# Derivation Capacitance - Series and Parallel Connection of Capacitors

Level 3 (with higher mathematics)
Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.
Updated by Alexander Fufaev on

Here we want to derive the total capacitance $$C$$ of a circuit in which two capacitors are connected in series or parallel. One capacitor has the capacitance $$C_1$$ and the other capacitor has the capacitance $$C_2$$. The total capacitance is also called equivalent capacitance because we can replace individual capacitances $$C_1$$ and $$C_2$$ with $$C$$ without changing the functionality of the circuit.

## Total capacitance of a series connection of capacitors

Let us consider a circuit with an alternating voltage. For this we take two capacitors with the capacitances $$C_1$$ and $$C_2$$. We apply an AC voltage $$U(t)$$ to these two capacitors, as shown in Illustration 1. In this way we have constructed a series of capacitors.

Charge amount on the capacitor plates:
Applied AC voltage causes an alternating electric current $$\class{red}{I(t)}$$, which charges and discharges the two capacitors. Since the series circuit has no nodes at which the current could split, the same current $$\class{red}{I}(t)$$ flows through the two capacitors. The current is defined as charge $$\class{red}{Q(t)}$$ per time $$t$$:

Definition of electric current
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At time $$t$$, in both capacitors the amount of charge $$\class{red}{Q(t)} ~=~ \class{red}{I(t)} \, t$$ is stored. Since the current through both capacitors is the same, the amount of charge $$\class{red}{Q(t)}$$ at time $$t$$ is the same on both capacitors.

Voltage on the capacitors:
The applied AC voltage $$U(t)$$ is the total voltage which drops across both capacitors. It is the sum of the voltage $$U_1(t)$$ applied between the electrodes of the first capacitor and the voltage $$U_2(t)$$ applied between the electrodes of the second capacitor:

Total voltage is sum of individual voltages
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We bring the capacitance into play by using the relationship between the charge and the voltage ($$\class{red}{Q} = C\, U$$). For the first and second capacitor we have the following equations:

Equations for charge on the capacitors
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The total capacitance $$C$$ of the series circuit is related to the total voltage $$U(t)$$ in the same way as the individual capacitances in 3:

Total charge is proportional to total voltage
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These equations state that the charge $$\class{red}{Q(t)}$$ on the capacitor plates is proportional to the corresponding voltage between the capacitor plates, where the constant of proportionality is the capacitance. Rearrange both equations 3 and4 for the voltage:

Equations for all voltages
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Now you can replace the voltages in 3 and 4 with those in 5:

Equations for total voltage and charge per total capacitance
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Divide both sides by the charge $$\class{red}{Q(t)}$$ to eliminate it:

Reciprocal total capacitance is reciprocal sum of two individual capacitances
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This equation can be rearranged for $$C$$:

Formula: Equivalent capacitance of two capacitors connected in series
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If you have more than two capacitors, you can repeat all the steps above to derive the following formula:

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Here $$C_n$$ denotes the capacitance of the $$n$$th capacitor.

## Total capacitance of a parallel connection of capacitors

Let's consider a slightly different circuit. Again we take two capacitors with the capacitances $$C_1$$ and $$C_2$$. We apply an AC voltage $$U(t)$$ to these two capacitors, but this time as shown in Illustration 2. In this way, we have constructed a parallel circuit of capacitors.

Current through capacitor:
In a parallel circuit, the total current $$\class{red}{I(t)}$$ splits at the nodes to the capacitors. Now we cannot assume that the current through both capacitors is the same. Therefore, we denote the current through the first capacitor as $$\class{red}{I_1(t)}$$ and through the second capacitor as $$\class{red}{I_2(t)}$$. The total current must of course be the sum of the two individual currents because of charge conservation:

Total current is sum of individual currents
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Charge on the capacitor plates:
We can use the definition of current (as charge per time) in 2 to get an equation for the total charge $$\class{red}{Q(t)}$$:

Total charge per time is sum of individual charges per time
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Divide both sides by the time $$t$$:

Total charge is the sum of individual charges
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In contrast to a series circuit, the charge on the capacitors is different in a parallel circuit.

Now use the relationship $$\class{red}{Q} = C\, U$$ between the charge and the voltage to bring the capacitance into play:

Equations for individual charges
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Replace the charges in 12 with those from 13:

Equation: Total capacitance times total voltage
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Divide both sides by the voltage $$U(t)$$ to eliminate it:

Formula: Total capacitance is the sum of two individual capacitances
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If you have more than two capacitors, you can follow the same steps above to derive the following formula:

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Here $$C_n$$ denotes the capacitance of the $$n$$th capacitor.

Next, let's look at how to derive the total inductance of coils connected in parallel and in series.