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# Power cable parameters calculation

QuickField, enforced by ActiveField technology may be effectively used for multi-physics analysis of various engineering tasks. This analysis could be highly automated.

This document displays the results of cable analysis based on specific modeling parameters. Pictures, tables and graphs below have been automatically calculated by QuickField Professional Edition, controlled by VBA code implemented as Microsoft Word macros.

1. Model description.

This high-voltage tetra-core cable has three triangle sectors with phase conductors and round neutral conductor in the lesser area of the cross-section above. All the conductors are made of aluminum. Each conductor is insulated and the cable as a whole has a three-layered insulation. The cable insulation consists of inner and outer insulators and a protective braiding (steel tape). The sharp corners of the phase conductors are chamfered to reduce the field crown. The corners of the conductors are rounded. Empty space between conductors is filled with some insulator, possibly with an air.

It is often required to design a cable according to parameters of the conductor section areas. Conductor section areas are defined in the Table 1. The tables 2 to 7 describe other input parameters.

2. Input parameters.

Table 1. Conductors' geometric parameters.
 Phase conductor area 120 mm² Neutral conductor area 35 mm² Thread rounding radius (R) 2 mm

Table 2. Insulator geometric parameters.
 Cable-core insulation thickness 2 mm Inner cable insulation thickness 1 mm Protective steel braiding thickness 1 mm Outer cable isolation thickness 3 mm

Table 3. The precision.
 Areas calculation reasonable error 0.001 mm².

 Current amplitude 200 A Voltage amplitude (electrostatics) 6500 V Frequency 50 Hz Current phase (for static problems) 0 deg

Table 5. Conductors' physical properties.
 Relative permeability 1 Conductivity 36000000 S/m Thermal conductivity 140 W/K-m Young's modulo 6.9e+10 N/m² Poisson's ratio 0.33 Coefficient of thermal expansion 2.33·10-5 1/K Specific density 2700 kg/m³

Table 6. Steel braiding physical properties.
 Relative permeability 1000 Conductivity 6000000 S/m. Thermal conductivity 85 W/K-m Young's modulo 2·1011 N/m² Poisson's ratio 0.3 Coefficient of thermal expansion 0.000012 1/K Specific density 7870 kg/m³

Table 7. Insulator physical properties.
 Core Inner Outer Relative permeability 1 1 1 Conductivity 0 0 0 S/m Relative electric permittivity 2.5 2.5 2.5 Thermal conductivity 0.04 0.04 0.04 W/K-m Young's modulo 10000000 10000000 10000000 N/m² Poisson's ratio 0.3 0.3 0.3 Coefficient of thermal expansion 0.0001 0.0001 0.0001 1/K Specific density 900 900 1050 kg/m³

3. Calculated cable parameters.

Cable physical parameters are presented in the next table.

Cable outer diameter is calculated using conductor and insulator geometrical parameters put into Table 1 and Table 2. Cable linear weight per meter is calculated from geometrical parameters and specific densities of the cable components. The whole cable specific density is a total density calculated by taking into account all cable components.

Table 8. Cable physical parameters
 Cable outer diameter 42.8 mm Weight (per meter) 2.74 kg Cable specific density 1900 kg/m²

"Conductors' capacitance" table holds self- and mutual-capacitances of the cable conductors. These values are calculated in the QuickField electrostatics problem.

 q1 = c11 * (U1 - 0) + c12 * (U1 - U2) + ... + c1n * (U1 - Un) q2 = c21 * (U2 - U1) + c22 * (U2 - 0) + ... + c2n * (U2 - Un) ...... qn = cn1 * (Un - U1) + cn2 * (Un - U2) + ... + cnn * (Un - 0)

Table 9. Conductors' capacitance, pF/m (lumped capacitance)
 Conductor1 Conductor2 Conductor3 Null-cord Conductor1 170 66.1 9.47 36.8 Conductor2 66.3 169 66.3 0.413 Conductor3 9.45 66.1 170 36.8 Neutral cord 37.0 0.534 37.0 64.5

Conductors' inductances are represented in the Table 10. Values in the columns 2-5 are calculated in the magnetostatics problem at the phase defined in the Table 4. Values in the columns 6-9 are calculated in AC magnetic problem. All values are calculated using the flux linkage approach by the formula: Lij = Fj / Ii. The table diagonal elements represent the self-inductance values.

Table 10. Conductors' inductance, uH/m
 In magnetostatic problem In AC magnetic problem C-1 C-2 C-3 0-cord C-1 C-2 C-3 0-cord Conductor1 11.5 11.2 11.1 11.3 8.73 8.47 8.41 8.51 Conductor2 11.2 11.5 11.2 11.1 8.47 8.73 8.47 8.38 Conductor3 11.1 11.2 11.5 11.3 8.41 8.47 8.73 8.51 Neutral cord 11.3 11.1 11.3 117 8.51 8.38 8.51 8.87

Table 11 includes the impedance and impedance-like values. In the magnetostatics problem the conductor's impedance (equal to the resistance) per meter is calculated by the formula: R = l / (ρ·S)
Joule heat per meter in magnetostatics problem is calculated by the formula: P = IA² · R, where IA is the root-mean-square current and R is the conductor impedance.
The conductors' impedances in AC magnetics problem are calculated using the Ohm's law as a complex ratio of the conductor's average potential divided by the conductor total current density. The real part of this ratio represents the resistance, imaginary part - reactance and the modulus - impedance. The Joule heat in the AC magnetic problem is calculated using the corresponding QuickField integral.

Table 11. Conductors' impedance.
 In electrostatics problem In AC magnetic problem Conductors Null cord Conductor1 Conductor2 Conductor3 Impedance, ω/m 2.31e-04 7.94e-04 2.40e-04 2.55e-04 2.80e-04 Resistance, ω/m 2.31e-04 7.94e-04 2.15e-04 2.37e-04 2.59e-04 Reactance, ω/m 0.00 0.00 1.08e-04 9.41e-05 1.06e-04 Joule heat, W/m 4.63 0.00 4.71 4.74 4.71

The generated heat field is exported from the AC magnetics problem into the heat transfer problem. As a result of QuickField simulation you can see the cable exterior surface average temperature, heat flow from the cable surface and the average temperatures of all conductors. Average temperatures are relative numbers presented in Celsius assumed that ambient space temperature is 20 °C.

Table 12. Cable heat parameters
 Exterior surface average temperature 23.5 °C Heat flow 14.2 W Conductors average temperature, °C Conductor1 Conductor2 Conductor3 Null-cord 45.9 46.8 45.9 39.3

Stress analysis problem is the utmost one that imports the temperature field from the heat transfer problem and the magnetic forces from the AC magnetic problem. Due to this magnetic and thermal loading the cable components become deformed. The numerical values of these deformations are presented in the next table.

Table 13. Stress analysis problem results.
 Maximal displacement 0.0514 Mm Maximal Mohr criteria value 8.16e+07 N/m²

The strength value is important for the cable fault analysis.

Table 14. The strength.
 Maximal peak strength value 8780 A/m

The "Strength" field is shown on a figure below as well as the "Total current density", "Energy density", "Momentary flux density", "Temperature" and "Displacement" field pictures.

4. Field pictures.

Current density in the cable Magnetic field energy distribution in the cable Magnetic field strength distribution in the cable Temperature distribution in the cable Cable mechanical deformation  