If the temperature of the electric potential-temperature parabolic curve exceeds the melting points of the contact wire and the contact strip, the molten range of a member is consistent with the range between α and β₁, β₂ which are the normalized potentials corresponding to the melting points of each member. Normalized potentials β₁, β₂ are calculated with (5) and (6) by substituting T_m1, T_m2 [K] which are the melting points of the contact wire and the contact strip into (1).

  

$${\beta }_1=\frac{1}{2}+\frac{1}{2V_{\mathrm{c}}}\sqrt{V_{\mathrm{c}}^2-4\left(T_{\mathrm{m}1}^2-300^2\right)L}$$（5）

  

$${\beta }_2=\frac{1}{2}-\frac{1}{2V_{\mathrm{c}}}\sqrt{V_{\mathrm{c}}^2-4\left(T_{\mathrm{m}2}^2-300^2\right)L}$$（6）

Next, the molten depths are calculated as the distance between the contact boundary and the molten range as follows. It is assumed that the molten range is along the equipotential surface as shown in Fig. 3. Each electric resistance R_m1, R_m2 [Ω] between α and β₁, β₂ is calculated with (7) and (8) on the basis of Holm [12].

  

$$R_{\mathrm{m}1}=\frac{{\rho }_1}{2\mathit{\pi a}}\arctan \frac{h_1}{a}$$（7）

  

$$R_{\mathrm{m}2}=\frac{{\rho }_2}{2\mathit{\pi a}}\arctan \frac{h_2}{a}$$（8）

where, h₁, h₂ [m] are the molten depths from the contact boundary. Since the current I does not change in the contact members, β₁, β₂ are calculated by dividing (7) and (8) by the whole contact resistance R [Ω] and α. Then, h₁, h₂ are calculated with (9) and (10).

  

$$\begin{array}{c}{\beta }_1=\alpha +\frac{R_{\mathrm{m}1}}{R}=\alpha +\frac{2}{\pi }\left(1-\alpha \right)\arctan \frac{h_1}{a}\\ h_1=a\tan \frac{\pi }{2}\left(\frac{{\beta }_1-\alpha }{1-\alpha }\right)\hfill \end{array}$$（9）

  

$$\begin{array}{c}{\beta }_2=\alpha -\frac{R_{\mathrm{m}2}}{R}=\alpha -\frac{2\alpha }{\pi }\arctan \frac{h_2}{a}\\ h_2=a\tan \frac{\pi }{2}\left(\frac{\alpha -{\beta }_2}{\alpha }\right)\hfill \end{array}$$（10）

Fig. 3　Molten range along equipotential surface [14]

Since the molten radius is different from the radius of the contact spot and the shape of an isothermal surface is spheroid as shown in Fig. 3, the molten radii of each member r₁, r₂ [m] are calculated with (11) and (12).

  

$$r_1=\sqrt{a^2+h_1^2}$$（11）

  

$$r_2=\sqrt{a^2+h_2^2}$$（12）

The molten volumes of each member V₁, V₂ [m³] when current flows through a contact spot are calculated with (13) and (14) by regarding the molten range as a semi-elliptical sphere.

  

$$V_1=\frac{2\pi }{3}{r}_1^2{h}_1$$（13）

  

$$V_2=\frac{2\pi }{3}{r}_2^2{h}_2$$（14）

The molten radius r [m] and the molten depth h [m] are functions of the contact radius a [m] as shown in (9) - (12), and the contact radius is a function of the current I [A] as shown in (3). Therefore, the molten volumes V [m³] calculated in (13) and (14) are functions proportional to the cube of the current I [A].

