The defining inductor equation is V = L(di/dt). It dictates that the voltage across an inductor is strictly proportional to the rate of change of current flowing through it. If the current is constant (DC steady-state), the rate of change (di/dt) is zero, and the inductor acts as a short circuit, limited only by its parasitic wire resistance. This single formula governs everything from flyback voltage spikes in relay coils to the ripple current in switch-mode power supplies.
The Core Inductor Equation and Symbol Definitions
At the bench, you will rarely see the calculus notation v(t) = L(di/dt) written out in full. Instead, we use the algebraic delta form for discrete time steps, which is far more practical for calculating switching transients and power supply ripple.
V = L × (Δi / Δt)
| Symbol | Parameter | Standard Unit | Practical Bench Unit |
|---|---|---|---|
| V | Voltage across the inductor | Volts (V) | Volts (V) or millivolts (mV) |
| L | Inductance | Henries (H) | Microhenries (μH) or Millihenries (mH) |
| Δi | Change in current | Amperes (A) | Amperes (A) or milliamperes (mA) |
| Δt | Change in time (duration of current change) | Seconds (s) | Microseconds (μs) or nanoseconds (ns) |
Rearranged Forms for Circuit Debugging
Depending on what you are trying to solve for on the bench, you will need to rearrange the inductor equation. Keep these forms handy:
- Solving for Inductance (L):
L = (V × Δt) / Δi(Used when selecting an inductor for a target ripple current in a buck/boost converter). - Solving for Current Ripple (Δi):
Δi = (V × Δt) / L(Used to verify if your inductor will saturate given a specific switching frequency). - Solving for Time (Δt):
Δt = (L × Δi) / V(Used to calculate the required ON-time for a given current ramp). - Solving for Voltage (V):
V = L × (Δi / Δt)(Used to calculate flyback spikes or required drive voltage).
Physical Inductance: Calculating L from Core Geometry
The circuit equation tells you how an inductor behaves, but the physical equation tells you how to build one. The inductance of a coil is determined by its physical geometry and the magnetic permeability of its core material. The formula is:
L = (μ0 × μr × N2 × A) / l
Notice that the number of turns (N) is squared. Doubling the number of turns quadruples the inductance, but it also increases the DC resistance (DCR) and parasitic capacitance, which is why core material selection is critical.
| Core Material | Relative Permeability (μr) | Saturation Flux (Bsat) | Typical Frequency Range | Common Applications |
|---|---|---|---|---|
| Air | 1 | None (Linear) | DC to GHz | RF tuning, high-frequency filters, crossover networks |
| Powdered Iron | 10 to 100 | ~1.2 Tesla | 50 kHz to 2 MHz | Switch-mode power supplies, PFC chokes |
| MnZn Ferrite | 800 to 15,000 | 0.3 to 0.5 Tesla | 1 kHz to 2 MHz | Transformers, common-mode chokes, low-freq inductors |
| NiZn Ferrite | 10 to 2,000 | 0.3 to 0.4 Tesla | 2 MHz to 500 MHz | EMI suppression beads, RF broadband transformers |
| MPP (Molypermalloy) | 14 to 550 | ~0.8 Tesla | DC to 500 kHz | High-Q filters, aerospace power filtering |
Data sourced from industry magnetics design guidelines. Always consult the specific manufacturer's datasheet (e.g., Ferroxcube, Micrometals) for exact B-H curves, as permeability drops sharply as the core approaches saturation.
Worked Examples: Tracking Units from Bench to Breadboard
The most common point of failure when using the inductor equation on the bench is unit misalignment. The standard unit for inductance is the Henry (H), but we almost always work in microhenries (μH) or millihenries (mH). Below are two real-world scenarios with explicit unit tracking.
Problem 1: Calculating Relay Flyback Voltage
Scenario: You are driving a 12V automotive relay coil using a MOSFET. The coil has an inductance of 50 mH and draws a steady-state current of 100 mA. When the MOSFET turns off, the current drops to zero in 2 μs. What is the voltage spike across the MOSFET drain?
