|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Use dmy dates|date=June 2013}}{{Infobox physical quantity
| | Myrtle Benny is how I'm called and I really feel comfortable when people use the complete name. Managing individuals is what I do and the salary has been really satisfying. Minnesota is where he's been residing for many years. Doing ceramics is what love performing.<br><br>My homepage :: [http://cityuc.com/index.php?do=/profile-9526/info/ over the counter std test] |
| |name =
| |
| |image =
| |
| |caption =
| |
| |unit = [[farad]]
| |
| |symbols = ''C''
| |
| |derivations =
| |
| }}
| |
| {{electromagnetism|cTopic=[[Electrical network|Electrical Network]]}}
| |
| '''Capacitance''' is the ability of a body to store an electrical [[electric charge|charge]]. Any object that can be electrically charged exhibits capacitance. A common form of energy storage device is a parallel-plate [[capacitor]]. In a parallel plate capacitor, capacitance is directly proportional to the surface area of the conductor plates and inversely proportional to the separation distance between the plates. If the charges on the plates are +''q'' and −''q'', and ''V'' gives the [[voltage]] between the plates, then the capacitance ''C'' is given by
| |
| | |
| :<math>C = \frac{q}{V}.</math>
| |
| which gives the voltage/current relationship
| |
| :<math>I(t) = C \frac{\mathrm{d}V(t)}{\mathrm{d}t}.</math>
| |
| The capacitance is a function only of the physical dimensions (geometry) of the conductors and the [[permittivity]] of the [[dielectric]]. It is independent of the potential difference between the conductors and the total charge on them.
| |
| | |
| The [[SI]] unit of capacitance is the [[farad]] (symbol: F), named after the English physicist [[Michael Faraday]]; a 1 farad capacitor when charged with 1 [[coulomb]] of electrical charge will have a potential difference of 1 [[volt]] between its plates.<ref>http://www.collinsdictionary.com/dictionary/english/farad</ref> Historically, a farad was regarded as an inconveniently large unit, both electrically and physically. Its subdivisions were invariably used, namely the microfarad, nanofarad and picofarad. More recently, technology has advanced such that capacitors of 1 farad and greater can be constructed in a structure little larger than a [[coin battery]] (so-called '[[Electric double-layer capacitor|supercapacitors]]'). Such capacitors are principally used for energy storage replacing more traditional batteries.
| |
| | |
| The [[energy]] (measured in [[joule]]s) stored in a capacitor is equal to the ''work'' done to charge it. Consider a capacitor of capacitance ''C'', holding a charge +''q'' on one plate and −''q'' on the other. Moving a small element of charge d''q'' from one plate to the other against the potential difference {{nowrap|''V'' {{=}} ''q/C''}} requires the work d''W'':
| |
| | |
| :<math> \mathrm{d}W = \frac{q}{C}\,\mathrm{d}q </math>
| |
| | |
| where ''W'' is the work measured in joules, ''q'' is the charge measured in coulombs and ''C'' is the capacitance, measured in farads.
| |
| | |
| The energy stored in a capacitor is found by [[integral|integrating]] this equation. Starting with an uncharged capacitance ({{nowrap|''q'' {{=}} 0}}) and moving charge from one plate to the other until the plates have charge +''Q'' and −''Q'' requires the work ''W'':
| |
| | |
| :<math> W_\text{charging} = \int_0^Q \frac{q}{C} \, \mathrm{d}q = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV = \frac{1}{2}CV^2 = W_\text{stored}.</math>
| |
| | |
| ==Capacitors==
| |
| {{Main|Capacitor}}
| |
| The capacitance of the majority of capacitors used in electronic circuits is generally several orders of magnitude smaller than the [[farad]]. The most common subunits of capacitance in use today are the [[micro-|micro]]farad (µF), [[nano-|nano]]farad (nF), [[pico-|pico]]farad (pF), and, in microcircuits, [[femto-|femto]]farad (fF). However, specially made [[supercapacitors]] can be much larger (as much as hundreds of farads), and parasitic capacitive elements can be less than a femtofarad.
