View source for Capacitor

{{short description|Passive two-terminal electronic component that stores electrical energy in an electric field}}
{{About|the electronic component|the physical phenomenon|Capacitance|an overview of types|Capacitor types}}
{{Redirect|Capacitive|the term used when referring to touchscreens|Capacitive sensing}}
{{Use dmy dates|date=June 2022|cs1-dates=y}}
{{Infobox electronic component
| name              = Capacitor
| image             = [[File:Capacitors (7189597135).jpg|230px]]
| type              = [[Electronic component]]
| working_principle = [[Capacitance]]
| invented          = [[Ewald Georg von Kleist]] (1745)<br>[[Pieter van Musschenbroek]] (1746)
| symbol            = [[File:Types of capacitor.svg|230px]]
}}

In [[electrical engineering]], a '''capacitor''' is a device that stores [[electrical energy]] by accumulating [[electric charge]]s on two closely spaced surfaces that are insulated from each other. The capacitor was originally known as the '''condenser''',<ref name="duff">{{cite book
|author-last=Duff
|author-first=Wilmer
|title=A Text-Book of Physics
|date=1916
|orig-date=1908
|publisher=P. Blakiston's Son & Co.
|location=Philadelphia
|page=361
|edition=4th |url=https://archive.org/stream/atextbookphysic00carmgoog#page/n378/mode/2up |access-date=1 December 2016
}}</ref> a term still encountered in a few compound names, such as the ''[[condenser microphone]]''. It is a [[passivity (engineering)|passive]] [[electronic component]] with two [[terminal (electronics)|terminals]].

The utility of a capacitor depends on its [[capacitance]]. While some capacitance exists between any two electrical conductors in proximity in a [[electric circuit|circuit]], a capacitor is a component designed specifically to add capacitance to some part of the circuit.

The physical form and construction of practical capacitors vary widely and many [[capacitor types|types of capacitor]] are in common use. Most capacitors contain at least two [[electrical conductor]]s, often in the form of metallic plates or surfaces separated by a [[dielectric]] medium. A conductor may be a foil, thin film, [[sintered]] bead of metal, or an [[electrolyte]]. The nonconducting dielectric acts to increase the capacitor's charge capacity. Materials commonly used as dielectrics include [[glass]], [[ceramic]], [[plastic film]], [[paper]], [[mica]], air, and [[oxide|oxide layers]]. When an [[electric potential]] difference (a [[voltage]]) is applied across the terminals of a capacitor, for example when a capacitor is connected across a battery, an [[electric field]] develops across the dielectric, causing a net positive [[electric charge|charge]] to collect on one plate and net negative charge to collect on the other plate. No current actually flows through a [[perfect dielectric]]. However, there is a flow of charge through the source circuit. If the condition is maintained sufficiently long, the current through the source circuit ceases. If a time-varying voltage is applied across the leads of the capacitor, the source experiences an ongoing current due to the charging and discharging cycles of the capacitor.

Capacitors are widely used as parts of [[electrical circuit]]s in many common electrical devices. Unlike a [[resistor]], an ideal capacitor does not dissipate energy, although real-life capacitors do dissipate a small amount (see [[#Non-ideal behavior|Non-ideal behavior]]).

The earliest forms of capacitors were created in the 1740s, when European experimenters discovered that electric charge could be stored in water-filled glass jars that came to be known as [[Leyden jar]]s. Today, capacitors are widely used in [[electronic circuit]]s for blocking [[direct current]] while allowing [[alternating current]] to pass. In [[analog filter]] networks, they smooth the output of [[power supply|power supplies]]. In [[LC circuit|resonant circuits]] they tune [[radio]]s to particular [[frequency|frequencies]]. In [[electric power transmission]] systems, they stabilize voltage and power flow.<ref>{{cite book |title=Electrical and Electronic Principles and Technology |author-last=Bird |author-first=John |url={{google books |plainurl=y |id=A1tAHm_5sl0C}} |date=2010 |publisher=Routledge |pages=63–76 |isbn=978-0-08089056-2 |access-date=2013-03-17}}</ref> The property of energy storage in capacitors was exploited as dynamic memory in early digital computers,<ref name="floyd">{{cite book |author-last=Floyd |author-first=Thomas |title=Electronic Devices |date=2005 |orig-date=1984 |publisher=[[Pearson Education]] |location=Upper Saddle River, New Jersey, USA |isbn=0-13-127827-4 |page=10 |edition=7th}}</ref> and still is in modern [[DRAM]].

==History==
{{See also|Leyden jar}}
Natural capacitors have existed since prehistoric times. The most common example of natural capacitance are the static charges accumulated between clouds in the sky and the surface of the Earth, where the air between them serves as the dielectric. This results in bolts of [[lightning]] when the breakdown voltage of the air is exceeded.<ref>{{Cite web|url=https://micro.magnet.fsu.edu/electromag/java/lightning/index.html#:~:text=An+Example+Of+A+Natural,ionizing+radiation,+and+each+other.|title=Molecular Expressions: Electricity and Magnetism - Interactive Java Tutorials: Lightning: A Natural Capacitor|website=micro.magnet.fsu.edu}}</ref>

[[File:Leidse flessen Museum Boerhave december 2003 2.jpg|left|upright|thumb|Battery of four [[Leyden jar]]s in [[Museum Boerhaave]], [[Leiden]], the [[Netherlands]]]]
In October 1745, [[Ewald Georg von Kleist]] of [[Pomerania]], Germany, found that [[Electric charge|charge]] could be stored by connecting a high-voltage [[electrostatic generator]] by a wire to a volume of water in a hand-held glass jar.<ref>{{cite web |url=http://www.worldwideschool.org/library/books/sci/history/AHistoryofScienceVolumeII/chap49.html |title=A History of Science Volume II, Part VI: The Leyden Jar Discovered |author-last=Williams |author-first=Henry Smith |access-date=2013-03-17 |archive-date=2007-10-24  |archive-url=https://web.archive.org/web/20071024034729/http://www.worldwideschool.org/library/books/sci/history/AHistoryofScienceVolumeII/chap49.html |url-status=dead }}</ref> Von Kleist's hand and the water acted as conductors and the jar as a [[dielectric]] (although details of the mechanism were incorrectly identified at the time). Von Kleist found that touching the wire resulted in a powerful spark, much more painful than that obtained from an electrostatic machine. The following year, the Dutch physicist [[Pieter van Musschenbroek]] invented a similar capacitor, which was named the [[Leyden jar]], after the [[Leiden University|University of Leiden]] where he worked.<ref>{{cite book |title=The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s |author-last=Keithley |author-first=Joseph F. |url={{google books|plainurl=y|id=uwgNAtqSHuQC|page=23}} |date=1999 |publisher=John Wiley & Sons |page=23 |isbn=978-0780311930 |access-date=2013-03-17}}</ref> He also was impressed by the power of the shock he received, writing, "I would not take a second shock for the kingdom of France."<ref>{{cite book |title=Electricity in Every-day Life |author-last=Houston |author-first=Edwin J. |url={{google books|plainurl=y|id=ko9BAAAAIAAJ|page=71}} |date=1905 |publisher=P. F. Collier & Son |page=71 |access-date=2013-03-17}}</ref>

[[Daniel Gralath]] was the first to combine several jars in parallel to increase the charge storage capacity.<ref name="Benjamin1895">{{cite book |author-last=Benjamin |author-first=Park |title=A History of Electricity: (The Intellectual Rise in Electricity) from Antiquity to the Days of Benjamin Franklin |url=https://books.google.com/books?id=hkMPAAAAMAAJ&pg=PA523 |date=1895 |publisher=[[J. Wiley & Sons]] |pages=522–524}}</ref> [[Benjamin Franklin]] investigated the [[Leyden jar]] and came to the conclusion that the charge was stored on the glass, not in the water as others had assumed. He also adopted the term "battery",<ref>{{cite book |title=Benjamin Franklin: An American Life |author-last=Isaacson |author-first=Walter |author-link=Walter Isaacson |url={{google books|plainurl=y|id=oIW915dDMBwC|page=135}}|date=2003 |publisher=Simon and Schuster |page=136 |isbn=978-0-74326084-8 |access-date=2013-03-17}}</ref><ref>{{cite web |title=Experiments & Observations on Electricity: Letter IV to Peter Collinson |url=http://www.chemteam.info/Chem-History/Franklin-1749/Franklin-1749-all.pdf |date=1749-04-29 |author-first=Benjamin |author-last=Franklin |page=28 |access-date=2009-08-09}}</ref> (denoting the increase of power with a row of similar units as in a [[Artillery battery|battery of cannon]]), subsequently applied to [[Battery (electricity)|clusters of electrochemical cells]].<ref>{{cite web |title=Franklin and Electrostatics&nbsp;– Ben Franklin as my Lab Partner |url=http://www.compadre.org/Repository/document/ServeFile.cfm?ID=3430&DocID=2402&DocFID=3925&Attachment=1 |author-last=Morse |author-first=Robert A. |page=23 |date=September 2004 |format=PDF |work=Wright Center for Science Education |publisher=Tufts University |quote=After Volta's discovery of the electrochemical cell in 1800, the term was then applied to a group of electrochemical cells |access-date=2009-08-10}}</ref> In 1747, Leyden jars were made by coating the inside and outside of jars with metal foil, leaving a space at the mouth to prevent arcing between the foils.<ref>{{cite book|last1=Wolf|first1=A|last2=McKie|first2=D.|title=A history of science, technology and philosophy in the 18th century|date=1962|publisher=George Allen & Unwin|location=London|page=224|edition=2nd|url=https://archive.org/stream/in.ernet.dli.2015.460063/2015.460063.A-History#page/n261/mode/2up}}</ref> The earliest unit of capacitance was the [[Jar (unit)|jar]], equivalent to about 1.11 [[Farad#Definition|nanofarads]].<ref>{{cite web |title=eFunda: Glossary: Units: Electric Capacitance: Jar |url=http://www.efunda.com/glossary/units/units--electric_capacitance--jar.cfm |publisher=eFunda |access-date=2013-03-17}}</ref>

Leyden jars or more powerful devices employing flat glass plates alternating with foil conductors were used exclusively up until about 1900, when the invention of [[Wireless telegraphy|wireless]] ([[radio]]) created a demand for standard capacitors, and the steady move to higher [[frequency|frequencies]] required capacitors with lower [[inductance]]. More compact construction methods began to be used, such as a flexible dielectric sheet (like oiled paper) sandwiched between sheets of metal foil, rolled or folded into a small package.

{{anchor|Condenser}}

[[File:Radio Times - 1923-12-28 - page 39 - Dubilier.png|thumb|Advert from the 28 December 1923 edition of [[The Radio Times]] for Dubilier condensers, for use in wireless receiving sets|393x393px]]

Early capacitors were known as ''condensers'', a term that is still occasionally used today, particularly in high power applications, such as automotive systems. The term ''condensatore'' was used by [[Alessandro Volta]] in 1780 to refer to a device, similar to his [[electrophorus]], he developed to measure electricity, and translated in 1782 as ''condenser'',<ref>{{cite book |last1=Pancaldi |first1=G. |title=Volta: Science and culture in the Age of Enlightenment |date=2003 |publisher=Princeton University Press |location=Princeton |isbn=0691096856 |pages=112–126}}</ref> where the name referred to the device's ability to store a higher density of electric charge than was possible with an isolated conductor.<ref>{{Cite journal |title=Sketch of Alessandro Volta |url={{google books|plainurl=y|id=eCADAAAAMBAJ|page=117}} |journal=The Popular Science Monthly |publisher=Bonnier Corporation |location=New York |pages=118–119 |issn=0161-7370 |date=May 1892}}</ref><ref name="duff"/>  The term became deprecated because of the ambiguous meaning of [[steam condenser]], with ''capacitor'' becoming the recommended term in the UK from 1926,<ref>British Engineering Standards Association, ''British Standard Glossary of Terms in Electrical Engineering'', C. Lockwood & Son, 1926</ref> while the change occurred considerably later in the United States.

Since the beginning of the study of [[electricity]], non-conductive materials like [[glass]], [[porcelain]], [[paper]] and [[mica]] have been used as [[Insulator (electricity)|insulators]]. Decades later, these materials were also well-suited for use as the dielectric for the first capacitors.
Paper capacitors, made by sandwiching a strip of impregnated paper between strips of metal and rolling the result into a cylinder, were commonly used in the late 19th century; their manufacture started in 1876,<ref name="Boggs">{{cite journal |author-last2=Jow |author-first2=T. Richard |author-last3=Boggs |author-first3=Steven |date=January 2010 |title=Historical Introduction to Capacitor Technology |journal=IEEE Electrical Insulation Magazine |volume=26 |issue=1 |pages=20–25 |doi=10.1109/mei.2010.5383924 |author-first1=Janet |author-last1=Ho |s2cid=23077215 |url=https://zenodo.org/record/1232215}}</ref> and they were used from the early 20th century as [[decoupling capacitor]]s in [[telephony]].

Porcelain was used in the first [[ceramic capacitor]]s. In the early years of [[Marconi]]'s wireless transmitting apparatus, porcelain capacitors were used for high voltage and high frequency application in the [[transmitter]]s. On the receiver side, smaller [[Silver mica capacitor|mica capacitors]] were used for [[LC circuit|resonant circuits]]. Mica capacitors were invented in 1909 by William Dubilier. Prior to World War II, mica was the most common dielectric for capacitors in the United States.<ref name="Boggs"/>

Charles Pollak (born [[Karol Pollak]]), the inventor of the first [[electrolytic capacitor]]s, found out that the oxide layer on an aluminum anode remained stable in a neutral or alkaline [[electrolyte]], even when the power was switched off. In 1896 he was granted U.S. Patent No. 672,913 for an "Electric liquid capacitor with aluminum electrodes". Solid electrolyte [[tantalum capacitor]]s were invented by [[Bell Laboratories]] in the early 1950s as a miniaturized and more reliable low-voltage support capacitor to complement their newly invented [[transistor]].

With the development of plastic materials by organic chemists during the [[Second World War]], the capacitor industry began to replace paper with thinner polymer films. One very early development in [[film capacitor]]s was described in British Patent 587,953 in 1944.<ref name="Boggs"/>

Electric double-layer capacitors (now [[supercapacitor]]s) were invented in 1957 when H. Becker developed a "Low voltage electrolytic capacitor with porous carbon electrodes".<ref name="Boggs"/><ref>{{cite patent |country=US |number=2800616 |title=Low voltage electrolytic capacitor |gdate=1957-07-23 |invent1=Becker, H. I.}}</ref><ref>A brief history of supercapacitors AUTUMN 2007 [http://www.cantecsystems.com/ccrdocs/brief-history-of-supercapacitors.pdf Batteries & Energy Storage Technology] {{webarchive|url=https://web.archive.org/web/20140106032318/http://www.cantecsystems.com/ccrdocs/brief-history-of-supercapacitors.pdf |date=2014-01-06}}</ref> He believed that the energy was stored as a charge in the carbon pores used in his capacitor as in the pores of the etched foils of electrolytic capacitors. Because the double layer mechanism was not known by him at the time, he wrote in the patent: "It is not known exactly what is taking place in the component if it is used for energy storage, but it leads to an extremely high capacity."

