Skin and Proximity Effects in Medium Voltage Cables: Thermal Impact and Standards

Electrodynamic Fundamentals and Poynting's Theorem in Medium-Voltage Conductors

Rigorous analysis of electromagnetic behavior in medium-voltage cables requires abandoning lumped-parameter circuit approximations and delving into the strict resolution of Maxwell's equations in linear, isotropic, and homogeneous material media. When an alternating current of angular frequency \omega flows through a massive cylindrical conductor, Faraday's laws of electromagnetic induction and the Ampère-Maxwell circuit law impose a non-uniform current density distribution \vec{J}. This phenomenon, known as the skin effect, is governed by the magnetic diffusion equation derived directly from the curl of the magnetic field \vec{H} and the electric field \vec{E}:

Neglecting the displacement current in front of the conduction current in high-conductivity metals such as annealed copper or electrical-grade aluminum, and applying Ohm's microscopic constitutive relation \vec{J} = \sigma \vec{E}, we obtain the Helmholtz equation for the harmonic electric field inside the conductor:

Where \mu = \mu_0 \mu_r represents the absolute magnetic permeability of the conductor and \sigma the electrical conductivity. The fundamental parameter governing the penetration depth of electromagnetic waves into the metal is the skin depth (\delta), mathematically defined as:

For a typical medium-voltage cable operating at an industrial frequency of $f = 50 Hz$ or $60 Hz$, with a copper conductor where \sigma \approx 5.8 \times 10^7 S/m at $20 °C$ and \mu_r \approx 1, the skin depth turns out to be on the order of $9.35 mm$ at $50 Hz$. As we increase the cross-sectional area of the conductor to satisfy high current capacities in underground distribution networks, the conductor radius $r_0$ far exceeds \delta. This causes the current density to decay exponentially from the periphery toward the conductor core according to the modified Bessel function of zero order:

This spatial confinement of alternating current energy generates a direct increase in the effective resistance of the conductor relative to the direct current ohmic resistance $Rdc$. The resistance increase factor due to the skin effect is usually denoted as $y_s$, such that the alternating current resistance is expressed by Rac = Rdc(1 + y_s). Likewise, the flow of electromagnetic energy, quantified by the Poynting vector \vec{S} = \vec{E} \times \vec{H}, experiences significant vector distortion, heterogeneously concentrating the flow of reactive and active power through the surrounding insulating medium (cross-linked polyethylene XLPE or ethylene propylene rubber EPR).

Multiconductor Electromagnetic Interaction and Proximity Effect

While the skin effect is an intrinsic phenomenon of an isolated conductor, in three-phase medium-voltage systems—whether in compact trefoil configurations of three single-core cables or in a three-core shielded cable—the proximity effect operates simultaneously. This phenomenon arises because the alternating magnetic field generated by the currents flowing in adjacent conductors induces eddy currents and redistributes the current lines within the cross-section of the analyzed conductor.

The analytical formulation of the proximity effect requires the use of bipolar coordinates and the expansion in series of cylindrical harmonic functions to solve the field equations in the presence of multiple sources. The resistance increase factor due to the proximity effect is denoted as $y_p$. According to the international standardized formulations of the IEC 60287 standard, the total alternating current supplementary loss factor in the conductor is calculated by summing both effects under specified operating temperature conditions:

Where $R_0$ is the direct current resistance of the conductor at the maximum operating temperature. The term $y_p$ strongly depends on the cable layout geometry (trefoil or flat horizontal arrangement), the distance between conductor centers $s$, the conductor radius $r$, and the penetration depth \delta. For tubular or solid circular conductors, the proximity factor is approximated by expressions of the form:

Where $x_p$ is the corrected skin-thickness parameter and F_p(x) is an analytical function tabulated in technical regulations. In flat horizontal arrangements with narrow spacing, the magnetic fields add up vectorially in an asymmetric manner, causing the current density to shift toward the outer or inner zones of the conductor depending on the relative phase of the surrounding currents. This generates extreme thermal gradients along the circumference of the conductor, accelerating the thermomechanical degradation of the extruded semiconductor screens and the main XLPE insulation.

Parametric Matrix and IEEE / IEC Normative Limits

The sizing and operational evaluation of medium-voltage conductors demand strict compliance with international standards IEC 60287 and IEEE Std 835 / IEEE Std 399. The following comparative matrix details the critical parameters, operational limits, and dielectric consequences associated with the behavior of skin and proximity effect losses.