| |
| | |
| Capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. For example, the capacitance of a ''parallel-plate'' capacitor constructed of two parallel plates both of area ''A'' separated by a distance ''d'' is approximately equal to the following:
| |
| :<math>\ C=\varepsilon_r\varepsilon_0\frac{A}{d}</math>
| |
| where
| |
| :''C'' is the capacitance, in Farads;
| |
| :''A'' is the area of overlap of the two plates, in square meters;
| |
| :''ε''<sub>r</sub> is the [[relative static permittivity]] (sometimes called the dielectric constant) of the material between the plates (for a vacuum, {{nowrap|''ε''<sub>r</sub> {{=}} 1}});
| |
| :''ε''<sub>0</sub> is the [[vacuum permittivity|electric constant]] (''ε''<sub>0</sub> ≈ {{val|8.854|e=-12|u=F m<sup>–1</sup>}}); and
| |
| :''d'' is the separation between the plates, in meters;
| |
| | |
| Capacitance is proportional to the area of overlap and inversely proportional to the separation between conducting sheets. The closer the sheets are to each other, the greater the capacitance.
| |
| The equation is a good approximation if ''d'' is small compared to the other dimensions of the plates so the field in the capacitor over most of its area is uniform, and the so-called ''fringing field'' around the periphery provides a small contribution. In [[Gaussian units|CGS units]] the equation has the form:<ref>''The Physics Problem Solver'', 1986, [http://books.google.com/books?id=KVM2onr8_QYC&pg=PA648 Google books link]</ref>
| |
| :<math>C=\varepsilon_r\frac{A}{4\pi d}</math>
| |
| where ''C'' in this case has the units of length.
| |
| Combining the SI equation for capacitance with the above equation for the energy stored in a capacitance, for a flat-plate capacitor the energy stored is:
| |
| | |
| :<math> W_\text{stored} = \frac{1}{2} C V^2 = \frac{1}{2} \varepsilon_{r}\varepsilon_{0} \frac{A}{d} V^2.</math>
| |
| | |
| where ''W'' is the energy, in joules; ''C'' is the capacitance, in farads; and ''V'' is the voltage, in volts.
| |
| | |
| ===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>\ dQ=C(V) \, dV</math>
| |
| | |
| where the voltage dependence of capacitance, ''C''(''V''), stems from the field, which in a large area parallel plate device is given by ''<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 field, which, in the case of a large area parallel plate device, translates into a capacitance that is a nonlinear function of the voltage causing the field.<ref name= Araujo>{{Cite book|page=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=http://books.google.com/books?id=QMlOkeJ4qN4C&pg=PA504&dq=nonlinear+capacitance+ferroelectric&lr=&as_brr=0 |isbn=90-5699-704-1 |author=Carlos Paz de Araujo, Ramamoorthy Ramesh, George W Taylor (Editors) |publisher=CRC Press |year=2001 |nopp=true}}</ref><ref name=Musikant>{{Cite book|title=What Every Engineer Should Know about Ceramics |author=Solomon Musikant |isbn=0-8247-8498-7 |year=1991 |publisher=CRC Press |page=Figure 3.9, p. 43 |url=http://books.google.com/books?id=Jc8xRdgdH38C&pg=PA44&dq=nonlinear+capacitance+ferroelectric&lr=&as_brr=0#PPA43,M1 |nopp=true}}</ref>
| |
| | |
| Corresponding to the voltage-dependent capacitance, to charge the capacitor to voltage ''V'' an integral relation is found:
| |
| | |
| :<math> Q=\int_0^VC(V) \, dV\ </math>
| |
| | |
| which agrees with ''Q'' = ''CV'' only when ''C'' is voltage independent.
