Question

drive an expression for the effective capacitance when capacitor are connected (a) series and (b) parallel

Answer 1

For series connected capacitors, the charging current ( iC ) flowing through the capacitors is THE SAME for all capacitors as it only has one path to follow.

Then, Capacitors in Series all have the same current flowing through them as iT = i1 = i2 = i3 etc. Therefore each capacitor will store the same amount of electrical charge, Q on its plates regardless of its capacitance. This is because the charge stored by a plate of any one capacitor must have come from the plate of its adjacent capacitor. Therefore, capacitors connected together in series must have the same charge.

QT = Q1 = Q2 = Q3 ….etc

Consider the following circuit in which the three capacitors, C1, C2 and C3  are all connected together in a series branch across a supply voltage between points A and B.

Capacitors in a Series Connection

In the previous parallel circuit we saw that the total capacitance, CT of the circuit was equal to the sum of all the individual capacitors added together. In a series connected circuit however, the total or equivalent capacitance CT is calculated differently.

In the series circuit above the right hand plate of the first capacitor, C1 is connected to the left hand plate of the second capacitor, C2 whose right hand plate is connected to the left hand plate of the third capacitor, C3. Then this series connection means that in a DC connected circuit, capacitor C2 is effectively isolated from the circuit.

The result of this is that the effective plate area has decreased to the smallest individual capacitance connected in the series chain. Therefore the voltage drop across each capacitor will be different depending upon the values of the individual capacitance’s.

Then by applying Kirchoff’s Voltage Law, ( KVL ) to the above circuit, we get:

Since Q = CV and rearranging for V = Q/C, substituting Q/C for each capacitor voltage VC in the above KVL equation will give us:

dividing each term through by Q gives

Series Capacitors Equation

When adding together Capacitors in Series, the reciprocal ( 1/C ) of the individual capacitors are all added together ( just like resistors in parallel ) instead of the capacitance’s themselves. Then the total value for capacitors in series equals the reciprocal of the sum of the reciprocals of the individual capacitances.