Inductance
In May 1884, Oliver Heaviside reached for a shorter way to say something cumbersome. The full phrase was "coefficient of self-induction." He coined the word inductance instead, and the name stuck. Behind that tidy term sits a stubborn behavior in every wire ever made. Push a current through a conductor, and the conductor pushes back against any change to that current. Speed the current up, and an opposing voltage appears to slow it. Slow it down, and a voltage appears to keep it going. This is the property at the heart of motors, transformers, and the wireless charging of devices across a gap. To understand why a coil of wire stores energy in empty space, and why a magnetic core can multiply that effect by thousands, you have to start with a current, a magnetic field, and a law that insists nothing in nature happens quietly.
In 1831, Michael Faraday wrapped two wires around opposite sides of an iron ring and waited for a wave. He expected that starting a current in one wire would send some effect traveling through the ring to the other side. Using a galvanometer, he saw a transient current flicker in the second coil every time a battery was connected to or disconnected from the first. The current was not constant. It appeared only at the moments of change, induced by the shifting magnetic flux as the battery was switched on and off. Faraday chased the same effect in other shapes. He slid a bar magnet quickly in and out of a coil of wire and caught the same fleeting currents. He rotated a copper disk near a bar magnet with a sliding electrical lead and produced a steady direct current, an arrangement remembered as Faraday's disk. The deeper history reached back to the ancients, who knew electric charge from rubbing silk on amber, electric current from lightning, and magnetic attraction from lodestone. Binding those scattered forces into one theory of electromagnetism was the work of the 19th century.
A current flowing through a conductor builds a magnetic field around it, a relationship set out by Ampere's circuital law. The total magnetic flux through a circuit is the perpendicular component of the flux density multiplied by the area of the surface the current path spans. Faraday's law of induction states that any change in that flux induces an electromotive force in the circuit, in proportion to how fast the flux is changing. A negative sign carries the meaning. The induced voltage always opposes the very change that created it, a rule named Lenz's law, and the opposing voltage earns the name back EMF. When the current rises, the voltage turns positive at the end where current enters and negative where it leaves, working to throttle the rise. When the current falls, the polarity flips to prop it up. Self-inductance, usually shortened to inductance, is the ratio of that induced voltage to the rate of change of the current. The symbol for inductance honors the physicist Heinrich Lenz. The unit honors Joseph Henry, who discovered inductance independently of Faraday.
Every conductor has some inductance, and the effect can be helpful or harmful depending on the device. Its size depends on the geometry of the current path and on the magnetic permeability of nearby materials. Ferromagnetic materials with high permeability, such as iron placed near a conductor, strengthen the magnetic field and raise the inductance. Anything that increases the flux a given current produces increases the inductance, since inductance also equals the ratio of magnetic flux to current. An inductor is a component built deliberately to do this, a wire wound into a coil or helix. A coiled wire holds more inductance than a straight wire of the same length, because the field lines thread the circuit many times over, creating multiple flux linkages. The inductance grows with the square of the number of turns, assuming full flux linkage. Drop a magnetic core of ferromagnetic material into the center of the coil and the effect deepens. The coil's field magnetizes the core, aligning its magnetic domains, and the core's own field adds to the coil's. A magnetic core can raise the inductance of a coil by thousands of times.
Charges climbing through a conductor whose current is increasing lose potential energy, fighting a polarity that opposes them on top of any ordinary resistance. The energy spent overcoming this potential hill does not vanish. It is stored in the swelling magnetic field around the conductor, which is to say an inductor banks energy in its own magnetic field. The power flowing into that field equals the product of the current and the voltage across the conductor. With no current there is no field and the stored energy is zero. Neglecting resistive losses, the energy stored, measured in joules, equals the work needed to build the current up from nothing. Hold the current steady and the energy sits there untouched. Let the current fall and the field collapses, inducing a voltage in the opposite direction that returns the stored energy to the external circuit. This neat accounting holds only while the inductance stays constant. Near a ferromagnetic core driven hard enough to saturate, the inductance starts to vary with current, and the simple constant-inductance equation gives way to the integral form.
