Capacitor

A capacitor is an electrical/electronic device that can store energy in the electric field between a pair of conductors (called "plates"). The process of storing energy in the capacitor is known as "charging", and involves electric charges of equal magnitude, but opposite polarity, building up on each plate.

Capacitors are often used in electrical circuit and electronic circuits as energy-storage devices. They can also be used to differentiate between high-frequency and low-frequency signals. This property makes them useful in electronic filters.

Capacitors are occasionally referred to as condensers. This is considered an antiquated term in English, but most other languages use an equivalent, like "Kondensator" in German.

[edit] History

Various types of capacitors. From left: multilayer ceramic, ceramic disc, multilayer polyester film, tubular ceramic, polystyrene, metallized polyester film, aluminium electrolytic. Major scale divisions are cm.

Various Capacitors. The large cylinders are high value electrolytic types

In October 1745, Ewald Georg von Kleist of Pomerania in Germany invented the first recorded capacitor: a glass jar coated inside and out with metal. The inner coating was connected to a rod that passed through the lid and ended in a metal sphere. By having this thin layer of glass insulation (a dielectric) between two large, closely spaced plates, von Kleist found the energy density could be increased dramatically compared with the situation with no insulator.

In January 1746, before Kleist's discovery became widely known, a Dutch physicist Pieter van Musschenbroek independently invented a very similar capacitor. It was named the Leyden jar, after the University of Leyden where van Musschenbroek worked. Daniel Gralath was the first to combine several jars in parallel into a "battery" to increase the total possible stored charge.

The earliest unit of capacitance was the 'jar', equivalent to about 1 nF.

Early capacitors were also known as condensers, a term that is still occasionally used today. It was coined by Alessandro Volta in 1782 (derived from the Italian condensatore), with reference to the device's ability to store a higher density of electric charge than a normal isolated conductor. Most non-English European languages still use a word derived from "condensatore".

Condensers patented by Nikola Tesla in U.S. Patent 567,818, Electrical Condenser, in 1896 on September 15.

[edit] Physics

Diagrammatic

A capacitor consists of two conductive electrodes, or plates, separated by a dielectric.

[edit] Capacitance

The capacitor's capacitance (C) is a measure of the amount of charge (Q) stored on each plate for a given potential difference or voltage (V) which appears between the plates:

$C = {Q \over V}$

In SI units, a capacitor has a capacitance of one farad when one coulomb of charge is stored due to one volt applied potential difference across the plates. Since the farad is a very large unit, values of capacitors are usually expressed in microfarads (µF), nanofarads (nF), or picofarads (pF).

When there is a difference in electric charge between the plates, an electric field is created in the region between the plates that is proportional to the amount of charge that has been moved from one plate to the other. This electric field creates a potential difference V = E·d between the plates of this simple parallel-plate capacitor.

The capacitance is proportional to the surface area of the conducting plate and inversely proportional to the distance between the plates. It is also proportional to the permittivity of the dielectric (that is, non-conducting) substance that separates the plates.

The capacitance of a parallel-plate capacitor is given by:

$C \approx \frac{\varepsilon A}{d}; A \gg d^2$ ^[1]

where ε is the permittivity of the dielectric (see Dielectric constant), A is the area of the plates and d is the spacing between them.

In the diagram, the rotated molecules create an opposing electric field that partially cancels the field created by the plates, a process called dielectric polarization.

[edit] Stored energy

As opposite charges accumulate on the plates of a capacitor due to the separation of charge, a voltage develops across the capacitor due to the electric field of these charges. Ever-increasing work must be done against this ever-increasing electric field as more charge is separated. The energy (measured in joules, in SI) stored in a capacitor is equal to the amount of work required to establish the voltage across the capacitor, and therefore the electric field. The energy stored is given by:

$E_\mathrm{stored} = {1 \over 2} C V^2 = {1 \over 2} {Q^2 \over C} = {1 \over 2} {V Q}$

where V is the voltage across the capacitor.

The maximum energy that can be (safely) stored in a particular capacitor is limited by the maximum electric field that the dielectric can withstand before it breaks down. Therefore, capacitors made with the same dielectric have about the same maximum energy density (joules of energy per cubic meter), if the dielectric volume dominates the total volume.

[edit] Hydraulic model

Main article: Hydraulic analogy

As electrical circuitry can be modeled by fluid flow, a capacitor can be modeled as a chamber with a flexible diaphragm separating the input from the output. As can be determined intuitively as well as mathematically, this provides the correct characteristics:

The pressure difference (voltage difference) across the unit is proportional to the integral of the flow (current)
A steady state current cannot pass through it because the pressure will build up across the diaphragm until it equally opposes the source pressure.
But a transient pulse or alternating current can be transmitted
The capacitance of units connected in parallel is equivalent to the sum of their individual capacitances

[edit] Electrical circuits

The electrons within dielectric molecules are influenced by the electric field, causing the molecules to rotate slightly from their equilibrium positions. The air gap is shown for clarity; in a real capacitor, the dielectric is in direct contact with the plates. Capacitors also allow AC current to flow and block DC current.

