Capacitance

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In electromagnetism and electronics, capacitance is the ability of a body to hold an electrical charge. Capacitance is also a measure of the amount of electric charge stored (or separated) for a given electric potential. A common form of charge storage device is a two-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 is given by

$C = \frac{Q}{V}.$

The SI unit of capacitance is the farad; 1 farad = 1 coulomb per volt.

The energy (measured in joules) stored in a capacitor is equal to the work done to charge it. Consider a 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 V = q/Crequires the work $d W$ :

$\mathrm{d}W = \frac{q}{C}\,\mathrm{d}q$

where W is the work measured in joules, q is the charge measured in coulombs and C is the capacitance, measured in farads.

We can find the energy stored in a capacitance by integrating this equation. Starting with an uncharged capacitance (q=0) and moving charge from one plate to the other until the plates have charge +Q and -Q requires the work W:

$W_\text{charging} = \int_{0}^{Q} \frac{q}{C} \, \mathrm{d}q = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}CV^2 = W_\text{stored}.$

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[edit]Capacitance and 'displacement current'

The physicist James Clerk Maxwell invented the concept of displacement current in his 1861 paper in connection with the displacement of electrical particles:^[1]

“

Bodies which do not permit a current of electricity to flow through them are called insulators. But though electricity does not flow through them, electrical effects are propagated through them … a dielectric is like an elastic membrane which may be impervious to the fluid, but transmits the pressure of the fluid on one side to that on the other.

”

“

Electromotive force acting on a dielectric produces a state of polarization of its parts...capable of being described as a state in which every particle has its poles in opposite conditions.

”

“

...we may conceive that the electricity in each molecule is so displaced that one side is rendered positively, and the other negatively electrical, but that the electricity remains entirety connected with the molecule, and does not pass from one molecule to another.

”

“

This displacement does not amount to a current, because when it attains a certain value it remains constant, but it is the commencement of a current, and its variations constitute currents in the positive or negative direction, according as the displacement is increasing or diminishing.

”

“

...when the electromotive force varies, the electric displacement also varies. But a variation of displacement is equivalent to a current, and this current must be taken into account...

”

He then added displacement current to Ampère's law.^[2] Maxwell's correction to Ampère's law remains valid today, and is expressed in the form:

$\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}\ ,$

with J_f the current density due to motion of free charges and the displacement current density given as ∂D/∂t with the electric displacement field D related to the electrical polarization density of the medium P as:

$\boldsymbol D = \varepsilon_0 \boldsymbol E + \boldsymbol P \ .$

Here ε₀ is the electric constant. The polarization is the contribution described by Maxwell in the quotations above, and is due to the separation and alignment of charge in the material that is not free to transport, but is free to align with an applied electric field, and to move over atomic dimensions, for example, by stretching of molecules. (This polarization in response to the field actually screensthe dielectric from the electric field, resulting in a lower field the greater the polarization of the medium. See the figure.) In simple materials, the polarization is proportional to the electric field and an adequate approximation is:

$\boldsymbol D = \varepsilon_r \varepsilon_0 \boldsymbol E \ ,$

with ε_r the relative static permittivity of the material. When there exists no material medium, ε_r = 1, so there still exists a displacement field when there is no medium present.

A capacitor with charges represented by blue and red circles. Left: Capacitor with unpolarized dipole (the two charges are not separated); electric field unscreened. Right: Capacitor with polarized dipole (the two charges are spread apart); electric field screened. The dotted box is a Gaussian surface Σ.

[edit]Gauss's law

Main article: Gauss's law

To connect the displacement to charge, Gauss's law is used, which in integral form relates the charge in a region to the surface integral over an enclosing surface Σ of the component of D normal to the surface:

$Q(t) = \oint_{\Sigma} \boldsymbol D (\boldsymbol r ,\ t)\ \boldsymbol{\cdot} \ d\boldsymbol {\Sigma} \ ,$

where a vector dot product is indicated by the "·".

