Capacitance

Capacitance is the ability of an object to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are two closely related notions of capacitance: self capacitance and mutual capacitance. An object that can be electrically charged exhibits self capacitance, for which the electric potential is measured between the object and ground. Mutual capacitance is measured between two components, and is particularly important in the operation of the capacitor, an elementary linear electronic component designed to add capacitance to an electric circuit.
The capacitance between two conductors depends only on the geometry; the opposing surface area of the conductors and the distance between them; and the permittivity of any dielectric material between them. For many dielectric materials, the permittivity, and thus the capacitance, is independent of the potential difference between the conductors and the total charge on them.
The SI unit of capacitance is the farad, named after the English physicist Michael Faraday. A 1 farad capacitor, when charged with 1 coulomb of electrical charge, has a potential difference of 1 volt between its plates. The reciprocal of capacitance is called elastance.

Self capacitance

In discussing electrical circuits, the term capacitance is usually a shorthand for the mutual capacitance between two adjacent conductors, such as the two plates of a capacitor. However, every isolated conductor also exhibits capacitance, here called self capacitance. It is measured by the amount of electric charge that must be added to an isolated conductor to raise its electric potential by one unit of measurement, e.g., one volt. The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, with the conductor centered inside this sphere.
Self capacitance of a conductor is defined by the ratio of charge and electric potential:
where

is the charge held,
is the electric potential,
is the surface charge density,
is an infinitesimal element of area on the surface of the conductor, over which the surface charge density is integrated,
is the length from to a fixed point M on the conductor,
is the vacuum permittivity.

Using this method, the self capacitance of a conducting sphere of radius in free space is:
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.

The inter-winding capacitance of a coil is sometimes called self capacitance, but this is a different phenomenon. It is actually mutual capacitance between the individual turns of the coil and is a form of stray or parasitic capacitance. This self capacitance is an important consideration at high frequencies: it changes the impedance of the coil and gives rise to parallel resonance. In many applications this is an undesirable effect and sets an upper frequency limit for the correct operation of the circuit.

Mutual capacitance

A common form is a parallel-plate capacitor, which consists of two conductive plates insulated from each other, usually sandwiching a dielectric material. In a parallel plate capacitor, capacitance is very nearly proportional to the surface area of the conductor plates and inversely proportional to the separation distance between the plates.
If the charges on the plates are and, and gives the voltage between the plates, then the capacitance is given by
which gives the voltage/current relationship
where is the instantaneous rate of change of voltage, and is the instantaneous rate of change of the capacitance. For most applications, the change in capacitance over time is negligible, so the formula reduces to:
The energy stored in a capacitor is found by integrating the work :

Capacitance matrix

The discussion above is limited to the case of two conducting plates, although of arbitrary size and shape. The definition does not apply when there are more than two charged plates, or when the net charge on the two plates is non-zero. To handle this case, James Clerk Maxwell introduced his coefficients of potential. If three conductors are given charges, then the voltage at conductor 1 is given by
and similarly for the other voltages. Hermann von Helmholtz and Sir William Thomson showed that the coefficients of potential are symmetric, so that, etc. Thus the system can be described by a collection of coefficients known as the elastance matrix, which is defined as:
Similarly, the charge can be written in terms of voltages :
The collection of coefficients is known as the capacitance matrix,.
In open systems which are not charge neutral, so that field lines can end at infinity, the capacitance and elastance matrices are inverses of each other:. In closed systems however the capacitance matrix is singular, and so formally the elastance matrix as the inverse of the capacitance matrix is ill defined; it would require independently varying the charges.
2×2 case — From this, the mutual capacitance between two objects can be defined by solving for the total charge and using.
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.

Capacitors

The capacitance of the majority of capacitors used in electronic circuits is generally several orders of magnitude smaller than the farad. The most common units of capacitance are the microfarad, nanofarad, picofarad, and, in microcircuits, femtofarad. Some applications also use supercapacitors that can be much larger, as much as hundreds of farads, and parasitic capacitive elements can be less than a femtofarad. Historical texts use other, obsolete submultiples of the farad, such as "mf" and "mfd" for microfarad ; "mmf", "mmfd", "pfd", "μμF" for picofarad.
The capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. Capacitance is proportional to the area of overlap and inversely proportional to the separation between conducting sheets. The closer the sheets are to each other, the greater the capacitance.
An example is the capacitance of a capacitor constructed of two parallel plates both of area separated by a distance. If is sufficiently small with respect to the smallest chord of, there holds, to a high level of accuracy:
where

is the capacitance, in farads;
is the area of overlap of the two plates, in square meters;
is the electric constant
is the relative permittivity ; and
is the separation between the plates, in meters.

The equation is a good approximation if d is small compared to the other dimensions of the plates so that the electric field in the capacitor area is uniform, and the so-called fringing field around the periphery provides only a small contribution to the capacitance.
Combining the equation for capacitance with the above equation for the energy stored in a capacitor, for a flat-plate capacitor the energy stored is:
where is the energy, in joules; is the capacitance, in farads; and is the voltage, in volts.

Stray capacitance

Any two adjacent conductors can function as a capacitor, though the capacitance is small unless the conductors are close together for long distances or over a large area. This capacitance is called parasitic or stray capacitance. Stray capacitance can allow signals to leak between otherwise isolated circuits, and it can be a limiting factor for proper functioning of circuits at high frequency.
Stray capacitance between the input and output in amplifier circuits can be troublesome because it can form a path for feedback, which can cause instability and parasitic oscillation in the amplifier. It is often convenient for analytical purposes to replace this capacitance with a combination of one input-to-ground capacitance and one output-to-ground capacitance; the original configuration – including the input-to-output capacitance – is often referred to as a pi-configuration. Miller's theorem can be used to effect this replacement: it states that, if the gain ratio of two nodes is, then an impedance of Z connecting the two nodes can be replaced with a impedance between the first node and ground and a impedance between the second node and ground. Since impedance varies inversely with capacitance, the internode capacitance, C, is replaced by a capacitance of KC from input to ground and a capacitance of 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 impedance.

Capacitance of conductors with simple shapes

Calculating the capacitance of a system amounts to solving the Laplace equation with a constant potential on the 2-dimensional surface of the conductors embedded in 3-space. This is simplified by symmetries. There is no solution in terms of elementary functions in more complicated cases.
For plane situations, analytic functions may be used to map different geometries to each other. See also Schwarz–Christoffel mapping.

Type	Capacitance	Diagram and definitions
Parallel-plate capacitor		125px : Permittivity
Concentric cylinders		130px : Permittivity
Eccentric cylinders		130px : Permittivity : Outer radius : Inner radius : Distance between center : Wire length
Pair of parallel wires		130px
Wire parallel to wall		: Wire radius : Distance, : Wire length
Two parallel coplanar strips		: Distance : Length : Strip width : Complete elliptic integral of the first kind
Concentric spheres		97px : Permittivity
Two spheres, equal radius		: Radius : Distance, : Euler's constant : the q-digamma function : the q-gamma function See also Basic hypergeometric series.
Sphere in front of wall		: Radius : Distance,
Sphere		: Radius
Circular disc		: Radius
Thin straight wire, finite length		: Wire radius : Length