Why does a wireless power link need compensation?
A loosely coupled pair of coils is mostly leakage inductance. With k = 0.2, only a small part of each coil’s reactance couples to the other; the rest is reactive power that the inverter would have to circulate for nothing. Adding a capacitor in series or in parallel with each coil, sized to resonate with it, cancels that reactance at the operating frequency. The source then sees an essentially resistive load, the inverter’s volt-ampere rating drops to close to the real power, and the induced voltage jωM·I₁ is delivered to the load instead of across the receiver’s own inductance.
The four basic networks are named by how the primary and secondary capacitors connect: series (S) or parallel (P). Wang, Covic and Stielau analysed all four; Zhang and Mi review them and the higher-order LCC family for high-power systems.
What are the design equations for SS, SP, PS and PP?
With ω = 2πf the operating angular frequency, the secondary capacitor resonates with L₂ in all four cases:
The primary capacitor depends on the topology:
| Topology | Primary capacitor C₁ | Depends on |
|---|---|---|
| SS | 1 / (ω² L₁) | L₁ only |
| SP | 1 / (ω² (L₁ − M²/L₂)) | L₁ and coupling |
| PS | load-dependent (Wang et al. 2004) | coupling and load |
| PP | load-dependent (Wang et al. 2004) | coupling and load |
SS is the only basic network whose capacitors depend on neither k nor the load, which is why it is popular for links whose gap or alignment varies. At resonance the input resistance is (ωM)²/R_L, and with a voltage-source inverter the output current is V₁/(ωM), independent of the load.
SP uses a parallel-tuned receiver that reflects −jωM²/L₂ back into the primary, so C₁ must resonate with L₁ − M²/L₂. With the primary current I₁ set, the load current is (M/L₂)·I₁: the receiver behaves like a current source, which suits battery charging.
PS and PP put the primary capacitor in parallel, which favours current-fed inverters. Their primary capacitor values depend on the load as well as the coupling, so the tuning shifts as the load changes; Wang, Covic and Stielau give the expressions.
Worked example: 24 µH coils at 85 kHz
Take two identical 24 µH coils with k = 0.2 (M = 4.8 µH), 85 kHz, a primary current of 5 A RMS and a 10 Ω load. These are the defaults of the compensation capacitor calculator, computed here with the same code.
| Quantity | SS | SP |
|---|---|---|
| C₁ | 146.1 nF | 152.2 nF |
| C₂ | 146.1 nF | 146.1 nF |
| Voltage across C₁ (RMS) | 64.1 V | 61.5 V |
| Load current (RMS) | 1.28 A | 1 A |
| Power to load | 16.4 W | 10 W |
| Input resistance at resonance | 0.657 Ω | 0.4 Ω |
The SP primary capacitor is 4.17 % larger than the SS value, which is exactly the factor 1/(1 − k²) for identical coils. Using the SS value in an SP design would leave the inverter driving a reactive load. Peak capacitor voltage is √2 times the RMS value, which sets the capacitor rating.
When should you use LCC compensation?
The double-sided LCC network adds a series inductor L_f and a parallel capacitor C_f on each side, with the series capacitor C₁ in line with the coil. Li et al. tune it so that
and symmetrically on the secondary. With this tuning the primary coil current is set by the input voltage and L_f1, independent of the load and of k, and the output current is independent of the load. The resonant frequency also does not depend on coupling, so zero-phase operation is easier to keep across misalignment. The price is two extra components per side and more design freedom to manage. For example, choosing L_f1 = 6 µH with the 24 µH coil above at 85 kHz gives C_f1 = 584.3 nF and C₁ = 194.8 nF.
Every compensation design starts from the same three numbers: L₁, L₂ and M at the operating point, plus how far M moves as the gap and alignment change. Getting those right matters more than the choice of formula, because a capacitor tuned to the wrong M or L stays wrong in hardware. AirInduct Sim computes L_tx, L_rx, M and k for its eight coil topologies from closed-form Biot–Savart and Neumann expressions, and its parameter sweeps show how they vary across the gap and misalignment range a design must tolerate. The SPICE export carries the coil model into a circuit simulator, where the compensation network, inverter and rectifier can be simulated together. For a quick check, the free compensation capacitor calculator sizes SS and SP capacitors and their voltage stress from L₁, L₂, M and frequency. For LCC, use the tuning equations above with the inductances from the coil model.
What are frequency splitting and bifurcation?
Two identical series resonators coupled by k have two natural frequencies instead of one. Writing the loop equations with the load removed gives
For the example above (f₀ = 85 kHz, k = 0.2) these are 77.59 kHz and 95.03 kHz. A heavily loaded receiver damps the split, and the link keeps a single resonance at f₀. A lightly loaded, high-Q receiver lets the split appear: the power transfer curve develops two peaks, and the input impedance becomes purely resistive at three frequencies instead of one. Wang, Covic and Stielau call this bifurcation; Sample, Meyer and Smith measured the corresponding frequency splitting in magnetically coupled resonators.
The table scans the input phase of the lossless example link over 0.6–1.4·f₀ and lists every zero-phase frequency:
| Topology and load | Secondary Q | Zero-phase frequencies |
|---|---|---|
| SS, R_L = 10 Ω | 1.28 | 85 kHz |
| SS, R_L = 3 Ω | 4.27 | 85 kHz |
| SS, R_L = 2 Ω | 6.41 | 80.55, 85, 91.54 kHz |
| SS, R_L = 1 Ω | 12.8 | 78.21, 85, 94.28 kHz |
| SP, R_L = 10 Ω | 0.78 | 85 kHz |
| SP, R_L = 50 Ω | 3.9 | 85 kHz |
| SP, R_L = 100 Ω | 7.8 | 78.52, 85, 92.02 kHz |
For a series-tuned secondary the loaded Q is ωL₂/R_L, so bifurcation appears as the load resistance falls; for a parallel-tuned secondary it is R_L/(ωL₂), so it appears as the load resistance rises. In both cases the split frequencies tend towards f₀/√(1 ± k). Two practical rules follow: keep the secondary Q below the bifurcation bound across the whole load and coupling range, and if the controller tracks zero phase, make sure it cannot lock onto a side frequency. Higher k lowers the Q at which splitting begins.
References
- C.-S. Wang, G. A. Covic and O. H. Stielau, “Power transfer capability and bifurcation phenomena of loosely coupled inductive power transfer systems,” IEEE Trans. Industrial Electronics 51(1), 148–157 (2004). doi.org/10.1109/TIE.2003.822038
- W. Zhang and C. C. Mi, “Compensation topologies of high-power wireless power transfer systems,” IEEE Trans. Vehicular Technology 65(6), 4768–4778 (2016). doi.org/10.1109/TVT.2015.2454292
- S. Li, W. Li, J. Deng, T. D. Nguyen and C. C. Mi, “A double-sided LCC compensation network and its tuning method for wireless power transfer,” IEEE Trans. Vehicular Technology 64(6), 2261–2273 (2015). doi.org/10.1109/TVT.2014.2347006
- A. P. Sample, D. A. Meyer and J. R. Smith, “Analysis, experimental results, and range adaptation of magnetically coupled resonators for wireless power transfer,” IEEE Trans. Industrial Electronics 58(2), 544–554 (2011). doi.org/10.1109/TIE.2010.2046002
- G. A. Covic and J. T. Boys, “Inductive power transfer,” Proceedings of the IEEE 101(6), 1276–1289 (2013). doi.org/10.1109/JPROC.2013.2244536