Wireless Power Coupling Coefficient & Q-Factor Simulator
Design a two-coil magnetic-resonance link (Kurs–Soljacic) as used in Qi phone charging and EV wireless power transfer. Vary coil diameter, turns, distance, frequency and Q to see the coupling coefficient k, kQ figure of merit and Kurs–Soljacic maximum efficiency update in real time, and reach the target efficiency on a clean design.
Parameters
Primary coil diameter D₁
cm
Secondary coil diameter D₂
cm
Primary turns N₁
turns
Secondary turns N₂
turns
Coil separation d
cm
Coaxial centre-to-centre gap between Tx and Rx coils
Air-core Litz wire Q ≈ 200–400; with ferrite 100–200
Results
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Primary self-inductance L₁ (μH)
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Mutual inductance M (μH)
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Coupling coefficient k
—
Combined Q √(Q₁Q₂)
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kQ figure of merit
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Max efficiency η (%)
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Coil pair and magnetic flux — resonant link diagram
Left: primary (Tx) / right: secondary (Rx) coil. Flux-line density represents the coupling coefficient k, and the gauge shows the maximum efficiency. Colour tracks the kQ figure of merit (green = high efficiency, red = low).
Efficiency η vs coil separation d
kQ figure of merit vs maximum efficiency (Kurs–Soljacic curve)
Coupling coefficient k and Kurs–Soljacic maximum efficiency. M: mutual inductance; L₁, L₂: self-inductances; Q: combined Q; kQ: figure of merit (≥10 is the practical sweet spot).
Combined Q and the critical coupling k_crit. k < k_crit: under-coupled (efficiency collapses); k ≈ k_crit: critical (optimum); k > 2 k_crit: over-coupled (frequency splitting).
$$\omega L = \frac{1}{\omega C},\qquad C = \frac{1}{\omega^{2} L}$$
Resonance condition and the tuning capacitance C with ω = 2πf. Primary and secondary must be tuned to the same resonance — any mismatch drops the efficiency sharply.
Wireless Power Transfer: Coupling Coefficient k & Q-Factor Design
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A Qi pad charges a phone "just by placing it on top", and now we have EVs that recharge from a pad on the ground. How does that work? There's no wire, but power gets across.
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Think of it as a transformer split into two halves with a much bigger air gap. The transmitter coil carries a high-frequency current, which throws a magnetic field into the surrounding space. The receiver coil catches that flux and Faraday's law induces a voltage in it. The "magnetic resonance" approach turns both coils into tuned resonators on the same frequency so that the energy hands across the air far more efficiently than a plain weakly-coupled transformer would. Kurs and Soljacic at MIT demonstrated this in 2007, and Qi and EV WPT both grew out of it.
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Right. But Qi already reaches 70-80% efficiency through air. Normal transformers depend on a magnetic core to keep the flux trapped — how can you go that high without one?
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That's the elegant part. Look at the "coupling coefficient k" in this tool: a regular cored transformer gives 0.95-0.99, a Qi pad with a flat coil and ferrite backing gives 0.5-0.7, and an EV pad with a 15 cm air gap drops to 0.1-0.3. Efficiency still gets high because the real driver is the kQ figure of merit. Push the Q-factor up to 200-500 and even a small k can give a kQ of 20-100, and the formula (kQ)²/(1+√(1+(kQ)²))² takes you above 90%. "Small k" doesn't mean "low efficiency"; "small kQ" does.
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So if I just pile on Q, can I make the link span 1 m or even 10 m?
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In theory yes, in practice it's hard. Raising Q means cutting the loss resistance R, and even with Litz wire (lots of fine strands to defeat the skin effect) an air-core coil tops out around Q = 500. Meanwhile k falls fast with distance d: in the efficiency-vs-distance chart you can see a "cliff" near d ≈ coil radius r. Once kQ falls under 1 the efficiency collapses. So past about 1 m you have to make the coils bigger (raise r) to push the cliff back. That is exactly why EV WPT uses 50-cm-class pads.
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When I drag the frequency slider from 85 kHz to 6.78 MHz, the efficiency barely changes. What is actually different?