The MOS capacitor was later widely adopted as a storage capacitor in [[memory chip]]s, and as the basic building block of the [[charge-coupled device]] (CCD) in [[image sensor]] technology.<ref>{{cite book |author-last1=Sze |author-first1=Simon Min |author-link1=Simon Sze |author-last2=Lee |author-first2=Ming-Kwei |chapter=MOS Capacitor and MOSFET |title=Semiconductor Devices: Physics and Technology |date=May 2012 |publisher=[[John Wiley & Sons]] |isbn=978-0-47053794-7 |url=https://www.oreilly.com/library/view/semiconductor-devices-physics/9780470537947/13_chap05.html |access-date=2019-10-06}}</ref> In 1966, Dr. [[Robert Dennard]] invented modern DRAM architecture, combining a single MOS transistor per capacitor.<ref name="ibm100">{{cite web |date=9 August 2017 |title=DRAM |url=https://www.ibm.com/ibm/history/ibm100/us/en/icons/dram/ |access-date=20 September 2019 |website=IBM100 |publisher=[[IBM]]}}</ref><ref>{{cite book |author-last=Sze |author-first=Simon M. |author-link=Simon Sze |url=http://www.fulviofrisone.com/attachments/article/453/Semiconductor.Devices_Physics.Technology_Sze.2ndEd_Wiley_2002.pdf |title=Semiconductor Devices: Physics and Technology |date=2002 |publisher=[[Wiley (publisher)|Wiley]] |isbn=0-471-33372-7 |edition=2nd |page=214 |archive-url=https://web.archive.org/web/20230123184804/http://www.fulviofrisone.com/attachments/article/453/Semiconductor.Devices_Physics.Technology_Sze.2ndEd_Wiley_2002.pdf |archive-date=January 23, 2023}}</ref>

==Theory of operation==
{{Main|Capacitance}}

===Overview===
[[File:Capacitor schematic with dielectric.svg|thumb|left|Charge separation in a parallel-plate capacitor causes an internal electric field. A dielectric (orange) reduces the field and increases the capacitance.]]
[[File:Plattenkondensator hg.jpg|right|thumb|A simple demonstration capacitor made of two parallel metal plates, using an air gap as the dielectric]]
A capacitor consists of two [[Electrical conductor|conductors]] separated by a non-conductive region.{{sfn|Ulaby|1999|p=168}} The non-conductive region can either be a [[vacuum]] or an electrical insulator material known as a [[dielectric]]. Examples of dielectric media are glass, air, paper, plastic, ceramic, and even a [[semiconductor]] [[depletion region]] chemically identical to the conductors. From [[Coulomb's law]] a charge on one conductor will exert a force on the [[charge carrier]]s within the other conductor, attracting opposite polarity charge and repelling like polarity charges, thus an opposite polarity charge will be induced on the surface of the other conductor. The conductors thus hold equal and opposite charges on their facing surfaces,{{sfn|Ulaby|1999|p=157}} and the dielectric develops an electric field.

An ideal capacitor is characterized by a constant [[capacitance]] ''C'', in [[farad]]s in the [[SI]] system of units, defined as the ratio of the positive or negative charge ''Q'' on each conductor to the voltage ''V'' between them:{{sfn|Ulaby|1999|p=168}}
<math display="block">C= \frac{Q}{V}</math>
A capacitance of one [[farad]] (F) means that one [[coulomb]] of charge on each conductor causes a voltage of one [[volt]] across the device.{{sfn|Ulaby|1999|p=69}} Because the conductors (or plates) are close together, the opposite charges on the conductors attract one another due to their electric fields, allowing the capacitor to store more charge for a given voltage than when the conductors are separated, yielding a larger capacitance.

In practical devices, charge build-up sometimes affects the capacitor mechanically, causing its capacitance to vary. In this case, capacitance is defined in terms of incremental changes:
<math display="block">C= \frac{\mathrm{d}Q}{\mathrm{d}V}</math>

===Hydraulic analogy===
[[File:Capacitor-animation.gif|thumb|In the [[hydraulic analogy]], a capacitor is analogous to an elastic diaphragm within a pipe. This animation shows a diaphragm being stretched and un-stretched, which is analogous to a capacitor being charged and discharged.]]
In the [[hydraulic analogy]], voltage is analogous to water pressure and electrical current through a wire is analogous to water flow through a pipe. A capacitor is like an elastic diaphragm within the pipe. Although water cannot pass through the diaphragm, it moves as the diaphragm stretches or un-stretches.
* Capacitance is analogous to diaphragm [[Elasticity (physics)|elasticity]]. In the same way that the ratio of charge differential to voltage would be greater for a larger capacitance value (<math>C=Q / V  </math>),  the ratio of water displacement to pressure would be greater for a diaphragm that flexes more readily.
* In an AC circuit, a capacitor behaves like a diaphragm in a pipe, allowing the charge to move on both sides of the dielectric while no electrons actually pass through. For DC circuits, a capacitor is analogous to a [[hydraulic accumulator]], storing the energy until pressure is released. Similarly, they can be used to smooth the flow of electricity in [[rectifier|rectified]] DC circuits in the same way an accumulator damps surges from a hydraulic pump. 
* Charged capacitors and stretched diaphragms both store [[potential energy]]. The more a capacitor is charged, the higher the voltage across the plates (<math>V = Q / C  </math>). Likewise, the greater the displaced water volume, the greater the elastic potential energy. 
* Electrical current affects the charge differential across a capacitor just as the flow of water affects the volume differential across a diaphragm.
* Just as capacitors experience [[dielectric breakdown]] when subjected to high voltages, diaphragms burst under extreme pressures.
* Just as capacitors block DC while passing AC, diaphragms displace no water unless there is a change in pressure.

=== Circuit equivalence at short-time limit and long-time limit ===
In a circuit, a capacitor can behave differently at different time instants. However, it is usually easy to think about the short-time limit and long-time limit:
* In the long-time limit, after the charging/discharging current has saturated the capacitor, no current would come into (or get out of) either side of the capacitor; Therefore, the long-time equivalence of capacitor is an open circuit.
* In the short-time limit, if the capacitor starts with a certain voltage V, since the voltage drop on the capacitor is known at this instant, we can replace it with an ideal voltage source of voltage V. Specifically, if V=0 (capacitor is uncharged), the short-time equivalence of a capacitor is a short circuit.

===Parallel-plate capacitor===
[[File:Parallel plate capacitor.svg|thumb|Parallel plate capacitor model consists of two conducting plates, each of area ''A'', separated by a gap of thickness ''d'' containing a dielectric.]]
[[File:Big SMD capacitor 2.jpg|thumb|A surface-mount capacitor. The plates, not visible, are layered horizontally between ceramic dielectric layers, and connect alternately to either end-cap, which are visible.]]
The simplest model of a capacitor consists of two thin parallel conductive plates each with an area of <math>A</math> separated by a uniform gap of thickness <math>d</math> filled with a dielectric of [[permittivity]] <math>\varepsilon</math>.  It is assumed the gap <math>d</math> is much smaller than the dimensions of the plates.  This model applies well to many practical capacitors which are constructed of metal sheets separated by a thin layer of insulating dielectric, since manufacturers try to keep the dielectric very uniform in thickness to avoid thin spots which can cause failure of the capacitor.

Since the separation between the plates is uniform over the plate area, the electric field between the plates <math>E</math> is constant, and directed perpendicularly to the plate surface, except for an area near the edges of the plates where the field decreases because the electric field lines "bulge" out of the sides of the capacitor. This "fringing field" area is approximately the same width as the plate separation, <math>d</math>, and assuming <math>d</math> is small compared to the plate dimensions, it is small enough to be ignored.  Therefore, if a charge of <math>+Q</math> is placed on one plate and <math>-Q</math> on the other plate (the situation for unevenly charged plates is discussed below), the charge on each plate will be spread evenly in a [[Surface charge#Conductors|surface charge]] layer of constant [[charge density]] <math>\sigma = \pm Q/A</math> coulombs per square meter, on the inside surface of each plate.  From [[Gauss's law]] the magnitude of the electric field between the plates is <math>E = \sigma / \varepsilon</math>.  The voltage(difference) <math>V</math> between the plates is defined as the [[line integral]] of the electric field over a line (in the z-direction) from one plate to another
<math display="block">V= \int_0^d E(z)\,\mathrm{d}z = Ed = \frac{\sigma}{\varepsilon}d = \frac{Qd}{\varepsilon A}</math>
The capacitance is defined as <math>C = Q/V</math>.  Substituting <math>V</math> above into this equation
{{Equation box 1 |indent =: |cellpadding = 5 |border = 1 |border colour = black |background colour = transparent
|equation = <math>C = \frac{\varepsilon A}{d}</math>
}}
Therefore, in a capacitor the highest capacitance is achieved with a high [[permittivity]] dielectric material, large plate area, and small separation between the plates.

Since the area <math>A</math> of the plates increases with the square of the linear dimensions and the separation <math>d</math> increases linearly, the capacitance scales with the linear dimension of a capacitor (<math>C \varpropto L</math>), or as the cube root of the volume.

A parallel plate capacitor can only store a finite amount of energy before [[dielectric breakdown]] occurs. The capacitor's dielectric material has a [[dielectric strength]] ''U''<sub>d</sub> which sets the [[Capacitor#Breakdown voltage|capacitor's breakdown voltage]] at {{math|1=''V'' = ''V''<sub>bd</sub> = ''U''<sub>d</sub>''d''}}. The maximum energy that the capacitor can store is therefore
<math display="block">E = \frac{1}{2} CV^2 = \frac{1}{2} \frac{\varepsilon A}{d} \left(U_d d\right)^2 = \frac{1}{2} \varepsilon A d U_d^2</math>

The maximum energy is a function of dielectric volume, [[permittivity]], and [[dielectric strength]]. Changing the plate area and the separation between the plates while maintaining the same volume causes no change of the maximum amount of energy that the capacitor can store, so long as the distance between plates remains much smaller than both the length and width of the plates. In addition, these equations assume that the electric field is entirely concentrated in the dielectric between the plates. In reality there are fringing fields outside the dielectric, for example between the sides of the capacitor plates, which increase the effective capacitance of the capacitor. This is sometimes called [[parasitic capacitance]]. For some simple capacitor geometries this additional capacitance term can be calculated analytically.<ref name="Pillai1970">{{Cite journal |author-last=Pillai |author-first=K. P. P. |title=Fringing field of finite parallel-plate capacitors |doi=10.1049/piee.1970.0232 |journal=Proceedings of the Institution of Electrical Engineers |volume=117 |issue=6 |pages=1201–1204 |date=1970}}</ref> It becomes negligibly small when the ratios of plate width to separation and length to separation are large.

For unevenly charged plates:
* If one plate is charged with <math>Q_1</math> while the other is charged with <math>Q_2</math>, and if both plates are separated from other materials in the environment, then the inner surface of the first plate will have <math display="inline">\frac{Q_1 - Q_2}{2}</math>, and the inner surface of the second plated will have <math display="inline">-\frac{Q_1 - Q_2}{2}</math> charge.{{citation needed|date=January 2020}} Therefore, the voltage <math>V</math> between the plates is <math display="inline">V = \frac{Q_1 - Q_2}{2C}</math>. Note that the outer surface of both plates will have <math display="inline">\frac{Q_1 + Q_2}{2}</math>, but those charges do not affect the voltage between the plates.
* If one plate is charged with <math>Q_1</math> while the other is charged with <math>Q_2</math>, and if the second plate is connected to ground, then the inner surface of the first plate will have <math>Q_1</math>, and the inner surface of the second plated will have <math>-Q_1</math>. Therefore, the voltage <math>V</math> between the plates is <math display="inline">V = \frac {Q_1} C</math>. Note that the outer surface of both plates will have zero charge.

===Interleaved capacitor===
[[File:Interleaved Capacitor.jpg|thumb|The interleaved capacitor can be seen as a combination of several parallel connected capacitors.]]
For <math>n</math> number of plates in a capacitor, the total capacitance would be
<math display="block">C = \varepsilon_o\frac{A}{d} (n-1)</math>
where <math>C = \varepsilon_o A / d</math> is the capacitance for a single plate and <math>n</math> is the number of interleaved plates.

As shown to the figure on the right, the interleaved plates can be seen as parallel plates connected to each other. Every pair of adjacent plates acts as a separate capacitor; the number of pairs is always one less than the number of plates, hence the <math>(n-1)</math> multiplier.

===Energy stored in a capacitor===
To increase the charge and voltage on a capacitor, [[Work (thermodynamics)|work]] must be done by an external power source to move charge from the negative to the positive plate against the opposing force of the electric field.<ref name="Purcell">{{cite book |author-last=Purcell |author-first=Edward |title=Electricity and Magnetism, 2nd Ed. |publisher=[[Cambridge University Press]] |date=2011 |pages=110–111 |url=https://books.google.com/books?id=Z3bkNh6h4WEC&pg=PA110 |isbn=978-1-13950355-6}}</ref><ref name="Serway">{{cite book |author-last1=Serway |author-first1=Raymond A. |author-last2=Vuille |author-first2=Chris |title=College Physics, 10th Ed. |publisher=Cengage Learning |date=2014 |pages=582 |url=https://books.google.com/books?id=xETAAgAAQBAJ&pg=PA582 |isbn=978-1-30514282-4}}</ref> If the voltage on the capacitor is <math>V</math>, the work <math>dW</math> required to move a small increment of charge <math>dq</math> from the negative to the positive plate is <math>dW = Vdq</math>. The energy is stored in the increased electric field between the plates.  The total energy <math>W</math> stored in a capacitor (expressed in [[joule]]s) is equal to the total work done in establishing the electric field from an uncharged state.<ref name="Hammond2013">{{cite book |author-last=Hammond |author-first=P. |title=Electromagnetism for Engineers: An Introductory Course |url={{google books |plainurl=y |id=-8QgBQAAQBAJ |page=44}} |date=2013 |publisher=Elsevier Science |isbn=978-1-48314978-3 |pages=44–45}}</ref><ref name="Serway"/><ref name="Purcell"/>
<math display="block">W = \int_0^Q V(q) \, \mathrm{d}q = \int_0^Q \frac{q}{C} \, \mathrm{d}q = \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} VQ = \frac{1}{2} C V^2</math>
where <math>Q</math> is the charge stored in the capacitor, <math>V</math> is the voltage across the capacitor, and <math>C</math> is the capacitance.  This potential energy will remain in the capacitor until the charge is removed. If charge is allowed to move back from the positive to the negative plate, for example by connecting a circuit with resistance between the plates, the charge moving under the influence of the electric field will do work on the external circuit.

If the gap between the capacitor plates <math>d</math> is constant, as in the parallel plate model above, the electric field between the plates will be uniform (neglecting fringing fields) and will have a constant value <math>E = V/d</math>.  In this case the stored energy can be calculated from the electric field strength
<math display="block">W = \frac{1}{2}CV^2 = \frac{1}{2}\frac{\varepsilon A}{d}\left(Ed\right)^2 = \frac{1}{2} \varepsilon AdE^2 = \frac{1}{2} \varepsilon E^2 (\text{volume of electric field})</math>
The last formula above is equal to the energy density per unit volume in the electric field multiplied by the volume of field between the plates, confirming that the energy in the capacitor is stored in its electric field.

