Capacitance

Common symbols	C
SI unit	farad
Other units	μF, nF, pF
In SI base units	F = A2 s4 kg−1 m−2
Derivations from; other quantities	C = charge / voltage
Dimension	Lua error in Module:Wd at line 621: attempt to index field 'wikibase' (a nil value).

Capacitance is the capability of a material object or device to store electric charge. It is measured by the charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are two closely related notions of capacitance: self capacitance and mutual capacitance.<ref name=Harrington_2003>Harrington, Roger F. (2003). Introduction to Electromagnetic Engineering (1st ed.). Dover Publications. p. 43. ISBN 0-486-43241-6.</ref>^{: 237–238} An object that can be electrically charged exhibits self capacitance, for which the electric potential is measured between the object and ground. Mutual capacitance is measured between two components, and is particularly important in the operation of the capacitor, an elementary linear electronic component designed to add capacitance to an electric circuit.

The capacitance between two conductors is a function only of the geometry; the opposing surface area of the conductors and the distance between them, and the permittivity of any dielectric material between them. For many dielectric materials, the permittivity, and thus the capacitance, is independent of the potential difference between the conductors and the total charge on them.

The SI unit of capacitance is the farad (symbol: F), named after the English physicist Michael Faraday. A 1 farad capacitor, when charged with 1 coulomb of electrical charge, has a potential difference of 1 volt between its plates.<ref>"Definition of 'farad'". Collins.</ref> The reciprocal of capacitance is called elastance.

Self capacitance

In discussing electrical circuits, the term capacitance is usually a shorthand for the mutual capacitance between two adjacent conductors, such as the two plates of a capacitor. However, every isolated conductor also exhibits capacitance, here called self capacitance. It is measured by the amount of electric charge that must be added to an isolated conductor to raise its electric potential by one unit of measurement, e.g., one volt.<ref>William D. Greason (1992). Electrostatic discharge in electronics. Research Studies Press. p. 48. ISBN 978-0-86380-136-5.</ref> The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, with the conductor centered inside this sphere.

Self capacitance of a conductor is defined by the ratio of charge and electric potential: <math display="block">C = \frac{q}{V},</math> where

<math display="inline">q</math> is the charge held,
<math display="inline">V = \frac{1}{4\pi\varepsilon_0}\int \frac{\sigma}{r}\,dS</math> is the electric potential,
<math display="inline">\sigma</math> is the surface charge density,
<math display="inline">dS</math> is an infinitesimal element of area on the surface of the conductor,
<math display="inline">r</math> is the length from <math display="inline">dS</math> to a fixed point M on the conductor,
<math>\varepsilon_0</math> is the vacuum permittivity.

Using this method, the self capacitance of a conducting sphere of radius <math display="inline">R</math> in free space (i.e. far away from any other charge distributions) is:<ref name=NSW>"Lecture notes: Capacitance and Dieletrics" (PDF). University of New South Wales. Archived from the original (PDF) on 26 February 2009.</ref> <math display="block">C = 4 \pi \varepsilon_0 R.</math>

Example values of self capacitance are:

for the top "plate" of a van de Graaff generator, typically a sphere 20 cm in radius: 22.24 pF,
the planet Earth: about 710 µF.<ref>Tipler, Paul; Mosca, Gene (2004). Physics for Scientists and Engineers (5th ed.). Macmillan. p. 752. ISBN 978-0-7167-0810-0.</ref>

The inter-winding capacitance of a coil is sometimes called self capacitance,<ref>Massarini, A.; Kazimierczuk, M. K. (1997). "Self capacitance of inductors". IEEE Transactions on Power Electronics. 12 (4): 671–676. Bibcode:1997ITPE...12..671M. CiteSeerX 10.1.1.205.7356. doi:10.1109/63.602562: example of the use of the term 'self capacitance'.{{cite journal}}: CS1 maint: postscript (link)</ref> but this is a different phenomenon. It is actually mutual capacitance between the individual turns of the coil and is a form of stray or parasitic capacitance. This self capacitance is an important consideration at high frequencies: it changes the impedance of the coil and gives rise to parallel resonance. In many applications this is an undesirable effect and sets an upper frequency limit for the correct operation of the circuit.^{[citation needed]}

Mutual capacitance

A common form is a parallel-plate capacitor, which consists of two conductive plates insulated from each other, usually sandwiching a dielectric material. In a parallel plate capacitor, capacitance is very nearly proportional to the surface area of the conductor plates and inversely proportional to the separation distance between the plates.

