Inductance · aiwiki.page

Inductance is a property of an electrical circuit that relates its electric current to the magnetic flux linked with it. A changing current produces a changing magnetic field, which can induce a voltage in the same circuit or in another circuit. These effects are described by self-inductance and mutual inductance, respectively. A component designed to provide appreciable self-inductance is called an inductor; inductance itself is the physical quantity, not the component. (openstax.org)

Definition and units

For a coil, flux linkage is the sum of the magnetic fluxes through its turns:

\lambda=\sum_{k=1}^{N}\Phi_k.

If all $N$ turns link the same flux $\Phi$ , then $\lambda=N\Phi$ . For a fixed circuit with a linear magnetic response and no independently imposed flux, the flux linkage produced by its own current is proportional to that current:

\lambda=Li, \qquad L=\frac{\lambda}{i},

where $L$ is the self-inductance. Its value depends on the conductor arrangement and the magnetic properties of the surrounding materials. (openstax.org)

The SI unit of inductance is the henry, symbol H:

1\ \mathrm{H} =1\ \mathrm{Wb/A} =1\ \mathrm{V\,s/A} =1\ \mathrm{kg\,m^2\,s^{-2}\,A^{-2}}.

Thus, a constant inductance of one henry corresponds to an induced-voltage magnitude of one volt when the current changes at one ampere per second. Common submultiples are the millihenry, microhenry, and nanohenry. (nist.gov)

Electromagnetic origin and voltage conventions

Inductance is a consequence of electromagnetic induction. Faraday’s law of induction gives the electromotive force associated with changing flux linkage:

\mathcal{E}=-\frac{d\lambda}{dt}.

For constant self-inductance,

\mathcal{E}=-L\frac{di}{dt}.

The minus sign expresses Lenz’s law: the induced electromotive force opposes the change that produces it. It does not necessarily oppose the existing current. When current decreases, the induced electromotive force acts to sustain it. (openstax.org)

In circuit analysis, an ideal inductor’s terminal voltage is usually defined using the passive sign convention, with current entering the terminal designated positive. Under this convention,

v=\frac{d\lambda}{dt}=L\frac{di}{dt}.

The positive sign in this terminal-voltage equation and the negative sign in the induced-emf equation reflect different reference conventions, not different physical laws. (openstax.org)

Geometry and magnetic materials

Inductance is not restricted to wound coils: conductor loops, cables, and other current-carrying arrangements also possess it. A coil concentrates flux linkage by arranging many turns so that they link a common magnetic field. (openstax.org)

For a long solenoid with $N$ turns, cross-sectional area $A$ , length $\ell$ , and a uniform linear medium of magnetic permeability $\mu$ ,

L\approx\frac{\mu N^2A}{\ell}.

This approximation neglects end effects and assumes an approximately uniform internal field. It shows that inductance increases with the square of the number of turns when the other quantities remain fixed. (openstax.org)

Magnetic cores can increase inductance, but their response need not remain linear. In particular, core saturation causes inductance to decrease as current increases. Practical component models therefore distinguish small-signal behavior from large-current behavior. (coilcraft.com)

Mutual inductance

Mutual inductance describes flux linkage in one circuit produced by current in another. If current $i_1$ produces flux linkage $\lambda_{21}$ in circuit 2,

\lambda_{21}=M_{21}i_1.

For fixed, linear, reciprocal arrangements, the mutual inductances in the two directions are equal:

M_{21}=M_{12}=M.

The induced electromotive forces are then

\mathcal{E}_2=-M\frac{di_1}{dt}, \qquad \mathcal{E}_1=-M\frac{di_2}{dt}.

