Inductor Fundamentals & V-I Relationship

The dual of the capacitor: it fights current change and stores energy in a magnetic field.

Circuit AnalysisElectrical Engineering Year 1
⏱️ About 16 min

Try to suddenly cut the current through an inductor and it will generate whatever voltage is necessary to keep that current flowing — the mirror image of the capacitor's voltage stubbornness.

💡
The big idea: An inductor stores energy in a magnetic field, and its voltage depends not on the current through it but on how fast that current is changing.
🎯 By the end, you'll be able to
  • Define inductance in terms of magnetic flux, current, and coil geometry
  • Apply the v=L di/dt relationship to time-varying currents
  • Explain why inductor current cannot change instantaneously
  • Use the integral form to recover current from voltage history
  • Identify inductor behavior in DC steady state as a short circuit
  • Recognize the duality between capacitors and inductors
📎 Helpful to know first

What Is an Inductor?

An inductor is a two-terminal device that stores energy in a magnetic field. Its simplest form is a coil of wire — when current flows through the coil, a magnetic field is established around and within it, and that field stores energy. The inductance L, measured in henries (H), quantifies how much magnetic flux linkage is produced per ampere of current.

The Defining V-I Relationship

By Faraday's law of induction, a changing magnetic flux induces a voltage. Since the flux is proportional to current, the induced voltage is proportional to the rate of change of current:

\[ v(t) = L\,\frac{di(t)}{dt} \]
Inductor voltage is proportional to the rate of change of current, not the current itself.
🔑 Voltage depends on di/dt, not i

Just as a capacitor's current depends on dv/dt (not v), an inductor's voltage depends on di/dt (not i). An inductor carrying 100 A of steady DC current has zero voltage across it; an inductor carrying 0 A whose current is rising rapidly can have a large voltage.

Current Cannot Change Instantaneously

If the inductor current were to jump instantaneously, di/dt would be infinite, requiring infinite voltage — a physical impossibility. Therefore:

✨ Current continuity rule

An inductor's current cannot change instantaneously. If i(0−) = 3 A just before a switching event, then i(0+) = 3 A immediately after. This is the exact dual of the capacitor's voltage-continuity rule and is equally central to transient analysis.

The Integral Form

When you know the voltage history and need the current, integrate the defining equation:

\[ i(t) = \frac{1}{L}\int_{0}^{t} v(\tau)\,d\tau + i(0) \]
Current at time t equals the initial current plus the accumulated volt-seconds (scaled by 1/L) from the voltage applied since.

DC Steady State: The Short-Circuit Analogy

In DC steady state, all currents are constant, so di/dt = 0 and therefore v = L di/dt = 0. An inductor with zero voltage across it behaves like a short circuit. To analyze inductors in DC steady-state circuits, replace them with a wire and solve the remaining network.

⚠️ Short circuit only at true steady state

An inductor acts as a short circuit only when all transients have settled and every current is truly constant. During a transient — while di/dt is nonzero — voltage appears across the inductor and it is actively storing or releasing energy.

Duality: Capacitor vs. Inductor

The capacitor and inductor are duals: every property of one maps to a property of the other with voltage and current roles swapped. Once you understand one, you understand the other — just swap V with I and C with L.

Capacitor: stores energy in an electric field; i = C dv/dt; voltage is continuous; acts as an open circuit at DC.
Inductor: stores energy in a magnetic field; v = L di/dt; current is continuous; acts as a short circuit at DC.

📝 Worked example: A 50 mH inductor has current i(t) = 2 sin(200t) A flowing through it. Find the voltage v(t).
  1. Start with the defining relationship: v = L di/dt.
  2. Differentiate i(t) = 2 sin(200t): di/dt = 2×200 cos(200t) = 400 cos(200t) A/s.
  3. Multiply by L = 50×10⁻³ H: v = 0.05×400 cos(200t) = 20 cos(200t) V.
✓ v(t) = 20 cos(200t) V
✏️ Practice: A 100 mH inductor has a current that increases linearly at 50 A/s. What is the voltage across the inductor (in volts)?
V
Solution
  1. Since current changes linearly, di/dt = 50 A/s (constant).
  2. Apply v = L di/dt = 0.1×50 = 5 V.

Check your understanding

1. An inductor carries 5 A of steady DC current. What is the voltage across it?
Steady DC current means di/dt = 0, so v = L di/dt = 0 regardless of the inductance or current magnitude.
2. Which statement correctly captures the duality between a capacitor and an inductor?
The duality pairs voltage continuity (capacitor) with current continuity (inductor). At DC steady state, the capacitor is an open circuit while the inductor is a short circuit — opposite behaviors.
🔎 Related lessons
Energy Storage in Capacitors & Inductors · soon
✅ Key takeaways
  • An inductor stores energy in a magnetic field created by current flowing through a coil of wire.
  • Inductance L (in henries) quantifies the flux linkage per ampere of current.
  • The defining V-I relationship is v = L di/dt — voltage depends on the rate of current change, not current itself.
  • Inductor current cannot change instantaneously because that would require infinite voltage.
  • In DC steady state (di/dt = 0), an inductor acts as a short circuit.
  • The integral form i(t) = (1/L)×∫v dt + i(0) recovers current from voltage history.
  • Capacitors and inductors are duals: swap V with I and C with L to map one's behavior to the other's.
➡️ With both V-I relationships in hand, the next step is quantifying the energy each element stores in its field.
Want to test yourself on this? Try the Electrical Aptitude test →