Capacitors are devices that can store electric charge and energy in an electric circuit. They consist of two conductive plates separated by a thin layer of insulating material, such as air, paper, or ceramic. The amount of charge that a capacitor can store per unit of voltage is called its capacitance, and it is measured in farads (F).
Capacitors can be connected in different ways to form a network of capacitors, which can have different effects on the overall capacitance and voltage distribution.
We will focus on two common types of connections: series and parallel. We will also learn how to calculate the equivalent capacitance between any two points in a network, using the example of points A and B in Figure 1.
Series Connection
A series connection of capacitors is when the capacitors are arranged in a row, such that the same current flows through each capacitor. Figure 2 shows an example of three capacitors connected in series.
The equivalent capacitance of a series connection of capacitors is given by the formula:
This formula can be derived by applying Kirchhoff’s voltage law, which states that the sum of the voltage drops across each capacitor must equal the total voltage applied to the series connection. The voltage drop across each capacitor is proportional to the charge stored on it, and inversely proportional to its capacitance. Therefore, we can write:
where Q is the charge on each capacitor, which is the same for all capacitors in series. Solving for Q, we get:
The equivalent capacitance of the series connection is defined as the ratio of the charge to the voltage, so we have:
This expression can be simplified by taking the reciprocal of both sides, and using the fact that the sum of the reciprocals of the capacitors is equal to the reciprocal of the equivalent capacitance. Thus, we obtain the formula given above.
The formula for the equivalent capacitance of a series connection shows that the equivalent capacitance is always smaller than the smallest individual capacitance in the series.
This is because the series connection increases the effective distance between the plates of the equivalent capacitor, which reduces the capacitance. The series connection also divides the total voltage among the capacitors, so that each capacitor has a smaller voltage drop than the total voltage.
Parallel Connection
A parallel connection of capacitors is when the capacitors are arranged side by side, such that the same voltage is applied across each capacitor. Figure 3 shows an example of three capacitors connected in parallel.
The equivalent capacitance of a parallel connection of capacitors is given by the formula:
This formula can be derived by applying Kirchhoff’s current law, which states that the sum of the currents entering and leaving a node must be zero. The current flowing through each capacitor is proportional to the rate of change of the charge stored on it, and proportional to its capacitance. Therefore, we can write:
where I is the total current entering or leaving the parallel connection, and V is the voltage across each capacitor, which is the same for all capacitors in parallel. Since V is constant, we can take it out of the derivatives, and get:
The equivalent capacitance of the parallel connection is defined as the ratio of the charge to the voltage, so we have:
This expression can be simplified by integrating both sides with respect to time, and using the fact that the sum of the charges on the capacitors is equal to the charge on the equivalent capacitor. Thus, we obtain the formula given above.
The formula for the equivalent capacitance of a parallel connection shows that the equivalent capacitance is always larger than the largest individual capacitance in the parallel.
This is because the parallel connection increases the effective area of the plates of the equivalent capacitor, which increases the capacitance.
The parallel connection also adds the charges stored on each capacitor, so that the equivalent capacitor has a larger charge than any of the individual capacitors.
Equivalent Capacitance between Points A and B
To find the equivalent capacitance between any two points in a network of capacitors, we can use the following steps:
- Identify the capacitors that are directly connected to the points of interest, and label them as C1, C2, C3, etc.
- Identify the type of connection (series or parallel) between each pair of adjacent capacitors, and apply the corresponding formula to find the equivalent capacitance of the pair.
- Repeat step 2 until only one equivalent capacitance remains, which is the equivalent capacitance between the points of interest.
Let us apply these steps to the network of capacitors shown in Figure 1, and find the equivalent capacitance between points A and B.
- The capacitors that are directly connected to points A and B are C1, C2, C3, and C4. We label them as shown in Figure 4.
- The type of connection between C1 and C2 is parallel, so we apply the formula for parallel connection and find the equivalent capacitance of C1 and C2:
The type of connection between C3 and C4 is series, so we apply the formula for series connection and find the equivalent capacitance of C3 and C4:
We can redraw the network of capacitors as shown in Figure 5, replacing C1 and C2 with C12, and C3 and C4 with C34.
- The type of connection between C12 and C34 is series, so we apply the formula for series connection and find the equivalent capacitance of C12 and C34:
This is the equivalent capacitance between points A and B, which is the final answer.
Conclusion
We learned how to determine the equivalent capacitance of capacitors in series and parallel combinations, and how to apply this knowledge to find the equivalent capacitance between any two points in a network of capacitors.
We also learned that the equivalent capacitance of a series connection is always smaller than the smallest individual capacitance in the series, and the equivalent capacitance of a parallel connection is always larger than the largest individual capacitance in the parallel. We hope that this article was helpful and informative, and that you enjoyed reading it. Thank you for your attention.
