Inductors Basics

Describing basic functionality of the inductors and how they are treated when connected in series or in parallel.

What is an inductor? How does it work? And how we handle inductors when they are connected in series or in parallel? Here are the answers.

An inductor is an electric device capable of storing energy in the form of a magnetic or electromagnetic field.

inductor

In its basic form, an inductor can be made of a single loop of wire, or several loops (solenoid). These loops can be arranged in air or on a ferromagnetic core.

When an inductor is connected to a battery, a current starts flowing in the circuit. The current that flows inside the inductor generates a magnetic field, like the one that would be generated by an actual magnet. This field stores an amount of energy, the same way an electric field does.

inductor_circuit

If the battery is suddenly disconnected, the energy that was accumulated in the inductor must be somehow released. but the energy cannot be released instantaneously, it needs to be released a little bit at a time. And since the energy depends on the current flowing in the inductor, the inductor tries to keep the it running, even if the battery is no more connected. To do so, it uses the energy stored into the magnetic field to generate a voltage at its terminals to keep the current going.

inductor_open_circuit

However, since the inductor is now connected nowhere, current cannot flow, unless the voltage is so high that the current can flow in the thin air. And that is exactly what happens: the voltage increases so much that there is a sudden discharge of current through the air, in the form of a spark, that dissipates all the energy that was stored in the inductor. This spark is the one you may sometimes notice when opening a switch that is powering a lamp or a motor, or when you pull the plug from a device that was working using a considerable amount of current.

Similarly to the case where the current is suddenly removed, an inductor generates a voltage also when the current is just changed in intensity. In this case, the voltage is created to react to the change in current, trying to keep it to the same value, so the energy can be conserved.

In both cases, the amount of voltage is proportional to the change in current (ΔI) and inversely proportional to the amount of time in which the current changes (Δt). In other words, the faster the current change, the higher is the voltage.

For a specific inductor, the ratio between the change of current and the interval in which that happens equals the voltage generated by the inductor divided by a constant that depends on the physics dimensions of the inductor. Such constant is called inductance, represented with the letter L, and can be calculated with the following experimental formula:

inductance_formula

where:

μ = permeability of the material inside the coil

N = number of turns making the coil

A = area of the cross section of the coil

l = length of the coil

L is measured in Henry.

μ is the product of the permeability of the void (or air) and the relative permeability of the material:mu

The voltage at the terminals of the inductor is therefore calculated as:

vdit

We can now calculate the energy stored in the magnetic field of an inductor as the integral of the power, which is obtained multiplying the voltage at the inductor and the current that flows through it:

inductor_energy

which, considering the value of the voltage previously calculated, can be solved as follows:

inductor_energy_value

where I is the current flowing through the inductor at the time the energy is calculated.

When choosing an inductor for a circuit, the following parameters must be considered:

  • the value of the inductance in Henry

  • the max current the inductor can sustain; failure to specify that could cause the inductor to overheat, since the wire could be too thin to deal with the required current;

  • the max voltage that can be applied to the inductor; an excessive voltage on the inductor could cause sparks due to insufficient insulation of the wire.

Inductors In Series

Let’s consider a series of inductors of different inductance values and let’s calculate the equivalent inductance.

inductors_in_series

All the inductors, being in series, are traversed by the same current. And since each inductor has its own inductance value, each one will store a different amount of energy:

inductors_series_energies.png

The total energy stored in the inductors is therefore:

inductors_series_total_energy.png

So, the equivalent inductance is clearly:

series_inductance.png

which we can generalize as:

series_inductance_gen.png

Inductors In Parallel

In the case of inductors in parallel, they are all subject to the same voltage and are traversed by a different current:

inductors_in_parallel.png

parallel_inductors_voltages.png

From these equations we can find the currents by integration:

parallel_inductors_currents.png

The total amount of current is therefore:

parallel_inductors_total_current.png

So we can say that the equivalent inductance of a parallel of inductors can be determined through the formula:

parallel_inductors_formula_1.png

or, more in general:

parallel_inductors_formula_2.png

All the formulas presented here are very general and can be applied to both DC and AC circuits. Note, however, that since AC circuits have a variable voltage and current, the application of the formulas in AC is a little more challenging then in DC. But this is a story for another time.