| |
| | |
| By the same token, the energy stored in the capacitor now is given by
| |
| | |
| :<math>dW =Q \, dV =\left[ \int_0^V\ dV' \ C(V') \right] \ dV \ . </math>
| |
| | |
| Integrating:
| |
| | |
| :<math>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''; R Waser, U Böttger & S Tiedke, editors |isbn=3-527-40532-1 |publisher=Wiley-VCH |year=2005 |url=http://books.google.com/books?id=wQ09DhMBJroC&pg=PA304&dq=nonlinear+capacitance+ferroelectric&lr=&as_brr=0#PPA303,M1 |page=Chapter 16 |nopp=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.<ref name=Sze0>{{Cite book|title=Physics of Semiconductor Devices |author=Simon M. Sze, Kwok K. Ng |isbn=0-470-06830-2 |publisher=Wiley |year=2006 |edition=3rd Edition |url=http://books.google.com/books?id=o4unkmHBHb8C&pg=PA217&dq=high+low+frequency+C-V&lr=&as_brr=0#PPA121,M1|page=Figure 25, p. 121 |nopp=true}}</ref> 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, then the polarization of the dielectric cannot follow the signal. 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>\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 ''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=Gabriele Giuliani, Giovanni Vignale |page=111 |url=http://books.google.com/books?id=kFkIKRfgUpsC&pg=PA538&dq=%22linear+response+theory%22+capacitance+OR+conductance&lr=&as_brr=0#PPA111,M1 |isbn=0-521-82112-6 |publisher=Cambridge University Press |year=2005}}</ref><ref name=Rammer>{{Cite book|title=Quantum Field Theory of Non-equilibrium States |author=Jørgen Rammer |page=158 |url=http://books.google.com/books?id=A7TbrAm5Wq0C&pg=PR6&dq=%22linear+response+theory%22+capacitance+OR+conductance&lr=&as_brr=0#PPA158,M1 |isbn=0-521-87499-8 |publisher=Cambridge University Press |year=2007}}</ref> The integral extends over the entire past history up to the present time. A [[Fourier_analysis#.28Continuous.29_Fourier_transform|Fourier transform]] in time then results in:
| |
| | |
| :<math>\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>I(\omega) = j\omega Q(\omega) = j\omega \oint_{\Sigma} \boldsymbol D (\boldsymbol r , \ \omega)\cdot d \boldsymbol{\Sigma} \ </math>
| |
| :::<math>=\left[ G(\omega) + j \omega C(\omega)\right] V(\omega) = \frac {V(\omega)}{Z(\omega)} \ , </math>
| |
| | |
| where ''j'' is the [[imaginary unit]], ''V''(''ω'') is the voltage component at angular frequency ''ω'', ''G''(''ω'') is the ''real'' part of the current, called the ''conductance'', and ''C''(''ω'') determines the ''imaginary'' part of the current and is the ''capacitance''. ''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> \varepsilon_r(\omega) = \varepsilon '_r(\omega) - j \varepsilon ''_r(\omega) = \frac{1}{j\omega Z(\omega) C_0} = \frac{C(\omega)}{C_0} \ , </math>
| |
| | |
| where a single ''prime'' denotes the real part and a double ''prime'' the imaginary part, ''Z''(''ω'') is the complex impedance with the dielectric present, ''C''(''ω'') is the so-called ''complex'' capacitance with the dielectric present, and ''C''<sub>0</sub> is the capacitance without the dielectric.<ref name=Handbook>{{Cite book|title=Springer Handbook of Materials Measurement Methods |author=Horst Czichos, Tetsuya Saito, Leslie Smith |page=475 |url=http://books.google.com/books?id=8lANaR-Pqi4C&pg=RA1-PA475&dq=%22the+dielectric+permittivity+is+defined%22&lr=&as_brr=0 |publisher=Springer |isbn=3-540-20785-6 |year=2006}}</ref><ref name=Coffey>{{Cite book|title=Fractals, diffusion and relaxation in disordered complex systems..Part A |author=William Coffey, Yu. P. Kalmykov |publisher=Wiley |year=2006 |isbn=0-470-04607-4 |page=17 |url=http://books.google.com/books?id=mgtQslaXBc4C&pg=PA18&dq=%22dielectric+relaxation+function%22&lr=&as_brr=0#PPA17,M1}}</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>J. Obrzut, A. Anopchenko and R. Nozaki, [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1604368 "Broadband Permittivity Measurements of High Dielectric Constant Films"], ''Proceedings of the IEEE: Instrumentation and Measurement Technology Conference, 2005'', pp. 1350–1353, 16–19 May 2005, Ottawa ISBN 0-7803-8879-8 {{doi|10.1109/IMTC.2005.1604368}}</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.