3. Verification of the proposed method by testing Method for estimating molten volume

3.1 Wear test apparatus and test conditions

The proposed method calculates the molten volume at the time of contact loss, i.e. when the contact members of the contact wire and the contact strip open. The method was then verified using the high-speed wear tester for current collecting materials shown in Fig. 4. In this apparatus, a real contact wire is installed on a disk and a contact strip specimen is pushed against the disk for current collection. The diameter of the disk is 2 m, and the circumference of the disk is 6.3 m. The method for estimating the molten volume uses the ϕ-θ theory used in steady state, so it is necessary that the sliding speed should be set as low as possible during the verification test. Since the fluctuation of contact force in this apparatus is larger than that in the linear wear tester which is reported in [9], arc discharge frequently occurs even at low sliding speed. In addition, since a larger current is able to set in this apparatus, it is expected to obtain effective results for determination the allowable current. From the above, we think that this apparatus is appropriate for the verification test.

Fig. 4　Schematic image of wear test apparatus [13]

Table 2 shows the material properties of the hard-drawn copper contact wire and the iron-based sintered alloy contact strip. In Table 2, the boiling points of each material are regarded as those of the base materials such as copper and iron [11]. Table 3 shows the test conditions. The verification test was carried out after running-in at a sliding speed condition of 300 km/h without current to suppress partial contact of the contact strip. The sliding surface of the contact wire was ground before the verification test to remove the transfer layer on the contact wire. The contact force of the contact strip was set to 50 N which is close to the contact force of a pantograph. The sliding speed was set to 5 km/h which is the lowest speed of this apparatus. The current was set to 100 A and 200 A. Since the molten spot of the contact wire was too small to be observed at a current below 100 A, we set the current to 100 A or higher. In order to preserve the molten spot and to measure its radius and depth, the test was stopped and the contact strip was unloaded when several arc discharges were observed. The radius and the depth of the molten spot only of the contact wire were measured after the test. The reason for this is that the molten spot of the contact strip was damaged by additional friction and another arc discharge because the contact strip continued to slide in the same surface during the test.

Table 2　Test specimens [13]

	Contact wire	Contact strip
Material	Hard-drawn copper	Iron-based sintered alloy
Melting point, K	1,334	1,646
Boiling point, K11)	2,853	3,027
Electric resistivity, Ωm	1.77×10−8	0.40×10−6
Heat conductivity, W/mK	373	25.3

Table 3　Test conditions [13]

Contact force, N	Approximately 50
Sliding speed, km/h	5
Current, A	100, 200
Sliding time, s	～60

After the test, the section profile around the molten spot of the contact wire was measured several times with a roughness meter (Mitsutoyo, SJ-310, load: 0.75 mN, distance: 5 mm). Figure 5 shows an example of the measured profile around the molten spot. The sliding surface was regarded as the reference surface, and the maximum depth of the molten spot from the reference surface was measured. Since the shape of the molten range was assumed to be spheroid as described above, the maximum value among several measurements was adopted as the molten depth. On the other hand, the molten spot was observed by a microscope (Keyence, VHX-100, lens: 175x). Figure 6 shows an example of the microscopic image of the molten spot. Although there were some cases where the molten spots were not perfect circular because of scattering of the molten copper and additional friction, the radius of the molten spot was measured by fitting a circle to three points on the circumference of the spot. The molten volume of the spot was calculated with (13) by substituting the measured depth and radius.

Fig. 5　Example of cross-sectional profile of a molten spot [13]

Fig. 6　Example of microscopic image of a molten spot [13]

3.2 Wear test results

Figure 7 shows a comparison of the experimental and the estimated values of the molten depth under varying current. The estimated values were calculated up to a current of 300 A because the current condition of iron-based sintered alloy contact strips used in Shinkansen trains is approximately 300 A. The boiling point of the contact strip is used for T_b in (4). The reason is that in the case of the combination of the hard-drawn copper contact wire and the iron-based sintered alloy contact strip, the maximum temperature in the electric potential-temperature parabolic curve would appear in the contact strip side because the normalized potential at the contact boundary α is thought to be 0.5 or more.

