#### Capacitors in Series and Parallel

### Capacitors in Series

Capacitors in series means two or more capacitors are connected in a single line i.e positive plate of the one capacitor is connected to the negative plate of the next capacitor. All the capacitors in series have equal charge (Q) and equal charging current (i_{C}).

Consider N- numbers of capacitors are connected in series, then

Q_{T} = Q_{1} = Q_{2} = Q_{3} = ---------- = Q_{N}

I_{C} = I_{1} = I_{2} = I_{3} = --------- = I_{N}

**Capacitors in a Series Connection:**

The following circuits show the series connection of group of capacitors.

Series connection of N-number of capacitors:

Series connection of 2 capacitors:

In this circuit the charge (Q) stored in all capacitors is same because every capacitor has the charge which is flowing from the adjacent capacitor. The voltage drop in all capacitors is different from each other. But the total voltage drop applied between input and output lines of the circuit is equal to the sum of all the individual voltage drops of each capacitor. The equivalent capacitance of the circuit is C_{eq} = Q/V.

Thus,

V_{T} = V_{1} + V_{2}

C_{eq} = Q/V_{1} + Q/V_{2}

1/C_{eq} = (V_{1}+ V_{2})/Q

V_{T} = Q/C_{eq} = Q/C_{1} + Q/C_{2}

**Series Capacitors Equation**

1/Ceq = 1/C1 + 1/C2 +......... + 1/CN_{}

When the capacitors are in series connection the reciprocal of the equivalent capacitance is equal to the sum of the reciprocals of the individual capacitances of the capacitors in the circuit.

From the figure 2, the reciprocal of equivalent capacitance value of the circuit is equal to the sum of reciprocal capacitances values of two capacitors C_{1} and C_{2}, the expression is given below.

1/C_{eq} = 1/C_{1} + 1/C_{2}

**Capacitors in Series Example No 1:**

We assume that the values of two capacitors in the above figure2 are 0.4uF and 0.5uF respectively. Now we can calculate the equivalent capacitance for the two capacitors in series as,

1/C_{eq} = 1/C_{1} + 1/C_{2}

1/C_{eq} = (C_{1} + C_{2})/ C_{1}C_{2}

C_{eq} = (C_{1}C_{2}) / (C_{1} + C_{2})

C_{eq} = (0.4uF*0.5uF) / (0.4uF+0.5uF)

C_{eq} = 0.22uF

We know that the equivalent capacitance of the capacitors in series connection is less than the value of the smallest capacitor in the circuit. In that way here also we get the equivalent value 0.22uF is less than the smallest capacitance 0.4uF in the series connection of the given two capacitors circuit.

**Capacitors in Series Example No 2**

Calculate the equivalent capacitance and the individual voltage drops across the set of two capacitors in series have 0.1uF and 0.2uF respectively when connected to a 12V a.c. supply.

1/C_{eq} = 1/C_{1} + 1/C_{2}

C_{eq} = (C_{1}C_{2}) / (C_{1} + C_{2})

C_{eq} = (0.1uF*0.2uF) / (0.1uF+0.2uF)

C_{eq} = 0.066uF = 66nF

Voltage drops across the two given capacitors in series are,

V_{1} = (C_{2}*V_{T})/ (C_{1}+C_{2}) = (0.2uF*12V)/ (0.1uF+0.2uF) = 8Volts

V_{2} = (C_{1}*V_{T})/ (C_{1}+C_{2}) = (0.1uF*12V)/ (0.1uF+0.2uF) = 4Volts

From these results we observed that the equivalent capacitance 66nF is less than the smallest capacitance 0.1uF of the given two capacitors. The individual voltage drops across the given two capacitors are different. But the sum of individual voltage drops of both the capacitors is equal to the total voltage. i.e 8V+4V=12V.

Now we calculate the charge stored in individual capacitor,

Q_{1} = V_{1}* C_{1} = 8V * 0.1uF = 0.8uC

Q_{2} = V_{2}* C_{2} = 4V * 0.2uF = 0.8uC

Here we observed that equal charge 0.8uC is stored in both the capacitors C_{1} and C_{2} which are connected in series.