<ref name=Schroder2>{{Cite book|title=Semiconductor Material and Device Characterization |author=Dieter K Schroder |page=347 ''ff''. |url=http://books.google.com/books?id=OX2cHKJWCKgC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA347,M1 |edition=3rd Edition |publisher=Wiley |year=2006 |isbn=0-471-73906-5}}</ref> Alternatively, the time response of the capacitance can be used directly, as in ''[[deep-level transient spectroscopy]]''.<ref name=Schroder>{{Cite book|title=Semiconductor Material and Device Characterization |author=Dieter K Schroder |page=270 ''ff''. |url=http://books.google.com/books?id=OX2cHKJWCKgC&pg=PA305&dq=capacitance+conductance+measurement+spectroscopy&lr=&as_brr=0#PPA271,M1 |edition=3rd Edition |publisher=Wiley |year=2006 |isbn=0-471-73906-5}}</ref>
| |
| | |
| 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.<ref name=Sze>{{Cite book|title=Physics of Semiconductor Devices |author=Simon M. Sze, Kwok K. Ng |isbn=0-470-06830-2 |publisher=Wiley |year=2006 |edition=3rd Edition |url=http://books.google.com/books?id=o4unkmHBHb8C&pg=PA217&dq=high+low+frequency+C-V&lr=&as_brr=0 |page=217}}</ref><ref name=Kasap>{{Cite book|title=Springer Handbook of Electronic and Photonic Materials |author=Safa O. Kasap, Peter Capper |year=2006 |publisher=Springer |page=Figure 20.22, p. 425 |url=http://books.google.com/books?id=rVVW22pnzhoC&pg=PA425&dq=high+low+frequency+C-V&lr=&as_brr=0 |nopp=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=http://books.google.com/books?id=W9pdJZoAeyEC&pg=PA244&dq=isbn=3-540-25470-6#PPA315,M1 |title=Fundamentals of Semiconductors |author=PY Yu and Manuel Cardona |isbn=3-540-25470-6 |year=2001 |edition=3rd Edition |publisher=Springer |page=§6.6 ''Modulation Spectroscopy}}</ref>
| |
| | |
| ==Capacitance matrix==
| |
| | |
| The discussion above is limited to the case of two conducting plates, although of arbitrary size and shape. The definition C=Q/V still holds for a single plate given a charge, in which case the field lines produced by that charge terminate as if the plate were at the center of an oppositely charged sphere at infinity.
| |
| | |
| <math>C = Q/V</math> does not apply when there are more than two charged plates, or when the net charge on the two plates is non-zero. To handle this case, Maxwell introduced his ''coefficients of potential''. If three plates are given charges <math>Q_1, Q_2, Q_3</math>, then the voltage of plate 1 is given by
| |
| | |
| :<math>V_1 = P_{11}Q_1 + P_{12} Q_2 + P_{13}Q_3, </math>
| |
| | |
| and similarly for the other voltages. [[Hermann von Helmholtz]] and [[Sir William Thomson]] showed that the coefficients of potential are symmetric, so that <math>P_{12}=P_{21}</math>, etc. Thus the system can be described by a collection of coefficients known as the ''elastance matrix'' or ''reciprocal capacitance matrix'', which is defined as:
| |
| | |
| :<math>P_{ij} = \frac{\partial V_{i}}{\partial Q_{j}}</math>
| |
| | |
| From this, the mutual capacitance <math>C_{m}</math> between two objects can be defined<ref name=Jackson1999>{{citation|last=Jackson|first=John David|title=Classical Electrodynamic|publisher = John Wiley & Sons, Inc. |year=1999|location=USA|edition=3rd.|pages=43|isbn=978-0-471-30932-1}}</ref> by solving for the total charge ''Q'' and using <math>C_{m}=Q/V</math>.
| |
| | |
| : <math>C_m = \frac{1}{(P_{11} + P_{22})-(P_{12} + P_{21})}</math>
| |
| | |
| Since no actual device holds perfectly equal and opposite charges on each of the two "plates", it is the mutual capacitance that is reported on capacitors.
| |
| | |
| The collection of coefficients <math>C_{ij} =\frac{\partial Q_{i}}{\partial V_{j}}</math> is known as the ''capacitance matrix'',<ref name=maxwell>{{Citation | last =Maxwell | first =James | author-link =James Clerk Maxwell | title = A treatise on electricity and magnetism, Volume 1 | publisher = Clarendon Press | year = 1873 | chapter =3 | pages =88ff | url = http://www.archive.org/details/electricandmagne01maxwrich}}</ref><ref>{{Cite web|title=Capacitance|url=http://www.av8n.com/physics/capacitance.htm|accessdate=20 September 2010}}</ref> and is the [[matrix inverse|inverse]] of the elastance matrix.
