The inductor voltage equation, fundamentally expressed as v = L · (di / dt), defines the instantaneous voltage induced across an inductor proportional to the rate of change of current flowing through it. Unlike resistors, which drop voltage based on absolute current, inductors only generate a voltage differential when the current is actively changing. If di/dt is zero (steady DC current), the ideal inductor voltage v drops to zero, acting as a short circuit. This principle is the governing law behind switch-mode power supplies, flyback transformers, and relay snubber circuits.
The Core Inductor Voltage Equation and Symbol Definitions
To apply the formula accurately on the bench, we must distinguish between the continuous calculus form (used for AC sine waves and transient analysis) and the discrete delta form (used for calculating ripple in DC-DC switching converters).
Continuous Form: v(t) = L · (di(t) / dt)
Discrete Form: V = L · (Δi / Δt)
| Symbol | Parameter | Standard SI Unit | Practical Bench Unit | Definition & Constraints |
|---|---|---|---|---|
| v or V | Induced Voltage | Volts (V) | Volts (V) | The electromotive force (EMF) opposing the change in current. Polarity follows Lenz's Law. |
| L | Inductance | Henries (H) | Microhenries (μH) | The component's ability to store magnetic energy. Assumes operation below saturation current. |
| di or Δi | Current Change | Amperes (A) | Amperes (A) or mA | The absolute difference between final and initial current (ifinal - iinitial). |
| dt or Δt | Time Interval | Seconds (s) | Nanoseconds (ns) or μs | The duration over which the current change occurs. Smaller dt yields higher v. |
Understanding the magnitude of di/dt in real circuits is critical for component selection. Below is a data-dense breakdown of standard surface-mount power inductors and the specific rate of current change required to induce a 50V spike across them—a common threshold that triggers avalanche breakdown in 30V-rated switching MOSFETs.
| Manufacturer Part | Inductance (L) | Saturation Current (Isat) | DCR (mΩ) | di/dt to induce 50V | Typical Application |
|---|---|---|---|---|---|
| Coilcraft XEL3530-471 | 470 nH (0.47 μH) | 33.0 A | 1.1 | 106.3 A/μs | High-current multi-phase VRMs |
| Wurth 74477410 | 10 μH | 2.8 A | 56.0 | 5.0 A/μs | Standard 12V-to-5V Buck converters |
| Bourns SRP1265A-220M | 22 μH | 9.0 A | 42.0 | 2.27 A/μs | Automotive LED drivers, SEPIC |
| TDK SPM6528T-100M | 10 μH | 5.9 A | 59.0 | 5.0 A/μs | Battery-powered boost converters |
| Coilcraft MSS1210-473 | 47 μH | 4.6 A | 78.0 | 1.06 A/μs | Low-frequency offline flybacks |
Rearranged Forms, Assumptions, and Unit Traps
On the bench, you rarely solve for v directly. More often, you are selecting an inductor to limit current ripple, or calculating the required on-time for a switch. Here are the rearranged forms of the discrete equation:
- Solving for Inductance: L = (V · Δt) / Δi (Used to size an inductor for a target ripple current)
- Solving for Current Change: Δi = (V · Δt) / L (Used to calculate peak-to-peak ripple current in a steady-state converter)
- Solving for Time Interval: Δt = (L · Δi) / V (Used to determine MOSFET on-time or off-time requirements)
When the Formula Applies (and Its Assumptions)
The equation V = L · (Δi / Δt) assumes an ideal, linear inductor. This means it assumes the core material's permeability remains constant regardless of the current. In reality, ferromagnetic cores (ferrite, powdered iron) exhibit non-linear B-H curves. As current approaches the saturation limit (Isat), the effective inductance L drops drastically. If you apply this formula using the nominal L value while the core is saturated, your calculated voltage will be dangerously wrong, often leading to destroyed switching transistors.
Furthermore, the formula calculates the pure inductive EMF. It ignores the voltage drop across the inductor's Equivalent Series Resistance (ESR or DCR) and ignores parasitic parallel capacitance, which causes high-frequency ringing during rapid di/dt events.
The 'Micro-Milli' Unit Trap
The most common way engineers and hobbyists break this formula is through unit misalignment. The SI base units are Henries, Amperes, and Seconds. However, bench components are specified in μH, mA, and ns. If you multiply 10 μH by a current change of 500 mA over 20 ns without converting to base units, your calculator will output garbage. The fix: Always convert to base scientific notation before calculating. 10 μH = 10 × 10-6 H. 500 mA = 0.5 A. 20 ns = 20 × 10-9 s. According to a comprehensive guide on power magnetics by All About Circuits, failing to track these exponents is the root cause of 90% of initial prototype switching regulator failures.