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Sharp eye. The efficiency formula (kQ)²/(...)² has no explicit f, so for the same k and Q the answer is the same. What changes is (1) the tuning capacitance C scales as 1/ω², so 85 kHz needs nF-class caps and 6.78 MHz needs pF-class — much harder to control; (2) higher frequency makes Q easier to reach, but parasitic capacitance, body absorption and EMC regulation get worse; (3) the band you can actually use is fixed by law — Qi/EV sit at 85-205 kHz, AirFuel at the 6.78 MHz ISM band. Watching the tuning-capacitance line in this tool gives a feel for the capacitor-selection problem.
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The verdict mentions over-coupled, critical and under-coupled. "Over-coupled" sounds stronger, but the message says "critical" is the best?
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That's the quirk of resonant WPT. At critical coupling k = 1/Q, efficiency peaks and the frequency response is flat. Below it kQ collapses and efficiency falls (under). Above it the single resonance splits into two peaks — frequency splitting — and at the original design frequency you end up in the dip between them. Qi and EV WPT run in the over-coupled regime and either track the new optimum frequency with PLL/PWM control or rematch impedance with an LCC compensation network. With the default k = 0.26 and Q = 200, k_crit = 1/200 = 0.005, so k is more than 50× larger — clearly over-coupled. The theoretical 96% is only reached with optimum load and active frequency tuning.
Frequently Asked Questions
k = M/√(L₁·L₂) is a geometric measure of how well two coils share magnetic flux — a dimensionless number between 0 and 1. Q = ωL/R is the loss inverse of each coil on its own and describes the sharpness of resonance. The product kQ is what governs the maximum achievable efficiency of a magnetic-resonance WPT link. As you separate two air-core coils, k drops, but with a high Q you can keep kQ large enough to maintain high efficiency even with weak coupling. This tool shows k, Q and kQ together so you can balance them.
This is the Kurs–Soljacic (MIT 2007) upper bound on efficiency for a resonant WPT link when the receiver is matched to its optimum load. At kQ = 1 efficiency is about 38%, at kQ = 3 about 75%, at kQ = 10 about 90%, and at kQ = 100 it exceeds 96%. Production Qi (85–205 kHz) and SAE J2954 EV WPT (85 kHz) systems target kQ between 20 and 100. Below kQ = 1 efficiency collapses quickly, so extending range becomes a fight to keep kQ above 1.
Qi runs in a near-coupled regime (k ≈ 0.5–0.7, gap much smaller than coil diameter), so even a modest Q with ferrite-backed flat coils gives a large enough kQ. EV WPT has a 100–250 mm air gap between ground assembly and vehicle assembly, where k falls to 0.1–0.3. To compensate, designers use Litz wire to push Q to 200–500 and tune the 85 kHz magnetic resonance tightly, reaching kQ of 30–60. The efficiency-vs-distance chart in this tool shows the difference clearly: Qi-type systems are flat then fall off a cliff; EV-type systems sag more gradually.
The critical-coupling point is k = 1/Q_combined, where efficiency peaks and the frequency response is flat. Below it (under-coupled, kQ < 1 region) efficiency collapses. Above it (over-coupled) the resonance splits into two peaks — frequency splitting — and at the original design frequency efficiency can dip. Over-coupled designs need frequency tracking or LCC compensation to chase the new optimum. This tool compares the current k with k_crit = 1/Q_combined and reports the regime in the verdict line. Qi and EV WPT normally operate in the over-coupled region with active frequency tuning.
Real-World Applications
Qi charging for phones and wearables: The Wireless Power Consortium's Qi standard operates at 100–205 kHz and 5–15 W and is now built into virtually every smartphone and smartwatch. With a 2–5 mm gap and k ≈ 0.5–0.7, ferrite-backed flat coils with relatively modest Q still reach 70–80% efficiency. Qi v2 and Apple's MagSafe add magnets for alignment, suppressing k variation and tightening the tuning match.
EV wireless charging (SAE J2954): BMW 530e iPerformance, Mercedes-Benz, Tesla and others have piloted EV WPT systems at 85 kHz and 3.7–11 kW (with 22–50 kW on the roadmap). With 100–250 mm between the ground assembly (GA) and the vehicle assembly (VA), k drops to 0.1–0.3. Designers compensate with high-Q Litz coils and LCC compensation networks to hit 90%-class grid-to-battery efficiency. WiTricity, InductEV (formerly Momentum Dynamics) and Plugless supply the underlying technology.