Send a sinusoidal alternating current through a linear inductance and the induced back-EMF comes out sinusoidal too. Inductive reactance is the opposition an inductor offers to alternating current, defined in parallel with resistance as the ratio of the peak alternating voltage to the peak current. Reactance is measured in ohms. It climbs in proportion to frequency, so for a given applied voltage an inductor passes less current as the frequency rises. Because the induced voltage is largest when the current is changing fastest, the two waveforms fall out of step. The voltage peaks arrive earlier in each cycle than the current peaks. The phase difference between current and induced voltage is 90 degrees, meaning that in an ideal inductor the current lags the voltage by 90 degrees.
Inductance can be derived from Maxwell's equations in full generality, but most useful cases yield to simplifications. For thin wires the self-inductance still depends on the wire radius and how the current spreads across it, a distribution that stays roughly constant when the radius is far smaller than every other length in the problem. Longer wires hold more inductance and thicker wires less, loosely echoing electrical resistance, though the relationships are nonlinear and different in kind. A single conductor of lamp cord 10 units long, made of 18 AWG wire, would show a low-frequency inductance of about 19.67 if stretched out straight. At high frequencies the skin effect drives interior currents to the surface, and the formula shifts by a single constant, from 0.75 down to 1. A solenoid, a coil far longer than it is wide, holds a nearly constant flux density inside, and its inductance becomes a matter of geometry and turn count alone, independent of current for an air core. The derivations behind these practical formulas trace back to Rosa in 1908.
Place two circuits close enough and the magnetic field of one passes through the other, and the pair is said to be inductively coupled. A change of current in one then induces a voltage in the other, the effect captured by mutual inductance, given the symbol M and defined as the ratio of voltage induced in one coil to the rate of change of current in its neighbor. This is the principle behind a transformer. For thin filamentary wires the mutual inductance follows the double-integral Neumann formula taken over the two curves. The same coupling that powers transformers can also leak unwanted signals between conductors that were never meant to talk. The coupling coefficient measures how tightly two inductors are linked, and most authors fix its range between 0 and 1. Wind a capacitor across a transformer winding and the winding becomes a tuned circuit. With a capacitor on each side it is a double tuned transformer, and the mutual inductance and the circuit's Q factor together shape its frequency response. Push the coupling past its critical value and the response peak splits into two peaks that drift apart, a state called overcoupling. The same strong coupling lets self-resonant coils carry power wirelessly across distances of up to two metres.
Common questions
Who coined the term inductance and when?
Oliver Heaviside coined the term inductance in May 1884 as a convenient way to refer to the coefficient of self-induction.
What is inductance in electrical conductors?
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. A changing current produces a changing magnetic field, which induces a voltage that opposes the change, as described by Lenz's law.
What is the SI unit of inductance?
The SI unit of inductance is the henry, abbreviated H. One henry is the amount of inductance that causes a voltage of one volt when the current changes at a rate of one ampere per second. It is named for Joseph Henry, who discovered inductance independently of Faraday.
Who discovered electromagnetic induction?
Michael Faraday first described electromagnetic induction in 1831. He wrapped two wires around opposite sides of an iron ring and observed a transient current in the second coil each time a battery was connected to or disconnected from the first.
How does a magnetic core affect the inductance of a coil?
A magnetic core of ferromagnetic material placed in the center of a coil can increase its inductance by thousands of times. The coil's field magnetizes the core and aligns its magnetic domains, and the core's field adds to the coil's, increasing the flux through the coil.
What is mutual inductance and how do transformers use it?
Mutual inductance is the ratio of the voltage induced in one coil to the rate of change of current in a neighboring coil, given the symbol M. It is the principle behind transformers, where a changing current in one winding induces a voltage in another.
How does inductive reactance change with frequency?
Inductive reactance increases proportionally with frequency, so an inductor conducts less current for a given applied AC voltage as frequency rises. In an ideal inductor the current lags the voltage by 90 degrees.
