In this section, the steady-state temperature rise experiments for the contact terminal and the equivalent resistance measurement experiments for the armor rod segment were designed. Subsequently, the three-dimensional thermal simulation model associated with the 3D-EM model of the contact terminal was established. Next, the conductive bridge radius r of the 3D-EM model was determined by the sum of squared error between the thermal simulation results and the measuring results of the steady-state temperature rise experiments. The Rz1 output by the 3D-EM model was computed based on the calculated conductive bridge radius r. Finally, the accuracy of the 3D-EM model was verified by comparing the measurement results of the equivalent resistance of the armor rod segment with Rz1.
4.1. The Steady-State Temperature Rise Experiments of the Contact Terminal
To investigate the characteristics of the steady-state temperature distribution in the contact terminal, a series of steady-state temperature rise experiments were performed on a contact terminal consisting of the 50 mm
2 ground wire with a length of 7 m and corresponding armor rod with a length of 1.3 m, as shown in
Figure 11. The experimental setup is shown in
Figure 11a, where the experimental system consists of three parts: the current generation loop, the feedback control loop, and the experimental load [
24,
25,
26,
27]. The current generation loop was comprised of a 380 V power supply, a voltage regulator, a current generator, and a compensation capacitor. The feedback loop was comprised of a current transformer, a PC, and a programmable logic controller (PLC). The PLC adjusted the load current according to the differences between the current values measured by the current transformer and the values preset by the PC. The experimental load was located indoors to avoid the effects of wind and sunshine on the measurement results of the temperature distribution in the contact terminal.
For the experimental contact terminal, eleven temperature measuring points were evenly set along the axial direction on the surface, and the axial distance between two adjacent points was 20 mm. Points 1–5 were located on the surface of the armor rod segment, and points 6–11 were located on the surface of the ground wire, as shown in
Figure 11b. In the experiments, the calibrated thermocouples (with an average measuring error less than 0.1 °C) were used to obtain the temperature data of eleven temperature measuring points. To improve the accuracy of the temperature measurements, two thermocouples were installed at each temperature measuring point. The average of the readings of each of the two thermocouples was adopted as the measured temperature. All thermocouples were connected to the data acquisition system for processing and recording the acquired data.
Under the initial condition without loading, step currents of 55 A, 68 A, and 76 A were applied to the experimental contact terminal, respectively, until the steady state was reached, and the steady state axial temperature distribution of the contact terminal under different currents are shown in
Figure 12. Ambient temperatures corresponding to the steady state axial temperature distribution curves under the step currents of 55 A, 68 A, and 76 A were 30.4 °C, 25.9 °C, and 26.1 °C, respectively. In
Figure 12, the highest surface temperature occurred on the bare ground wire segment at the contact surface. With the increase in the axial distance from the contact surface, both the surface temperature of the ground wire and the armor rod exhibited a decreasing trend. The decreasing rate declined as the axial distance from the contact surface increased. Under the condition of a load current less than 76 A, for the ground wire part of which the distance from the contact surface exceeded 60 mm or the armor rod part of which the distance from the contact surface exceeded 80 mm, the axial distribution of the surface temperature had already tended to be stable.
4.2. The Equivalent Resistance Measuring Experiments for the Armor Rod Segment in the Contact Terminal
To provide data support for the verification of the 3D-EM model of the contact terminal, equivalent resistance measurement experiments for the armor rod segment of the contact terminal were designed. If the widely used DC bridge (i.e., the Wheatstone bridge) was adopted to measure the equivalent resistance of the armor rod segment, the skin effect would not occur in the contact terminal. Thus, the resistance result measured by the DC bridge did not correspond to the
Rz1 output by the 3D-EM model of the contact terminal. In this paper, to measure the equivalent resistance of the armor rod segment, the approach that combined the alternating current voltage drop method with the digital DC bridge measuring method was adopted. The system of the measurement experiment is shown in
Figure 13c.
Figure 13a,b show the actual wiring diagrams of the alternating current voltage drop method and the digital DC bridge measuring method, respectively.
