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Capacitors in Series and Parallel
OpenStax College
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Abstract
- Derive expressions for total capacitance in series and in parallel.
- Identify series and parallel parts in the combination of connection of capacitors.
- Calculate the eective capacitance in series and parallel given individual capacitances. Several capacitors may be connected together in a variety of applications. Multiple connections of capac- itors 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.
1 Capacitance in Series
Figure 1(a) 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 C = QV. Note in Figure 1 that opposite charges of magnitude Q ow to either side of the originally uncharged com- bination 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 eec- tive plate separation greater than that of the individual capacitors alone. (See Figure 1(b).) 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.
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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 nd an expression for the total capacitance by considering the voltage across the individual capacitors shown in Figure 1. Solving C = QV for V gives V = QC. The voltages across the individual
2 Capacitors in Parallel
Figure 2(a) shows a parallel connection of three capacitors with a voltage applied. Here the total capacitance is easier to nd than in the series case. To nd the equivalent total capacitance Cp, we rst 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 = Q 1 + Q 2 + Q 3. (8)
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 = CpV , and the individual charges are Q 1 = C 1 V , Q 2 = C 2 V , and Q 3 = C 3 V. Entering these into the previous equation gives
CpV = C 1 V + C 2 V + C 3 V. (9) Canceling V from the equation, we obtain the equation for the total capacitance in parallel Cp:
Cp = C 1 + C 2 + C 3 + .... (10)
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 the example above were connected in parallel, their capacitance would be
Cp = 1.000 μF + 5.000 μF + 8.000 μF = 14. 000 μF. (11) The equivalent capacitor for a parallel connection has an eectively larger plate area and, thus, a larger capacitance, as illustrated in Figure 2(b).
: Total capacitance in parallel Cp = C 1 + C 2 + C 3 + ...
More complicated connections of capacitors can sometimes be combinations of series and parallel. (See Figure 3.) To nd the total capacitance of such combinations, we identify series and parallel parts, compute their capacitances, and then nd the total.
Figure 3: (a) This circuit contains both series and parallel connections of capacitors. See Example 2 (A Mixture of Series and Parallel Capacitance) for the calculation of the overall capacitance of the circuit. (b) C 1 and C 2 are in series; their equivalent capacitance CS is less than either of them. (c) Note that CS is in parallel with C 3. The total capacitance is, thus, the sum of CS and C 3.
Example 2: A Mixture of Series and Parallel Capacitance Find the total capacitance of the combination of capacitors shown in Figure 3. Assume the capacitances in Figure 3 are known to three decimal places ( C 1 = 1. 000 μF , C 2 = 3. 000 μF , and C 3 = 8. 000 μF ), and round your answer to three decimal places. Strategy To nd the total capacitance, we rst identify which capacitors are in series and which are in parallel. Capacitors C 1 and C 2 are in series. Their combination, labeled CS in the gure, is in parallel with C 3. Solution Since C 1 and C 2 are in series, their total capacitance is given by (^) C^1 S = (^) C^11 + (^) C^12 + (^) C^13. Entering their values into the equation gives
1 CS
C 1
C 2
1 .000 μF
5 .000 μF
μF
Figure 4: A combination of series and parallel connections of capacitors.
Exercise 3 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 hook together to achieve your goal, and how would you connect them?
Exercise 4 (Solution on p. 9.) What total capacitances can you make by connecting a 5 .00 μF and an 8 .00 μF capacitor together?
Exercise 5 (Solution on p. 9.) Find the total capacitance of the combination of capacitors shown in Figure 5.
Figure 5: A combination of series and parallel connections of capacitors.
Exercise 6 Find the total capacitance of the combination of capacitors shown in Figure 6.
Figure 6: A combination of series and parallel connections of capacitors.
Exercise 7 (Solution on p. 9.) Unreasonable Results (a) An 8 .00 μF capacitor is connected in parallel to another capacitor, producing a total ca- pacitance of 5 .00 μF. What is the capacitance of the second capacitor? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
