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The Inductor

The Inductor

An Inductor is a passive electrical component consisting of a coil of wire which is designed to take advantage of the relationship between magentism and electricity as a result of an electric current passing through the coil

In this tutorial we will see that the inductor is an electrical component used to introduce inductance into a circuit which opposes the change of current flow, both magnitude and direction, and that even a straight piece of conductive wire can have some amount of inductance in it.

In our tutorials about Electromagnetism we saw that when an electrical current flows through a wire conductor, a magnetic flux is developed around that conductor. This affect produces a relationship between the direction of the magnetic flux, which is circulating around the conductor, and the direction of the current flowing through the same conductor. This results in a 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.

an inductor choke

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, (  ) around it that is proportional to this flow of electrical current.

An Inductor, also called a choke, is another passive type electrical component consisting of a coil of wire designed to take advantage of this relationship by inducing a magnetic field in itself or within its core as a result of the current flowing through the wire coil. Forming a wire coil into an inductor 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 Symbol

inductor construction

 

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, 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.
  • 100μH = 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 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:

faradays law of self induced emf

  • 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 L denoting 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: NΦ = 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:

Back emf Generated by an Inductor

back emf of an inductor

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

an inductor coil

Inductor Coil

So from this equation we can say that the “Self-induced emf equals Inductance times the 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.

inductor circuit

Consider the circuit of a pure 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 back-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 a steady state condition. In other words, in a real coil only the coils DC resistance exists to oppose the flow of current through itself.

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 another 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

current 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.

Tutorial Example No1

A steady state direct current of 4 ampere passes through a solenoid coil of 0.5H. What would be the average 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.

induced voltage in an inductor

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:

generated back emf

 

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

power absorbed

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

The Energy Stored

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

energy stored

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 = IR ) 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.

183 Comments

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  • Irshad Ahmad khanday

    Excellent topic

  • David Mac Farlane

    I am completely lost on why sometimes the equations for the inductor is written v = -L*di/dt and other times written as v = L*di/dt throughout the different sections. I must be misunderstanding something because the equation on back emf seems self contradictory to me. I agree with V = d(phi)/dt = d(L*I)/dt. But I disagree with the part that says d(L*I)/dt = -L*di/dt. Pulling a constant L out of the derivative should not negate the derivative. There are many examples where the minus signs is included as well as many examples where it is absent.

    The Fundamentals of Electric Circuits textbook By Alexander & Sadiku never included a negative sign that I recall, although it could sometimes possible to get a negative value provided that the reference polarity of the voltage was opposite the direction of current flow by the passive sign convention. Has that text pulled the wool over my eyes and done some hand waving with the minus signs.

    I really like how the text is presented overall and some of the content is explained more naturally than Alexander and Sadiku’s text. I want to use this article to teach someone I know about Mutual inductance. But if the inconsistency with minus signs are throwing off someone who is experienced, it will definitely throw off the person I intend to explain it to.

    • Wayne Storr

      Its quite clear, as explained in the tutorial. The induced emf opposes the flow of current through an inductor (Lenz’s Law). Thus if the flux, Φ generated by the inductor increases by a small amount Φ in a fraction of a second dt, so that the rate of increase is dΦ/dt. Then the back EMF induced in the single turn coil is: -v = dΦ/dt or -NdΦ/dt, in the case of an N-turn coil. Since the back emf v = -NdΦ/dt it therefore follows that v = -LdΦ/dt as it tries to cancel out those current changes. That is the rate of voltage change in an inductor is proportional to the current flowing through it. Thus, it can be viewed as a voltage on its own with a positive sign (+v), or as a back emf created as a direct result of current change with a negative sign (-v). The choice is yours.

  • Neeraj Kumar

    Thanks for your deep explanation of inductor I really influenced by it

    • Bruce W Kidd

      Neeraj, I just know an inductor can be a solid state as well…

      • johnny brize

        don’t send anymore e-mails

        • Bruce W Kidd

          So this has led to a diffusion equation…

          I have my L, R or Z circuit mapped out… Maybe it’s a diffusion equation too

          And if so, that many diffusion equations are, the consider the Schrodinger equation… Because it’s diffusion equation form too…
          So I put my uncertainty relationship on my Facebook page
          Bruce
          See it there…
          Bruce Kidd

      • johnny

        don’t send anymore e-mails to this address, and i mean don’t send any more email

        • Bruce W Kidd

          I have a similar quotient as for the diffusion equation of R and C, I have one for the magnetic field… I assume L, C circuit… And a differential equation and a unit less quotient…

  • Bruce W Kidd

    I’m wanting to parallel the resistor-capacitor derivation of the diffusion equation…

  • chiara martelli

    I really like your descriptions! it is so helpful for me!

