The fundamental inductor voltage formula is V = L × (di / dt). This equation dictates that the voltage across an inductor is directly proportional to its inductance and the rate at which the current through it changes over time. If the current is stable, the induced voltage is zero; if the current changes instantly, the theoretical voltage approaches infinity. Below, we break down the derivation, rearrange the formula for practical circuit design, and solve real-world engineering problems with strict unit tracking.

The Core Inductor Voltage Formula and Symbol Definitions

The relationship between voltage and current in an inductor is defined by Faraday's Law of Induction, simplified for a discrete component as:

V = L × (di / dt)

To use this formula correctly on the bench or in simulation, every variable must be mapped to its base SI unit. Mixing practical units (like milliHenries and microseconds) without conversion is the primary cause of calculation errors in power electronics.

Symbol Parameter Standard SI Unit Common Practical Unit
V (or vL) Induced Voltage (Electromotive Force) Volts (V) Millivolts (mV), Kilovolts (kV)
L Inductance Henries (H) MilliHenries (mH), MicroHenries (µH)
di Change in Current (ΔI) Amperes (A) MilliAmperes (mA)
dt Change in Time (Δt) Seconds (s) Milliseconds (ms), Microseconds (µs)
Critical Unit Trap: The most common way engineers destroy switching MOSFETs on the bench is by failing to convert practical units to base SI units. If you multiply 50 mH by (200 mA / 5 µs) directly in a calculator without converting to Henries, Amperes, and Seconds, your output will be off by a factor of 1,000,000. Always convert to base SI units (H, A, s, V) before executing the math.

Rearranged Forms for Circuit Design

While finding the voltage spike is common for snubber design, power electronics engineers frequently need to rearrange the inductor voltage formula to size components for DC-DC converters. According to Electronics Tutorials, understanding these algebraic variations is essential for selecting the right magnetics.

  • Solving for Inductance (L): L = (V × dt) / di
    Use case: Sizing the output inductor for a buck or boost converter when you know your allowable current ripple (di), switching frequency (which gives dt), and input/output voltages.
  • Solving for Current Ripple (di): di = (V × dt) / L
    Use case: Determining the peak-to-peak ripple current in an existing power supply to ensure the inductor won't saturate and the output capacitor can handle the RMS ripple.
  • Solving for Time (dt): dt = (L × di) / V
    Use case: Calculating the required on-time for a switch to ramp an inductor's current to a specific peak threshold, commonly used in peak-current-mode control loops.

Worked Examples with Strict Unit Tracking

Let's apply the formula to two distinct, real-world scenarios: a destructive transient spike and a controlled steady-state switching ripple.

Problem 1: Calculating a Relay Flyback Voltage Spike

Scenario: A 12V automotive relay coil has an inductance of 50 mH and draws a steady 200 mA. The mechanical contacts open, interrupting the current in 5 µs. What is the induced voltage spike across the coil?

  1. Identify and convert to base SI units:
    L = 50 mH = 0.050 H
    di = 200 mA - 0 A = 0.200 A
    dt = 5 µs = 0.000005 s (5 × 10-6 s)
  2. Calculate the rate of current change (di/dt):
    di / dt = 0.200 A / 0.000005 s = 40,000 A/s
  3. Apply the formula:
    V = L × (di / dt)
    V = 0.050 H × 40,000 A/s
  4. Final Result:
    V = 2,000 Volts.

Bench Insight: This 2,000V spike explains why relay contacts arc and pit over time, and why a flyback diode is mandatory when driving relay coils with solid-state transistors. Without the diode, this spike will instantly punch through the drain-source junction of a standard 60V MOSFET.

Problem 2: Sizing Current Ripple in a Buck Converter

Scenario: You are designing a buck converter. During the switch's ON time, 12V is applied across a 10 µH inductor for 2.5 µs. What is the peak-to-peak current ripple (di)?

  1. Identify and convert to base SI units:
    V = 12 V (Net voltage across the inductor during ON time)
    L = 10 µH = 0.000010 H (10 × 10-6 H)
    dt = 2.5 µs = 0.0000025 s (2.5 × 10-6 s)
  2. Rearrange the formula to solve for di:
    di = (V × dt) / L
  3. Substitute the values:
    di = (12 V × 0.0000025 s) / 0.000010 H
    di = 0.00003 / 0.000010
  4. Final Result:
    di = 3.0 Amperes.