If the charges on the plates are <math display="inline">+q</math> and <math display="inline">-q</math>, and <math display="inline">V</math> gives the voltage between the plates, then the capacitance <math display="inline">C</math> is given by <math display="block">C = \frac{q}{V},</math> which gives the voltage/current relationship <math display="block">i(t) = C \frac{dv(t)}{dt} + V\frac{dC}{dt},</math> where <math display="inline">\frac{dv(t)}{dt}</math> is the instantaneous rate of change of voltage, and <math display="inline">\frac{dC}{dt}</math> is the instantaneous rate of change of the capacitance. For most applications, the change in capacitance over time is negligible, so you can reduce to: <math display="block">i(t) = C \frac{dv(t)}{dt},</math>

The energy stored in a capacitor is found by integrating the work <math display="inline">W</math>: <math display="block"> W_\text{charging} = \frac{1}{2}CV^2.</math>

Capacitance matrix

The discussion above is limited to the case of two conducting plates, although of arbitrary size and shape. The definition <math>C = Q/V</math> does not apply when there are more than two charged plates, or when the net charge on the two plates is non-zero. To handle this case, James Clerk Maxwell introduced his coefficients of potential. If three (nearly ideal) conductors are given charges <math>Q_1, Q_2, Q_3</math>, then the voltage at conductor 1 is given by <math display="block">V_1 = P_{11}Q_1 + P_{12} Q_2 + P_{13}Q_3, </math> and similarly for the other voltages. Hermann von Helmholtz and Sir William Thomson showed that the coefficients of potential are symmetric, so that <math>P_{12} = P_{21}</math>, etc. Thus the system can be described by a collection of coefficients known as the elastance matrix or reciprocal capacitance matrix, which is defined as: <math display="block">P_{ij} = \frac{\partial V_{i}}{\partial Q_{j}}.</math>

From this, the mutual capacitance <math>C_{m}</math> between two objects can be defined<ref name=Jackson1999>Jackson, John David (1999). Classical Electrodynamic (3rd ed.). John Wiley & Sons. p. 43. ISBN 978-0-471-30932-1.</ref> by solving for the total charge <math display="inline">Q</math> and using <math>C_{m}=Q/V</math>.

Since no actual device holds perfectly equal and opposite charges on each of the two "plates", it is the mutual capacitance that is reported on capacitors.

The collection of coefficients <math>C_{ij} = \frac{\partial Q_{i}}{\partial V_{j}}</math> is known as the capacitance matrix,<ref name=maxwell>Maxwell, James (1873). "3". A treatise on electricity and magnetism. Vol. 1. Clarendon Press. p. 88ff.</ref><ref>"Capacitance: Charge as a Function of Voltage". Av8n.com. Retrieved 20 September 2010.</ref><ref>Smolić, Ivica; Klajn, Bruno (2021). "Capacitance matrix revisited". Progress in Electromagnetics Research B. 92: 1–18. arXiv:2007.10251. doi:10.2528/PIERB21011501. Retrieved 4 May 2021.</ref> and is the inverse of the elastance matrix.

Capacitors

The capacitance of the majority of capacitors used in electronic circuits is generally several orders of magnitude smaller than the farad. The most common units of capacitance are the microfarad (µF), nanofarad (nF), picofarad (pF), and, in microcircuits, femtofarad (fF). Some applications also use supercapacitors that can be much larger, as much as hundreds of farads, and parasitic capacitive elements can be less than a femtofarad. Historical texts use other, obsolete submultiples of the farad, such as "mf" and "mfd" for microfarad (µF); "mmf", "mmfd", "pfd", "µµF" for picofarad (pF).<ref>"Capacitor MF-MMFD Conversion Chart". Just Radios.</ref><ref>Fundamentals of Electronics. Vol. 1b – Basic Electricity – Alternating Current. Bureau of Naval Personnel. 1965. p. 197.</ref>

The capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. Capacitance is proportional to the area of overlap and inversely proportional to the separation between conducting sheets. The closer the sheets are to each other, the greater the capacitance.

