Capacitance Capacitance is the ability of a body to store
an electrical 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 which gives the voltage/current relationship
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, has a
potential difference of 1 volt between its plates. 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 'supercapacitors'). Such capacitors are principally used for energy
storage replacing more traditional batteries. The energy (measured in joules) 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
dq from one plate to the other against the potential difference V = q/C requires the
work dW: where W is the work measured in joules, q
is the charge measured in coulombs and C is the capacitance, measured in farads.
The energy stored in a capacitor is found by integrating this equation. Starting with
an uncharged capacitance (q = 0) and moving charge from one plate to the other until the
plates have charge +Q and −Q requires the work W:
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 subunits of capacitance in use today are the microfarad (µF), nanofarad
(nF), picofarad (pF), and, in microcircuits, femtofarad (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:
where 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 CGS units the equation has the form:
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: 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: 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. Corresponding to the voltage-dependent capacitance,
to charge the capacitor to voltage V an integral relation is found:
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
Integrating: 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. 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. This effect is intentionally exploited in
diode-like devices known as varicaps. 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):
indicating the lag in response by the time dependence of εr, calculated in principle
from an underlying microscopic analysis, for example, of the dipole behavior in the dielectric.
See, for example, linear response function. The integral extends over the entire past
history up to the present time. A Fourier transform in time then results in:
where εr(ω) is now a complex function, with an imaginary part related to absorption of
energy from the field by the medium. See 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: 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: 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, Ccmplx(ω) is the so-called complex capacitance with the dielectric present, and
C0 is the capacitance without the dielectric. (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.)
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. Alternatively, the time response of the capacitance can be
used directly, as in deep-level transient spectroscopy.
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.
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. 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.
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, then the voltage of plate 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 or reciprocal capacitance
matrix, which is defined as: From this, the mutual capacitance between
two objects can be defined by solving for the total charge Q 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.
The collection of coefficients is known as the capacitance matrix, and is the 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). 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: 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, which changes its impedance at high frequencies
and gives rise to parallel resonance, is variously called self-capacitance, stray capacitance,
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
Any two adjacent conductors can be considered a capacitor, though the capacitance is 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), 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 feedback
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 impedance of Z connecting the two
nodes can be replaced with a Z/(1 − k) impedance between the first node and ground
and a KZ/(K − 1) impedance between the second node and ground. Since impedance varies
inversely with capacitance, the internode capacitance, C, is replaced by a capacitance
of KC from input to ground and a capacitance of (K − 1)C/K from output to ground. When
the input-to-output gain is very large, the equivalent input-to-ground impedance is very
small while the output-to-ground impedance is essentially equal to the original (input-to-output)
impedance. Capacitance of simple systems
Calculating the capacitance of a system amounts to solving the Laplace equation ∇2φ = 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. Capacitance of nanoscale systems
The capacitance of nanoscale dielectric capacitors such as quantum dots may differ from conventional
formulations of larger capacitors. In particular, the electrostatic potential difference experienced
by electrons in conventional capacitors is spatially well-defined and fixed by the shape
and size of metallic electrodes in addition to the statistically large number of electrons
present in conventional capacitors. In nanoscale capacitors, however, the electrostatic potentials
experienced by electrons are determined by the number and locations of all electrons
that contribute to the electronic properties of the device. In such devices, the number
of electrons may be very small, however, the resulting spatial distribution of equipotential
surfaces within the device are exceedingly complex.
Single-electron devices The capacitance of a connected, or "closed",
single-electron device is twice the capacitance of an unconnected, or "open", single-electron
device. This fact may be traced more fundamentally to the energy stored in the single-electron
device whose "direct polarization" interaction energy may be equally divided into the interaction
of the electron with the polarized charge on the device itself due to the presence of
the electron and the amount of potential energy required to form the polarized charge on the
device (the interaction of charges in the device's dielectric material with the potential
due to the electron). Few-electron devices
The derivation of a "quantum capacitance" of a few-electron device involves the thermodynamic
chemical potential of an N-particle system given by
whose energy terms may be obtained as solutions of the Schrödinger equation. The definition
of capacitance, with the potential difference
may be applied to the device with the addition or removal of individual electrons,
Then is the "quantum capacitance" of the device.
This expression of "quantum capacitance" may be written as
which differs from the conventional expression described in the introduction where, the stored
electrostatic potential energy, by a factor of 1/2 with.
However, within the framework of purely classical electrostatic interactions, the appearance
of the factor of 1/2 is the result of integration in the conventional formulation,
which is appropriate since for systems involving either many electrons or metallic electrodes,
but in few-electron systems,. The integral generally becomes a summation. One may trivially
combine the expressions of capacitance and electrostatic interaction energy,
respectively, to obtain, which is similar to the quantum capacitance.
A more rigorous derivation is reported in the literature. In particular, to circumvent
the mathematical challenges of the spatially complex equipotential surfaces within the
device, an average electrostatic potential experiences by each electron is utilized in
the derivation. The reason for the apparent mathematical differences
is understood more fundamentally as the energy,, of an isolated device (self-capacitance) is
twice that stored in a "connected" device in the lower limit N=1. As N grows large,.
Thus, the general expression of capacitance is
In nanoscale devices such as quantum dots, the "capacitor" is often an isolated, or partially
isolated, component within the device. The primary differences between nanoscale capacitors
and macroscopic (conventional) capacitors are the number of excess electrons (charge
carrier, or electrons that contribute to the device's electronic behavior) and the shape
and size of metallic electrodes. In nanoscale devices, nanowires consisting of metal atoms
typically do not exhibit the same conductive properties as their macroscopic, or bulk material,
counterparts.