8.2 Capacitors in Series and in Parallel
3 min read•june 24, 2024
in and are key to understanding electrical circuits. Series connections reduce overall , while parallel connections increase it. This impacts how charge and voltage distribute across the network.
Knowing how to calculate helps predict circuit behavior. Series capacitors share charge but have different voltages, while parallel capacitors share voltage but have different charges. This affects energy storage and distribution in the network.
Capacitors in Series and Parallel
Equivalent capacitance in circuits
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- Capacitors connected in series have an equivalent () that is always less than the smallest individual capacitance in the series
- Calculated using the formula:
- Example: Two capacitors with capacitances of 2 µF and 4 µF connected in series have an equivalent capacitance of 1.33 µF
- Capacitors connected in parallel have an equivalent capacitance that is always greater than the largest individual capacitance in the parallel network
- Calculated using the formula:
- Example: Two capacitors with capacitances of 2 µF and 4 µF connected in parallel have an equivalent capacitance of 6 µF
Charge and voltage across capacitors
- Capacitors in series have the same charge () across all capacitors
- Voltage () across each capacitor calculated using the formula: , where is the capacitance of the -th capacitor
- Total voltage across the series network is the sum of voltages across individual capacitors
- Example: Two capacitors with capacitances of 2 µF and 4 µF connected in series with a total charge of 10 µC will have voltages of 5 V and 2.5 V, respectively
- Capacitors in parallel have the same voltage across all capacitors
- Charge on each capacitor calculated using the formula: , where is the capacitance of the -th capacitor
- Total charge in the parallel network is the sum of charges on individual capacitors
- Example: Two capacitors with capacitances of 2 µF and 4 µF connected in parallel with a voltage of 10 V will have charges of 20 µC and 40 µC, respectively
- The across capacitors in parallel is the same, while in series, it varies based on capacitance
Effects of capacitor networks
- Total stored energy () in a capacitor network calculated using the formula: , where is the equivalent capacitance and is the voltage across the network
- For capacitors in series, the total stored energy is less than the sum of the energies that would be stored in each capacitor individually
- For capacitors in parallel, the total stored energy is equal to the sum of the energies stored in each capacitor
- Example: Two capacitors with capacitances of 2 µF and 4 µF connected in series with a voltage of 10 V will have a total stored energy of 33.3 µJ, while the same capacitors connected in parallel will have a total stored energy of 100 µJ
- Charge distribution in capacitor networks
- In series, the charge is the same across all capacitors, but the voltage divides according to the inverse of the capacitance values
- In parallel, the voltage is the same across all capacitors, but the charge distributes according to the capacitance values
- Total charge in a parallel network is the sum of the charges on each capacitor, while in a series network, the total charge is equal to the charge on any individual capacitor
- Example: Three capacitors with capacitances of 1 µF, 2 µF, and 3 µF connected in parallel with a voltage of 12 V will have charges of 12 µC, 24 µC, and 36 µC, respectively, while the same capacitors connected in series with a total charge of 6 µC will have equal charges of 6 µC on each capacitor
- The stored in a capacitor network depends on the arrangement of capacitors and their individual capacitances
Circuit Analysis with Capacitors
- are essential for analyzing complex circuits containing capacitors
- Capacitance affects the behavior of circuits, influencing current flow and voltage distribution
- techniques can be applied to determine equivalent capacitance, charge distribution, and potential differences in capacitor networks
Key Terms to Review (18)
C_eq = C1 + C2 + C3: The equation C_eq = C1 + C2 + C3 represents the equivalent capacitance of capacitors connected in series. In this arrangement, the total capacitance decreases as more capacitors are added, which is contrary to how capacitance behaves in parallel connections. The formula highlights that the reciprocal of the total capacitance in a series circuit is the sum of the reciprocals of the individual capacitances, showcasing the unique relationship between capacitance and circuit configuration.
Capacitance: Capacitance is the ability of a system to store charge per unit voltage. It is measured in farads (F).
Capacitance: Capacitance is a measure of the ability of a capacitor to store electric charge. It is a fundamental quantity in the study of electricity and electronics, and it plays a crucial role in various topics related to electrostatic equilibrium, electric potential, and energy storage.
Capacitors: Capacitors are passive electronic components that store electrical energy in an electric field. They are fundamental building blocks in electronic circuits, used for various purposes such as filtering, timing, and energy storage.
Ceramic Capacitor: A ceramic capacitor is a type of capacitor that uses a ceramic material as the dielectric, or insulating layer, between its two conductive plates. Ceramic capacitors are widely used in electronic circuits due to their small size, high capacitance, and reliability.
Circuit Analysis: Circuit analysis is the process of studying and understanding the behavior of electrical circuits, including the flow of current, the distribution of voltages, and the power dissipation within the circuit. It is a fundamental concept in electrical engineering and physics that helps engineers design, troubleshoot, and optimize electronic systems.
Electric Field Energy: Electric field energy is the potential energy stored in an electric field created by the presence of electric charges. It represents the work required to assemble a distribution of electric charges against the repulsive or attractive electric forces between them, resulting in the creation of an electric field.
Electric potential difference: Electric potential difference is the work done to move a unit charge between two points in an electric field. It is measured in volts (V).
Electric Potential Energy: Electric potential energy is the potential energy possessed by an electric charge due to its position in an electric field. It is the work done by an external force in moving a charge from an infinite distance to a specific location within the electric field.
Electrolytic Capacitor: An electrolytic capacitor is a type of capacitor that uses an electrolytic solution as one of its plates, allowing it to achieve a much higher capacitance per unit volume compared to other capacitor types. This makes it particularly useful in applications where space and size are at a premium.
Equivalent capacitance: Equivalent capacitance is the total capacitance of a combination of capacitors that can be replaced by a single capacitor without changing the overall behavior in a circuit. This concept is essential when analyzing circuits with multiple capacitors, as it simplifies calculations and helps in understanding how capacitors interact within series and parallel configurations.
Kirchhoff's Laws: Kirchhoff's laws are two fundamental principles that describe the conservation of electric charge and energy in electrical circuits. These laws provide a framework for analyzing and understanding the behavior of electric currents and voltages in complex circuits.
Michael Faraday: Michael Faraday was a pioneering scientist known for his groundbreaking work in electromagnetism and electrochemistry during the 19th century. His contributions, particularly in discovering electromagnetic induction and formulating Faraday's Law, laid the foundation for modern electrical engineering and technology.
Parallel: Parallel refers to a relationship between two or more entities that exist or occur side-by-side, with a consistent distance or spacing maintained between them. In the context of electrical circuits, parallel describes a configuration where components are connected to the same set of terminals, allowing for multiple paths for current to flow.
Parallel combination: A parallel combination of capacitors involves connecting multiple capacitors in such a way that each capacitor's terminals are connected to the same pair of nodes. The total capacitance in a parallel combination is the sum of the individual capacitances.
Potential Difference: Potential difference, also known as voltage, is the measure of the work done per unit charge in moving an electric charge between two points in an electric field. It represents the potential energy difference between two locations and is a fundamental concept in the study of electric circuits and the behavior of charged particles.
Series: In electrical circuits, a series configuration refers to a connection where components are arranged one after the other, so the same current flows through each component sequentially. This setup is crucial in understanding how capacitors behave when connected in this manner, as it affects the overall capacitance and voltage distribution across the components. In a series arrangement, the total capacitance decreases, making it an essential concept for analyzing how energy is stored in electrical systems.
Series combination: A series combination of capacitors occurs when multiple capacitors are connected end-to-end, with the same charge passing through each capacitor. The total capacitance in a series combination is less than any individual capacitor's capacitance in the circuit.