An example is the capacitance of a capacitor constructed of two parallel plates both of area <math display="inline">A</math> separated by a distance <math display="inline">d</math>. If <math display="inline">d</math> is sufficiently small with respect to the smallest chord of <math display="inline">A</math>, there holds, to a high level of accuracy: <math display="block">\ C=\varepsilon\frac{A}{d};</math>

<math display="block">\varepsilon=\varepsilon_0 \varepsilon_r,</math>

where

<math display="inline">C</math> is the capacitance, in farads;
<math display="inline">A</math> is the area of overlap of the two plates, in square meters;
<math display="inline">\varepsilon_0</math> is the electric constant {{{1}}}</math>);}}
<math display="inline">\varepsilon_r</math> is the relative permittivity (also dielectric constant) of the material in between the plates {{{1}}} for air); and
<math display="inline">d</math> is the separation between the plates, in meters.

The equation is a good approximation if d is small compared to the other dimensions of the plates so that the electric field in the capacitor area is uniform, and the so-called fringing field around the periphery provides only a small contribution to the capacitance.

Combining the equation for capacitance with the above equation for the energy stored in a capacitor, for a flat-plate capacitor the energy stored is: <math display="block"> W_\text{stored} = \frac{1}{2} C V^2 = \frac{1}{2} \varepsilon \frac{A}{d} V^2.</math> where <math display="inline">W</math> is the energy, in joules; <math display="inline">C</math> is the capacitance, in farads; and <math display="inline">V</math> is the voltage, in volts.

Stray capacitance

Any two adjacent conductors can function as a capacitor, though the capacitance is small unless the conductors are close together for long distances or over a large area. This (often unwanted) capacitance is called parasitic or stray capacitance. Stray capacitance can allow signals to leak between otherwise isolated circuits (an effect called crosstalk), and it can be a limiting factor for proper functioning of circuits at high frequency.

Stray capacitance between the input and output in amplifier circuits can be troublesome because it can form a path for feedback, which can cause instability and parasitic oscillation in the amplifier. It is often convenient for analytical purposes to replace this capacitance with a combination of one input-to-ground capacitance and one output-to-ground capacitance; the original configuration – including the input-to-output capacitance – is often referred to as a pi-configuration. Miller's theorem can be used to effect this replacement: it states that, if the gain ratio of two nodes is 1/K, then an impedance of Z connecting the two nodes can be replaced with a Z/1 − K impedance between the first node and ground and a KZ/K − 1 impedance between the second node and ground. Since impedance varies inversely with capacitance, the internode capacitance, C, is replaced by a capacitance of KC from input to ground and a capacitance of (K − 1)C/K from output to ground. When the input-to-output gain is very large, the equivalent input-to-ground impedance is very small while the output-to-ground impedance is essentially equal to the original (input-to-output) impedance.

Capacitance of conductors with simple shapes

Calculating the capacitance of a system amounts to solving the Laplace equation <math display="inline">\nabla^2\varphi=0</math> with a constant potential <math display="inline">\varphi</math> on the 2-dimensional surface of the conductors embedded in 3-space. This is simplified by symmetries. There is no solution in terms of elementary functions in more complicated cases.

For plane situations, analytic functions may be used to map different geometries to each other. See also Schwarz–Christoffel mapping.

