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While Ohm's Law applies directly to resistors in DC or in AC circuits, the form of the current-voltage relationship in AC circuits in general is modified to the form:

where I and V are the rms or "effective" values. The quantity Z is called impedance. For a pure resistor, Z = R. Because the phase affects the impedance and because the contributions of capacitors and inductors differ in phase from resistive components by 90 degrees, a process like vector addition (phasors) is used to develop expressions for impedance. More general is the complex impedance method.

Series and parallel combination of any two impedances

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Impedance Combinations

Combining impedances has similarities to the combining of resistors, but the phase relationships make it practically necessary to use the complex impedance method for carrying out the operations. Combining series impedances is straightforward:


Combining parallel impedances is more difficult and shows the power of the complex impedance approach. The expressions must be rationalized and are lengthy algebraic forms.


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Parallel Impedance Expressions

The complex impedance of the parallel circuit takes the form

when rationalized, and the components have the form


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Impedance Calculation

Impedances may be combined using the complex impedance method.

= + j
= + j

the series combination is

= + j= at phase.

The parallel combination is

= + j= at phase.

The units for all quantities are ohms. A negative phase angle implies that the impedance is capacitive, and a positive phase angle implies net inductive behavior.

Parallel impedance expressions Inductive reactance Capacitive reactance

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1)     Electromagnet

       2) Magnetic Fields: History

      3) Electricity Electronics

          4) Induction and Inductance

           5) Self-Inductance and Inductive Reactance

           6) Mutual Inductance
(The Basis for Eddy Current Inspection)  

           7) Ferromagnetism


1) Electromagnet

An electromagnet is a device in which magnetism is produced by an electric current.

British electrician, William Sturgeon invented the electromagnet in 1825. The first electromagnet was a horseshoe-shaped piece of iron that was wrapped with a loosely wound coil of several turns. When a current was passed through the coil; the electromagnet became magnetized and when the current was stopped the coil was de-magnetized. Sturgeon displayed its power by lifting nine pounds with a seven-ounce piece of iron wrapped with wires through which the current of a single cell battery was sent. The first electromagnet was a horseshoe-shaped piece of iron that was wrapped with a loosely wound coil of several turns. When a current was passed through the coil; the electromagnet became magnetized and when the current was stopped the coil was de-magnetized. Sturgeon displayed its power by lifting nine pounds with a seven-ounce piece of iron wrapped with wires through which the current of a single cell battery was sent.

Sturgeon could regulate his electromagnet; this was the beginning of using electrical energy for making useful and controllable machines and laid the foundations for large-scale electronic communications.                                                                               

Five year later an inventor called Joseph Henry - made a far more powerful version of the electromagnet.  American, Joseph Henry (1797-1878), demonstrated the potential of Sturgeon's device for long distance communication by sending an electronic current over one mile of wire to activate an electromagnet which caused a bell to strike. Thus the electric telegraph was born.

 Joseph Henry's Contributions to the Electromagnet and the Electric Motor

By Roger Sherman
Museum Specialist, National Museum of American History


Click on portrait of Henry
Henry as a young man. From miniature,
ca. 1829, attributed to Julius Rubens
Ames. Smithsonian neg. no. 69,029.

At the beginning of his career as an investigator of electromagnetism, in the fall of 1827, Joseph Henry took up a simple idea, and soon found that it led him to some remarkable results. He was starting his second academic year as Professor of Mathematics and Natural Philosophy at the Albany Academy, a school for boys in Albany, New York, offering instruction extending from what we would now call elementary grades up to and overlapping with the college level. Henry took his teaching responsibilities seriously, but he also had an ambition to make original scientific contributions.

His first paper on electromagnetism, presented on October 10, 1827, shows that at this early stage his research was guided by the didactic concerns of his science classes and the experimental demonstrations that he considered an essential element of effective teaching.

Henry began by pointing out that the introduction of electromagnetism as a subject of instruction had been hampered because of the expense and awkwardness of the large batteries and delicate apparatus needed to show the effects. Recently, however, the English experimenter William Sturgeon had eliminated much of the difficulty by showing that the use of strong permanent magnets allowed many of the experiments to be done on a larger scale, and with a smaller battery, than was previously thought possible. But some electromagnetic experiments depend on the earth's magnetic field, or the interaction between two current-carrying wires. For these experiments permanent magnets could not be used, and the difficulties remained.