===Current–voltage relation===
[[File:Capacitor-positive-voltage-and-current-functions-of-time.svg|thumb|233x233px|Schematic showing polarity of voltage and direction of current for this current–voltage relation.]]
The current ''I''(''t'') through any component in an electric circuit is defined as the rate of flow of a charge ''Q''(''t'') passing through it. Actual charges&nbsp;– electrons&nbsp;– cannot pass through the dielectric of an ''ideal'' capacitor.{{NoteTag|Most real capacitors may have a small dielectric leakage current that passes through the resistive dielectric layer in between the plates.}} Rather, one electron accumulates on the negative plate for each one that leaves the positive plate, resulting in an electron depletion and consequent positive charge on one electrode that is equal and opposite to the accumulated negative charge on the other. Thus the charge on the electrodes is equal to the [[integral]] of the current as well as proportional to the voltage, as discussed above. As with any [[antiderivative]], a [[constant of integration]] is added to represent the initial voltage ''V''(''t''<sub>0</sub>). This is the integral form of the capacitor equation:{{sfn|Dorf|Svoboda|2001|p=263}}
<math display="block">V(t) = \frac{Q(t)}{C} = V(t_0) + \frac{1}{C}\int_{t_0}^t I(\tau) \, \mathrm{d}\tau</math>

Taking the derivative of this and multiplying by ''C'' yields the derivative form:{{sfn|Dorf|Svoboda|2001|p=260}}
<math display="block">I(t) = \frac{\mathrm{d}Q(t)}{\mathrm{d}t} = C\frac{\mathrm{d}V(t)}{\mathrm{d}t}</math>
for {{mvar|C}} independent of time, voltage and electric charge.

The [[duality (electrical circuits)|dual]] of the capacitor is the [[inductor]], which stores energy in a [[magnetic field]] rather than an electric field. Its current-voltage relation is obtained by exchanging current and voltage in the capacitor equations and replacing {{mvar|C}} with the inductance&nbsp;{{mvar|L}}.

===DC circuits===
{{See also|RC circuit}}
[[File:RC switch.svg|A simple resistor–capacitor circuit demonstrates charging of a capacitor.|thumb]]
A series circuit containing only a [[resistor]], a capacitor, a switch and a constant DC source of voltage {{math|''V''<sub>0</sub>}} is known as a ''charging circuit''.<ref name="ChargingCircuit">{{cite web |title=Capacitor charging and discharging |url=http://www.allaboutcircuits.com/vol_6/chpt_3/17.html |work=All About Circuits |access-date=2009-02-19}}</ref> If the capacitor is initially uncharged while the switch is open, and the switch is closed at {{math|1=''t'' = 0}}, it follows from [[Kirchhoff's voltage law]] that
<math display="block">V_0 = v_\text{resistor}(t) + v_\text{capacitor}(t) = i(t) R + \frac{1}{C} \int_{t_0}^t i(\tau) \, \mathrm{d}\tau</math>

Taking the derivative and multiplying by ''C'', gives a [[first-order differential equation]]:
<math display="block">RC \frac{\mathrm{d}i(t)}{\mathrm{d}t} + i(t) = 0</math>

At {{math|1=''t'' = 0}}, the voltage across the capacitor is zero and the voltage across the resistor is ''V''<sub>0</sub>. The initial current is then {{math|1=''I''(0) = ''V''<sub>0</sub>/''R''}}. With this assumption, solving the differential equation yields
<math display="block">\begin{align}
  I(t) &= \frac{V_0}{R} e^{-t / \tau_0} \\
  V(t) &= V_0 \left( 1 - e^{-t / \tau_0}\right) \\
  Q(t) &= C V_0 \left( 1 - e^{-t / \tau_0}\right)
\end{align}</math>
where {{math|1=''τ''<sub>0</sub> = ''RC''}} is the ''[[time constant]]'' of the system. As the capacitor reaches equilibrium with the source voltage, the voltages across the resistor and the current through the entire circuit [[exponential decay|decay exponentially]]. In the case of a ''discharging'' capacitor, the capacitor's initial voltage ({{math|''V''<sub>Ci</sub>}}) replaces {{math|''V''<sub>0</sub>}}. The equations become
<math display="block">\begin{align}
  I(t) &= \frac{V_{Ci}}{R} e^{-t / \tau_0} \\
  V(t) &= V_{Ci} \, e^{-t / \tau_0} \\
  Q(t) &= C \, V_{Ci} \, e^{-t / \tau_0}
\end{align}</math>

===AC circuits===
{{See also|reactance (electronics)|electrical impedance#Deriving the device-specific impedances}}
[[Electrical impedance|Impedance]], the vector sum of [[Electrical reactance|reactance]] and [[Electrical resistance|resistance]], describes the phase difference and the ratio of amplitudes between sinusoidally varying voltage and sinusoidally varying current at a given frequency. [[Fourier analysis]] allows any signal to be constructed from a [[spectrum]] of frequencies, whence the circuit's reaction to the various frequencies may be found. The reactance and impedance of a capacitor are respectively
<math display="block">\begin{align}
  X &= -\frac{1}{\omega C}  = -\frac{1}{2\pi f C} \\
  Z &=  \frac{1}{j\omega C} = -\frac{j}{\omega C} = -\frac{j}{2\pi f C}
\end{align}</math>
where {{math|''j''}} is the [[imaginary unit]] and {{mvar|ω}} is the [[angular frequency]] of the sinusoidal signal. The {{math|−''j''}} phase indicates that the AC voltage {{math|1=''V'' = ''ZI''}} lags the AC current by 90°: the positive current phase corresponds to increasing voltage as the capacitor charges; zero current corresponds to instantaneous constant voltage, etc.

Impedance decreases with increasing capacitance and increasing frequency.<ref>[https://www.researchgate.net/publication/313799141_Universality_of_the_emergent_scaling_in_finite_random_binary_percolation_networks Current percolation through resistors and capacitors] PLoS one 2017</ref> This implies that a higher-frequency signal or a larger capacitor results in a lower voltage amplitude per current amplitude&nbsp;– an AC "short circuit" or [[AC coupling]]. Conversely, for very low frequencies, the reactance is high, so that a capacitor is nearly an open circuit in AC analysis&nbsp;– those frequencies have been "filtered out".

Capacitors are different from resistors and inductors in that the impedance is ''inversely'' proportional to the defining characteristic; i.e., [[capacitance]].

A capacitor connected to an alternating voltage source has a displacement current to flowing through it. In the case that the voltage source is ''V''<sub>0</sub>cos(ωt), the displacement current can be expressed as:
<math display="block"> I = C \frac{\text{d}V}{\text{d}t} = -\omega {C}{V_0}\sin(\omega t)</math>

At {{math|1=sin(''ωt'') = −1}}, the capacitor has a maximum (or peak) current whereby {{math|1=''I''<sub>0</sub> = ''ωCV''<sub>0</sub>}}. The ratio of peak voltage to peak current is due to [[Electrical reactance#Capacitive reactance|capacitive reactance]] (denoted X<sub>C</sub>).
<math display="block"> X_C = \frac{V_0}{I_0} = \frac{V_0}{\omega C V_0} = \frac{1}{\omega C} </math>

X<sub>C</sub> approaches zero as {{mvar|ω}} approaches infinity. If X<sub>C</sub> approaches 0, the capacitor resembles a short wire that strongly passes current at high frequencies. X<sub>C</sub> approaches infinity as ω approaches zero. If X<sub>C</sub> approaches infinity, the capacitor resembles an open circuit that poorly passes low frequencies.

The current of the capacitor may be expressed in the form of cosines to better compare with the voltage of the source:
<math display="block"> I = - I_0 \sin({\omega t}) = I_0 \cos({\omega t} + {90^\circ})</math>

In this situation, the current is out of [[Phase (waves)|phase]] with the voltage by +π/2 radians or +90 degrees, i.e. the current leads the voltage by 90°.

===Laplace circuit analysis (s-domain)===
When using the [[Laplace transform]] in circuit analysis, the impedance of an ideal capacitor with no initial charge is represented in the {{mvar|s}} domain by:
<math display="block">Z(s) = \frac{1}{sC}</math>
where
* {{mvar|C}} is the capacitance, and
* {{mvar|s}} is the complex frequency.

===Circuit analysis===
{{See also|Series and parallel circuits}}
;For capacitors in parallel
[[File:capacitors in parallel.svg|right|thumb|Several capacitors in parallel]]
[[File:Kondensator C1 plus C2.svg|thumb|Illustration of the parallel connection of two capacitors]]
:Capacitors in a parallel configuration each have the same applied voltage. Their capacitances add up. Charge is apportioned among them by size. Using the schematic diagram to visualize parallel plates, it is apparent that each capacitor contributes to the total surface area. <math display="block">C_\mathrm{eq} = \sum_{i=1}^n C_i = C_1 + C_2 + \cdots + C_n</math> {{Clear}}
;For capacitors in series
[[File:capacitors in series.svg|right|thumb|Several capacitors in series]]
[[File:Kondensator C1 C2 Reihe.svg|thumb|Illustration of the serial connection of two capacitors]]
:Connected in series, the schematic diagram reveals that the separation distance, not the plate area, adds up. The capacitors each store instantaneous charge build-up equal to that of every other capacitor in the series. The total voltage difference from end to end is apportioned to each capacitor according to the inverse of its capacitance. The entire series acts as a capacitor ''smaller'' than any of its components. <math display="block">C_\mathrm{eq} = \left(\sum_{i=1}^n\frac{1}{C_i}\right)^{-1} = \left({1\over C_1} + {1\over C_2} + {1\over C_3} + \dots + {1\over C_n}\right)^{-1}</math>
:Capacitors are combined in series to achieve a higher working voltage, for example for smoothing a high voltage power supply. The voltage ratings, which are based on plate separation, add up, if capacitance and leakage currents for each capacitor are identical. In such an application, on occasion, series strings are connected in parallel, forming a matrix. The goal is to maximize the energy storage of the network without overloading any capacitor. For high-energy storage with capacitors in series, some safety considerations must be applied to ensure one capacitor failing and leaking current does not apply too much voltage to the other series capacitors.
:Series connection is also sometimes used to adapt polarized [[electrolytic capacitor]]s for bipolar AC use. {{clear}}
;Voltage distribution in parallel-to-series networks.
:To model the distribution of voltages from a single charged capacitor <math> \left( A \right)</math> connected in parallel to a chain of capacitors in series <math> \left( B_\text{n} \right) </math>: <math display="block">\begin{align}
  \text{(volts)} A_\mathrm{eq} &= A\left(1 - \frac{1}{n + 1}\right) \\
  \text{(volts)} B_\text{1..n} &= \frac{A}{n} \left(1 - \frac{1}{n + 1}\right) \\
  A - B &= 0
\end{align}</math>
:'''Note:''' This is only correct if all capacitance values are equal.
:The power transferred in this arrangement is: <math display="block">P = \frac{1}{R} \cdot \frac{1}{n + 1} A_\text{volts} \left( A_\text{farads} + B_\text{farads} \right)</math>

==Non-ideal behavior==
{{anchor|Non-ideal behavior}}
<!-- NB. Section header used in various redirects to this page -->
In practice, capacitors deviate from the ideal capacitor equation in several aspects. Some of these, such as leakage current and parasitic effects are linear, or can be analyzed as nearly linear, and can be accounted for by adding virtual components to form [[Capacitor#Equivalent circuit|an equivalent circuit]]. The usual methods of [[network analysis (electrical circuits)|network analysis]] can then be applied.<ref>{{Cite web|url=https://www.researchgate.net/publication/313799141|title=(PDF) Universality of the emergent scaling in finite random binary percolation networks}}</ref> In other cases, such as with breakdown voltage, the effect is non-linear and ordinary (normal, e.g., linear) network analysis cannot be used, the effect must be considered separately. Yet another group of artifacts may exist, including temperature dependence, that may be linear but invalidates the assumption in the analysis that capacitance is a constant. Finally, combined parasitic effects such as inherent inductance, resistance, or dielectric losses can exhibit non-uniform behavior at varying frequencies of operation.

==={{anchor|sparking}}Breakdown voltage===
{{Main|Breakdown voltage}}
Above a particular electric field strength, known as the dielectric strength ''E<sub>ds</sub>'', the dielectric in a capacitor becomes conductive. The voltage at which this occurs is called the breakdown voltage of the device, and is given by the product of the dielectric strength and the separation between the conductors,{{sfn|Ulaby|1999|p=170}}
<math display="block">V_{\text{bd}}= E_{\text{ds}} d</math>

The maximum energy that can be stored safely in a capacitor is limited by the breakdown voltage. Exceeding this voltage can result in a short circuit between the plates, which can often cause permanent damage to the dielectric, plates, or both. Due to the scaling of capacitance and breakdown voltage with dielectric thickness, all capacitors made with a particular dielectric have approximately equal maximum [[energy density]], to the extent that the dielectric dominates their volume.<ref>{{cite book |title=Introduction to High Power Pulse Technology |url={{google books |plainurl=y |id=spZ_H4nwIN0C |page=47}} |author-last1=Pai |author-first1=S. T. |author-first2=Qi |author-last2=Zhang |publisher=World Scientific |date=1995 |series=Advanced Series in Electrical and Computer Engineering |volume=10 |isbn=978-9810217143 |access-date=2013-03-17}}</ref>

For air dielectric capacitors the breakdown field strength is of the order 2–5&nbsp;MV/m (or kV/mm); for [[mica]] the breakdown is 100–300&nbsp;MV/m; for oil, 15–25&nbsp;MV/m; it can be much less when other materials are used for the dielectric.<ref>{{cite book |title=Wiley Survey of Instrumentation and Measurement |url={{google books|plainurl=y|id=Wr6l42rEizUC|page=397}} |author-last=Dyer |author-first=Stephen A. |date=2004 |publisher=[[John Wiley & Sons]] |isbn=978-0-47122165-4 |page=397 |access-date=2013-03-17}}</ref> The dielectric is used in very thin layers and so absolute breakdown voltage of capacitors is limited. Typical ratings for capacitors used for general [[electronics]] applications range from a few volts to 1&nbsp;kV. As the voltage increases, the dielectric must be thicker, making high-voltage capacitors larger per capacitance than those rated for lower voltages.