Capacitance of simple systems
Type	Capacitance	Comment
Parallel-plate capacitor	<math>\ \mathcal{C} = \frac{\ \varepsilon A\ }{d}\ </math>	<math display="inline">\varepsilon</math>: Permittivity
Concentric cylinders	<math>\ \mathcal{C} = \frac{2\pi \varepsilon \ell}{\ \ln \left( R_{2}/R_{1}\right)\ }\ </math>	<math display="inline">\varepsilon</math>: Permittivity
Eccentric cylinders<ref>Dawes, Chester L. (1973). "Capacitance and potential gradients of eccentric cylindrical condensers". Physics. 4 (2): 81–85. doi:10.1063/1.1745162.</ref>	<math>\ \mathcal{C} = \frac{2\pi \varepsilon \ell}{\ \operatorname{arcosh}\left(\frac{R_{1}^2 + R_{2}^2 - d^2}{2 R_{1} R_{2}}\right)\ }\ </math>	<math display="inline">\varepsilon</math>: Permittivity <math display="inline">R_1</math>: Outer radius <math display="inline">R_2</math>: Inner radius <math display="inline">d</math>: Distance between center <math display="inline">\ell</math>: Wire length
Pair of parallel wires<ref name="Jackson 1975 80">Jackson, J. D. (1975). Classical Electrodynamics. Wiley. p. 80.</ref>	<math>\ \mathcal{C} = \frac{\pi \varepsilon \ell}{\ \operatorname{arcosh}\left( \frac{d}{2a}\right)\ } = \frac{\pi \varepsilon \ell}{\ \ln \left( \frac{d}{\ 2a\ } + \sqrt{\frac{d^2}{\ 4a^2\ } -1\ }\right)\ }\ </math>
Wire parallel to wall<ref name="Jackson 1975 80"/>	<math>\ \mathcal{C} = \frac{2\pi \varepsilon \ell}{\ \operatorname{arcosh}\left( \frac{d}{a}\right)\ } = \frac{2\pi \varepsilon \ell}{\ \ln \left( \frac{\ d\ }{a}+\sqrt{\frac{\ d^2\ }{a^2} - 1\ }\right)\ }\ </math>	<math display="inline">a</math>: Wire radius <math display="inline">d</math>: Distance, <math display="inline">d > a</math> <math display="inline">\ell</math>: Wire length
Two parallel coplanar strips<ref>Binns; Lawrenson (1973). Analysis and computation of electric and magnetic field problems. Pergamon Press. ISBN 978-0-08-016638-4.</ref>	<math>\ \mathcal{C} = \varepsilon \ell\ \frac{\ K\left( \sqrt{1-k^2\ } \right)\ }{ K\left( k \right) }\ </math>	<math display="inline">d</math>: Distance <math display="inline">\ell</math>: Length <math display="inline">w_1, w_2</math>: Strip width <math display="inline">\ k_1 = \left( \tfrac{\ 2 w_1\ }{d} + 1 \right)^{-1}\ </math> <math>\ k_2 = \left( \tfrac{\ 2 w_2\ }{d} + 1 \right)^{-1}\ </math><math>\ k = \sqrt{ k_1\ k_2\ }\ </math> <math display="inline">K</math>: Complete elliptic integral of the first kind
Concentric spheres	<math>\ \mathcal{C} = \frac{4\pi \varepsilon}{\ \frac{1}{R_1} - \frac{1}{R_2}\ }\ </math>	<math display="inline">\varepsilon</math>: Permittivity
Two spheres, equal radius<ref name="Maxwell 1873 266 ff">Maxwell, J.;C. (1873). A Treatise on Electricity and Magnetism. Dover. p. 266 ff. ISBN 978-0-486-60637-8.`{{cite book}}`: CS1 maint: multiple names: authors list (link)</ref><ref>Rawlins, A.D. (1985). "Note on the capacitance of two closely separated spheres". IMA Journal of Applied Mathematics. 34 (1): 119–120. doi:10.1093/imamat/34.1.119.</ref>	<math>\begin{align} \ \mathcal{C}\ = &\ {} 2 \pi \varepsilon a\ \sum_{n=1}^{\infty }\frac{\sinh \left( \ln \left( D+\sqrt{D^2-1}\right) \right) }{\sinh \left( n\ln \left( D+\sqrt{ D^2-1}\right) \right) } \\ ={}&{}2\pi \varepsilon a\left[ 1+\frac{1}{2D}+\frac{1}{4D^2}+\frac{1}{8D^3}+\frac{1}{8D^4}+\frac{3}{32D^5}+ \mathcal{O}\left( \frac{1}{D^6} \right) \right] \\ ={}&{} 2\pi \varepsilon a\left[ \ln 2+\gamma -\frac{1}{2}\ln \left( 2D-2\right) + \mathcal{O}\left( 2D-2\right) \right] \\ ={}&{} 2\pi \varepsilon a \,\frac{\sqrt{D^2 - 1}}{\log(q)}\left[\psi_q\left(1+\frac{i\pi}{\log(q)}\right) - i\pi - \psi_q(1)\right] \end{align}\ </math>	<math display="inline">a</math>: Radius <math display="inline">d</math>: Distance, <math display="inline">d > 2a</math> <math display="inline">D = d/2a, D > 1</math> <math display="inline">\gamma</math>: Euler's constant <math>q = D + \sqrt{D^2 - 1}</math> <math>\psi_q(z)=\frac{\partial_z\Gamma_q(z)}{\Gamma_q(z)}</math>: the q-digamma function <math>\Gamma_q(z)</math>: the q-Gamma function<ref>Gasper; Rahman (2004). Basic Hypergeometric Series. Cambridge University Press. p.20-22. ISBN 978-0-521-83357-8.</ref> See also Basic hypergeometric series.
Sphere in front of wall<ref name="Maxwell 1873 266 ff"/>	<math>\ \mathcal{C} = 4\pi \varepsilon a\sum_{n=1}^{\infty }\frac{\sinh \left( \ln \left( D+\sqrt{D^{2}-1}\right) \right) }{\sinh \left( n\ln \left( D+\sqrt{ D^{2}-1}\right) \right) }\ </math>	<math>\ a\ </math>: Radius <math>\ d\ </math>: Distance, <math>d > a</math> <math>D=d/a</math>
Sphere	<math>\ \mathcal{C} = 4 \pi \varepsilon a\ </math>	<math>a</math>: Radius
Circular disc<ref name="Jackson 1975 128">Jackson, J.D. (1975). Classical Electrodynamics. Wiley. p. 128, problem 3.3.</ref>	<math>\ \mathcal{C} = 8 \varepsilon a\ </math>	<math>a</math>: Radius
Thin straight wire, finite length<ref>Maxwell, J. C. (1878). "On the electrical capacity of a long narrow cylinder and of a disk of sensible thickness". Proceedings of the London Mathematical Society. IX: 94–101. doi:10.1112/plms/s1-9.1.94.</ref><ref>Vainshtein, L. A. (1962). "Static boundary problems for a hollow cylinder of finite length. III Approximate formulas". Zhurnal Tekhnicheskoi Fiziki. 32: 1165–1173.</ref><ref>Jackson, J. D. (2000). "Charge density on thin straight wire, revisited". American Journal of Physics. 68 (9): 789–799. Bibcode:2000AmJPh..68..789J. doi:10.1119/1.1302908.</ref>	<math>\ \mathcal{C} = \frac{2\pi \varepsilon \ell}{\Lambda }\left[ 1+\frac{1}{\Lambda }\left( 1-\ln 2\right) +\frac{1}{\Lambda ^{2}}\left( 1+\left( 1-\ln 2\right)^2 - \frac{\pi ^{2}}{12}\right) + \mathcal{O}\left(\frac{1}{\Lambda ^{3}}\right) \right]\ </math>	<math>a</math>: Wire radius <math>\ell</math>: Length <math>\ \Lambda = \ln \left( \ell/a \right)\ </math>

