Capacitors
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Wikipedia-Article "Capacitors"
- See Capacitor (component) for a discussion of specific types.
A capacitor is a device that stores energy in the electric field created between a pair of conductors on which equal but opposite electric charges have been placed. A capacitor is occasionally referred to using the older term condenser.
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History
In circa 600 BC, Thales of Miletus recorded that the Ancient Greeks could generate sparks by rubbing balls of amber on spindles. This is the triboelectric effect, the mechanical separation of charge in a dielectric. This effect is the basis of the capacitor.
In October 1745, Ewald Georg von Kleist of Pomerania 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 layering the insulator between two metal plates, von Kleist dramatically increased charge density.
Before Kleist's discovery became widely known, a Dutch physicist Pieter van Musschenbroek independently invented a very similar capacitor in January 1746. It was named the Leyden jar, after the University of Leyden where van Musschenbroek worked.
Benjamin Franklin investigated the Leyden jar, and proved that the charge was stored on the glass, not in the water as others had assumed. Originally, the units of capacitance were in 'jars'. A jar is equivalent to about 1 nF.
Early capacitors were also known as condensers, a term that is still occasionally used today. It was coined by 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 languages still use a word derived from "condensatore", like the French condensateur or the German Kondensator.
Physics
Overview
A capacitor consists of two electrodes or plates, each of which stores an opposite charge. These two plates are conductive and are separated by an insulator or dielectric. The charge is stored at the surface of the plates, at the boundary with the dielectric. Because each plate stores an equal but opposite charge, the total charge in the capacitor is always zero.
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:
In SI units, a capacitor has a capacitance of one farad when one coulomb of charge causes a potential difference of one volt 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).
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:
where ε is the permittivity of the dielectric, A is the area of the plates and d is the spacing between them.
Stored energy
As opposite charges accumulate on the plates of a capacitor due to the separation of charge, a voltage develops across the capacitor owing 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:
where V is the voltage across the capacitor.
Hydraulic model
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 across the unit is proportional to the integral of the current, a steady-state current cannot pass through it but a pulse or alternating current can be transmitted, the capacitance of units connected in parallel is equivalent to the sum of their individual capacitances; etc.
In electric circuits
Circuits with DC sources
Electrons cannot directly pass across the dielectric from one plate of the capacitor to the other. When there is a current through a capacitor, electrons accumulate on one plate and electrons are removed from the other plate. This process is commonly called 'charging' the capacitor even though the capacitor is at all times electrically neutral. In fact, the current through the capacitor results in the separation rather than the accumulation of electric charge. This separation of charge causes an electric field to develop between the plates of the capacitor giving rise to voltage across the plates. This voltage V is directly proportional to the amount of charge separated Q. But Q is just the time integral of the current I through the capacitor. This is expressed mathematically as:
where
- I is the current flowing in the conventional direction, measured in amperes
- dV/dt is the time derivative of voltage, measured in volts / second.
- C is the capacitance in farads
For circuits with a constant (DC) voltage source, 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 current.
Circuits with AC sources
The capacitor current due to an AC voltage or current source reverses direction periodically. That is, the AC current alternately charges the plates 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 current. However, at no time do electrons actually cross between the plates.
Since the voltage across a capacitor is 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 AC 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. The ratio of the voltage amplitude to the current amplitude is called the reactance of the capacitor. This capacitive reactance is given by:
where
- ω = 2πf, the angular frequency measured in radians per second
- XC = capacitive reactance, measured in ohms
- f = frequency of AC in hertz
- C = capacitance in farads
and is analogous to the resistance of a resistor. Clearly, the reactance 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.
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 energy that enters them, ultimately by electromagnetic emission (see Black body radiation), while reactive loads (analogous to a spring or frictionless moving object) retain the energy.
The impedance of a capacitor is given by:
where 
.
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°.
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 shunt, conductances sum.
In a tuned circuit such as a radio receiver, the frequency selected is a function of the inductance (L) and the capacitance (C) in series, and is given by:
This is the frequency at which resonance occurs in an RLC series circuit.
For an ideal capacitor, the capacitor current is proportional to the time rate of change of the voltage across the capacitor where the constant of proportionality is the capacitance, C:
The impedance in the frequency domain can be written as
- .
This shows that a capacitor has a high impedance to low-frequency signals (when ω is small) and a low impedance to high-frequency signals (when ω is large). This frequency dependent behaviour accounts for most uses of the capacitor (see "Applications", below).
