Kinetic inductance is the manifestation of the inertial mass of mobilecharge carriers in alternating electric fields as an equivalent seriesinductance. Kinetic inductance is observed in high carrier mobility conductors (e.g.superconductors) and at very high frequencies.

A change inelectromotive force (emf) will be opposed by theinertia of the charge carriers since, like all objects with mass, they prefer to be traveling at constant velocity and therefore it takes a finite time to accelerate the particle. This is similar to how a change in emf is opposed by the finite rate of change of magnetic flux in an inductor. The resulting phase lag in voltage is identical for both energy storage mechanisms, making them indistinguishable in a normal circuit.

Kinetic inductance (${\displaystyle L_K}$) arises naturally in theDrude model ofelectrical conduction considering not only the DC conductivity but also the finite relaxation time (collision time)${\displaystyle \tau }$ of the mobile charge carriers when it is not tiny compared to the wave period 1/f. This model defines acomplex conductance at radian frequency ω=2πf given by${\displaystyle \sigma (\omega )={\sigma }_1-i{\sigma }_2}$. The imaginary part, -σ₂, represents the kinetic inductance. The Drude complex conductivity can be expanded into its real and imaginary components:

${\displaystyle \sigma =\frac{n{e}^2\tau }{m(1+i\omega \tau )}=\frac{n{e}^2\tau }{m}\left(\frac{1}{1+{\omega }^2{\tau }^2}-i\frac{\omega \tau }{1+{\omega }^2{\tau }^2}\right)}$

where${\displaystyle m}$ is the mass of the charge carrier (i.e. the effectiveelectron mass in metallicconductors) and${\displaystyle n}$ is the carrier number density. In normal metals the collision time is typically${\displaystyle \approx {10}^{-14}}$ s, so for frequencies < 100 GHz${\displaystyle \omega \tau }$ is very small and can be ignored; then this equation reduces to the DC conductance${\displaystyle {\sigma }_0=n{e}^2\tau /m}$. Kinetic inductance is therefore only significant at optical frequencies, and in superconductors whose${\displaystyle \tau \to \mathrm{\infty }}$.

For a superconducting wire of cross-sectional area${\displaystyle A}$, the kinetic inductance of a segment of length${\displaystyle l}$ can be calculated by equating the total kinetic energy of theCooper pairs in that region with an equivalent inductive energy due to the wire's current${\displaystyle I}$:^[1]

${\displaystyle \frac{1}{2}(2m_e{v}^2)(n_{s}lA)=\frac{1}{2}{L}_K{I}^2}$

where${\displaystyle m_e}$ is the electron mass (${\displaystyle 2m_e}$ is the mass of a Cooper pair),${\displaystyle v}$ is the average Cooper pair velocity,${\displaystyle n_s}$ is the density of Cooper pairs,${\displaystyle l}$ is the length of the wire,${\displaystyle A}$ is the wire cross-sectional area, and${\displaystyle I}$ is the current. Using the fact that the current${\displaystyle I=2ev{n}_{s}A}$, where${\displaystyle e}$ is the electron charge, this yields:^[2]

${\displaystyle L_K=\left(\frac{m_e}{2n_s{e}^2}\right)\left(\frac{l}{A}\right)}$

The same procedure can be used to calculate the kinetic inductance of a normal (i.e. non-superconducting) wire, except with${\displaystyle 2m_e}$ replaced by${\displaystyle m_e}$,${\displaystyle 2e}$ replaced by${\displaystyle e}$, and${\displaystyle n_s}$ replaced by the normal carrier density${\displaystyle n}$. This yields:

${\displaystyle L_K=\left(\frac{m_e}{n{e}^2}\right)\left(\frac{l}{A}\right)}$

The kinetic inductance increases as the carrier density decreases. Physically, this is because a smaller number of carriers must have a proportionally greater velocity than a larger number of carriers in order to produce the same current, whereas their energy increases according to thesquare of velocity. Theresistivity also increases as the carrier density${\displaystyle n}$ decreases, thereby maintaining a constant ratio (and thus phase angle) between the (kinetic) inductive and resistive components of a wire'simpedance for a given frequency. That ratio,${\displaystyle \omega \tau }$, is tiny in normal metals up toterahertz frequencies.

Kinetic inductance is the principle of operation of the highly sensitivephotodetectors known askinetic inductance detectors (KIDs). The change in theCooper pair density brought about by the absorption of a singlephoton in a strip of superconducting material produces a measurable change in its kinetic inductance.

Kinetic inductance is also used in a design parameter for superconductingflux qubits:${\displaystyle \beta }$ is the ratio of the kinetic inductance of theJosephson junctions in the qubit to the geometrical inductance of the flux qubit. A design with a low beta behaves more like a simple inductive loop, while a design with a high beta is dominated by the Josephson junctions and has morehysteretic behavior.^[3]

• Drude model
• Electrical conduction
• Electron mobility
• Superconductivity

• YouTube video on kinetic inductance from MIT

• Electrodynamics
• Superconductivity

• Articles with short description
• Short description matches Wikidata