Wireless chargers, which are widely advertised today, are conventional air transformers that may (or may not) contain magnetic paths. Normally, the transmitting part (primary coil) is fixed while the receiving part (secondary coil) is located in a mobile device and can take different positions relative to a primary coil. Coils can be of a different shape, but most commonly they are either square or circular (as the one in the picture). In this example the influence of capacitances in series with primary and secondary coils on the energy transfer is studied.

Problem type:
Axisymmetric problem of AC magnetics.

Geometry:

All dimensions are in millimeters.

Given: Core has a linear relative permeability of μ_{r} = 1000;
Current in coils has an electrical frequency f = 10 kHz;
Primary coil is fed from a current source Î_{1} = 4 A;
Load is modeled with resistance R_{load} = 1 Ω.

Problem:
To obtain the values of coils self-inductances and to solve the model coupled with external circuit to study the effect of adding capacitances.

Solution:
The most effective energy transfer happens when primary and secondary parts work in resonance. In order to operate circuits in resonance at defined frequency a certain capacitance value shall be added to both circuits. This value directly depends on the self-inductance of the circuit as C = 1/(ω^{2}·L).
Inductance non-linearly depends on the core magnetic properties, eddy effects in the coils and core, mutual orientation of the coils. In order to simplify the model and omit non-linear effects, the core permeability is assumed to be constan. Also, dimensions of conductors are small and core is non-conductive, therefore, the eddy current effects are negligible.
With stated assumptions self-inductances of the coils are L_{1} = L_{2} = 22.65 μH. Thus, capacitances are C_{1} = C_{2} = 11.20 μF.

Results:
Without capacitances: primary part takes S_{1} = 6.88 VA and power in load resistance is P_{load} = 4.46 W.
With capacitances: primary part takes S_{1} = 13.68 VA and power in load resistance is P_{load} = 12.68 W.

Magnetic flux density in the cross-section of the wireless charger: