Inductance
Electromagnetism 


In electromagnetism and electronics, inductance is the property of an electrical conductor by which a change in current through it induces an electromotive force in both the conductor itself^{[1]} and in any nearby conductors by mutual inductance.^{[1]}
These effects are derived from two fundamental observations of physics: a steady current creates a steady magnetic field described by Oersted's law,^{[2]} and a timevarying magnetic field induces an electromotive force (EMF) in nearby conductors, which is described by Faraday's law of induction.^{[3]} According to Lenz's law,^{[4]} a changing electric current through a circuit that contains inductance induces a proportional voltage, which opposes the change in current (selfinductance). The varying field in this circuit may also induce an EMF in neighbouring circuits (mutual inductance).
The term inductance was coined by Oliver Heaviside in 1886.^{[5]} It is customary to use the symbol L for inductance, in honour of the physicist Heinrich Lenz.^{[6]}^{[7]} In the SI system, the measurement unit for inductance is the henry, with the unit symbol H, named in honor of Joseph Henry, who discovered inductance independently of, but not before, Faraday.^{[8]}
Circuit analysis
An electronic component that is intended to add inductance to a circuit is called an inductor. Inductors are typically manufactured from coils of wire. This design delivers two desired properties, a concentration of the magnetic field into a small physical space and a linking of the magnetic field into the circuit multiple times.
The relationship between the selfinductance, L, of an electrical circuit, the voltage, v(t), and the current, i(t), through the circuit is
 .
A voltage is induced across an inductor (back EMF), that is equal to the product of the inductor's inductance and the rate of change of current through the inductor.
All circuits have, in practice, some inductance, which may have beneficial or detrimental effects. For a tuned circuit, inductance is used to provide a frequencyselective circuit. Practical inductors may be used to provide filtering, or energy storage, in a given network. The inductance per unit length of a transmission line is one of the properties that determines its characteristic impedance; balancing the inductance and capacitance of cables is important for distortionfree telegraphy and telephony. The inductance of long AC power transmission lines affects the power capacity of the line. Sensitive circuits, such as microphone and computer network cables, may utilize special cabling construction, limiting the inductive coupling between circuits.
The generalization to the case of K electrical circuits with currents, {i_{m}}, and voltages, {v_{m}}, reads
Here, inductance L is a symmetric matrix. The diagonal coefficients L_{m,m} are called coefficients of selfinductance, the offdiagonal elements are called coefficients of mutual inductance. The coefficients of inductance are constant, as long as no magnetizable material with nonlinear characteristics is involved. This is a direct consequence of the linearity of Maxwell's equations in the fields and the current density. The coefficients of inductance become functions of the currents in the nonlinear case.
Derivation from Faraday's law of inductance
The inductance equations above are a consequence of Maxwell's equations. There is a straightforward derivation in the important case of electrical circuits consisting of thin wires.
In a system of K wire loops, each with one or several wire turns, the flux linkage of loop m, λ_{m}, is given by
Here N_{m} denotes the number of turns in loop m; Φ_{m}, the magnetic flux through loop m; and L_{m,n} are some constants. This equation follows from Ampere's law  magnetic fields and fluxes are linear functions of the currents. By Faraday's law of induction, we have
where v_{m} denotes the voltage induced in circuit m. This agrees with the definition of inductance above if the coefficients L_{m,n} are identified with the coefficients of inductance. Because the total currents N_{n}i_{n} contribute to Φ_{m} it also follows that L_{m,n} is proportional to the product of turns N_{m}N_{n}.
Inductance and magnetic field energy
Multiplying the equation for v_{m} above with i_{m}dt and summing over m gives the energy transferred to the system in the time interval dt,
This must agree with the change of the magnetic field energy, W, caused by the currents.^{[9]} The integrability condition
requires L_{m,n} = L_{n,m}. The inductance matrix, L_{m,n}, thus is symmetric. The integral of the energy transfer is the magnetic field energy as a function of the currents,
This equation also is a direct consequence of the linearity of Maxwell's equations. It is helpful to associate changing electric currents with a buildup or decrease of magnetic field energy. The corresponding energy transfer requires or generates a voltage. A mechanical analogy in the K = 1 case with magnetic field energy (1/2)Li^{2} is a body with mass M, velocity u and kinetic energy (1/2)Mu^{2}. The rate of change of velocity (current) multiplied with mass (inductance) requires or generates a force (an electrical voltage).
Coupled inductors and mutual inductance
Mutual inductance occurs when the change in current in one inductor induces a voltage in another nearby inductor. It is important as the mechanism by which transformers work, but it can also cause unwanted coupling between conductors in a circuit.
The mutual inductance, M, is also a measure of the coupling between two inductors. The mutual inductance by circuit i on circuit j is given by the double integral Neumann formula, see calculation techniques
The mutual inductance also has the relationship:
where
 is the mutual inductance, and the subscript specifies the relationship of the voltage induced in coil 2 due to the current in coil 1.
