Electronics/Inductors

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Inductors are a basic electronic component, and therfore there is a lot of information and techniques pertaining to their use. This topic is sub-divided into sections to make it easier to find the section you need. The basics can be found below this table of contents.

Inductors Contents

• Introdution
• Basics. This section deals with how the inductor works, and how inductance is defined.
• Inductive Networks. How inductors behave in series and parallel. For inductors connected to other components, see Passive Networks.
• Inductors vs. Capacitors. The similarities and differeces between inductors and capacitors. This includes derivations of the power and energy in an inductor.
• Real Inductors. How non-ideal inductors behave in the real world.
• Transient Analysis. How inductors behave when a DC voltage is suddenly applied. This includes the DC steady state model.
• Special Cases. A list of equations for quickly determining inductance in special cases where no derivation is required.

Introduction

A simple inductor made from a coil of conductor with the magnetic field associated with the current i

The inductor is another example of a passive element that stores energy. Inductors do this by storing energy in the form of a magnetic field, or B-field. Since capacitors store energy using electric fields, it is reasonable to assume that we will see many similarites between the two, and this is indeed the case as we shall see.

Inductors are, to use the hydraulic analogy, somewhat similar to a flywheel with mass driven by a water current. They cannot be stopped and started instantly, as to do so would require an infinite acceleration and therefore an infinite force. In the same way, the current through an inductor cannot be changed instantaneously, as to do so would require infinite voltages to be present. If the current is suddenly removed from an inductor, a very high voltage can be generated, which can produce a spark and perhaps damage something. This is somewhat analogous to stopping the flywheel suddenly - something will likely break or be damaged with the release of the kinetic energy of the wheel.

Inductors are commonly made by winding wire or other conductive material into a coil. When a current passes through this coil, a magnetic field is set up as shown in the diagram to the left. The following explanation of the functioning of an inductor assumes that the reader has a basic knowledge of magnetic fields. If not, it is advisable to read about these first, possibly on the Wikipedia article or elsewhere.

Normally, inductors are made from copper wire (copper has a very high conductivity, so little parasitic resistance - this is important, and we will look at it later), but not always. Inductors can be made using different wires, or can even be etched as a spiral pattern on a printed circuit board or even in the silicon substrate of a microchip. The nature of the material enclosed by the inductor (i.e. inside the coil) affect the behaviour of the inductor. The permeability, μ, of this material governs how strong the magnetic field will be for a given current. Common core materials are air (this is just a coil of wire, and generally gives a low inductance), iron or ferrite. Iron and ferrite types are more efficient because they conduct the magnetic field much better than air; of the two, ferrite is more efficient because stray electricity cannot flow through it so easily - it has a high resistance. Ferrite is more expensive but operates at much higher frequencies than iron cores.

Some inductors have more than one core, which is just a rod the coil is formed about. Some are formed like transformers, using two E-shaped pieces facing each other, the wires wound about the central leg of the E's. The E's are made of laminated iron/steel or ferrite. Toroidal inductors are most efficient of all, they are wound around a doughnut-shaped core. They are more difficult to make, because the formed coil cannot be placed on the toroid - it must be wound around it turn by turn. This makes then more expensive than E-core or rod inductors.

The circuit symbol for an inductor is shown on the right. The unit of inductance is the henry, which is equal to one weber per amp (Wb A^-1). Most inductors used in common circuits have inductances in the range of 1mH to 1H, but others certainly exist.

Important Qualities of Inductors

There are several important properties for an inductor that may need to be considered when choosing one for use in an electronic circuit. the following are the basic properties of a coil inductor. Other factors may be important for other kinds of inductor, but these are outside the scope of this article.

• Current carrying capacity is determined by wire thickness and resistivity.
• The quality factor, or Q-factor, describes the energy loss in an inductor due to imperfection in the manufacturing.
• The inductance of the coil is probably most important, as it is what makes the inductor useful. The inductance is the responce of the inductor to a changing current.

The inductance is determined by several factors.

• Coil shape: short and squat is best
• Core material
• The number of turns in the coil. These must be in the same direction, or they will cancel out, and you will have a resistor.
• Coil diameter. The larger the diameter (core area)the less induction.

Inductance

The following illustrates the properties of inductors using the example of a coil. Let this coil have the following properties:

• Area enclosed by each turn of the coil is A
• Length of the coil is 'l'
• Number of turns in the coil is N
• Permeability of the core is μ. μ is given by the permeability of free space, μ₀ multiplied by a factor, the relative permeability, μ_r
• The current in the coil is 'i'

The magnetic flux density, B, inside the coil is given by:

$B=\frac{N\mu i}{l},$

We know that the flux linkage in the coil, λ, is given by;

$\lambda =NBA.$

Thus,

$\lambda =\frac{N^{2}A\mu }{l}i$

The flux linkage in an inductor is therefore proportional to the current, assuming that A, N, l and μ all stay constant. The constant of proportionality is given the name inductance (measured in henries) and the symbol L. :

$\lambda =Li$

Taking the derivative with repect to time, we get:

$\frac{d\lambda }{dt}=L\frac{di}{dt}+i\frac{dL}{dt}$

Since L is time-invariant in nearly all cases, we can write:

$\frac{d\lambda }{dt}=L\frac{di}{dt}.$

Now, Faraday's Law of Induction states that:

$-ℰ=N\frac{d\mathrm{\Phi }}{dt}=\frac{d\lambda }{dt}.$

We call $-ℰ$ the electromotive force (emf) of the coil, and this is opposite to the voltage v across the inductor, giving

$v=L\frac{di}{dt}.$

This means that the voltage across an inductor is equal to the rate of change of the current in the inductor multiplied by a factor, the inductance. note that for a constant current, the voltage is zero, and for an instantaneous change in current, the voltage is infinite (or rather, undefined). This applies only to ideal inductors which do not exist in the real world.

This equation implies that

• The voltage across an inductor is proportional to the derivative of the current through the inductor.
• In inductors, voltage leads current.
• Inductors have a high resistance to high frequencies, and a low resistance to low frequencies. This property allows their use in filtering signals.

An inductor works by opposing current change. Whenever an electron is accelerated, some of the energy that goes into "pushing" that electron goes into the electron's kinetic energy, but much of that energy is stored in the magnetic field. Later when that or some other electron is decelerated (or accelerated the opposite direction), energy is pulled back out of the magnetic field.

If you want to find the current in an inductor after a voltage has been applied for a certain time, you can use the following method. Take the equation

$v=L\frac{di}{dt}$

Divide through by L and integrate with respect to time. Remember that voltage and current are both functions of time.

$i=\frac{1}{L}\underset{t_1}{\overset{t_2}{\int }}v(t)dt+i\left(t_1\right)$

Note that usually, the voltage will fall off to zero as the magnetic field is set up, reaching zero at an infinite time after the voltage is applied, so the integral of voltage tends to a finite limit, meaning that the current is also a finite quantity.

Quality factor: Q

Inductors and capacitors store energy.

The Q (quality factor) is the ratio of its ability to store energy to the sum total of all energy losses within the component.

$Q=\frac{X}{R}$

where

• Q = quality factor (no units)
• R = total resistance associated with energy losses (in ohms)
• X = reactance (in ohms). ( $X_L=2\pi fL$ for inductors; $X_C=\frac{1}{2\pi fC}$ for capacitors; f is the frequency of interest ).

Impedance

The Impedance of an inductor at any given angular frequency $\omega$ is given by:

$Z=j\omega L$

where j is $\sqrt{-1}$ and L is the inductance.