When using the Laplace transform in circuit analysis, the capacitive impedance is represented in the s domain by:
Capacitors and displacement current
The physicist James Clerk Maxwell invented the concept of displacement current, dD/dt, to make Ampere's law consistent with conservation of charge in cases where charge is accumulating as in a capacitor. He interpreted this as a real motion of charges, even in vacuum, where he supposed that it corresponded to motion of dipole charges in the ether. Although this interpretation has been abandoned, Maxwell's correction to Ampere's law remains valid.
Capacitor networks
A capacitor can be used to block the DC Current flowing within the circuit and therefore have important applications in coupling of ac signals between amplifier stages, whilst preventing dc from passing.
Series or parallel arrangements
Capacitors in a parallel configuration each have the same potential difference (voltage). To find their total equivalent capacitance (Ceq):
The current through capacitors in series stays the same, but the voltage across each capacitor can be different. The sum of the potential differences (voltage) is equal to the total voltage. To find their total capacitance:
In parallel, the total charge stored is the sum of the charge in each capacitor. While in series, the charge on each capacitor is the same.
One possible reason to connect capacitors in series is to increase the overall voltage rating. In practice, a very large resistor might be connected across each capacitor to ensure that the total voltage is divided appropriately for the individual ratings, rather than by minute differences in the capacitance values. Another application is for use of polarized capacitors in alternating current circuits; the capacitors are connected in series, in reverse polarity, so that at any given time one of the capacitors is not conducting.
Capacitor/inductor duality
In mathematical terms, the ideal capacitor can be considered as an inverse of the ideal inductor, because the voltage-current equations of the two devices can be transformed into one another by exchanging the voltage and current terms. Just as two or more inductors can be magnetically coupled to make a transformer, two or more charged conductors can be electrostatically coupled to make a capacitor. The mutual capacitance of two conductors is defined as the current that flows in one when the voltage across the other changes by unit voltage in unit time.
Applications
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Variable Capacitor |
Capacitors have very many uses in electronic and electrical systems.
Energy storage
A capacitor can store electric energy when disconnected from its charging circuit, so it can be used like a temporary battery. The recent commercial availability of very large value capacitors, one farad in size and larger, has enabled such components to allow batteries to be changed in electronic devices without the memory being lost, for instance, or for energy storage for delivery during extreme peak demands, as often found in the enormously powerful car audio systems now seen.
Signal processing
The energy stored in a capacitor can be used to represent information, either in binary form, as in computers, or in analogue form, as in switched-capacitor circuits and bucket-brigade delay lines. Capacitors can be used in analog circuits as components of integrators or more complex filters and in negative feedback loop stabilization. Signal processing circuits also use capacitors to integrate a current signal.
Power supply applications
Capacitors are commonly used in power supplies where they smooth the output of a full or half wave rectifier. They can also be used in charge pump circuits as the energy storage element in the generation of higher voltages than the input voltage. Capacitors are connected in parallel with the power circuits of most electronic devices and larger systems (such as factories) to shunt away and conceal current fluctuations from the primary power source to provide a "clean" power supply for signal or control circuits. Audio equipment, for example, uses several capacitors in this way, to shunt away power line hum before it gets into the signal circuitry. The capacitors act as a local reserve for the DC power source, and bypass AC currents from the power supply.
Capacitors are used in power factor correction. Such capacitors often come as three capacitors connected as a three phase load. Usually, the values of these capacitors are given not in farads but rather as a reactive power in Volt-Amperes reactive (VAr). The purpose is to match the inductive loading of machinery which contains motors, to make the load appear to be mostly resistive.
Capacitors are also used in parallel to interrupt units of a high-voltage circuit breaker in order to distribute the voltage between these units. In this case they are called grading capacitors. In schematic diagrams, a capacitor used primarily for DC charge storage is often drawn vertically in circuit diagrams with the lower, more negative, plate drawn as an arc. The straight plate indicates the positive terminal of the device, if it is polarized (see electrolytic capacitor).
Non-polarized electrolytic capacitors used for signal filtering are typically drawn with two curved plates. Other non-polarized capacitors are drawn with two straight plates.
Tuned circuits
Capacitors and inductors are applied together in tuned circuits to select information in particular frequency bands. For example, radio receivers rely on variable capacitors to tune the station frequency. Speakers use passive analog crossovers, and analog equalizers use capacitors to select different audio bands.
