INDUCTOR

The Inductor

In our tutorials about Electromagnetism we saw that when an electrical current flows through a wire conductor, a magnetic flux is developed around the conductor.

                                                                       

This produces a relationship between the direction of this magnetic flux which is circulating around the conductor and the direction of the current flowing through the same conductor resulting in a well known relationship between current and magnetic flux direction called, “Fleming’s Right Hand Rule”.

But there is also another important property relating to a wound coil that also exists, which is that a secondary voltage is induced into the same coil by the movement of the magnetic flux as it opposes or resists any changes in the electrical current flowing it.

A Typical Inductor

In its most basic form, an Inductor is nothing more than a coil of wire wound around a central core. For most coils the current, ( i ) flowing through the coil produces a magnetic flux, ( NΦ ) around it that is proportional to this flow of electrical current.

The Inductor, also called a choke, is another passive type electrical component which is just a coil of wire that is designed to take advantage of this relationship by inducing a magnetic field in itself or in the core as a result of the current passing through the coil. This results in a much stronger magnetic field than one that would be produced by a simple coil of wire.

Inductors are formed with wire tightly wrapped around a solid central core which can be either a straight cylindrical rod or a continuous loop or ring to concentrate their magnetic flux.

The schematic symbol for a inductor is that of a coil of wire so therefore, a coil of wire can also be called an Inductor. Inductors usually are categorised according to the type of inner core they are wound around, for example, hollow core (free air), solid iron core or soft ferrite core with the different core types being distinguished by adding continuous or dotted parallel lines next to the wire coil as shown below.

Inductor Symbols

 

The current, i that flows through an inductor produces a magnetic flux that is proportional to it. But unlike a Capacitor which oppose a change of voltage across their plates, an inductor opposes the rate of change of current flowing through it due to the build up of self-induced energy within its magnetic field.

In other words, inductors resist or oppose changes of current but will easily pass a steady state DC current. This ability of an inductor to resist changes in current and which also relates current, i with its magnetic flux linkage, NΦ as a constant of proportionality is called Inductance which is given the symbol L with units of Henry, (H) after Joseph Henry.

Because the Henry is a relatively large unit of inductance in its own right, for the smaller inductors sub-units of the Henry are used to denote its value. For example:

Inductance Prefixes

Prefix	Symbol	Multiplier	Power of Ten
milli	m	1/1,000	10-3
micro	µ	1/1,000,000	10-6
nano	n	1/1,000,000,000	10-9

So to display the sub-units of the Henry we would use as an example:

• 1mH = 1 milli-Henry  –  which is equal to one thousandths (1/1000) of an Henry.
• 100uH = 100 micro-Henries  –  which is equal to 100 millionth’s (1/1,000,000) of a Henry.

Inductors or coils are very common in electrical circuits and there are many factors which determine the inductance of a coil such as the shape of the coil, the number of turns of the insulated wire, the number of layers of wire, the spacing between the turns, the permeability of the core material, the size or cross-sectional area of the core etc, to name a few.

An inductor coil has a central core area, ( A ) with a constant number of turns of wire per unit length, ( l ). So if a coil of N turns is linked by an amount of magnetic flux, Φ then the coil has a flux linkage of NΦ and any current, ( i ) that flows through the coil will produce an induced magnetic flux in the opposite direction to the flow of current. Then according to Faraday’s Law, any change in this magnetic flux linkage produces a self-induced voltage in the single coil of:

• Where:
•     N is the number of turns
•     A is the cross-sectional Area in m2
•     Φ is the amount of flux in Webers
•     μ is the Permeability of the core material
•     l is the Length of the coil in meters
•     di/dt is the Currents rate of change in amps/second

A time varying magnetic field induces a voltage that is proportional to the rate of change of the current producing it with a positive value indicating an increase in emf and a negative value indicating a decrease in emf. The equation relating this self-induced voltage, current and inductance can be found by substituting the μN2A / l with Ldenoting the constant of proportionality called the Inductance of the coil.

The relation between the flux in the inductor and the current flowing through the inductor is given as: Φ = Li. As an inductor consists of a coil of conducting wire, this then reduces the above equation to give the self-induced emf, sometimes called the back emf induced in the coil too:

The back emf Generated by an Inductor

Where: L is the self-inductance and di/dt the rate of current change.

