Engineering Guide
Air-Core Coil Design: Wheeler Formula, Wire Loss, Q, Self-Resonance, and Measurement
An air-core formula provides a geometric starting point. A usable RF coil also requires buildable turns, conductor-loss estimates, self-resonance margin, environmental parasitics, and measurement in the final mounting geometry.
Coil Design Workflow
| Step | Action | Engineering Check |
|---|---|---|
| Define electrical targets | Set inductance, operating frequency, required Q, current, tuning range, and allowable size. | |
| Choose the coil model | Use the single-layer Wheeler approximation only for a cylindrical solenoid within its geometry assumptions. | |
| Choose practical geometry | Select finished winding diameter, axial length, wire diameter, pitch, and enough space for leads and adjustment. | |
| Solve turns | Calculate a fractional estimate, then test neighboring whole-turn counts because a physical coil cannot use the displayed fraction directly. | |
| Estimate conductor loss | Calculate wire length and DC resistance, then account for skin and proximity effects at operating frequency. | |
| Check frequency behavior | Calculate reactance, approximate Q, intended LC resonance, and margin below the coil's self-resonant frequency. | |
| Include surroundings | Model leads, pads, shields, enclosure metal, PCB copper, nearby coils, and the measurement fixture. | |
| Build, measure, and trim | Measure at the intended frequency and fixture reference plane; adjust spacing, turns, or a tuning element. |
Wheeler Geometry Model
Formula reference
Single-layer cylindrical coil
L(µH) = r²N² / (9r + 10l)N = √(L(9r + 10l) / r²)r = finished winding diameter / 2Variable definitions
- r and l are in inches for this coefficient set
- l is the occupied axial winding length
- N is the turn count
- Use another model for multilayer, toroidal, or planar geometry
Use the Air Core Inductor Calculator to calculate inductance or reverse-solve turns. Preserve full internal precision, then evaluate practical whole-turn alternatives.
Definitions and Model Boundaries
| Quantity | Definition | Boundary |
|---|---|---|
| Finished diameter | Diameter through the winding centerline | Do not substitute bare former diameter without wire thickness. |
| Axial length | Span occupied by the winding | Pitch and turn spacing directly change this value. |
| Turns | Complete magnetic turns | Lead routing and partial end geometry are outside the simple model. |
| Wire resistance | ρl/A at reference temperature | RF resistance is higher due to skin and proximity effects. |
| Q | Approximately XL/Rseries | Use total frequency-dependent series loss, not DCR alone. |
| Self-resonance | Resonance of L with distributed winding capacitance | A lumped Cp estimate is only an approximation. |
| Nearby material | Metal, magnetic material, and dielectric around the coil | Can change L, loss, capacitance, radiation, and coupling. |
Wire Loss, Reactance, and Q
Formula reference
First-order loss model
RDC = ρlwire/AwireXL = 2πfLQ ≈ XL/Rseriesδ = √(2ρ/(2πfµ))Variable definitions
- Rseries must represent the operating frequency
- Skin and proximity effects increase AC resistance
- Lead and connection loss reduce measured Q
- Current capability remains a thermal and insulation problem
The Inductive Reactance Calculator supplies XL. Q requires measured or modeled AC resistance rather than DC resistance alone.
Intended Resonance and Self-Resonance
Formula reference
Two different resonances
ftank = 1/(2π√(LCexternal))fSRF ≈ 1/(2π√(LCparasitic))Variable definitions
- External capacitance creates the intended tank
- Distributed capacitance creates coil self-resonance
- Use measured impedance near SRF
- Nearby conductors and fixtures change parasitic capacitance
Use the LC Resonance Calculator for an intended lumped LC tank. It does not predict distributed winding capacitance or the coil's measured SRF.
Worked Examples
Wheeler estimate
For 10 mm finished diameter, 20 mm winding length, and 10 turns, the single-layer formula estimates about 0.402 µH.
Turn scaling
With fixed geometry, inductance scales approximately with N². A 5% turn increase suggests roughly 10.25% more inductance before geometry changes.
Whole-turn check
If reverse solving gives 12.4 turns, evaluate 12 and 13 turns, then trim spacing or parallel capacitance after measurement.
Reactance
A 400 nH coil at 10 MHz has ideal XL ≈ 25.13 Ω.
Q estimate
If total series resistance at 10 MHz is 0.8 Ω, Q ≈ 25.13/0.8 ≈ 31.4.
Intended tank
A 400 nH coil with 100 pF resonates ideally near 25.16 MHz before coil and layout parasitics.
Self-resonance margin
If the measured coil SRF is 60 MHz, operation at 50 MHz may be too close for an ideal-inductor assumption.
Temperature
Copper resistance rises roughly with temperature, reducing Q even when geometric inductance changes little.
Lead effect
Long leads add series inductance and distributed capacitance; measure with the same lead and mounting geometry used in circuit.
Shield effect
A nearby conductive shield can reduce inductance and Q through eddy-current coupling, so final enclosure measurement matters.
Construction and Measurement
Control winding pitch with a former or spacer, secure turns without adding uncontrolled dielectric, and keep leads short and repeatable. Calibrate the fixture at the coil reference plane. Record L, Q, resistance, and impedance across frequency rather than trusting one scalar reading. Measure again after installing shields, covers, PCB mounts, and adjacent components.
Common Mistakes
Related Calculators
Support reference
FAQ
What is an air-core inductor?
It is a coil without a ferromagnetic core. Its inductance comes primarily from winding geometry and the permeability of air and surrounding nonmagnetic materials.
What is Wheeler's air-core formula?
For one common single-layer form, L in microhenries equals r squared times N squared divided by 9r plus 10l, with radius and winding length in inches.
Should I use coil radius or diameter?
The Wheeler expression uses radius. Measure the finished winding diameter through the conductor centerline and divide by two.
Why does measured inductance differ from the formula?
Winding pitch, wire size, lead geometry, nearby metal, distributed capacitance, fixture parasitics, frequency, and approximation limits all shift the result.
How do I calculate required turns?
Rearrange the selected geometry formula to solve N, then evaluate neighboring whole-turn counts and measure the built coil.
How is air-core coil Q estimated?
A first estimate is inductive reactance divided by total series resistance at the operating frequency. That resistance includes skin, proximity, radiation, and connection losses.
What is coil self-resonant frequency?
Distributed inter-turn and environmental capacitance resonates with coil inductance. Above that region, the coil no longer behaves as a simple inductor.
Does an air-core coil saturate?
It avoids magnetic-core saturation, but current is still limited by conductor heating, insulation, mechanical forces, voltage stress, and the surrounding circuit.
Does wire diameter change inductance?
It changes finished diameter, pitch, winding length, loss, and parasitic capacitance, so it affects both inductance and frequency behavior.
How far below SRF should a coil operate?
There is no universal ratio. Required margin depends on acceptable impedance and Q error; use measured impedance across the complete operating band.
Can I use the same formula for multilayer or flat spiral coils?
No. Those geometries require their own validated approximations or field models.
How should I verify an air-core coil?
Measure impedance or L and Q at the intended frequency with calibrated fixtures and the final mounting environment, then trim turns or spacing as needed.
Summary
Use a geometry equation for the starting coil, then close the design with practical turns, AC conductor loss, Q, self-resonance margin, construction tolerance, environmental parasitics, and frequency-specific measurement.