- Identify and convert to base SI units:
- L = 50 mH = 0.050 H
- Δi = 100 mA - 0 mA = 100 mA = 0.1 A
- Δt = 2 × 10-6 s
- Apply the formula:
- V = L × (Δi / Δt)
- V = 0.050 × (0.1 / 0.000002)
- Calculate the intermediate ratio (di/dt):
- 0.1 / 0.000002 = 50,000 A/s
- Final multiplication:
- V = 0.050 × 50,000 = 2,500 V
Problem 2: Sizing a Buck Converter Inductor
Scenario: You are designing a 12V to 5V buck converter. You want to limit the peak-to-peak inductor ripple current (Δi) to 1.0 A. The switch ON-time is 2.0 μs. What inductance value do you need?
- Identify the voltage across the inductor during ON-time:
- VL = Vin - Vout = 12V - 5V = 7 V
- Convert units to base SI:
- Δi = 1.0 A
- Δt = 2.0 μs = 2.0 × 10-6 s
- Rearrange the formula to solve for L:
- L = (V × Δt) / Δi
- L = (7 × 2.0 × 10-6) / 1.0
- Calculate:
- L = 14 × 10-6 H = 14 μH
Component Selection: You would select a standard 15 μH shielded ferrite power inductor (e.g., Coilcraft MSS1278 or Wurth WE-PD series) rated for at least the peak current (Iout + Δi/2) plus a 20% derating margin to prevent core saturation.
Assumptions, Unit Traps, and Realistic Magnitudes
When the Formula Applies (and Its Assumptions)
The equation V = L(di/dt) assumes an ideal, linear inductor. In reality, every physical inductor has parasitic elements:
- DCR (DC Resistance): The copper wire has resistance. The true voltage equation is V = L(di/dt) + I×DCR. For power inductors, DCR can be 10mΩ to 100mΩ, which matters for thermal calculations but is negligible for high-speed transient spike calculations.
- EPC (Equivalent Parallel Capacitance): The windings act as a capacitor. At very high frequencies (or extremely fast di/dt edges), the EPC dominates, and the inductor behaves like a capacitor. This is why fast-switching GaN FETs require careful PCB layout to manage parasitic ringing.
- Core Saturation: The formula assumes μr is constant. If the current exceeds the core's saturation limit (Bsat), the permeability drops toward that of air (1). When this happens, L collapses, di/dt spikes massively, and the component acts like a short circuit, often destroying the driving switch.
The 'Micro-Micro' Unit Trap
The Trap: This shortcut ONLY works if your current (Δi) is in Amperes. If you measure your current ripple in milliamperes (mA) and plug that raw number into the shortcut formula, your calculated voltage will be off by a factor of 1,000. Always convert mA to A before applying the micro-cancellation shortcut.
What a Realistic Answer Magnitude Looks Like
When you punch numbers into the inductor equation, use these sanity checks to verify your results:
- Power Supply Ripple (di): In a typical 100kHz to 1MHz switch-mode supply, realistic Δi values are between 0.1 A and 3.0 A. If you calculate a ripple of 50 A for a 5V logic rail, you forgot to convert μH to H.
- Inductance Values (L): RF circuits use nH to low μH. Switch-mode power supplies use 1 μH to 1 mH. Audio crossovers and line-frequency filters use 1 mH to 100 mH. If your buck converter calculation yields 400 H, your decimal is in the wrong place.
- Flyback Spikes (V): When a mechanical switch or fast MOSFET interrupts an inductive load, Δt is incredibly small (nanoseconds to low microseconds). Realistic flyback voltages range from 50 V to over 2,000 V. If you calculate a 5V spike from a relay coil turning off in 10 ns, your math is wrong; the physics demands a massive voltage to force the current to change that quickly.
References and Further Reading
For deeper dives into magnetics design, core loss calculations (Steinmetz equation), and practical component selection, consult these authoritative resources:
- Electronics Tutorials: Inductors and Inductive Circuits - Excellent foundational theory and AC impedance extensions.
- Coilcraft Inductor Tutorial - Industry-standard guide on physical core materials, saturation curves, and parasitic effects.
- All About Circuits: DC Theory Chapter on Inductors - Great breakdown of energy storage and time-constant behavior in RL circuits.
