Induced Voltage: Faraday's Law

If you need to design a practical inductor then knowledge of Faraday's law leads to understanding of how voltage is induced in a coil and how you can relate that to the magnetic flux which its core must be able to cope with.

  • The basis of Faraday's law
  • The effect of coil current
  • The 'flip side to Faraday'
  • Inductor with AC applied
  • Conclusion: why Faraday is cool
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    See also ...
    [Up sign Producing wound components] [ Air coils] [A guide to the terminology of magnetism] [ Power loss in wound components]

    The basis of Faraday's law

    Michael Faraday's greatest contribution to physics was to show that a voltage, E, is generated by a coil of wire when the magnetic flux, greek letter
phi, enclosed by it changes
    E = N×dgreek letter phi/dt   volts

    where N is the number of turns.

    (Vector quantities express this more rigorously, but Faraday was even less of a maths fan than me.) The flux may change because In electric motors and generators you will usually have more than one of these causes at the same time. It doesn't matter what causes the change; the result is an induced voltage, and the faster the flux changes the greater the voltage.

    Take an example of an inductor having 7 turns and in which the flux varies according to

    Φ = 0.008 sin(300t)   webers
    Substituting this into Faraday's law
    E = 7 × d(0.008 sin(300t))/dt

    E= 7 × 0.008 × 300 cos(300t)

    E= 16.8 cos(300t)   volts

    So our winding will have a peak voltage of 16.8 volts. You can easily generalize this for any flux varying sinusoidally at a frequency f to show
    ERMS = 4.44 N × f × Φpk   volts
    This is called the transformer equation.

    The effect of coil current

    OK, that's all clear enough, but there is one other reason for alteration to the flux: current flow in the coil. Hans Christian Oersted discovered that an electric current can produce a magnetic field. The more current you have the more flux you generate. That, too, is easy enough to grasp. What needs a firm intellectual grip is to appreciate that Faraday's Law does not stop operating just because you have current flowing in the coil. When the coil current varies then that will alter the flux and, says Faraday, if the flux changes then you get an induced voltage. This merry-go-round between current, flux and voltage lies at the heart of electromagnetism. Calculating the sequence of operation just described is quite easy.

    The 'flip side to Faraday'

    Now let's spin our fairground ride in the other direction. Instead of getting an induced voltage by putting in a current we'll put a voltage across the coil and see what happens. Normally, if you put several volts across any randomly arranged bit of wire then what will happen will be a flash and a bang; the current will follow Ohm's law and (unless the wire is very long and thin) there won't be enough resistance to prevent fireworks. It's a different story when the wire is wound into a coil. If the current increases then we get flux build up which induces a voltage of its own. The sign of this induced voltage is always such that the voltage will be positive if the current into the coil increases. We say that the induced voltage will oppose the externally applied voltage which made the current change (Lenz's law). This creates a limit to the rate of rise of the current and prevents (at least temporarily) the melt-down we get without coiling.

    Calculating this 'flip side to Faraday' is also easy; we take his law in its differential form above and integrate:

    greek letter phi = ( Time integralE.dt ) / N   webers

    where E is the externally applied voltage and N is the number of turns.

    We'll call this the integral form of Faraday's law.

    Inductor with AC applied

    Let's apply a sinusoidal voltage, frequency f, RMS amplitude a -
    E = (√2)a.sin(2Greek character pi.f.t)

    Substituting this into the integral form of Faraday:
    greek letter phi = ( Time integral(√2)a.sin(2Greek character pi.f.t) .dt ) / N

    greek letter phi = ((√2)a/N) Time integralsin(2Greek character pi.f.t) .dt

    greek letter phi = (-(√2)a/(2Greek character pi.f.N)) [cos(2.Greek character pi.f.t)]0t

    The expression with the limits of integration will always be between -1 and +1 so that the peak value of flux is given by
    greek letter phipk = (√2)a/(2Greek character pi.f.N)

    greek letter phipk = a / (4.44 f.N)   Wb

    Compare this with the transformer equation above.

    Example: If 230 volts at 50 Hz is applied to an inductor having 200 turns then what is the peak value of magnetic flux?

    greek letter phipk = 230 / (4.44 × 50 × 200)

    greek letter phipk = 5.18   mWb

    One important general point: if your winding has to cope with a given signal amplitude then the core flux is proportional to the inverse of the frequency. This means, for example, that mains transformers operating at 50 or 60 Hz are larger than transformers in switching supplies (capable of handling the same power) working at 50 kHz.

    Using inductance

    If the material permeability is constant then the relation between flux and current is linear and, by the definition of inductance :
    greek letter phi = L×I/N   amps

    where L is the inductance of the coil.

    Substituting into the integral form of the law:
    L×I/N = ( Time integralE.dt ) / N

    I = ( Time integralE.dt ) / L   amps

    If E is a constant then this formula for the current simplifies to:
    I = E×t/L   amps
    Example: If a 820 mH coil has 2 volts applied then find the current at the end of three seconds.

    I = 2×3/0.82 = 7.32  amps
    Current increase in 820mH

    Conclusion: why Faraday is cool

    Satisfy yourself that this result above is consistent with our original formulation of Faraday's law: voltage is proportional to the rate of change of flux. Consider also how useful this integration method is in practical inductor design; if you know the number of turns on a winding and the voltage waveform on it then you integrate wrt time and voilá you have found the amount of flux. What's more is that you found it without knowing about the inductance or the core: its permeabilty, size, shape, or even whether there was a core at all.

    Gee - thanks Mike!

    Caveat (there's always one): The coils decribed here are idealised. See the paragraph on flux linkage for a more complete picture.

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    Last modified: 2001 April 29th.