The signs depend on the chosen winding and current orientations. Each circuit also has its own self-inductance, so a winding’s total induced voltage may contain both self-inductive and mutually induced contributions. (openstax.org)

Mutual inductance enables an electrical transformer to transfer energy between windings through a shared magnetic field. It can also be unwanted: changing currents in one circuit may induce interfering voltages in nearby circuits. (openstax.org)

Stored energy

An ideal linear inductor stores energy in its magnetic field. With the passive sign convention, the instantaneous power entering it is

p=vi=Li\frac{di}{dt}.

Integrating as the current rises from zero to $I$ gives

U=\int_0^I Li\,di=\frac12 LI^2.

An ideal inductor can return this energy to the circuit; it does not dissipate it as a resistor does. The expression assumes constant inductance. (openstax.org)

In vacuum, the corresponding magnetic energy density is

u_B=\frac{B^2}{2\mu_0},

where $B$ is magnetic flux density and $\mu_0$ is vacuum permeability. Integrating this density over space gives the total magnetic-field energy. This field description and the circuit expression $U=\tfrac12 LI^2$ describe the same stored energy at different levels. (openstax.org)

Behavior in electrical circuits

Transient response. In a series circuit containing a constant inductance $L$ , resistance $R$ , and an applied constant voltage $V$ ,

L\frac{di}{dt}+Ri=V.

Starting from zero current,

i(t)=\frac{V}{R}\left(1-e^{-t/\tau}\right), \qquad \tau=\frac{L}{R}.

The time constant $\tau$ characterizes the current’s approach to its steady value. If the source is removed while a closed resistive path remains, the current decays as $i(t)=I_0e^{-t/\tau}$ . (openstax.org)

Steady direct current. Once current is constant, $di/dt=0$ , so an ideal inductor has zero terminal voltage even though its magnetic field may still store energy. A physical winding retains electrical resistance and therefore need not have zero voltage drop. (openstax.org)

Sinusoidal alternating current. For alternating current at angular frequency $\omega=2\pi f$ , an ideal inductor has impedance

Z_L=j\omega L,

where $j^2=-1$ . Its inductive reactance is $X_L=\omega L$ , and current lags voltage by $90^\circ$ . Increasing frequency therefore increases the opposition to sinusoidal current for a fixed ideal inductance. (openstax.org)

Practical limitations and measurement

A physical inductor is not described completely by a single constant $L$ . Important additional properties include winding resistance, magnetic-core losses, distributed capacitance, and dependence on frequency and current. Resistance causes heating, while parasitic capacitance produces self-resonance. Near resonance, the simple ideal-inductor model becomes inadequate; above the first resonance, the component may exhibit predominantly capacitive behavior. (coilcraft.com)

Inductance measurements commonly use impedance analyzers or related instruments. In a suitable series-equivalent model,

L_{\mathrm{s}}=\frac{\operatorname{Im}(Z)}{\omega}.

The reported value depends on test frequency, signal level, bias current, fixture effects, and the equivalent-circuit model. Consequently, a measured inductance is meaningful only together with its measurement conditions, especially near resonance or when magnetic saturation is significant. (coilcraft.com)

Historical development

The experimental foundation of inductance lies in the nineteenth-century discovery of electromagnetic induction. Michael Faraday and Joseph Henry independently discovered induced electromotive force. The henry unit commemorates Henry; the distinction between self-induction and induction between separate circuits provides the basis for the modern concepts of self-inductance and mutual inductance. (openstax.org)

References

2 Self-Inductance and Inductors — University Physics Volume 2openstax.org
1 Mutual Inductance — University Physics Volume 2openstax.org
NIST Guide to the SI, Chapter 4: The Two Classes of SI Units and the SI Prefixesnist.gov
Ch. 14 Summary — University Physics Volume 2openstax.org
9 Inductance — College Physics 2eopenstax.org
2 Simple AC Circuits — University Physics Volume 2openstax.org
3 Energy in a Magnetic Field — University Physics Volume 2openstax.org
4 RL Circuits — University Physics Volume 2openstax.org
Selecting the Best Inductor for Your DC-DC Converter — Coilcraftcoilcraft.com