The breakdown voltage is critically affected by factors such as the geometry of the capacitor conductive parts; sharp edges or points increase the electric field strength at that point and can lead to a local breakdown. Once this starts to happen, the breakdown quickly tracks through the dielectric until it reaches the opposite plate, leaving carbon behind and causing a short (or relatively low resistance) circuit. The results can be explosive, as the short in the capacitor draws current from the surrounding circuitry and dissipates the energy.<ref>{{cite book |title=Practical Electronics for Inventors |edition=2nd |url={{google books |plainurl=y |id=C9pL3iL6eSMC |page=100}} |author-last=Scherz |author-first=Paul |date=2006 |publisher=[[McGraw Hill Professional]] |page=100 |isbn=978-0-07177644-8 |access-date=2013-03-17}}</ref> However, in capacitors with particular dielectrics<ref>{{cite journal |author-last1=Inuishi |author-first1=Y. |author-last2=Powers |author-first2=D. A. |title=Electric breakdown and conduction through Mylar films |journal=Journal of Applied Physics |date=1957 |volume=28 |issue=9 |pages=1017–1022 |bibcode=1957JAP....28.1017I |doi=10.1063/1.1722899}}</ref><ref>{{cite journal |author-last1=Reed |author-first1=C. W. |author-last2=Cichanowski |author-first2=S. W. |title=The fundamentals of aging in HV polymer-film capacitors |journal=[[IEEE Transactions on Dielectrics and Electrical Insulation]] |volume=1 |issue=5 |date=1994 |pages=904–922 |doi=10.1109/94.326658}}</ref> and thin metal electrodes, shorts are not formed after breakdown. It happens because a metal melts or evaporates in a breakdown vicinity, isolating it from the rest of the capacitor.<ref>{{cite journal |author-last1=Klein |author-first1=N. |author-last2=Gafni |author-first2=H. |title=The maximum dielectric strength of thin silicon oxide films |journal=IEEE Transactions on Electron Devices |date=1966 |volume=13 |issue=2 |pages=281–289 |bibcode=1966ITED...13..281K |doi=10.1109/T-ED.1966.15681}}</ref><ref>{{cite journal |author-last=Belkin |author-first=A. |display-authors=et al |title=Recovery of alumina nanocapacitors after high voltage breakdown |journal=Scientific Reports |date=2017 |volume=7 |issue=1 |pages=932 |doi=10.1038/s41598-017-01007-9 |pmid=28428625 |pmc=5430567 |bibcode=2017NatSR...7..932B}}</ref>

The usual breakdown route is that the field strength becomes large enough to pull electrons in the dielectric from their atoms thus causing conduction. Other scenarios are possible, such as impurities in the dielectric, and, if the dielectric is of a crystalline nature, imperfections in the crystal structure can result in an [[avalanche breakdown]] as seen in semi-conductor devices. Breakdown voltage is also affected by pressure, humidity and temperature.<ref>{{cite book |title=Electrical Circuit Theory and Technology |url={{google books|plainurl=y|id=Q9zpWdgQeM4C|page=501}} |author-last=Bird |author-first=John |date=2007 |publisher=Routledge |page=501 |isbn=978-0-75068139-1 |access-date=2013-03-17}}</ref>

===Equivalent circuit===
[[File:Real capacitor model adding inductance and series and parallel resistance.svg|thumb|Real capacitor model that adds an inductance and resistance in series and a conductance in parallel to its capacitance. Its total impedance is: <math>\begin{align}
Z_\Sigma &{=} Z_\text{ESL} + R_\text{lead} + (Z_\text{C} \parallel G_\text{dielectric}) \\
&{=} j\omega \cdot \text{ESL} + R_\text{lead} + \frac{1}{j\omega \cdot C + G_\text{dielectric}}.
\end{align}</math>
|261x261px]]
An ideal capacitor only stores and releases electrical energy, without dissipation. In practice, capacitors have imperfections within the capacitor's materials that result in the following parasitic components:<ref>{{Cite journal |last1=Bisquert |first1=J. |last2=Garcia-Belmonte |first2=G. |last3=Fabregat-Santiago |first3=F. |date=2000 |title=The role of instrumentation in the process of modeling real capacitors |url=https://ieeexplore.ieee.org/document/883355 |journal=IEEE Transactions on Education |volume=43 |issue=4 |pages=439–442 |doi=10.1109/13.883355 |bibcode=2000ITEdu..43..439F |issn=1557-9638}}</ref>

* <math>\text{ESL}</math>, the ''[[equivalent series inductance]],'' due to the leads. This is usually significant only at relatively high frequencies.
* Two resistances that add a [[Real number|real-valued]] component to the total impedance, which wastes power:
** <math>R_\text{lead}</math>, a small series resistance in the [[Lead (electronics)|leads]]. Becomes more relevant as frequency increases.
** <math>G_\text{dielectric}</math>, a small [[Electrical Conductance|conductance]] (or reciprocally, a large resistance) in parallel with the capacitance, to account for imperfect dielectric material. This causes a small leakage current across the dielectric (see {{Slink|2=Leakage|nopage=y}}){{sfn|Ulaby|1999|p=169}} that slowly discharges the capacitor over time. This conductance dominates the total resistance at very low frequencies. Its value varies greatly depending on the capacitor material and quality.{{Citation needed|date=September 2023|reason=Probably should find citation giving numbers for different types of capacitors.}}

==== Simplified RLC series model ====
[[File:ESL ESR capacitor model.svg|thumb|Simplified [[RLC circuit#Series circuit|RLC series]] capacitor model. Its total equivalent impedance is: <math>
j\omega \cdot \text{ESL} + \text{ESR} - \frac{j}{\omega \cdot C}.
</math>]]
[[File:RLC Series Circuit Bode Magnitude Plot, relative to natural frequency.svg|thumb|[[Bode magnitude plot]] of voltages in an RLC circuit. Frequency is relative to the natural frequency ''ω''<sub>0</sub>. (Its [[Damping#Damping ratio definition|damping ratio]] ''ζ'' and ''ω''<sub>0</sub> would depend on the particular capacitor.) Lower frequencies are more capacitive. Around ''ω''<sub>0</sub>, the total impedance and voltage drop is primarily resistive. Higher frequencies are more inductive.]]
As frequency increases, the capacitive impedance (a negative reactance) reduces, so the dielectric's conductance becomes less important and the series components become more significant. Thus, a simplified [[RLC circuit#Series circuit|RLC series]] model valid for a large frequency range simply treats the capacitor as being in series with an equivalent series inductance <math>\text{ESL}</math> and a frequency-dependent ''[[equivalent series resistance]]'' <math>\text{ESR}</math>, which varies little with frequency. Unlike the previous model, this model is not valid at [[DC component|DC]] and very low frequencies where <math>G_\text{dielectric}</math> is relevant.

Inductive reactance increases with frequency. Because its sign is positive, it counteracts the capacitance.

At the RLC circuit's [[natural frequency]] <math>\omega_0 {=} \tfrac{1}{\sqrt{\text{ESL} \cdot \text{C}}}</math>, the inductance perfectly cancels the capacitance, so total reactance is zero. Since the total impedance at <math>\omega_0</math> is just the real-value of <math>\text{ESR}</math>, [[Root mean square#Average power|average power]] dissipation reaches its maximum of {{Sfrac|V{{sub|RMS}}{{sup|2}}|ESR}}, where V{{sub|RMS}} is the [[root mean square voltage|root mean square (RMS) voltage]] across the capacitor.

At even higher frequencies, the inductive impedance dominates, so the capacitor undesirably behaves instead like an inductor. High-frequency engineering involves accounting for the inductance of all connections and components.

===== Q factor =====
{{See also|Dielectric loss#Discrete circuit perspective}}
For a simplified model of a capacitor as an ideal capacitor in series with an [[equivalent series resistance]] <math>\text{ESR}</math>, the capacitor's [[Q factor|quality factor]] (or ''Q'') is the ratio of the magnitude of its [[Electrical reactance#Capacitive reactance|capacitive reactance]] <math>X_C</math> to its resistance at a given [[angular frequency|frequency]] <math>\omega</math>:

<math display="block">Q(\omega) = \frac{|X_{C}(\omega)|}{\text{ESR}}=\frac{1}{\omega C \cdot \text{ESR}} \, .</math>

The Q factor is a measure of its efficiency: the higher the Q factor of the capacitor, the closer it approaches the behavior of an ideal capacitor. [[Dissipation factor]] is its reciprocal.

===Ripple current===
[[Ripple (electrical)|Ripple]] current is the AC component of an applied source (often a [[switched-mode power supply]]) whose frequency may be constant or varying. Ripple current causes heat to be generated within the capacitor due to the dielectric losses caused by the changing field strength together with the current flow across the slightly resistive supply lines or the electrolyte in the capacitor. The equivalent series resistance (ESR) is the amount of internal series resistance one would add to a perfect capacitor to model this.

Some [[types of capacitor]]s, primarily [[tantalum]] and [[aluminum]] [[electrolytic capacitor]]s, as well as some [[film capacitor]]s have a specified rating value for maximum ripple current.
* Tantalum electrolytic capacitors with solid manganese dioxide electrolyte are limited by ripple current and generally have the highest ESR ratings in the capacitor family. Exceeding their ripple limits can lead to shorts and burning parts.
* Aluminum electrolytic capacitors, the most common type of electrolytic, suffer a shortening of life expectancy at higher ripple currents. If ripple current exceeds the rated value of the capacitor, it tends to result in explosive failure.
* [[Ceramic capacitor]]s generally have no ripple current limitation{{Citation needed|date=October 2020|reason=everything has a current limitation, even if it is not specified.}} and have some of the lowest ESR ratings.
* [[Film capacitor]]s have very low ESR ratings but exceeding rated ripple current may cause degradation failures.

===Capacitance instability===
The capacitance of certain capacitors decreases as the component ages. In [[ceramic capacitor]]s, this is caused by degradation of the dielectric. The type of dielectric, ambient operating and storage temperatures are the most significant aging factors, while the operating voltage usually has a smaller effect, i.e., usual capacitor design is to minimize voltage coefficient. The aging process may be reversed by heating the component above the [[Curie point]]. Aging is fastest near the beginning of life of the component, and the device stabilizes over time.<ref>{{cite web |url=http://www.johansondielectrics.com/technical-notes/general/ceramic-capacitor-aging-made-simple.html |title=Ceramic Capacitor Aging Made Simple |publisher=Johanson Dielectrics |date=2012-05-21 |access-date=2013-03-17 |archive-url=https://web.archive.org/web/20121226160520/http://www.johansondielectrics.com/technical-notes/general/ceramic-capacitor-aging-made-simple.html |archive-date=2012-12-26 |url-status=dead}}</ref> Electrolytic capacitors age as the [[Electrolytic capacitor#Electrical behavior of electrolytics|electrolyte evaporates]]{{Broken anchor|date=2024-06-17|bot=User:Cewbot/log/20201008/configuration|target_link=Electrolytic capacitor#Electrical behavior of electrolytics|reason= The anchor (Electrical behavior of electrolytics) [[Special:Diff/550131941|has been deleted]].}}. In contrast with ceramic capacitors, this occurs towards the end of life of the component.

Temperature dependence of capacitance is usually expressed in parts per million (ppm) per °C. It can usually be taken as a broadly linear function but can be noticeably non-linear at the temperature extremes. The temperature coefficient may be positive or negative, depending mostly on the dielectric material. Some, designated C0G/NP0, but called '''NPO''', have a somewhat negative coefficient at one temperature, positive at another, and zero in between. Such components may be specified for temperature-critical circuits.<ref>{{cite web|url=https://www.allaboutcircuits.com/technical-articles/x7r-x5r-c0g...-a-concise-guide-to-ceramic-capacitor-types/ |title=Concise Guide to Capacitor Types |publisher=EETech Media LLC |access-date=7 September 2023}}</ref>

Capacitors, especially ceramic capacitors, and older designs such as paper capacitors, can absorb sound waves resulting in a [[microphonic]] effect. Vibration moves the plates, causing the capacitance to vary, in turn inducing AC current. Some dielectrics also generate [[piezoelectricity]]. The resulting interference is especially problematic in audio applications, potentially causing feedback or unintended recording. In the reverse microphonic effect, the varying electric field between the capacitor plates exerts a physical force, moving them as a speaker. This can generate audible sound, but drains energy and stresses the dielectric and the electrolyte, if any.

===Current and voltage reversal===
Current reversal occurs when the current changes direction. Voltage reversal is the change of polarity in a circuit. Reversal is generally described as the percentage of the maximum rated voltage that reverses polarity. In DC circuits, this is usually less than 100%, often in the range of 0 to 90%, whereas AC circuits experience 100% reversal.

In DC circuits and pulsed circuits, current and voltage reversal are affected by the [[Damping ratio|damping]] of the system. Voltage reversal is encountered in [[RLC circuits]] that are [[underdamped]]. The current and voltage reverse direction, forming a [[harmonic oscillator]] between the [[inductance]] and capacitance. The current and voltage tends to oscillate and may reverse direction several times, with each peak being lower than the previous, until the system reaches an equilibrium. This is often referred to as [[ringing (signal)|ringing]]. In comparison, [[critically damped]] or [[overdamped]] systems usually do not experience a voltage reversal. Reversal is also encountered in AC circuits, where the peak current is equal in each direction.

For maximum life, capacitors usually need to be able to handle the maximum amount of reversal that a system may experience. An AC circuit experiences 100% voltage reversal, while underdamped DC circuits experience less than 100%. Reversal creates excess electric fields in the dielectric, causes excess heating of both the dielectric and the conductors, and can dramatically shorten the life expectancy of the capacitor. Reversal ratings often affect the design considerations for the capacitor, from the choice of dielectric materials and voltage ratings to the types of internal connections used.<ref>{{cite web |url=http://www.ga.com/websites/ga/images/products/ep/tech-bulletins/voltage-reversal.pdf |title=The Effect of Reversal on Capacitor Life |publisher=Sorrento Electronics |work=Engineering Bulletin 96-004 |date=November 2003 |access-date=2013-03-17 |url-status=dead |archive-url=https://web.archive.org/web/20140714134127/http://www.ga.com/websites/ga/images/products/ep/tech-bulletins/voltage-reversal.pdf |archive-date=2014-07-14}}</ref>

===Dielectric absorption===
Capacitors made with any type of dielectric material show some level of "[[dielectric absorption]]" or "soakage". On discharging a capacitor and disconnecting it, after a short time it may develop a voltage due to hysteresis in the dielectric. This effect is objectionable in applications such as precision [[sample and hold]] circuits or timing circuits. The level of absorption depends on many factors, from design considerations to charging time, since the absorption is a time-dependent process. However, the primary factor is the type of dielectric material. Capacitors such as tantalum electrolytic or [[polysulfone]] film exhibit relatively high absorption, while [[polystyrene]] or [[Teflon]] allow very small levels of absorption.<ref name="Kaiser2012">{{cite book |author-last=Kaiser |author-first=Cletus J. |title=The Capacitor Handbook |url={{google books|plainurl=y|id=FOatBAAAQBAJ}} |date=2012-12-06 |publisher=Springer Science & Business Media |isbn=978-9-40118090-0}}</ref> In some capacitors where dangerous voltages and energies exist, such as in [[flashtube]]s, [[television set]]s, [[microwave oven]]s and [[defibrillator]]s, the dielectric absorption can recharge the capacitor to hazardous voltages after it has been shorted or discharged. Any capacitor containing over 10 joules of energy is generally considered hazardous, while 50 joules or higher is potentially lethal. A capacitor may regain anywhere from 0.01 to 20% of its original charge over a period of several minutes, allowing a seemingly safe capacitor to become surprisingly dangerous.<ref>''Electronics''. McGraw-Hill 1960 p. 90</ref><ref>[http://donklipstein.com/xesafe.html Xenon Strobe and Flash Safety Hints]. donklipstein.com. 2006-05-29</ref><ref name="Prutchi2012">{{cite book |author-last=Prutchi |author-first=David |title=Exploring Quantum Physics through Hands-on Projects |url={{google books |plainurl=y |id=LLxsnx74KEYC |page=10}} |page=10 |date=2012 |publisher=[[John Wiley & Sons]] |isbn=978-1-11817070-0}}</ref><ref name="DixitYadav2010">{{cite book |author-last1=Dixit |author-first1=J. B. |author-last2=Yadav |author-first2=Amit |title=Electrical Power Quality |url={{google books |plainurl=y |id=lxLWYe5sGMsC |page=63}} |date=2010 |page=63 |publisher=Laxmi Publications, Ltd. |isbn=978-9-38038674-4}}</ref><ref name="ReferenceA">{{cite book |author=Winburn |title=Practical Laser Safety, Second Edition |url={{google books |plainurl=y |id=xx1I5eQybE8C |page=189}} |date=1989 |page=189 |publisher=[[CRC Press]] |isbn=978-0-82478240-5}}</ref>

===Leakage===
No material is a perfect insulator, thus all dielectrics allow some small level of current to leak through, which can be measured with a [[megohmmeter]].<ref>''Robinson's Manual of Radio Telegraphy and Telephony by S.S. Robinson -- US Naval Institute 1924 Pg. 170</ref> Leakage is equivalent to a resistor in parallel with the capacitor. Constant exposure to factors such as heat, mechanical stress, or humidity can cause the dielectric to deteriorate resulting in excessive leakage, a problem often seen in older vacuum tube circuits, particularly where oiled paper and foil capacitors were used. In many vacuum tube circuits, interstage coupling capacitors are used to conduct a varying signal from the plate of one tube to the grid circuit of the next stage. A leaky capacitor can cause the grid circuit voltage to be raised from its normal bias setting, causing excessive current or signal distortion in the downstream tube. In power amplifiers this can cause the plates to glow red, or current limiting resistors to overheat, even fail. Similar considerations apply to component fabricated solid-state (transistor) amplifiers, but, owing to lower heat production and the use of modern polyester dielectric-barriers, this once-common problem has become relatively rare.