To relate this expression to a capacitor, the surface Σ is made to enclose the dielectric medium and one of the two electrodes of the capacitor. The electrode contains the net charge upon the capacitor, and the dielectric medium is charge neutral. Referring to the figure, suppose initially the dipoles in the dielectric are unpolarized, as on the left side of the figure. The electric field due to the charge on the capacitor plates is the same as though the dielectric were not present. Next, suppose the dipoles are able to respond to the applied field and become polarized, as on the right side of the figure. Then the field from the extended dipole opposes that of the electrodes and the electric field inside the dielectric decreases. Suppose the left panel corresponds to an initial time just after the field is applied and the dipole has not had time to respond, while on the right is a later time when the dipoles are in the process of becoming extended. During this extension of the dipoles, a displacement current flows across the Gaussian surface. The more polarizable the medium, the more current for a given voltage, and the greater the capacitance. The net displacement current I through the region Σ is related to the displacement current density through the equation:

$I = \frac{dQ}{dt} = \oint_{\Sigma} \frac{\partial}{\partial t}\boldsymbol D \cdot d \boldsymbol {\Sigma} \ .$

(The partial time derivative is meant to emphasize that the spatial variables in D(r, t) are held fixed.) This equation includes current through the region Σ related to polarization of the medium, and is connected to capacitance and an applied voltage:

$I= \frac{dQ}{dt} = C\ \frac{dV}{dt} \ ,$

where C is capacitance, Q is charge, and V is the applied voltage responsible for the field causing the polarization of the medium inside the capacitor. For some materials represented by complicated behavior of D, the capacitance can be a function of voltage and may exhibit time dependence related to the ability of the medium to respond to the signal (see subsections below).

It should be mentioned that when there is no material medium in the capacitor, the displacement is not zero, but D = ε₀E. Consequently, a capacitance still is present. For example, a system of metal electrodes in free space may possess a capacitance.

Maxwell never used the term electric charge, but he did refer to the "distribution of electricity in a body" and to the "quantity of electricity". Capacity C was stated in his equation (138) for two surfaces bearing equal and opposite quantities of electricity e and electric tensions or potentials ψ₁ and ψ₂ as the ratio C = e / (ψ₁ - ψ₂). Then the effect upon C of inserting a dielectric between the plates was determined.^[3]

Today, capacitance is viewed primarily in terms of the capacity for storage of charge, whereas Maxwell's paper stressed the current that flowed through a capacitor. He calculated this current focusing upon the specific calculation of polarization for an "elastic sphere" distorting under an applied field and resisting deformation by virtue of its elastic properties, and the current that flowed when this state of polarization altered. The modern approach attempts to treat the polarization of materials by modeling the microscopic events contributing to the displacement field using quantum theory: for example, see below.

[edit]Capacitors

Main article: Capacitor

The capacitance of the majority of capacitors used in electronic circuits is several orders of magnitude smaller than the farad. The most common subunits of capacitance in use today are the millifarad (mF), microfarad (µF), the nanofarad (nF) and the picofarad(pF) (also known as a "puff")

The 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 Aseparated by a distance d is approximately equal to the following:

$C = \epsilon_{r}\epsilon_{0} \frac{A}{d}$ (in SI units)

where

C is the capacitance in farads, F

A is the area of overlap of the two plates measured in square metres. Capacitance is directly proportional to the surface area of the conductor plates or sheets.

ε_r is the relative static permittivity (sometimes called the dielectric constant) of the material between the plates, (for vacuum ε_r=1)

ε₀ is the permittivity of free space where ε₀ = 8.854x10^-12 F/m

d is the separation between the plates, measured in metres. Capacitance is inversely proportional to the separation between conducting sheets; in other words, 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 CGS units the equation has the form:

$C = \epsilon_{r} \frac{A}{d}$

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:

$W_{stored} = \frac{1}{2} C V^2 = \frac{1}{2} \epsilon_{r}\epsilon_{0} \frac{A}{d} V^2$ .

where

W is the energy measured in joules

C is the capacitance, measured in farads

V is the voltage measured in volts

[edit]Voltage dependent capacitors

The dielectric constant for a number of very useful dielectrics changes as a function of the applied electrical field, for exampleferroelectric 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:

