Resonator balancing in wireless power transfer systems

1. A system for wireless power transfer, comprising:

a resonator comprising a coil with at least two windings, each of the at least two windings comprising a plurality of loops formed by a conductive material and extending in a plane, wherein corresponding portions of each of the at least two windings are oriented in parallel, wherein at least one of the windings has a length that differs from a length of another one of the windings, and wherein the at least two windings are electrically connected in parallel; and

at least one inductor having an inductance value, wherein the at least one inductor is connected in series to at least one of the windings,

wherein the inductance value is selected so that when the coil carries a current during operation of the system, the at least one inductor maintains a distribution of current flows among the at least two windings such that for each of the at least two windings, an actual current flow in the winding differs from a target current flow for the winding by 10% or less.

2. The system of claim 1 , wherein the at least one inductor comprises an adjustable inductance value.

3. The system of claim 1 , wherein corresponding portions of each of the at least two windings are oriented in parallel along at least 80% of a length of at least one of the windings.

4. The system of claim 1 , wherein the loops of each winding are interleaved.

5. The system of claim 1 , wherein the loops of each winding are concentric and form a spiral.

6. The system of claim 1 , further comprising an electronic processor coupled to the at least two windings and configured to control electrical currents in each of the windings based on the target current flows for the at least two windings.

7. The system of claim 6 , wherein the electronic processor is configured to control electrical currents in each of the windings by:

determining a target inductance value for the at least one inductor based on a figure of merit related to the target current flows; and

adjusting the inductance value of the at least one inductor to match the target inductance value.

8. The system of claim 7 , wherein the electronic processor is configured to determine the target inductance value by:

(i) for each one of the windings:

determining a self-inductance value of the one winding based on a measurement of inductance of the one winding when it is electrically disconnected from all other windings; and

determining a plurality of cross-inductance values of the one winding, wherein each cross-inductance value is based on a measurement of inductance of the one winding when it is electrically disconnected from another one of the windings;

(ii) determining the target current flows for each of the windings based on the self-inductance values and the cross-inductance values; and

(iii) determining the target inductance value based on the target current flows for each of the windings.

9. The system of claim 8 , wherein the electronic processor is configured to determine the target current flows by:

constructing an inductance matrix based on the self-inductance values and the cross-inductance values of each of the windings;

calculating an adjusted inductance matrix by adding to the inductance matrix an inductance modification matrix comprising elements that correspond to changes in inductance of each of the windings due to the at least one inductor;

calculating an inverse matrix of the adjusted inductance matrix; and

determining the target current flows based on the inverse matrix.

10. The system of claim 9 , wherein the inductance modification matrix is a diagonal matrix, and wherein diagonal elements of the inductance modification matrix are inductance values of respective members of the at least one inductor connected to the windings.