Electric Current, Ohm's Law, and Electro-Thermal FEM

1. Electricity in Modern CAE: Beyond Structural Analysis

With the rapid growth of electric vehicles, power electronics, and electrified systems, CAE engineers increasingly need to analyze electrical current flow and its thermal consequences — not just mechanical stress. Electro-thermal FEM couples electrical analysis (finding the voltage and current distribution) with thermal analysis (computing the resulting Joule heat and temperature field) to design safe, reliable electrical components.

The mathematical structure is remarkably similar to structural analysis — Ohm's law has the same form as Hooke's law, and the FEM equations for electrical conduction are structurally identical to thermal conduction. An engineer fluent in thermal FEM can quickly learn electro-thermal coupling.

2. Ohm's Law: V = IR

Ohm's Law relates voltage (electric potential difference), current, and resistance:

This is the most powerful tool for quick circuit analysis. In FEM, the continuum form of Ohm's Law:

Where $\mathbf{J}$ is current density (A/m²), $\sigma_e$ is electrical conductivity (S/m), $\mathbf{E}$ is electric field (V/m), and $\phi$ is electric potential (V). Combined with current continuity ($\nabla \cdot \mathbf{J} = 0$ for steady state), this gives the governing equation for electrical FEM:

This is mathematically identical to the heat equation ($\nabla \cdot (k\nabla T) = 0$) — replace conductivity $\sigma_e$ with thermal conductivity $k$, and electric potential $\phi$ with temperature $T$. Every FEM thermal solver can solve electrical conduction problems with trivial material property substitution.

3. Joule Heating: P = I²R

When current flows through a resistor, electrical energy is dissipated as heat (Joule heating or I²R loss):

In the continuum form, the volumetric heat generation rate (W/m³) from Joule heating:

This Joule heat source term feeds directly into the thermal FEM equation — the source $\dot{q}'''$ from the electrical analysis drives the temperature rise in the thermal analysis. This is the coupling between the two physics.

4. Resistivity and Current Density

Electrical resistivity $\rho_e$ (Ω·m) is a material property characterizing how strongly a material opposes current flow. Resistance of a conductor:

Current density $J = I/A$ (A/m²) must be kept below safe limits in every conductor. Maximum safe current density depends on cooling conditions — typically 1–5 A/mm² for air-cooled conductors, 5–20 A/mm² for liquid-cooled.

Temperature Dependence of Resistivity

Resistivity increases with temperature for metals (phonon scattering): $\rho_e(T) = \rho_0[1 + \alpha_R(T-T_0)]$. This positive temperature coefficient creates thermal runaway risk: higher temperature → higher resistance → more Joule heating → even higher temperature. FEM must model this nonlinear coupling to correctly predict maximum operating temperature.

5. Skin Effect at High Frequency

At DC, current distributes uniformly over the conductor cross-section. At high frequency (AC), the electromagnetic skin effect concentrates current near the conductor surface. The skin depth:

Where $\omega = 2\pi f$ is angular frequency, $\mu$ is magnetic permeability. At 50 Hz in copper: $\delta = \sqrt{1.72 \times 10^{-8} / (\pi \times 50 \times 4\pi \times 10^{-7})} \approx 9.3$ mm. At 10 kHz: $\delta \approx 0.66$ mm. At 1 MHz: $\delta \approx 66$ µm.

For high-frequency power electronics (EV inverters, DC-DC converters operating at 10–100 kHz), skin effect significantly reduces the effective conductor cross-section and increases AC resistance. Litz wire (multiple thin strands, each thinner than skin depth) is used to recover low AC resistance.

6. EV Fast-Charging Cable: Worked Example

Design a DC fast-charging cable capable of continuously carrying 250A at 800V (200 kW charging). Maximum conductor temperature: 90°C. Ambient temperature: 40°C. Cable length: 5m.

Step 1: Required Conductor Area

Assuming copper ($\rho_e = 1.72 \times 10^{-8}$ Ω·m) and maximum current density 3.5 A/mm² for air-cooled flexible cable:

Step 2: Joule Heat Calculation

Step 3: Temperature Rise Check (lumped)

Cable surface area: $A_s = \pi D L = \pi \times 0.022 \times 5 = 0.346$ m². With $h = 10$ W/m²·K (natural convection): $\Delta T = P/(hA_s) = 56.6/(10 \times 0.346) = 16.4$°C above ambient. At 40°C ambient: conductor reaches ~56°C — well below 90°C limit. FEM verification confirms this is conservative (internal heat must conduct through insulation), but the hand calc gives confidence.

7. FEM Electro-Thermal Coupling

Electro-thermal FEM simultaneously solves for voltage/current distribution and temperature field, with Joule heating coupling them:

The coupling is: electrical conductivity $\sigma_e$ depends on temperature (material nonlinearity), and the Joule heating term $\sigma_e|\nabla\phi|^2$ drives thermal analysis. Sequential decoupling is common: solve electrical first, extract Joule heat, input to thermal. Fully coupled iteration is needed when $\sigma_e(T)$ variation is significant.

Abaqus Electro-Thermal Setup

*MATERIAL, NAME=COPPER
*ELECTRICAL CONDUCTIVITY
5.8E7, 20.   ! Conductivity S/m at 20°C
5.0E7, 100.  ! Temperature-dependent
*THERMAL CONDUCTIVITY
385.,

*COUPLED THERMAL-ELECTRICAL, DELTMX=10.
1., 1000., 0.1, 10.
*CFILM  (surface convection)
CABLE_OUTER, 1, 10., 40.  ! h=10, T_ambient=40°C

8. EV Battery Pack: Electrical and Thermal Co-Simulation

An EV battery pack combines hundreds or thousands of cells in series/parallel configurations. The electrical and thermal behavior are tightly coupled: cell resistance increases with aging and temperature, affecting current distribution; current distribution affects local heat generation; heat affects aging rate. This creates a full electro-thermal-electrochemical coupling problem.

Key analysis objectives:

• Current distribution: Which cells/modules carry the most current due to resistance variation?
• Thermal hot spots: Which cells reach highest temperature during fast charging?
• Cell-to-cell temperature variation: Gradient > 5°C between cells accelerates differential aging
• Thermal runaway propagation: If one cell enters thermal runaway, does it propagate?

9. Cross-Topics