$dQ = C(V) dV \ ,$

where the voltage dependence of capacitance C(V) stems from the field, which in a large area parallel plate device is given by ε = 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.^[4]^[5]

Corresponding to the voltage-dependent capacitance, to charge the capacitor to voltage V an integral relation is found:

$Q =\int_0^VdV\ C(V) \ ,$

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

$dW =Q dV =\left[ \int_0^V\ dV' \ C(V') \right] \ dV \ .$

Integrating:

$W = \int_0^V\ dV\ \int_0^V \ dV' \ C(V')$ $= \int_0^V \ dV' \ \int_{V'}^V \ dV \ C(V')$ $= \int_0^V\ dV' \left(V-V'\right) C(V') \ ,$

where interchange of the 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.^[6]

Another example of voltage dependent capacitance occurs in semiconductor devices such as semiconductor diodes, 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.^[7]

[edit]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):

$\boldsymbol D (t) = \varepsilon_0 \int_{-\infty}^t dt' \ \varepsilon_r (t-t') \boldsymbol E (t')\ ,$

indicating the lag in response by the time dependence of ε_r, calculated in principle from an underlying microscopic analysis, for example, of the dipole behavior in the dielectric. See, for example, linear response function.^[8]^[9] The integral extends over the entire past history up to the present time. A Fourier transform in time then results in:

$\boldsymbol D(\omega) = \varepsilon_0 \varepsilon_r(\omega)\boldsymbol E (\omega)\ ,$

where ε_r(ω) is now a complex function, with an imaginary part related to absorption of energy from the field by the medium. Seepermittivity. The capacitance, being proportional to the dielectric constant, also exhibits this frequency behavior. Fourier transforming Gauss's law with this form for displacement field:

$I(\omega) = j\omega Q(\omega) = j\omega \oint_{\Sigma} \boldsymbol D (\boldsymbol r , \ \omega)\cdot d \boldsymbol{\Sigma} \$

$=\left[ G(\omega) + j \omega C(\omega)\right] V(\omega) = \frac {V(\omega)}{Z(\omega)} \ ,$

where j = √−1, 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. Symbol 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:

$\varepsilon_r(\omega) = \varepsilon '_r(\omega) - j \varepsilon ''_r(\omega) = \frac{1}{j\omega Z(\omega) C_0} = \frac{C(\omega)}{C_0} \ ,$

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 C₀ is the capacitance without the dielectric.^[10]^[11] (Measurement "without the dielectric" in principle means measurement in free space, an unattainable goal inasmuch as even the 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 C₀, is sufficiently accurate.^[12] )

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.^[13] Alternatively, the time response of the capacitance can be used directly, as in deep-level transient spectroscopy.^[14]

Another example of frequency dependent capacitance occurs with 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.^[15]^[16]

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.^[17]

[edit]Coefficients of potential

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.

$C = \frac{Q}{V}$ 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 $Q 1, Q 2, Q 3$ , then the voltage of plate 1 is given by

V 1 = p 11 Q 1 + p 12 Q 2 + p 13 Q 3

and similarly for the other voltages. Maxwell showed that the coefficients of potential are symmetric, so that $p 12 = p 21$ , etc.

[edit]Capacitance/inductance duality

In mathematical terms, the ideal capacitance can be considered as an inverse of the ideal inductance, because the voltage-current equations of the two phenomena can be transformed into one another by exchanging the voltage and current terms.

[edit]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. 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 volt. 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:^[18]

$C=4\pi\epsilon_0R \,$

Typical values of self-capacitance are:

for the top "plate" of a van de Graaf generator, typically a sphere 20 cm in radius: 20 pF
the planet Earth: about 0,709 mF

[edit]Elastance

The inverse of capacitance is called elastance. The unit of elastance is the daraf.