Signal coupling
Because capacitors pass AC but block DC signals (when charged up to the applied dc voltage), they are often used to separate the AC and DC components of a signal. This method is known as AC coupling. (Sometimes transformers are used for the same effect.) Here, a large value of capacitance, whose value need not be accurately controlled, but whose reactance is small at the signal frequency, is employed. Capacitors for this purpose designed to be fitted through a metal panel are called feed-through capacitors, and have a slightly different schematic symbol.
Transducer applications
Although capacitors usually maintian a fixed physical structure and utilization varies the electrical voltage and current, the effects of varying the physical and/or electrical characteristics of the dielectric with a fixed electrical supply can also be of use. Capacitors with an exposed and porous dielectric can be used to measure humidity in air. Capacitors with a flexible plate can be used to measure strain or pressure. Capacitors are used as the transducer in condenser microphones, where one plate is moved by air pressure, relative to the fixed position fo the other plate.
Noise filters, motor starters, and snubbers
When an inductive circuit is opened, the energy stored in the magnetic field of the inductance collapses quickly, creating a large voltage across the open circuit of the switch or relay. If the inductance is large enough, the energy will generate a spark, causing the contact points to oxidize, deteriorate, or sometimes weld together, or destroying a solid-state switch. A snubber capacitor across the newly opened circuit creates a path for this impulse to bypass the contact points, thereby preserving their life; these were commonly found in contact breaker ignition systems, for instance. Similarly, in smaller scale circuits, the spark may not be enough to damage the switch but will still radiate undesirable radio frequency interference (RFI), which a filter capacitor absorbs. Snubber capacitors are usually employed with a low-value resistor in series, to dissipate energy more slowly and minimize RFI. Such resistor-capacitor combinations are available in a single package.
In an inverse fashion, to initiate current quickly through an inductive circuit requires a greater voltage than required to maintain it; in uses such as large motors, this can cause undesirable startup characteristics, and a motor starting capacitor is used to store enough energy to give the current the initial push required to start the motor up.
Weapons applications
An obscure military application of the capacitor is in an EMP weapon. A plastic explosive is used for the dielectric. The capacitor is charged up and the explosive is detonated. The capacitance becomes smaller, but the charge on the plates stays the same. This creates a high-energy electromagnetic shock wave capable of destroying unprotected electronics for miles around. These devices are rumored to have been employed by the US in the 2003 invasion of Iraq, though this is highly unlikely. See Explosively pumped flux compression generator.
Large high-voltage low-inductance capacitors are used as energy sources for the exploding-bridgewire detonators or slapper detonators in nuclear weapons and other specialty weapons, and are also used as power supplies for electromagnetic guns such as railguns or coilguns.
Ideal and nonideal capacitors
In practice, this ideal model of the capacitor often has to be modified to reflect real world capacitor construction and operation. The most obvious example is electrolytic capacitors, where the capacitor is polarized such that when the voltage is connected in reversed fashion, the capacitor acts as a resistor. Similar problems of dielectric leakage are a constant complication of all capacitor design however, and have led to constant improvements in capacitor design, as the material used for dielectrics has changed from oiled paper to mylar and from ceramic to Teflon. This also addresses the related problem of dielectric stability; oiled or electrolyte soaked paper dries out over time, reducing the capacitance and increasing leakage, a problem reduced in modern components.
On the other hand, the requirements of large plate area for reasonably useful capacitor values as well as reasonable packaging resulted in the universal practice of rolling the plate/dielectric sandwich into a cylinder, which was then encapsulated. However, this process also creates an inductance in series with the capacitance, just as introducing a coiled wire of similar characteristics in series with the flat capacitor would; in sensitive circuits, this inductance must be taken into account, either by using a capacitor designed to have lower inductance, or by bypassing a large capacitor with a smaller, noninductive one. This practice has become more common in audiophile-oriented products recently, as inductive problems in low-cost capacitors were demonstrated to degrade high-frequency fidelity.
Dielectric materials can produce unwanted side effects. For example, the dielectric constant of barium titanate used in ceramic capacitors changes with temperature and pressure. Such capacitors are sensitive to vibration and flexing, and can cause a type of signal modulation in electronic circuits call microphonics.
Capacitor hazards and safety
Capacitors may retain a charge long after power is removed from a circuit; this charge can cause shocks (up to and including electrocution) or damage to connected equipment. Since capacitors have such low ESRs, they have the capacity to deliver large currents into short circuits; this can be dangerous. Care must be taken to ensure that any large or high-voltage capacitor is properly discharged before servicing the containing equipment. For safety purposes, all large capacitors should be discharged before handling. F