Science Popularization Mini-Lesson: Parallel Equivalent Circuits

The figure shows the equivalent circuit of a power cable.
In the figure, L represents the equivalent inductance of the cable, with a value of 0.1-0.7 mH/km, which can generally be calculated using a formula.

Power Cable Equivalent Circuit

(a) Theoretical equivalent circuit; (b) Actual equivalent circuit
L = 0.05 + 0.2 * ln(ks/r)

Where k is a coefficient (k=1 for triangular structure, k=1.26 for parallel structure, see cable structure diagram);
S is the distance between the centers of the conductors in mm;
R is the radius of the conductor in mm.

Cable Structure Diagrams
(a) Triangular structure; (b) Parallel structure
Due to the relatively small equivalent value of L, it can be considered a short circuit in practical analysis.  The figure shows the equivalent inductance values ​​of some XLPE power cables.

Figure RX shows the DC resistance of the power cable core conductor, which is related to the conductor material, core cross-sectional area, and cable length, and conforms to the following relationship:

where ρ is the resistivity; at an ambient temperature of 20℃, ρCU = 1.75 × 10⁻⁸ Ω·m, ρAL = 2.83 × 10⁻⁸ Ω·m, and ρsteel strip = 8.79 × 10⁻⁸ Ω·m;
LC is the cable length in meters;
S is the cable conductor cross-sectional area in mm².
The table shows the maximum DC resistance values ​​of some XLPE power cable cores (at 20℃).

In the figure, R0 and C0 are two parameters related to the cable insulation material and its structure. R0 is generally small, but much larger than RX, i.e., R’0 ≈ R0; C0 can be considered as the equivalent capacitance between the cable core and the cable metal shield, with a value of 0.1-0.7 μF/km.  It can generally be calculated using a formula.

In the formula, ε is the dielectric constant of the insulating medium, such as the dielectric constant of XLPE power cables ε1=2.3, and the dielectric constant of oil-impregnated paper insulated cables ε2=3.3;
d1 is the diameter of the conductor core, including the semiconductor shielding layer;
d2 is the outer diameter of the insulation.

The table shows the single-phase equivalent capacitance of some XLPE power cables.
Series equivalent circuits are generally used when analyzing AC circuits.