[edit]Stray capacitance

Any two adjacent conductors can be considered as a capacitor, although the capacitance will be small unless the conductors are close together for long. This (unwanted) effect is termed "stray capacitance". Stray capacitance can allow signals to leak between otherwise isolated circuits (an effect called crosstalk), and it can be a limiting factor for proper functioning of circuits at high frequency.

Stray capacitance 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. Miller's theorem states that, if the gain ratio of two nodes is 1:K, then an impedance of Z connecting the two nodes can be replaced with a Z/(1-k) impedance between the first node and ground and a KZ/(K-1) impedance between the second node and ground. (Since impedance varies inversely with capacitance, the internode capacitance, C, 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.

[edit]Footnotes and in-line references

^ See On Physical Lines of Force: Part III–The Theory of Molecular Vortices applied to Statical Electricity
^ See Eq. (112), p. 19 in Maxwell's paper.
^ See The theory of molecular Vortices applied to Statical Electricity, p. 21
^ Carlos Paz de Araujo, Ramamoorthy Ramesh, George W Taylor (Editors) (2001). Science and Technology of Integrated Ferroelectrics: Selected Papers from Eleven Years of the Proceedings of the International Symposium on Integrated Ferroelectrics. CRC Press. Figure 2, p. 504. ISBN 9056997041.
^ Solomon Musikant (1991). What Every Engineer Should Know about Ceramics. CRC Press. Figure 3.9, p. 43. ISBN 0824784987.
^ Yasuo Cho (2005). Scanning Nonlinear Dielectric Microscope (in Polar Oxides; R Waser, U Böttger & S Tiedke, editors ed.). Wiley-VCH. Chapter 16. ISBN 3527405321.
^ Simon M. Sze, Kwok K. Ng (2006). Physics of Semiconductor Devices (3rd Edition ed.). Wiley. Figure 25, p. 121. ISBN 0470068302.
^ Gabriele Giuliani, Giovanni Vignale (2005). Quantum Theory of the Electron Liquid. Cambridge University Press. p. 111. ISBN 0521821126.
^ Jørgen Rammer (2007). Quantum Field Theory of Non-equilibrium States. Cambridge University Press. p. 158. ISBN 0521874998.
^ Horst Czichos, Tetsuya Saito, Leslie Smith (2006). Springer Handbook of Materials Measurement Methods. Springer. p. 475. ISBN 3540207856.
^ William Coffey, Yu. P. Kalmykov (2006). Fractals, diffusion and relaxation in disordered complex systems..Part A. Wiley. p. 17. ISBN 0470046074.
^ See, for example, J. Obrzut, A. Anopchenko and R. Nozaki Broadband Permittivity Measurements of High Dielectric Constant Films
^ Dieter K Schroder (2006). Semiconductor Material and Device Characterization (3rd Edition ed.). Wiley. p. 347 ff.. ISBN 0471739065.
^ Dieter K Schroder (2006). Semiconductor Material and Device Characterization (3rd Edition ed.). Wiley. p. 270 ff.. ISBN 0471739065.
^ Simon M. Sze, Kwok K. Ng (2006). Physics of Semiconductor Devices (3rd Edition ed.). Wiley. p. 217. ISBN 0470068302.
^ Safa O. Kasap, Peter Capper (2006). Springer Handbook of Electronic and Photonic Materials. Springer. Figure 20.22, p. 425.
^ PY Yu and Manuel Cardona (2001). Fundamentals of Semiconductors (3rd Edition ed.). Springer. p. §6.6 Modulation Spectroscopy. ISBN 3540254706.
^ Lecture notes; University of New South Wales

[edit]Further reading

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.

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Capacitance

Capacitance

From Wikipedia, the free encyclopedia

Contents

[edit]Capacitance and 'displacement current'

[edit]Gauss's law

[edit]Capacitors

[edit]Voltage dependent capacitors

[edit]Frequency dependent capacitors

[edit]Coefficients of potential

[edit]Capacitance/inductance duality

[edit]Self-capacitance

[edit]Elastance

[edit]Stray capacitance

[edit]Footnotes and in-line references

[edit]Further reading

[edit]External links

[edit]See also