Click on diagram of multiplier
Diagram of Schweigger's multiplier.
From Journal fr Chemie und
31 (Neue Reihe, Bd. I,
1821), Plate I (after p. 114), Fig. 10.
Smithsonian neg. no. 46,825.

Here is where Henry made his first contribution. He had read that Johann S. C. Schweigger, professor of chemistry at the University of Halle, had invented what came to be called a "galvanic multiplier" for augmenting the deflecting action of an electric current on a compass needle. This effect (the first discovery linking electricity with magnetism) had been announced in 1820 by Hans C. Oersted. Oersted used in his experiments a single straight wire passing close to the compass; Schweigger, a few months later, showed that if the wire was formed into a vertical coil of several turns around the compass, the effect would be greatly increased. Henry, in turn, now described in his 1827 paper how Schweigger's coil idea could be applied to other standard electromagnetic demonstration devices to make them more sensitive or powerful. Henry's versions of these devices embodied no new discovery, but were simply more dramatic and effective as educational aids.

Click on Sturgeon's electromagnet
Sturgeon's electromagnet. From
Transactions of the Society for
the Encouragement of the Arts

43 (1824), Plate 3, Fig. 13.
Smithsonian neg. no. 46,761-D.

His next step was to apply the coil principle "to a development of magnetism in soft iron, much more extensively, than to my knowledge had been previously effected by a small galvanic element." He did this by winding an electromagnet with about 400 tight turns of a wire 35 feet long, "instead of loosely coiling around it a few feet of wire, as is usually described." This is probably an indirect reference to the electromagnet described by Sturgeon, who is generally credited with its invention. Sturgeon used un insulated wire (insulated wire for electrical use was not then commercially available); to prevent short-circuiting of the windings, he varnished the iron core and separated the turns of wire to keep them from touching. The illustration of his magnet, in fact, shows only 18 loose turns. Henry insulated the wire itself with silk thread and so could apply a large number of tight turns.

Click on photograph of magnet
Henry's Albany magnet. Image copied
from old photograph, N.M.A.H. Cat. No.
181,451c. Smithsonian neg. no. 39,040.

So far, he had not done anything really new, but just extended and combined known principles using simple techniques. He now saw, however, a new line of investigation opening before him: the determination of the principles for designing powerful, efficient electromagnets, which would develop the greatest possible lifting force with a given small battery. Systematically he explored different methods of winding, using various lengths of wire in various arrangements and trying increasingly large iron cores. From these experiments Henry discovered that if a cell of a single pair of electrodes is to be used with a given magnet, the magnet should be wound with several coils of wire in parallel; on the other hand, if a battery of many cells is to be used, the magnet winding should form a single long wire. Henry was the first person to understand this idea. It later became a fundamental basis for much of electrical technology, and, in particular, made Samuel F. B. Morse's telegraph feasible.

Click on sketch of Albany magnet
Henry's Albany magnet with its
battery and apparatus for measuring
its strength. From Silliman's American
Journal of Science
19 (January 1831):
408. Smithsonian neg. no. 46,797-F.

Applying this principle (together with the valuable but less easily described practical skill in magnet-making he had acquired in the course of his experiments), Henry, with the assistance of a colleague, Philip Ten Eyck, went on to build a 21-pound "experimental magnet on a large scale." With a modest battery, this "Albany magnet" supported 750 pounds, making it, Henry claimed, "probably, therefore, the most powerful magnet ever constructed." Quickly he wrote a paper describing these experiments and his magnet-winding principle, and sent it off to Benjamin Silliman, Professor of Chemistry and Natural History at Yale College and editor of the American Journal of Science, a widely read and influential publication. Silliman readily accepted what he called Henry's "highly important & interesting paper" and published it in the issue of January 1831.

Click on photograph of Yale magnet
Henry's Yale magnet, mounted in
frame constructed under Silliman's
direction. N.M.A.H. Cat. No. 181,343.
Smithsonian neg. no. 13,346.

In addition to the paper, Henry sent Silliman an offer to "superintend the construction for your lecture room of a Galvanic magnet on my pla[n] which will support 1000 or 1200 lbs." Silliman agreed, and in a few months Henry built a magnet that exceeded his own projection. This "Yale magnet" embodied no principles not already explained in Henry's paper, but it represented a big step beyond the Albany magnet in size and power. With a core weighing 59 pounds, it supported the unprecedented weight of 2,063 pounds. Silliman published Henry's detailed description of this latest and most highly developed product of his magnet-building skills and in an editor's note said of Henry, "He has the honor of having constructed by far, the most powerful magnets that have ever been known, and his last ... is eight times more powerful than any magnet hitherto known in Europe."