| |
| | |
| ==Self-capacitance==
| |
| In electrical circuits, the term ''capacitance'' is usually a shorthand for the ''[[mutual capacitance]]'' between two adjacent conductors, such as the two plates of a capacitor. However, for an isolated conductor there also exists a property called ''self-capacitance'', which is the amount of electrical charge that must be added to an isolated conductor to raise its [[electrical potential]] by one unit (i.e. one volt, in most measurement systems).<ref>{{cite book|author=William D. Greason|title=Electrostatic discharge in electronics|url=http://books.google.com/books?id=404fAQAAIAAJ|accessdate=4 December 2011|year=1992|publisher=Research Studies Press|isbn=978-0-86380-136-5|page=48}}</ref> The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, centered on the conductor. Using this method, the self-capacitance of a conducting sphere of radius ''R'' is given by:<ref name=NSW>[http://www.phys.unsw.edu.au/COURSES/FIRST_YEAR/pdf%20files/5Capacitanceanddielectr.pdf Lecture notes]; University of New South Wales</ref>
| |
| | |
| :<math>C=4\pi\varepsilon_0R \,</math>
| |
| | |
| Example values of self-capacitance are:
| |
| *for the top "plate" of a [[van de Graaff generator]], typically a sphere 20 cm in radius: 22.24 pF
| |
| *the planet [[Earth]]: about 710 µF<ref>{{Citation | last = Tipler | first = Paul | last2 = Mosca | first2 = Gene | title = Physics for scientists and engineers | publisher = Macmillan | year = 2004 | edition = 5th | page =752 | isbn =978-0-7167-0810-0 }}</ref>
| |
| | |
| The capacitative component of a [[electromagnetic coil|coil]], which reduces its [[Electrical impedance|impedance]] at high frequencies and can lead to [[resonance]] and self-oscillation, is also called self-capacitance<ref>{{cite journal|title=Self-capacitance of inductors|doi=10.1109/63.602562 |author=Massarini, A.; Kazimierczuk, M.K.|year=1997|volume=12 |issue=4|pages= 671–676|journal=IEEE Transactions on Power Electronics}}: example of use of term self-capacitance</ref> as well as stray or [[parasitic capacitance]].
| |
| | |
| ==Elastance==
| |
| The reciprocal of capacitance is called ''elastance''. The unit of elastance is the [[daraf]], but is not recognised by SI.
| |
| | |
| ==Stray capacitance==
| |
| {{Main|Parasitic capacitance}}
| |
| Any two adjacent conductors can be considered a capacitor, although the capacitance will be small unless the conductors are close together for long distances or over a large area. This (often unwanted) effect is termed "stray capacitance". Stray capacitance can allow signals to leak between otherwise isolated circuits (an effect called [[Crosstalk (electronics)|crosstalk]]), and it can be a limiting factor for proper functioning of circuits at [[high frequency]].
| |
| | |
| Stray capacitance is often encountered in amplifier circuits in the form of "feedthrough" capacitance that interconnects the input and output nodes (both defined relative to a common ground). It is often convenient for analytical purposes to replace this capacitance with a combination of one input-to-ground capacitance and one output-to-ground capacitance; the original configuration — including the input-to-output capacitance — is often referred to as a pi-configuration. Miller's theorem can be used to effect this replacement: it states that, if the gain ratio of two nodes is 1/''K'', then an [[electrical impedance|impedance]] of ''Z'' connecting the two nodes can be replaced with a ''Z''/(1 − ''k'') impedance between the first node and ground and a ''KZ''/(''K'' − 1) impedance between the second node and ground. Since impedance varies inversely with capacitance, the internode capacitance, ''C'', will be seen to have been replaced by a capacitance of KC from input to ground and a capacitance of (''K'' − 1)''C''/''K'' from output to ground. When the input-to-output gain is very large, the equivalent input-to-ground impedance is very small while the output-to-ground impedance is essentially equal to the original (input-to-output) impedance.
| |
| | |
| ==Capacitance of simple systems==
| |
| Calculating the capacitance of a system amounts to solving the [[Laplace equation]] ''∇<sup>2</sup>φ = 0'' with a constant potential ''φ'' on the surface of the conductors. This is trivial in cases with high symmetry. There is no solution in terms of elementary functions in more complicated cases.
| |
| | |
| For quasi-two-dimensional situations analytic functions may be used to map different geometries to each other. See also [[Schwarz–Christoffel mapping]].
