27 February, 2007

Capacitors in Electrical Networks

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 of electric charge, 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. 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 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, the voltage across the capacitor cannot exceed the voltage of the source. (Unless the circuit includes a switch and an inductor, as in SMPS, or a switch and some diodes, as in a charge pump). 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.

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.

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


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.

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°.


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

Ceq = C1 + C2+ . . . . . . + CnThe reason for putting capacitors in parallel is to increase the total amount of charge stored. In other words, increasing the capacitance also increases the amount of energy that can be stored. Its expression is:

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 effective area of the combined capacitor has increased, increasing the overall capacitance. While in series, the distance between the plates has effectively been increased, reducing the overall capacitance.

In practice capacitors will be placed in series as a means of economically obtaining very high voltage capacitors, for example for smoothing ripples in a high voltage power supply. Three "600 volt maximum" capacitors in series, will increase their overall working voltage to 1800 volts. This is of course offset by the capacitance obtained being only one third of the value of the capacitors used. This can be countered by connecting 3 of these series set-ups in parallel, resulting in a 3x3 matrix of capacitors with the same overall capacitance as an individual capacitor but operable under three times the voltage. In this application, a large resistor would be connected across each capacitor to ensure that the total voltage is divided equally across each capacitor and also to discharge the capacitors for safety when the equipment is not in use.
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.

For more details visit the following sites:
http://www.aplac.hut.fi/courses/bee/exercises.pdf ---- exercise problems on capacitor

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