The fundamental inductor current formula defines how current changes over time when a voltage is applied across an inductive component. In its most practical integral form, the formula is i(t) = (1/L) ∫ v(t) dt + i(t₀). In its differential form, which is more commonly used to calculate instantaneous voltage or rate of change, it is expressed as v(t) = L × (di/dt). These equations are the bedrock of designing switch-mode power supplies, RF filters, and transient protection circuits.
The Core Formula, Symbols, and Real-World Baselines
To use the inductor current formula correctly on the bench, you must first map the mathematical symbols to physical circuit parameters. The differential form v(t) = L × (di/dt) tells us that the voltage across an inductor is proportional to the rate of change of current through it. The integral form i(t) = (1/L) ∫ v(t) dt + i(t₀) tells us that the current at any given moment is the accumulated integral of the applied voltage, scaled by the inverse of the inductance, plus the initial current.
Symbol Definition Table
| Symbol | Parameter | Standard SI Unit | Practical Engineering Unit |
|---|---|---|---|
| v(t) | Instantaneous voltage across the inductor | Volts (V) | Volts (V) |
| i(t) | Instantaneous current through the inductor | Amperes (A) | Amperes (A) or milliamperes (mA) |
| L | Inductance value | Henries (H) | Microhenries (µH) or millihenries (mH) |
| t | Time variable | Seconds (s) | Microseconds (µs) or nanoseconds (ns) |
| t₀ | Initial time (start of observation) | Seconds (s) | Seconds (s) |
| i(t₀) | Initial current at time t₀ | Amperes (A) | Amperes (A) |
| di/dt | Rate of change of current over time | Amperes per second (A/s) | Amps per microsecond (A/µs) |
Real-World Inductor Baselines (Data-Dense Reference)
Abstract formulas mean little without physical context. The table below maps real, purchasable inductor models to their typical applications and the realistic current magnitudes you should expect when applying the formula. This data helps you sanity-check your calculations before powering up a prototype.
| Manufacturer Part Number | Inductance (L) | I_sat / I_rms | Typical Application | Realistic Δi Magnitude |
|---|---|---|---|---|
| Coilcraft XEL3520-101 | 100 nH | 15 A / 12 A | High-frequency GaN buck converters (1MHz+) | 2.0 A to 5.0 A ripple |
| Bourns SRP1265A-150M | 15 µH | 14 A / 8 A | Standard 12V-to-5V DC-DC buck stages | 0.5 A to 1.5 A ripple |
| Würth Elektronik 744774210 | 1 mH | 1.2 A / 1.0 A | Low-frequency EMI filtering, audio crossovers | 50 mA to 200 mA transient |
| TDK VLF5045T-100M | 10 µH | 3.2 A / 2.2 A | Compact point-of-load (PoL) regulators | 0.3 A to 0.8 A ripple |
Source: Manufacturer datasheets from Coilcraft and Electronics Tutorials for general inductor classifications.
Rearranged Forms, Assumptions, and Unit Traps
On the workbench, you rarely solve for just one variable. Depending on whether you are selecting a component, sizing a current sense resistor, or debugging a switching node, you need the formula rearranged. Here are the practical algebraic forms derived from v = L(di/dt):
- Solve for Inductance (L):
L = (v × dt) / di— Used when selecting an inductor for a target ripple current in a switching regulator. - Solve for Current Change (di):
di = (v × dt) / L— Used to calculate peak-to-peak ripple current. - Solve for Time Interval (dt):
dt = (L × di) / v— Used to determine the required on-time for a MOSFET switch. - Solve for Voltage (v):
v = L × (di / dt)— Used to calculate flyback voltage spikes when a switch opens abruptly.
Critical Assumptions and Limits
The standard inductor current formula assumes an ideal component. In reality, three physical phenomena break the ideal math:
- DC Resistance (DCR): Real inductors have copper wire resistance. In DC steady-state, the inductor acts as a resistor, and current is limited by Ohm's Law (I = V/R), not the inductance formula.
- Core Saturation: If the current exceeds the saturation current (I_sat), the magnetic core cannot store more flux. The effective inductance (L) drops precipitously, causing di/dt to spike and potentially destroying the switching MOSFET.
- Parasitic Capacitance: At very high frequencies (typically >10 MHz for power inductors), inter-winding capacitance creates a self-resonant frequency (SRF). Above the SRF, the component behaves like a capacitor, rendering the inductor formula invalid.
The 'Micro-Milli' Unit Trap
The most common mistake that breaks the inductor current formula is mixing prefixes without converting to base SI units. If you plug v = 5V, L = 10 (meaning 10 µH), and dt = 2 (meaning 2 µs) directly into di = (v × dt) / L, you will calculate di = 1 A. This is mathematically correct for the numbers entered, but physically wrong for the units intended.