**
Capacitors in Series Summary:
**

The equivalent capacitance of the capacitors connected in series chain is, the reciprocal of the equivalent capacitance is equal to the sum of the reciprocals of all the individual capacitances of the capacitors connected in series. All the capacitors which are connected in series have the equal charge (Q). The charging current (i_{C}) is also same for all the individual capacitors which are connected in series like i_{C}=i_{1}= i_{2} etc.

The voltage drops for individual capacitors in series connection are different. But the sum of all the individual voltage drops is equal to the total voltage across the circuit. i.e. V_{T} = V_{1}+V_{2} etc. A large value of capacitance will result in a smaller voltage drop while a small capacitance value will result in a larger voltage drop.

### Capacitors in Parallel Circuits

Capacitors in parallel means two or more capacitors are connected in parallel way, i.e. both of their terminals are connected to each terminal of the other capacitor or capacitors respectively. All the capacitors which are connected in parallel have the same voltage and is equal to the V_{T} applied between the input and output terminals of the circuit. Then, parallel capacitors have a ‘common voltage’ supply across them. i.e. V_{T} = V_{1} = V_{2} etc.

The equivalent capacitance, C_{eq} of the circuit where the capacitors are connected in parallel is equal to the sum of all the individual capacitances of the capacitors added together. This is because the top plate of each capacitor in the circuit is connected to the top plate of adjacent capacitors. In the same way the bottom plate of each capacitor in the circuit is connected to the bottom plate of adjacent capacitors.

The following circuits show parallel connection between groups of capacitors.

Parallel connection of n number of capacitors:

Parallel connection of 2 capacitors:

In figure 4 the total charge (Q) across the circuit is divided between the two capacitors, means the charge Q distributes itself between the capacitors connected in parallel. Because the voltage drop across individual capacitors is equal and also it is equal to the total voltage applied to the circuit. But the total charge Q is equal to the sum of all the individual capacitor charges connected in parallel. i.e. From the above figure 4 the two different capacitors C1 and C2 have two different charges Q_{1} and Q_{2} respectively. Here Q=Q_{1}+Q_{2}

Now we see the equivalent capacitance of the capacitors C1 and C2 connected in parallel which shown in the above figure 4.

We know the formula,

Q=C_{eq} V_{T}

Here, Q = Q_{1}+Q_{2}

And V_{T} = V_{1} = V_{2}

C_{eq}=Q/V_{T} = (Q_{1}+Q_{2})/V_{T} = (Q_{1}/V_{T}) + (Q_{2}/V_{T})

**Parallel Capacitors Equation:**

C_{eq =} C_{1}+C_{2}+C_{3}+ ------------ +C_{N}

The equivalent capacitance of the capacitors which are connected in parallel is equal to the sum of the individual capacitances of the capacitors in the circuit.

From the figure 4, the equivalent capacitance (C_{eq}) value is equal to the sum of both the capacitance values of C_{1} and C_{2}, the expression is shown below.

Ceq = C1+C2_{}

**Capacitors in Parallel Example No 1:**

Consider the capacitance values of the two capacitors C_{1} =0.2uF and C_{2} =0.3uF which are shown in above figure 4. Now calculate the equivalent capacitance of the circuit.

We know that the Equivalent capacitance

C_{eq} = C_{1} + C_{2}

C_{eq} =0.2uF + 0.3uF

C_{eq} =0.5uF

One important point to remember about parallel connected capacitor circuits, the equivalent capacitance (C_{eq}) of any two or more capacitors connected together in parallel will always be greater than the value of the largest capacitor in the circuit as we are adding together values. So in our example above C_{eq} = 0.5uF whereas the largest value capacitor in the circuit is only 0.3uF.

**Capacitors in Parallel Example No 2:**

Calculate the equivalent capacitance across the set of two capacitors in parallel shown in figure 4.

(a) One capacitor of 0.1uF connected in parallel with 0.2uF.

(b) One capacitor of 750nF connected in parallel with 0.5uF.

