Inductors and the Farady emf

Inductors and the Farady emf

An inductor is usually a coil of wire. In an ideal inductor, the resistance of this wire is negligibile, as is its capacitance. The voltage that appears across an inductor is due to its own magnetic field and Faraday's law of electromagnetic induction. The current i(t) in the coil sets up a magnetic field, whose magnetic flux φ_B is proportional to the field strength, which is proportional to the current flowing. (Do not confuse the phase φ with the flux φ_B.) So we define the (self) inductance of the coil thus:

φ_B(t) = L.i(t)

Faraday's law gives the emf E_L = - dφ_B/dt. Now this emf is a voltage rise, so for the voltage drop v_L across the inductor, we have:

Again we define the inductive reactance X_L as the ratio of the magnitudes of the voltage and current, and from the equation above we see that X_L = ωL. Again we note the analogy to Ohm's law: the voltage is proportional to the current, and the peak voltage and currents are related by

V_m = X_L.I_m.

Remembering that the derivative is the local slope of the curve (the purple line), we can see in the next animation why voltage and current are out of phase in an inductor.

Again, there is a difference in phase: the derivative of the sinusoidal current is a cos function: it has its maximum (largest voltage across the inductor) when the current is changing most rapidly, which is when the current is intantaneously zero. The animation should make this clear. The voltage across the ideal inductor is 90° ahead of the current, (ie it reaches its peak one quarter cycle before the current does). Note how this is represented on the phasor diagram.
Again we note that the reactance is frequency dependent X_L = ωL. This is shown in the next animation: when the frequency is halved but the current amplitude kept constant, the current is varying only half as quickly, so its derivative is half as great, as is the Faraday emf. For an inductor, the ratio of voltage to current increases with frequency, as the next animation shows.