===Electrolytic failure from disuse===
[[Aluminum electrolytic capacitor]]s are ''conditioned'' when manufactured by applying a voltage sufficient to initiate the proper internal chemical state. This state is maintained by regular use of the equipment. If a system using electrolytic capacitors is unused for a long period of time it can [[Aluminum electrolytic capacitor#Capacitor behavior after storage or disuse|lose its conditioning]]. Sometimes they fail with a short circuit when next operated.

===Lifespan===
All capacitors have varying lifespans, depending upon their construction, operational conditions, and environmental conditions. Solid-state ceramic capacitors generally have very long lives under normal use, which has little dependency on factors such as vibration or ambient temperature, but factors like humidity, mechanical stress, and [[fatigue (material)|fatigue]] play a primary role in their failure. Failure modes may differ. Some capacitors may experience a gradual loss of capacitance, increased leakage or an increase in [[equivalent series resistance]] (ESR), while others may fail suddenly or even [[catastrophic failure|catastrophically]]. For example, metal-film capacitors are more prone to damage from stress and humidity, but will self-heal when a breakdown in the dielectric occurs. The formation of a [[glow discharge]] at the point of failure prevents arcing by vaporizing the metallic film in that spot, neutralizing any short circuit with minimal loss in capacitance. When enough pinholes accumulate in the film, a total failure occurs in a metal-film capacitor, generally happening suddenly without warning.

Electrolytic capacitors generally have the shortest lifespans. Electrolytic capacitors are affected very little by vibration or humidity, but factors such as ambient and operational temperatures play a large role in their failure, which gradually occur as an increase in ESR (up to 300%) and as much as a 20% decrease in capacitance. The capacitors contain electrolytes which will eventually diffuse through the seals and evaporate. An increase in temperature also increases internal pressure, and increases the reaction rate of the chemicals. Thus, the life of an electrolytic capacitor is generally defined by a modification of the [[Arrhenius equation]], which is used to determine chemical-reaction rates:
<math display="block"> L = B e^{\frac{e_A}{k T_o}}</math>

Manufacturers often use this equation to supply an expected lifespan, in hours, for electrolytic capacitors when used at their designed operating temperature, which is affected by both ambient temperature, ESR, and ripple current. However, these ideal conditions may not exist in every use. The rule of thumb for predicting lifespan under different conditions of use is determined by:
<math display="block"> L_a = L_0 2^{\frac{T_0 - T_a}{10}}</math>

This says that the capacitor's life decreases by half for every 10 degrees Celsius that the temperature is increased,<ref>{{cite web |url=https://www.nrel.gov/docs/fy19osti/71386.pdf |title=A Review of Degradation Behavior and Modeling of Capacitors |author-last1=Gupta |author-first1=Anunay |author-first2=Om Prakash |author-last2=Yadav |author-first3=Douglas |author-last3=DeVoto |author-first4=Joshua |author-last4=Major |publisher=National Renewable Energy Laboratory |date=October 2018 |access-date=2021-07-23 |url-status=live |archive-url=https://web.archive.org/web/20200605220416/https://www.nrel.gov/docs/fy19osti/71386.pdf |archive-date=2020-06-05}}</ref> where:

* <math>L_0</math> is the rated life under rated conditions, e.g. 2000 hours
* <math>T_0</math> is the rated max/min operational temperature
* <math>T_a</math> is the average operational temperature
* <math>L_a</math> is the expected lifespan under given conditions

==Capacitor types==
{{Main|Capacitor types}}
Practical capacitors are available commercially in many different forms. The type of internal dielectric, the structure of the plates and the device packaging all strongly affect the characteristics of the capacitor, and its applications.

Values available range from very low (picofarad range; while arbitrarily low values are in principle possible, stray (parasitic) capacitance in any circuit is the limiting factor) to about 5&nbsp;kF [[Electric double-layer capacitor|supercapacitors]].

Above approximately 1 microfarad electrolytic capacitors are usually used because of their small size and low cost compared with other types, unless their relatively poor stability, life and polarised nature make them unsuitable. Very high capacity supercapacitors use a porous carbon-based electrode material.

===Dielectric materials===
[[File:Condensators.JPG|thumb|An assortment of capacitor types. From left: multilayer ceramic, ceramic disc, multilayer polyester film, tubular ceramic, polystyrene, metalized polyester film, aluminum electrolytic. Major scale divisions are in centimetres.]]

Most capacitors have a dielectric spacer, which increases their capacitance compared to air or a vacuum. In order to maximise the charge that a capacitor can hold, the dielectric material needs to have as high a [[permittivity]] as possible, while also having as high a [[breakdown voltage]] as possible. The dielectric also needs to have as low a loss with frequency as possible.

However, low value capacitors are available with a high vacuum between their plates to allow extremely high voltage operation and low losses. [[Variable capacitor]]s with their plates open to the atmosphere were commonly used in radio tuning circuits. Later designs use polymer foil dielectric between the moving and stationary plates, with no significant air space between the plates.

Several solid dielectrics are available, including [[paper]], [[plastic]], [[glass]], [[mica]] and [[ceramic]].<ref name="Boggs"/>

Paper was used extensively in older capacitors and offers relatively high voltage performance. However, paper absorbs moisture, and has been largely replaced by plastic [[film capacitor]]s.

Most of the plastic films now used offer better stability and ageing performance than such older dielectrics such as oiled paper, which makes them useful in timer circuits, although they may be limited to relatively low [[operating temperature]]s and frequencies, because of the limitations of the plastic film being used. Large plastic film capacitors are used extensively in suppression circuits, motor start circuits, and [[power-factor correction]] circuits.

Ceramic capacitors are generally small, cheap and useful for high frequency applications, although their capacitance varies strongly with voltage and temperature and they age poorly. They can also suffer from the piezoelectric effect. Ceramic capacitors are broadly categorized as [[EIA Class 1 dielectric|class 1 dielectrics]], which have predictable variation of capacitance with temperature or [[EIA Class 2 dielectric|class 2 dielectrics]], which can operate at higher voltage.  Modern multilayer ceramics are usually quite small, but some types have inherently wide value tolerances, microphonic issues, and are usually physically brittle.

Glass and mica capacitors are extremely reliable, stable and tolerant to high temperatures and voltages, but are too expensive for most mainstream applications.

Electrolytic capacitors and [[supercapacitor]]s are used to store small and larger amounts of energy, respectively, ceramic capacitors are often used in [[LC circuit|resonators]], and [[parasitic capacitance]] occurs in circuits wherever the simple conductor-insulator-conductor structure is formed unintentionally by the configuration of the circuit layout.

[[File:Electronic-Component-Elec-Capacitors.jpg|thumb|right|Three aluminum electrolytic capacitors of varying capacity]]
[[File:Capacitor3Dmodel.png|thumb|right|3D model of capacitor]]

[[Electrolytic capacitor]]s use an [[aluminum]] or [[tantalum]] plate with an oxide dielectric layer. The second electrode is a liquid [[electrolyte]], connected to the circuit by another foil plate. Electrolytic capacitors offer very high capacitance but suffer from poor tolerances, high instability, gradual loss of capacitance especially when subjected to heat, and high leakage current. [[Capacitor plague|Poor quality capacitors]] may leak electrolyte, which is harmful to printed circuit boards. The conductivity of the electrolyte drops at low temperatures, which increases equivalent series resistance. While widely used for power-supply conditioning, poor high-frequency characteristics make them unsuitable for many applications. Electrolytic capacitors suffer from self-degradation if unused for a period (around a year), and when full power is applied may short circuit, permanently damaging the capacitor and usually blowing a fuse or causing failure of rectifier diodes. For example, in older equipment, this may cause arcing in rectifier tubes. They can be restored before use by gradually applying the operating voltage, often performed on antique [[vacuum tube]] equipment over a period of thirty minutes by using a variable transformer to supply AC power.  The use of this technique may be less satisfactory for some solid state equipment, which may be damaged by operation below its normal power range, requiring that the power supply first be isolated from the consuming circuits. Such remedies may not be applicable to modern high-frequency power supplies as these produce full output voltage even with reduced input.{{citation needed|reason=how-to|date=November 2016}}

Tantalum capacitors offer better frequency and temperature characteristics than aluminum, but higher [[dielectric absorption]] and leakage.<ref>{{cite web |url=http://www.analog.com/library/analogDialogue/Anniversary/21.html |title=Ask The Applications Engineer – 21 |author-last=Guinta |author-first=Steve |publisher=Analog Devices |access-date=2013-03-17}}</ref>

[[Polymer capacitor]]s (OS-CON, OC-CON, KO, AO) use solid conductive polymer (or polymerized organic semiconductor) as electrolyte and offer longer life and lower [[equivalent series resistance|ESR]] at higher cost than standard electrolytic capacitors.

A [[Feedthrough|feedthrough capacitor]] is a component that, while not serving as its main use, has capacitance and is used to conduct signals through a conductive sheet.

Several other types of capacitor are available for specialist applications. [[Supercapacitor]]s store large amounts of energy. Supercapacitors made from carbon [[aerogel]], carbon nanotubes, or highly porous electrode materials, offer extremely high capacitance (up to 5&nbsp;kF {{As of|2010|lc=on}}) and can be used in some applications instead of [[rechargeable battery|rechargeable batteries]]. [[Alternating current]] capacitors are specifically designed to work on line (mains) voltage AC power circuits. They are commonly used in [[electric motor]] circuits and are often designed to handle large currents, so they tend to be physically large. They are usually ruggedly packaged, often in metal cases that can be easily grounded/earthed. They also are designed with [[direct current]] breakdown voltages of at least five times the maximum AC voltage.

===Voltage-dependent capacitors===
The dielectric constant for a number of very useful dielectrics changes as a function of the applied electrical field, for example [[ferroelectric]] materials, so the capacitance for these devices is more complex. For example, in charging such a capacitor the differential increase in voltage with charge is governed by:
<math display="block"> dQ = C(V) \, dV</math>
where the voltage dependence of capacitance, {{math|''C''(''V'')}}, suggests that the capacitance is a function of the electric field strength, which in a large area parallel plate device is given by {{math|1=''<big>ε</big>'' = ''V''/''d''}}. This field polarizes the dielectric, which polarization, in the case of a ferroelectric, is a nonlinear ''S''-shaped function of the electric field, which, in the case of a large area parallel plate device, translates into a capacitance that is a nonlinear function of the voltage.<ref name="Araujo">{{Cite book |at=Figure 2, p. 504 |title=Science and Technology of Integrated Ferroelectrics: Selected Papers from Eleven Years of the Proceedings of the International Symposium on Integrated Ferroelectrics |url={{google books |plainurl=y |id=QMlOkeJ4qN4C |page=504}} |isbn=90-5699-704-1 |editor-first1=Carlos Paz |editor-last1=de Araujo |editor-first2=Ramamoorthy |editor-last2=Ramesh |editor-first3=George W. |editor-last3=Taylor |publisher=[[CRC Press]] |date=2001}}</ref><ref name="Musikant">{{Cite book |title=What Every Engineer Should Know about Ceramics |author-first=Solomon |author-last=Musikant |isbn=0-8247-8498-7 |date=1991 |publisher=[[CRC Press]] |at=Figure 3.9, p. 43 |url={{google books |plainurl=y |id=Jc8xRdgdH38C |page=44}}}}</ref>

Corresponding to the voltage-dependent capacitance, to charge the capacitor to voltage {{mvar|V}} an integral relation is found:
<math display="block"> Q = \int_0^V C(V) \, dV </math>
which agrees with {{math|1=''Q'' = ''CV''}} only when {{mvar|C}} does not depend on voltage {{mvar|V}}.

By the same token, the energy stored in the capacitor now is given by
<math display="block">dW = Q \, dV =\left[ \int_0^V dV' \, C(V') \right] dV \, . </math>

Integrating:
<math display="block">W = \int_0^V dV \int_0^V dV' \, C(V') = \int_0^V dV' \int_{V'}^V dV \, C(V') = \int_0^V dV' \left(V-V'\right) C(V') \, , </math>
where interchange of the [[Order of integration (calculus)|order of integration]] is used.

The nonlinear capacitance of a microscope probe scanned along a ferroelectric surface is used to study the domain structure of ferroelectric materials.<ref name="Cho">{{Cite book |title=Scanning Nonlinear Dielectric Microscope |author=Yasuo Cho |edition=in ''Polar Oxides''; [[Rainer Waser|R. Waser]], U. Böttger & S. Tiedke, editors |isbn=3-527-40532-1 |publisher=Wiley-VCH |date=2005 |url={{google books |plainurl=y |id=wQ09DhMBJroC |page=304}} |page=Chapter 16 |no-pp=true}}</ref>

Another example of voltage dependent capacitance occurs in [[semiconductor devices]] such as semiconductor [[diode]]s, where the voltage dependence stems not from a change in dielectric constant but in a voltage dependence of the spacing between the charges on the two sides of the capacitor.{{sfn|Sze|Ng|2006|p=217}} This effect is intentionally exploited in diode-like devices known as [[varicap]]s.

===Frequency-dependent capacitors===
If a capacitor is driven with a time-varying voltage that changes rapidly enough, at some frequency the polarization of the dielectric cannot follow the voltage. As an example of the origin of this mechanism, the internal microscopic dipoles contributing to the dielectric constant cannot move instantly, and so as frequency of an applied alternating voltage increases, the dipole response is limited and the dielectric constant diminishes. A changing dielectric constant with frequency is referred to as [[dielectric dispersion]], and is governed by [[dielectric relaxation]] processes, such as [[Debye relaxation]]. Under transient conditions, the displacement field can be expressed as (see [[electric susceptibility]]):
<math display="block">\boldsymbol{D(t)}=\varepsilon_0\int_{-\infty}^t \varepsilon_r (t-t') \boldsymbol E (t')\, dt' , </math>

indicating the lag in response by the time dependence of {{math|''ε<sub>r</sub>''}}, calculated in principle from an underlying microscopic analysis, for example, of the dipole behavior in the dielectric. See, for example, [[linear response function]].<ref name="Giuliani">{{Cite book |title=Quantum Theory of the Electron Liquid |author-first1=Gabriele |author-last1=Giuliani |author-first2=Giovanni |author-last2=Vignale |page=111 |url={{google books |plainurl=y |id=kFkIKRfgUpsC |page=538}} |isbn=0-521-82112-6 |publisher=[[Cambridge University Press]] |date=2005}}</ref><ref name="Rammer">{{Cite book |title=Quantum Field Theory of Non-equilibrium States |author-first=Jørgen |author-last=Rammer |page=158 |url={{google books |plainurl=y |id=A7TbrAm5Wq0C |page=6}} |isbn=978-0-52187499-1 |publisher=[[Cambridge University Press]] |date=2007}}</ref> The integral extends over the entire past history up to the present time. A [[Fourier analysis#(Continuous) Fourier transform|Fourier transform]] in time then results in:
<math display="block">\boldsymbol D(\omega) = \varepsilon_0 \varepsilon_r(\omega) \boldsymbol E (\omega)\, , </math>

where ''ε''<sub>r</sub>(''ω'') is now a [[Complex function#Complex functions|complex function]], with an imaginary part related to absorption of energy from the field by the medium. See [[Permittivity#Complex permittivity|permittivity]]. The capacitance, being proportional to the dielectric constant, also exhibits this frequency behavior. Fourier transforming Gauss's law with this form for displacement field:

<math display="block">\begin{align}
I(\omega) &= j\omega Q(\omega) = j\omega \oint_{\Sigma} \boldsymbol D (\boldsymbol r , \omega)\cdot d \boldsymbol{\Sigma} \\
&=\left[ G(\omega) + j \omega C(\omega)\right] V(\omega) = \frac {V(\omega)}{Z(\omega)} \, , 
\end{align}</math>
where {{math|''j''}} is the [[imaginary unit]], {{math|''V''(''ω'')}} is the voltage component at angular frequency {{mvar|ω}}, {{math|''G''(''ω'')}} is the ''real'' part of the current, called the ''conductance'', and {{math|''C''(''ω'')}} determines the ''imaginary'' part of the current and is the ''capacitance''. {{math|''Z''(''ω'')}} is the complex impedance.

