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Derivation Capacitance - series and parallel connection of capacitors

Parallel connection of two capacitors - currents
Level 3 (for advanced students)
Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.
Updated by Alexander Fufaev on
Table of contents
  1. Total capacitance of a series connection of capacitors Here we derive a formula for the total capacitance (equivalent capacitance) when capacitors are connected in series.
  2. Total capacitance of a parallel connection of capacitors Hier leiten wir eine Formel für die Gesamtkapazität (Ersatzkapazität) her, wenn Kondensatoren parallel geschaltet sind.

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

Illustration : Series connection of two 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:

Total capacitance of a series circuit
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Here \(C_n\) denotes the capacitance of the \(n\)th capacitor.

Total capacitance of a parallel connection of capacitors

Illustration : Parallel connection of two 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:

Total capacitance of a parallel circuit
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Here \(C_n\) denotes the capacitance of the \(n\)th capacitor.