Electrical GENIUS: CAPACITOR

A capacitor is a passive element designed to store energy in its electric field. It generally consists of two conducting plates separated by an insulator (or dielectric).

WHAT HAPPENS IN A CAPACITOR?

When a voltage source ‘v’ is connected to a capacitor, the source deposits positive charge q on one plate and a negative charge –q on the other plate. The capacitor is said to store the electric charge. The amount of charge stored is given by

\[q=Cv\]

Where, C – constant of proportionality

v – Voltage applied

The unit of capacitance is farad (F), in honor of English physicist Michael Faraday (1791-1867). For a parallel plate capacitor, capacitance is given by

\[C=\in \frac{A}{d}\]

Where, A- surface area of the plates

d- Distance between the plates

$\in $- Permittivity of the material

TYPES OF CAPACITORS –

Capacitors are of two types based on their functioning:-

Fixed
Variable

The symbols for fixed and variable type capacitors are shown in the figure below.

CIRCUIT SYMBOLS FOR FIXED AND VARIABLE CAPACITOR

The figure below shows common types of fixed value capacitors. Polyester capacitors are light in weight, stable and their change with temperature is predictable. Other dielectrics such as mica or polystyrene may be used. Film capacitors are rolled and housed in metal or plastic films. Electric capacitors produce very high capacitances.

FIXED CAPACITORS:- (a) Polyester (b) Ceramic (c) Electrolytic

The figure below shows some common types of variable capacitors. The capacitance of a trimmer capacitor is varied by turning the screw. It is often placed in parallel with another capacitor so that equivalent capacitance can be varied slightly. The capacitance of the variable capacitor is varied by turning the shaft.

VARIABLE CAPACITORS:- (a) Trimmer (b) Filmtrim

USES OF CAPACITOR –

Besides resistors, capacitors are the most common electrical components. They are used extensively in electronics, communication, computers, and power systems. They are used in tuning circuits and as dynamic memory elements in computer.

Capacitors are also used to block dc, pass ac, shift phase, start motors and suppress noise.

ENERGY STORED IN A CAPACITOR –

We know that, $q=Cv$

Differentiating this, we get $\frac{dq}{dt}=C\frac{dv}{dt}$ or $i=C\frac{dv}{dt}$ (Since, $i=\frac{dq}{dt}$ )

Integrating this, $v=\frac{1}{C}\int\limits_{-\infty }^{t}{idt}$

\[v=\frac{1}{C}\int\limits_{{{t}_{0}}}^{t}{idt+v({{t}_{0}})}\]

V(t₀) is the voltage across the capacitor at time t₀. This shows that the capacitor voltage depends on the past history of the capacitor current.

Instantaneous power delivered to the capacitor is $p=vi=Cv\frac{dv}{dt}$

Therefore, energy stored is

\[w=\int\limits_{-\infty }^{t}{pdt}=C\int\limits_{-\infty }^{t}{v\frac{dv}{dt}dt}=C\int\limits_{-\infty }^{t}{vdv}=\frac{1}{2}C{{v}^{2}}\]

V ($-\infty $) =0, because the capacitor was not charged at t=$-\infty $

Thus,

\[w=\frac{1}{2}C{{v}^{2}}\text{ or }w=\frac{{{q}^{2}}}{2C}\]

PROPERTIES OF A CAPACITOR –

When the voltage across a capacitor is not changing with time (i.e. dc voltage), the current through the capacitor is zero.A capacitor is an open circuit to dc. With dc voltage connected across a capacitor, it charges.
The voltage on the capacitor must be continuous. The voltage on the capacitor cannot change abruptly because an abrupt change would mean an infinite current, which is physically impossible. Conversely, the current through a capacitor can change instantaneously.

VOLTAGE ACROSS A CAPACITOR (a) possible (b) not possible
An ideal capacitor does not dissipate energy.
A real, non-ideal capacitor has a parallel-model leakage resistance as shown in figure. The leakage resistance may be as high as 100M$\Omega $ and can be neglected for most practical applications.

CIRCUIT MODEL OF A NON-IDEAL CAPACITOR