When a parallel-plate capacitor is filled with a dielectric, the measurement of dielectric properties of the medium is based upon the relation:
<math display="block"> \varepsilon_r(\omega) = \varepsilon '_r(\omega) - j \varepsilon ''_r(\omega) = \frac{1}{j\omega Z(\omega) C_0} = \frac{C_{\text{cmplx}}(\omega)}{C_0} \, , </math>
where a single ''prime'' denotes the real part and a double ''prime'' the imaginary part, {{math|''Z''(''ω'')}} is the complex impedance with the dielectric present, {{math|''C''<sub>cmplx</sub>(''ω'')}} is the so-called ''complex'' capacitance with the dielectric present, and {{math|''C''<sub>0</sub>}} is the capacitance without the dielectric.<ref name="Handbook">{{Cite book |title=Springer Handbook of Materials Measurement Methods |author-first1=Horst |author-last1=Czichos |author-first2=Tetsuya |author-last2=Saito |author-first3=Leslie |author-last3=Smith |page=475 |url={{google books |plainurl=y |id=8lANaR-Pqi4C |page=1}} |publisher=Springer |isbn=3-540-20785-6 |date=2006}}</ref><ref name="Coffey">{{Cite book |title=Fractals, diffusion and relaxation in disordered complex systems. Part A |author-first1=William |author-last1=Coffey |author-first2=Yu. P. |author-last2=Kalmykov |publisher=Wiley |date=2006 |isbn=0-470-04607-4 |page=17 |url={{google books |plainurl=y |id=mgtQslaXBc4C |page=18}}}}</ref> (Measurement "without the dielectric" in principle means measurement in [[free space]], an unattainable goal inasmuch as even the [[Vacuum state|quantum vacuum]] is predicted to exhibit nonideal behavior, such as [[dichroism]]. For practical purposes, when measurement errors are taken into account, often a measurement in terrestrial vacuum, or simply a calculation of ''C''<sub>0</sub>, is sufficiently accurate.<ref>{{cite book |title=2005 IEEE Instrumentation and Measurement Technology Conference (IMTC): Ottawa, Ontario Canada, 16–19 May 2005 |url={{google books |plainurl=y |id=85I5nQAACAAJ |page=1350}} |date=2005 |pages=1350–1353 |publisher=IEEE |isbn=978-0-78038879-6 |doi=10.1109/IMTC.2005.1604368 |s2cid=37739028}}</ref>)

Using this measurement method, the dielectric constant may exhibit a [[resonance]] at certain frequencies corresponding to characteristic response frequencies (excitation energies) of contributors to the dielectric constant. These resonances are the basis for a number of experimental techniques for detecting defects. The ''conductance method'' measures absorption as a function of frequency.{{sfn|Schroder|2006|p=347}} Alternatively, the time response of the capacitance can be used directly, as in ''[[deep-level transient spectroscopy]]''.{{sfn|Schroder|2006|p=305}}

Another example of frequency dependent capacitance occurs with [[MOSFET#Metal–oxide–semiconductor structure|MOS capacitors]], where the slow generation of minority carriers means that at high frequencies the capacitance measures only the majority carrier response, while at low frequencies both types of carrier respond.{{sfn|Sze|Ng|2006|p=217}}<ref name="Kasap">{{Cite book |title=Springer Handbook of Electronic and Photonic Materials |author-first1=Safa O. |author-last1=Kasap |author-first2=Peter |author-last2=Capper |date=2006 |publisher=Springer |page=Figure 20.22, p. 425 |url={{google books |plainurl=y |id=rVVW22pnzhoC |page=425}} |no-pp=true}}</ref>

At optical frequencies, in semiconductors the dielectric constant exhibits structure related to the band structure of the solid. Sophisticated modulation spectroscopy measurement methods based upon modulating the crystal structure by pressure or by other stresses and observing the related changes in absorption or reflection of light have advanced our knowledge of these materials.<ref name="Cardona">{{Cite book |url={{google books |plainurl=y |id=W9pdJZoAeyEC |page=244}} |title=Fundamentals of Semiconductors |author1=PY Yu |author-first2=Manuel |author-last2=Cardona |isbn=3-540-25470-6 |date=2001 |edition=3rd |publisher=Springer |at=§6.6 "Modulation Spectroscopy"}}</ref>

===Styles===
[[File:Photo-SMDcapacitors.jpg|thumb|left|Capacitor packages: [[Surface-mount technology|SMD]] ceramic at top left; SMD tantalum electrolytic at bottom left; [[through-hole]] ceramic at top right; through-hole aluminium electrolytic at bottom right. Major scale divisions are cm.]]

The arrangement of plates and dielectric has many variations in different styles depending on the desired ratings of the capacitor. For small values of capacitance (microfarads and less), ceramic disks use metallic coatings, with wire leads bonded to the coating. Larger values can be made by multiple stacks of plates and disks. Larger value capacitors usually use a metal foil or metal film layer deposited on the surface of a dielectric film to make the plates, and a dielectric film of impregnated [[Electrical insulation paper|paper]] or plastic{{spaced ndash}}these are rolled up to save space. To reduce the series resistance and inductance for long plates, the plates and dielectric are staggered so that connection is made at the common edge of the rolled-up plates, not at the ends of the foil or metalized film strips that comprise the plates.

The assembly is encased to prevent moisture entering the dielectric{{spaced ndash}}early radio equipment used a cardboard tube sealed with wax. Modern paper or film dielectric capacitors are dipped in a hard thermoplastic. Large capacitors for high-voltage use may have the roll form compressed to fit into a rectangular metal case, with bolted terminals and bushings for connections. The dielectric in larger capacitors is often impregnated with a liquid to improve its properties.

[[File:Axial electrolytic capacitors.jpg|thumb|right|Several axial-lead [[electrolytic capacitor]]s]]
Capacitors may have their connecting leads arranged in many configurations, for example axially or radially. "Axial" means that the leads are on a common axis, typically the axis of the capacitor's cylindrical body{{spaced ndash}}the leads extend from opposite ends. Radial leads are rarely aligned along radii of the body's circle, so the term is conventional. The leads (until bent) are usually in planes parallel to that of the flat body of the capacitor, and extend in the same direction; they are often parallel as manufactured.

Small, cheap discoidal [[ceramic capacitor]]s have existed from the 1930s onward, and remain in widespread use. After the 1980s, [[surface mount]] packages for capacitors have been widely used. These packages are extremely small and lack connecting leads, allowing them to be soldered directly onto the surface of [[printed circuit boards]]. Surface mount components avoid undesirable high-frequency effects due to the leads and simplify automated assembly, although manual handling is made difficult due to their small size.

Mechanically controlled variable capacitors allow the plate spacing to be adjusted, for example by rotating or sliding a set of movable plates into alignment with a set of stationary plates. Low cost variable capacitors squeeze together alternating layers of aluminum and plastic with a [[trimmer (electronics)|screw]]. Electrical control of capacitance is achievable with [[varactor]]s (or varicaps), which are [[reverse-biased]] [[semiconductor diode]]s whose depletion region width varies with applied voltage. They are used in [[Phase locked loop|phase-locked loops]], amongst other applications.

==Capacitor markings==
===Marking codes for larger parts===
Most capacitors have designations printed on their bodies to indicate their electrical characteristics. Larger capacitors, such as electrolytic types usually display the capacitance as value with explicit unit, for example, ''220 μF''.

For typographical reasons, some manufacturers print ''MF'' on capacitors to indicate microfarads (μF).<ref name="KaplanWhite2003">{{cite book |title=Hands-On Electronics: A Practical Introduction to Analog and Digital Circuits  |author-last1=Kaplan |author-first1=Daniel M. |author-last2=White |author-first2=Christopher G. |date=2003 |publisher=[[Cambridge University Press]] |isbn=978-0-52189351-0 |page=19 |url={{google books |plainurl=y |id=kNbGGpuJyjMC |page=19}}}}</ref>

===Three-/four-character marking code for small capacitors===
Smaller capacitors, such as ceramic types, often use a shorthand-notation consisting of three digits and an optional letter, where the digits (''XYZ'') denote the capacitance in [[picofarad]] (pF), calculated as ''XY''&nbsp;× 10<sup>''Z''</sup>, and the letter indicating the tolerance. Common tolerances are ±5%, ±10%, and ±20%, denotes as J, K, and M, respectively.

A capacitor may also be labeled with its working voltage, temperature, and other relevant characteristics.

Example:
A capacitor labeled or designated as ''473K 330V'' has a capacitance of {{val|47|e=3|u=pF}} = 47&nbsp;nF (±10%) with a maximum working voltage of 330&nbsp;V. The working voltage of a capacitor is nominally the highest voltage that may be applied across it without undue risk of breaking down the dielectric layer.

==={{anchor|2-char-capacitor-marking-code}}Two-character marking code for small capacitors===
For capacitances following the [[E3 series (preferred numbers)|E3]], [[E6 series (preferred numbers)|E6]], [[E12 series (preferred numbers)|E12]] or [[E24 series (preferred numbers)|E24 series]] of preferred values, the former ANSI/EIA-198-D:1991, ANSI/EIA-198-1-E:1998 and ANSI/EIA-198-1-F:2002 as well as the amendment IEC&nbsp;60062:2016/AMD1:2019 to [[IEC&nbsp;60062]] define a ''special two-character marking code for capacitors'' for very small parts which leave no room to print the above-mentioned three-/four-character code onto them. The code consists of an uppercase letter denoting the two significant digits of the value followed by a digit indicating the multiplier. The EIA standard also defines a number of lowercase letters to specify a number of values not found in E24.<ref name="SIST-EN60062-A1_2019">{{cite book |title=SLOVENSKI STANDARD SIST EN 60062:2016/A1:2019 |chapter=Annex B: Special two-character code system for capacitors |date=2019-12-01 |type=preview |pages=3–4 |url=https://cdn.standards.iteh.ai/samples/67713/7c42aed17a454cdaaff49b69445f0ee8/SIST-EN-60062-2016-A1-2019.pdf |access-date=2022-06-17 |url-status=live |archive-url=https://web.archive.org/web/20220617174351/https://cdn.standards.iteh.ai/samples/67713/7c42aed17a454cdaaff49b69445f0ee8/SIST-EN-60062-2016-A1-2019.pdf |archive-date=2022-06-17}}</ref>