[edit] DC sources

The dielectric between the plates is an insulator and blocks the flow of electrons. A steady current through a capacitor deposits electrons on one plate and removes the same quantity of electrons from the other plate. This process is commonly called 'charging' the capacitor. The current through the capacitor results in the separation of electric charge within the capacitor, which develops an electric field between the plates of the capacitor, equivalently, developing a voltage difference between the plates. This voltage V is directly proportional to the amount of charge separated Q. Since the current I through the capacitor is the rate at which charge Q is forced through the capacitor (dQ/dt), this can be expressed mathematically as:

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

where

I is the current flowing in the conventional direction, measured in amperes,

dV/dt is the time derivative of voltage, measured in volts per second, and

C is the capacitance in farads.

For circuits with a constant (DC) voltage source and consisting of only resistors and capacitors, the voltage across the capacitor cannot exceed the voltage of the source. Thus, an equilibrium is reached where the voltage across the capacitor is constant and the current through the capacitor is zero. For this reason, it is commonly said that capacitors block DC.

[edit] AC sources

The current through a capacitor due to an AC source reverses direction periodically. That is, the alternating current alternately charges the plates: first in one direction and then the other. With the exception of the instant that the current changes direction, the capacitor current is non-zero at all times during a cycle. For this reason, it is commonly said that capacitors "pass" AC. However, at no time do electrons actually cross between the plates, unless the dielectric breaks down. Such a situation would involve physical damage to the capacitor and likely to the circuit involved as well.

Since the voltage across a capacitor is proportional to the integral of the current, as shown above, with sine waves in AC or signal circuits this results in a phase difference of 90 degrees, the current leading the voltage phase angle. It can be shown that the AC voltage across the capacitor is in quadrature with the alternating current through the capacitor. That is, the voltage and current are 'out-of-phase' by a quarter cycle. The amplitude of the voltage depends on the amplitude of the current divided by the product of the frequency of the current with the capacitance, C.

[edit] Impedance

The ratio of the phasor voltage across a circuit element to the phasor current through that element is called the impedance $Z$ . For a capacitor, the impedance is given by

$Z_C = \frac{V_C}{I_C} = \frac{-j}{2 \pi f C} = -j X_C ,$

where

$X_C = \frac{1}{\omega C}$ is the capacitive reactance,

$\omega = 2 \pi f \,$ is the angular frequency,

f is the frequency),

C is the capacitance in farads, and

j is the imaginary unit.

While this relation (between the frequency domain voltage and current associated with a capacitor) is always true, the ratio of the time domain voltage and current amplitudes is equal to $X C$ only for sinusoidal (AC) circuits in steady state.

See derivation Deriving capacitor impedance.

Hence, capacitive reactance is the negative imaginary component of impedance. The negative sign indicates that the current leads the voltage by 90° for a sinusoidal signal, as opposed to the inductor, where the current lags the voltage by 90°.

The impedance is analogous to the resistance of a resistor. The impedance of a capacitor is inversely proportional to the frequency -- that is, for very high-frequency alternating currents the reactance approaches zero -- so that a capacitor is nearly a short circuit to a very high frequency AC source. Conversely, for very low frequency alternating currents, the reactance increases without bound so that a capacitor is nearly an open circuit to a very low frequency AC source. This frequency dependent behaviour accounts for most uses of the capacitor (see "Applications", below).

Reactance is so called because the capacitor doesn't dissipate power, but merely stores energy. In electrical circuits, as in mechanics, there are two types of load, resistive and reactive. Resistive loads (analogous to an object sliding on a rough surface) dissipate the energy delivered by the circuit as heat, while reactive loads (analogous to a spring or frictionless moving object) store this energy, ultimately delivering the energy back to the circuit.

Also significant is that the impedance is inversely proportional to the capacitance, unlike resistors and inductors for which impedances are linearly proportional to resistance and inductance respectively. This is why the series and shunt impedance formulae (given below) are the inverse of the resistive case. In series, impedances sum. In parallel, conductances sum.

[edit] Laplace equivalent (s-domain)

When using the Laplace transform in circuit analysis, the capacitive impedance is represented in the s domain by:

$Z(s)=\frac{1}{Cs}$

where C is the capacitance, and s (= σ+jω) is the complex frequency.