Energy storage

The energy (measured in joules) stored in a capacitor is equal to the work required to push the charges into the capacitor, i.e. to charge it. Consider a capacitor of capacitance C, holding a charge +q on one plate and −q on the other. Moving a small element of charge dq from one plate to the other against the potential difference V = q/C requires the work dW: <math display="block"> \mathrm{d}W = \frac{q}{C}\,\mathrm{d}q,</math> where W is the work measured in joules, q is the charge measured in coulombs and C is the capacitance, measured in farads.

The energy stored in a capacitor is found by integrating this equation. Starting with an uncharged capacitance (q = 0) and moving charge from one plate to the other until the plates have charge +Q and −Q requires the work W: <math display="block"> W_\text{charging} = \int_0^Q \frac{q}{C} \, \mathrm{d}q = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV = \frac{1}{2}CV^2 = W_\text{stored}.</math>

Nanoscale systems

The capacitance of nanoscale dielectric capacitors such as quantum dots may differ from conventional formulations of larger capacitors. In particular, the electrostatic potential difference experienced by electrons in conventional capacitors is spatially well-defined and fixed by the shape and size of metallic electrodes in addition to the statistically large number of electrons present in conventional capacitors. In nanoscale capacitors, however, the electrostatic potentials experienced by electrons are determined by the number and locations of all electrons that contribute to the electronic properties of the device. In such devices, the number of electrons may be very small, so the resulting spatial distribution of equipotential surfaces within the device is exceedingly complex.

