Potential Gradient in the Cable

Since the cable is a form of a cylindrical condenser, the electric intensity at a distance x from the center O of the cable is calculated using the following formula:

E_x = (q) / (2π ε₀ ε_r x) [V/m]

Where:

• E_x: Electric intensity at a distance x from the center [V/m].
• q: Charge density on the cable.
• ε₀: Permittivity of free space.
• ε_r: Relative permittivity of the dielectric material.
• x: Radial distance from the center of the cable [m].

Formula for Potential Gradient:
The potential gradient g is equal to the electric intensity:
g = q / (2π ε₀ ε_r x) [V/m]

From earlier derivations:
V = (q / (2π ε₀ ε_r)) * ln(D / d)

Where:

• V: Voltage between inner and outer conductors.
• D: Outer diameter of the insulation.
• d: Diameter of the conductor.

Rearranging for q:
q = (2π ε₀ ε_r V) / ln(D / d)

Substituting q into the formula for g:
g = (V * ln(D / x)) / (x * ln(D / d)) [V/m]

Maximum and Minimum Potential Gradients:
The potential gradient will be:

• Maximum when x = d.
• Minimum when x = D.

Maximum gradient:
g_max = (2V) / (d * ln(D / d)) [V/m]

Minimum gradient:
g_min = (2V) / (D * ln(D / d)) [V/m]

Applications:

• This calculation is essential for determining the dielectric stress in cable insulation.
• Helps in identifying the critical voltage points for safe cable operation.
• Useful in the design of high-voltage cables where electric field distribution needs to be managed effectively.

Conclusion:
By calculating the maximum and minimum potential gradients, engineers can assess whether the cable insulation is sufficient to handle the voltage stress, ensuring reliable performance and preventing breakdown.