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.
Electric circuits
When a capacitor is connected to a current source, charge is transfered between its plates at a rate i(t) = dq(t) / dt. As the voltage between the plates is proportional to the charge, it follows that
.
Conversely, if a capacitor is connected to a voltage source, the resulting displacement current is given by
.
For example, if one were to connect a 1000 µF capacitor to a voltage source, then increase the sourced voltage at a rate of 2.5 Volts per second, the current flowing through the capacitor would be
.
DC sources
A simple resistor-capacitor circuit demonstrates charging of a capacitor.
A circuit containing only a resistor, a capacitor, a switch and a constant (DC) voltage source vsrc(t) = V0 in series is known as a charging circuit. From Kirchhoff's voltage law it follows that
,
where vr(t) and vc(t) are the voltages across the resistor and capacitor respectively. This reduces to a first order differential equation
Assuming that the capacitor is initially uncharged, there is no internal electric field, and the initial current is I0 = V0 / R. This initial condition allows solution of the differential equation as
.
The corresponding voltage drop across the capacitor is
.
Therefore, as charge increases on the capacitor plates, the voltage across the capacitor increases, until it reaches a steady-state value of V0, and the current drops to zero. Both the current, and the difference between the source and capacitor voltage decay exponentially with respect to time. The time constant of the decay is given by τ = RC.
AC sources
When connected to an alternating current (AC) voltage source, the plates on a capacitor repeatedly charge and discharge relative to each other. The current varies sinusoidally, with a nonzero amplitude. For this reason, capacitors effectively conduct AC although charge ideally never passes directly through the dielectric. Since the current is proportional to the time derivative of the voltage, a sinusoidal current leads the voltage by a 90 degree phase shift, or equivalently 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.
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
where is the capacitive reactance, 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 XC 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 does not 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.
Laplace equivalent (s-domain)
When using the Laplace transform in circuit analysis, the capacitive impedance is represented in the s domain by:
where C is the capacitance, and s (= σ+jω) is the complex frequency.
Capacitor types
Main article: capacitor (component)
Practical capacitors are available commercially in many different forms. The type of internal dielectric, the structure of the plates and the device packaging all strongly affect the characteristics of the capacitor, and its applications.
A 12 pF 20 kV fixed vacuum capacitor
Dielectric materials
Most types of capacitor include a dielectric spacer, which increases their capacitance. However, low capacitance devices are available with a vacuum between their plates, which allows extremely high voltage operation and low losses. Air filled variable capacitors are also commonly used in radio tuning circuits.
Several solid dielectrics are available, including paper, plastic, glass, mica and ceramic materials. Paper was used extensively in older devices and offers relatively high voltage performance. However, it is susceptible to water absorption, and has been largely replaced by plastic film capacitors. Plastics offer better stability and aging performance, which makes them useful in timer circuits, although they may be limited to low operating temperatures and frequencies. Ceramic capacitors are generally small, cheap and useful for high frequency applications, although their capacitance varies strongly with voltage, and they age poorly. They are broadly categorized as Class 1 dielectrics, which have predictable variation of capacitance with temperature or Class 2 dielectrics, which can operate at higher voltage. Glass and mica capacitors are extremely reliable, stable and tolerant to high temperatures and voltages, but are too expensive for most mainstream applications.
Capacitor materials. From left: multilayer ceramic, ceramic disc, multilayer polyester film, tubular ceramic, polystyrene, metalized polyester film, aluminum electrolytic. Major scale divisions are in centimetres.
Electrolytic capacitors use an aluminum or tantalum plate with an oxide dielectric layer. The second electrode is a liquid electrolyte. Electrolytic capacitors offer very high capacitance but suffer from poor tolerances, high instability, gradual loss of capacitance especially when subjected to heat, and high leakage current. The conductivity of the electrolyte drops at low temperatures, which increases equivalent series resistance. While widely used for power-supply conditioning, poor high-frequency characteristics make them unsuitable for many applications. Tantalum capacitors offer better frequency and temperature characteristics than aluminum, but higher dielectric absorption and leakage.[3] OS-CON (or OC-CON) capacitors are a polymerized organic semiconductor solid-electrolyte type that offer longer life at higher cost than standard electrolytic capacitors.
Several other types of capacitor are available for specialist applications. Supercapacitors made from carbon aerogel, carbon nanotubes, or highly porous electrode materials offer extremely high capacity and can be used in some applications instead of rechargeable batteries. Alternating current capacitors are specifically designed to work on line (mains) voltage AC power circuits. They are commonly used in electric motor circuits and are often designed to handle large currents, so they tend to be physically large. They are usually ruggedly packaged, often in metal cases that can be easily grounded/earthed. They also tend to have rather high direct current breakdown voltages.
Capacitor packages: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. Major scale divisions are cm.
Structure
Capacitors may have their plates arranged in many configurations, for example axially or radially. Small, cheap discoidal ceramic capacitors have existed since the 1930s, and remain in widespread use. Since the 1980s, surface mount packages for capacitors have been widely used. These packages are extremely small and lack connecting leads, allowing them to be soldered directly onto the surface of printed circuit boards. Surface mount components avoid undesirable high-frequency effects due to the leads and simplify automated assembly, although manual handling is made difficult due to their small size.
Variable capacitors are available in various forms. Mechanically controlled variable capacitors allow the plate spacing to be adjusted, for example by rotating or sliding a set of movable plates into alignment with a set of stationary plates. Very cheap variable capacitors squeeze together alternating layers of aluminum and plastic with a screw, but the resulting capacitance is unstable, and unreproducible. Electrical control of capacitance is achievable with varactors (or varicaps), which are reverse-biased semiconductor diodes whose depletion region width varies with applied voltage. They are used in phase-locked loops, amongst other applications.