| |
| | |
| {| class="wikitable"
| |
| |+ Capacitance of simple systems
| |
| ! Type !! Capacitance !! Comment
| |
| |-
| |
| ! Parallel-plate capacitor
| |
| | <math> \varepsilon A /d </math>
| |
| | [[Image:Plate CapacitorII.svg|125px]]
| |
| ''ε'': [[Permittivity]]
| |
| |-
| |
| ! Coaxial cable
| |
| | <math> \frac{2\pi \varepsilon l}{\ln \left( R_{2}/R_{1}\right) } </math>
| |
| | [[Image:Cylindrical CapacitorII.svg|130px]]
| |
| ''ε'': [[Permittivity]]
| |
| |-
| |
| ! Pair of parallel wires<ref name="Jackson 1975 80">{{Cite book|last=Jackson |first=J. D. |title=Classical Electrodynamics |year=1975|publisher=Wiley |page=80}}</ref>
| |
| | <math>\frac{\pi \varepsilon l}{\operatorname{arcosh}\left( \frac{d}{2a}\right) }=\frac{\pi \varepsilon l}{\ln \left( \frac{d}{2a}+\sqrt{\frac{d^{2}}{4a^{2}}-1}\right) }</math>
| |
| |[[Image:Parallel Wire Capacitance.svg|130px]]
| |
| |-
| |
| ! Wire parallel to wall<ref name="Jackson 1975 80"/>
| |
| | <math>\frac{2\pi \varepsilon l}{\operatorname{arcosh}\left( \frac{d}{a}\right) }=\frac{2\pi \varepsilon l}{\ln \left( \frac{d}{a}+\sqrt{\frac{d^{2}}{a^{2}}-1}\right) }</math>
| |
| | ''a'': Wire radius <br/>''d'': Distance, ''d > a'' <br/>''l'': Wire length
| |
| |-
| |
| ! Two parallel<br/>coplanar strips<ref>{{Cite book| last1 = Binns | last2 = Lawrenson | title = Analysis and computation of electric and magnetic field problems | publisher = Pergamon Press | year = 1973 | accessdate = 2010-06-04 | isbn = 978-0-08-016638-4}}</ref>
| |
| | <math>\varepsilon l \frac{ K\left( \sqrt{1-k^{2}} \right) }{ K\left(k \right) }</math>
| |
| | ''d'': Distance<br/>''w<sub>1</sub>, w<sub>2</sub>'': Strip width<br/>''k<sub>m</sub>'': ''d/(2w<sub>m</sub>+d)''<br/>
| |
| ''k<sup>2</sup>'': ''k<sub>1</sub>k<sub>2</sub>''<br/>''K'': [[Elliptic integral]]<br/>''l'': Length
| |
| |-
| |
| ! Concentric spheres
| |
| | <math> \frac{4\pi \varepsilon}{\frac{1}{R_1}-\frac{1}{R_2}} </math>
| |
| | [[Image:Spherical Capacitor.svg|97px]]
| |
| ''ε'': [[Permittivity]]
| |
| |-
| |
| ! Two spheres,<br/>equal radius<ref name="Maxwell 1873 266 ff">{{Cite book|last=Maxwell |first=J. C. |title=A Treatise on Electricity and Magnetism |year=1873|publisher=Dover |pages=266 ff |isbn=0-486-60637-6}}</ref><ref>{{Cite journal|last=Rawlins |first=A. D. |title=Note on the Capacitance of Two Closely Separated Spheres |journal=IMA Journal of Applied Mathematics |volume=34 |issue=1 |pages=119–120 |year=1985 |doi=10.1093/imamat/34.1.119}}</ref>
| |
| | <math>2\pi \varepsilon a\sum_{n=1}^{\infty }\frac{\sinh \left( \ln \left( D+\sqrt{D^2-1}\right) \right) }{\sinh \left( n\ln \left( D+\sqrt{ D^2-1}\right) \right) } </math><br/><math>=2\pi \varepsilon a\left\{ 1+\frac{1}{2D}+\frac{1}{4D^2}+\frac{1}{8D^{3}}+\frac{1}{8D^{4}}+\frac{3}{32D^{5}}+O\left( \frac{1}{D^{6}}\right) \right\}</math><br/><math>=2\pi \varepsilon a\left\{ \ln 2+\gamma -\frac{1}{2}\ln \left( \frac{d}{a}-2\right) +O\left( \frac{d}{a}-2\right) \right\}</math>
| |
