In electromagnetism and electronics, inductance is the property of an electrical conductor by which a change in current through it induces an electromotive force in both the conductor itself[ and in any nearby conductors by mutual inductance.

The term inductance was coined by Oliver Heaviside in 1886. It is customary to use the symbol L for inductance, in honour of the physicist Heinrich Lenz. In the SI system, the measurement unit for inductance is the henry, with the unit symbol H, named in honor

of Joseph Henry, who discovered inductance independently of, but not before, Faraday.

The henry is a derived unit based on four of the seven base units of the International System of Units: kilogram (kg), meter (m),second (s), and ampere (A). Expressed in combinations of SI units, the henry is:[4]

where

Wb = weber,T = tesla,J = joule,m = meter,s = second,A = ampere,V = volt,C = coulomb,F = farad,Hz = hertz,Ω = ohm

Inductors do not behave the same as resistors. Whereas resistors simply oppose the flow of electrons through them (by dropping a voltage directly proportional to the current), inductors oppose changes in current through them, by dropping a voltage directly proportional to the rate of change of current. In accordance with Lenz’s Law, this induced voltage is always of such a polarity as to try to maintain current at its present value. That is, if current is increasing in magnitude, the induced voltage will “push against” the electron flow; if current is decreasing, the polarity will reverse and “push with” the electron flow to oppose the decrease. This opposition to current change is called reactance, rather than resistance.

Expressed mathematically, the relationship between the voltage dropped across the inductor and rate of current change through the inductor is as such:

The expression di/dt is one from calculus, meaning the rate of change of instantaneous current (i) over time, in amps per second. The inductance (L) is in Henrys, and the instantaneous voltage (e), of course, is in volts. Sometimes you will find the rate of instantaneous voltage expressed as “v” instead of “e” (v = L di/dt), but it means the exact same thing. To show what happens with alternating current, let’s analyze a simple inductor circuit.

Inductor in Series Circuit

The current, ( I ) that flows through the first inductor, L1 has no other way to go but pass through the second inductor and the third and so on. Then, series inductors have aCommon Current flowing through them, for example:

 IL1 = IL2 = IL3 = IAB …etc.

 In the example above, the inductors L1, L2 and L3 are all connected together in series between points A and B. The sum of the individual voltage drops across each inductor can be found using Kirchoff’s Voltage Law (KVL) where, VT = V1 + V2 + V3 and we know from the previous tutorials on inductance that the self-induced emf across an inductor is given as: V = L di/dt.

So by taking the values of the individual voltage drops across each inductor in our example above, the total inductance for the series combination is given as:

 By dividing through the above equation by di/dt we can reduce it to give a final expression for calculating the total inductance of a circuit when connecting inductors together in series and this is given as:

Ltotal = L1 + L2 + L3 + ….. + Ln etc.Then the total inductance of the series chain can be found by simply adding together the individual inductances of the inductors in series just like adding together resistors in series. However, the above equation only holds true when there is “NO” mutual

inductance or magnetic coupling between two or more of the inductors, (they are magnetically isolated from each other).

One important point to remember about inductors in series circuits, the total inductance ( LT ) of any two or more inductors connected together in series will always be GREATER than the value of the largest inductor in the series chain.

Inductors in Series Example No1Three inductors of 10mH, 40mH and 50mH are connected together in a series combination with no mutual inductance between them. Calculate the total inductance of the series combination.

Inductors in Parallel

Inductors are said to be connected together in “Parallel” when both of their terminals

are respectively connected to each terminal of the other inductor or inductors.

The voltage drop across all of the inductors in parallel will be the same. Then, Inductors in Parallel have a Common Voltage across them and in our example below the voltage across the inductors is given as:

VL1 = VL2 = VL3 = VAB …etc In the following circuit the inductors L1, L2 and L3 are all connected together in parallel between the two points A and B.

Inductors in Parallel Circuit

In the previous series inductors tutorial, we saw that the total inductance, LT of the circuit was equal to the sum of all the individual inductors added together. For inductors in parallel the equivalent circuit inductance LT is calculated differently.

