16 Capacitors in Series and Parallel

Capacitors in Series

In many electrical and electronic devices, two or more capacitors are connected together rather than used individually. By combining capacitors, engineers can obtain a desired capacitance, increase the maximum operating voltage of a circuit, or tailor how electrical energy is stored and released. Regardless of how many capacitors are used, the combination behaves like a single equivalent capacitor whose capacitance depends both on the individual capacitors and on how they are connected.

The two most common ways of connecting capacitors are series and parallel. More complicated circuits can usually be analyzed by identifying portions that are connected in series or parallel and replacing them with equivalent capacitors.

Before working with equations, it helps to keep two important physical ideas in mind.

• Capacitance measures how much charge a device stores per volt. A larger capacitance means more charge can be stored for the same applied voltage.
• The wiring determines which quantity is shared. In a series connection, every capacitor stores the same magnitude of charge. In a parallel connection, every capacitor has the same voltage across it.

These two constraints—same charge in series and same voltage in parallel—are the reason the mathematical expressions for equivalent capacitance are different.

Figure 16.1(a) shows three capacitors connected in series to a voltage source. Because the capacitors are connected end-to-end, there is only one path through which charge can be separated. As with any capacitor, the relationship between charge, voltage, and capacitance is

The single-path connection forces every capacitor in the series chain to store the same magnitude of charge.

Notice in Figure 16.1 that equal and opposite charges of magnitude [latex]Q[/latex] appear on every capacitor after the voltage is applied. Conservation of charge explains why this must happen. Suppose a small amount of negative charge is pushed onto the leftmost plate. Because charge cannot cross the insulating gap inside a capacitor, it induces an equal amount of positive charge on the opposite plate. Since adjacent plates are connected by conducting wires that cannot accumulate excess charge, this process continues through the entire chain. As a result, every capacitor ends up storing the same magnitude of charge.

This constraint has a useful physical interpretation. Each capacitor stores charge across a gap between its plates. Connecting capacitors in series is equivalent to placing several gaps one after another, increasing the effective plate separation. Since larger plate separation reduces capacitance, the equivalent capacitance of capacitors connected in series is always smaller than the capacitance of any individual capacitor in the combination.

We can now derive an expression for the equivalent capacitance. Starting from the definition

the voltage across a single capacitor is

Because every capacitor in the series connection stores the same charge, the voltages across the individual capacitors are

The total voltage supplied by the source equals the sum of the individual voltage drops:

Define the equivalent series capacitance, [latex]C_{\mathrm S}[/latex], as the capacitance of a single capacitor that stores the same charge [latex]Q[/latex] at the same total voltage. By definition,

Substituting the individual voltages gives

Since [latex]Q[/latex] appears in every term, it cancels, yielding the equivalent capacitance for any number of capacitors connected in series:

This expression immediately explains why the equivalent capacitance is always smaller than the smallest individual capacitance. Adding positive reciprocals makes [latex]1/C_{\mathrm S}[/latex] larger than any individual reciprocal, which means [latex]C_{\mathrm S}[/latex] must be smaller. Physically, placing capacitors in series increases the effective plate separation, making it more difficult for the combination to store charge at a given voltage.

Capacitors in Parallel

Figure 16.2(a) shows three capacitors connected in parallel to the same voltage source. In a parallel connection, each capacitor is connected directly across the same two nodes (or "rails") of the circuit. Because conductors are equipotential surfaces, the potential difference between those two nodes is the same for every capacitor. Therefore, each capacitor has the same voltage across it, equal to the source voltage V.

This shared voltage is the defining characteristic of a parallel connection. Since every capacitor experiences the same voltage, each stores the amount of charge predicted by

A capacitor with a larger capacitance stores more charge at the same voltage. The battery supplies charge to every branch of the circuit, so the total charge delivered is simply the sum of the charges stored on the individual capacitors:

 

To derive the expression for the equivalent capacitance, write the total charge and the individual charges using

 

[latex]Q=CV[/latex]

 

The equivalent capacitor stores

 

[latex]Q=C_PV[/latex]

 

while the individual capacitors store

 

[latex]Q_1=C_1V,\qquad Q_2=C_2V,\qquad Q_3=C_3V[/latex]

 

Substituting these expressions into the total charge equation gives

 

[latex]C_PV=C_1V+C_2V+C_3V[/latex]

 

Since every capacitor has the same voltage, the common factor V cancels, leaving

 

[latex]C_P=C_1+C_2+C_3+\cdots[/latex]

 

This expression applies to any number of capacitors connected in parallel. Unlike a series combination, the equivalent capacitance is always greater than any individual capacitance. Physically, connecting capacitors in parallel is equivalent to increasing the effective plate area. Since a larger plate area allows more charge to be stored at the same voltage, the equivalent capacitance increases.

