Inductor Equations- Electronics Formulas

The Core Inductor Equations You Actually Need

Inductors are deceptively simple components. A coil of wire, right? Wrong. They're frequency-dependent energy storage devices that obey some very specific math. Get these equations wrong and your circuit either oscillates wildly or does nothing at all.

Defining Inductance

The fundamental equation:

V = L × (dI/dt)

Where:

• V = Voltage across the inductor (volts)
• L = Inductance (henries)
• dI/dt = Rate of current change (amperes/second)

This is Faraday's Law in action. A changing current creates a opposing voltage. The bigger the coil, the bigger the L, the bigger the voltage spike when you try to stop current flow fast.

Calculating Inductance from Physical Properties

For a solenoid inductor:

L = (μ₀ × N² × A) / l

Where:

• μ₀ = Permeability of free space = 4π × 10⁻⁷ H/m
• N = Number of turns
• A = Cross-sectional area (m²)
• l = Coil length (m)

If you add a magnetic core, multiply by the core's relative permeability (μr). Ferrite cores with μr of 1000+ turn a tiny coil into a serious inductor.

Energy Stored in an Inductor

Inductors store energy in their magnetic field:

W = ½ × L × I²

Where:

• W = Energy (joules)
• L = Inductance (H)
• I = Current (A)

A 10mH inductor carrying 2A stores 0.02 joules. That doesn't sound like much until you open the circuit and it all dumps into your switch contacts as an arc.

Inductive Reactance

In AC circuits, inductors resist current flow based on frequency:

X_L = 2πfL

Where:

• X_L = Inductive reactance (ohms)
• f = Frequency (Hz)
• L = Inductance (H)

At DC (f=0), reactance is zero. At high frequencies, it shoots up toward infinity. This is why inductors make decent RF chokes — they block high frequencies while passing DC.

Complex Impedance

In AC analysis, inductors have complex impedance:

Z_L = jωL

Where ω = 2πf and j = √(-1)

The "j" accounts for the 90° phase shift between voltage and current. Voltage leads current by 90° in a pure inductor. Remember this or your phasor analysis will be backwards.

Time Constants and Transient Response

Inductor charging and discharging follow predictable exponential curves:

τ = L / R

Where τ = Time constant (seconds)

Current During Charging

I(t) = I_max × (1 - e^(-t/τ))

Current During Discharge

I(t) = I_0 × e^(-t/τ)

After 5τ, the circuit is essentially at steady state (99% of final value). A 10mH inductor with 100Ω series resistance has τ = 0.1ms. It reaches steady state in 0.5ms.

Series and Parallel Combinations

Series Connection

L_total = L₁ + L₂ + L₃ + ...

Just add them up. The magnetic fields add constructively. Total inductance increases.

Parallel Connection

1/L_total = 1/L₁ + 1/L₂ + 1/L₃ + ...

Same formula as resistors. Two 10mH inductors in parallel give 5mH total. Current splits between parallel branches.

Quality Factor (Q)

Q measures inductor efficiency:

Q = ωL / R_ac

Where R_ac is the AC resistance at the operating frequency.

• High Q = Low losses = Sharp resonance
• Low Q = High losses = Broad, damped response
• Air core inductors: Q of 100-500
• Ferrite core inductors: Q of 10-50

Skin effect and proximity effect increase R_ac at high frequencies, killing your Q. This is why HF inductors often use litz wire — many thin strands twisted together to reduce skin effect.

Mutual Inductance and Transformers

When magnetic fields from one coil link to another:

V₁/V₂ = N₁/N₂ (ideal transformer)

M = k√(L₁ × L₂)

Where k = Coupling coefficient (0 to 1)

• k = 1: Perfect coupling (rare in practice)
• k < 0.5: Loose coupling
• k > 0.9: Tight coupling (toroidal transformers)

Comparing Common Inductor Types

Getting Started: Calculating Your First Inductor

Step 1: Identify your requirements

• What current will flow through it?
• What frequency are you working at?
• Do you need high Q or just energy storage?

Step 2: Choose your core type

• Power supply filtering → Iron or ferrite core
• RF circuits → Air core or ceramic core
• DC-DC converters → Powdered iron or ferrite

Step 3: Calculate turns needed

For a ferrite toroid with Al value of 3000nH/turn²:

N = √(L / Al)

Need 10mH? N = √(10,000,000 / 3000) = 58 turns

Step 4: Check wire size

Current density of ~500 circular mils per ampere keeps heating reasonable. 1A needs ~500cmil wire (AWG 20). 3A needs ~1500cmil (AWG 14).

Step 5: Verify it fits

58 turns of AWG 20 on a small toroid might not physically fit. Either use a larger core or thinner wire (if you can tolerate the resistance increase).

Common Mistakes That Will Burn You

• Ignoring DC resistance — Your "10mH" inductor might have 2Ω of DC resistance, completely changing your circuit behavior
• Forgetting saturation current — Push too much current through an iron core inductor and L drops dramatically
• Using DC resistance for Q calculations — Use AC resistance at your actual operating frequency
• Assuming ideal coupling in transformers — Real transformers have leakage inductance that kills performance
• Mixing up reactance and resistance — They look similar in Ohm's Law but behave completely differently

Quick Reference Equations

Bookmark this page. You'll be back.