In
Figure 13c, two constant-current source leads (i.e., L1 and L2) were connected to the experimental contact terminal via the parallel groove clamp. To guarantee consistency between the armor rod segment measured in the experiment and that corresponding to
Rz1 in the 3D-EM model, L1 was connected to the bare ground wire segment in a position adjacent to the contact surface, and L2 was connected to the armor rod segment in a position at an axial distance of 70 mm away from the contact surface. The constant current
I for the equivalent resistance measurement loop was provided by the adjustable constant alternating current source and was set to 75 A in the experiment. The measurement method of the equivalent resistance for the armor rod segment shown in
Figure 13c was implemented as follows:
- 1)
The output voltage
U of the adjustable constant alternating current source, i.e., total voltage of the constant-current source leads, the parallel groove clamps, and the measurement segment of the experimental contact terminal, was measured. Then, the equivalent output AC resistance of the adjustable constant alternating current source (i.e.,
R) was computed with Equation (7). In Equation (7), cos
φ is the power factor of the measuring loop.
R consisted of three parts: the AC resistance of the measurement segment of the experimental contact terminal (i.e.,
Rd1), the total AC resistance of L1 and the corresponding parallel groove clamp (i.e.,
Rf1), and the total AC resistance of L2 and the corresponding parallel groove clamp (i.e.,
Rf2), as shown in Equation (8):
- 2)
Under the condition of no load on the equivalent resistance measurement loop, the total DC resistance of L1 (L2) and the corresponding parallel groove clamp, i.e., Rf1′ (Rf2′), was measured by the digital DC bridge PC36C (accuracy of 0.01 μΩ). The materials of the constant-current source leads and the parallel groove clamps were copper and aluminum alloy, respectively, which belonged to the non-ferromagnetic materials. Therefore, the differences between the AC resistance and the DC resistance of the constant-current source leads and the parallel groove clamps were very slight. Rf1′ and Rf2′ were adopted to approximately substitute Rf1 and Rf2.
- 3)
Rd1 was obtained using Equation (8) on the basis of the measurement results of the alternating current voltage drop method and the digital DC bridge measuring method.
Based on the above procedure, when the ambient temperature was 22.3 °C, the measurement value of Rd1 was 655.9 μΩ.
4.3. Determination of the Conductive Bridge Radius r
Based on the analysis in
Section 2.2, for the 3D-EM model of the contact terminal, different contact cases between the ground wire and the armor rod can be simulated by changing the conductive bridge radius
r. Therefore, determining the value of
r in the 3D-EM model of the contact terminal is the premise of using the model for computing
Rz1. In this paper, based on the coupling of the thermal field and the electromagnetic field, a method for determining the value of
r in the 3D-EM model of the contact terminal was proposed. First, the three-dimensional thermal simulation model associated with the 3D-EM model of the contact terminal was established. Then, the value of
r was determined by comparing the computational results of the thermal simulation model and the measurement results of the steady-state temperature rise experiments.
For the three-dimensional thermal simulation model of the contact terminal, part of the geometric parameters are shown in
Table 1, and the axial lengths of the armor rod segment and the bare ground wire segment were both set to 100 mm. Then, the established three-dimensional thermal simulation model of the contact terminal is shown in
Figure 14a. Combined with the analysis of
Figure 12 in
Section 4.1, it was inferred that in the case of a load less than 76 A, the radial sections of the ground wire and the armor rod at an axial distance of 100 mm from the contact surface could be considered as the axial adiabatic plane in the three-dimensional thermal simulation model of the contact terminal.
As the surfaces of the bare ground wire segment and the armor rod segment came into contact with air, the third type of thermal boundary condition was applied to the surfaces. The controlling equation of the third type of thermal boundary condition is shown in Equation (9). In Equation (9),
TW is the surface temperature of the contact terminal;
Tf is the temperature of the surrounding air;
h is the natural heat transfer coefficient;
n is the outer normal of the heat exchange surface; and
W represents the external surface of the contact terminal:
To visualize the specific settings of the thermal boundary conditions in the thermal simulation model of the contact terminal, the two-dimensional axial section of the contact terminal (
Figure 14b) was presented for further explanation:
For Boundary I1 and I3, which belong to the horizontal boundaries, their natural heat transfer coefficients, h1 and h3, respectively, were automatically computed by COMSOL after inputting the ambient temperature and the diameters of the ground wire and the armor rod to COMSOL.
For Boundary I2, which belongs to the vertical boundaries, the natural heat transfer coefficient h2 was automatically computed by COMSOL after inputting the ambient temperature and the vertical height of Boundary I2 to COMSOL.
Boundary I4 was set as the axial adiabatic plane, i.e., the second type of thermal boundary condition.