Click on photograph of Yale magnet
Yale magnet, showing windings
and conductors for connection
to the battery. N.M.A.H. Cat. No.
181,343. Smithsonian neg. no. 74-4407.

Henry's papers on his electromagnets attracted considerable attention. Before long requests started coming in for magnets like the one made for Silliman. He turned down most of these but did provide helpful practical information to his correspondents. Henry made an exception for Parker Cleaveland of Bowdoin College, furnishing him with a magnet similar to Silliman's while incorporating some recent refinements of construction. In the meantime, however, having worked out and published the fundamental principles of the design of these magnets, he was considering the next stages of his research: "At the conclusion of the series of experiments which I described in Silliman's Journal, there were two applications of the electro-magnet in my mind: one the production of a machine to be moved by electro-magnetism, and the other the transmission of or calling into action power at a distance."

Click on sketch of Henry's motor

Henry's oscillating electromagnet
motor. From Silliman's American
Journal of Science
20 (July 1831): 342.
Smithsonian neg. no. 46,797-E.

In the summer of 1831 Henry described the first of these applications in a short paper, "On a Reciprocating Motion Produced by Magnetic Attraction and Repulsion."  It was a simple device whose moving part was a straight electromagnet rocking on a horizontal axis. Its polarity was reversed automatically by its motion as two pairs of wires projecting from its ends made connections alternately with two electrochemical cells. Two vertical permanent magnets alternately attracted and repelled the ends of the electromagnet, making it rock back and forth at 75 vibrations per minute.


Click on photograph of reconstructed motor
Reconstruction of Henry's original
oscillating electromagnet motor.
N.M.A.H. Cat. No. 244,904.
Smithsonian neg. no. 24,976-A.

Henry at this time considered his little machine merely a "philosophical toy," but nevertheless believed it was important as the first demonstration of continuous motion produced by magnetic attraction and repulsion. Furthermore, "in the progress of discovery and invention, it is not impossible that the same principle, or some modification of it on a more extended scale, may hereafter be applied to some useful purpose." Indeed, one authority has stated, "Henry's apparatus was the first clear-cut instance of a motor capable of further mechanical development. It had the essentials of a modern DC motor: a magnet to provide the field, an electromagnet as armature, and a commutator to apply the mechanical forces at the right time." Other inventors did later develop motors of various designs based on similar reciprocating actions, but it is not clear whether these inventors knew of Henry's device, or created theirs independently. In any case, reciprocating motors never became commercially successful; continuous rotary motion proved to be a more efficient and useful principle.

In 1831 the future development and importance of what we now call the electric motor could scarcely be foreseen. Henry at that time was striving to build a solid reputation as an original scientist while conscientiously discharging his teaching responsibilities at the Albany Academy. At the start of his career, a few years before, his research interests had been dominated by a desire to develop effective, compelling demonstration apparatus for his classes. His 1827 paper on extending the "galvanic multiplier" principle, discussed above, exemplifies this motivation. Large equipment and dramatic experimental effects, he knew from experience, attract the attention and hold the interest of students. Certainly some of the original motivation for the construction of his powerful electromagnets can also be ascribed to this desire. But the process of developing those magnets led Henry to make a real discovery, and this, along with the magnets themselves, was beginning to make his name known in scientific circles.

His next project, the oscillating electromagnet, was not built on the grand scale of the large magnets. But it, like them, exemplifies Henry's continuing desire not only to make original discoveries, but also to embody them in the form of didactically useful devices. And, as with his original idea for developing the electromagnet, Henry could not leave his motor alone after he had conceived the principle, embodied it in a working model, and described it to the scientific community. Although he published no more papers about it, there is ample evidence that he continued to ponder and develop the idea after he had submitted his article to Silliman. A few months later, for example, he wrote to Parker Cleaveland that he had "lately improved the form of the little machine" there described, but he gave no details of any change in the design.

Click on photograph of Henry motor
Henry's modified oscillating
electromagnet motor, at Princeton.
Smithsonian neg. no. 16,188.

