Design a wireless power transfer (WPT) link that sends power across an air gap with no wires, using the magnetic field shared between two coils. Adjust the coil distance, radius, frequency and Q factor to see the coupling coefficient k, the figure of merit kQ and the real transfer efficiency update in real time.
Parameters
Coil distance d
mm
Air gap between the transmitting and receiving coils
Coil radius r
mm
Radius of the coils (assumed identical)
Operating frequency f
kHz
Frequency at which the link resonates
Coil Q factor
Quality factor of the coil resonator (same for both coils)
Load match
How well the receiver load is matched. 1.0 = optimum match
Results
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Coupling coefficient k
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Figure of merit kQ
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Max efficiency η_max (%)
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Actual efficiency (%)
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Distance / radius ratio
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Transfer verdict
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WPT link diagram — magnetic coupling animation
Magnetic field loops leaving the transmitting coil on the left thread through the receiving coil on the right across the gap d. The fraction of field lines linking the receiver represents the coupling coefficient k.
Coupling coefficient k of two coaxial coils (d: coil distance, r: coil radius) and the maximum efficiency η_max of a resonant WPT link. The achievable efficiency is set not by k alone but by the figure of merit kQ — the product of k and the quality factor Q.
Figure of merit FOM = kQ (both coils assumed to have the same Q). The actual transfer efficiency η_actual is the maximum efficiency η_max multiplied by the load match m (0.3 to 1.0); at a match of 1 you obtain η_max directly.
What is the Wireless Power Transfer Simulator?
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You can charge a phone just by setting it on a charging pad, right? How does the electricity get across when there are no wires?
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In short, it is "a transformer with the iron core replaced by air". The charging pad holds a transmitting coil, the phone holds a receiving coil. Running AC current through the transmitting coil creates a magnetic field, and that field threads the receiving coil. A voltage is then induced in the receiving coil — that is electromagnetic induction. Even though the coils are not physically connected, power crosses over the "invisible bridge" of the magnetic field.
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I see, the same principle as a transformer. But a normal transformer has both coils wound tightly on an iron core. Does it still work well with just air?
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That is exactly where it gets hard. In an iron-core transformer the two coils are "tightly coupled" and the coupling coefficient k is close to 1. But pull air-cored coils apart and they become only "loosely coupled" — k collapses fast. Try raising the "coil distance" on the left. You will see k drop steeply as the distance/radius ratio grows. Once the gap is several times the radius, most of the field loops sail straight past the receiving coil.
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Weak coupling sounds like poor efficiency... yet Qi charging is clearly practical. How?
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Good question. It is actually wrong to assume "loose coupling = poor efficiency". What truly sets the efficiency is not k alone but the product of k and the Q factor — the "figure of merit kQ". Q says how good a resonator the coil is; a well-made air-cored coil reaches 100 to several hundred. Even if k is only 0.5, a Q of 100 gives kQ = 50. When kQ is well above 1, the efficiency approaches 1. That is the breakthrough of resonant WPT.
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So a small k can be rescued by Q. Looking at the "efficiency vs kQ" chart below, the efficiency shoots up right around kQ = 1.
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Exactly. With η_max = (kQ)²/(1+√(1+(kQ)²))², efficiency is below 50% when kQ is under 1, but climbs into the 90s once kQ passes about 10. So WPT design becomes a contest of either "raise k" or "raise Q" to win kQ. That is why a phone held a centimetre off a Qi pad still charges efficiently — its kQ comfortably exceeds 1.
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I have heard about EV charging pads and powering medical implants inside the body. Is that the same mechanism?
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It is the exact same world of kQ. An electric vehicle sends kilowatts from a large transmitting coil in the ground to a receiving coil on the underside of the car; the big coils keep k high even across a gap. Medical pacemakers and cochlear implants are powered from outside the body to a coil inside, without breaking the skin — the magnetic field passes through living tissue almost unchanged. The scales differ wildly, but designers are all watching the same single point: how to keep kQ large.
Frequently Asked Questions
For two coaxial coils, this tool uses the ratio of coil distance d to coil radius r, ratio = 2d/r, and approximates k = 1/(1+ratio²)^1.5. k is a dimensionless number between 0 and 1: the closer the coils or the larger their radius, the closer k gets to 1. The tool uses this k as the starting point for the figure of merit kQ and the transfer efficiency. The fact that k collapses rapidly once the gap exceeds the coil radius is the very essence of why wireless power is hard.
The key is the figure of merit kQ. Separating air-cored coils makes the coupling coefficient k small, but the achievable efficiency is set not by k alone but by the product of k and the coil quality factor Q. If each coil is made a high-Q resonator tuned to the same frequency, even a small k can be multiplied by a large Q to push kQ well above 1. The maximum transfer efficiency is η_max = (kQ)² / (1+√(1+(kQ)²))², which approaches 1 (100%) as kQ grows. This is the founding principle of resonant WPT.
The coupling coefficient k decays as 1/(1+(2d/r)²)^1.5 with distance d, so efficiency falls steeply with the gap. When the distance approaches the coil radius, k drops sharply, and once kQ falls below about 1 the efficiency starts to collapse. The 'efficiency vs coil distance' chart in this tool shows a cliff-like characteristic: nearly flat at short range, then a sharp fall beyond a certain distance. Making the coils larger raises k at the same gap and pushes that cliff farther out.