Inductor Coil

So from this equation we can say that the “self-induced emf = inductance x rate of current change” and a circuit has an inductance of one Henry will have an emf of one volt induced in the circuit when the current flowing through the circuit changes at a rate of one ampere per second.

One important point to note about the above equation. It only relates the emf produced across the inductor to changes in current because if the flow of inductor current is constant and not changing such as in a steady state DC current, then the induced emf voltage will be zero because the instantaneous rate of current change is zero, di/dt = 0.

With a steady state DC current flowing through the inductor and therefore zero induced voltage across it, the inductor acts as a short circuit equal to a piece of wire, or at the very least a very low value resistance. In other words, the opposition to the flow of current offered by an inductor is very different between AC and DC circuits.

The Time Constant of an Inductor

We now know that the current can not change instantaneously in an inductor because for this to occur, the current would need to change by a finite amount in zero time which would result in the rate of current change being infinite, di/dt = ∞, making the induced emf infinite as well and infinite voltages do no exist. However, if the current flowing through an inductor changes very rapidly, such as with the operation of a switch, high voltages can be induced across the inductors coil.

Consider the circuit of the inductor on the right. With the switch, ( S1 ) open, no current flows through the inductor coil. As no current flows through the inductor, the rate of change of current (di/dt) in the coil will be zero. If the rate of change of current is zero there is no self-induced emf, ( VL = 0 ) within the inductor coil.

If we now close the switch (t = 0), a current will flow through the circuit and slowly rise to its maximum value at a rate determined by the inductance of the inductor. This rate of current flowing through the inductor multiplied by the inductors inductance in Henry’s, results in some fixed value self-induced emf being produced across the coil as determined by Faraday’s equation above, VL = Ldi/dt.

This self-induced emf across the inductors coil, ( VL ) fights against the applied voltage until the current reaches its maximum value and a steady state condition is reached. The current which now flows through the coil is determined only by the DC or “pure” resistance of the coils windings as the reactance value of the coil has decreased to zero because the rate of change of current (di/dt) is zero in steady state. In other words, only the coils DC resistance now exists to oppose the flow of current.

Likewise, if switch, (S1) is opened, the current flowing through the coil will start to fall but the inductor will again fight against this change and try to keep the current flowing at its previous value by inducing a voltage in the other direction. The slope of the fall will be negative and related to the inductance of the coil as shown below.

Current and Voltage in an Inductor

 

How much induced voltage will be produced by the inductor depends upon the rate of current change. In our tutorial about Electromagnetic Induction, Lenz’s Law stated that: “the direction of an induced emf is such that it will always opposes the change that is causing it”. In other words, an induced emf will always OPPOSE the motion or change which started the induced emf in the first place.

So with a decreasing current the voltage polarity will be acting as a source and with an increasing current the voltage polarity will be acting as a load. So for the same rate of current change through the coil, either increasing or decreasing the magnitude of the induced emf will be the same.

Inductor Example No1

A steady state direct current of 4 ampere passes through a solenoid coil of 0.5H. What would be the back emf voltage induced in the coil if the switch in the above circuit was opened for 10mS and the current flowing through the coil dropped to zero ampere.

Power in an Inductor

We know that an inductor in a circuit opposes the flow of current, ( i ) through it because the flow of this current induces an emf that opposes it, Lenz’s Law. Then work has to be done by the external battery source in order to keep the current flowing against this induced emf. The instantaneous power used in forcing the current, ( i ) against this self-induced emf, ( VL ) is given from above as:

 

Power in a circuit is given as, P = V.I therefore:

An ideal inductor has no resistance only inductance so R = 0 Ω’s and therefore no power is dissipated within the coil, so we can say that an ideal inductor has zero power loss.

Energy in an Inductor

When power flows into an inductor, energy is stored in its magnetic field. When the current flowing through the inductor is increasing and di/dt becomes greater than zero, the instantaneous power in the circuit must also be greater than zero, ( P > 0 ) ie, positive which means that energy is being stored in the inductor.

