Reflected impedance

Xun (Ken) Liu, ConvenientPower
A good tool to analyze Resonant Coupling is 'reflected impedance'. Fig.1(a) shows the coupled circuit model with capacitor, Cs, added in series with secondary winding to form resonant tank. Rp, Lp, Rs and Ls are the resistance, and inductance of the primary and secondary winding, respectively. M is the mutual inductance between the primary and the secondary. RL represents the equivalent resistance of the load. Fig.1(b) is the equivalent primary circuit model with reflected impedance.

The reflected impedance, Zr, can be expressed with following equations:

in which ReZr is the real-part of the reflected impedance. It needs to be maximized for highest primary efficiency.
By analyzing the above equations, it can be found that when the secondary works under resonant condition

the reflected resistance, ReZr, exhibits maximum property and is equal to



. (if

, there isn’t maximum point in finite frequency range.)
It can also be found that said maximum ReZr can be further increased with the increase of frequency, the increase of mutual inductance, or the reduction of load resistance and secondary winding resistance. But it must be noted that the substantial decrease of load resistance may influence the secondary efficiency, because the secondary efficiency equals

Indeed, other resonant topologies (like parallel resonance, or the combination of series and parallel) can also be employed at secondary side. They can be analyzed and optimized with similar approach described above.

[Ref]: X. Liu, W. M. Ng, C. K. Lee and S. Y. R. Hui, "Optimal operation of contactless transformers with resonance at secondary circuit", in APEC'08, pp. 645-650, Feb. 2008, USA


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