created at June 11, 2021

Induction and Inductance


When the magnetic field around a conductive loop is changing, the electromotive force (EMF) is being generated in the loop, and the current starts to flow in the loop if it is closed. This is called Induction. The EMF is a voltage in essence, and is measured in Volts.

The magnetic flux through the planar loop:

Φ=BAcos(α)(1)\Phi = B\cdot A\cdot cos(\alpha)\quad (1)

where B - magnetic flux density [Tesla], A - the planar area enclosed by the loop, α - angle between magnetic flux density vector and plane normal vector. One can provide an analogy with the water flow through the hole, where B - water velocity, A - hole's area. The bigger the velocity or the area are, the bigger the water consumption through the hole is.

If any of quantities from equation (1) do change, the total flux Φ also changes, which produces the EMF in the loop:

e=dΦdt(2)e=-\frac{\mathrm{d} \Phi }{\mathrm{d} t}\quad (2)

The EMF arise in such direction so that the generated current's magnetic flux is trying to resist the change of external magnetic flux. That's why there is a "minus" sign in the formula.


The current flowing through the loop also produces its own magnetic field. When the current changes, the field changes as well, thus producing the EMF in the loop. This is called Self-induction.

The conductor (coil) and inductance

A coil is used to generate magnetic field. The coil is basically a wire, cyclically coiled around a ferromagnetic material.

The total magnetic flux generated around the coil is also called flux linkage:

Ψ=LI\Psi = L\cdot I

L - is Inductance of the coil [Henry], which describes coil's ability to produce magnetic fields. According to (2), the EMF produced in the coil is:

eL=LdIdte_{L} = -L\frac{\mathrm{d} I }{\mathrm{d} t}

The faster the current changes, the bigger produced voltage is. The "minus" is here again. It shows that EMV voltage direction is such that it will try to resist the change of the current. This is a kind of electric inertia, which in effect is equivalent to resistance. So, this is also called as inductive resistance.

Inductive Resistance

When the alternating sinusoidal current i(t) = Im·sin(ω·t) flows through the coil, the voltage on the coil is:

uL=Ld(Imsin(ωt))dt=LImωcos(ωt)=Umcos(ωt)u_{L} = -L\frac{\mathrm{d} (I_{m}\cdot sin(\omega\cdot t)) }{\mathrm{d} t} = L\cdot I_{m}\cdot \omega \cdot cos(\omega\cdot t) = U_{m}\cdot cos(\omega\cdot t)


LImω=Um=ImXLL\cdot I_{m}\cdot \omega = U_{m} = I_{m}\cdot X_{L}

where XL = ω·L is inductive resistance of the coil in alternating current circuit. The angular frequency ω = 2·π·f, where f - the frequency, measured in Hertz. One can also write:

Umcos(ωt)=Umsin(ωt+π2)U_{m}\cdot cos(\omega\cdot t) = U_{m}\cdot sin(\omega\cdot t + \frac{\pi}{2})

So we can see that the voltage on the coil outstrips the current by the angle π/2. One can show it on the circular diagram:

The ideal coil was described here. A real coil has also an active resistance (because of wires). The Q factor shows the quality of the coil:

Q=XLRQ = \frac{X_{L}}{R}

In the real coil the voltage phase outstrips the current by the angle a bit smaller than π/2.