The fundamental inductor current equation in its integral form is i(t) = (1/L) ∫ v(t) dt + i(t₀). For practical DC and switching power supply design, this simplifies to the linear ramp equation: ΔI = (V · Δt) / L. This tells you exactly how much current builds up in the coil when a specific voltage is applied for a set time. Whether you are designing a buck converter or debugging a saturated core on the bench, this algebraic form is the primary tool for predicting inductor behavior.
The Core Inductor Current Equation & Symbol Definitions
Inductors resist changes in current by storing energy in a magnetic field. The calculus-based definition of an inductor relates the voltage across its terminals to the rate of change of current through it: v(t) = L · (di/dt). By integrating both sides with respect to time, we derive the equation for the current at any given moment.
While the integral form is necessary for AC waveform analysis (like sine or triangle waves), bench engineers and power electronics designers almost exclusively use the discrete step form for DC-DC switching applications. Assuming a constant applied voltage over a specific time window, the equation becomes:
ΔI = (VL · Δt) / L
Here is the complete symbol definition table for both the integral and discrete forms:
| Symbol | Parameter | SI Unit | Practical Bench Unit |
|---|---|---|---|
| i(t) | Instantaneous current at time t | Amperes (A) | A or mA |
| i(t₀) | Initial current at start of interval | Amperes (A) | A or mA |
| ΔI | Change in current (ripple or ramp) | Amperes (A) | A or mA |
| v(t) / VL | Voltage across the inductor | Volts (V) | V |
| L | Inductance | Henries (H) | µH or nH |
| t / Δt | Time / Time interval (pulse width) | Seconds (s) | µs or ns |
Rearranged Forms for Bench & Design Work
On the bench, you rarely solve for current in isolation. You are usually selecting an inductor value to limit ripple, or calculating the maximum on-time before the core saturates. Here are the rearranged forms of ΔI = (V · Δt) / L solving for each variable:
- Solve for Inductance (L): L = (VL · Δt) / ΔI
Use case: Sizing the inductor for a target ripple current in a SMPS. - Solve for Voltage (VL): VL = (L · ΔI) / Δt
Use case: Calculating the flyback voltage spike when a switch opens (snubber design). - Solve for Time (Δt): Δt = (L · ΔI) / VL
Use case: Finding the maximum MOSFET on-time before the inductor hits saturation current. - Solve for Current Change (ΔI): ΔI = (VL · Δt) / L
Use case: Verifying peak-to-peak ripple current on an oscilloscope.
Real-World Component Data & Magnitude Expectations
Before calculating, you must understand the physical limits of real components. The inductor current equation assumes L is constant. In reality, inductance drops as current approaches the saturation limit (Isat). Furthermore, what does a 'realistic' answer look like? In DC-DC converters, peak-to-peak ripple current (ΔI) is typically designed to be 20% to 40% of the maximum DC load current. Absolute currents range from 50mA in IoT sensors to 50A+ in GPU VRMs.
Below is a data-dense table of real-world power inductors commonly used in 2026 power supply designs, showing the critical thresholds where the linear equation breaks down.
| Manufacturer / Series | Part Number | Inductance (L) | Isat (Saturation) | Irms (Thermal) | Typical Application |
|---|---|---|---|---|---|
| Coilcraft XEL | XEL4030-471ME | 470 nH (0.47 µH) | 32.4 A | 23.5 A | High-frequency GPU Buck (1-2 MHz) |
| Würth Elektronik WE-PD | 74477420 | 1.0 mH (1000 µH) | 0.42 A | 0.31 A | Low-power offline flyback / LED driver |
| TDK SRN6045TA | SRN6045TA-100M | 10 µH | 3.2 A | 2.2 A | Standard 12V to 3.3V Buck (500 kHz) |
| Abracon AIRD | AIRD-02-101K | 100 µH | 1.8 A | 1.2 A | Boost converter input filtering |
Note: Isat is the current where inductance typically drops by 20-30%. Irms is the current that causes a 40°C temperature rise due to wire resistance (DCR). For more on selecting magnetics, refer to the Coilcraft Power Inductor Finder and design guides.
Worked Examples with Unit Tracking
The most common point of failure for junior engineers is unit mismanagement. Below are two step-by-step solved problems demonstrating strict unit tracking.