Likewise, if the current through the inductor is decreasing and di/dt is less than zero then the instantaneous power must also be less than zero, ( P < 0 ) ie, negative which means that the inductor is returning energy back into the circuit. Then by integrating the equation for power above, the total magnetic energy which is always positive, being stored in the inductor is therefore given as:

Energy stored by an Inductor

Where:  W is in joules, L is in Henries and i is in Amperes

The energy is actually being stored within the magnetic field that surrounds the inductor by the current flowing through it. In an ideal inductor that has no resistance or capacitance, as the current increases energy flows into the inductor and is stored there within its magnetic field without loss, it is not released until the current decreases and the magnetic field collapses.

Then in an alternating current, AC circuit an inductor is constantly storing and delivering energy on each and every cycle. If the current flowing through the inductor is constant as in a DC circuit, then there is no change in the stored energy as P = Li(di/dt) = 0.

So inductors can be defined as passive components as they can both stored and deliver energy to the circuit, but they cannot generate energy. An ideal inductor is classed as loss less, meaning that it can store energy indefinitely as no energy is lost.

However, real inductors will always have some resistance associated with the windings of the coil and whenever current flows through a resistance energy is lost in the form of heat due to Ohms Law, ( P = I2 R ) regardless of whether the current is alternating or constant.

Then the primary use for inductors is in filtering circuits, resonance circuits and for current limiting. An inductor can be used in circuits to block or reshape alternating current or a range of sinusoidal frequencies, and in this role an inductor can be used to “tune” a simple radio receiver or various types of oscillators. It can also protect sensitive equipment from destructive voltage spikes and high inrush currents.

In the next tutorial about Inductors, we will see that the effective resistance of a coil is called Inductance, and that inductance which as we now know is the characteristic of an electrical conductor that “opposes a change in the current”, can either be internally induced, called self-inductance or externally induced, called mutual-inductance.

Inductance of a Coil

Inductance is the name given to the property of a component that opposes the change of current flowing through it and even a straight piece of wire will have some inductance.

                                                               

Inductors do this by generating a self-induced emf within itself as a result of their changing magnetic field. In an electrical circuit, when the emf is induced in the same circuit in which the current is changing this effect is called Self-induction, ( L ) but it is sometimes commonly called back-emf as its polarity is in the opposite direction to the applied voltage.

When the emf is induced into an adjacent component situated within the same magnetic field, the emf is said to be induced by Mutual-induction, ( M ) and mutual induction is the basic operating principal of transformers, motors, relays etc. Self inductance is a special case of mutual inductance, and because it is produced within a single isolated circuit we generally call self-inductance simply, Inductance.

The basic unit of measurement for inductance is called the Henry, ( H ) after Joseph Henry, but it also has the units of  Webers per Ampere ( 1 H = 1 Wb/A ).

Lenz’s Law tells us that an induced emf generates a current in a direction which opposes the change in flux which caused the emf in the first place, the principal of action and reaction. Then we can accurately define Inductance as being: “a coil will have an inductance value of one Henry when an emf of one volt is induced in the coil were the current flowing through the said coil changes at a rate of one ampere/second”.

In other words, a coil has an inductance, ( L ) of one Henry, ( 1H ) when the current flowing through it changes at a rate of one ampere/second, ( A/s ) inducing a voltage of one volt, ( VL ) in it. This mathematical representation of the rate of change in current through a coil per unit time is given as:

 

Where: di is the change in the current in Amperes and dt is the time taken for this current to change in seconds. Then the voltage induced in a coil, ( VL ) with an inductance of L Henries as a result of this change in current is expressed as:

 

Note that the negative sign indicates that voltage induced opposes the change in current through the coil per unit time ( di/dt ).

From the above equation, the inductance of a coil can therefore be presented as:

Inductance of a Coil

 

Where: L is the inductance in Henries, VL is the voltage across the coil and di/dt is the rate of change of current in Amperes per second, A/s.

Inductance, L is actually a measure of an inductors “resistance” to the change of the current flowing through the circuit and the larger is its value in Henries, the lower will be the rate of current change.

We know from the previous tutorial about the Inductor, that inductors are devices that can store their energy in the form of a magnetic field. Inductors are made from individual loops of wire combined to produce a coil and if the number of loops within the coil are increased, then for the same amount of current flowing through the coil, the magnetic flux will also increase.