In November 1832, a year after his letter to Cleaveland, Henry moved from Albany to the college at Princeton, New Jersey, where he continued conducting experimental researches whenever he could find time away from his teaching duties. In 1834, he lent the machine (apparently in its original configuration) to another colleague, Jacob Green of Philadelphia, who wanted to use it in a popular lecture. In their correspondence about this loan, both men called the device the "sheep's tail," presumably referring to the wagging motion of the projecting wires. The familiar use of such a nickname suggests that Green and Henry had discussed the machine at some length. In his letter to Green, Henry remarked that he was constructing a new version "on a some what different plan" using C-shaped magnets in place of straight ones. This promised to give "a moving force double of that in the other plan." He also said, "Many different forms of the instrument have suggested themselves to my mind." It is clear that the device had become more than a mere "philosophical toy" in Henry's imagination.

Click on photograph of replica of Henry motor
Smithsonian replica of Henry's Prince-
ton oscillating electromagnet motor.
N.M.A.H. Cat. No. 181,324.
Smithsonian neg. no. 29,682.

Concrete evidence of this continuing interest is a machine long preserved in the Princeton apparatus collection. In its general form it is similar to Henry's 1831 invention, but it differs in having, instead of the two vertical bar magnets under the rocking electromagnet, a single horizontal bar magnet. With this configuration, the ends of the electromagnet are acted on by two different magnetic poles, north and south; in the original device both were north poles. In consequence of this change, an ordinary electromagnet would not oscillate--both ends would be attracted or repelled at the same time. An apparently accurate replica of the Princeton device made for the Smithsonian suggests that Henry accordingly wound the electromagnet in opposite directions on its two ends--in effect, creating two electromagnets end to end, with like poles together at the middle. No written record of this development appears to survive; it is known to us only through the three-dimensional evidence of the demonstration device.

Click on diagram of Ritchie's motor
Diagram of Ritchie's revolving electro-
magnet motor. From Ritchie, "Experimental
Researches in electro-magnetism and
magneto-electricity," Philosophical Trans-
123 (1833), Plate 7 (opposite p. 316),
Fig. 2. Smithsonian neg. no. 45,393-H.

In 1833, not long after Henry invented his oscillating electromagnet, William Ritchie, a clergyman, educator, and experimentalist in England, contrived a way to make an electromagnet revolve continuously. He caused its polarity to reverse twice in each revolution by an arrangement of wires grazing across two semicircular troughs of mercury.  It is unlikely that Ritchie had heard of Henry's invention; he probably devised his scheme independently. Nevertheless, when Henry heard of it, he suspected that Ritchie had "lately reinvented my machine," implying that he had neglected to give Henry due credit for the idea.

Click on photograph of Richie's motor
Demonstration motor according to
Ritchie's design, showing cup for
mercury commutator. N.M.A.H. Cat. No.
315,532. Smithsonian neg. no. 47,797-C.

On several later occasions extending over twelve years Henry went to some lengths to see that his claim was acknowledged in published accounts on the subject, and in his notes and correspondence he repeatedly asserted his priority. In 1840, for example, he described to the American Philosophical Society yet another variant of his machine. Even in the brief official note of his talk, his pique is perceptible: "Prof. Henry described an apparatus for producing a reciprocating motion by the repulsion in the consecutive parts of a conductor, through which a galvanic current is passing, and made some remarks in reference to the electromagnetic engine invented by him in 1831, and subsequently described by Dr. Ritchie, of London."

His touchiness in this matter is an example of a characteristic he displayed at other times in his career. The most notable instance was his dispute with Samuel Morse about his contributions to the electromagnetic telegraph. Like his remarks in that dispute, Henry's expressions of concern about the credit due him for his motor reveal his strongly held beliefs about the value of his contributions and the importance of scrupulous fairness in scholarly publications.

Click on sketch of Page's magnet
Page's Revolving Electro-
magnet. From Daniel Davis,
Jr., Manual of Magnetism
(Boston, 1842), Fig. 71.
Smithsonian neg. no. 72-5048.

