# Calculation of inductance of coils: formula

Each of us had problems with objects at school. Someone had problems with chemistry, someone - with physics. But even if everything was always good with these items, you probably do not remember everything that you were given at school. One of these topics is electromagnetism in general and the calculation of the inductance of the coils in particular.

To begin, let us dive a little into the history of such a phenomenon as magnetism.

## History

Magnetism begins its history from ancient China and ancient Greece. Magnetic iron discovered in China was then used as a compass needle pointing north. There are mentions that the Chinese emperor used him during the battle.

However, until 1820, magnetism was considered only as a phenomenon. All its practical application was concluded in pointing the compass needle to the north. However, in 1820, Oersted conducted his experiment with a magnetic needle, showing the effect of an electric field on a magnet. This experience was the impetus for some scientists who took it seriously to develop a theory of the magnetic field.

After only 11 years, in 1831, Faraday discovered the law of electromagnetic induction and introduced the concept of "magnetic field" to physicists. It was this law that served as the basis for the creation of inductors, which will be discussed today.

And before starting to consider the device itself of these coils, let's refresh the concept of a magnetic field in our head.

## A magnetic field

This phrase is familiar to us from school. But many have already forgotten what it means. Although each of us remembers that the magnetic field can affect objects, attracting or pushing them away. But, besides this, it has other features: for example, a magnetic field can affect electrically charged objects, which means that electricity and magnetism are closely interconnected, and one phenomenon can smoothly flow into another. Scientists have understood this for a long time and therefore began to call all these processes together in one word - “electromagnetic phenomena”. In fact, electromagnetism is a rather interesting and not yet fully studied field of physics. It is very extensive, and the knowledge that we can present to you here is a very small part of what mankind knows about magnetism today.

And now let's go directly to the subject of our article.The next section will be devoted to examining the inductance device itself.

## What is an inductor?

We are confronted with these items all the time, but we hardly give them any special meaning. This is a routine for us. Actually inductors are found today in almost every instrument, but the most striking example of their use is transformers. If you think that transformers are only at power substations, then you are greatly mistaken: your charger from a laptop or smartphone is also a kind of transformer, only smaller than those used in power plants and distribution substations.

Any inductor consists of a core and a winding. The core is a core of a dielectric or ferromagnetic material on which the winding is wound. The latter is most often made of copper wire. The number of turns of the winding is directly related to the magnitude of the magnetic induction of the resulting coil.

Now, before we consider the calculation of the inductance of the coils and the formulas necessary for it, let's talk about what parameters and properties we will calculate.

## What parameters does a coil have?

The coil has several physical characteristics that reflect its quality and suitability for a particular job. One of them is inductance. It is numerically equal to the ratio of the magnetic field flux created by the coil to the magnitude of this current. Inductance is measured in Henry (H) and in most cases takes values from units of microgens to tens of Henry.

Inductance is perhaps the most important parameter of a coil. It is therefore not surprising that most people do not even think that there are other quantities that can describe the behavior of a coil and reflect its suitability for a particular application.

When choosing inductors, professionals also pay attention to loss resistance. As can be understood from this phrase, it reflects the magnitude of the loss of electricity resulting from parasitic effects, such as, for example, heating of wires, which occurs according to the Joule-Lenz law. It is easy to understand that the lower this value for the coil, the better it is.

Another parameter that needs to be considered is the quality of the circuit.It is closely related to the previous parameter and is the ratio of reactance to active (loss resistance). Accordingly, the higher the quality factor - the better. Its increase is achieved by choosing the optimal wire diameter, material and core diameter, the number of windings.

Now let's take a closer look at the most important and most exciting parameter to us - the coil inductance.

## A little more about inductance

We have already dismantled this concept, and now it remains to talk about it in a little more detail. What for? After all, we have to calculate the inductance of the coils, which means we need to understand what it is and why we need to calculate it.

The inductance coil is designed to create a magnetic field, and therefore has parameters that describe its strength. This parameter is the magnetic flux. But different coils have different losses when current passes through them and, accordingly, different efficiency. Depending on the diameter of the wires and the number of turns, the coil can produce a different magnetic field. Therefore, it is necessary to introduce such a value that would reflect the relationship between the magnitude of the magnetic flux and the current strength passed through the coil. This parameter is inductance.