(a)Equivalent capacitance,

C_{eq} = C_{1} + C_{2}

_{}C_{eq} =0.1uF + 0.2uF

C_{eq} =0.3uF

(b)Equivalent capacitance,

C_{eq} = C_{1} + C_{2}

C_{eq} =750nF + 0.5uF

C_{eq} =750nF + 500nF

C_{eq =} 1250nF =1.25uF

In the above two cases we observed that the equivalent capacitance (C_{eq}) value of the two capacitors connected together in parallel is greater than the value of the largest capacitor in the circuit as we are adding together values. In section (a) the value of C_{eq} = 0.3uF whereas the largest value capacitor is only 0.2uF. In section (b) the value of C_{eq} = 1.25uF whereas the largest value capacitor in the circuit is 0.5uF only.

**
Capacitors in Parallel Summary
**

The equivalent capacitance of the group of capacitors which are connected in parallel is equal to the sum of capacitances of the individual capacitors. i.e. C_{eq =} C_{1}+C_{2}. Because the voltage drop across the individual capacitor is equal to the total voltage applied between the input and output terminals of the circuit. i.e. V_{T} = V_{1}=V_{2}. The charges stored in individual capacitors are different but the sum of all the individual capacitor charges is equal to the total charge flows in the circuit. i.e. Q = Q_{1}+Q_{2}.

The charging current flow in circuit is distributed to all the capacitors in the circuit. But the total charging current is equal to the sum of all the individual charging currents of the capacitors in the circuit. i.e. i_{C}=i_{1}+i_{2} etc. The equivalent capacitance value of the group of capacitors which are connected in parallel is always greater than the value of the largest capacitor in the circuit.

Capacitors are the standard components in electronic circuits. Different combinations of capacitors are used in circuits practically.This article explains about series and parallel combinations of capacitors.

## Capacitors in series

### How to connect capacitors in Series?

Capacitors in series means two or more capacitors connected in a single line. Positive plate of the one capacitor is connected to the negative plate of the next capacitor.

Here,

QT =Q1 = Q2 = Q3 = ———- = Q

IC = I1 = I2 = I3 = ——— = IN

When the capacitors are connected in series Charge and current is same on all the capacitors.

### Why is the charge of capacitors in series the same?

For series capacitors same quantity of electrons will flow through each capacitor because the charge on each plate is coming from the adjacent plate. So, coulomb charge is same.As current is nothing but flow of electrons , current is also same.

### What is the equivalent capacitance?

Equivalent capacitance is the overall capacitance of the capacitors. Let us see how to calculate the capacitance when they are in series.

Below is the figure showing three capacitors connected in series to the battery. When the capacitors are connected in series the adjacent plates get charged due to electrostatic induction.

Each plate will have different potential. But the magnitude of charge on the plates is same.

First plate of the C1 will have potential V1 which is equal to the voltage of the battery and second plate will have potential less than V1. Let it be V2.

Now the first plate of C2 will have potential equal to V2 and second plate will have potential less than V3 let it be V4.

First plate of C3 will have potential V5 (V5=V4) and the potential of second plate is less than V5. Let it be V6.

But the overall potential difference between the plates is equal to the emf of the battery.

So VT = V1+V2+V3

But we know that, Q=CV

C=Q/V

Ceq = Q/V1 + Q/V2 + Q/V3 (As charge is same)

1/Ceq = (V1+ V2+ V3 )/Q

VT = Q/Ceq = Q/C1 + Q/C2 + Q/C3

Hence,1/Ceq = 1/C1 + 1/C2 + 1/C3

If N capacitors are connected in series then equivalent capacitance can be given as below.

1/Ceq = 1/C1 + 1/C2 +……… + 1/CN

Thus when the capacitors are in series connection, the reciprocal of the equivalent capacitance is equal to the sum of the reciprocals of the individual capacitance of the capacitors in the circuit.

### Capacitors in Series Example

Calculate the equivalent capacitance and the individual voltage drops across the set of two capacitors in series have 0.1uF and 0.2uF respectively when connected to a 12V a.c. supply.

Equivalent capacitance,

1/Ceq = 1/C1 + 1/C2

Ceq = (C1C2) / (C1 + C2)

Ceq = (0.1uF*0.2uF) / (0.1uF+0.2uF)

Ceq = 0.066uF = 66nF

Voltage drops across the two given capacitors in series are,

V1 = (C2*VT)/ (C1+C2) = (0.2uF*12V)/ (0.1uF+0.2uF) = 8Volts

V2 = (C1*VT)/ (C1+C2) = (0.1uF*12V)/ (0.1uF+0.2uF) = 4Volts

From these results we observed that the equivalent capacitance 66nF is less than the smallest capacitance 0.1uF of the given two capacitors. The individual voltage drops across the given two capacitors are different.