So by increasing the number of loops or turns within a coil, increases the coils inductance. Then the relationship between self-inductance, ( L ) and the number of turns, ( N ) and for a simple single layered coil can be given as:

Self Inductance of a Coil

• Where:
•         L is in Henries
•         N is the Number of Turns
•         Φ is the Magnetic Flux Linkage
•         Ι  is in Amperes

This expression can also be defined as the flux linkage divided by the current flowing through each turn. This equation only applies to linear magnetic materials.

Inductance Example No1

A hollow air cored inductor coil consists of 500 turns of copper wire which produces a magnetic flux of 10mWb when passing a DC current of 10 amps. Calculate the self-inductance of the coil in milli-Henries.

 

 

Inductance Example No2

Calculate the value of the self-induced emf produced in the same coil after a time of 10mS.

 

The self-inductance of a coil or to be more precise, the coefficient of self-inductance also depends upon the characteristics of its construction. For example, size, length, number of turns etc. It is therefore possible to have inductors with very high coefficients of self induction by using cores of a high permeability and a large number of coil turns. Then for a coil, the magnetic flux that is produced in its inner core is equal to:

 

Where: Φ is the magnetic flux linkage, B is the flux density, and A is the area.

If the inner core of a long solenoid coil with N number of turns per metre length is hollow, “air cored”, then the magnetic induction within its core will be given as:

 

Then by substituting these expressions in the first equation above for Inductance will give us:

 

By cancelling out and grouping together like terms, then the final equation for the coefficient of self-inductance for an air cored coil (solenoid) is given as:

• Where:
•         L is in Henries
•         μο is the Permeability of Free Space (4.π.10-7)
•         N is the Number of turns
•         A is the Inner Core Area (π.r 2) in m2
•         l is the length of the Coil in metres

As the inductance of a coil is due to the magnetic flux around it, the stronger the magnetic flux for a given value of current the greater will be the inductance. So a coil of many turns will have a higher inductance value than one of only a few turns and therefore, the equation above will give inductance L as being proportional to the number of turns squared N2.

As well as increasing the number of coil turns, we can also increase inductance by increasing the coils diameter or making the core longer. In both cases more wire is required to construct the coil and therefore, more lines of force exists to produce the required back emf. The inductance of a coil can be increased further still if the coil is wound onto a ferromagnetic core, that is one made of a soft iron material, than one wound onto a non-ferromagnetic or hollow air core.

Ferrite Core

If the inner core is made of some ferromagnetic material such as soft iron, cobalt or nickel, the inductance of the coil would greatly increase because for the same amount of current flow the magnetic flux generated would be much stronger. This is because the material concentrates the lines of force more strongly through the the softer ferromagnetic core material as we saw in the Electromagnets tutorial.

So for example, if the core material has a relative permeability 1000 times greater than free space, 1000μο such as soft iron or steel, then the inductance of the coil would be 1000 times greater so we can say that the inductance of a coil increases proportionally as the permeability of the core increases.

Then for a coil wound around a former or core the inductance equation above would need to be modified to include the relative permeability μr of the new former material.

If the coil is wound onto a ferromagnetic core a greater inductance will result as the cores permeability will change with the flux density. However, depending upon the ferromagnetic material the inner cores magnetic flux may quickly reach saturation producing a non-linear inductance value and since the flux density around the coil depends upon the current flowing through it, inductance, L also becomes a function of current flow, i.

In the next tutorial about inductors, we will see that the magnetic field generated by a coil can cause a current to flow in a second coil that is placed next to it. This effect is called Mutual Inductance, and is the basic operating principle of transformers, motors and generators.

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 a 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.

Coupling Coefficient

Generally, the amount of inductive coupling that exists between the two coils is expressed as a fractional number between 0 and 1 instead of a percentage (%) value, where 0 indicates zero or no inductive coupling, and 1 indicating full or maximum inductive coupling.

