## EMBEDDED ADVENTURES: CONNECT THE WORLD AROUND YOU

Tutorials in this section:

Introduction to Electronics
Inductors

The inductor is one of the three basic passive components, which include resistors and capacitors. A passive component is one in which a limited amount of energy can be stored and delivered. Active components, such as a bipolar transistors or FETs are different in that they can generate energy through amplification.

An inductor is similar in some ways to a capacitor, but opposite in other ways. A capacitor stores energy in the form of an electric field between two plates in a dielectric material, whereas an inductor stores its energy in a magnetic field around a coil with a core material. Time is an important factor for inductors, as the rate of change of current determines the voltage; while in a capacitor, the rate of change of voltage determines the current.

The theory of an inductor is very complex, and goes beyond the scope of this article, as it get into electromagnetic theory, left and right hand rules, etc. This article will attempt to explain an inductor's behavior in a circuit, and how it produces a voltage as the current is changed.

From a historical perspective, Ampere made an important discovery in France in the early 1800's that there is a linear relationship between a magnetic field and the current in a conductor that produced it. About twenty years later, Michael Faraday postulated an expression for induced electromotive force (EMF) associated with a rate of change of magnetic flux Ø with time:

v = -dΦ/dt,  where Φ is the Magnetic Flux Linkage

This showed that a changing magnetic field could produce a voltage in a neighboring circuit, and this voltage was proportional to the time rate of change of the current that produced the varying magnetic field (although this change of flux could be caused by the motion of a conductor through a static magnetic field). The constant of proportionality is called inductance, symbolized by L, so a prediction for "back EMF" could be made by the formula:

v = L di/dt, where  Φ = i ·  L

The symbols v and i are both functions of time, and could have been indicated by v(t) and i(t). From this equation, it is seen that the basic action of inductance is to resist any change in currents flowing through it. As a voltage is applied to an inductor, it is like an open circuit, as the current through it remains at zero. Then after a certain time, the inductor looks like a short circuit, as the current steadily rises to its maximum value, limited only by any series resistance.

As the above equation verifies, there is a “self-induced EMF” or “back EMF” on a coil as it is energized or de-energized. The inductor generates a self-induced EMF within itself as a result of its changing magnetic field. When the EMF is induced in the same circuit where the current is changing the surrounding magnetic field, it is called Self-induction, ( L ), and it is also commonly called back-EMF, as its polarity is in the opposite direction to the applied voltage.

When the EMF is induced into an adjacent coil sharing the same magnetic field, the EMF is said to be induced by Mutual induction.

A good example of an inductor is the distributor coil of a car, where the current through the coil is suddenly cut off, and then appears across the gap in a spark plug as an arc. This is because the stored energy has to be dissipated somewhere, and inductor current cannot change instantaneously from one value to another.

The unit in which inductance is measured is the Henry (H), and has the units of a volt-second per ampere. An inductor of one Henry will have an EMF of one volt induced in the coil when the current flowing through the coil changes at a rate of one ampere per second. Similar to the Farad for capacitance, one Henry is too large to be used in practical circuits, so smaller inductance values measured in mill-henries are used instead.

A physical inductor can be made be winding any length of wire into a coil (although a wire of any size, shape, or form will have a certain "stray" inductance, which is usually very small). Winding a coil would increase inductance in two ways:  1) it increases the number of current "loops", which creates a larger internal magnetic flux field through the coil, and 2) there is an increased number of neighboring current loops into which Faraday's voltage may be induced. The result of this twofold effect is that the inductance of a coil is approximately proportional to the square of the number of complete turns of wound conducting wire.

If a coil of N turns is linked by an amount of magnetic flux, Φ, then the coil would have a flux linkage of NΦ, and as a current flows through the coil, it will produce an induced magnetic flux in the opposite direction to the flow of current.

An inductor that is wound as a coil around a core material is found to have an inductance of μN 2A/s, where A is the cross sectional area, s is the axial length of the form, N is the number of complete turns of wire, and μ is a constant of the material inside the form, and is called permeability. For air, μ = μo =  4π · 10-7 H/m.