Problem 1: Buck Converter Ripple Current
Scenario: You are designing a synchronous buck converter stepping 12V down to 3.3V. The switching frequency is 500 kHz, and you have selected a 4.7 µH inductor. Calculate the peak-to-peak inductor ripple current (ΔI).
- Identify Knowns:
Vin = 12V, Vout = 3.3V, L = 4.7 µH (4.7 × 10-6 H), fsw = 500,000 Hz. - Calculate Inductor Voltage (VL) during the ON-time:
When the high-side MOSFET is on, VL = Vin - Vout = 12V - 3.3V = 8.7V. - Calculate ON-time (Δt):
Duty Cycle (D) = Vout / Vin = 3.3 / 12 = 0.275.
Δt = D / fsw = 0.275 / 500,000 = 5.5 × 10-7 seconds (550 ns). - Apply Equation:
ΔI = (VL · Δt) / L
ΔI = (8.7 V · 5.5 × 10-7 s) / (4.7 × 10-6 H)
ΔI = 4.785 × 10-6 / 4.7 × 10-6 = 1.018 A.
Result: The peak-to-peak ripple current is 1.018 A. If your maximum DC load is 5A, this ripple is ~20%, which is an excellent, stable design margin.
Problem 2: Time to Core Saturation
Scenario: A fault condition in a boost converter causes the control loop to fail, leaving the MOSFET stuck 'ON'. The input voltage is 5V, the inductor is 100 µH, and its saturation current (Isat) is 2.5A. How long until the inductor saturates and the MOSFET likely explodes from overcurrent?
- Identify Knowns:
VL = 5V (assuming ideal switch, 0V drop), L = 100 µH (100 × 10-6 H), ΔI = 2.5A (starting from 0A). - Rearrange Equation for Time:
Δt = (L · ΔI) / VL - Calculate:
Δt = (100 × 10-6 H · 2.5 A) / 5 V
Δt = 250 × 10-6 / 5 = 50 × 10-6 seconds = 50 µs.
Result: The core saturates in just 50 microseconds. This highlights why hardware overcurrent protection (like cycle-by-cycle current limiting in modern PWM controllers) must react in under 100ns to 500ns to prevent catastrophic failure. For deeper theory on magnetic saturation, consult the All About Circuits inductor textbook chapter.
Assumptions, Limitations, and Fatal Unit Mistakes
The equation ΔI = (V · Δt) / L is powerful, but it relies on strict assumptions. Violating these assumptions or mismanaging units will yield physically impossible or destructive results.
When the Formula Applies (and When it Fails)
- Linear B-H Curve: The formula assumes the core material is operating in its linear region. Once current exceeds Isat, the permeability (µ) of the core drops toward that of air. L is no longer constant; it plummets. When L drops, ΔI spikes exponentially, rendering the linear equation invalid.
- Ignoring Parasitic DCR: The equation uses VL, the voltage across the ideal inductance. Real inductors have DC Resistance (DCR). If you apply 5V to an inductor with 0.5Ω DCR carrying 2A, the actual VL is 5V - (2A · 0.5Ω) = 4V. Using 5V in your calculation will overestimate the current ramp rate.
- AC vs DC: This algebraic form is strictly for rectangular pulses (DC steps). For sinusoidal AC, you must use the calculus integral form or rely on inductive reactance (XL = 2πfL) and Ohm's law.
The 'Micro' Cancellation Trick (Pro-Tip)
The most common unit mistake is forgetting to convert microhenries (µH) to Henries, or microseconds (µs) to seconds. However, bench engineers use a mathematical shortcut. Look at the units in the equation:
Amps = (Volts · microseconds) / microHenries
Because 'micro' (10-6) appears in both the numerator (time) and the denominator (inductance), they cancel each other out perfectly. If your time is in µs and your inductance is in µH, you can plug the raw numbers directly into the equation, and the result will be in exact Amperes.
Example: V = 8.7V, Δt = 0.55 µs, L = 4.7 µH.
ΔI = (8.7 · 0.55) / 4.7 = 1.018 A.
Warning: This trick only works if both are in 'micro'. If your time is in nanoseconds (ns) and inductance is in microhenries (µH), you must convert one of them, or your answer will be off by a factor of 1,000. Always verify your oscilloscope timebase and your component datasheet before trusting the shortcut. For comprehensive component parameters, cross-reference datasheets on Electronics Tutorials or manufacturer portals.
