## Inductance: Loop and Partial

In electromagnetism and electronics , inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor whose strength depends on the magnitude of the current. Any change in the magnet field strength, caused by a change in current, induces an electromotive force EMF in the conductor that opposes the voltage creating the change of current. This is known as electromagnetic induction , and is described by the laws of Faraday and Lenz. This induced voltage in the circuit is called back EMF , to reflect its opposing character.

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Inductance is defined as the product of the ratio of the induced voltage to the rate of change of current causing it and a proportionality factor that depends on the geometry of circuit conductors and the magnetic permeability of nearby materials. A device that is used as an electronic component to intentionally add inductance to a circuit is called an inductor. It typically consists of a coil or helix of wire. The term inductance was coined by Oliver Heaviside in It is named for Joseph Henry , who discovered inductance independently of Faraday.

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The history of electromagnetic induction, a facet of electromagnetism, began with observations of the ancients: electric charge or static electricity rubbing silk on amber , electric current lightning , and magnetic attraction lodestone. Understanding the unity of these forces of nature, and the scientific theory of electromagnetism began in the late 18th century.

Electromagnetic induction was first described by Michael Faraday in He expected that, when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. Using a galvanometer , he observed a transient current flow in the second coil of wire each time that a battery was connected or disconnected from the first coil.

For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady DC current by rotating a copper disk near the bar magnet with a sliding electrical lead "Faraday's disk". The negative sign in the equation indicates that the induced voltage is in a direction which opposes the change in current that created it; this is called Lenz's law.

The potential is therefore called a back EMF. If the current is increasing, the voltage is positive at the end of the conductor through which the current enters and negative at the end through which it leaves, tending to reduce the current. If the current is decreasing, the voltage is positive at the end through which the current leaves the conductor, tending to maintain the current. Thus, inductance is a property of a conductor or circuit, due to its magnetic field, which tends to oppose changes in current through the circuit.

The unit of inductance in the SI system is the henry H , named after American scientist Joseph Henry , which is the amount of inductance which generates a voltage of one volt when the current is changing at a rate of one ampere per second.

All conductors have some inductance, which may have either desirable or detrimental effects in practical electrical devices. The inductance of a circuit depends on the geometry of the current path, and on the magnetic permeability of nearby materials; ferromagnetic materials with a higher permeability like iron near a conductor tend to increase the magnetic field and inductance.

Any alteration to a circuit which increases the flux total magnetic field through the circuit produced by a given current increases the inductance, because inductance is also equal to the ratio of magnetic flux to current    . An inductor is an electrical component consisting of a conductor shaped to increase the magnetic flux, to add inductance to a circuit.

Typically it consists of a wire wound into a coil or helix. A coiled wire has a higher inductance than a straight wire of the same length, because the magnetic field lines pass through the circuit multiple times, it has multiple flux linkages. The inductance is proportional to the square of the number of turns in the coil.

The inductance of a coil can be increased by placing a magnetic core of ferromagnetic material in the hole in the center. The magnetic field of the coil magnetizes the material of the core, aligning its magnetic domains , and the magnetic field of the core adds to that of the coil, increasing the flux through the coil. This is called a ferromagnetic core inductor. A magnetic core can increase the inductance of a coil by thousands of times.

If multiple electric circuits are located close to each other, the magnetic field of one can pass through the other; in this case the circuits are said to be inductively coupled. Due to Faraday's law of induction , a change in current in one circuit can cause a change in magnetic flux in another circuit and thus induce a voltage in another circuit. This is the principle behind a transformer. The property describing the effect of one conductor on itself is more precisely called self-inductance , and the properties describing the effects of one conductor with changing current on nearby conductors is called mutual inductance.

The charges flowing through the circuit lose potential energy moving from the higher voltage to the lower voltage end. The energy from the external circuit required to overcome this "potential hill" is being stored in the increased magnetic field around the conductor.

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6. Therefore, any inductance with a current through it stores energy in its magnetic field. When there is no current, there is no magnetic field and the stored energy is zero. This is given by:. So therefore inductance is also proportional to how much energy is stored in the magnetic field for a given current. This energy is stored as long as the current remains constant.

If the current decreases, the magnetic field will decrease, inducing a voltage in the conductor in the opposite direction, negative at the end through which current enters and positive at the end through which it leaves. This will return stored magnetic energy to the external circuit. If ferromagnetic materials are located near the conductor, such as in an inductor with a magnetic core , the constant inductance equation above is only valid for linear regions of the magnetic flux, at currents below the level at which the ferromagnetic material saturates , where the inductance is approximately constant.

If the magnetic field in the inductor approaches the level at which the core saturates, the inductance will begin to change with current, and the integral equation must be used. When a sinusoidal alternating current AC is passing through a linear inductance, the induced back-EMF will also be sinusoidal. Inductive reactance is the opposition of an inductor to an alternating current. Reactance has units of ohms. Because the induced voltage is greatest when the current is increasing, the voltage and current waveforms are out of phase ; the voltage peaks occur earlier in each cycle than the current peaks.

In the most general case, inductance can be calculated from Maxwell's equations. Many important cases can be solved using simplifications. Where high frequency currents are considered, with skin effect , the surface current densities and magnetic field may be obtained by solving the Laplace equation. Where the conductors are thin wires, self-inductance still depends on the wire radius and the distribution of the current in the wire.

This current distribution is approximately constant on the surface or in the volume of the wire for a wire radius much smaller than other length scales.

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As a practical matter, longer wires have more inductance, and thicker wires have less, analogous to their electrical resistance although the relationships aren't linear, and are different in kind from the relationships that length and diameter bear to resistance. As an essential component of coils and circuits, understanding what the inductance of a wire is, is essential. Yet, there is no simple, unambiguous definition of the inductance of a wire. The reasons are both practical and epistemological. There are practical difficulties in measuring wire inductance.

A short straight segment of single-conductor wire has some inductance, which in our ordinary experience is intangible because it is so small that it is effectively undetectable: It is too small to be readily be measured at low frequencies, and at high frequencies becomes entangled with the inductance of the wires of any meter connected to measure it. A long straight wire like an electric transmission line hundreds of kilometers long has substantial inductance, and there is no problem at all measuring it.

Difficulties with epistemology also trouble the calculations: There is no unambiguous definition of the inductance of a straight wire.

## Inductance: Loop and Partial - Clayton R. Paul - Google книги

If we consider the wire in isolation we ignore the question of how the current gets into the wire. That current will affect the flux which is developed in the vicinity of the wire. But this flux should also be a part of the definition. A consequence of Maxwell's equations is that we cannot define the inductance of only a portion of a circuit, we can only define the inductance of a whole circuit, which includes how the current gets to the wire and how it returns to the source. The magnetic flux incident to the whole circuit determines the inductance of the circuit and of any part of it.

The magnetic flux is an indivisible entity, yet we wish to consider only a part of it, the part incident to the wire, between whatever we define to be the "ends" of the wire. For derivation of the formulas below, see Rosa The constant 0.

## ISBN 13: 9780470461884

Disks or thick cylinders have slightly different formulas. Currents in the wires need not be equal, though they often are, as in the case of a complete circuit, where one wire is the source and the other the return. This is the generalized case of the paradigmatic two-loop cylindrical coil carrying a uniform low frequency current; the loops are independent closed circuits that can have different lengths, any orientation in space, and carry different currents.

None-the-less, the error terms, which are not included in the integral will only be small if the geometries of the loops are mostly smooth and convex: they do not have too many kinks, sharp corners, coils, crossovers, parallel segments, concave cavities or other topological "close" deformations.