Single-electron devices

The capacitance of a connected, or "closed", single-electron device is twice the capacitance of an unconnected, or "open", single-electron device.<ref name= Tsu>Raphael Tsu (2011). Superlattice to Nanoelectronics. Elsevier. pp. 312–315. ISBN 978-0-08-096813-1.</ref> This fact may be traced more fundamentally to the energy stored in the single-electron device whose "direct polarization" interaction energy may be equally divided into the interaction of the electron with the polarized charge on the device itself due to the presence of the electron and the amount of potential energy required to form the polarized charge on the device (the interaction of charges in the device's dielectric material with the potential due to the electron).<ref name= LaFave-DCD>T. LaFave Jr. (2011). "Discrete charge dielectric model of electrostatic energy". J. Electrostatics. 69 (6): 414–418. arXiv:1203.3798. doi:10.1016/j.elstat.2011.06.006. S2CID 94822190.</ref>

Few-electron devices

The derivation of a "quantum capacitance" of a few-electron device involves the thermodynamic chemical potential of an N-particle system given by <math display="block">\mu(N) = U(N) - U(N-1),</math>

whose energy terms may be obtained as solutions of the Schrödinger equation. The definition of capacitance, <math display="block">{1\over C} \equiv {\Delta V\over\Delta Q},</math> with the potential difference <math display="block">\Delta V = {\Delta \mu \,\over e} = {\mu(N + \Delta N) -\mu(N) \over e}</math>

may be applied to the device with the addition or removal of individual electrons, <math display="block">\Delta N = 1</math> and <math display="block">\Delta Q = e.</math>

The "quantum capacitance" of the device is then<ref>G. J. Iafrate; K. Hess; J. B. Krieger; M. Macucci (1995). "Capacitive nature of atomic-sized structures". Phys. Rev. B. 52 (15): 10737–10739. Bibcode:1995PhRvB..5210737I. doi:10.1103/physrevb.52.10737. PMID 9980157.</ref> <math display="block">C_Q(N) = \frac{e^2}{\mu(N+1)-\mu(N)} = \frac{e^2}{E(N)}.</math>

This expression of "quantum capacitance" may be written as <math display="block">C_Q(N) = {e^2\over U(N)},</math> which differs from the conventional expression described in the introduction where <math>W_\text{stored} = U</math>, the stored electrostatic potential energy, <math display="block">C = {Q^2\over 2U},</math> by a factor of 1/2 with <math>Q = Ne</math>.

However, within the framework of purely classical electrostatic interactions, the appearance of the factor of 1/2 is the result of integration in the conventional formulation involving the work done when charging a capacitor, <math display="block"> W_\text{charging} = U = \int_0^Q \frac{q}{C} \, \mathrm{d}q,</math>

which is appropriate since <math>\mathrm{d}q = 0</math> for systems involving either many electrons or metallic electrodes, but in few-electron systems, <math>\mathrm{d}q \to \Delta \,Q= e</math>. The integral generally becomes a summation. One may trivially combine the expressions of capacitance <math display="block">Q=CV</math> and electrostatic interaction energy, <math display="block">U = Q V ,</math> to obtain <math display="block">C = Q{1\over V} = Q {Q \over U} = {Q^2 \over U},</math>

which is similar to the quantum capacitance. A more rigorous derivation is reported in the literature.<ref>T. LaFave Jr; R. Tsu (March–April 2008). "Capacitance: A property of nanoscale materials based on spatial symmetry of discrete electrons" (PDF). Microelectronics Journal. 39 (3–4): 617–623. doi:10.1016/j.mejo.2007.07.105. Archived from the original (PDF) on 22 February 2014. Retrieved 12 February 2014.</ref> In particular, to circumvent the mathematical challenges of spatially complex equipotential surfaces within the device, an average electrostatic potential experienced by each electron is utilized in the derivation.