Fig. 7　Experimental and estimated values of molten depth of contact wire [13]

From Fig. 7, it is found that the estimated values increase as the value of α decreases, and the experimental values are in the range where α is estimated to be 0.90 to 0.94. The value of α in no film resistance condition is calculated to be 0.96 from (2) and Table 2. The decrease of α from 0.96 indicates an increase in the film resistance of the contact wire because it is unlikely that the resistivity of the contact strip decreases. In addition, the estimated value of the molten depth became negative at α of 0 .96. The reason for this is that the maximum temperature of the contact wire in the electric potential-temperature parabolic curve at the time of contact strip boiling is lower than the melting point of the wire T_m1. In this case, it means that the contact wire does not melt even when arc discharge occurs in no film resistance condition.

Figure 8 shows a comparison of the experimental and the estimated values of the molten radius under varying current. It is found that the estimated values increase as the value of α decreases, but the influence of α on the estimated radius is not as great as that on the estimated depth. Although some of the experimental values are out of the estimated range, we think that the trend of the increasing melt radius and the order of the values are generally consistent with the estimation. It should be noted that the experimental depths are in the range where α is estimated to be 0.90 to 0.94 as shown in Fig. 7, but some experimental radii are in the range where α is estimated to be 0.84 or less as shown in Fig. 8. Hayasaka [6] reported that the scattered volume was 1/5 -1/100 of the molten volume at the time of arc discharge. From (11) - (14), since the molten radius is also a function of the molten depth, the molten volume is proportional to the cube of the molten depth. In this case, it is thought that the experimental depth is 1/2 -1/5 of the true molten depth. In Fig. 7, if the true molten depth is twice the experimental value, some of them should be in the range where α is estimated to be 0.84 or less. Then the tendency of the molten depth would be similar to that shown in Fig. 8.

Fig. 8　Experimental and estimated values of molten radius of contact wire [13]

Figure 9 shows the experimental and the estimated values of the molten volume under varying current. As mentioned in Chapter 2, the relationship between the current and the estimated molten volume is not liner, but the molten volume is proportional to the cube of the current. And it is also found that the estimated values increase as the value of α decreases, and the experimental values are in the range where α is estimated to be 0.84 to 0.94. As mentioned above, it is quite possible that the true molten volume is larger than the experimental value because not all of the molten portions were scattered. Therefore, when considering the wear of the contact wire, it is necessary to take into account the softening of the remaining molten portions.

Fig. 9　Experimental and estimated values of molten volume of contact wire [13]

These tests have not verified the validity of α because the film resistance of the contact wire was not measured. Since the tests were carried out immediately after the surface of the contact wire was ground, it is unlikely that there was a large film resistance on the contact wire. Therefore, it is reasonable that the experimental values are in the range where α is close to 0.96.

We think that it is useful to formulate the molten volume of the contact wire and confirm the consistency between the estimated value and the experimental value. On the other hand, this proposed method of estimating the molten volume is applicable only under the low sliding speed condition close to the steady state where the ϕ-θ theory holds. Therefore, in order to use this method in the railway field, it is necessary to improve the method so that the molten volume can properly be estimated under the unsteady state such as the high sliding speed condition. In the future, the estimation of the molten volume of the current collecting materials due to Joule heat without using the sliding speed, will contribute to the innovation of maintenance, such as the prediction of wear in usage conditions and the determination of the allowable contact loss ratio in usage current. In addition, it contributes to the determination of the allowable current of pantographs.

4. Conclusions

In order to estimate the molten volume of the contact wire at the contact loss point, we focused on the relationship between the electric potential and the temperature in the vicinity of the contact spot to propose a method for estimating the molten volume taking into account the film resistance. In addition, we carried out verification tests and measured the molten volume at the contact loss point. The results obtained in this study are as follows.

(1) Using the electric potential-temperature parabolic curve in the vicinity of the contact spot, we determine the area enclosed by the electric potential corresponding to the melting point and that corresponding to the contact boundary. Then the estimation formulas of the molten volume are proposed by estimating the molten depth and radius at the time of contact boiling.

(2) The estimated results show that the molten depth and the molten radius are proportional to the current, and the molten volume is proportional to the cube of current. In addition, the results also show that the molten volume of the contact wire increases as the normalized potential α, which is the contact boundary in the electric potential-temperature parabolic curve, decreases.

