Air Core Coil Inductance Calculator
Calculate the inductance of a flat spiral air core coil from its inner diameter, outer diameter, and number of turns. Enter the coil dimensions in millimeters and read the result in nanohenries.
Flat Spiral Coil Inductance Tool
Air Core Flat Spiral Inductance Formula
Where:
- \(L\) — inductance of the coil (nH)
- \(N\) — total number of spiral turns
- \(d_1\) — inner diameter of the spiral (mm)
- \(d_2\) — outer diameter of the spiral (mm)
This is a Wheeler-type empirical approximation for flat (planar) spiral coils wound on air. Both diameters are entered in millimeters, and the formula returns the inductance directly in nanohenries. The expression is only meaningful when the denominator \(15d_1 - 7d_2\) is positive.
Air Core Flat Spiral Inductance – Calculation Example
Example: a flat spiral coil on a PCB has an inner diameter of 10 mm, an outer diameter of 20 mm, and 5 turns. What is its inductance?
\[ L = \frac{4.921 \times 5^2 \times (10 + 20)^2}{15 \times 10 - 7 \times 20} = \frac{4.921 \times 25 \times 900}{150 - 140} = \frac{110\,722.5}{10} \approx 11\,072.25 \text{ nH} \]
Result: \(L \approx 11\,072\) nH, which is about 11.07 µH.
How to use the calculator
- Enter the Outer diameter (\(d_2\)) of the spiral in millimeters. This is the diameter at the widest point of the winding.
- Enter the Inner diameter (\(d_1\)) in millimeters — the diameter of the open center of the spiral.
- Enter the Number of turns (\(N\)) of the spiral winding.
- Read the result, shown as Air Core Flat Spiral Inductance in nanohenries (nH). Divide by 1000 if you need the value in microhenries (µH).
The calculation runs instantly as you type, so you can quickly sweep different turn counts or diameters to hit a target inductance.
What is an air core flat spiral coil?
A flat spiral coil is a conductor wound in a single plane, like a clock spring, so that every turn lies flat instead of stacking in a cylinder. When there is no ferromagnetic core inside — just air, or the FR-4 of a printed circuit board — the coil is called an air core coil. Because the inductance comes purely from the geometry of the winding, this calculator needs only the two diameters and the number of turns to estimate it.
Air core coils are popular at radio frequencies because they remove nearly all core-related problems: no hysteresis, no saturation, no ferrite losses, and a highly linear inductance over a wide current range. Flat spirals in particular are easy to manufacture — they can be etched directly as copper traces on a PCB — which is why they appear in NFC and RFID antennas, wireless charging receivers, RF tank circuits, and metal detector search coils.
The inductance \(L\) tells you how much magnetic flux the coil links for a given current, and it scales with the square of the turn count: doubling \(N\) roughly quadruples \(L\). Larger coils (bigger diameters) also produce more inductance. Keep in mind that at high frequencies the coil also behaves as a resonant circuit due to its parasitic capacitance and winding resistance, so the geometric inductance calculated here is only part of the full picture. When a flat spiral is combined with other inductors in a circuit, use the Series Inductance or Parallel Inductance calculator to find the total, and don't forget the contribution of connecting traces, which you can estimate with the Straight Wire Inductance calculator.
Accuracy, limitations and practical tips
- The formula is an empirical approximation. It assumes the turns are evenly spaced across the radial width and that the conductor is thin compared to the coil diameter. Real coils can deviate by several percent or more.
- The denominator must be positive: \(15d_1 > 7d_2\). If your inner diameter is too small relative to the outer diameter, the result becomes meaningless — redesign the spiral so the winding has a reasonable radial width.
- Conductor width, thickness, and skin effect are not included in the model. At very high frequencies, skin effect and proximity effect raise the effective resistance even though the inductance changes only slightly.
- Nearby metal objects, ground planes, or shielding alter the magnetic field and shift the actual inductance away from the calculated value. Keep the coil at least one coil-diameter away from large conductive surfaces when possible.