Apparent mathematical differences may be understood more fundamentally. The potential energy, <math>U(N)</math>, of an isolated device (self-capacitance) is twice that stored in a "connected" device in the lower limit <math>N = 1</math>. As <math>N</math> grows large, <math>U(N)\to U</math>.<ref name=LaFave-DCD/> Thus, the general expression of capacitance is <math display="block">C(N) = {(Ne)^2 \over U(N)}.</math>

In nanoscale devices such as quantum dots, the "capacitor" is often an isolated or partially isolated component within the device. The primary differences between nanoscale capacitors and macroscopic (conventional) capacitors are the number of excess electrons (charge carriers, or electrons, that contribute to the device's electronic behavior) and the shape and size of metallic electrodes. In nanoscale devices, nanowires consisting of metal atoms typically do not exhibit the same conductive properties as their macroscopic, or bulk material, counterparts.

Capacitance in electronic and semiconductor devices

In electronic and semiconductor devices, transient or frequency-dependent current between terminals contains both conduction and displacement components. Conduction current is related to moving charge carriers (electrons, holes, ions, etc.), while displacement current is caused by a time-varying electric field. Carrier transport is affected by electric fields and by a number of physical phenomena - such as carrier drift and diffusion, trapping, injection, contact-related effects, impact ionization, etc. As a result, device admittance is frequency-dependent, and a simple electrostatic formula for capacitance <math>C = q/V,</math> is not applicable. A more general definition of capacitance, encompassing electrostatic formula, is:<ref name=LauxCapacitance>Laux, S.E. (October 1985). "Techniques for small-signal analysis of semiconductor devices". IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. 4 (4): 472–481. doi:10.1109/TCAD.1985.1270145. S2CID 13058472.</ref> <math display="block">C = \frac{\operatorname{Im}(Y(\omega))}{\omega} ,</math> where <math>Y(\omega)</math> is the device admittance, and <math>\omega</math> is the angular frequency.

In general, capacitance is a function of frequency. At high frequencies, capacitance approaches a constant value, equal to "geometric" capacitance, determined by the terminals' geometry and dielectric content in the device. A paper by Steven Laux<ref name=LauxCapacitance /> presents a review of numerical techniques for capacitance calculation. In particular, capacitance can be calculated by a Fourier transform of a transient current in response to a step-like voltage excitation: <math display="block">C(\omega) = \frac{1}{\Delta V} \int_0^\infty [i(t)-i(\infty)] \cos (\omega t) dt.</math>

Negative capacitance in semiconductor devices

Usually, capacitance in semiconductor devices is positive. However, in some devices and under certain conditions (temperature, applied voltages, frequency, etc.), capacitance can become negative. Non-monotonic behavior of the transient current in response to a step-like excitation has been proposed as the mechanism of negative capacitance.<ref name=JonscherNegCap>Jonscher, A.K. (1986). "The physical origin of negative capacitance". J. Chem. Soc. Faraday Trans. II. 82: 75–81. doi:10.1039/F29868200075.</ref> Negative capacitance has been demonstrated and explored in many different types of semiconductor devices.<ref>Ershov, M.; Liu, H.C.; Li, L.; Buchanan, M.; Wasilewski, Z.R.; Jonscher, A.K. (October 1998). "Negative capacitance effect in semiconductor devices". IEEE Trans. Electron Devices. 45 (10): 2196–2206. arXiv:cond-mat/9806145. Bibcode:1998ITED...45.2196E. doi:10.1109/16.725254. S2CID 204925581.</ref>

Measuring capacitance

A capacitance meter is a piece of electronic test equipment used to measure capacitance, mainly of discrete capacitors. For most purposes and in most cases the capacitor must be disconnected from circuit.

Many DVMs (digital volt meters) have a capacitance-measuring function. These usually operate by charging and discharging the capacitor under test with a known current and measuring the rate of rise of the resulting voltage; the slower the rate of rise, the larger the capacitance. DVMs can usually measure capacitance from nanofarads to a few hundred microfarads, but wider ranges are not unusual. It is also possible to measure capacitance by passing a known high-frequency alternating current through the device under test and measuring the resulting voltage across it (does not work for polarised capacitors).

More sophisticated instruments use other techniques such as inserting the capacitor-under-test into a bridge circuit. By varying the values of the other legs in the bridge (so as to bring the bridge into balance), the value of the unknown capacitor is determined. This method of indirect use of measuring capacitance ensures greater precision. Through the use of Kelvin connections and other careful design techniques, these instruments can usually measure capacitors over a range from picofarads to farads.

References

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Self capacitance