In other words, if k = 1 the two coils are perfectly coupled, if k > 0.5 the two coils are said to be tightly coupled and if k < 0.5 the two coils are said to be loosely coupled. Then the equation above which assumes a perfect coupling can be modified to take into account this coefficient of coupling, k and is given as:

Coupling Factor Between Coils

or

 

When the coefficient of coupling, k is equal to 1, (unity) such that all the lines of flux of one coil cuts all of the turns of the second coil, that is the two coils are tightly coupled together, the resulting mutual inductance will be equal to the geometric mean of the two individual inductances of the coils.

Also when the inductances of the two coils are the same and equal, L 1 is equal to L 2, the mutual inductance that exists between the two coils will equal the value of one single coil as the square root of two equal values is the same as one single value as shown.

Mutual Inductance Example No1

Two inductors whose self-inductances are given as 75mH and 55mH respectively, are positioned next to each other on a common magnetic core so that 75% of the lines of flux from the first coil are cutting the second coil. Calculate the total mutual inductance that exists between the two coils.

Mutual Inductance Example No2

When two coils having inductances of 5H and 4H respectively were wound uniformly onto a non-magnetic core, it was found that their mutual inductance was 1.5H. Calculate the coupling coefficient that exists between.

 

In the next tutorial about Inductors, we look at connecting together Inductors in Seriesand the affect this combination has on the circuits mutual inductance, total inductance and their induced voltages.

Inductors in Series

Inductors can be connected together in either a series connection, a parallel connection or combinations of both series and parallel together.

                             

These interconnections of inductors produce more complex networks whose overall inductance is a combination of the individual inductors. However, there are certain rules for connecting inductors in series or parallel and these are based on the fact that no mutual inductance or magnetic coupling exists between the individual inductors.

Inductors are said to be connected in “Series” when they are daisy chained together in a straight line, end to end. In the Resistors in Series tutorial we saw that the different values of the resistances connected together in series just “add” together and this is also true of inductance. Inductors in series are simply “added together” because the number of coil turns is effectively increased, with the total circuit inductance LT being equal to the sum of all the individual inductances added together.

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 a Common 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:

Inductors in Series Equation

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 No1

Three 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.

Mutually Connected Inductors in Series

When inductors are connected together in series so that the magnetic field of one links with the other, the effect of mutual inductance either increases or decreases the total inductance depending upon the amount of magnetic coupling. The effect of this mutual inductance depends upon the distance apart of the coils and their orientation to each other.

Mutually connected series inductors can be classed as either “Aiding” or “Opposing” the total inductance. If the magnetic flux produced by the current flows through the coils in the same direction then the coils are said to be Cumulatively Coupled. If the current flows through the coils in opposite directions then the coils are said to be Differentially Coupled as shown below.

Cumulatively Coupled Series Inductors

 

While the current flowing between points A and D through the two cumulatively coupled coils is in the same direction, the equation above for the voltage drops across each of the coils needs to be modified to take into account the interaction between the two coils due to the effect of mutual inductance. The self inductance of each individual coil, L1 and L2 respectively will be the same as before but with the addition of Mdenoting the mutual inductance.

Then the total emf induced into the cumulatively coupled coils is given as:

 

Where: 2M represents the influence of coil L1 on L2 and likewise coil L2 on L1.

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 the inductors are cumulatively connected and this is given as:

Ltotal = L 1 + L 2 + 2M

If one of the coils is reversed so that the same current flows through each coil but in opposite directions, the mutual inductance, M that exists between the two coils will have a cancelling effect on each coil as shown below.

Differentially Coupled Series Inductors

 

The emf that is induced into coil 1 by the effect of the mutual inductance of coil 2 is in opposition to the self-induced emf in coil 1 as now the same current passes through each coil in opposite directions. To take account of this cancelling effect a minus sign is used with M when the magnetic field of the two coils are differentially connected giving us the final equation for calculating the total inductance of a circuit when the inductors are differentially connected as:

Ltotal = L 1 + L 2 – 2M

Then the final equation for inductively coupled inductors in series is given as:

Inductors in Series Example No2

Two inductors of 10mH respectively are connected together in a series combination so that their magnetic fields aid each other giving cumulative coupling. Their mutual inductance is given as 5mH. Calculate the total inductance of the series combination.

Inductors in Series Example No3

Two coils connected in series have a self-inductance of 20mH and 60mH respectively. The total inductance of the combination was found to be 100mH. Determine the amount of mutual inductance that exists between the two coils assuming that they are aiding each other.

