Several capacitors may be connected together in a variety of applications. Multiple connections of capacitors act like a single equivalent capacitor. The total capacitance of this equivalent single capacitor depends both on the individual capacitors and how they are connected. There are two simple and common types of connections, called series and parallel, for which we can easily calculate the total capacitance. Certain more complicated connections can also be related to combinations of series and parallel.

Capacitance in Series

Figure 1a shows a series connection of three capacitors with a voltage applied. As for any capacitor, the capacitance of the combination is related to charge and voltage by [latex]C=\frac{Q}{V}\\[/latex].

Note in Figure 1 that opposite charges of magnitude Q flow to either side of the originally uncharged combination of capacitors when the voltage V is applied. Conservation of charge requires that equal-magnitude charges be created on the plates of the individual capacitors, since charge is only being separated in these originally neutral devices. The end result is that the combination resembles a single capacitor with an effective plate separation greater than that of the individual capacitors alone. (See Figure 1b.) Larger plate separation means smaller capacitance. It is a general feature of series connections of capacitors that the total capacitance is less than any of the individual capacitances.

Figure 1. (a) Capacitors connected in series. The magnitude of the charge on each plate is Q. (b) An equivalent capacitor has a larger plate separation d. Series connections produce a total capacitance that is less than that of any of the individual capacitors.

We can find an expression for the total capacitance by considering the voltage across the individual capacitors shown in Figure 1. Solving [latex]C=\frac{Q}{V}\\[/latex] for V gives [latex]V=\frac{Q}{C}\\[/latex]. The voltages across the individual capacitors are thus [latex]V_1=\frac{Q}{C_1},V_2=\frac{Q}{C_2},\text{ and }V_3=\frac{Q}{C_3}\\[/latex].

The total voltage is the sum of the individual voltages:

V = V₁ + V₂ +V₃.

Now, calling the total capacitance C_S for series capacitance, consider that

[latex]V=\frac{Q}{C_{\text{S}}}=V_1+V_2+V_3\\[/latex].

Entering the expressions for V₁, V₂, and V₃, we get

[latex]\frac{Q}{C_{\text{S}}}=\frac{Q}{C_{1}}+\frac{Q}{C_{2}}+\frac{Q}{C_{3}}\\[/latex].

Canceling the Qs, we obtain the equation for the total capacitance in series C_S to be

[latex]\frac{1}{C_{\text{S}}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}+\dots,\\[/latex]

where “…” indicates that the expression is valid for any number of capacitors connected in series. An expression of this form always results in a total capacitance C_S that is less than any of the individual capacitances C₁, C₂, …, as Example 1 illustrates.

Capacitors in Parallel

Figure 2a shows a parallel connection of three capacitors with a voltage applied. Here the total capacitance is easier to find than in the series case. To find the equivalent total capacitance C_p, we first note that the voltage across each capacitor is V, the same as that of the source, since they are connected directly to it through a conductor. (Conductors are equipotentials, and so the voltage across the capacitors is the same as that across the voltage source.) Thus the capacitors have the same charges on them as they would have if connected individually to the voltage source. The total charge Q is the sum of the individual charges: Q = Q₁ + Q₂ + Q₃.

Figure 2. (a) Capacitors in parallel. Each is connected directly to the voltage source just as if it were all alone, and so the total capacitance in parallel is just the sum of the individual capacitances. (b) The equivalent capacitor has a larger plate area and can therefore hold more charge than the individual capacitors.

Using the relationship Q = CV, we see that the total charge is Q = C_pV, and the individual charges are Q₁ = C₁V, Q₂ = C₂V, and Q₃ = C₃V. Entering these into the previous equation gives

C_pV = C₁V + C₂V + C₃V.

Canceling V from the equation, we obtain the equation for the total capacitance in parallel

C_p: C_p = C₁ + C₂ + C₃ + ….

Total capacitance in parallel is simply the sum of the individual capacitances. (Again the “…” indicates the expression is valid for any number of capacitors connected in parallel.) So, for example, if the capacitors in Example 1 were connected in parallel, their capacitance would be

C_p = 1.000 µF + 5.000 µF + 8.000 µF = 14.000 µF.

The equivalent capacitor for a parallel connection has an effectively larger plate area and, thus, a larger capacitance, as illustrated in Figure 2b.

More complicated connections of capacitors can sometimes be combinations of series and parallel. (See Figure 3.) To find the total capacitance of such combinations, we identify series and parallel parts, compute their capacitances, and then find the total.

Figure 3. (a) This circuit contains both series and parallel connections of capacitors. See Example 2 for the calculation of the overall capacitance of the circuit. (b) C₁ and C₂ are in series; their equivalent capacitance C_S is less than either of them. (c) Note that C_S is in parallel with C₃. The total capacitance is, thus, the sum of C_S and C₃.

Section Summary

• Total capacitance in series [latex]\frac{1}{C_{\text{S}}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}+\dots\\[/latex]
• Total capacitance in parallel C_p = C₁ + C₂ + C₃ + …
• If a circuit contains a combination of capacitors in series and parallel, identify series and parallel parts, compute their capacitances, and then find the total.