Bench Insight: A 3A ripple means if your average load current is 5A, the inductor current swings from 3.5A to 6.5A. You must select an inductor with a saturation current rating (Isat) well above 6.5A to prevent core saturation and subsequent converter failure.

Assumptions, Edge Cases, and Realistic Magnitudes

The formula V = L(di/dt) is an idealized model. As detailed in All About Circuits, real-world magnetics introduce parasitic elements that alter the math at the extremes.

When the Formula Applies (and When it Doesn't)

  • Assumption 1: Linear Core Material. The formula assumes 'L' is a constant. In reality, ferrite and powdered iron cores saturate. As current increases, the magnetic domains align, and 'L' drops precipitously. If 'L' drops by 50% due to saturation, your 'di/dt' will double for a given voltage, leading to runaway current spikes.
  • Assumption 2: Ideal Component. The formula ignores DC Resistance (DCR) and parasitic winding capacitance. At high frequencies (fast 'dt'), the inter-winding capacitance creates a parallel resonant tank, causing high-frequency ringing that the basic derivative formula cannot predict.

What Does a Realistic Answer Magnitude Look Like?

Beginners often expect inductors to always generate massive voltages. The realistic magnitude depends entirely on the operating state:

  • DC Steady State: When current is constant, di = 0, so V = 0. The inductor acts as a short circuit. The only voltage present is the resistive drop (V = I × DCR). For a 10A load on a typical power inductor with 15mΩ DCR, the realistic voltage is just 150 mV.
  • Controlled Switching (SMPS): In a well-designed DC-DC converter, dt is controlled by the PWM controller. Voltages across the inductor typically alternate between the input voltage and the reflected output voltage, yielding realistic magnitudes of 5V to 48V.
  • Uncontrolled Transients (Flyback): When a circuit is broken mechanically or a MOSFET turns off without a clamp, dt approaches zero. Realistic magnitudes here range from 100V to over 10,000V, limited only by parasitic capacitance and the breakdown voltage of the surrounding air or silicon.

Frequently Asked Questions

Why is the inductor voltage formula negative in some textbooks?

You will often see the formula written as V = -L(di/dt). This negative sign represents Lenz's Law, which states that the induced electromotive force (EMF) will always oppose the change in current that created it. In physics and rigorous circuit analysis, the negative sign indicates the polarity of the 'back-EMF'. However, in practical power electronics and nodal analysis using the passive sign convention, engineers typically drop the negative sign and assign the voltage polarity based on the direction of current flow through the component to keep Kirchhoff's Voltage Law (KVL) loop equations straightforward.

How do you calculate inductor voltage in AC circuits?

The time-domain derivative formula V = L(di/dt) becomes cumbersome when dealing with sinusoidal AC waveforms. Instead, engineers transition to the phasor (frequency) domain. By taking the derivative of a sine wave, the formula transforms into Ohm's Law for inductors: V = I × XL, where XL is the inductive reactance. The reactance is calculated as XL = 2πfL (where 'f' is the frequency in Hertz). In complex notation, this is written as V = jωLI, indicating that the voltage leads the current by exactly 90 degrees.

What happens to the inductor voltage when the current is constant?

When the current through an inductor is perfectly constant (pure DC steady state), the change in current (di) is zero. Consequently, di/dt = 0, and the induced voltage V = L × 0 = 0V. In this state, the inductor behaves exactly like a piece of wire (a short circuit). The only voltage you will measure across its terminals with a multimeter is the minor voltage drop caused by the wire's inherent DC Resistance (DCR), calculated via standard Ohm's Law (V = I × R).

Does core saturation change the inductor voltage?

Core saturation does not directly change the applied voltage, but it drastically alters the inductance value (L) in the formula. If you force a constant voltage across an inductor (like in a switching power supply) and the core saturates, 'L' drops toward the value of an air-core coil (near zero). According to the rearranged formula di/dt = V/L, if 'L' approaches zero while 'V' remains constant, the rate of current change (di/dt) spikes toward infinity. This results in a massive, uncontrolled current spike that usually destroys the driving transistor. Therefore, saturation limits the current-handling capability, not the voltage-handling capability, of the component.