 N_{1} is the number of turns in coil 1,
 N_{2} is the number of turns in coil 2,
 P_{21} is the permeance of the space occupied by the flux.
The mutual inductance also has a relationship with the coupling coefficient. The coupling coefficient is always between −1 and 1, and is a convenient way to specify the relationship between a certain orientation of inductors with arbitrary inductance:
where
 k is the coupling coefficient and −1 ≤ k ≤ 1,
 L_{1} is the inductance of the first coil, and
 L_{2} is the inductance of the second coil.
Once the mutual inductance, M, is determined from this factor, it can be used to predict the behavior of a circuit:
where
 v_{1} is the voltage across the inductor of interest,
 L_{1} is the inductance of the inductor of interest,
 di_{1}/dt is the derivative, with respect to time, of the current through the inductor of interest,
 di_{2}/dt is the derivative, with respect to time, of the current through the inductor that is coupled to the first inductor, and
 M is the mutual inductance.
The minus sign arises because of the sense the current i_{2} has been defined in the diagram. With both currents defined going into the dots the sign of M will be positive (the equation would read with a plus sign instead).^{[10]}
Matrix representation
The circuit can be described by any of the twoport network parameter matrix representations. The most direct are the z parameters, which are given by
where s is the complex frequency variable.
Equivalent circuit
Mutually coupled inductors can equivalently be represented by a Tcircuit of inductors as shown. If the coupling is strong and the inductors are of unequal values then the series inductor on the stepdown side may take on a negative value.
This can be analyzed as a two port network. With the output terminated with some arbitrary impedance, Z, the voltage gain, A_{v}, is given by,
For tightly coupled inductors where k = 1 this reduces to
which is independent of the load impedance. If the inductors are wound on the same core and with the same geometry, then this expression is equal to the turns ratio of the two inductors because inductance is proportional to the square of turns ratio.
The input impedance of the network is given by,
For k = 1 this reduces to
Thus, the current gain, A_{i} is not independent of load unless the further condition
is met, in which case,
and
Tuned transformer
When either side of the transformer is a tuned circuit, the amount of mutual inductance between the two windings, together with the Q factor of the circuit, determine the shape of the frequency response curve. The tuned circuit together with the transformer load form an RLC circuit with a definite peak in the frequency response. When both sides of the transformer are tuned, it is described as doubletuned. The coupling of doubletuned circuits is described as loose, critical, or overcoupled depending on the value of k. When two tuned circuits are loosely coupled through mutual inductance, the bandwidth will be narrow. As the amount of mutual inductance increases, the bandwidth continues to grow. When the mutual inductance is increased beyond a critical point, the peak in the response curve begins to drop, and the center frequency will be attenuated more strongly than its direct sidebands. This is known as overcoupling.
Ideal transformers
When k = 1, the inductor is referred to as being closely coupled. If in addition, the selfinductances go to infinity, the inductor becomes an ideal transformer. In this case the voltages, currents, and number of turns can be related in the following way:
where
 V_{s} is the voltage across the secondary inductor,
 V_{p} is the voltage across the primary inductor (the one connected to a power source),
 N_{s} is the number of turns in the secondary inductor, and
 N_{p} is the number of turns in the primary inductor.
Conversely the current:
where
 I_{s} is the current through the secondary inductor,
 I_{p} is the current through the primary inductor (the one connected to a power source),
 N_{s} is the number of turns in the secondary inductor, and
 N_{p} is the number of turns in the primary inductor.
Note that the power through one inductor is the same as the power through the other. Also note that these equations don't work if both inductors are forced (with power sources).
Calculation techniques
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, selfinductance 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.
Mutual inductance of two wire loops
The mutual inductance by a filamentary circuit m on a filamentary circuit n is given by the double integral Neumann formula^{[11]}
The symbol μ_{0} denotes the magnetic constant (4π × 10^{−7} H/m), C_{m} and C_{n} are the curves spanned by the wires. See a derivation of this equation.
Selfinductance of a wire loop
Formally, the selfinductance of a wire loop would be given by the above equation with m = n. The problem, however, is that 1/x − x' now becomes infinite, leading to a logarithmically divergent integral. This necessitates taking the finite wire radius a and the distribution of the current in the wire into account. There remain the contribution from the integral over all points with x − x' > a/2 and a correction term,^{[12]}
Here a and l denote the radius and length of the wire, and Y is a constant that depends on the distribution of the current in the wire: Y = 0 when the current flows in the surface of the wire (skin effect), Y = 1/2 when the current is homogeneous across the wire. The error O(μ_{0}a) is small when the wire is long compared to its radius.
