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.

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.

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.

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:

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:

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

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:

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

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.

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:

The total energy stored in the inductors is therefore:

So, the equivalent inductance is clearly:

which we can generalize as:

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:

From these equations we can find the currents by integration:

The total amount of current is therefore:

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

or, more in general:

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.

Conductors, Insulators, and Semiconductors

Everybody knows that an electric wire, usually made of copper, is a conductor. And everybody knows that all metals are conductors.

Everybody also knows that plastic is a good electrical insulator, as well as other materials such as glass and rubber.

But how about semiconductors? What are they? And how do we really distinguish among conductors, semiconductors and insulators?

To answer all these question we need to look deeper inside the materials.We know that matter is made of atoms, and atoms are made of protons, neutrons and electrons. Protons and neutrons reside at the center of the atom structure, called the nucleus. Electrons are allocated all around the nucleus, at a long distance from it, relatively to the scale of the nucleus itself.

Electrons have a certain amount of energy, that is always an integral value of a certain amount that is called quantum of energy. Depending on the amount of energy they posses, they are locate closer or farther away from the nucleus. The more energy, the farther they are.

Based on quantum mechanics, which we are not going to talk in details in this context, electrons occupy bands of energy. The farther bands in the atom are the so called Valence Band and Conduction Band.

The valence band contains all those electrons that allow the atoms to stick together and forming molecules by bonding with other atoms of the same or a different substance.

For certain materials, rather than having molecules, the atoms form what is called a crystalline lattice, which is the case we are more interested in this context. It is worth noting that a new theory, highly based on quantum mechanics, is also distinguishing between actual crystalline lattices and material networks. However, for all the scope and purpose of this context, we will make a simplification and name both of them as crystalline lattices.

(This picture, courtesy of Wikipedia)

In certain conditions, electrons in the valence band can jump to a higher level of energy, thus moving in what is called the conduction band. In the conduction band, electrons are no more stuck to their own atoms, but can start moving freely in the lattice that makes up the material. When that happens, we can control their movement by applying an electric field by the means, for example, of a battery. The voltage of the battery, applied to the two ends of the same block of material (for example a wire), produces the electric field inside the material and forces the electrons to move toward the positive electrode of the battery while, in the mean time, the negative electrode of the battery provides electrons to the material, to replace those that have entered the positive electrode of the battery.

Don’t think that electrons move very fast when they do that. Electrons, in fact, move very slowly, but it is the huge amount of them that help creating a measurable current.

Then you would ask: but when I turn the switch on, the light comes out of a lamp instantly. If the electrons move slowly, shouldn’t a lamp emit light only after a while?

Well, yes and no. In fact, the lamp does not light up immediately. It takes a certain amount of time to do that. But that time is so small that for us the event happens instantly.

Also, when you turn the switch on, the electrons closest to the positive electrode move into it almost immediately, not because they are fast, but because they are so close to it. At the same time, new electrons are fed to the material from the negative electrode. So, at the end, all the electrons in the wire start moving simultaneously inside of it, causing the current to start flowing immediately.

But I am digressing. Let’s go back to our primary subject.

We have talked about electrons in the valence band and the possibility they have to jump to the conduction band and, thus, helping creating a current if we apply a voltage.

But how much energy do the electrons need to jump to the conduction band?

Here it comes the definition of conductors, semiconductors and insulators.

In the materials called conductors, the level of energy that the valence electrons need to jump to the conduction band is basically ‘0’. In fact, valence band and conduction band overlap each other and, therefore, some or all the electrons in the valence band are also, already, in the conduction band. This happens mostly with metals, like copper, iron, aluminum, and so forth. Depending on the particular metal, the conduction and valence bands are more or less overlapped. Those material where there is more overlap are those that conduct electricity better. Those material where there is less overlap, are those that are worst to conduct current, although still conductors.

In the materials called insulators, the gap between the valence band and the conduction band is so high that electrons cannot jump from the valence band to the conduction band, and so they cannot generate an electrical current.
Of course, if we apply a voltage high enough, we can still provide them the energy to make the jump. However, in that case, because of the very high voltage, electrons jump from one band to the other in a disruptive way, causing the material to break. Once that happened, the insulator loose its property and it is no good anymore as such.

Finally, in the materials called semiconductors, the valence band and the conduction band are separated, but close together. It is relatively easy for an electron in the valence band to jump to the conduction band if we only heat a little bit the semiconductor, maybe just with our bare hands. The heat provides enough energy to the electrons to jump to the conduction band. However, not many electrons will do so, unless we keep heating the semiconductor. So, at the end, although capable of conducting some current, semiconductors are not good in doing so. Thus the name of their category.

In this article, we have talked about energy bands in materials, and how materials behave based on the position of the two highest energy band levels.

We have said that conductors are those were valence and conduction bands are partially overlapping.

Conversely, insulators are those that have a high gap in between the valence and the conduction bands.

And finally, semiconductors are those somewhat in between. For them, the valence and conduction bands are separated with a gap, but that gap is small enough to allow, under certain conditions, for the electrons to jump from the valence to the conduction band. That’s why they perform poorly both as conductors and insulators. However, we will see later on how semiconductors can be of great advantage for us, as long as they are treated in a certain way. They are those that allow us to build all the wonders of modern electronics.