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Diodes

Diodes

A capacitor is a device that stores energy in the electric field created between a pair of conductors on which equal magnitude but opposite sign electric charges have been placed. A capacitor is occasionally referred to using the older term condenser.


SMD capacitors: electrolytic at the bottom line, ceramic above them; through-hole ceramic and electrolytic capacitors at the right side for comparison. Major scale divisions are cm


Various types of capacitors. Major scale divisions are cm

History

About 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", the German "Kondensator", or the Spanish "condensador".

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

Capacitance in a capacitor

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 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 (F), 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:

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

where ? is the permittivity of the dielectric, A is the area of the plates and d is the spacing between them.


When electric charge accumulates on the plates, an electric field is created in the region between the plates that is proportional to the amount of accumulated charge. This electric field creates a potential difference V = Ed between the plates of this simple parallel-plate capacitor.


The electrons within dielectric molecules are influenced by the electric field, causing the molecules to rotate slightly from their equillibrium positions. The air gap is shown for clarity; in a real capacitor, the dielectric is in direct contact with the plates.

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:

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

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.

Capacitors in electric circuits

Circuits with DC sources

Electrons cannot easily pass directly across the dielectric from one plate of the capacitor to the other as the dielectric is carefully chosen so that it is a good insulator. 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:

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.

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, unless the dielectric breaks down or becomes excessively 'leaky' in which case it would probably malfunction, burn out, or even explode.

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.

Impedance

The ratio of the phasor voltage to the phasor current is called the impedance of a capacitor and is given by: Z_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 = input frequency,

C = capacitance in farads, and

j=\sqrt{-1} and 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. Clearly, the impedance 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, ultimately by electromagnetic emission (see Black body radiation), 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 parrallel, conductances sum.

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}{sC}

where C is the capacitance, and s(= ?+jw) is the complex frequency

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

Series or parallel arrangements

Main article: Series and parallel circuits

Capacitors in a parallel configuration each have the same potential difference (voltage). Their total capacitance (Ceq) is given by:

The reason for putting capacitors in parallel is to increase the total amount of charge stored. In other words, increasing the capacitance we also increase the amount of energy that can be stored as its expression is

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

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. Their total capacitance is given by:

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.

What is the reason to put capacitors in series?. We get less capacitance and less charge storage than with either alone (the total voltage is divided between the number of capacitors). It is sometimes done in electronics practice because capacitors have maximum working voltages, and with two "600 volt maximum" capacitors in series, you can increase the working voltage to 1200 volts. Thus, one possible reason to connect capacitors in series is to increase the overall voltage rating.

Also, 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 conductors 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.

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