Capacitor in series and parallel combination: The capacitor connection

To achieve the desired capacitance for your application, you can connect it different combinations like a series and a parallel combination of a capacitor. Just like resistor combination, the resultant capacitance is bigger or smaller depending upon the connection.

capacitors in series :

Capacitors are called to be connected in series if the there is only one path for the flow of current. OR, If the head of the second capacitor is connected to the tail of first, it is called series combination as shown in the following circuit.

Apply the KVL to the following circuit will give us

$E=v_{1}+v_{2}+v_{3} \quad \dots(1)$

Where we know that

$v=\frac{Q}{C}$

By putting it to the equation (1), we get

$
\frac{Q_{T}}{C_{T}}=\frac{Q_{1}}{C_{1}}+\frac{Q_{2}}{C_{2}}+\frac{Q_{3}}{C_{3}}
\quad \dots(2)$

There is only one path for current to flow, so the current for series circuit same i.e.

$i_{T}=i_{1}=i_{2}=\ldots =i_{n}$

Multiply the above equation by time t will give us

$i_{T}t=i_{1}t=i_{2}t=\ldots =i_{n}t$

Where, we know from the basic current definition, that

$it=Q$

So

$Q_{T}=Q_{1}=Q_{2}=\ldots =Q_{n} \quad \dots (3)$

Now put it to the equation (2), so we get

$\frac{Q}{C_{T}}=\frac{Q}{C_{1}}+\frac{Q}{C_{2}}+\frac{Q}{C_{3}}$

Dividing the equation by Q

$\frac{1}{C_{T}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}} \quad \dots (4)$

capacitors in series summary :

  • Provide only one path for charging and discharging current.
  • Head of the second capacitor is connected to the tail of the first capacitor.
  • The charge of all the capacitor connected in series is the same.
  • Adding more capacitors in series will reduce the resultant capacitance.
  • The voltage across each capacitor is different.

capacitor in series and parallel calculator :

Connection type?

First Capacitor

Second Capacitor

Third Capacitor

Resultant Capacitance CT

capacitors in parallel :

If the heads of multiple capacitors are connected together, such combination is called parallel combination. Just like resistor parallel combination, it provides multiple paths for flow of current. The voltage across each capacitor is the same for each voltage source and the total charge is the summation of all individual charge of capacitors.

As the voltage of each capacitor is the same, so mathematically

$I_{T}=i_{1}+i_{2}+i_{3}\ldots +i_{n} \quad \dots (5)$

By the definition of current, the current is

$i=\frac{Q}{t}$

The equation (5) will become

$\frac{Q_{T}}{t}=\frac{Q_{1}}{t}+\frac{Q_{2}}{t}+\ldots
+\frac{Q_{3}}{t} \quad \dots (6)$

Suppose the whole parallel combination of capacitor is connected for specific duration of time, t. Now dividing the equation (6) by t, we will get

$Q_{T}=Q_{1}+Q_{2}+\ldots +Q_{n} \quad \dotsĀ (7)$

We know that charge of capacitor depends upon the capacitance and voltage, as follow

$Q=CV$

Put it to the equation (7), to get

$C_{T}E=C_{1}V_{1}+C_{2}V_{2}+\ldots +C_{n}V_{n} \quad \dots (8)$

As the connection is parallel and we know that voltage across each component is will be same as source voltage i.e.

$E=v_{1}+v_{2}+v_{3}\ldots +v_{n} $

Now put it to the equation (8), and we get

$C_{T}v=C_{1}v+C_{2}v+\ldots +C_{n}v -(9) $

Finally, dividing the equation (9) by the voltage v to get the total capacitance of parallel combination i.e.

$C_{T}=C_{1}+C_{2}+\ldots +C_{n}$

capacitors in parallel summary :

  • Provides multiple paths for charging and discharging current.
  • Heads and tails of all capacitors are connected together.
  • The charge of each capacitor depends upon the capacitance and may be different.
  • Adding more capacitors in series will increase the resultant capacitance.
  • The voltage across each capacitor remains the same.
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