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 respectively, 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 geometry (including their distance) of the conductors and the permittivity of the dielectric. For many dielectrics, 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 (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. [1] 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 d q from one plate to the other against the potential difference V = q/C requires the work d W: 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 Main article: 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 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. A qualitative explanation for this can be given as follows. Once a positive charge is put unto a conductor, this charge creates an electrical field, repelling any other positive charge to be moved onto the conductor. I.e. increasing the necessary voltage. But if nearby there is another conductor with a negative charge on it, the electrical field of the positive conductor repelling the second positive charge is weakened (the second positive charge also feels the attracting force of the negative charge). So due to the second conductor with a negative charge, it becomes easier to put a positive charge on the already positive charged first conductor, and vice versa. I.e. the necessary voltage is lowered. As a quantitative example consider the capacitance of a parallel-plate capacitor constructed of two parallel plates both of area A separated by a distance d : where C is the capacitance, in Farads; A is the area of overlap of the two plates, in square meters; εr is the relative static permittivity (sometimes called the dielectric constant) of the material between the plates (for a vacuum, εr = 1); ε0 is the electric constant (ε 0 ≈ 8.854 × 10 −12 F m–1 ); 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 CGS units the equation has the form: [2] 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. [3][4] 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. [5] 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. [6] 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 .[7][8] 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 Gausss law with this form for displacement field:
Posted on: Wed, 24 Dec 2014 20:13:48 +0000
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