Electromagnetic-Thermal Coupled Simulation of Wireless Charging

Good question. When transmitting power via magnetic coupling, heat is always generated from copper loss in the coils and iron loss in the ferrite core. Even with an 80% transmission efficiency for the Qi standard 15W, 3W of loss turns entirely into heat. If this heat is trapped in a smartphone's thin casing, the battery temperature can exceed 45°C, shortening its lifespan.

Exactly. For 11kW wireless EV charging (SAE J2954 WPT3 class), even with 90% efficiency, that's 1.1kW of loss. Cooling design to keep the charging pad surface temperature below 60°C is essential. Without predicting the temperature distribution using coupled electromagnetic-thermal FEM, you'd end up repeating prototyping many times.

Simply put, it's a mechanism where high-frequency current flows through the transmitting coil (Tx) to generate an alternating magnetic field, and the receiving coil (Rx) picks up that magnetic flux to generate an electromotive force. Think of it like spatially separating the primary and secondary windings of a transformer.

The coupling coefficient $k$ represents the degree of magnetic flux coupling between the transmitting and receiving coils. Using mutual inductance $M$ and the self-inductance of each coil $L_1, L_2$:

$k=1$ is the ideal state where all magnetic flux links the other coil (like inside a transformer core). In actual wireless charging, $k$ is around $0.1\sim0.6$. $k$ decreases as the coil distance increases or if they are misaligned.

Exactly. The total loss relative to input power is:

For example, for 11kW EV charging with $\eta=0.90$, $P_{\text{loss}} = 1.1\,\text{kW}$. If $k$ halves from 0.3 to 0.15, $\eta$ drops from about 0.9 to 0.7, causing $P_{\text{loss}}$ to jump to $3.3\,\text{kW}$. That's why allowable misalignment and thermal design must be considered together.

Losses can be largely broken down into three parts. This is the breakdown of heat sources:

Here $F_r$ is the AC resistance factor, which depends on the skin depth $\delta = \sqrt{2\rho / (\omega \mu)}$. For copper at 100kHz, $\delta \approx 0.21\,\text{mm}$, so with a single round wire, current doesn't flow in the center, reducing efficiency. That's why litz wire (structure of twisted thin strands) is used.

$C_m$, $\alpha$, $\beta$ are constants obtained from the ferrite material's datasheet (typically $\alpha \approx 1.5$, $\beta \approx 2.5$ for MnZn ferrite). $\hat{B}$ is the maximum magnetic flux density in the core, $V_{\text{core}}$ is the core volume.

Let's summarize practical loss ratio estimates in a table:

The electromagnetic field is governed by Maxwell's equations, and the temperature field by Fourier's heat conduction equation. For WPT, we couple quasi-static magnetic field frequency-domain analysis (with magnetic vector potential $\mathbf{A}$ as the unknown) with steady-state or transient heat conduction equations.

Here $\mu$ is permeability (temperature-dependent for ferrite), $\sigma$ is electrical conductivity (copper's $\sigma$ also changes with temperature: $\sigma(T) = \sigma_0 / [1+\alpha_R(T-T_0)]$), $\mathbf{J}_s$ is the source current density.

$Q_{\text{em}}(T)$ is the volumetric heat generation rate obtained from electromagnetic analysis, providing the spatial distribution of the sum of copper, iron, and eddy current losses. This $Q_{\text{em}}$ becomes a function of $T$ through temperature-dependent electrical resistivity and permeability, so bidirectional coupling (strong coupling) is inherently necessary.

Exactly. However, in practice, we often solve using weak coupling (one-way iteration) by leveraging the fact that "the electromagnetic field response is sufficiently fast compared to temperature changes." We'll explain in detail later.

This is called the FOD (Foreign Object Detection/Debris) problem and is a very important safety issue. If metal pieces like coins, clips, or aluminum foil get caught between Tx and Rx, large eddy currents are induced by the alternating magnetic field.

Of course. To analyze FOD eddy current loss, calculate the loss distribution using electromagnetic FEM with the object's shape, material, and position as parameters, then input the result into thermal FEM. Maximum loss occurs when the object size is comparable to the skin depth $\delta$, so mesh requires element sizes of $\delta/3$ or smaller. Qi standard verification of FOD detection algorithms also uses these analysis results.

The key point in electromagnetic-thermal coupling for WPT is the large difference in time scales between the two physical fields:

When the ferrite core is near its Curie temperature (temperature region where permeability changes rapidly) or for high-power WPT (50kW and above) where temperature rise reaches several hundred °C, property changes due to temperature significantly affect solution stability, so strong coupling (Monolithic) or weak coupling with sub-iterations becomes necessary.

In structural analysis, nodal displacements are scalar or vector degrees of freedom, but in electromagnetic FEM, the magnetic vector potential $\mathbf{A}$ is the unknown. For 3D problems, using edge elements (Edge Element, Nedelec element) is standard, offering the advantage of naturally satisfying gauge conditions, unlike nodal elements.

Here $\mathbf{w}$ is the basis function for edge elements. Discretization yields a complex system of equations $[K_{\text{em}}]\{\mathbf{A}