Simple Capacitors in Series and Parallel Examples and Equations

capacitors in series and parallel examples

This time we will learn about capacitors in series and parallel examples. This thing is crucial for us like learning the series and parallel resistors. We know from resistive circuits that a series-parallel combination is a powerful tool for reducing circuits.

This technique can be extended to series-parallel connections of capacitors, which are sometimes encountered.

Parallel Capacitors Formula

In order to obtain the equivalent capacitor Ceq of N capacitors in parallel, consider the circuit in Figure.(1a).

capacitors in series and parallel examples
Figure 1. (a) Parallel-connected N capacitors, (b) the equivalent circuit for the parallel capacitors.

The equivalent circuit is in Figure.(1b). Note that the capacitors have the same voltage v across them. Applying KCL to Figure.(1a),

capacitors in series and parallel examples
(1)

But ik = Ck dv/dt. Hence,

capacitors in series and parallel examples
(2)

where

capacitors in series and parallel examples
(3)

The equivalent capacitance of N parallel-connected capacitors is the
sum of the individual capacitances.

We observe that capacitors in parallel combine in the same manner as resistors in series.

Series Capacitors Formula

We now obtain Ceq of N capacitors connected in series by comparing the circuit in Figure.(2a) with the equivalent circuit in Figure.(2b).

capacitors in series and parallel examples
Figure 2. (a) Series-connected N capacitors, (b) equivalent circuit for the series capacitor.

Note that the same current i flows (and consequently the same charge) through the capacitors. Applying KVL to the loop in Figure.(2a),

capacitors in series and parallel examples
(4)

But,

capacitors in series and parallel examples

Therefore,

capacitors in series and parallel examples
(5)

where

capacitors in series and parallel examples
(6)

The initial voltage v(t0) across Ceq is required by KVL to be the sum of
the capacitor voltages at t0. Or according to Equation.(5),

capacitors in series and parallel examples

Thus, according to Equation.(6),

The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.

Note that capacitors in series combine in the same manner as resistors in parallel. For N = 2 (i.e., two capacitors in series), Equation.(6) becomes

capacitors in series and parallel examples
(7)

Capacitors in Series and Parallel Examples

1. Find the equivalent capacitance seen between terminals a and b of the circuit in Figure.(3).

capacitors in series and parallel examples
Figure 3

Solution:
The 20- μF and 5- μF capacitors are in series; their equivalent capacitance is

capacitors in series and parallel examples

This 4- μF capacitor is in parallel with the 6- μF and 20- μF capacitors; their combined capacitance is

capacitors in series and parallel examples

This 30- μF capacitor is in series with the 60- μF capacitor. Hence, the equivalent capacitance for the entire circuit is

capacitors in series and parallel examples

 

2. For the circuit in Figure.(4), find the voltage across each capacitor.

capacitors in series and parallel examples
Figure 4

Solution:
We fist fid the equivalent capacitance Ceq, shown in Figure.(5). The two
parallel capacitors in Figure.(4) can be combined to get
40 + 20 = 60 mF.

capacitors in series and parallel examples
Figure 5

This 60-mF capacitor is in series with the 20-mF and 30-mF capacitors. Thus,

capacitors in series and parallel examples

The total charge is

capacitors in series and parallel examplesThis is the charge on the 20-mF and 30-mF capacitors because they are in series with the 30-V source. (A crude way to see this is  to imagine that charge acts like current, since i = dq/dt.) Therefore,

capacitors in series and parallel examples Having determined v1 and v2, we now use KVL to determine v3 by

capacitors in series and parallel examplesAlternatively, since the 40-mF and 20-mF capacitors are in parallel, they have the same voltage v3 and their combined capacitance is
40 + 20 =
60 mF.
This combined capacitance is in series with the 20-mF and 30-mF
capacitors and consequently has the same charge on it. Hence,

capacitors in series and parallel examples