But the sum of individual voltage drops of both the capacitors is equal to the total voltage. i.e 8V+4V=12V.

Now we calculate the charge stored in individual capacitor,

Q1 = V1* C1 = 8V * 0.1uF = 0.8uC

Q2 = V2* C2 = 4V * 0.2uF = 0.8uC

Here we observed that equal charge 0.8uC is stored in both the capacitors C1 and C2 which are connected in series.

### Capacitors in Series Summary

• Charge on the capacitors is same in series connection.

• Equivalent capacitance of the capacitors is less than the smallest capacitance in series.

• Equivalent capacitance of n capacitors in series is given as

1/Ceq = 1/C1 + 1/C2 +……… + 1/CN

## Capacitors in Parallel Circuits

### Why do we connect capacitors in parallel?

There is an advantage of connecting capacitors in parallel than in series. When the capacitors are connected in parallel the total capacitance value is increased. There are some applications where higher capacitance values are required.

### How to connect capacitors in Parallel?

Below figure shows the connection of capacitors in parallel. All the positive terminals are connected to one point and negative terminals are connected to another point.

What is Equivalent capacitance of the capacitors in parallel?

- All the capacitors which are connected in parallel have the same voltage and is equal to the VT applied between the input and output terminals of the circuit.
- Then, parallel capacitors have a ‘common voltage’ supply across them .i.e. VT = V1 = V2 etc.
- The equivalent capacitance, Ceq of the circuit where the capacitors are connected in parallel is equal to the sum of all the individual capacitance of the capacitors added together.
- This is because the top plate of each capacitor in the circuit is connected to the top plate of adjacent capacitors.In the same way the bottom plate of each capacitor in the circuit is connected to the bottom plate of adjacent capacitors.

Let us see how to calculate the equivalent capacitance of capacitors when connected in parallel.Consider two capacitors connected as shown in the below circuit.

The total charge (Q) across the circuit is divided between the two capacitors, means the charge Q distributes itself between the capacitors connected in parallel. charge Q is equal to the sum of all the individual capacitor charges.

Thus Q=Q1+Q2

Where Q1,Q2 are charges at capacitors C1 And C2.

We know that,

Q=Ceq VT

Here, Q = Q1+Q2

Ceq VT = C1xV1+C2xV2

Since VT = V1 = V2 = V

Ceq VT = C1xV+C2xV

Ceq VT = (C1+C2)V

Hence Ceq=C1+c2

If N capacitors are connected in parallel then Ceq=C1+C2+C3+——Cn

Thus equivalent capacitance of the capacitors which are connected in parallel is equal to the sum of the individual capacitance of the capacitors in the circuit.

### Capacitors in Parallel Example No1

Consider the capacitance values of the two capacitors C1 =0.2uF and C2 =0.3uF which are shown in above figure 4. Now calculate the equivalent capacitance of the circuit.

We know that the Equivalent capacitance,

Ceq = C1 + C2

Ceq =0.2uF + 0.3uF

Ceq =0.5uF

One important point to remember about parallel connected capacitor circuits, the equivalent capacitance (Ceq) of any two or more capacitors connected together in parallel will always be greater than the value of the largest capacitor in the circuit as we are adding together values. So in our example above Ceq =0.5uF whereas the largest value capacitor in the circuit is only 0.3uF.

### When capacitors are connected in parallel?

Here are some applications where capacitors are connected in parallel.

- In some DC supplies for better filtering small capacitors with superior ripple factor are used. These are connected in parallel to increase the capacitance value.
- This can be used in automotive industries in large vehicles like trams for regenerative braking. These application may require large capacitance values than the capacitance usually available in the market.

### Capacitors in parallel Summary

- Voltage on the capacitors is same when connected in parallel. The equivalent voltage of the parallel capacitors is equal to the smallest voltage rating capacitor in parallel.
- The overall capacitance value of the capacitors is sum of all the capacitance values connected in parallel.
- Equivalent capacitance of n capacitors in parallel is Ceq=C1+C2+C3…Cn.

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