    A question related to the formulas above.
    I understood that :
    L = u N^2 A / l

    and

    V(t) = dPH / dt = di / dt L / N

    But then the formula following the sentence <> has a “N” which should not be there, right?
    The formula written above is V = N dPH / dt , however it should only be V = dPH / dt, right?

    If this is not the case, could you please be so kind to explain the difference of V w.r.t. V(t) in the two cases? Thank you so much!

    • Wayne Storr

      The self-induced emf of an inductive coil is commonly given as the inductance of the coil times the rate of change of current through the same coil. This is presented mathematically for an ideal inductor as being: V(t) = -Ldi/dt

      If the number of turns of the coil is increased, then for the same intensity of current flowing through the coil the magnetic flux generated by the current is also increased. Thus increasing the number of coil turns also increases the self-inductance of the coil. That is: L is proportional to N. But also if we leave the number of coil windings (N) the same and double the intensity of the current, (i) flowing through the coil we will also double the generated magnetic flux (phi). Then we can correctly say that phi = Li

      Adding together the formulas for the voltage induced in the coil gives: V = Nphi/i x di/dt = Ndphi/dt

  • Mohan

    Very pure way of explaining concepts. Loved it.

  • Faruk

    Dear Sir
    Thank you for your explanation of inductor. i want to use a LC filter at a full wave bridge rectifier of 50 volt and 50 ampere. i used capacitor 3mF and need inductor of 12mH. My question how many turn and wire size i will use here??

  • Chi bui duc

    Thank you very much

  • Johnny

    Suburb tutorial, could you please tell what 12 volt regulator has a 1 amp capability your the best. thank you

    • Jeffrey Stroman

      JOHNNY, The three pin series of solid state regulators comes to mind, 7801 is for positive and 7901 is for negative regulation both are good for one amp. The pinout is in, ground and out, looking at the print and leads pointing down. These things are ubiquitous and you probably have one in the same room with you, albeit already used in something

  • Sandeep Singh

    So easy method use to explain, thanks Electronics Tutorials.

  • student

    Hello. First of all hats off for doing this tutorial and for your patience answering all the questions. thanks, really.
    Q1: This is a basic question about circuits. We have always been taught that in a single-loop circuit , each and every point (cross-section) of the circuit is experiencing identical amount of current flowing in the direction from higher to lower electrical potential. Is it possible that this assumption is not true / does not apply here?
    Q2: If it is true, then how can there be any current at all anywhere (shown in purple as linearly decreasing in the last diagram) after the the switch is turned off? One comment below points this out, but I haven’t found any resolution / explanation of where the current goes in A) ideal circuit / coil / materials, and B) actual circuits / coil / materials.
    Thanks again for all your efforts!

    • Wayne Storr

      1. Your assumption is correct for a single closed-loop circuit. As explained many times, and in this tutorial. An Inductor opposes or resists the rate of change of current flowing through it as it stores energy supplied by the source voltage within it’s magnetic field (being a coil of wire). Then inductors DO NOT respond instantly to step currents (when the switch is initially closed), but the current flowing around the single loop inductive circuit will build up slowly (exponentially) to it’s maximum value over a period of time as shown. This ability of an inductor to oppose current flow is called Inductance.

      2. Assuming initial steady state conditions. When the switch is opened (OFF) there is no voltage supply available to maintain the magnetic field created around the inductor, so it collapses (decay’s). As it does the change in current flow creates and induces an emf back into the same coil (Lenz’s Law) winding which opposes this decay of current exponentially down from its steady state value to it’s minimum value, (which will be zero) in zero time.

      The direction of this self-induced emf in the coil is the reverse of the supply voltage which created the magnetic field, (which is why it’s called a “back-emf”) so the decaying current is absorbed by the coil as it tries to convert the energy of the decaying current into a magnetic field. This self-induced voltage can be very high, which is why breaking a magnetic circuit results in arcing or sparking across the switch contacts.

      Then as explained many times: Inductors resist the rate of change in current, while Capacitors resist the rate of changes in voltage.

  • Bryan Watkins

    Nice job

  • Johnny

    Superior tutorial. I appreciate you taking the time and effort to produce these lessons

  • Kryštof Sirový

    Hello,
    want to ask anyone, how to cite this webpage,
    thanks

  • Pintu Prasad

    Much Informative post. I have learned much more about inductance. To learn about inductor in Hindi you can visit

  • Shashank

    Thank you so much
    Well explanation this really help to write good answers n also its uses

  • Tyassin

    Hi
    There is some serious errors in this tutorial.
    One example is: Inductor example nr1: Stady state direct current (DC) through an inductor and it gives 200V!!
    It is 0V.!!
    V=L*di/dt, di=0 so no voltage.
    Please remove or correct these statements. Can see people are reading this and thinking it is correct.

    • Wayne Storr

      Please make an effort to read the question and tutorial correctly. Initially 4 amperes is flowing through the solenoid coil with the switch closed. This current is constant and steady so there is no self-induced emf, but creates a magnetic field around the same coil. Energy is stored within this magnetic field. This could be calculated if you want at:  0.5LI2 joules. (Should be 4J if you do).