{| class="wikitable" style="float:left; margin-right:2em;"
!Code
!Series
!colspan="10"|Digit
|-
!Letter<ref group="nb" name="NB_AmbiguousLetters"/>
!E24
!9
!0
!1
!2
!3
!4
!5
!6
!7
!8
|-
!A
!1.0
|0.10&nbsp;pF||1.0&nbsp;pF||10&nbsp;pF||100&nbsp;pF||1.0&nbsp;nF||10&nbsp;nF||100&nbsp;nF||1.0&nbsp;μF||10&nbsp;μF||100&nbsp;μF
|- 
!B
!1.1
|0.11&nbsp;pF||1.1&nbsp;pF||11&nbsp;pF||110&nbsp;pF||1.1&nbsp;nF||11&nbsp;nF||110&nbsp;nF||1.1&nbsp;μF||11&nbsp;μF||110&nbsp;μF
|-
!C
!1.2
|0.12&nbsp;pF||1.2&nbsp;pF||12&nbsp;pF||120&nbsp;pF||1.2&nbsp;nF||12&nbsp;nF||120&nbsp;nF||1.2&nbsp;μF||12&nbsp;μF||120&nbsp;μF
|-
!D
!1.3
|0.13&nbsp;pF||1.3&nbsp;pF||13&nbsp;pF||130&nbsp;pF||1.3&nbsp;nF||13&nbsp;nF||130&nbsp;nF||1.3&nbsp;μF||13&nbsp;μF||130&nbsp;μF
|-
!E
!1.5
|0.15&nbsp;pF||1.5&nbsp;pF||15&nbsp;pF||150&nbsp;pF||1.5&nbsp;nF||15&nbsp;nF||150&nbsp;nF||1.5&nbsp;μF||15&nbsp;μF||150&nbsp;μF
|-
!F
!1.6
|0.16&nbsp;pF||1.6&nbsp;pF||16&nbsp;pF||160&nbsp;pF||1.6&nbsp;nF||16&nbsp;nF||160&nbsp;nF||1.6&nbsp;μF||16&nbsp;μF||160&nbsp;μF
|-
!G
!1.8
|0.18&nbsp;pF||1.8&nbsp;pF||18&nbsp;pF||180&nbsp;pF||1.8&nbsp;nF||18&nbsp;nF||180&nbsp;nF||1.8&nbsp;μF||18&nbsp;μF||180&nbsp;μF
|-
!H
!2.0
|0.20&nbsp;pF||2.0&nbsp;pF||20&nbsp;pF||200&nbsp;pF||2.0&nbsp;nF||20&nbsp;nF||200&nbsp;nF||2.0&nbsp;μF||20&nbsp;μF||200&nbsp;μF
|-
!J
!2.2
|0.22&nbsp;pF||2.2&nbsp;pF||22&nbsp;pF||220&nbsp;pF||2.2&nbsp;nF||22&nbsp;nF||220&nbsp;nF||2.2&nbsp;μF||22&nbsp;μF||220&nbsp;μF
|-
!K
!2.4
|0.24&nbsp;pF||2.4&nbsp;pF||24&nbsp;pF||240&nbsp;pF||2.4&nbsp;nF||24&nbsp;nF||240&nbsp;nF||2.4&nbsp;μF||24&nbsp;μF||240&nbsp;μF
|-
!L
!2.7
|0.27&nbsp;pF||2.7&nbsp;pF||27&nbsp;pF||270&nbsp;pF||2.7&nbsp;nF||27&nbsp;nF||270&nbsp;nF||2.7&nbsp;μF||27&nbsp;μF||270&nbsp;μF
|-
!M
!3.0
|0.30&nbsp;pF||3.0&nbsp;pF||30&nbsp;pF||300&nbsp;pF||3.0&nbsp;nF||30&nbsp;nF||300&nbsp;nF||3.0&nbsp;μF||30&nbsp;μF||300&nbsp;μF
|-
!N
!3.3
|0.33&nbsp;pF||3.3&nbsp;pF||33&nbsp;pF||330&nbsp;pF||3.3&nbsp;nF||33&nbsp;nF||330&nbsp;nF||3.3&nbsp;μF||33&nbsp;μF||330&nbsp;μF
|-
!P
!3.6
|0.36&nbsp;pF||3.6&nbsp;pF||36&nbsp;pF||360&nbsp;pF||3.6&nbsp;nF||36&nbsp;nF||360&nbsp;nF||3.6&nbsp;μF||36&nbsp;μF||360&nbsp;μF
|-
!Q
!3.9
|0.39&nbsp;pF||3.9&nbsp;pF||39&nbsp;pF||390&nbsp;pF||3.9&nbsp;nF||39&nbsp;nF||390&nbsp;nF||3.9&nbsp;μF||39&nbsp;μF||390&nbsp;μF
|-
!R
!4.3
|0.43&nbsp;pF||4.3&nbsp;pF||43&nbsp;pF||430&nbsp;pF||4.3&nbsp;nF||43&nbsp;nF||430&nbsp;nF||4.3&nbsp;μF||43&nbsp;μF||430&nbsp;μF
|-
!S
!4.7
|0.47&nbsp;pF||4.7&nbsp;pF||47&nbsp;pF||470&nbsp;pF||4.7&nbsp;nF||47&nbsp;nF||470&nbsp;nF||4.7&nbsp;μF||47&nbsp;μF||470&nbsp;μF
|-
!T
!5.1
|0.51&nbsp;pF||5.1&nbsp;pF||51&nbsp;pF||510&nbsp;pF||5.1&nbsp;nF||51&nbsp;nF||510&nbsp;nF||5.1&nbsp;μF||51&nbsp;μF||510&nbsp;μF
|-
!U
!5.6
|0.56&nbsp;pF||5.6&nbsp;pF||56&nbsp;pF||560&nbsp;pF||5.6&nbsp;nF||56&nbsp;nF||560&nbsp;nF||5.6&nbsp;μF||56&nbsp;μF||560&nbsp;μF
|-
!V
!6.2
|0.62&nbsp;pF||6.2&nbsp;pF||62&nbsp;pF||620&nbsp;pF||6.2&nbsp;nF||62&nbsp;nF||620&nbsp;nF||6.2&nbsp;μF||62&nbsp;μF||620&nbsp;μF
|-
!W
!6.8
|0.68&nbsp;pF||6.8&nbsp;pF||68&nbsp;pF||680&nbsp;pF||6.8&nbsp;nF||68&nbsp;nF||680&nbsp;nF||6.8&nbsp;μF||68&nbsp;μF||680&nbsp;μF
|-
!X
!7.5
|0.75&nbsp;pF||7.5&nbsp;pF||75&nbsp;pF||750&nbsp;pF||7.5&nbsp;nF||75&nbsp;nF||750&nbsp;nF||7.5&nbsp;μF||75&nbsp;μF||750&nbsp;μF
|-
!Y
!8.2
|0.82&nbsp;pF||8.2&nbsp;pF||82&nbsp;pF||820&nbsp;pF||8.2&nbsp;nF||82&nbsp;nF||820&nbsp;nF||8.2&nbsp;μF||82&nbsp;μF||820&nbsp;μF
|-
!Z
!9.1
|0.91&nbsp;pF||9.1&nbsp;pF||91&nbsp;pF||910&nbsp;pF||9.1&nbsp;nF||91&nbsp;nF||910&nbsp;nF||9.1&nbsp;μF||91&nbsp;μF||910&nbsp;μF
|}
{| class="wikitable" style="float:left; margin-right:2em;"
!Code
!Series
!colspan="10"|Digit
|-
!Letter
!EIA
!9
!0
!1
!2
!3
!4
!5
!6
!7
!8
|-
!a
!2.5
|0.25&nbsp;pF||2.5&nbsp;pF||25&nbsp;pF||250&nbsp;pF||2.5&nbsp;nF||25&nbsp;nF||250&nbsp;nF||2.5&nbsp;μF||25&nbsp;μF||250&nbsp;μF
|- 
!b?<ref name="Zabkar_2011">{{cite web |title=EIA-198-D capacitance codes |author-first=Franc |author-last=Zabkar |date=2011-09-15 |url=https://www.electronicspoint.com/forums/threads/eia-198-d-capacitance-codes.238921/ |access-date=2022-06-18 |url-status=live |archive-url=https://web.archive.org/web/20220617225054/https://www.electronicspoint.com/forums/threads/eia-198-d-capacitance-codes.238921/ |archive-date=2022-06-17}}</ref>
!3.0?<ref name="Zabkar_2011"/>
|0.30&nbsp;pF||3.0&nbsp;pF||30&nbsp;pF||300&nbsp;pF||3.0&nbsp;nF||30&nbsp;nF||300&nbsp;nF||3.0&nbsp;μF||30&nbsp;μF||300&nbsp;μF
|-
!b?<ref name="SIST-EN60062-A1_2019"/>/c?<ref name="Zabkar_2011"/>
!3.5
|0.35&nbsp;pF||3.5&nbsp;pF||35&nbsp;pF||350&nbsp;pF||3.5&nbsp;nF||35&nbsp;nF||350&nbsp;nF||3.5&nbsp;μF||35&nbsp;μF||350&nbsp;μF
|-
!d
!4.0
|0.40&nbsp;pF||4.0&nbsp;pF||40&nbsp;pF||400&nbsp;pF||4.0&nbsp;nF||40&nbsp;nF||400&nbsp;nF||4.0&nbsp;μF||40&nbsp;μF||400&nbsp;μF
|-
!e
!4.5
|0.45&nbsp;pF||4.5&nbsp;pF||45&nbsp;pF||450&nbsp;pF||4.5&nbsp;nF||45&nbsp;nF||450&nbsp;nF||4.5&nbsp;μF||45&nbsp;μF||450&nbsp;μF
|-
!f
!5.0
|0.50&nbsp;pF||5.0&nbsp;pF||50&nbsp;pF||500&nbsp;pF||5.0&nbsp;nF||50&nbsp;nF||500&nbsp;nF||5.0&nbsp;μF||50&nbsp;μF||500&nbsp;μF
|-
!m
!6.0
|0.60&nbsp;pF||6.0&nbsp;pF||60&nbsp;pF||600&nbsp;pF||6.0&nbsp;nF||60&nbsp;nF||600&nbsp;nF||6.0&nbsp;μF||60&nbsp;μF||600&nbsp;μF
|-
!n
!7.0
|0.70&nbsp;pF||7.0&nbsp;pF||70&nbsp;pF||700&nbsp;pF||7.0&nbsp;nF||70&nbsp;nF||700&nbsp;nF||7.0&nbsp;μF||70&nbsp;μF||700&nbsp;μF
|-
!t
!8.0
|0.80&nbsp;pF||8.0&nbsp;pF||80&nbsp;pF||800&nbsp;pF||8.0&nbsp;nF||80&nbsp;nF||800&nbsp;nF||8.0&nbsp;μF||80&nbsp;μF||800&nbsp;μF
|-
!g
!9.0
|0.90&nbsp;pF||9.0&nbsp;pF||90&nbsp;pF||900&nbsp;pF||9.0&nbsp;nF||90&nbsp;nF||900&nbsp;nF||9.0&nbsp;μF||90&nbsp;μF||900&nbsp;μF
|}
{{clear}}

===RKM code===
The [[RKM code]] following [[IEC 60062]] and [[BS 1852]] is a notation to state a capacitor's value in a circuit diagram. It avoids using a [[decimal separator]] and replaces the decimal separator with the SI prefix symbol for the particular value (and the letter {{mono|F}} for weight 1). The code is also used for part markings. Example: {{mono|4n7}} for 4.7&nbsp;nF or {{mono|2F2}} for 2.2&nbsp;F.

===Historical===
{{See also|Farad#Informal and deprecated terminology}}
In texts prior to the 1960s and on some capacitor packages until more recently,<ref name="Boggs"/> obsolete capacitance units were utilized in electronic books,<ref name="FoE-1965">{{Cite web |url=http://archive.org/details/FundamentalsOfElectronics93400A1b |title=Fundamentals of Electronics, Volume 1b: Basic Electricity, Alternating Current, NAVPERS 93400A-1b |date=1965-04-12 |via=Internet Archive}}</ref> magazines, and electronics catalogs.<ref>{{cite web |title=1930 Catalog – Capacitors (Condensers) |url=http://www.alliedcatalogs.com/html/catalogs_additional/1930/hr139.html |website=[[Allied Electronics|Allied Radio]] |page=139 |access-date=2017-07-11 |archive-url=https://web.archive.org/web/20170711160331/http://www.alliedcatalogs.com/html/catalogs_additional/1930/hr139.html |archive-date=2017-07-11}}</ref> The old units "mfd" and "mf" meant ''microfarad'' (μF); and the old units "mmfd", "mmf", "uuf", "μμf", "pfd" meant ''picofarad'' (pF); but they are rarely used any more.<ref>{{Cite web |url=https://www.justradios.com/MFMMFD.html |title=Capacitor MF - MMFD Conversion Chart |website=www.justradios.com}}</ref>  Also, "Micromicrofarad" or "micro-microfarad" are obsolete units that are found in some older texts that is equivalent to ''picofarad'' (pF).<ref name="FoE-1965"/>

Summary of obsolete capacitance units: (upper/lower case variations are not shown)
* μF (microfarad) = mf, mfd
* pF (picofarad) = mmf, mmfd, pfd, μμF

==Applications==
{{Main|Applications of capacitors}}
[[File:24 Million Watt high speed flash through welding lens.jpg|thumb|A capacitor discharging its stored energy through a [[flashtube]]. The mylar-film capacitor has very low inductance and low resistance, producing a 3.5 microsecond pulse with 24 million watts of power, to operate a [[dye laser]].]]

===Energy storage===
A capacitor can store electric energy when disconnected from its charging circuit, so it can be used like a temporary [[Battery (electricity)|battery]], or like other types of [[rechargeable energy storage system]].<ref>{{cite book |author-last=Miller |author-first=Charles |url={{google books |plainurl=y |id=RSsJAAAAQBAJ |page=445}} |title= Illustrated Guide to the National Electrical Code |page=445 |publisher=Cengage Learning |date=2011}}</ref> Capacitors are commonly used in electronic devices to maintain power supply while batteries are being changed. (This prevents loss of information in volatile memory.)

A capacitor can facilitate conversion of kinetic energy of charged particles into electric energy and store it.<ref>{{cite journal |author-last1=Shinn |author-first1=Eric |display-authors=et al |title=Nuclear energy conversion with stacks of graphene nanocapacitors |journal=Complexity |volume=18 |issue=3 |pages=24–27 |date=2012 |doi=10.1002/cplx.21427 |bibcode=2013Cmplx..18c..24S}}</ref>

There are tradeoffs between capacitors and batteries as storage devices. Without external resistors or inductors, capacitors can generally release their stored energy in a very short time compared to batteries. Conversely, batteries can hold a far greater charge per their size. Conventional capacitors provide less than 360 [[joule]]s per kilogram of [[specific energy]], whereas a conventional [[alkaline battery]] has a density of 590 kJ/kg. There is
an intermediate solution: [[supercapacitor]]s, which can accept and deliver charge much faster than batteries, and tolerate many more charge and discharge cycles than rechargeable batteries. They are, however, 10 times larger than conventional batteries for a given charge. On the other hand, it has been shown that the amount of charge stored in the dielectric
layer of the thin film capacitor can be equal to, or can even exceed, the amount of charge stored on its plates.<ref>{{cite journal |author-last1=Bezryadin |author-first1=A. |author-last2=Belkin |author-first2=A. |display-authors=et al |title=Large energy storage efficiency of the dielectric layer of graphene nanocapacitors |journal=Nanotechnology |date=2017 |volume=28 |issue=49 |pages=495401 |doi=10.1088/1361-6528/aa935c |pmid=29027908 |bibcode=2017Nanot..28W5401B |arxiv=2011.11867 |s2cid=44693636}}</ref>

In [[car audio]] systems, large capacitors store energy for the [[amplifier]] to use on demand. Also, for a [[flash tube]], a capacitor is used to hold the [[high voltage]].

===Digital memory===
In the 1930s, [[John Vincent Atanasoff|John Atanasoff]] applied the principle of energy storage in capacitors to construct dynamic digital memories for the first binary computers that used electron tubes for logic.<ref name="Floyd2017">{{cite book |author-last=Floyd |author-first=Thomas L. |title=Electronic Devices |url={{google books |plainurl=y |id=vg41vgAACAAJ |page=10}} |date=2017 |publisher=Pearson |isbn=978-0-13441444-7 |page=10}}</ref>

===Pulsed power and weapons===
[[Pulsed power]] is used in many applications to increase the power intensity (watts) of a volume of energy (joules) by releasing that volume within a very short time. Pulses in the nanosecond range and powers in the gigawatts are achievable. Short pulses often require specially constructed, low-inductance, high-voltage capacitors that are often used in large groups (''capacitor banks'') to supply huge pulses of current for many pulsed power applications. These include [[electromagnetic forming]], [[Marx generator]]s, pulsed [[laser]]s (especially [[TEA laser]]s), [[pulse forming network]]s, [[radar]], [[Fusion power|fusion]] research, and [[particle accelerator]]s.<ref>''Pulsed Power'' by Gennady A. Mesyats -- Springer 2005 Page 1--5</ref>

Large capacitor banks (reservoir) are used as energy sources for the [[exploding-bridgewire detonator]]s or [[slapper detonator]]s in [[nuclear weapon]]s and other specialty weapons. Experimental work is under way using banks of capacitors as power sources for [[electromagnetic armour]] and electromagnetic [[railgun]]s and [[coilgun]]s.

===Power conditioning===
[[File:Capacitor.jpg|thumb|A 10,000&nbsp;[[microfarad]] capacitor in an amplifier power supply]]

[[Reservoir capacitor]]s are used in [[Power supply|power supplies]] where they smooth the output of a full or half wave [[rectifier]]. They can also be used in [[charge pump]] circuits as the energy storage element in the generation of higher voltages than the input voltage.

Capacitors are connected in parallel with the power circuits of most electronic devices and larger systems (such as factories) to shunt away and conceal current fluctuations from the primary power source to provide a "clean" power supply for signal or control circuits. Audio equipment, for example, uses several capacitors in this way, to shunt away power line hum before it gets into the signal circuitry. The capacitors act as a local reserve for the DC power source, and [[bypass capacitor|bypass]] AC currents from the power supply. This is used in car audio applications, when a stiffening capacitor compensates for the inductance and resistance of the leads to the [[lead–acid battery|lead–acid]] [[car battery]].

====Power-factor correction====
[[File:Condensor bank 150kV - 75MVAR.jpg|thumb|left|upright|A high-voltage capacitor bank used for [[power-factor correction]] on a power transmission system]]
In electric power distribution, capacitors are used for [[power-factor correction]]. Such capacitors often come as three capacitors connected as a [[three-phase electric power|three phase]] [[Electrical load|load]]. Usually, the values of these capacitors are not given in farads but rather as a [[reactive power]] in volt-amperes reactive (var). The purpose is to counteract inductive loading from devices like [[Induction motor|electric motors]] and [[transmission line]]s to make the load appear to be mostly resistive. Individual motor or lamp loads may have capacitors for power-factor correction, or larger sets of capacitors (usually with automatic switching devices) may be installed at a load center within a building or in a large utility [[electrical substation|substation]].

===Suppression and coupling===

====Signal coupling====
{{Main|capacitive coupling}}
[[File:Polyester film capacitor.jpg|thumb|right|Polyester [[film capacitor]]s are frequently used as coupling capacitors.]]
Because capacitors pass AC but block DC [[Signal (information theory)|signals]] (when charged up to the applied DC voltage), they are often used to separate the AC and DC components of a signal. This method is known as ''AC coupling'' or "capacitive coupling". Here, a large value of capacitance, whose value need not be accurately controlled, but whose [[Reactance (electronics)|reactance]] is small at the signal frequency, is employed.

