How to Calculate Inductance: Solenoid Formula, Worked Examples, and Core Selection

The fundamental formula to calculate inductance for a standard solenoid is L = (μ × N² × A) / l. This equation bridges the physical geometry of your coil with the magnetic properties of its core, allowing you to design custom chokes, transformers, and filter inductors from scratch. Below, we break down the exact mathematics, track units through real-world bench problems, and provide a concrete decision path for sourcing power inductors.

The Core Formula to Calculate Inductance

For a long, tightly wound solenoid, the inductance is determined by the physical dimensions of the coil and the magnetic permeability of the material inside it. The governing equation is:

L = (μ × N² × A) / l

Table 1: Symbol Definitions and Standard SI Units

Symbol	Parameter	SI Unit	Typical Bench Unit
L	Inductance	Henrys (H)	μH, mH, nH
μ	Absolute Permeability of the core (μ₀ × μᵣ)	Henrys per meter (H/m)	N/A
N	Number of turns of wire	Unitless (count)	Turns
A	Cross-sectional area of the core	Square meters (m²)	mm², cm²
l	Length of the magnetic path / coil	Meters (m)	mm, cm

When This Formula Applies (and Its Assumptions)

• Uniform Magnetic Field: Assumes the coil is a 'long solenoid' where the length l is significantly greater than the square root of the area A (typically l > 10× diameter). Fringing flux at the ends is ignored.
• Linear Core Material: Assumes the core permeability (μ) is constant. In reality, ferromagnetic cores saturate at high currents, causing μ to drop and inductance to roll off. This formula calculates the unsaturated (small-signal) inductance.
• Tightly Wound: Assumes the wire diameter is negligible compared to the coil length, meaning all turns occupy roughly the same geometric space.

Rearranged Forms: Solving for Turns, Area, and Core Length

On the bench, you rarely solve for L directly; you usually have a target inductance and need to figure out how many turns to wind on a specific core. Here are the algebraic rearrangements solving for each variable:

• Solve for Turns (N): N = √( (L × l) / (μ × A) )
• Solve for Cross-Sectional Area (A): A = (L × l) / (μ × N²)
• Solve for Magnetic Path Length (l): l = (μ × N² × A) / L
• Solve for Absolute Permeability (μ): μ = (L × l) / (N² × A)

Worked Examples with Strict Unit Tracking

The most common point of failure in inductor design is unit mismatch. Below are two solved problems demonstrating strict SI unit tracking.

Problem 1: Air-Core RF Choke

Goal: Calculate the number of turns needed to achieve 100 nH on a non-magnetic (air) form with a 4 mm diameter and a 10 mm winding length.

1. Convert all inputs to base SI units:
   • L = 100 nH = 100 × 10⁻⁹ H
   • l = 10 mm = 0.01 m
   • Diameter = 4 mm → Radius (r) = 2 mm = 0.002 m
   • A = π × r² = π × (0.002)² = 1.2566 × 10⁻⁵ m²
   • μ (air) = μ₀ = 4π × 10⁻⁷ H/m ≈ 1.2566 × 10⁻⁶ H/m
2. Apply the rearranged formula for N:
   • N = √( (L × l) / (μ × A) )
   • Numerator: (100 × 10⁻⁹ H) × (0.01 m) = 1.0 × 10⁻⁹ H·m
   • Denominator: (1.2566 × 10⁻⁶ H/m) × (1.2566 × 10⁻⁵ m²) = 1.579 × 10⁻¹¹ H·m
3. Calculate final value:
   • N = √( 1.0 × 10⁻⁹ / 1.579 × 10⁻¹¹ ) = √( 63.33 ) = 7.95 turns
4. Result: Wind 8 turns. In RF applications, you would stretch or compress these 8 turns slightly while monitoring on a VNA or grid dip meter to hit exactly 100 nH.

Problem 2: Ferrite Rod Audio Crossover Inductor

Goal: Find the turns required for a 4.7 mH inductor using a ferrite rod with a relative permeability (μᵣ) of 2000, a cross-sectional area of 50 mm², and a magnetic path length of 20 mm.