Worked Examples with Strict Unit Tracking
Let's apply the formula to two common bench scenarios, tracking every unit conversion and intermediate step.
Problem 1: Calculating the Flyback Voltage Spike
Scenario: You are driving a 4.7 μH inductor with a MOSFET. The inductor is carrying 3.0 A of steady current. The MOSFET turns off, dropping the current to 0 A in 15 nanoseconds. What is the magnitude of the induced voltage spike across the inductor?
Step 1: Identify and convert variables to SI base units.
- L = 4.7 μH = 4.7 × 10-6 H
- Δi = ifinal - iinitial = 0 A - 3.0 A = -3.0 A (Magnitude is 3.0 A)
- Δt = 15 ns = 15 × 10-9 s
Step 2: Calculate the rate of change (di/dt).
- di/dt = 3.0 A / (15 × 10-9 s)
- di/dt = 200,000,000 A/s (or 200 A/μs)
Step 3: Multiply by inductance to find V.
- V = (4.7 × 10-6 H) × (200,000,000 A/s)
- V = 940 Volts
Bench Reality Check: A 940V spike will instantly punch through the drain-source junction of a standard 600V switching MOSFET. This is exactly why snubber circuits (RC networks) or clamping diodes are mandatory in flyback topologies to artificially increase Δt or clamp V.
Problem 2: Sizing an Inductor for a Buck Converter
Scenario: You are designing a buck converter stepping 12V down to 3.3V. The internal MOSFET switch turns on for 2.5 μs. You want to limit the peak-to-peak inductor current ripple (Δi) to 0.8 A to maintain continuous conduction mode (CCM). What inductance value do you need?
Step 1: Identify the voltage across the inductor during the on-time.
When the high-side switch is ON, the voltage at the switch node is Vin (12V). The output is 3.3V. Therefore, the voltage across the inductor is:
- VL = Vin - Vout = 12V - 3.3V = 8.7 V
Step 2: Identify and convert remaining variables.
- Δt = 2.5 μs = 2.5 × 10-6 s
- Δi = 0.8 A
Step 3: Rearrange formula to solve for L and calculate.
- L = (VL · Δt) / Δi
- L = (8.7 V × 2.5 × 10-6 s) / 0.8 A
- L = (21.75 × 10-6) / 0.8
- L = 27.1875 × 10-6 H
- L ≈ 27.2 μH
Bench Reality Check: 27.2 μH is not a standard E-series value. You would select the next standard value up, such as 33 μH, which would slightly reduce your ripple current to ~0.66 A, improving efficiency at the cost of a slightly larger physical footprint and higher DCR. For deeper insights into standard component selection, refer to the Electronics Tutorials inductor guide.
Realistic Magnitudes and Bench Reality
When measuring the inductor voltage equation in practice, the numbers on your oscilloscope rarely match the pure theoretical calculation perfectly due to parasitics. Here is what realistic magnitudes look like and how to interpret them.
Parasitic Ringing and Capacitance
If you calculate a 50V spike using v = L(di/dt), your oscilloscope might show the voltage ringing up to 80V or 100V before settling. This is not a failure of the equation; it is the result of the inductor's parasitic parallel capacitance interacting with the inductance to form an LC tank circuit. The initial di/dt injects energy into this parasitic capacitance. To measure this accurately on the bench, you must use a low-inductance ground spring on your scope probe rather than the standard 6-inch alligator ground clip, which will artificially add nanohenries of inductance to your measurement loop and exaggerate the ringing.
The Saturation Cliff
The most dangerous edge case for this formula is core saturation. If you push 15A through an inductor rated for 10A saturation, the ferrite core can no longer support additional magnetic flux. The relative permeability drops toward 1 (the permeability of air). If L drops from 10 μH to 0.05 μH during saturation, and your circuit attempts to force a current change, the di/dt term will skyrocket to maintain the energy balance. This massive, uncontrolled spike in di/dt manifests as a near-instantaneous current surge that exceeds the Safe Operating Area (SOA) of your driving MOSFET, resulting in a shorted drain-source junction and a dead board. Always design your Δi such that the absolute peak current (Iload + Δi/2) remains at least 20% below the manufacturer's Isat specification at maximum operating temperature.