====Decoupling====
{{Main|decoupling capacitor}}
A [[decoupling capacitor]] is a capacitor used to protect one part of a circuit from the effect of another, for instance to suppress noise or transients. Noise caused by other circuit elements is shunted through the capacitor, reducing the effect they have on the rest of the circuit. It is most commonly used between the power supply and ground.
An alternative name is ''[[bypass capacitor]]'' as it is used to bypass the power supply or other high impedance component of a circuit.

Decoupling capacitors need not always be discrete components. Capacitors used in these applications may be built into a [[printed circuit board]], between the various layers. These are often referred to as embedded capacitors.<ref>{{cite journal |author-last1=Alam |author-first1=Mohammed |author-first2=Michael H. |author-last2=Azarian |author-first3=Michael |author-last3=Osterman |author-first4=Michael |author-last4=Pecht |title=Effectiveness of embedded capacitors in reducing the number of surface mount capacitors for decoupling applications |journal=Circuit World |date=2010 |volume=36 |issue=1 |page=22 |doi=10.1108/03056121011015068}}</ref> The layers in the board contributing to the capacitive properties also function as power and ground planes, and have a dielectric in between them, enabling them to operate as a parallel plate capacitor.

====High-pass and low-pass filters====
{{Further|High-pass filter|Low-pass filter}}

====Noise suppression, spikes, and snubbers====
{{Further|High-pass filter|Low-pass filter}}
When an inductive circuit is opened, the current through the inductance collapses quickly, creating a large voltage across the open circuit of the switch or relay. If the inductance is large enough, the energy may generate a spark, causing the contact points to oxidize, deteriorate, or sometimes weld together, or destroying a solid-state switch. A [[snubber]] capacitor across the newly opened circuit creates a path for this impulse to bypass the contact points, thereby preserving their life; these were commonly found in [[contact breaker]] [[ignition system]]s, for instance. Similarly, in smaller scale circuits, the spark may not be enough to damage the switch but may still [[Spark-gap transmitter|radiate]] undesirable [[radio frequency interference]] (RFI), which a [[filter capacitor]] absorbs. Snubber capacitors are usually employed with a low-value resistor in series, to dissipate energy and minimize RFI. Such resistor-capacitor combinations are available in a single package.

Capacitors are also used in parallel with interrupting units of a high-voltage [[circuit breaker]] to equally distribute the voltage between these units. These are called "grading capacitors".

In schematic diagrams, a capacitor used primarily for DC charge storage is often drawn vertically in circuit diagrams with the lower, more negative, plate drawn as an arc. The straight plate indicates the positive terminal of the device, if it is polarized (see [[electrolytic capacitor]]).

===Motor starters===
{{Main|Motor capacitor}}
In single phase [[Squirrel-cage rotor|squirrel cage]] motors, the primary winding within the motor housing is not capable of starting a rotational motion on the rotor, but is capable of sustaining one. To start the motor, a secondary "start" winding has a series non-polarized ''[[starting capacitor]]'' to introduce a lead in the sinusoidal current. When the secondary (start) winding is placed at an angle with respect to the primary (run) winding, a rotating electric field is created. The force of the rotational field is not constant, but is sufficient to start the rotor spinning. When the rotor comes close to operating speed, a centrifugal switch (or current-sensitive relay in series with the main winding) disconnects the capacitor. The start capacitor is typically mounted to the side of the motor housing. These are called capacitor-start motors, that have relatively high starting torque. Typically they can have up-to four times as much starting torque as a split-phase motor and are used on applications such as compressors, pressure washers and any small device requiring high starting torques.

Capacitor-run induction motors have a permanently connected phase-shifting capacitor in series with a second winding. The motor is much like a two-phase induction motor.

Motor-starting capacitors are typically non-polarized electrolytic types, while running capacitors are conventional paper or plastic film dielectric types.

===Signal processing===
The energy stored in a capacitor can be used to represent [[information]], either in binary form, as in [[DRAM]]s, or in analogue form, as in [[analog sampled filter]]s and [[Charge-coupled device|CCDs]]. Capacitors can be used in [[analog circuit]]s as components of integrators or more complex filters and in [[negative feedback]] loop stabilization. Signal processing circuits also use capacitors to [[integral|integrate]] a current signal.

====Tuned circuits====
Capacitors and inductors are applied together in [[RLC circuit|tuned circuits]] to select information in particular frequency bands. For example, [[radio receiver]]s rely on variable capacitors to tune the station frequency. Speakers use passive analog [[Audio crossover|crossovers]], and analog equalizers use capacitors to select different audio bands.

The [[resonant frequency]] ''f'' of a tuned circuit is a function of the inductance (''L'') and capacitance (''C'') in series, and is given by:
<math display="block">f = \frac{1}{2 \pi \sqrt{LC}}</math>
where {{mvar|L}} is in [[henry (unit)|henries]] and {{mvar|C}} is in farads.

===Sensing===
{{main|capacitive sensing|Capacitive displacement sensor}}
Most capacitors are designed to maintain a fixed physical structure. However, various factors can change the structure of the capacitor, and the resulting change in capacitance can be used to [[Sensor|sense]] those factors.

;Changing the dielectric:
:The effects of varying the characteristics of the '''dielectric''' can be used for sensing purposes. Capacitors with an exposed and porous dielectric can be used to measure humidity in air. Capacitors are used to accurately measure the fuel level in [[airplane]]s; as the fuel covers more of a pair of plates, the circuit capacitance increases. Squeezing the dielectric can change a capacitor at a few tens of bar pressure sufficiently that it can be used as a pressure sensor.<ref>Downie, Neil A and Mathilde Pradier, 'Method and apparatus for monitoring fluid pressure", US Patent 7526961 (2009)</ref> A selected, but otherwise standard, polymer dielectric capacitor, when immersed in a compatible gas or liquid, can work usefully as a very low cost pressure sensor up to many hundreds of bar.
;Changing the distance between the plates:
:Capacitors with a flexible plate can be used to measure strain or pressure. Industrial pressure transmitters used for [[process control]] use pressure-sensing diaphragms, which form a capacitor plate of an oscillator circuit. Capacitors are used as the [[sensor]] in [[condenser microphone]]s, where one plate is moved by air pressure, relative to the fixed position of the other plate. Some [[accelerometer]]s use [[MEMS]] capacitors etched on a chip to measure the magnitude and direction of the acceleration vector. They are used to detect changes in acceleration, in tilt sensors, or to detect free fall, as sensors triggering [[airbag]] deployment, and in many other applications. Some [[Fingerprint authentication#Fingerprint sensors|fingerprint sensors]] use capacitors. Additionally, a user can adjust the pitch of a [[theremin]] musical instrument by moving their hand since this changes the effective capacitance between the user's hand and the antenna.
;Changing the effective area of the plates:
:Capacitive [[touch switch]]es are now{{when|date=May 2018}} used on many consumer electronic products.

===Oscillators===
{{Further|Hartley oscillator}}
[[File:Garner oscillator.svg|alt=|thumb|198x198px|Example of a simple oscillator incorporating a capacitor]]
A capacitor can possess spring-like qualities in an oscillator circuit. In the image example, a capacitor acts to influence the biasing voltage at the npn transistor's base. The resistance values of the voltage-divider resistors and the capacitance value of the capacitor together control the oscillatory frequency.

=== Producing light ===
{{main | light emitting capacitor }}
A light-emitting capacitor is made from a dielectric that uses [[phosphorescence]] to produce light. If one of the conductive plates is made with a transparent material, the light is visible. Light-emitting capacitors are used in the construction of electroluminescent panels, for applications such as backlighting for laptop computers. In this case, the entire panel is a capacitor used for the purpose of generating light.

==Hazards and safety==
The hazards posed by a capacitor are usually determined, foremost, by the amount of energy stored, which is the cause of things like electrical burns or heart [[fibrillation]]. Factors such as voltage and chassis material are of secondary consideration, which are more related to how easily a shock can be initiated rather than how much damage can occur.<ref name="ReferenceA"/> Under certain conditions, including conductivity of the surfaces, preexisting medical conditions, the humidity of the air, or the pathways it takes through the body (i.e.: shocks that travel across the core of the body and, especially, the heart are more dangerous than those limited to the extremities), shocks as low as one joule have been reported to cause death, although in most instances they may not even leave a burn. Shocks over ten joules will generally damage skin, and are usually considered hazardous. Any capacitor that can store 50 joules or more should be considered potentially lethal.<ref>{{Cite web|url=http://donklipstein.com/xesafe.html|title=Some Xenon Strobe and Flash Safety Hints|website=donklipstein.com}}</ref><ref name="ReferenceA"/>

Capacitors may retain a charge long after power is removed from a circuit; this charge can cause dangerous or even potentially fatal [[Electric shock|shocks]] or damage connected equipment. For example, even a seemingly innocuous device such as the [[Flash (photography)|flash]] of a [[disposable camera]], has a [[photoflash capacitor]] which may contain over 15 joules of energy and be charged to over 300 volts. This is easily capable of delivering a shock. Service procedures for electronic devices usually include instructions to discharge large or high-voltage capacitors, for instance using a [[Brinkley stick]]. Larger capacitors, such as those used in [[microwave oven]]s, [[HVAC]] units and medical [[defibrillator]]s may also have built-in discharge resistors to dissipate stored energy to a safe level within a few seconds after power is removed. High-voltage capacitors are stored with the terminals [[short circuit|shorted]], as protection from potentially dangerous voltages due to [[Permittivity#Lossy medium|dielectric absorption]] or from transient voltages the capacitor may pick up from static charges or passing weather events.<ref name="ReferenceA"/>

Some old, large oil-filled paper or plastic film capacitors contain [[polychlorinated biphenyls]] (PCBs). It is known that waste PCBs can leak into [[groundwater]] under [[landfill]]s. Capacitors containing PCBs were labelled as containing "Askarel" and several other trade names. PCB-filled paper capacitors are found in very old (pre-1975) [[fluorescent lamp]] ballasts, and other applications.

Capacitors may [[catastrophic failure|catastrophically fail]] when subjected to voltages or currents beyond their rating, or in case of polarized capacitors, applied in a reverse polarity. Failures may create arcing that heats and vaporizes the dielectric fluid, causing a build up of pressurized gas that may result in swelling, rupture, or an explosion. Larger capacitors may have vents or 
similar mechanism to allow the release of such pressures in the event of failure. Capacitors used in [[Radio frequency|RF]] or sustained high-current applications can overheat, especially in the center of the capacitor rolls. Capacitors used within high-energy capacitor banks can violently explode when a short in one capacitor causes sudden dumping of energy stored in the rest of the bank into the failing unit. High voltage vacuum capacitors can generate soft X-rays even during normal operation. Proper containment, fusing, and preventive maintenance can help to minimize these hazards.

High-voltage capacitors may benefit from a [[pre-charge]] to limit in-rush currents at power-up of high voltage direct current (HVDC) circuits. This extends the life of the component and may mitigate high-voltage hazards.

<gallery mode="packed">
File:Defekte Kondensatoren.jpg|Swollen electrolytic capacitors. The vent on the tops allows the release of pressurized gas build-up in the event of failure, preventing it from exploding.
File:High-energy capacitor from a defibrillator 42 MFD @ 5000 VDC.jpg|This high-energy capacitor from a [[defibrillator]] has a resistor connected between the terminals for safety, to dissipate stored energy.
File:Exploded Electrolytic Capacitor.jpg|An exploded electrolytic capacitor, showing fragments of paper and metallic foil
</gallery>

==See also==
{{Portal|Electronics}}
* [[Capacitance meter]]
* [[Capacitor plague]]
* [[Electric displacement field]]
* [[Electroluminescence]]
* [[List of capacitor manufacturers]]

==Notes==
{{NoteFoot}}{{Reflist|group="nb"|refs=
<ref group="nb" name="NB_AmbiguousLetters">In order to reduce the risk for read errors, the letters <code>I</code> and <code>O</code> are not used as their glyphs look similar to other letters and digits.</ref>
}}

==References==
{{Reflist|30em}}

===Bibliography===
* {{cite book |title=Introduction to Electric Circuits |url={{google books|plainurl=y|id=l-weAQAAIAAJ}} |author-last1=Dorf |author-first1=Richard C. |author-last2=Svoboda |author-first2=James A. |edition=5th |publisher=John Wiley & Sons |location=New York |date=2001 |isbn=978-0-47138689-6}}
* Philosophical Transactions of the Royal Society LXXII, Appendix 8, 1782 (Volta coins the word ''condenser'')
* {{cite book |title=Fundamentals of Applied Electromagnetics |url={{google books |plainurl=y |id=a_C8QgAACAAJ}} |author-last=Ulaby |author-first=Fawwaz Tayssir |publisher=[[Prentice Hall]] |date=1999 |edition=2nd |location=Upper Saddle River, New Jersey, USA |isbn=978-0-13011554-6}}
* {{Cite book |title=Semiconductor Material and Device Characterization |author-first=Dieter K. |author-last=Schroder |page=270 ''ff'' |url={{google books |plainurl=y |id=OX2cHKJWCKgC |page=305}} |edition=3rd |publisher=Wiley |date=2006 |isbn=978-0-47173906-7}}
* {{Cite book |title=Physics of Semiconductor Devices |author-first1=Simon M. |author-last1=Sze |author-first2=Kwok K. |author-last2=Ng |isbn=978-0-47006830-4 |publisher=Wiley |date=2006 |edition=3rd |url={{google books |plainurl=y |id=o4unkmHBHb8C}}}}

==Further reading==
* ''Tantalum and Niobium-Based Capacitors – Science, Technology, and Applications''; 1st Ed; Yuri&nbsp;Freeman; Springer; 120 pages; 2018; {{ISBN|978-3-31967869-6}}.
* ''Capacitors''; 1st Ed; R. P. Deshpande; McGraw-Hill; 342 pages; 2014; {{ISBN|978-0-07184856-5}}.
* ''The Capacitor Handbook''; 1st Ed; Cletus Kaiser; Van Nostrand Reinhold; 124 pages; 1993; {{ISBN|978-9-40118092-4}}.
* ''Understanding Capacitors and their Uses''; 1st Ed; William Mullin; Sams Publishing; 96 pages; 1964. <small>[https://worldradiohistory.com/BOOKSHELF-ARH/Sams-Books/Sams-Understanding-Capacitors-And-Their-Uses-1964-Mullin.pdf (archive)]</small>
* ''Fixed and Variable Capacitors''; 1st Ed; G. W. A. Dummer and Harold Nordenberg; Maple Press; 288 pages; 1960. <small>[https://archive.org/details/FixedAndVariableCapacitors/ (archive)]</small>
* ''The Electrolytic Capacitor''; 1st Ed; Alexander Georgiev; Murray Hill Books; 191 pages; 1945. <small>[https://archive.org/details/TheElectrolyticCapacitor/ (archive)]</small>

==External links==
{{Commons category multi|Capacitors|Capacitors (SMD)}}
{{Wikibooks
|1=Electronics
|2=Capacitors
|3=Capacitors
}}
{{Wiktionary}}
* [http://www.sparkmuseum.com/BOOK_LEYDEN.HTM The First Condenser – A Beer Glass] – SparkMuseum
* [http://electronics.howstuffworks.com/capacitor.htm/printable How Capacitors Work] – Howstuffworks
* [http://www.robotplatform.com/electronics/capacitor/capacitor.html Capacitor Tutorial]

{{Electronic component}}
{{Digital electronics}}
{{Authority control}}

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