Inductor math is the applied physics of calculating inductance, reactance, energy storage, and time constants to select or design the correct magnetic component for a circuit. Whether you are sizing a choke for a 5V buck converter or calculating the cutoff frequency of an RL filter, the math dictates the physical geometry, core material, and wire gauge you need. Relying on guesswork leads to saturated cores, overheated windings, and destroyed switching MOSFETs.
The Core Inductor Math Equations
Before selecting a physical part from a catalog like Coilcraft's design tools, you must define the electrical requirements using four foundational equations. Assume standard copper windings and a 20°C ambient baseline unless otherwise noted.
1. Inductance of a Coil (Physics)
The base inductance ($L$) in Henries is determined by the core geometry and material:
$L = \frac{N^2 \cdot \mu \cdot A}{l}$
Where $N$ is the number of turns, $\mu$ is the core permeability, $A$ is the cross-sectional area, and $l$ is the magnetic path length. This equation matters when you are winding your own toroids or debugging why a custom transformer is missing its target value.
2. Inductive Reactance (AC Circuits)
In AC or switching circuits, an inductor opposes changes in current. The reactance ($X_L$) in Ohms is:
$X_L = 2 \pi f L$
Worked Example: A 10µH inductor in a 100kHz switching regulator presents a reactance of $2 \cdot \pi \cdot 100,000 \cdot 0.000010 = 6.28 \Omega$. This limits the AC ripple current flowing through the coil.
3. Energy Storage (SMPS and Converters)
Switch-mode power supplies rely on inductors to store and transfer energy. The energy ($E$) in Joules is:
$E = \frac{1}{2} L I^2$
Worked Example: If a 4.7µH inductor carries a peak current of 5A, it stores $0.5 \cdot 0.0000047 \cdot 25 = 58.75 \mu J$ per cycle. If the core cannot physically hold this energy without saturating, the inductance collapses.
4. RL Time Constant (Filters and Snubbers)
The time ($\tau$) it takes for current to reach 63.2% of its final value in a DC RL circuit is:
$\tau = \frac{L}{R}$
This is critical for designing snubber networks where you need to dampen voltage spikes within a specific microsecond window.
Physical Inductor Types and Selection Criteria
Translating your math into a bill of materials (BOM) requires choosing the right physical construction. Here is how the primary types compare based on real-world bench characteristics.
| Type | Construction | Tolerance | Tempco | Typical Use Case |
|---|---|---|---|---|
| Shielded Drum Core | Ferrite drum with magnetic shield can | ±20% | Low drift | High-density SMPS, buck/boost converters (e.g., Coilcraft MSS1210) |
| Unshielded Bobbin | Open ferrite bobbin, exposed windings | ±20% | Moderate | Low-cost, low-current filtering where EMI is not critical |
| Toroidal (Iron Powder) | Wire wound on a continuous ring core | ±10% | Highly stable | High-current DC chokes, EMI filtering, audio crossovers |
| Air Core | Wire wound on non-magnetic ceramic/plastic | ±5% | Near zero | RF tuning, high-frequency resonant circuits (no saturation risk) |
Which type for which job? Use shielded drum cores for modern switching regulators to prevent magnetic flux from coupling into nearby sensitive traces. Use toroids for high-current, low-frequency DC filtering where you need to avoid saturation at high amp loads. Reserve air cores strictly for RF applications above 10MHz where core losses would destroy efficiency.
Decoding Inductor Markings and SMD Codes
Unlike resistors, SMD inductors do not use standard color bands. Reading the laser-etched markings on the top of the component is essential for verifying your BOM during assembly. The industry standard uses a three-digit or four-digit code based on microhenries (µH).
- Three-Digit Code: The first two digits are the significant figures, and the third digit is the multiplier (number of zeros). Example:
470means 47 followed by zero zeros = 47µH.101means 10 followed by one zero = 100µH. - The 'R' Decimal Indicator: When the value is less than 10µH, the letter 'R' acts as the decimal point. Example:
4R7= 4.7µH.R22= 0.22µH. - Four-Digit Code (Precision): Used for tight-tolerance RF inductors. Example:
1001= 1000nH (or 1.0µH).