      When the switch contacts open there becomes a change in current flowing through the coil, but this change is not instant from 4 amperes to 0, but incremental over time as the self-inductance of the coil (0.5H in this example) causes an voltage to be induced in itself which opposes any current change (Lenz’s Law). The rate-of-change of current from 4 amperes to 0 in this simple example is given as being 10mS, that is the current (and also the magnetic field) decays from its old value to its new value over a time period of 10mS, thus dt = 0.01 seconds as given.

      The rate-of-change in current value is therefore: 4 – 0 = 4A, thus di = 4. Then the average back-emf induced in the solenoid coil as a result of energy being released from the decaying magnetic field surrounding the coil when the switch contacts are opened is given as being: V = L(di/dt) = 0.5 x (4/0.01) = 200 volts, the same value given in the tutorial. As the current is decaying, (switch open) then the induced emf is positive in value, unlike the growth of current, (switch closed) which would be negative in value. Again as given in the tutorial.

      One final point. For inductors, coils, chokes or any inductive circuit, the rate-of-change of current is never instant as energy is created, stored and released within its magnetic field, and unlike a capacitor which stores its energy as an electrostatic charge on its plates.

      • Alan

        Hi Wayne,

        Your tutorial is excellent and your patience in answering so many of the comments is admirable. As a EE, I think your approach helps many students who struggle with difficult concepts like inductance.

        The only thing I have trouble understanding is how you can tolerate some of the nonsense written by those who think they are right and you are wrong. You are spot on so good luck in keeping up the good work.

        Alan

        • Wayne Storr

          Hello Alan, thank you for your kind comments, they are much appreciated. We accept the negative comments as well as the positive comments, we do not filter them as everyone is entitled to their own opinion, whether it is right, wrong or misunderstood. Everyone is an expert these days, but we continue to provide free online content and answer the comments of those who wish to pursue a career in Electrical or Electronic Engineering. That is our driving force 🙂

      • sashi

        Help me to solve this please. A coil carries a current of 60 mA ans has an indicator of 300 micro H. Determine the flux in the coil. How come the answer is 360 Wb? 🙁

      • tyassin

        You say there is a steady current of 4A. In the given circuit there is no resistance, so is the steady current(if we can say that) not infinite?

        You open the switch and the current goes to zero over time? Where does this current flow? Through the switch which is open???

        Do for example a circuit simulation and see what result you get.

  • mahdi Boukerdja

    Thank you for this explanation, it is very readable. I want to know if the short-circuits in the inductor affect the internal resistance.

    • Wayne Storr

      The DC resistance of an inductor is that of the wire used to wind it and may be affected by excessive currents flowing through it.

  • B.P.A.D.Perera

    Very useful

  • Luis Franco

    In the “Current and Voltage in an Inductor” figures, the inductor voltage waveform is wrong. The correct waveform should be exactly the opposite.

    • Rupert UK

      I am afraid that Luis is CORRECT. Wayne – you have used the wrong formula for the induced Voltage. It is; v(t) = +L di(t)/dt. (Pls refer to Wikipedia Inductor).

      Wayne you have also corrected quoted Lenz’s Law, which will give the correct polarity.

      • Wayne Storr

        Your comment is flawed as you are quoting Wikipedia, but if you would like to quote Lenz’s Law then it basically states that: a self-induced voltage appears in an inductive coil whenever there is a change in the amount of current flowing within the same coil and that the polarity of this self-induced voltage will always be opposite and opposing the current change through the same coil. That is V(t) = –L(di/dt) as the action of opposing the current is called: self-inductance (L), Then the tutorial is correct as given.

    • Wayne Storr

      When a step input voltage (V) is applied to an inductive coil as shown, the rate of change of current (i) now flowing through the coil is not instant but varies with time (di/dt) as the magnetic flux created by the flow of current through the coil grows linearly as shown, up to a maximum value restricted by the coils resistance, R. This changing current induces a “back-emf”, that is a revese voltage, (-VL) in the same coil which opposes the rate at which the current changes, known commonly as self-inductance. Thus this self-induced back-emf opposes di/dt (Lenz’s law) by an amount determined by the design of the inductive coil.

      When the current flowing through the coil reaches its “steady-state” maximum value, there is no di/dt current change, so no generated back-emf, and VL reduces to zero volts, as shown. However, the magnetic field generated around the coil still exists as long as a steady state current flows, (electromagnet). When the supply voltage is removed, current stops flowing, the inductive coils magnetic field collapses as there is no steady state current to support it, and as there is a negative (reducing) di/dt occuring, a back-emf in the opposite direction (+VL) is created as shown, slowing down the rate of di/dt decay as shown and explained in the tutorial. Thus the tutorial is correct as given.