Ritchie's device, like Henry's, was a didactic instrument, with no practical application beyond its demonstration of electromagnetic principles. The first manufacturer of educational electromagnetic apparatus in the United States, Daniel Davis of Boston, seems to have ignored Henry's machine. In his catalogue of 1838, Davis did, however, include Ritchie's, as well as a similar device, "Page's Revolving Magnet." In the latter, Ritchie's awkward mercury commutator was replaced with a "pole-changer," identical with the present-day commutator. This device, which had previously been used by others in somewhat different applications, consisted of two wires, each connected to a pole of the battery, and making contact with a pair of insulated half-cylinders on the rotating shaft. Each half-cylinder was soldered to one end of the winding of the electromagnet, so the polarity reversed with every half revolution. Davis attributed this improved version of Ritchie's instrument to Charles Grafton Page, an experimenter and prolific inventor of electromagnetic devices, many of which Davis offered for sale. But in his 1848 catalogue Davis changed its name to "Revolving Electro-Magnet," probably to acknowledge that Page did not deserve exclusive credit for it. Even so, Page later did claim credit for it.

Click on photograph of Page's electromagnet
Page's revolving electromagnet
(far right). Said to have belonged
to Henry. The two brushes and
binding posts are missing.
N.M.A.H. Cat. No. 181,743.
Smithsonian neg. no. 13,367.

It is clear that all along Henry believed that much of the honor belonged neither to Ritchie nor to Page, but to himself. Yet whatever Henry may have felt about the lack of recognition for his contribution to the Revolving Electro-Magnet, Davis's commercial version of it was just the kind of impressive demonstration device that he liked to put into action for his classes and lectures. And indeed an example survives at the Smithsonian among a collection of apparatus said to have once belonged to Henry. The date he acquired it is unknown. It agrees closely with Daniel Davis's illustration in his catalogues of 1842 and 1848, however, and may have been made by Davis about that time. It is impossible to be certain, however, because the device is not signed, and other makers copied Davis's handsome designs.

Here in one instrument, then, are embodied the contributions of several electrical investigators. With regard to Henry, Ritchie, and Page, perhaps it would do justice to all three to say that Henry was the first to show how polarity could be automatically reversed and Ritchie the first to produce rotary motion of an electromagnet, while Page introduced into Ritchie's device the simple and effective "pole-changer," which remains in use today.



2) Magnetic Fields : History

Edmond Halley
Edmond Halley

Until 1820, the only magnetism known was that of iron magnets and of "lodestones", natural magnets of iron-rich ore. It was believed that the inside of the Earth was magnetized in the same fashion, and scientists were greatly puzzled when they found that the direction of the compass needle at any place slowly shifted, decade by decade, suggesting a slow variation of the Earth's magnetic field.

How can an iron magnet produce such changes? Edmond Halley (of comet fame) ingeniously proposed that the Earth contained a number of spherical shells, one inside the other, each magnetized differently, each slowly rotating in relation to the others.
Hans Christian Oersted
Hans Christian Oersted was a professor of science at Copenhagen University. In 1820 he arranged in his home a science demonstration to friends and students. He planned to demonstrate the heating of a wire by an electric current, and also to carry out demonstrations of magnetism, for which he provided a compass needle mounted on a wooden stand.

While performing his electric demonstration, Oersted noted to his surprise that every time the electric current was switched on, the compass needle moved. He kept quiet and finished the demonstrations, but in the months that followed worked hard trying to make sense out of the new phenomenon.
Hans Christian Oersted Experiment
But he couldn't! The needle was neither attracted to the wire nor repelled from it. Instead, it tended to stand at right angles (see drawing right). In the end he published his findings (in Latin!) without any explanation.

Andre-Marie Ampere in France felt that if a current in a wire exerted a magnetic force on a compass needle, two such wires also should interact magnetically. In a series of ingenious experiments he showed that this interaction was simple and fundamental--parallel (straight) currents attract, anti-parallel currents repel. The force between two long straight parallel currents was inversely proportional to the distance between them and proportional to the intensity of the current flowing in each.
Andre-Marie Ampere


There thus existed two kinds of forces associated with electricity--electric and magnetic. In 1864, James Clerk Maxwell demonstrated a subtle connection between the two types of force, unexpectedly involving the velocity of light. From this connection sprang the idea that light was an electric phenomenon, the discovery of radio waves, the theory of relativity and a great deal of present-day physics.


3) Electricity - Electronics

Electricity is a form of energy.                                                                             

Electricity is the flow of electrons. All matter is made up of atoms, and an atom has a center, called a nucleus. The nucleus contains positively charged particles called protons and uncharged particles called neutrons. The nucleus of an atom is surrounded by negatively charged particles called electrons. The negative charge of an electron is equal to the positive charge of a proton, and the number of electrons in an atom is usually equal to the number of protons. When the balancing force between protons and electrons is upset by an outside force, an atom may gain or lose an electron. When electrons are "lost" from an atom, the free movement of these electrons constitutes an electric current.

