The Core Inductance Formula and Symbol Definitions

When you need to wind your own choke, design an RF matching network, or build a custom SMPS transformer, you cannot rely on guessing. The fundamental equation for calculating the inductance of a long solenoid or toroidal coil is derived from Ampere’s and Faraday’s laws. Before we plug in numbers, here is the master formula:

L = (μ0 × μr × N2 × A) / l

This formula applies under specific assumptions: the magnetic field must be relatively uniform inside the coil, and the coil's length should ideally be much greater than its diameter (the "long solenoid" approximation). For short, stubby air-core coils, Wheeler’s empirical approximations are more accurate, but for toroids and iron-core chokes, the equation above is your daily driver.

Table 1: Inductance Formula Symbol Definitions
Symbol Parameter Standard Unit Notes & Constants
L Inductance Henries (H) The target value. 1 H is massive; most bench work is in μH or mH.
μ0 Vacuum Permeability T·m/A or H/m Constant: 4π × 10-7 (approx. 1.2566 × 10-6).
μr Relative Permeability Dimensionless Core material multiplier. Air = 1. Ferrite = 20 to 10,000+.
N Number of Turns Dimensionless Total loops of wire. Note that N is squared in the formula.
A Cross-Sectional Area Square meters (m2) Area of the core or coil form. Must be converted from cm2 or mm2.
l Magnetic Path Length Meters (m) Length of the coil or the effective magnetic path (le) of a toroid.

Rearranged Forms: Solving for Turns, Area, and Length

On the bench, you rarely solve for L from scratch. Usually, you know the inductance you need (e.g., a 47 μH buck converter inductor) and the core you have in your parts bin, and you need to find N. Here are the algebraically rearranged forms of the master equation:

  • Solve for Turns (N): N = √((L × l) / (μ0 × μr × A))
  • Solve for Area (A): A = (L × l) / (μ0 × μr × N2)
  • Solve for Length (l): l = (μ0 × μr × N2 × A) / L
  • Solve for Core Permeability (μr): μr = (L × l) / (μ0 × N2 × A)

Worked Examples with Strict Unit Tracking

The most common reason hobbyists get garbage numbers from this formula is failing to convert millimeters and centimeters into base SI units (meters). Let’s walk through two distinct builds with rigorous unit tracking.

Problem 1: Air-Core RF Choke

Scenario: You are winding an air-core solenoid for an AM radio antenna matching network. You use a 10 mm diameter plastic form, wind 20 turns of enameled copper wire tightly, resulting in a coil length of 30 mm. What is the inductance?

  1. Identify Knowns & Convert:
    N = 20
    Radius (r) = 5 mm = 0.005 m
    Length (l) = 30 mm = 0.03 m
    μr = 1 (air/plastic)
  2. Calculate Area (A):
    A = π × r2 = 3.14159 × (0.005 m)2 = 7.854 × 10-5 m2
  3. Calculate Numerator (μ0 × μr × N2 × A):
    (1.2566 × 10-6) × 1 × (202) × (7.854 × 10-5)
    = (1.2566 × 10-6) × 400 × (7.854 × 10-5) = 3.947 × 10-8
  4. Divide by Length (l):
    L = 3.947 × 10-8 / 0.03 = 1.315 × 10-6 H
  5. Final Answer: 1.32 μH

Problem 2: Iron Powder Toroid for EMI Filtering

Scenario: You need a differential mode choke. You select a Micrometals T50-26 iron powder toroid (μr = 75). The datasheet lists an effective area (Ae) of 1.33 cm2 and an effective magnetic path length (le) of 6.35 cm. You wind 45 turns of 22 AWG wire.

  1. Convert to Base Units:
    A = 1.33 cm2 = 1.33 × 10-4 m2 (Remember: 1 cm2 = 10-4 m2)
    l = 6.35 cm = 0.0635 m
    N = 45 (N2 = 2025)
  2. Calculate Numerator:
    (1.2566 × 10-6) × 75 × 2025 × (1.33 × 10-4) = 2.536 × 10-5
  3. Divide by Path Length:
    L = 2.536 × 10-5 / 0.0635 = 3.99 × 10-4 H
  4. Final Answer: 399 μH

Real-World Bench Scenario: The Buck Converter Inductor Trap

Math on paper is clean; the workbench is not. Here is a classic failure mode that burns hobbyists and junior engineers when calculating inductance for switched-mode power supplies (SMPS).

