How to Compute Inductance: Core Formulas, Toroid Design, and Real-World Coil Math

Inductance is a circuit's inherent resistance to changes in electrical current, measured in henries (H), caused by the magnetic field generated when current flows through a conductor. When you are designing a switching power supply, an RF filter, or a custom sensor, you cannot rely on off-the-shelf parts alone; you must understand how to compute inductance from physical geometry and core material properties. This guide strips away the abstract physics and gives you the exact formulas, a real-world toroid winding example, and the practical benchmarks you need on the bench.

The Core Formulas to Compute Inductance

The inductance of a coil is dictated by four physical factors: the number of turns of wire (N), the cross-sectional area of the core (A), the length of the coil (l), and the magnetic permeability of the core material (μ). Before you pick up your magnet wire, you need to select the right geometric formula. Below is the reference table for the most common coil geometries you will encounter in power electronics and RF design.

Worked Numeric Example: Designing a 47μH Buck Converter Inductor

Let's move from theory to the workbench. Suppose you are building a custom 12V-to-5V buck converter running at 100 kHz, and your control loop calculations dictate a 47 μH inductor. You have chosen a standard Micrometals T50-2 (red) iron powder toroid core because its distributed air gap prevents hard saturation at high DC currents.

Step 1: Identify the Core's A_L Value

According to the manufacturer datasheet, the T50-2 material (Material -2) has an A_L value of 49 nH/N² (which is equivalent to 4.9 μH per 100 turns squared).

Step 2: Compute the Required Turns

Rearranging the toroid formula L = N² × A_L to solve for N:

• N = √(L / A_L)
• Convert target inductance to nanohenries: 47 μH = 47,000 nH
• N = √(47,000 nH / 49 nH/N²)
• N = √(959.18)
• N ≈ 30.97 turns

You will wind exactly 31 turns of wire around the core.

Step 3: Select Wire Gauge and Check Fill Factor

Your buck converter outputs 3A continuous. At 100 kHz, skin effect begins to matter, but for 3A, standard 24 AWG enameled magnet wire (rated for ~3.5A at 60°C temperature rise) is sufficient. The T50 core has an inner diameter of 7.62mm. Thirty-one turns of 24 AWG (0.51mm bare diameter, ~0.56mm with enamel) will easily pass through the center window multiple times without jamming, keeping your fill factor well below the practical 40% limit for toroidal winding.

Where You Meet Inductance in Practice

Understanding how to compute inductance is only half the battle; you must understand what that inductance actually changes in a live circuit. In practical installations and PCB layouts, inductance dictates three primary behaviors:

1. Energy Storage in Switching Regulators: In a buck, boost, or flyback converter, the inductor acts like a mechanical flywheel. It stores energy in its magnetic field during the MOSFET's 'on' time and releases it to the load during the 'off' time. If your computed inductance is too low, the converter enters discontinuous conduction mode (DCM), causing massive output voltage ripple and potential overvoltage faults on light loads.
2. Phase Shift and Reactive Power: In AC circuits, inductance causes the current waveform to lag the voltage waveform by up to 90 degrees. This phase shift creates reactive power (VARs). In industrial motor installations, uncorrected inductive loads drag down the power factor, leading to utility penalties and undersized upstream transformers.
3. EMI Filtering and Chokes: Common-mode chokes rely on high inductance to block high-frequency noise while passing DC or 50/60Hz power. The impedance of an inductor scales linearly with frequency (X_L = 2πfL), meaning a 10mH choke presents negligible resistance to 60Hz mains but acts as a massive roadblock to 10MHz switching noise.

What People Commonly Confuse It With

On the bench, misidentifying these three parameters will lead to blown components and failed inspections:

• Inductance (L) vs. Inductive Reactance (X_L): Inductance is the physical property of the coil (measured in Henries). Reactance is the effective AC resistance that results from that inductance at a specific frequency (measured in Ohms). A 10μH inductor has the same inductance at 1kHz and 1MHz, but its reactance is 100 times higher at 1MHz.
• Inductance vs. DC Resistance (DCR): Every real inductor is made of wire, which has resistance. DCR is the parasitic series resistance (measured in milliohms) that causes I²R heat losses. When measuring a coil with a standard multimeter, you are reading the DCR, not the inductance. You must use an LCR meter to measure inductance.
• Self-Inductance vs. Mutual Inductance: Self-inductance (L) is a single coil's reaction to its own changing current. Mutual inductance (M) is the magnetic coupling between two separate coils, which is the governing principle behind transformers and wireless power transfer.

Practical Measurement and Common Pitfalls

Once you have computed and wound your inductor, you must verify it. Here are the most frequent questions and failure modes encountered when validating custom magnetics.

Why does my LCR meter read a different value than my calculation?

Inductance is not a static number; it varies with test frequency and AC voltage amplitude. If you compute an inductance based on a ferrite core's initial permeability, but measure it with a benchtop LCR meter at 1V / 100kHz, the core's complex permeability might yield a reading 10-15% off. Always set your LCR meter test frequency to match the actual operating frequency of your circuit (e.g., 100kHz for a switching regulator, 1kHz for an audio crossover).

Does the physical arrangement of the turns matter on a toroid?

Yes. To minimize parasitic capacitance (which lowers the self-resonant frequency), wind the turns as tightly and evenly as possible, covering exactly one full layer around the core. If you must stack turns, wind them in a single direction and avoid crossing wires over each other, which creates localized hot spots and increases high-frequency AC losses due to the proximity effect.

How do I prevent core saturation in high-current DC circuits?

Saturation occurs when the magnetic flux density (B) exceeds the core material's limit, causing the inductance to plummet toward zero and the DCR to act as a dead short across your power supply. To prevent this:

• Use iron powder cores (like the -2 or -26 materials) for high DC bias applications; their distributed air gaps provide a soft, predictable roll-off.
• If using ferrite cores (like 3C90 or N87), you must physically grind an air gap into the center leg of an E-core, or purchase pre-gapped versions. An ungapped ferrite core will saturate at just a few hundred milliamps of DC current.
• Always calculate the peak current (I_peak = I_DC + ΔI/2) and verify it against the manufacturer's B-H curve or saturation current (I_sat) specification.