All sources
36 references cited across the entry
- 1BookPhysics for global scientists and engineersA. Raymond Serway et al. — Cengage AU — 2017
- 2BookSir William Preece, F.R.S.: Victorian Engineer ExtraordinaryEdward Cecil Baker — Hutchinson — 1976
- 3BookElectrical Papers, Vol. 1Oliver Heaviside — Macmillan — 1894
- 4The Physics Hypertextbook: InductanceGlenn Elert
- 5Molecular Expressions: Electricity and Magnetism Introduction: InductanceMichael W. Davidson — 1995–2008
- 7BookFundamentals of applied electromagneticsFawwaz Ulaby — Pearson / Prentice Hall — 2007
- 9BookMichael Faraday: A BiographyL. Pearce Williams — Simon and Schuster — 1971
- 10BookPhysics: Principles with ApplicationsDouglas C. Giancoli — 1998
- 11BookElectro Magnetic Field TheoryYaduvir Singh — Pearson Education India — 2011
- 12BookElectrical Power SystemsC.L. Wadhwa — New Age International — 2005
- 13BookBarron's AP Physics CRobert A. Pelcovits et al. — Barron's Educational Series — 2007
- 14BookElectricity and MagnetismEdward M. Purcell et al. — Cambridge Univ. Press — 2013
- 15BookPrinciples of Physics: A Calculus-Based Text, 5th Ed.Raymond A. Serway et al. — Cengage Learning — 2012
- 16BookEngineering Electromagnetics, 2nd Ed.Nathan Ida — Springer Science and Business Media — 2007
- 17BookElectricity and Magnetism, 2nd Ed.Edward Purcell — Cambridge University Press — 2011
- 18BookIntroduction to ElectronicsEarl D. Gates — Cengage Learning — 2001
- 19JournalThe self and mutual inductances of linear conductorsE.B. Rosa — U.S. Bureau of Standards — 1908
- 20JournalSelf inductance of a wire loop as a curve integralR. Dengler — 2016
- 21JournalAllgemeine Gesetze der inducirten elektrischen StrömeF.E. Neumann — Wiley — 1846
- 22BookClassical ElectrodynamicsJ. D. Jackson — Wiley — 1975
- 23BookSchaum's outline of theory and problems of electric circuitsMahmood Nahvi et al. — McGraw-Hill Professional — 2002
- 24BookHigh-speed Circuit Board Signal IntegrityStephen C. Thierauf — Artech House — 2004
- 25JournalDesign of a Reliable Broadband I/O Employing T-coilSeok Kim et al. — 2009
- 26BookNetwork Theory and Filter DesignVasudev K. Aatre — John Wiley & Sons — 1981
- 27BookLinear and Nonlinear CircuitsLeon O. Chua et al. — McGraw-Hill, Inc. — 1987
- 28BookCircuit Analysis FundamentalsMansour Eslami — Agile Press — May 24, 2005
- 29JournalSimultaneous 6-Gb/s Data and 10-mW Power Transmission Using Nested Clover Coils for Noncontact Memory CardAndrzej Radecki et al. — 2012
- 30JournalWireless Power Transfer via Strongly Coupled Magnetic ResonancesA. Kurs et al. — 6 July 2007
- 31JournalAnalysis, Experimental Results, and Range Adaptation of Magnetically Coupled Resonators for Wireless Power TransferAlanson P. Sample et al. — 2011
- 32Book2022 IEEE 35th International Conference on Micro Electro Mechanical Systems Conference (MEMS)Adrian A. Rendon-Hernandez et al. — 2022
- 33JournalFormulas for the skin effectH.A. Wheeler — 1942
- 34JournalSimple inductance formulas for radio coilsH.A. Wheeler — 1928
- 35BookElectromagneticsR.S. Elliott — IEEE Press — 1993
- 36BookInductance Calculations: Working formulas and tablesFrederick W. Grover — Dover Publications, Inc. — 1946