Inductive Reactance Theory

The theory behind the inductive reactance and the proofs of the commonly used equations are relatively straightforward and provide some interesting insights.

Inductance and Transformer Tutorial Includes:
Inductance Symbols Lenz's law Self inductance Inductive reactance calculations Inductive reactance theory Inductance of wire & coils Mutual inductance Transformers

Although it is good to be able to use the equations to calculate inductive reactance, and to add inductive reactance and resistance together, it is also useful to see how the equations come about with a little of the theory of inductance.

Understanding how the inductance equations have come about from the theory gives further understanding, a topic which is not always fully understood.

To understand how inductive reactance occurs, it helps to take the case of an inductor with a sinusoidal alternating current passing through it. As inductors are widely used in electronic cicuits of all sorts, this can be very useful.

The sinusoidal current will cause the magnetic field to vary and this will induce a back EMF which will also be sinusoidal.

If the current passing through the inductor is i(t) = I_p sin(ωt) then using some simple theory, the voltage across the inductor can be expressed as:

v (t) = L \frac{d i}{d t} = L \frac{d}{d t} (I_{p} \sin (ω t))

v (t) = ω L I_{p} \cos (ω t)

v (t) = ω L I_{p} \sin (ω t + \frac{π}{2})

This is where:
I_p is the peak amplitude of the sinusoidal current in amperes
ω is 2 π f the angular frequency of the alternating current
f is the frequency of the alternating current in Hz
L is th inductance of the inductor in Henries.

Therefore the peak amplitude of the voltage across the inductor will be:

V_{p} = ω L I_{p}

V_{p} = 2 π f L I_{p}

Inductive reactance can be defined as the opposition cause by an inductor to the flow of current from an alternating waveform. It can be treated in a similar way to electrical resistance. It is therefore possible to calculate the reactance of an inductance.

X_{L} = \frac{V_{p}}{I_{p}} = 2 π f L

Just like resistance, the unit of reactance is the Ohm. It can also be seen from the equation, that the reactance of an inductor increases linearly with increasing frequency.

An often overlooked fact is that because the voltage induced is greatest when the current change is at its maximum, the voltage and current are ninety degrees out of phase. The voltage peaks occurs earlier in the cycle than the current peak.

The phase difference between the current and the induced voltage is equal to φ = π / 2 radians or 90 degrees. Thus in an ideal inductor the current lags the voltage by 90°.

Current waveform lags the voltage in an ideal inductor

The important element to take from the proof is that the inductive reactance of an inductor is equal to the angular frequency times the inductance. Also the current lags the voltage by 90°.