Various materials are used as core materials to provide greater magnetic flux for a given magnetic field and also increase the associated inductance. This is analogous to the permittivity of a dielectric material in a capacitor.

The actual permeability µ of an inductor depends on the relative permeability µr of material introduced into the magnetic field:  µ = µr · µ0. For example, Nickel has a relative permeability of 99.5; Steel is 696; and certain alloys have values of several thousands. The permeability, µ, can vary with temperature and other factors, so it can be quite difficult to calculate accurately.

Some coil forms have "air gaps" intentionally placed in them at the cost of lower inductance in order to prevent magnetic saturation, which improves the voltage-current characteristics. This is common for magnetically tuned inductors and transformer cores. Inductors are usually designed in such a way that its flux density would never approach saturation levels, so they would operate in a more linear portion of the B/H curve.

The energy w L stored in the magnetic field around the coil is expressed by the integral of the power over the desired time interval as follows:

This calculation assumes that the current, i(to), is zero at t=0. The reference for zero energy is taken at this point, although there could be several points in the integration interval where the current is zero in the integration. The reference could be taken at any one of these points. Whenever the current is not zero, and regardless of its direction, energy is stored in the inductance. From this, it follows that power is delivered to the inductor for part of the time and recovered later. For example, when using alternating current (AC), an inductor is constantly storing and delivering energy on each and every cycle.

These calculations work out for an ideal inductor, but for a real coil, there are resistances and magnetic losses associated with it, so energy cannot be stored and recovered without a loss. Magnetic losses can be minimized by winding the coil in a continuous loop or ring instead of a straight cylindrical rod to contain the magnetic flux.

The term “reactance“ is used as a measure of AC resistance, while “resistance” is a measure of DC resistance.  Like resistance, reactance is considered an impedance, and is measured in Ohm's, but it has the symbol "X" to distinguish it from a purely resistive value (R).

Inductive Reactance, which is given the symbol XL, is the property in an AC circuit which opposes the change in the current. It was shown in a previous tutorial that XC is the capacitive reactance, and opposes the change in voltage. In a purely capacitive circuit, the current "leads" the voltage by 90o. In a purely inductive AC circuit, the exact opposite is true, as the current "lags" the applied voltage by 90o, or (π/2 rads). It can also be shown that an inductor has an impedance (XL) on the imaginary axis (j) in the positive direction on a complex number plane, while a resistor has an impedance in the positive direction along the real axis (the capacitor has an impedance in the negative direction on the j axis).

From a frequency domain perspective, this action is seen as a reactive impedance that is proportional to frequency:

ZL = j · XL, where  XL = w · L = 2 · π · F · L

The imaginary operator j = √ (-1) is included to represent a 90 degree phase lag between inductive current and applied AC voltage.

The Q-factor is often used to indicate the quality of an inductor. The Q-factor is the ratio of the back-voltage to the applied voltage. This can be as low as 1 or 2, and up to100 or more for higher quality inductors.. The quality depends on how much magnetic flux is produced and how effectively it is contained within the coil. The Reactive Power, (Q) of a coil can be given as: I2 ·  XL (similar to I2R in a DC circuit).

There are many disadvantages associated with designing inductors into a circuit that cause circuit designers to use capacitors instead whenever they can to avoid these unwanted effects:

• Unlike capacitors, which are relatively easy to manufacture with negligible stray effects, inductors are known to have substantial amounts of series resistance and measurable amounts of stray capacitance because of the close spacing of wire from one coil turn to another.
• If there are two or more inductors used in a circuit, they could interfere with one another through the effects of mutual inductance. Capacitors tend to contain their respective electric fields within their component package, and therefore do not typically generate these undesired effects with other components.
• Inductors usually have a much larger physical size than capacitors when storing equivalent amounts of energy. An engineer would typically design circuits with the smallest possible package size, and choose capacitors over inductors if possible.