Capacitor

Capacitor is a passive element. capacitance (C) is a property of the capacitor by which it stores electrical energy by means of electrostatic stress.Capacitance is measured in Farad. The charge across the capacitor is expressed by

q=cv                                (20)

As the current is defined by

$\small \mathrm i=\frac{\mathrm{dq}}{\mathrm{dt}}$                                (21)

Substituting (20) in equation (21) gives

$\mathrm i=\mathrm C\frac{\mathrm{dv}}{\mathrm{dt}}$                                (22)

And   $\mathrm v=\frac1{\mathrm C}\int\mathrm{idt}$                                (23)

Equation (22) and (23) are the parametric equations of an capacitor.

Series and parallel connections of capacitors: Two or more capacitors can be connected in series or parallel as shown in figure 1.7.

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Fig 1.7 (a) Capacitors connected in series (b) Capacitors connected in parallel

If ,  and  capacitors  are connected in series as shown in figue 1.7 (a), the equivalent capacitance C is expressed by

$\frac1{\mathrm C}=\frac1{{\mathrm C}_1}+\frac1{{\mathrm C}_2}+...+\frac1{{\mathrm C}_{\mathrm n}}$                                (24)

If two inductors C1 and C2 are connected in series, then the equivalent capacitance is expressed by

$\frac1{\mathrm C}=\frac1{{\mathrm C}_1}+\frac1{{\mathrm C}_2}$
$C=\frac{C_1C_2}{C_1+C_2}$                                (25)

If C1, C2 and Cn inductors are connected in parallel as shown in figue 1.7 (b), the equivalent capacitance C is expressed by

$C=C_1+C_2+...+C_n$                                (26)

Behaviour of Capacitor for AC and DC Sources:

Ideally, the capacitors behaves as an open circuit for the dc and behaves as a short circuit for ac. Generally, it allows ac and opposes dc. Figure 1.8 shows the behaviour of the capacitor at ideal condition.

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Fig 1.8 (a) Capacitor  (b) capacitor behaves as an open circuit for dc (c) capacitor behaves as a short circuit for ac.

The current through a capacitor is represented in time domain by the equation (22). Similarly, in the frequency domain the current is represented by

$i=j\omega Cv$                          (27)

From (27)   $\frac vi=-JX_C$  and  $X_C=\frac1{\omega C}=\frac1{2\pi fC}$        (28)

Where XC is the reactance of the capacitor measured in ohms and f is the frequency in Hz.