Worked Examples with Strict Unit Tracking
Let's apply the formula to two common scenarios: calculating switching ripple in a DC-DC converter, and analyzing a DC turn-on transient. For deeper theoretical background on transient responses, refer to the All About Circuits textbook chapter on inductors and calculus.
Example 1: Buck Converter Inductor Ripple Current
Scenario: You are designing a buck converter stepping 24V down to 5V. The switching frequency is 500 kHz, and you have selected a 15 µH inductor (Bourns SRP1265A-150M). What is the peak-to-peak inductor ripple current (Δi)?
Step 1: Identify knowns and convert to consistent units.
- V_in = 24 V
- V_out = 5 V
- L = 15 µH = 15 × 10⁻⁶ H
- f_sw = 500,000 Hz
Step 2: Calculate the switch ON time (dt).
The duty cycle (D) for an ideal buck converter is V_out / V_in.
- D = 5 V / 24 V = 0.2083
- dt (t_on) = D / f_sw = 0.2083 / 500,000 Hz = 4.166 × 10⁻⁷ seconds (416.6 ns)
Step 3: Determine the voltage across the inductor during t_on (v).
When the high-side switch is ON, the inductor sees V_in minus V_out.
- v_L = 24 V - 5 V = 19 V
Step 4: Apply the rearranged formula to solve for di.
- di = (v_L × dt) / L
- di = (19 V × 4.166 × 10⁻⁷ s) / (15 × 10⁻⁶ H)
- di = (7.915 × 10⁻⁶) / (15 × 10⁻⁶)
- di = 0.527 A (or 527 mA peak-to-peak ripple)
Sanity Check: A 527 mA ripple on a 15 µH inductor in a 5V/8A output supply is roughly 6.5% of the max load, which aligns perfectly with the standard 20-40% design rule when operating at full load.
Example 2: RL Circuit DC Turn-on Transient
Scenario: A 48V DC source is connected to a series circuit containing a 100 mH inductor and a 12 Ω resistor. What is the current exactly 5 milliseconds after the switch closes?
Step 1: Identify knowns.
- V = 48 V
- R = 12 Ω
- L = 100 mH = 0.1 H
- t = 5 ms = 0.005 s
Step 2: Calculate the time constant (τ).
The inductor current formula for an RL step response integrates to an exponential curve defined by τ = L / R.
- τ = 0.1 H / 12 Ω = 0.00833 seconds (8.33 ms)
Step 3: Apply the integrated transient formula.
The specific integral solution for an RL step is i(t) = (V/R) × (1 - e^(-t/τ)).
- Steady-state current (V/R) = 48 V / 12 Ω = 4.0 A
- Exponent term = -t / τ = -0.005 / 0.00833 = -0.60
- e^(-0.60) ≈ 0.5488
- i(5ms) = 4.0 A × (1 - 0.5488)
- i(5ms) = 4.0 A × 0.4512
- i(5ms) = 1.805 A
Realistic Answer Magnitudes and Bench Verification
Knowing what a 'correct' answer looks like prevents you from chasing ghosts on the oscilloscope. If your calculation for a power inductor yields a ripple current of 0.002 A, you likely dropped a decimal or missed a micro-prefix. Conversely, if your RF choke calculation yields 15 A, your core would instantly saturate and vaporize the trace.
Magnitude Expectations by Domain
- Power Conversion (Buck/Boost): Expect Δi (ripple) to be 20% to 40% of the maximum DC load current. For a 10A supply, a realistic Δi is 2A to 4A.
- Motor Drive / Solenoids: Expect massive initial di/dt spikes limited only by parasitic resistance, settling into a steady-state DC current dictated by the coil's DCR.
- RF / Signal Filtering: Expect currents in the 5 mA to 50 mA range. Inductance values here are typically in the nanohenry (nH) range, and the formula is applied using AC phasor math (impedance Z = jωL) rather than time-domain step responses.
Verifying the Math on the Bench
To verify your calculated di/dt or Δi, you must measure the current waveform. Do not rely solely on voltage node probing. Use a Hall-effect AC/DC current probe (like the Tektronix TCP0030A) clamped around a dedicated U-shaped wire loop on your PCB. This provides galvanic isolation and captures the true triangular ripple waveform without introducing ground loops.
If a current probe is unavailable, use a low-inductance shunt resistor (e.g., a 10 mΩ Bourns CSS series SMD resistor) placed in the low-side return path. Measure the voltage across the shunt with an oscilloscope and apply Ohm's law (I = V_shunt / 0.01). For advanced topology selection and inductor sizing methodologies, the Analog Devices technical article on inductor selection provides excellent supplementary design frameworks.
By strictly tracking your units, respecting the physical limits of core saturation, and verifying your mathematical models with proper bench equipment, the inductor current formula transitions from a textbook abstraction to a reliable daily design tool.