Inductors in Series Summary

We now know that we can connect together inductors in series to produce a total inductance value, LT equal to the sum of the individual values, they add together, similar to connecting together resistors in series. However, when connecting together inductors in series they can be influenced by mutual inductance.

Mutually connected series inductors are classed as either “aiding” or “opposing” the total inductance depending whether the coils are cumulatively coupled (in the same direction) or differentially coupled (in opposite direction).

In the next tutorial about Inductors, we will see that the position of the coils when connecting together Inductors in Parallel also affects the total inductance, LT of the circuit.

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.

Mutually Coupled Inductors in Parallel

When inductors are connected together in parallel so that the magnetic field of one links with the other, the effect of mutual inductance either increases or decreases the total inductance depending upon the amount of magnetic coupling that exists between the coils. The effect of this mutual inductance depends upon the distance apart of the coils and their orientation to each other.

Mutually connected inductors in parallel can be classed as either “aiding” or “opposing” the total inductance with parallel aiding connected coils increasing the total equivalent inductance and parallel opposing coils decreasing the total equivalent inductance compared to coils that have zero mutual inductance.

Mutual coupled parallel coils can be shown as either connected in an aiding or opposing configuration by the use of polarity dots or polarity markers as shown below.

Parallel Aiding Inductors

 

The voltage across the two parallel aiding inductors above must be equal since they are in parallel so the two currents, i1 and i2 must vary so that the voltage across them stays the same. Then the total inductance, LT for two parallel aiding inductors is given as:

 

Where: 2M represents the influence of coil L 1 on L 2 and likewise coil L 2 on L 1.

If the two inductances are equal and the magnetic coupling is perfect such as in a toroidal circuit, then the equivalent inductance of the two inductors in parallel is L as LT = L1 = L2 = M. However, if the mutual inductance between them is zero, the equivalent inductance would be L ÷ 2 the same as for two self-induced inductors in parallel.

If one of the two coils was reversed with respect to the other, we would then have two parallel opposing inductors and the mutual inductance, M that exists between the two coils will have a cancelling effect on each coil instead of an aiding effect as shown below.

Parallel Opposing Inductors

 

Then the total inductance, LT for two parallel opposing inductors is given as:

 

This time, if the two inductances are equal in value and the magnetic coupling is perfect between them, the equivalent inductance and also the self-induced emf across the inductors will be zero as the two inductors cancel each other out.

This is because as the two currents, i1 and i2 flow through each inductor in turn the total mutual flux generated between them is zero because the two flux’s produced by each inductor are both equal in magnitude but in opposite directions.

Then the two coils effectively become a short circuit to the flow of current in the circuit so the equivalent inductance, LT becomes equal to ( L ± M ) ÷ 2.

Inductors in Parallel Example No2

Two inductors whose self-inductances are of 75mH and 55mH respectively are connected together in parallel aiding. Their mutual inductance is given as 22.5mH. Calculate the total inductance of the parallel combination.

Inductors in Parallel Example No3

Calculate the equivalent inductance of the following inductive circuit.

 

Calculate the first inductor branch LA, (Inductor L5 in parallel with inductors L6 and L7)

 

Calculate the second inductor branch LB, (Inductor L3 in parallel with inductors L4 and LA)

 

Calculate the equivalent circuit inductance LEQ, (Inductor L1 in parallel with inductors L2and LB)

 

Then the equivalent inductance for the above circuit was found to be: 15mH.

Inductors in Parallel Summary

As with the resistor, inductors connected together in parallel have the same voltage, Vacross them. Also connecting together inductors in parallel decreases the effective inductance of the circuit with the equivalent inductance of “N” inductors connected in parallel being the reciprocal of the sum of the reciprocals of the individual inductances.

As with series connected inductors, mutually connected inductors in parallel are classed as either “aiding” or “opposing” this total inductance depending whether the coils are cumulatively coupled (in the same direction) or differentially coupled (in opposite direction).

Thus far we have examined the inductor as a pure or ideal passive component. In the next tutorial about Inductors, we will look at non-ideal inductors that have real world resistive coils producing the equivalent circuit of an inductor in series with a resistance and examine the time constant of such a circuit.