Always verify the physical size against the datasheet. A 100 (10µH) marking on a 0402 package will have a vastly different saturation current and DCR (DC Resistance) than the same marking on a 1210 package.
Safe Substitution: When the Exact Part is Missing
Supply chain shortages frequently force engineers to substitute inductors. Swapping a Würth Elektronik part for a TDK equivalent is fine, but you cannot just match the inductance value. Follow these four rules to substitute safely:
- Inductance Tolerance: The substitute must be within ±20% of the original target value. Dropping from 4.7µH to 3.3µH in a buck converter will increase your ripple current, potentially tripping the IC's overcurrent protection.
- Saturation Current ($I_{sat}$): This is the most critical parameter. $I_{sat}$ is the current at which inductance drops by 20% to 30%. Your substitute's $I_{sat}$ must exceed the peak inductor current of your circuit by at least a 20% safety margin.
- DC Resistance (DCR): Choose a part with an equal or lower DCR. A higher DCR will increase $I^2R$ copper losses, causing the part to run hotter and reducing overall converter efficiency.
- Self-Resonant Frequency (SRF): The SRF of the substitute must be at least 10 times higher than your circuit's switching frequency. If the SRF is too low, the parasitic capacitance of the coil will dominate, and it will act like a capacitor instead of an inductor.
Never substitute flyback or forward transformer inductors based solely on primary inductance math. The turns ratio, interleaving, and leakage inductance dictate the voltage spikes on the switching node. A mismatched substitute will cause avalanche breakdown of the primary MOSFET.
Failure Modes and Visual Diagnostics
Inductors are generally robust, but they do fail when pushed beyond their physical limits. According to foundational texts like All About Circuits, magnetic components fail in predictable ways. Here is what to look for on the bench:
- Core Saturation (Thermal Runaway of the FET):
Symptom: The inductor looks perfectly fine, but the switching MOSFET is charred or shorted. Cause: Peak current exceeded $I_{sat}$, causing inductance to drop to near zero. The inductor acted as a short wire, dumping massive current through the FET. - Thermal Overload (Copper Loss):
Symptom: Discolored epoxy coating, melted solder joints at the pads, or a distinct burnt-varnish smell. Cause: RMS current exceeded the part's thermal rating, or the DCR was too high for the continuous load. - Insulation Breakdown (High dV/dt):
Symptom: Visible arcing marks between windings, or a hairline crack in the ferrite core material. Cause: Extreme voltage spikes (common in poorly snubbed relay coils or flyback circuits) punctured the thin enamel insulation on the magnet wire, creating a shorted turn that destroys the inductance.
Inductor Math FAQ
How do I calculate the inductor math for a buck converter?
To size an inductor for a buck converter, use the volt-second balance equation: $L = \frac{V_{out} \cdot (V_{in} - V_{out})}{V_{in} \cdot f_{sw} \cdot \Delta I_L}$. First, define your target ripple current ($\Delta I_L$), which is typically 20% to 30% of your maximum DC load current. Plug in your maximum input voltage ($V_{in}$), output voltage ($V_{out}$), and the IC's switching frequency ($f_{sw}$). The result gives you the minimum required inductance in Henries. Always round up to the nearest standard value and verify the saturation current rating.
Does inductor math change at high frequencies due to skin effect?
Yes. The standard DC resistance (DCR) math becomes inaccurate above 100kHz due to the skin effect and proximity effect. At high frequencies, AC current flows only on the outer 'skin' of the copper wire, effectively reducing the cross-sectional area and increasing the AC resistance ($R_{AC}$). For high-frequency math, you must use Litz wire (multiple individually insulated thin strands) to mitigate this, and consult the manufacturer's core loss curves (usually measured in $mW/cm^3$) to calculate total thermal dissipation.
How do I measure unknown inductance without an LCR meter?
If you lack a dedicated LCR meter, you can use the RL time constant method. Wire the unknown inductor in series with a known, precise power resistor (e.g., 100Ω, 1%). Apply a square wave from a function generator and measure the voltage across the resistor using an oscilloscope. Measure the time ($\tau$) it takes for the voltage to rise to 63.2% of its peak value. Since $\tau = L / R$, you can rearrange the math to solve for inductance: $L = \tau \cdot R$. Ensure your square wave frequency is low enough to allow the current to fully settle before the next edge.