The sum of the individual currents flowing through each inductor can be found using Kirchoff’s Current Law (KCL) where, IT = I1 + I2 + I3 and we know from the previous tutorials on inductance that the self-induced emf across an inductor is given as:V = L di/dt

Then by taking the values of the individual currents flowing through each inductor in our circuit above, and substituting the current i for i1 + i2 + i3 the voltage across the parallel combination is given as:

By substituting di/dt in the above equation with v/L gives:

 We can reduce it to give a final expression for calculating the total inductance of a circuit when connecting inductors in parallel and this is given as:

Parallel Inductor Equation

Here, like the calculations for parallel resistors, the reciprocal ( 1/Ln ) value of the individual inductances are all added together instead of the inductances themselves. But again as with series connected inductances, the above equation only holds true when there is “NO” mutual inductance or magnetic coupling between two or more of the inductors, (they are magnetically isolated from each other). Where there is coupling between coils, the total inductance is also affected by the amount of coupling.

This method of calculation can be used for calculating any number of individual inductances connected together within a single parallel network. If however, there are only two individual inductors in parallel then a much simpler and quicker formula can be used to find the total inductance value, and this is:

One important point to remember about inductors in parallel circuits, the total inductance ( LT ) of any two or more inductors connected together in parallel will always be LESS than the value of the smallest inductance in the parallel chain.

Inductors in Parallel Example No1

Three inductors of 60mH, 120mH and 75mH respectively, are connected together in a parallel combination with no mutual inductance between them. Calculate the total inductance of the parallel combination in millihenries.

Inductor Circuit in Series and In Parallel

We'll now do an inductor circuit in which inductors are both in series and in parallel in the same circuit.

Below is a circuit which has inductors in both series and parallel:

So how do we add them to find the total inductance value?

First, we can start by finding the resistance of the resistors in series. In the first branch, containing the 20H and 40H inductors, the series resistance is 60H. And in the second branch, containing the 30H and 60H inductors, the series inductance is 90H. Now in total, the circuit has 3 inductances in parallel, 10H, 60H, and 90h. Now, we plug these 3 values into the parallel inductance formula and we get a total inductance of 7.83H. 

If you want to test the above series and parallel connections out practically, get 1mH inductor or whatever inductors you have, but let them be of the same value. In this example, I'll stick with 2 1mH inductors. Take the inductors and place them in series. Now take a multimeter and place the multimeter in the inductance setting (if available) and place the probes over the 2 inductors You should read just about 2mH, which is double the value of both inductors. This proves that inductors add when connected in series. Now place the inductors in parallel. Take the multimeter probes and place one end on one side of a inductor (either one) and place the other probe on the other side of that inductor. You should now read about 0.5mH, or half the value, because inductance decreases in parallel. This is a practical, real-life test you can do to show how inductors add.

Energy Stored in an Inductor

Suppose that an inductor of inductance   is connected to a variable DC voltage supply. The supply is adjusted so as to increase the current   flowing through the inductor from zero to some final value  . As the current through

the inductor is ramped up, an emf   is generated, which acts to oppose the increase in the current. Clearly, work must be done against this emf by the voltage source in order to establish the current in the inductor. The work done by the voltage source during a time interval   is 

(247)

Here,   is the instantaneous rate at which the voltage source performs work. To find the total work   done in establishing the final current   in the inductor, we must integrate the above expression. Thus, 

(248)

giving 

(249)

This energy is actually stored in the magnetic field generated by the current flowing through the inductor. In a pure inductor, the energy is stored without loss, and is returned to the rest of the circuit when the current through the inductor is ramped down, and its associated magnetic field collapses.