Medical implants and industrial robots: Transcutaneous energy transfer (TET) to pacemakers and LVAD heart pumps avoids surgical battery replacement by transferring a few watts across the skin. Low frequencies (around 200 kHz) keep tissue heating in check, and even Q ≈ 100 is enough to get a usable kQ. Factory AGVs and industrial robots (Powermat, Yank Technology) use WPT to dodge connector wear and arcing.
Multi-coil and dynamic WPT (DWPT): The KAIST OLEV bus in Korea and the European ELECTREON trials charge vehicles on the move by burying coil arrays in the road surface. Coil switching, time-varying k and multi-receiver behaviour all need detailed modelling — a single-pair tool like this one is the first sanity check, after which SPICE and FEM (COMSOL, ANSYS Maxwell) take over.
Common Misconceptions and Pitfalls
The most common pitfall is thinking "large k = high efficiency". As this tool makes clear, the real driver is kQ, not k alone. k = 0.5 with Q = 20 gives kQ = 10 and ≈ 90% efficiency; k = 0.05 with Q = 400 gives kQ = 20 and ≈ 95%. EV WPT in fact operates at k ≈ 0.2 and still reaches 90%-class efficiency. A "just bring the coils closer" mindset that neglects Q falls off a cliff for tiny distance changes. Getting Q up — typically with Litz wire and clean magnetic backing — is the real heart of resonant WPT.
Next, assuming the resonant frequency can be tuned once and left alone. In practice L and C drift with temperature, coil deformation, foreign metallic objects and, for EVs, with vehicle ride height (which moves 1–2 cm with passenger load). In the over-coupled regime this tool warns about, frequency splitting then drops the original design point into a dip and efficiency collapses. Real systems sweep the PWM frequency, PLL-lock onto the optimum, or actively rematch impedance with LCC compensation. The "kQ = 51 → 96%" reading here is only the upper bound under optimum load and perfect matching.
Finally, "just crank up the frequency" is too simple. Higher frequency does help shrink coils and tuning capacitors, but it also (1) raises losses through skin and proximity effects until Q stops improving, (2) drastically increases eddy-current losses in nearby metal, (3) tightens human SAR (Specific Absorption Rate) limits, and (4) leaves you with only a handful of legal bands (ISM 6.78 MHz, 13.56 MHz, …) under ICNIRP, FCC and Japanese radio regulations. The fact that Qi and EV WPT cluster around 85–205 kHz is a balance of all of these — sliding f freely in this tool is fine, but real designs always stay inside the regulatory bands.
How to Use
Enter primary coil diameter (cm) and turn count; repeat for secondary coil (typical Qi charger: 40–80 mm diameter, 8–15 turns each).
The simulator calculates self-inductance L₁ and L₂ using L = μ₀N²A/l, then mutual inductance M from measured coil separation (fixed at 5 mm for standard phone charging gap).
Observe coupling coefficient k = M/√(L₁L₂) and geometric Q-factor √(Q₁Q₂); kQ product above 100 ensures η_max > 90% at matched load impedance.
Worked Example
Design a Qi-compatible link: primary coil 50 mm diameter, 12 turns (copper wire 0.8 mm, ρ = 1.68 µΩ·cm); secondary 48 mm, 10 turns; 5 mm air gap. Calculated L₁ ≈ 3.2 μH, L₂ ≈ 2.4 μH, M ≈ 1.8 μH, yielding k = 0.73. At 125 kHz operating frequency, Q₁ ≈ 280, Q₂ ≈ 240, so √(Q₁Q₂) ≈ 259. kQ product = 189 delivers η_max ≈ 92% power transfer efficiency, meeting Qi v1.2 certification minimum.
Practical Notes
Coil separation dominates coupling: increase gap from 3 mm to 10 mm drops k by ~40%, severely reducing kQ. Phone cases (1–2 mm) have minimal impact; metallic phone frames reduce k by 15–25%.
Asymmetric coil diameters degrade coupling; pairing 80 mm primary with 40 mm secondary yields k ≈ 0.55 versus k ≈ 0.78 for matched 60 mm coils—use concentric near-field alignment for EV charging pads (200 mm coils).
High-Q coils (litz wire, ferrite cores) boost √(Q₁Q₂) but increase cost; standard copper achieves Q ≈ 200–300 at 100–200 kHz, sufficient for consumer Qi and AirFuel inductive standards.