Electricity is a basic part of nature and it is one of our most widely used forms of energy. We get electricity, which is a secondary energy source, from the conversion of other sources of energy, like coal, natural gas, oil, nuclear power and other natural sources, which are called primary sources. Many cities and towns were built alongside waterfalls (a primary source of mechanical energy) that turned water wheels to perform work. Before electricity generation began slightly over 100 years ago, houses were lit with kerosene lamps, food was cooled in iceboxes, and rooms were warmed by wood-burning or coal-burning stoves. Beginning with Benjamin Franklin's experiment with a kite one stormy night in Philadelphia, the principles of electricity gradually became understood. In the mid-1800s, everyone's life changed with the invention of the electric Light bulb. Prior to 1879, electricity had been used in arc lights for outdoor lighting. The light bulb's invention used electricity to bring indoor lighting to our homes.

An electric generator (Long ago, a machine that generated electricity was named "dynamo" today's preferred term is "generator".) is a device for converting mechanical energy into electrical energy. The process is based on the relationship between magnetism and electricity . When a wire or any other electrically conductive material moves across a magnetic field, an electric current occurs in the wire. The large generators used by the electric utility industry have a stationary conductor. A magnet attached to the end of a rotating shaft is positioned inside a stationary conducting ring that is wrapped with a long, continuous piece of wire. When the magnet rotates, it induces a small electric current in each section of wire as it passes. Each section of wire constitutes a small, separate electric conductor. All the small currents of individual sections add up to one current of considerable size. This current is what is used for electric power.

An electric utility power station uses either a turbine, engine, water wheel, or other similar machine to drive an electric generator or a device that converts mechanical or chemical energy to electricity. Steam turbines, internal-combustion engines, gas combustion turbines, water turbines, and wind turbines are the most common methods to generate electricity. 


Electricity is measured in units of power called watts. It was named to honor James Watt, the inventor of the Steam engine. One watt is a very small amount of power. It would require nearly 750 watts to equal one horsepower. A kilowatt represents 1,000 watts. A kilowatt-hour (kWh) is equal to the energy of 1,000 watts working for one hour. The amount of electricity a power plant generates or a customer uses over a period of time is measured in kilowatt hours (kWh). Kilowatt hours are determined by multiplying the number of kW's required by the number of hours of use. For example, if you use a 40-watt light bulb 5 hours a day, you have used 200 watts of power, or .2 kilowatt hours of electrical energy. 


4) Induction and Inductance


In 1824 Oersted discovered that current passing though a coil created a magnetic field capable of shifting a compass needle. Seven years later Faraday and Henry discovered just the opposite. They noticed that a moving magnetic field would induce current in an electrical conductor. This process of generating electrical current in a conductor by placing the conductor in a changing magnetic field is called electromagnetic induction or just induction. It is called induction because the current is said to be induced in the conductor by the magnetic field.

Faraday also noticed that the rate at which the magnetic field changed also had an effect on the amount of current or voltage that was induced. Faraday's Law for an uncoiled conductor states that the amount of induced voltage is proportional to the rate of change of flux lines cutting the conductor. Faraday's Law for a straight wire is shown below.


VL = the induced voltage in volts
d/dt = the rate of change in magnetic flux in webers/second

Induction is measured in unit of Henries (H) which reflects this dependence on the rate of change of the magnetic field. One henry is the amount of inductance that is required to generate one volt of induced voltage when the current is changing at the rate of one ampere per second. Note that current is used in the definition rather than magnetic field. This is because current can be used to generate the magnetic field and is easier to measure and control than magnetic flux..


When induction occurs in an electrical circuit and affects the flow of electricity it is called inductance, L. Self-inductance, or simply inductance is the property of a circuit whereby a change in current causes a change in voltage in the same circuit. When one circuit induces current flow in a second nearby circuit, it is known as mutual-inductance. The image to the right shows an example of mutual-inductance. When an AC current is flowing through a piece of wire in a circuit, an electromagnetic field is produced that is constantly growing and shrinking and changing direction due to the constantly changing current in the wire. This changing magnetic field will induce electrical current in another wire or circuit that is brought close to the wire in the primary circuit. The current in the second wire will also be AC and in fact will look very similar to the current flowing in the first wire. An electrical transformer uses inductance to change the voltage of electricity into a more useful level. In nondestructive testing, inductance is used to generate eddy currents in the test piece.