The electromagnetic loss power of various components computed by the 3D-EM model of the contact terminal was adopted as the heat sources of corresponding components in the thermal simulation model of the contact terminal after processing, which was achieved by MATLAB. When using the above method to load heat source in the thermal simulation model of the contact terminal, it was necessary to note that the electromagnetic loss power output by the 3D-EM model was computed at 20 °C. Therefore, the relationship between the electromagnetic loss power and temperature should be taken into consideration. A conversion relationship between the electromagnetic loss power computed by the 3D-EM model of the contact terminal and the heat source loading in the thermal simulation model of the contact terminal existed, as shown in Equation (10):
In Equation (10), PT is the heat source power at the temperature T; i is the loading current; r20 is the resistance at 20 °C; α is the temperature coefficient of steel; T is the conductor temperature; and P20 is the heat source power at 20 °C, i.e., the electromagnetic loss power output by the 3D-EM model.
In summary, the correlations between the electromagnetic field simulation model and the thermal field simulation model of the contact terminal can be mainly reflected in three points, as shown in
Figure 15:
- 1)
The geometric models of the contact terminal in the electromagnetic field simulation model and the thermal field simulation model were the same.
- 2)
In the thermal field simulation model, the third type of thermal boundary condition was used to simulate the air domain wrapped around the ground wire and the armor rod in the 3D-EM model.
- 3)
There was a conversion relationship between the electromagnetic loss power of various components computed by the 3D-EM model and the heat source of corresponding components in the thermal simulation model.
Combined with
Figure 15, this indicated that the variation of the conductive bridge radius
r led to the variation of the electromagnetic loss power output by the 3D-EM model of the contact terminal, which further led to the variation of the temperature distribution computed by the thermal simulation model of the contact terminal. Thus, the conductive bridge radius
r and the temperature distribution of the contact terminal were correlated by the electromagnetic simulation model and the thermal simulation model. Based on the two simulation models of contact terminal, the surface temperature at the eleven temperature measuring points in
Figure 11b could be computed under the premise that the conductive bridge radius was
r. To evaluate the degree of fit between the surface temperature distribution curve calculated by simulation models and measured surface temperature distribution curve, the sum of square for the simulation result error at various temperature measuring points (i.e.,
Ssqu) was chosen as the index, which can be calculated by Equation (11). Obviously, the value of
Ssqu was associated with
r. The minimum value of
Ssqu could be found by adjusting the value of
r, and for the case of minimum
Ssqu occurring, the simulation curve and the measured curve fit well. Thus,
r corresponding to minimum
Ssqu was regarded as the conductive bridge radius that enabled the temperature distribution in the thermal simulation model of the contact terminal to coincide with the actual distribution:
where
Tai is the simulated surface temperature at temperature measuring point
i, and
Tbi is the measured surface temperature at temperature measuring point
i. For the simulation model in this paper, if the variation of
r was less than 0.0005 mm, the changes in temperature field of contact terminal could be negligible. Therefore, the minimum variation of
r was set to 0.0005 mm when adjusting the value of
r in this paper. Then, the conductive bridge radii under different loads were calculated, as shown in
Table 3.
It was inferred from
Table 3 that under the condition that the steady-state temperature change of contact terminal was not significant, the conductive bridge radius
r hardly varied with the load. This phenomenon can be explained by the following. The deformation of contact terminal arising from the inapparent temperature change of the contact terminal could be ignored. Thus, the load variation had little influence on the contact condition between the ground wire and armor rod, which was reflected by the conductive bridge radius
r. In the remaining part of
Section 4, the conductive bridge radius
r was set to 0.0340 mm for the experimental contact terminal.
To verify whether the conductive bridge radius
r determined by the above method could make the temperature distribution in the thermal simulation model of contact terminal coincide with the actual distribution, the simulation results and experimental results of the steady state temperature of contact terminal under different loads were compared in
Figure 16a. With reference to the experimental results, an error analysis of the thermal simulation model was conducted in
Figure 16b. Based on
Figure 16a,b, it was concluded that the axial surface temperature distribution curves of the contact terminal calculated by the thermal simulation model basically matched the actual measured axial surface temperature distribution curves. Moreover, the absolute error in the thermal simulation model of the steady state surface temperature was less than 9%. Thus, the conductive bridge radii determined by Equation (11) had sufficient accuracy to be applied in the 3D-EM model and the thermal simulation model of contact terminal.