2.4.2.1 Introduction

Inductors are a fundamental building block of electrical circuits especially AC circuits.

As their impedance increases with frequency, they are commonly used limit the flow of higher frequency currents. They can also be used to transfer energy from one part of a circuit to another via magnetic coupling.

2.4.2.2 Ohms Law

Ohms law was discussed in the section on resistors. Inductors follow the same law but we need to modify our understanding of how the inductor works.

The inductor has a frequency dependent "resistance" to the flow of current. But since resistance is reserved for resistors, a new term is introduced for components such as inductors and capacitors that impede the flow of current. This term is impedance. Inductors have an impedance that varies proportional to the frequency of current (voltage).

The inductor when current is flowing, produces a magnetic flux ϕ which envelopes the loops formed by the inductor's wiring. In fact any piece of wire has inductance. A single, straight cable from point A to point B will have an inductance albeit this will be extremely low. Inductance is increased by simply coiling this wire many times.

$$\lambda =N·\varphi$$

Where:

λ = flux linkage through the inductor (weber - Wb).

N = Number of turns or coils (unitless).

ϕ = magnetic flux (weber - Wb).

Each turn or coil or wire will produce magnetic flux and the total flux is that is created by N coils or turns.

Flux linkage is proportional to the current flowing through the inductor. Inductance is webers per amp and the unit assigned is henry

$$L=\frac{\mathrm{Wb}}{A}$$

Where:

L = inductance (henry - H).

Wb = flux (weber - Wb) = λ.

A = current (ampere - A).

Furthermore:

$$\lambda =L·i$$

If the current is increased in the inductor, a voltage is produced across the inductor. I.e, a changing current produces a voltage across the inductor and this is the basis of electromagnetic induction. This was first observed by Joseph Henry but he failed to publish his findings and Michael Faraday is therefore credited with the discovery of the electromagnetic law of induction.

$$v=\frac{\mathrm{d\lambda }}{\mathrm{dt}}=L·\frac{\mathrm{di;}}{\mathrm{dt}}$$

We now look at the impedance of the inductor. From the above formula we can see that voltage is dependent on changes in current over time. For DC, the change in current = 0. Therefore the voltage across the inductor is 0 and hence it appears as a short circuit to DC current.

Conversely, if we have an infinitely large change in current, the voltage will be infinitely high which we can conclude that frequencies approaching infinity, the impedance of the inductor will start to behave like an open circuit. This is why current in an inductor does not step up or down.

From these two extremes, we can observe that inductors work well in AC circuits where there is a spectrum of frequencies a circuit may operate under. A classic example is blocking high frequency currents. Inductors can act as filters, passing lower frequency currents whilst attenuating higher frequency currents.

2.4.2.3 Energy Storage

Inductors store their energy in the magnetic field.

$$w_{\mathrm{L}}\mathrm{(t)}=\frac{1}{2}·L·i^2\mathrm{(t)}J$$

Where:

w(t) = energy (Joules).

L = inductance (henry).

i = current (ampere).

If flux linkage (λ) is known then we can also calculate the energy stored.

$$w_{\mathrm{L}}\mathrm{(t)}=\frac{{\lambda }^2}{2·L}J$$

2.4.2.4 Parallel and Series

Inductors wired in series are treated like resistors. The total inductance is the sum of the individual inductance values.

$$L_{\mathrm{s}}=\underset{n=1}{\overset{\mathrm{N}}{\Sigma }}{L}_{\mathrm{N}}$$

Simplified for 2 inductors in series:

$$L_{\mathrm{s}}=L_1+L_2$$

Inductors wired in parallel are also treated like resistors. The total inductance is the inverse of the sum of the reciprical values of each individual inductor.

$$\frac{1}{L_{\mathrm{p}}}=\underset{n=1}{\overset{\mathrm{N}}{\Sigma }}\frac{1}{L_{\mathrm{n}}}$$

Simplified for 2 inductors in parallel:

$$L_{\mathrm{p}}=\frac{L_1·L_2}{L_1+L_2}$$