Inductors: Storing Energy and Voltage-Current Relationship

Hello!

In our last lesson, we explored our first energy storage device, the capacitor. We established that it stores energy in an electric field created by separated charges, with the core relationship being $Q=CV$.

Today, we introduce the capacitor's counterpart: the inductor. While capacitors resist changes in voltage, inductors resist changes in current. They do this by storing energy, but in a magnetic field. This lesson will build the physical intuition behind this behavior, connecting it to the core equation that governs inductors. Your interest in the flow of energy and the physical "why" behind circuit laws will be central to our exploration.

This lesson covers the second learning outcome in our "Energy Storage Elements" module: Explain how an inductor stores energy in a magnetic field and use $V=L(di/dt)$ to relate its voltage to the rate of change of current.

1. The Inductor's Core Behavior: Electrical Inertia

At its simplest, an inductor is just a coil of wire. When current flows through this wire, it generates a magnetic field. The coil's shape concentrates this field, and it's this field that gives the inductor its unique property: it opposes any change in the current flowing through it.

Let's watch a short video that introduces this fundamental behavior.

This resistance to change is a form of electrical inertia. To build a strong mental model for this, a physical analogy is incredibly useful.

The key takeaway from the analogy is this:

• An inductor does not resist steady, constant current. In a DC circuit where the current is stable, an ideal inductor behaves just like a simple piece of wire.
• An inductor only resists a change in current. This is where its unique properties emerge.

2. The "Why": Energy in the Magnetic Field

So, what provides this "inertia"? The answer lies in the magnetic field.

When current flows through the inductor's coil, it creates a magnetic field. This field contains energy. To increase the current, you must supply extra energy to build up the magnetic field. Conversely, if you try to decrease the current, the magnetic field collapses and releases its stored energy, pushing the current along to try and keep it flowing.

This process of storing and releasing energy from the magnetic field is the physical mechanism behind the inductor's opposition to current changes.

This behavior is governed by Lenz's Law, a fundamental principle in physics which states that an induced effect will always oppose the cause that produced it. In an inductor:

• When current increases, the inductor generates a voltage that pushes against the incoming current.
• When current decreases, the inductor generates a voltage that pushes in the same direction as the current, trying to keep it going.

3. Quantifying the Behavior: The Inductor's V-I Equation

We now have a solid conceptual model. Let's connect this physical behavior to a mathematical relationship. We've seen that the inductor's effect is tied not to the current itself, but to how it changes. This suggests a relationship involving a rate of change.

The voltage across an inductor is directly proportional to the rate of change of the current through it. This is the fundamental law for inductors:

$$v=L\frac{di}{dt}$$

Where:

• $v$ is the instantaneous voltage across the inductor in Volts (V).
• $L$ is the inductance of the coil in Henries (H). Inductance is a physical property determined by the coil's geometry (number of turns, length, area) and the material of its core.
• $\frac{di}{dt}$ is the instantaneous rate of change of current in Amperes per second (A/s). It's a term from calculus representing the slope of the current-time graph.

4. Applying the Equation: Key Scenarios

Let's use the equation $v=L\frac{di}{dt}$ to solidify your intuition for three critical scenarios.

Scenario 1: Constant Current (DC Steady-State)

In a DC circuit that has been running for a while, the current is constant.

• If the current $i$ is constant, its rate of change $\frac{di}{dt}$ is zero.
• Therefore, the voltage across the inductor is $v=L\cdot 0=0$ Volts.

This is a crucial insight: In a DC steady-state, an ideal inductor acts like a short circuit (a zero-ohm wire).

Scenario 2: Linearly Changing Current

What if we force the current to increase at a constant rate (a straight, sloped line on a graph)?

• The rate of change $\frac{di}{dt}$ is a constant, positive value (the slope of the line).
• Therefore, the voltage across the inductor is $v=L\cdot (\text{constant})=\text{constant}$. A constantly changing current produces a constant voltage.

Scenario 3: Rapidly Changing Current

What happens if we try to change the current very quickly, for example by opening a switch?

• Opening a switch causes the current to drop from some value to zero in a tiny amount of time.
• This makes $\frac{di}{dt}$ a very large negative number.
• The resulting voltage $v=L\cdot (\text{very large negative number})$ becomes a very large negative voltage spike. This is often called "inductive kickback" and can be powerful enough to create a spark or damage other circuit components.

5. Thought Experiment: Applying Your Knowledge

Let's test this new mental model. Imagine you connect an inductor to a voltage source that produces a square wave, flipping between +1V and -1V.

• During the +1V phase: The voltage v is constant and positive. According to $v=L\frac{di}{dt}$, what must the current be doing?
• During the -1V phase: The voltage v is constant and negative. What must the current be doing now?

Based on your answers, can you sketch the shape of the current waveform $i(t)$? This is a great exercise in logical thinking about the component's behavior.

Click here to see the solution and explanation.

This is a classic problem that demonstrates the inductor's behavior.

• During the +1V phase: Since $v$ is constant and positive, $\frac{di}{dt}$ must also be constant and positive. A constant rate of change means the current $i$ is increasing linearly—it's a straight-line ramp upwards.
• During the -1V phase: Since $v$ is constant and negative, $\frac{di}{dt}$ must be constant and negative. This means the current $i$ is decreasing linearly—a straight-line ramp downwards.

Putting it together, the current waveform will be a triangle wave. The inductor "integrates" the voltage over time. When the voltage is constant positive, the current ramps up. When the voltage is constant negative, the current ramps down.

For a deeper dive into this specific example, you can review the "Example" section in the Inductor i-v equation in action article from Spinning Numbers (resource ID LINK).

Conclusion

In this lesson, we demystified the inductor, establishing it as a component that stores energy in a magnetic field and, as a result, opposes any change in current.

Key Takeaways:

• An inductor's primary characteristic is its resistance to changes in current, a property analogous to mechanical inertia.
• This behavior is due to energy being stored in and released from the magnetic field surrounding the coiled wire.
• The relationship between an inductor's voltage and current is defined by the rate of change: $v=L\frac{di}{dt}$.
• Because of this relationship, a constant current (DC steady-state) produces zero voltage, making an ideal inductor behave like a short circuit.
• A rapid change in current can induce a very high voltage across an inductor.

Preview of the Next Lesson:
We have now established the fundamental behaviors of both capacitors and inductors. In the next lesson, we will quantify the energy they store by deriving and applying the formulas $U=\frac{1}{2}C{V}^2$ for capacitors and $U=\frac{1}{2}L{I}^2$ for inductors. We will also explore how to calculate the equivalent inductance and capacitance when these components are combined in series and parallel.