Method of images
In some cases, different current distributions generate the same magnetic field in some section of space. This fact may be used to relate self inductances (method of images). As an example, consider the two systems:
 A wire at distance d/2 in front of a perfectly conducting wall (which is the return)
 Two parallel wires at distance d, with opposite current
The magnetic field of the two systems coincides (in a half space). The magnetic field energy and the inductance of the second system thus are twice as large as that of the first system.
Relation between inductance and capacitance
Inductance per length L' and capacitance per length C' are related to each other in the special case of transmission lines consisting of two parallel perfect conductors of arbitrary but constant cross section,^{[13]}
Here ε and µ denote the dielectric constant and magnetic permeability of the medium that the conductors are embedded in. There is no electric and no magnetic field inside the conductors (complete skin effect, high frequency). Current flows down on one line and returns on the other. Signals will propagate along the transmission line at the speed of electromagnetic radiation in the nonconductive medium enveloping the conductors.
Selfinductance of simple electrical circuits in air
The selfinductance of many types of electrical circuits can be given in closed form. Examples are listed in the table.
Type  Inductance  Comment 

Single layer solenoid^{[14]} 
 
Coaxial cable, high frequency 
a_{1}: Outer radius a: Inner radius l: Length  
Circular loop^{[15]}  r: Loop radius a: Wire radius  
Rectangle^{[16]}  b, d: Border length d ≫ a, b ≫ a a: Wire radius  
Pair of parallel wires 
a: Wire radius d: Distance, d ≥ 2a l: Length of pair  
Pair of parallel wires, high frequency 
a: Wire radius d: Distance, d ≥ 2a l: Length of pair  
Wire parallel to perfectly conducting wall 
a: Wire radius d: Distance, d ≥ a l: Length  
Wire parallel to conducting wall, high frequency 
a: Wire radius d: Distance, d ≥ a l: Length 
The symbol μ_{0} denotes the magnetic constant (4π×10^{−7} H/m) in SI units.
The purpose of the constant Y : For high frequencies, the electric current flows in the conductor surface (skin effect) and, depending on the geometry, it is sometimes necessary to distinguish low and high frequency inductances. For high frequencies, Y = 0 when the current is uniformly distributed over the surface of the wire (skin effect), Y = 0.5 when the current is uniformly distributed over the cross section of the wire (very low frequencies). In the high frequency case, if conductors are within one wire diameter of each other, an additional screening current flows in their surfaces, and expressions containing Y become invalid.
Inductance with physical symmetry
Inductance of a solenoid
A solenoid is a long, thin coil; i.e., a coil whose length is much greater than its diameter. Under these conditions, and without any magnetic material used, the magnetic flux density within the coil is practically constant and is given by
where is the magnetic constant, the number of turns, the current and the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density by the crosssection area :
When this is combined with the definition of inductance,
it follows that the inductance of a solenoid is given by:
A table of inductance for short solenoids of various diameter to length ratios has been calculated by Dellinger, Whittmore, and Ould.^{[17]}
This, and the inductance of more complicated shapes, can be derived from Maxwell's equations. For rigid aircore coils, inductance is a function of coil geometry and number of turns, and is independent of current.
Similar analysis applies to a solenoid with a magnetic core, but only if the length of the coil is much greater than the product of the relative permeability of the magnetic core and the diameter. That limits the simple analysis to lowpermeability cores, or extremely long thin solenoids. Although rarely useful, the equations are,
where the relative permeability of the material within the solenoid,
from which it follows that the inductance of a solenoid is given by:
where N is squared because of the definition of inductance.
Note that, since the permeability of ferromagnetic materials changes with applied magnetic flux, the inductance of a coil with a ferromagnetic core will generally vary with current.
Inductance of a coaxial line
Let the inner conductor have radius and permeability , let the dielectric between the inner and outer conductor have permeability , and let the outer conductor have inner radius , outer radius , and permeability . Assume that a DC current flows in opposite directions in the two conductors, with uniform current density. The magnetic field generated by these currents points in the azimuthal direction and is a function of radius ; it can be computed using Ampère's law:
The flux per length in the region between the conductors can be computed by drawing a surface containing the axis:
Inside the conductors, L can be computed by equating the energy stored in an inductor, , with the energy stored in the magnetic field:
For a cylindrical geometry with no dependence, the energy per unit length is
where is the inductance per unit length. For the inner conductor, the integral on the righthandside is ; for the outer conductor it is
Solving for and summing the terms for each region together gives a total inductance per unit length of:
However, for a typical coaxial line application, we are interested in passing (nonDC) signals at frequencies for which the resistive skin effect cannot be neglected. In most cases, the inner and outer conductor terms are negligible, in which case one may approximate
Phasor circuit analysis and impedance
If signals of current and voltage are sine, using phasors, the equivalent impedance of an inductance is given by:
where
 j is the imaginary unit,
 L is the inductance,
 ω = 2πf is the angular frequency,
 f is the frequency and
 ωL = X_{L} is the inductive reactance.