It should be noted that since it is the changing magnetic field that is responsible for inductance, it is only present in AC circuits and that high frequency AC will result in greater inductive reactance since the magnetic field is changing more rapidly.


5) Self-Inductance and Inductive Reactance

The property of self-inductance is a particular form of electromagnetic induction. Self- inductance is defined as the induction of a voltage in a current-carrying wire when the current in the wire itself is changing. In the case of self-inductance, the magnetic field created by a changing current in the circuit itself induces a voltage in the same circuit. Therefore, the voltage is self-induced.

The term inductor is used to describe a circuit element possessing the property of inductance and a coil of wire is a very common inductor. In circuit diagrams, a coil or wire is usually used to indicate an inductive component. Taking a closer look at a coil will help understand the reason that a voltage is induced in a wire carrying a changing current. The alternating current running through the coil creates a magnetic field in and around the coil that is increasing and decreasing as the current changes. The magnetic field forms concentric loops that surrounds the wire and joins up to form larger loops that surround the coil as shown in the image below. When the current increases in one loop the expanding magnetic field will cut across some or all of the neighboring loops of wire, inducing a voltage in these loops. This causes a voltage to be induced in the coil when the current is changing.

By studying this image of a coil, it can be seen that the number of turns in the coil will have an effect on the amount of voltage that is induced into the circuit. Increasing the number of turns or the rate of change of magnetic flux increases the amount of induced voltage. Therefore, Faraday's Law must be modified for a coil of wire and becomes the following.


VL = the induced voltage in volts
N = the number of turns in the coil
d/dt = the rate of change in magnetic flux in webers per second

The equation simply states that the amount of induced voltage (VL) is proportional to the number of turns in the coil and the rate of change of the magnetic flux (d/dt). In other words, when the frequency of the flux is increased or the number of turns in the coil is increased, the amount of induced voltage will also increase.

In a circuit, it is much easier to measure current than it is to measure magnetic flux so the following equation can be used to determine the induced voltage if the inductance and frequency of the current are known. This equation can also be reorganized to allow the inductance to be calculated when the amount of inducted voltage can be determined and the current frequency is known.


VL = the induced voltage in volts
L = the value of inductance in henries
di/dt = the rate of change in current in amperes per second

Lenz's Law

Soon after Faraday proposed his law of induction, Heinrich Lenz developed a rule for determining the direction of the induced current in a loop. Basically,  Lenz's law states that an induced current has a direction such that its magnetic field opposes the change in magnetic field that induced the current. This means that the current induced in a conductor will oppose the change in current that is causing the flux to change. Lenz's law is important in understanding the property of inductive reactance, which is one of the properties measured in eddy current testing.

Inductive Reactance

The reduction of current flow in a circuit due to induction is called inductive reactance.

It should be noted that inductive reactance will increase if the number of winds in the coil is increased since the magnetic field from one coil will have more coils to interact with.

Since inductive reactance reduces the flow of current in a circuit, it appears as an energy loss just like resistance. However, it is possible to distinguish between resistance and inductive reactance in a circuit by looking at the timing between the sine waves of the voltage and current of the alternating current. In an AC circuit that contains only resistive components, the voltage and the current will be in-phase, meaning that the peaks and valleys of their sine waves will occur at the same time. When there is inductive reactance present in the circuit, the phase of the current will be shifted so that its peaks and valleys do not occur at the same time as those of the voltage. This will be discussed in more detail in the section on circuits.

6) Mutual Inductance
(The Basis for Eddy Current Inspection)

The magnetic flax through a circuit can be related to the current in that circuit and the currents in other nearby circuits, assuming that there are no nearby permanent magnets. Consider the following two circuits.

The magnetic field produced by circuit 1 will intersect the wire in circuit 2 and create current flow. The induced current flow in circuit 2 will have its own magnetic field which will interact with the magnetic field of circuit 1. At some point P on the magnetic field consists of a part due to i1 and a part due to i2. These fields are proportional to the currents producing them.

Self Inductance:
The property of an electric circuit or component that caused an e.m.f. to be generated in it as a result of a change in the current flowing through the circuit.

