AC Inductors: UNIZOR.COM - Physics4Teens - Electromagnetism
Channel: Zor Shekhtman on UNIZOR Education
Date: 02/10/2020

UNIZOR.COM - Physics through a Looking Glass of Mathematician


Alternating Current and Inductors

For the purpose of this lecture it's important to be familiar with the concept of a self-induction explained in "Electromagnetism - Self-Induction" chapter of this course.

In this lecture we will discuss the AC circuit that contains an inductor - a wire, optionally wound around some core in a reel or a solenoid, thus making multiple loops, schematically presented on the following picture.


Both direct and alternating current go through an inductor, but, while direct current goes with very little resistance through a wire, whether it's in a shape of a loop or not, alternating current meets some additional resistance when this wire is wound into a loop.

Consider the following experiment.


Here the AC circuit includes a lamp, an inductor in a shape of a solenoid and an iron rod fit to be inserted into a solenoid.
While the rod is not inside a solenoid, the lamp lights with normal intensity.
The cause of this resistance is self-induction. This concept was explained earlier in this course and its essence is that variable magnetic field flux, going through a wire loop, creates electromotive force (EMF) directed against the original EMF that drives electric current through a loop.

Any current that goes along a wire creates a magnetic field around this wire. Since the current in our wire loop is alternating, the magnetic field that goes through this loop is variable. According to the Faraday's Law, the variable magnetic field going through a wire loop generates EMF equal to a rate of change of the magnetic field flux and directed opposite to the EMF that drives the current through a wire, thus resisting it.

Magnetic flux Φ(t) going through inductor, as a function of time t, is proportional to an electric current I(t) going through its wire
Φ(t) = L·I(t)
where L is a coefficient of proportionality that depends on physical properties of the inductor (number of loop in a reel, type of its core etc.) called inductance of the inductor.

If the current is alternating as
I(t) = Imax·sin(ωt)
the flux will be
Φ(t) = L·Imax·sin(ωt)

According to Faraday's Law, self-induction EMF Ei is equal in magnitude to a rate of change of magnetic flux and opposite in sign (see chapter "Electromagnetism - "Self-Induction" in this course)
Ei(t) = −dΦ/dt =
= −L·dI(t)/dt =
= −L·ω·Imax·cos(ωt) =
= −L·ω·Imax·sin(ωt+π/2) =
= −Eimax·sin(ωt+π/2)
where
Eimax = L·ω·Imax

The unit of measurement of inductance is henry (H) with 1H being an inductance of an inductor that generates 1V electromotive force, if the rate of change of current is 1A/sec.
That is,
henry = volt·sec/ampere = ohm·sec

An expression XL=L·ω in the above formula for Ei is called inductive reactance. It plays the same role for an inductor as resistance for resistors.
The units of the inductive reactance is Ohm (Ω) because
henry/sec = ohm·sec/sec = ohm.

Using this concept of inductive reactance XL of an inductor, the time dependent induced EMF is
Ei(t) = −XL·Imax·sin(ωt+π/2) = −Eimax·sin(ωt+π/2)
and
Eimax = XL·Imax,
which for inductors in AC circuit is an analogue of the Ohm's Law for resistors.

What's most important in the formula
Ei(t) = −Eimax·sin(ωt+π/2)
and the most important property of an inductor in an AC circuit is that, while the electric current in a circuit oscillates with angular speed ω, the voltage drop on an inductor oscillates with the same angular speed ω as the current, but its period is shifted in time by π/2 relative to the current.