Electrotechnical Parameter	IEC 60287 Standard Limit	IEEE Standard Limit (Std 835 / 399)	Critical Failure Condition	Operational and Dielectric Consequences
Skin Increase Factor ($y_s$)	Max. 0.15 for sections \le 630 mm ^2 at $50 Hz$	Max. 0.18 under nominal power factor	Cumulative harmonic overload (THD_i > 8\%)	Unaccounted Joule losses, local hot-spots, and XLPE shrinkage.
Proximity Increase Factor ($y_p$)	Dependent on spacing s/d \ge 2.5	Strict restriction in closed ducts and flat trefoils	Flat horizontal arrangements without proper spacing	Asymmetric electric field deformation, tangential stress in semiconductor screen.
Maximum Conductor Temperature	$90 °C$ (XLPE) / $105 °C$ (EPR) permanent regime	$90 °C$ normal, $130 °C$ emergency	Thermal excursion due to short-circuit or high harmonics	Thermal fusion of semiconductor screens, loss of dielectric rigidity of the polymer.
Critical Cross-Section	$> 500 mm ^2$ in copper / $> 800 mm ^2$ in aluminum	$> 1000 kcmil$	Use of standard unsegmented solid or compacted cable	Erroneous positive sequence impedance, instability in distance protections.

Forensic Engineering Failure Analysis in Medium-Voltage Systems

Catastrophic failures in underground medium-voltage networks (typically operating at voltage levels of $13.8 kV$, $23 kV$, $34.5 kV$, or up to $69 kV$) frequently find their root in an inadequate analysis of high-frequency phenomena associated with the skin and proximity effects. When an industrial substation powers highly non-linear loads (medium-voltage frequency converters, arc furnaces, large-scale photovoltaic or wind inverter systems), the harmonic current spectrum introduces multiple-frequency components of the fundamental (h \cdot f).

Considering the skin depth equation, it is evident that for harmonic order $h$, the penetration depth is drastically reduced:

For a harmonic of order $h = 25$ ($1250 Hz$ in a $50 Hz$ system), the penetration depth is reduced by a factor of 5. This confines the harmonic current to an extremely thin layer at the periphery of the conductor. The effective resistance of the conductor for this harmonic component shoots up exponentially, causing severe localized heating (Localized Joule Effect). This phenomenon triggers the following forensic failure mechanisms:

• Thermal Degradation of XLPE Insulation: The heat internally generated in the conductor cannot dissipate quickly enough due to the low thermal conductivity of the polymeric insulation itself (\lambda \approx 0.4 W/m \cdot K ). This generates an abrupt radial thermal gradient, raising the temperature at the interface between the conductor and the inner semiconductor screen above the polymer cross-linking design limits.
• Micropore Formation and Electrical Treeing: Continuous thermal cycles induced by supplementary losses generate thermomechanical stresses of differential expansion and contraction between the conductor metal and the extruded insulation. This causes micro-mechanical separation of the semiconductor screen, creating air voids where partial discharges (PD) initiate. Partial discharges act as the precursor mechanism for the growth of electrical trees that culminate in dielectric breakdown and frank earth-fault failure.
• Joint and Termination Failures: Mechanical or compression connectors used in medium-voltage joints are designed under the premise of a relatively homogeneous current distribution. Severe skin effect and proximity effect asymmetry at the joint ends concentrate current on the inner walls of the compression sleeves, causing the fusion of inhibitory grease, oxidation of contact surfaces, increased contact resistance (cascading effect), and thermal destruction of the accessory.

Advanced Design Strategies, Mitigation, and Segmented Conductors

To mitigate the adverse effects of skin and proximity phenomena at high current capacities in medium-voltage systems, modern engineering implements advanced constructive solutions at the conductor design level and network geometric configurations.

Milliken-Type Segmented Conductors

When cross-sections exceed $800 mm ^2$ in copper or $1200 mm ^2$ in aluminum, the use of standard compact cylindrical conductors becomes unfeasible due to the high $y_s + y_p$ factor. The industry standard solution is the employment of Milliken-type segmented conductors. This design consists of dividing the total cross-sectional area of the conductor into four or eight sectors electrically isolated from each other by means of a thin layer of semiconductor paper or polymeric tape. Furthermore, each sector is cabled with a helical pitch that periodically inverts the position of the sectors along the length of the cable.

Electrodynamically, this segmentation forces each sector to carry an equal fraction of the total current, eliminating the preferential paths of eddy currents induced by the skin effect and the proximity magnetic field. Mathematically, the equivalent alternating current resistance of the Milliken conductor is drastically reduced, asymptotically approximating the direct current resistance $Rdc$:

Where $n$ is the number of independent sectors (typically 4 or 6). This technology reduces Joule effect losses by more than 40% in large-gauge cables.