Mutual Inductance:
The property of an electric circuit or component that caused an e.m.f. to be generated in it as a result of a change in the current flowing through a neighboring circuit with which it is magnetically linked.

The coils in the circuits are labeled L1 and L2 and this term represents the self inductance of each of the coils. The values of L1 and L2 depend on the geometrical arrangement of the circuit (i.e. number of turns in the coil) and the conductivity of the material. The constant M, called the mutual inductance of the two circuits and it is dependent on the geometrical arrangement of both circuits. In particular, if the circuits are far apart, the magnetic flux through circuit 2 due to the current i1 will be small and the mutual inductance will be small. L2 and M are constants.

We can write the flux, B through circuit 2 as the sum of two parts.

B2 = L2i2 + i1M

An equation similar to the one above can be written for the flux through circuit 1.

B1 = L1i1 + i2M

Though it is certainly not obvious, it can be shown that the mutual inductance is the same for both circuits. Therefore, it can be written as follows:

M1,2 = M2,1

How is mutual induction used in eddy current inspection?

In eddy current inspection, the eddy currents are generated in the test material due to mutual induction. The test probe is basically a coil of wire through which alternating current is passed. Therefore, when the probe is connected to an eddyscope instrument, it is basically represented by circuit one above. The second circuit can be any piece of conductive material.


Eddy Current:
A current induced in a conductor situated in a changing magnetic field or moving in a fixed one.

When alternating current is passed through the coil, a magnetic field is generated in and around the coil. When the probe is brought in close proximity to a conductive material, such as aluminum, the probes changing magnetic field generates current flow in the material. The induced current flows in closed loops in planes perpendicular to the magnetic flux. They are named eddy currents because they are thought to resemble the eddy currents that can be seen swirling in streams.

The eddy currents produce their own magnetic fields that interact with the primary magnetic field of the coil. By measuring changes in the resistance and inductive reactance of the coil, information can be gathered about the test material. This information includes the electrical conductivity and magnetic permeability of the material, the amount of material cutting through the coils magnetic field, and the condition of the material (i.e. whether it contains cracks or other defects.) The distance that the coil is from the conductive material is called ,

Magnetic Permeability:
The ratio of the magnetic flux density, B, in a substance to the external field strength.

A term used to describe materials, such as iron, nickel, and cobalt, which have a high magnetic permeability.

It should be noted that if a sample is ferromagnetic , the magnetic flux is concentrated and strengthened despite opposing eddy current affects. The increase inductive reactance due to the magnetic permeability of ferromagnetic materials makes it easy to distinguish these materials from non ferromagnetic materials.

In the applet below, the probe and the sample are shown in cross-section. The boxes represent a the cross-sectional area of a group of turns in the coil. The liftoff distance and the drive current of the probe can be varied to see the effects of the shared magnetic field. The liftoff value can be set to 0.1 or less and the current value can be varied from 0.01 to 1.0. The strength of the magnetic field is shown by the darkness of the lines.


7) Ferromagnetism

Iron, nickel, cobalt and some of the rare earths (gadolinium, dysprosium) exhibit a unique magnetic behavior, which is called ferromagnetism because iron (ferric) is the most common and most dramatic example. Ferromagnetic materials exhibit a long-range ordering phenomenon at the atomic level, which causes the unpaired electron spins to line up parallel with each other in a region called a domain. Within the domain, the magnetic field is intense, but in a bulk sample the material will usually be un magnetized because the many domains will themselves be randomly oriented with respect to one another. Ferromagnetism manifests itself in the fact that a small externally imposed magnetic field, say from a solenoid, can cause the magnetic domains to line up with each other and the material is said to be magnetized. The driving magnetic field will then be increased by a large, factor, which is usually expressed as a relative permeability for the material. There are many practical applications of ferromagnetic materials, such as the electromagnet.

Ferro magnets will tend to stay magnetized to some extent after being subjected to an external magnetic field. This tendency to "remember their magnetic history" is called hysteresis. The fraction of the saturation magnetization, which is retained when the driving field is removed is called the remanence of the material, and is an important factor in permanent magnets.

All ferro magnets have a maximum temperature where the ferromagnetic property disappears as a result of thermal agitation. This temperature is called the Curie temperature.

Ferro magnetic materials will respond mechanically to an impressed magnetic field, changing length slightly in the direction of the applied field. This property, called magnetostriction, leads to the familiar hum of transformers as they respond mechanically to 60 Hz AC voltages.