(3) The experimental results show that the measured values of the molten volume of the contact wire were spread the range where α is estimated to be 0.90 to 0.94. In the experiment, the true molten volume was thought to be larger than the measured value because not all of the molten volume was scattered, but the magnitude and tendency of the measured values are generally consistent with the estimated value.

(This paper is the updated version of the reference [13] [14].)

References

• [1]  Iwase, M., “Current Collecting by the Pantograph and its Wear (I),” Railway Technical Research Report, No. 53, 1959 (in Japanese).
• [2]  Iwase, M., “Capacity of Overhead Wire and Current-Collecting Capacity of Pantograph,” Railway Technical Research Document, Vol. 14, No. 1, pp. 283-287, 1957 (in Japanese).
• [3]  Kohno, A. Ohyabu, H. and Soda, N., “Effect of Discontact Arc on Wear of Materials for Current Collection (Part 1),” Journal of Japan Society of Lubrication Engineers, Vol. 27, No. 4, pp. 283-287, 1982 (in Japanese).
• [4]  Kohno, A., Ohyabu, H. and Soda, N., “Effect of Discontact Arc on Wear of Materials for Current Collection (Part 2),” Journal of Japan Society of Lubrication Engineers, Vol. 27, No. 7, pp. 527-532, 1982 (in Japanese).
• [5]  Oda, O., “Study on Wear of Contact Wire and Contact Strip on Shinkansen,” Railway Technical Research Report, No. 1323, 1986 (in Japanese).
• [6]  Hayasaka, T., Akagi, Y., “Influence of Arc Discharge on the Surface of a Contact Wire,” IEEJ Transactions on Industry Applications, Vol. 135, No. 4, pp. 327-334, 2015.
• [7]  Sasamoto, T., “The Theoretical Analysis of Bridge Transfer of the Opening Contacts,” IEICE Transactions on Electronics, J58-C, 3, pp. 101-108, 1975 (in Japanese).
• [8]  Kubono, T. and Suzuki, J., “A Theoretical Investigation of the Dimension of Molten Metal Bridge and Its Transfer,” IEICE Transactions on Electronics, J63-C, 10, pp. 667-674, 1980 (in Japanese).
• [9]  Yamashita, C., Adachi, K., “Influence of Current on Wear Modes and Transition Condition of Current Collecting Materials,” Journal of Japanese Society of Tribologist, Vol. 58, No. 7, pp. 496-503, 2013 (in Japanese).
• [10]  Yamashita, C., Adachi, K., “Wear Mode Map of Current Collecting Materials,” Journal of Japanese Society of Tribologist, Vol. 62, No. 2, pp. 129-136, 2017 (in Japanese).
• [11]  Takagi, T., Arc Discharge Phenomena of Electric Contacts, Corona, pp. 56-67, 1995 (in Japanese).
• [12]  Holm, R., Electric Contacts: Theory and Applications, 4th ed., Springer, pp. 64-68, 2000.
• [13]  Yamashita, C., Nemoto, K., Ohara, T., “Estimation Method of Melting Volume of Current Collecting Materials under Current Flowing Condition in Electric Railway,” Journal of Japanese Society of Tribologist, Vol. 67, No. 5, pp. 366-372, 2022. (in Japanese).
• [14]  Yamashita, C., Nemoto, K., Ohara, T., “Estimation Method of Melting Volume of Current Collecting Materials at Contact loss Point by using Theory,” RTRI Report, Vol. 38, No. 7, pp. 23-28, 2024. (in Japanese).

Authors

	Chikara YAMASHITA, Ph.D. Senior Chief Researcher, Head of Current Collection Maintenance Laboratory, Power Supply Technology Division Research Areas: Overhead Contact Line, Tribology, Electric Contact, Fatigue
	Koki NEMOTO Researcher, Current Collection Maintenance Laboratory, Power Supply Technology Division Research Areas: Overhead Contact Line, Tribology
	Takuya OHARA Assistant Senior Researcher, Collection Maintenance Laboratory, Power Supply Technology Division Research Areas: Overhead Contact Line, Fatigue