| | ''a'': Radius<br/>''d'': Distance, ''d'' > 2''a''<br/>''D'' = ''d''/2''a''<br/> ''γ'': [[Euler–Mascheroni constant|Euler's constant]]
| |
| |-
| |
| ! Sphere in front of wall<ref name="Maxwell 1873 266 ff"/>
| |
| | <math>4\pi \varepsilon a\sum_{n=1}^{\infty }\frac{\sinh \left( \ln \left( D+\sqrt{D^{2}-1}\right) \right) }{\sinh \left( n\ln \left( D+\sqrt{ D^{2}-1}\right) \right) } </math>
| |
| | ''a'': Radius<br/>''d'': Distance, ''d > a''<br/>''D = d/a''
| |
| |-
| |
| ! Sphere
| |
| | <math> 4\pi \varepsilon a </math>
| |
| | ''a'': Radius
| |
| |-
| |
| ! Circular disc<ref name="Jackson 1975 128">{{Cite book|last=Jackson |first=J. D. |title=Classical Electrodynamics |year=1975|publisher=Wiley |page=128, problem 3.3}}</ref>
| |
| | <math> 8\varepsilon a </math>
| |
| | ''a'': Radius
| |
| |-
| |
| ! Thin straight wire,<br/> finite length<ref>{{Cite journal|last=Maxwell |first=J. C. |title=On the electrical capacity of a long narrow cylinder and of a disk of sensible thickness |doi=10.1112/plms/s1-9.1.94 |journal=Proc. London Math. Soc. |volume=IX|pages=94–101 |year=1878}}</ref><ref>{{Cite journal|last=Vainshtein |first=L. A. |title=Static boundary problems for a hollow cylinder of finite length. III Approximate formulas |journal=Zh. Tekh. Fiz. |volume=32 |pages=1165–1173 |year=1962}}</ref><ref>{{Cite journal|last=Jackson |first=J. D. |title=Charge density on thin straight wire, revisited |journal=Am. J. Phys |volume=68 |issue=9 |pages=789–799 |year=2000|doi=10.1119/1.1302908|bibcode = 2000AmJPh..68..789J }}</ref>
| |
| | <math> \frac{2\pi \varepsilon l}{\Lambda }\left\{ 1+\frac{1}{\Lambda }\left( 1-\ln 2\right) +\frac{1}{\Lambda ^{2}}\left[ 1+\left( 1-\ln 2\right) ^{2}-\frac{\pi ^{2}}{12}\right] +O\left(\frac{1}{\Lambda ^{3}}\right) \right\} </math>
| |
| | ''a'': Wire radius<br>''l'': Length<br/>''Λ: ln(l/a)''
| |
| |}
| |
| | |
| ==See also==
| |
| {{colbegin|3}}
| |
| *[[Ampère's law]]
| |
| *[[Capacitor]]
| |
| *[[Capacitive Displacement Sensors]]
| |
| *[[Electrical conductance|Conductance]]
| |
| *[[Electrical conductor|Conductor]]
| |
| *[[Displacement current]]
| |
| *[[Electromagnetism]]
| |
| *[[Electricity]]
| |
| *[[Electronics]]
| |
| *[[Hydraulic analogy]]
| |
| *[[Inductor]]
| |
| *[[Inductance]]
| |
| *[[Orders of magnitude (capacitance)]]
| |
| *[[Quantum capacitance]]
| |
| *[[LCR meter]]
| |
| {{colend}}
| |
| | |
| ==References==
| |
| {{Reflist|33em}}
| |
| | |
| ==Further reading==
| |
| {{Refbegin}}
| |
| *Tipler, Paul (1998). ''Physics for Scientists and Engineers: Vol. 2: Electricity and Magnetism, Light'' (4th ed.). W. H. Freeman. ISBN 1-57259-492-6
| |
| *Serway, Raymond; Jewett, John (2003). ''Physics for Scientists and Engineers'' (6 ed.). Brooks Cole. ISBN 0-534-40842-7
| |
| *Saslow, Wayne M.(2002). ''Electricity, Magnetism, and Light''. Thomson Learning. ISBN 0-12-619455-6. See Chapter 8, and especially pp. 255–259 for coefficients of potential.
| |
| {{Refend}}
| |
| | |
| [[Category:Concepts in physics]]
| |
| [[Category:Physical quantities]]
| |
| [[Category:Electricity]]
| |
| [[Category:Electronics]]
| |