 

Total Capacitance in Parallel, [latex]{C}_{\text{p}}[/latex]

 

Total capacitance in parallel: [latex]{C}_{\text{p}}={C}_{1}+{C}_{2}+{C}_{3}+\ldots[/latex]

 

More complicated connections of capacitors can sometimes be combinations of series and parallel. (See Figure 16.3.) To find the total capacitance of such combinations, we identify series and parallel parts, compute their equivalent capacitances step by step, and then combine those equivalents until the entire network reduces to one capacitance.

 

Example 16.2: Finding the Equivalent Capacitance of a Mixed Capacitor Network

Three capacitors are connected as shown in Figure 16.3. The capacitances are

[latex]C_1=1.000\ \mu\text{F},\qquad C_2=5.000\ \mu\text{F},\qquad C_3=8.000\ \mu\text{F}[/latex]

Determine the equivalent capacitance of the entire combination. Report your answer to three significant figures.

Strategy

When analyzing a network containing both series and parallel connections, identify the simplest series or parallel group first and replace it with an equivalent capacitor. Repeat this process until only one equivalent capacitor remains.
In this circuit, C₁ and C₂ are connected in series, so we first replace them with an equivalent capacitance C_S. That equivalent capacitor is then connected in parallel with C₃, allowing us to calculate the total capacitance.

Solution

Since C₁ and C₂ are in series,

[latex]\frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2}[/latex]

Substituting the given values,

[latex]\frac{1}{C_S}=\frac{1}{1.000\ \mu\text{F}}+\frac{1}{5.000\ \mu\text{F}}=\frac{1.200}{\mu\text{F}}[/latex]

Taking the reciprocal,

[latex]C_S=0.833\ \mu\text{F}[/latex]

This equivalent capacitor is now connected in parallel with C₃, so the total capacitance is

[latex]C_{\text{tot}}=C_S+C_3[/latex]

 

[latex]C_{\text{tot}}=0.833\ \mu\text{F}+8.000\ \mu\text{F}=8.833\ \mu\text{F}[/latex]

 

Discussion

Reducing a complex capacitor network step by step is the standard approach for solving these problems. At each stage, replace a group of capacitors that is clearly connected in series or clearly connected in parallel with its equivalent capacitance. Repeating this process eventually reduces the entire network to a single equivalent capacitor.
As a useful check on your work, remember these rules:

• • A series combination always has an equivalent capacitance that is smaller than the smallest capacitor in that series group.

 

• • A parallel combination always has an equivalent capacitance that is greater than the largest capacitor in that parallel group.

These simple observations can help you recognize calculation errors before moving on to the next step.

Section Summary

• For capacitors connected in series, each capacitor stores the same magnitude of charge, while the total voltage is divided among them. The equivalent capacitance is given by
  [latex]\frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots[/latex]

  The equivalent capacitance is always smaller than the smallest individual capacitor.

• For capacitors connected in parallel, each capacitor has the same voltage across it, while the total stored charge is the sum of the individual charges. The equivalent capacitance is
  [latex]C_P=C_1+C_2+C_3+\cdots[/latex]

  The equivalent capacitance is always greater than the largest individual capacitor.

• More complex capacitor networks can be analyzed by identifying groups connected in series or in parallel, replacing each group with its equivalent capacitance, and repeating the process until only a single equivalent capacitor remains.

Conceptual Questions

21. If you wish to store a large amount of energy in a capacitor bank, would you connect capacitors in series or parallel? Explain.

Problems & Exercises

57. Find the total capacitance of the combination of capacitors in Figure 16.4.

    

58. Suppose you want a capacitor bank with a total capacitance of 0.750 F and you possess numerous 1.50 mF capacitors. What is the smallest number you could connect together to achieve your goal, and how would you connect them?
59. What total capacitances can you make by connecting a [latex]5.00\ \mu\text{F}[/latex] capacitor and an [latex]8.00\ \mu\text{F}[/latex] capacitor together?
60. Find the total capacitance of the combination of capacitors shown in Figure 16.5.

    

61. Find the total capacitance of the combination of capacitors shown in Figure 16.6.

    

62. Unreasonable Results (a) An [latex]8.00\ \mu\text{F}[/latex] capacitor is connected in parallel to another capacitor, producing a total capacitance of [latex]5.00\ \mu\text{F}[/latex]. What is the capacitance of the second capacitor? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Glossary

capacitors in series

A set of capacitors connected end-to-end so that each stores the same magnitude of charge while the total voltage is divided among them.

capacitors in parallel

A set of capacitors connected across the same two nodes of a circuit so that each has the same voltage while the total charge is shared among them.

equivalent capacitance

The capacitance of a single capacitor that stores the same amount of charge at the same applied voltage as an entire combination of capacitors.