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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. It even can be implemented as a Microsoft Word document equipped by the set of VBA macros for automatic creation of QuickField problem, solving, postprocessing and report generation. Rather complicated example - analysis of tetra-core cable - is available as ActiveField Cable Example. If you have working knowledge of Visual Basic, and understanding of QuickField Object model - you are welcome to analyze source code of these macros.
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 MS Word macros. Corresponding QuickField problems can be analyzed by the Student Edition.

  1. Model description
  2. Input parameters
  3. Calculated cable parameters
  4. Field pictures

1. Model description.

3-phase tetra-core cable

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.

Table 4. Conductors' loading.

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

1.90e+03

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 magnetostatic 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 magnetostatic 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 magnetostatic 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

5.14e-02

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

8.78e+03

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
current density in the cable

Magnetic field energy distribution in the cable
magnetic field energy distribution in the cable

Magnetic field strength distribution in the cable
magnetic field strength distribution in the cable

Temperature distribution in the cable
temperature distribution in the cable

Cable mechanical deformation
cable mechanical deformation

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