- Above the coil's self-resonant frequency (SRF) it no longer behaves as an inductor at all, so always check that your operating frequency is well below the SRF of the physical coil.
- For critical RF designs, verify the result with an LCR meter or a vector network analyzer, and fine-tune the number of turns experimentally.
Quick reference values
The table below shows calculated inductances for coils with \(d_1 = 10\) mm and \(d_2 = 20\) mm at various turn counts, so you can sense how strongly \(N\) influences the result:
| Turns \(N\) | Inductance (nH) | Inductance (µH) |
|---|---|---|
| 3 | 3,986.01 | 3.99 |
| 5 | 11,072.25 | 11.07 |
| 8 | 28,344.96 | 28.34 |
| 10 | 44,289.00 | 44.29 |
Once you know the target inductance, you can plug it into an impedance-matching stage such as a PI Network Impedance Matching network to complete your RF front-end design.
Frequently Asked Questions
Can I calculate flat spiral coil inductance by hand?
Yes. Measure the inner diameter \(d_1\), outer diameter \(d_2\) (both in mm), and count the turns \(N\). Then compute \(L = \frac{4.921\,N^2(d_1+d_2)^2}{15d_1-7d_2}\), which gives the inductance directly in nanohenries. Work through the numerator first, then the denominator, and divide. The calculator uses this same Wheeler-type formula, so you can use it to check your hand calculation.
How do I convert the result from nH to µH or µH to mH?
Divide nanohenries by 1,000 to get microhenries: 11,072 nH ≈ 11.07 µH. To go further, divide microhenries by 1,000 for millihenries (11.07 µH = 0.01107 mH). Flat PCB spirals typically land between a few hundred nH and a few tens of µH. If a datasheet quotes an inductor in µH, multiply by 1,000 before comparing it with the nH output of this calculator.
Why does the calculator give a strange or zero result for my coil?
The formula only works when \(15d_1 > 7d_2\), i.e. the inner diameter must be at least about 47% of the outer diameter. If your spiral has a very small center hole relative to its width, the denominator goes to zero or negative and the answer is meaningless. Redesign the coil with a wider inner opening, or the code returns 0 when the denominator is exactly 0.
How accurate is this Wheeler formula for a real PCB spiral?
Expect within roughly 5–10% for evenly spaced turns with thin, widely spaced conductors. Deviation grows with thick or tightly packed traces, square vs. circular layout, and nearby ground planes or metal. For square spirals, dedicated modified-Wheeler expressions give better accuracy. For anything critical, such as an RF tank circuit or NFC antenna, verify the final coil with an LCR meter or VNA and trim the turn count.
How do I design a coil for a target inductance?
Since \(L\) scales with \(N^2\), start from any prototype coil and scale the turns: \(N_{new} = N_{old}\sqrt{L_{target}/L_{old}}\), then round and re-check with the calculator. Alternatively, fix the diameters and sweep turn counts until you hit the target. Changing the diameters has a weaker, roughly linear-to-quadratic effect, so adjusting turns is usually the fastest way to converge on the value you need.
What are common mistakes when entering coil dimensions?
The most frequent error is entering radii instead of diameters — always measure across the full center-to-center width of the winding, edge to edge of the conductor path. Also make sure both values are in millimeters, not inches or centimeters, since the constant 4.921 assumes mm in and nH out. Finally, count only complete spiral turns: a 4.5-turn spiral should be entered as about 4, not rounded up to 5, because the formula squares \(N\) and errors amplify quickly.
Does this formula work for cylindrical (helical) air core coils?
No. This Wheeler variant is specifically for flat planar spirals, where all turns lie in one plane. For a solenoid wound on a cylindrical former, use the helical air core formula instead, which takes coil length, diameter, and turn count. Both are air core and both scale with \(N^2\), but the geometry terms differ, so swapping formulas can produce errors of tens of percent or more.