Consider a simple solenoid. Equations  (244), (246), and (249) can be combined to give 

(250)

which reduces to 

(251)

This represents the energy stored in the magnetic field of the solenoid. However, the volume of the field-filled core of the solenoid is  , so the magnetic energy density (i.e., the energy per unit volume) inside the solenoid

is  , or 

Mutual Inductance

In the previous tutorial we saw that an inductor generates an induced emf within itself as a result of the changing magnetic field around its own turns.

When this emf is induced in the same circuit in which the current is changing this effect is called Self-induction, ( L ). However, when the emf is induced into an adjacent coil situated within the same magnetic field, the emf is said to be induced magnetically, inductively or by Mutual induction, symbol ( M ). Then when two or more coils are magnetically linked together by a common magnetic flux they are said to have the property of Mutual Inductance.

Mutual Inductance is the basic operating principal of the transformer, motors, generators and any other electrical component that interacts with another magnetic field. Then we can define mutual induction as the current flowing in one coil that induces an voltage in an adjacent coil.

But mutual inductance can also be a bad thing as “stray” or “leakage” inductance from a coil can interfere with the operation of another adjacent component by means of electromagnetic induction, so some form of electrical screening to a ground potential may be required.

The amount of mutual inductance that links one coil to another depends very much on the relative positioning of the two coils. If one coil is positioned next to the other coil so that their physical distance apart is small, then nearly all of the magnetic flux generated by the first coil will interact with the coil turns of the second coil inducing a relatively large emf and therefore

producing a large mutual inductance value.

Likewise, if the two coils are farther apart from each other or at different angles, the amount of induced magnetic flux from the first coil into the second will be weaker producing a much smaller induced emf and therefore a much smaller mutual inductance value. So the effect of mutual inductance is very much dependant upon the relative positions or spacing, ( S ) of the two coils and this is demonstrated below.

Mutual Inductance between Coils

The mutual inductance that exists between the two coils can be greatly increased by positioning them on a common soft iron core or by increasing the number of turns of either coil as would be found in a transformer.

If the two coils are tightly wound one on top of the other over a common soft iron core unity coupling is said to exist between them as any losses due to the leakage of flux will be extremely small. Then assuming a perfect flux linkage between the two coils the mutual inductance that exists between them can be given as.

Where:

        µo is the permeability of free space (4.π.10-7)

        µr is the relative permeability of the soft iron core

        N is in the number of coil turns

        A is in the cross-sectional area in m2

        l is the coils length in meters

Mutual Induction

Here the current flowing in coil one, L1 sets up a magnetic field around itself with some of these magnetic field lines passing through coil two, L2 giving us mutual inductance. Coil one has a current of I1 and N1 turns while, coil two has N2 turns. Therefore, the mutual inductance, M12 of coil two that exists with respect to coil one depends on their position with respect to each other and is given as:

Likewise, the flux linking coil one, L1 when a current flows around coil two, L2 is exactly the same as the flux linking coil two when the same current flows around coil one above, then the mutual inductance of coil one with respect of coil two is defined as M21. This mutual inductance is true irrespective of the size, number of turns, relative position or orientation of the two coils. Because of this, we can write the mutual inductance between the two coils as: M12 = M21 = M.

Then we can see that self inductance characterises an inductor as a single circuit element, while mutual inductance signifies some form of magnetic coupling between two inductors or coils, depending on their distance and arrangement, an hopefully we remember from our tutorials on Electromagnets that the self inductance of each individual coil is given as:

and

By cross-multiplying the two equations above, the mutual inductance, M that exists between the two coils can be expressed in terms of the self inductance of each coil.

giving us a final and more common expression for the mutual inductance between the two coils of:

Mutual Inductance Between Coils

However, the above equation assumes zero flux leakage and 100% magnetic coupling between the two coils, L 1 and L 2. In reality there will always be some loss due to leakage and position, so the magnetic coupling between the two coils can never reach or exceed 100%, but can become very close to this value in some special inductive coils.

If some of the total magnetic flux links with the two coils, this amount of flux linkage can be defined as a fraction of the total possible flux linkage between the coils. This fractional value is called the coefficient of coupling and is given the letter k.