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.

Conductors, Insulators, and Semiconductors

cpu-3061923_1280

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

copper-72062_1280

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

insulators-3838730_1280

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.

water-40708_1280

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.

lattice(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?

lamp-18869_1280

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.

conductors_semiconductors_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.

Let’s Talk About Voltage

What is voltage? Where the name comes from? How it makes electric current flow?

transmission_lines_scaled

Voltage is a very common word in the context of electrical and electronics engineering. However, common as it is, it is a concept that is often not fully understood.

How many people know that voltage is a concept related with potential energy? The same kind of energy that is so often used in mechanical physics!

When we talked about electrical current in the previous post, we said that current moves from a higher electrical potential energy to a lower one. We also defined the current as the flow of positive charges moving from the positive electrical potential energy to the negative one.

charges_in_wire

In order for the charges to move, there has to be a force that puts them in motion in a specific direction. Such force will have to do some sort of work on the charges and the work done by the force can be translated in a change in energy of the charges.

We define Electromotive Force the ratio between the energy change of the charges and the amount of charges. This can be represented with the following formula:

emf

Note that, despite the name, the Electromotive Force is not a force in the mechanical sense. Instead it is a potential energy per unit of charge, which we also call electric potential, or just potential.

The unit for the emf is called Volt, and is the ratio between one unit of energy, or 1 Joule, and one unit of charge, or 1 Coulomb:

volt

And that’s where the name voltage comes from: it is a name derived from the measurement unit of the electric potential. Although the correct name is electric potential, to make things easier in the day to day talk, we just call it voltage.

So, again, what is voltage? The voltage is the difference of electrical potential energy that allows a unit of charge to move and produce a current.

Now, do you think it is correct to say that when there is voltage we have a current? Do the two things always go together? Well, the answer is no. You can actually have one without the other.

An example of voltage without a current is the battery. If you don’t connect the battery to a circuit, there is no current involved!

batteries

And what about the current? This example is definitively less intuitive. So, let’s just say for now that if we force a current in a ring of a special material called superconductor and then we remove the cause that generated the current, the current keeps flowing in the superconductor, even in the absence of a voltage. Maybe we can explore this concept a little more in a future post, if I see there is an interest for it. Translation: let me know what you think! Comments are always welcome!

superconductor_ring

Let’s now go back to the battery. The battery provides a voltage between its positive and negative poles. With that, if we connect the battery to an electric load, current starts to flow immediately and the higher the voltage the greater the amount of current (more on that in a future post).

current_with battery

Like the current, there are two forms of voltage:

  1. The DC voltage

  2. The AC voltage

The DC voltage is the one typically provided by batteries. It is a voltage of a fixed value and of fixed polarity. The plus and minus on the electrodes of the battery never change. One electrode will always be the positive one, and the other electrode will be the negative one.

dc

When we connect a battery to an electric circuit, we have a flow of DC current.

The AC voltage is the one created for example in the power plants and provided at the wall outlets in your house.

power_plant_scaled

outlet_scaled

The voltage at the outlet is not constant as the one in the batteries. Instead, it changes continuously following the shape of a sine wave. Because of that, the polarity at each electrode of the outlet changes over time from positive to negative and vice versa, following the shape of the sine wave.

sine_wave

When we connect a device to the electric outlet, the current that will flow through that device will be an AC current as well.

The sinusoidal shape of the AC voltage depends on the way the electricity is generated. In the power plants there are devices called alternators, a much bigger version of those that you can find inside your car to recharge the battery, or on a bike, to provide electricity to turn on the lights at night.

Depending on the power plant, a different kind of energy is used to put in motion the alternator. It could be fossil fuel or nuclear energy that heat a reservoir of water and create the steam that makes the alternator rotate.

big_alternator_scaled

Or it could be the rotation of a propeller-like device that is put in motion by the wind.

propeller

Whatever the source is of the mechanical energy, the alternator converts that energy in electrical energy. But, since the rotation translates into a sine wave when described on a Cartesian reference system, the resulting electrical energy acquires that shape too.

DC and AC are both necessary to us to power devices that make our lives easier. Some devices need to be powered with DC, others need the AC.

DC voltage is necessary, for example, to power electronic devices: your smartphone or radio or computer, for example.

AC voltage is used to transmit the electrical energy from the places where it is created to the places where it is used. When AC reaches our home, it can be used as is to power electric motors like those in the refrigerator or the washing machine, or other household appliance. Or it can be converted to DC to power other devices, like your TV set.

Summarizing:

  1. To generate a current, we need to provide some energy to the charges in the conductor.

  2. The potential energy per unit of charge is called emf, or electromotive force. That measures the capability of the generator (a battery, for example) of generating a current.

  3. Most used synonyms of the emf are: voltage, difference of potential, and potential. Somebody also uses the name “tension”.

  4. The measurement unit of the potential is the Volt, which is the potential energy of 1 Joule per unit of charge, or 1 Coulomb.

  5. The Volt is a differential measure, not an absolute one. You always measure Volts with respect to a point that we arbitrarily define as 0V, or ground.

  6. Emf can be DC or AC and, correspondingly, it can generate a DC or an AC current when connected to an electrical circuit.

  7. DC voltage is normally generated through chemical reactions in a battery, or can be obtained from the AC through a process called “rectification”.

  8. AC voltage is normally generated with alternators like those used in power plants.

If you want to know more on this topic, I suggest you to watch the companion video of this article, which I posted on my YouTube channel:

https://www.youtube.com/watch?v=vPX-B4xAdtk

Thank you for reading this article and, as always,

Happy Experiments!