The maximum efficiency η_max is the theoretical ceiling obtained when the receiver load is matched to its optimum value. In real devices the load impedance can drift away from the optimum, and this tool represents that with a load match (0.3 to 1.0). The actual transfer efficiency is η_actual = η_max × load match: at a match of 1.0 you get η_max directly, and efficiency falls as the match degrades. Real systems use impedance-matching networks and dynamic load tracking to keep the match close to 1.
Real-World Applications
Charging smartphones and small devices: Drop-and-charge pads, exemplified by the Qi standard, are the most familiar use of wireless power transfer. A transmitting coil sits in the pad and a receiving coil in the phone, and a few watts cross a gap of just a few millimetres. The short distance and close coils give a fairly high coupling coefficient k, and high-Q coils raise kQ further, achieving practical efficiency with no cable to plug in. Earbud cases and electric toothbrushes work on the same principle.
Contactless charging of electric vehicles: A large transmitting coil buried in the ground sends kilowatts — to tens of kilowatts — to a receiving coil on the underside of the car. Because the coils are large, k can be kept adequate even across a gap of more than ten centimetres, and resonant operation maintains a high kQ. Charging starts simply by parking, so no cable has to be inserted or removed; it is being developed as a method that works well in bad weather and pairs naturally with self-driving cars.
Powering medical implants: Implanted devices such as pacemakers, cochlear implants and nerve stimulators are powered from outside the body without breaking the skin. Because the magnetic field passes through living tissue with almost no attenuation, transcutaneous energy transfer (TET) is possible. It can reduce repeat surgery to replace batteries, and even in layouts where coupling tends to be weak, the design aims to keep kQ high for stable power delivery.
Industrial equipment, robots and sensors: WPT is used to power places where wiring cannot be routed — rotating parts, moving joints, and the inside of sealed enclosures. It is effective for sensors on a rotary table, underwater robots and equipment in explosion-proof environments, where connector wear and sparking must be avoided. Here it is important to design coils whose kQ does not collapse against distance and positional misalignment, while preserving freedom of placement.
Common Misconceptions and Pitfalls
The biggest misconception is the belief that "a small coupling coefficient k must mean poor efficiency". A small k is indeed "loose coupling" from a transformer's viewpoint, but in resonant WPT the efficiency is set not by k alone but by the figure of merit kQ. Even with k of only 0.1, a Q of 200 gives kQ = 20 and η_max above 95%. Conversely, even k of 0.5 yields poor efficiency if Q is low. "Raising Q" carries the same value as "raising k" — without this view, designs drift toward forcibly bringing the coils closer together.
Next is the leap that "high transfer efficiency means high overall system efficiency". The η_max and η_actual this tool computes are strictly the "coil-to-coil" efficiency from the transmitting coil to the receiving coil. A real system adds the losses of the transmit-side inverter (DC-to-AC conversion), the rectifier circuit, and wiring losses outside the coils. Even at 95% coil-to-coil efficiency, the overall wall-socket-to-device efficiency including these is lower. When discussing the energy savings of WPT, you must be clear about which segment's efficiency you mean.
Finally, the misconception that "raising the frequency improves everything". A higher operating frequency makes it easier to obtain a high Q even with a small coil, but skin and proximity effects increase conductor losses, radiation and EMC (electromagnetic compatibility) issues arise, and human-exposure regulations become stricter. Qi charging uses the 100-200 kHz band and some methods use the several-MHz band as the balance point of these trade-offs. This tool treats frequency as a premise of resonant operation, but in real hardware you must weigh the "Q rises" benefit against the "losses and regulations increase" cost when choosing the frequency.
How to Use
Set transmission distance (mm) using distNum slider; typical values range 5–50mm for near-field inductive coupling
Define coil radius (mm) with radNum; larger coils (25–40mm) improve coupling but increase size
Adjust operating frequency (kHz) via freqNum; 100–200kHz balances efficiency and EMI compliance for consumer devices
Input quality factor Q using qNum; higher Q (50–150) reduces losses in resonant circuits
Read coupling coefficient k and maximum efficiency η_max to validate your WPT link design
Worked Example
Design a smartphone charging coil pair: Set distance=8mm, primary coil radius=20mm, secondary radius=18mm, frequency=125kHz (Qi standard), Q=80 for ferrite-cored inductors. Simulator returns coupling coefficient k=0.42, kQ figure of merit=33.6, and maximum efficiency η_max=87%. Actual efficiency under 5W load reaches 81%, confirming practical viability for 15W wireless charging pads.
Practical Notes
Coupling coefficient k drops as fourth power of distance; exceeding 15mm gap with 20mm coils yields k<0.25 and efficiency below 60%
Foreign object detection requires monitoring impedance shift; ferrite shielding reduces stray coupling to <3% of primary power
Resonant frequency tuning compensates for load variation; detuning by ±5% causes efficiency loss of 8–12%
Inductive losses scale with frequency squared; 200kHz systems require 30–40% more copper area than 100kHz designs for equivalent performance