Geometric Optimization of Underground Triplex/Single-Core Circuits

In installations of medium-voltage single-core cables in ducts or trenches, the geometric phase arrangement defines the magnitude of the proximity effect. Trefoil arrangements (compact triangular) minimize asymmetric concatenated magnetic flux compared to flat horizontal or vertical arrangements. However, the trefoil arrangement increases mutual thermal coupling, reducing the heat dissipation capacity toward the surrounding soil. The ampacity calculation must simultaneously solve the coupled Neher-McGrath thermal equation system:

Where $T_1, T_2, T_3$ are the thermal resistances of the insulation, inner jacket/filling, and soil surroundings respectively, and $W_d$ are the dielectric losses.

Practical Application and Computational Analysis via Vexten Suite

To illustrate the application of the exposed theoretical principles, an engineering case solved through the calculation routines of the Vexten Suite engineering platform is presented, applying international standards IEC 60909 for short-circuit currents and IEC 60287 / NEC 310 for cable sizing with harmonic derating factors.

Definition of the Industrial Scenario

An industrial underground distribution network at a nominal three-phase voltage of $34.5 kV$, fundamental frequency $60 Hz$, is analyzed. The main substation supplies a massive high-power rectifier load that introduces significant harmonic current content into the system. The critical circuit design parameters are as follows:

• Nominal Operating Voltage: $V_n = 34.5 kV$ (Phase-Phase)
• Fundamental Load Current: $I_1 = 650 A$
• Current Harmonic Spectrum: h_5 = 18\%,\, h_7 = 12\%,\, h11 = 7\%,\, h13 = 5\%
• Total Current Harmonic Distortion ($THD_i$): 23.47\%
• Selected Cable Type: Single-core cable with Milliken-type Segmented Copper conductor, XLPE insulation, copper wire screen, and HDPE outer jacket.
• Initial Evaluated Cross-Section: $630 mm ^2$
• Ground Ambient Temperature: $30 °C$
• Soil Thermal Resistivity: \rho_t = 1.0 K \cdot m/W

Execution of the Calculation Algorithm in Vexten Suite

The electromagnetic and thermal calculation engine of Vexten Suite processes individual harmonics by calculating the specific alternating current resistance for each frequency component according to the Bessel formulation:

Subsequently, the software evaluates the total supplementary loss factor weighted by the harmonic spectrum. For the standard solid $630 mm ^2$ conductor, the skin factor at $60 Hz$ is $y_{s,1} = 0.12$, but for harmonic $h = 13$ ($780 Hz$), the penetration depth drops to \delta13 = 2.59 mm , raising the harmonic skin factor to $y_{s,13} = 0.89$.

The Vexten Suite algorithm applies the Harmonic Derating Factor (HDF) derived from IEEE Std 539 and IEC 60287-2-2:

When introducing a conventional compact copper conductor of $630 mm ^2$, the HDF calculated by the platform yields a critical value of $0.78$, which means that the current-carrying capacity (ampacity) of the cable is reduced by 22\% exclusively due to the skin and proximity effects exacerbated by harmonics. The estimated conductor temperature under this operating regime reaches $96.4 °C$, exceeding the normative limit of $90 °C$ for XLPE.

Corrective Action Implemented in Vexten Suite

Faced with the thermal normative non-compliance, the Vexten Suite optimization engine automatically redesigns the technical specification toward an $800 mm ^2$ Milliken-type Segmented Copper Conductor. With this new geometry:

• The skin increase factor at fundamental frequency is reduced to $y_{s,1} = 0.03$.
• The impact of higher harmonics is substantially mitigated due to sectoral subdivision, reducing the harmonic supplementary loss factor by 65\%.
• The Harmonic Derating Factor (HDF) increases up to $0.95$.
• The estimated operating temperature in steady-state permanent regime stabilizes at $82.1 °C$, comfortably complying with IEC 60287 requirements and guaranteeing an estimated useful life exceeding 40 years without risk of electrical tree formation or premature degradation of the medium-voltage polymeric insulation.

This procedure demonstrates the indispensability of integrating advanced electromagnetic models based on electromagnetic fields and diffusion equations into electrical system design software, overcoming the limitations of traditional methods based on simplified tables and guaranteeing maximum operational reliability in complex medium-voltage distribution networks.