⚠️ Bench Warning: If an SMPS inductor saturates, its inductance drops to near-zero. The current spikes uncontrollably, often destroying the switching MOSFET and taking your oscilloscope probe ground clip with it. Always verify inductance under actual DC load, not just with a small-signal LCR meter.

The Setup: I was designing a 5A continuous, 12V-to-5V buck converter. The controller required a 400 μH inductor to maintain continuous conduction mode (CCM) at light loads. I grabbed the same T50-26 toroid from Problem 2 (μr = 75) and used the rearranged formula to solve for N. I wound 45 turns, which calculated out to 399 μH.

The Numbers: I hooked the inductor up to my Keysight E4980A LCR meter. At 1 kHz and 1V RMS, the meter read 395 μH. Perfect. I soldered it into the PCB.

The Outcome: Upon applying a 5A resistive load, the output voltage ripple was massive (over 400mV peak-to-peak), the inductor was screaming audibly, and its temperature shot to 95°C in three minutes. The converter had fallen into discontinuous conduction mode (DCM).

What Went Wrong: The formula assumes μr is a static constant. In reality, magnetic cores suffer from DC bias roll-off. As DC current increases, the magnetic domains in the core align and saturate. Iron powder cores like the T50-26 are particularly susceptible. According to the Micrometals design software, at 5 Amps of DC bias, the effective permeability of that specific core drops by roughly 60%. My 395 μH inductor was actually operating at ~158 μH under load. The fix was to switch to a gapped ferrite core (like a TDK B66317 with a 1mm distributed air gap). The air gap dominates the magnetic reluctance, drastically lowering the initial μr but keeping the inductance rock-solid regardless of DC current.

Unit Mistakes That Break the Math and Realistic Magnitudes

If your calculated inuctance is off by a factor of 10,000, you almost certainly made a unit conversion error. Here are the specific traps to avoid:

  • The Area Squaring Trap: Converting cm2 to m2 requires multiplying by 10-4, not 10-2. If you just divide by 100, your calculated inductance will be 100x too high.
  • Confusing μ and μr: μ0 is the physical constant of free space. μr is the material multiplier. If you accidentally use the absolute permeability of ferrite (e.g., 0.0025) in the spot reserved for μr (which should be ~2000), your math will collapse.
  • Radius vs. Diameter: Datasheets often give core diameter. The area formula requires radius. Forgetting to halve the diameter before squaring it will inflate your area (and inductance) by a factor of 4.

What Does a Realistic Answer Look Like?

When you finish your calculation, sanity-check the magnitude against this cheat sheet. If your math says you need 5 Henries for an RF filter, you dropped a decimal somewhere.

Table 2: Realistic Inductance Magnitudes by Application
Magnitude Typical Range Common Applications Physical Form Factor
NanoHenries (nH) 1 nH – 900 nH PCB trace routing, VHF/UHF RF matching, GHz decoupling. Surface mount 0402/0603 ceramics, straight wire stubs.
MicroHenries (μH) 1 μH – 900 μH SMPS buck/boost inductors, differential EMI chokes, IF transformers. Toroids, drum cores, shielded SMD power inductors.
MilliHenries (mH) 1 mH – 900 mH Audio crossover networks, common-mode line filters, low-frequency ballasts. Large laminated iron E-cores, bulky multi-layer bobbins.
Henries (H) 1 H – 100+ H Power factor correction (PFC), massive motor starting chokes, tube amplifier plate loads. Heavy gapped steel laminations, potting compound, weighs several kilos.

For deeper theoretical derivations and the underlying magnetic circuit analogies (where Reluctance acts like Resistance), the Electronics Tutorials inductor guide provides excellent foundational reading. Remember: the formula gets you in the ballpark, but an LCR meter and a thermal camera under real load are what actually validate the design.