1. Convert to SI and calculate absolute permeability (μ):
   • L = 4.7 mH = 4.7 × 10⁻³ H
   • l = 20 mm = 0.02 m
   • A = 50 mm² = 50 × 10⁻⁶ m²
   • μ = μ₀ × μᵣ = (4π × 10⁻⁷) × 2000 = 2.513 × 10⁻³ H/m
2. Apply the formula for N:
   • Numerator: (4.7 × 10⁻³ H) × (0.02 m) = 9.4 × 10⁻⁵ H·m
   • Denominator: (2.513 × 10⁻³ H/m) × (50 × 10⁻⁶ m²) = 1.2565 × 10⁻⁷ H·m
3. Calculate final value:
   • N = √( 9.4 × 10⁻⁵ / 1.2565 × 10⁻⁷ ) = √( 748.1 ) = 27.35 turns
4. Result: Wind 27 turns of heavy-gauge enameled copper wire to minimize DCR for the audio crossover.

Unit Traps That Will Break Your Calculation

Decision Path: Selecting a 47µH Buck Inductor

When designing a switching power supply (e.g., a 12V to 5V buck converter), calculating the physical winding is only half the battle. You must select a core material that won't saturate at your peak current or overheat at your switching frequency. Use this decision tree to specify your part.

Condition / Constraint	Material Choice	Action / Next Step
Switching Frequency (f_sw) > 500 kHz	Manganese-Zinc Ferrite (e.g., N87, N97)	Ferrites have low core loss at high frequencies but sharp saturation curves. Proceed to gap calculation.
Switching Frequency (f_sw) < 150 kHz	Iron Powder (e.g., Micrometals -2 or -26)	Iron powder handles high DC bias without sharp saturation but suffers high core loss above 200 kHz.
Peak Current > 10A AND Space is constrained	Metal Alloy Powder (e.g., Sendust, Kool Mµ)	Offers the best DC bias characteristics and high saturation flux density in a small footprint.
Design Time is limited / Prototyping phase	Off-the-shelf Shielded SMD	Skip custom winding. Select a pre-gapped, tested component with guaranteed saturation specs.

The Concrete Pick: If you are building a modern high-frequency buck converter (500kHz to 2MHz) utilizing GaN or SiC FETs and need a reliable 47µH inductor that handles 10A+ without saturating, do not wind your own. The parasitic capacitance of hand-wound coils will destroy your high-frequency efficiency.

Default Recommendation: Specify the Coilcraft XEL5030-473MEB. It is a 47µH shielded metal-alloy SMD inductor rated for 11.5A saturation current (Isat) and 11.8A RMS current (Irms), optimized for the high di/dt edges of modern wide-bandgap semiconductors. You can verify its exact impedance curve using the Coilcraft Inductor Finder tool.

Realistic Magnitudes and Bench Verification

Knowing what a 'normal' answer looks like prevents you from chasing decimal errors. According to standard electromagnetic theory references, inductance scales exponentially with turns, meaning small geometry changes yield massive value shifts.

• Nanohenrys (nH): 1 nH to 900 nH. Typical for RF matching networks, VHF/UHF chokes, and PCB trace inductance. A 1-inch straight piece of hookup wire has roughly 20 nH of parasitic inductance.
• Microhenrys (µH): 1 µH to 900 µH. The standard domain for switch-mode power supplies (buck/boost), EMI filtering, and motor drive chokes. A 47µH or 100µH inductor is the most common bench stock item.
• Millihenrys (mH): 1 mH to 100+ mH. Found in audio crossovers, 50/60Hz line filters, and low-frequency ballasts. Requires high-permeability cores (ferrite or laminated silicon steel) and hundreds of turns.

Verification on the Bench

Never trust a hand calculation blindly. Core permeability varies by ±20% from batch to batch. Always verify your wound inductor using an LCR meter. However, test at the correct frequency. If you measure a 47µH power inductor at 1 kHz, you might read 55µH due to the core's low-frequency permeability peak. Measure it at your circuit's actual switching frequency (e.g., 100 kHz or 1 MHz) to see the true operational inductance. For high-current applications, you must also use a DC bias fixture to measure how much the inductance drops when your actual operating current is applied, referencing core material datasheets from manufacturers like Micrometals to predict the roll-off.