Series and Parallel Combination of Capacitors


Sometimes we need a particular value Capacitor. Suppose the Capacitor value we need is not available, but have different Capacitors of different value available. In this case we arrange the available Capacitors in different ways to get the particular value. There are two ways of combination of Capacitors to get the required value i.e. series combination and parallel combination.


Series Combination of Capacitors. The capacitors are said to be connected in series when negative plate of one capacitor is connected to the positive plated of another capacitor. The capacitors connected in series have equal charge displacement. The capacitors connected in series is shown below in figure (1)
Applying Kirchhoff’s Voltage law to the circuit shown in figure (1)
V = V1 + V2 + V3                              (1)
Where V is the supply voltage and V1, V2, and V3 are voltage drops across capacitor having capacitances C1, C2 and C3 respectively




Putting the value of V1, V2, and Vin equation (1)





Now Consider the equivalent circuit of the figure (1) as shown in figure (2)
         




V = Q / Ceq                    (6)   








Putting the value of “V” from equation (6) into equation (5). Then



Where Ceq is the equivalent capacitance of capacitors having capacitance C1, C2, and C3 Connected in series as shown in the figure (1).
If we have “n” Capacitors connected in series having capacitances C1, C2, C3 and soon up-to Cn
Then 




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Parallel combination of Capacitors: - The capacitors are said to be connected in parallel when positive plate of each capacitor is to each other and negative plated of each capacitor is connected to each other but the charge through each capacitor is different. The voltage across each capacitor is same. The capacitor connected in parallel are shown in figure (3).
Applying Kirchhoff’s current law at junction “a”
          Q = Q1 + Q2 + Q3                   (8)
We know
          Q = CV
Similarly
          Q1 = C1V                                 (9)
          Q2 = C2V                                 (10)
          Q3 = C3V                                 (11)

Where Q1, Q2 and Q3 are charge through the capacitors having capacitance C1, C2 and C3 respectively.
    Now putting the value of Q1, Q2 and Q3 from equation (9), (10) and (11) into equation (8).
Then
          Q = C1V + C2V + C3V
          Q = (C1 + C2 + C3) V              (12)
Now consider the equivalent circuit of the figure (3) shown in figure (4)
         
 



Q = CeqV                                 (13)

                                                                                                          


Where Ceq is the equivalent capacitance of capacitors C1, C2 and C3
Putting the value of “Q” from equation (13) into equation (12)
Then
          CeqV = (C1 + C2 + C3) V
         
Ceq = C1 + C2 + C3
If we have “n” Capacitors connected in series having capacitances C1, C2, C3 and soon up-to Cn
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