## Why do you need inductance calculation?

There are a lot of coils of different types in the world. They differ in their properties, and therefore in their applications. Some are used in transformers, others, solenoids, perform the role of electromagnets of great power. In addition to these, there will be plenty of applications for inductors. And all of them need different types of coils. They differ in their properties. But most of these properties can be combined with the concept of inductance.

We have come close to explaining what the formula for calculating the inductance of a coil includes. But it is necessary to make a reservation that it will not be about the "formula", but about the "formulas", since all the coils can be divided into several large groups, each of which has its own separate formula.

## Types of coils

Functionally, they distinguish contour coils, which are used in radiophysics, communication coils used in transformers, and variometers, i.e. coils, whose performance can be varied by changing the relative position of the coils.

There is also a type of coils, such as chokes. Inside this class there is also a division into regular and dual.They have a high resistance to alternating current and a very low resistance to direct current, due to which they can serve as a good filter that allows direct current and delays alternating current. Dual chokes are more efficient at high currents and frequencies compared to conventional ones.

## Formula Calculations

It is time we move on to the main topic of the article. We begin by telling you how to calculate the inductance of a coil without a core. This is the easiest type of calculation. But here, too, has its own subtleties. Take, for simplicity, the coil, the winding of which lies in one layer. For her, the calculation of a single-layer inductance is valid:

L = D2* n2/ (45D + 100l*).*

Here L is the inductance, D is the diameter of the coil in centimeters, n is the number of turns, l is the length of winding in centimeters. A single-layer coil assumes that the winding thickness will be no more than one layer, which means that the calculation of a flat inductance coil is valid for it. In general, the majority of formulas for inductance calculations are very similar: significant differences only in the coefficients of the variables in the numerator and denominator. The simplest is to calculate the inductance of a coil without a core.

Of interest is also the formula for calculating the inductance of a coil with a large number of turns:

L = 0.08 * D2* n2/ (3 * D + 9 * b + 10 * c).

Here b is the width of the wire, c is its height. This formula is effective in order to calculate the multilayer inductance coil. It is used in practice a little less often than the one that will be discussed below.

The most relevant, perhaps, will be the calculation of the inductance of the coil with the core. There is a special formula that shows that this inductance is determined by the material from which the core is made, or rather, by its magnetic permeability. This formula looks like this:

L = m * m0* n2* S / l,

where m is the magnetic permeability of the core material, m0- magnetic constant (it is equal to 12.56 · 10-7 GN / m), S is the cross-sectional area of the coil, l is the length of the winding.

The calculation of the turns of the inductor is very simple: it is the number of conductor layers wound on the core.

We figured out the formulas, and now a little about where exactly these formulas and calculations can be useful to us.

## Practical use

These formulas are very widely used because of the ubiquity of inductors.As we have already found out, there are different types of coils, each of which corresponds to its application. In this regard, it becomes necessary to somehow separate them according to their characteristics, because each industry needs its own specific inductance and good quality.

Basically, the calculation of the inductance of the coils is used in production and in electrical engineering. Every amateur radio operator should know how to calculate inductance, otherwise how can he determine which coil of a huge array is suitable for his purpose, and which coil is not.

## You are interested?

Today there are a lot of scientists interested in magnetism and magnetic phenomena. They study both magnetic and electrical aspects of substances, trying to identify patterns and synthesize powerful magnets with certain necessary properties: for example, with a high melting point or superconductivity. All these materials can be used in a huge number of industries.

Let us give an example with the aerospace industry: promising for long-range interstellar flights are rockets with ionic engines that create thrust by ejecting ionized gas from a nozzle.The thrust force in such an engine depends on the temperature of the gas and the speed of its movement. Accordingly, in order to give the gas a maximum force for acceleration, we need a very strong magnet, which accelerates charged particles and also has a very high melting point in order not to melt when the gases exit the nozzle.

## Conclusion

Knowledge is never superfluous and always somewhere, but it will come in handy. Now, if you get a program for calculating the inductance of a coil, you can easily tell why there are such formulas and what variables in them what they mean. This article is intended only for your reference, and if you want to know more, you should read the specialized literature (the benefit of many years of studying magnetic phenomena accumulated a lot).