Nonlinear inductance
Many inductors make use of magnetic materials. These materials over a large enough range exhibit a nonlinear permeability with such effects as saturation. In turn, the saturation makes the resulting inductance a function of the applied current. Faraday's Law still holds but inductance is ambiguous and is different whether you are calculating circuit parameters or magnetic fluxes.
The secant or largesignal inductance is used in flux calculations. It is defined as:
The differential or smallsignal inductance, on the other hand, is used in calculating voltage. It is defined as:
The circuit voltage for a nonlinear inductor is obtained via the differential inductance as shown by Faraday's Law and the chain rule of calculus.
There are similar definitions for nonlinear mutual inductances.
See also
 Alternating current
 Dot convention
 Eddy current
 Electromagnetic induction
 Electricity
 Faraday's law of induction
 Gyrator
 Hydraulic analogy
 Inductor
 Leakage inductance
 LC circuit
 Magnetomotive force (MMF)
 RLC circuit
 RL circuit
 SI electromagnetism units
 Solenoid
 Transformer
 Kinetic inductance
 LCR meter
References
 1 2 Sears and Zemansky 1964:743
 ↑ Sears and Zemansksy 1964:671
 ↑ Sears and Zemansky 1964:671 – "The work of Oersted thus demonstrated that magnetic effects could be produced by moving electric charges, and that of Faraday and Henry that currents could be produced by moving magnets."
 ↑ Sears and Zemansky 1964:731 – "The direction of an induced current is such as to oppose the cause producing it".
 ↑ Heaviside, Oliver (1894). Electrical Papers. Macmillan and Company. p. 271.
 ↑ Glenn Elert. "The Physics Hypertextbook: Inductance". Retrieved 20160730.
 ↑ Michael W. Davidson (1995–2008). "Molecular Expressions: Electricity and Magnetism Introduction: Inductance".
 ↑ "A Brief History of Electromagnetism" (PDF).
 ↑ The kinetic energy of the drifting electrons is many orders of magnitude smaller than W, except for nanowires.
 ↑ Mahmood Nahvi; Joseph Edminister (2002). Schaum's outline of theory and problems of electric circuits. McGrawHill Professional. p. 338. ISBN 0071393072.
 ↑ Neumann, F. E. (1847). "Allgemeine Gesetze der inducirten elektrischen Ströme". Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin, aus dem Jahre 1845: 1–87.
 ↑ Dengler, R. (2016). "Self inductance of a wire loop as a curve integral". Advanced Electromagnetics. 5 (1): 1–8. Bibcode:2016AdEl....5....1D. doi:10.7716/aem.v5i1.331.
 ↑ Jackson, J. D. (1975). Classical Electrodynamics. Wiley. p. 262.
 ↑ Lorenz, L. (1879). "Über die Fortpflanzung der Elektrizität". Annalen der Physik. VII: 161–193. (The expression given is the inductance of a cylinder with a current around its surface). Bibcode:1879AnP...243..161L. doi:10.1002/andp.18792430602.
 ↑ Elliott, R. S. (1993). Electromagnetics. New York: IEEE Press. Note: The constant −3/2 in the result for a uniform current distribution is wrong.
 ↑ Rosa, E.B. (1908). "The Self and Mutual Inductances of Linear Conductors". Bulletin of the Bureau of Standards. 4 (2): 301–344. doi:10.6028/bulletin.088.
 ↑ D. Howard Dellinger; L. E. Whittmore; R. S. Ould (1924). "Radio Instruments and Measurements". NBS Circular. National Bureau of Standards. C74. Retrieved 20090907.
General references
 Frederick W. Grover (1952). Inductance Calculations. Dover Publications, New York.
 Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 013805326X.
 Wangsness, Roald K. (1986). Electromagnetic Fields (2nd ed.). Wiley. ISBN 0471811866.
 Hughes, Edward. (2002). Electrical & Electronic Technology (8th ed.). Prentice Hall. ISBN 058240519X.
 Küpfmüller K., Einführung in die theoretische Elektrotechnik, SpringerVerlag, 1959.
 Heaviside O., Electrical Papers. Vol.1. – L.; N.Y.: Macmillan, 1892, p. 429560.
 Fritz LangfordSmith, editor (1953). Radiotron Designer's Handbook, 4th Edition, Amalgamated Wireless Valve Company Pty., Ltd. Chapter 10, "Calculation of Inductance" (pp. 429–448), includes a wealth of formulas and nomographs for coils, solenoids, and mutual inductance.
 F. W. Sears and M. W. Zemansky 1964 University Physics: Third Edition (Complete Volume), AddisonWesley Publishing Company, Inc. Reading MA, LCCC 6315265 (no ISBN).