Glossary

6.3 Series and parallel combinations

2 min read•august 6, 2024

Capacitors and inductors can be combined in series or parallel, changing how they behave in circuits. Understanding these combinations is key to analyzing and designing electrical systems effectively.

Series connections reduce overall capacitance but increase inductance. Parallel connections do the opposite. These principles help engineers control voltage and current distribution in complex circuits.

Series and Parallel Capacitors

Calculating Equivalent Capacitance in Series and Parallel Circuits

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  • Series capacitors are connected end-to-end, with only one path for current to flow through all capacitors
  • of series-connected capacitors is always less than the smallest individual capacitance value in the series
    • Calculated using: 1Ceq=1C1+1C2+...+1Cn\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}
  • Parallel capacitors are connected with common nodes, allowing multiple paths for current to flow
  • Equivalent capacitance of parallel-connected capacitors is the sum of all individual capacitance values
    • Calculated using: Ceq=C1+C2+...+CnC_{eq} = C_1 + C_2 + ... + C_n

Voltage Division in Series Capacitor Circuits

  • Voltage divides across series-connected capacitors inversely proportional to their capacitance values
    • Larger capacitance results in a smaller voltage drop, while smaller capacitance results in a larger voltage drop
  • Individual voltages can be calculated using formula: VCi=Vtotal×CeqCiV_{C_i} = V_{total} \times \frac{C_{eq}}{C_i}
    • Where VCiV_{C_i} is the voltage across the ii-th capacitor, VtotalV_{total} is the total voltage applied to the series, and CeqC_{eq} is the equivalent capacitance of the series
  • Sum of individual capacitor voltages in a series always equals the total applied voltage ()

Series and Parallel Inductors

Calculating Equivalent Inductance in Series and Parallel Circuits

  • Series inductors are connected end-to-end, with only one path for current to flow through all inductors
  • of series-connected inductors is the sum of all individual inductance values
    • Calculated using: Leq=L1+L2+...+LnL_{eq} = L_1 + L_2 + ... + L_n
  • Parallel inductors are connected with common nodes, allowing multiple paths for current to flow
  • Equivalent inductance of parallel-connected inductors is always less than the smallest individual inductance value in the parallel
    • Calculated using: 1Leq=1L1+1L2+...+1Ln\frac{1}{L_{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + ... + \frac{1}{L_n}

Current Division in Parallel Inductor Circuits

  • Current divides among parallel-connected inductors inversely proportional to their inductance values
    • Larger inductance results in a smaller current, while smaller inductance results in a larger current
  • Individual currents can be calculated using formula: ILi=Itotal×LeqLiI_{L_i} = I_{total} \times \frac{L_{eq}}{L_i}
    • Where ILiI_{L_i} is the current through the ii-th inductor, ItotalI_{total} is the total current entering the parallel, and LeqL_{eq} is the equivalent inductance of the parallel
  • Sum of individual inductor currents in a parallel always equals the total current entering the parallel ()

Key Terms to Review (12)

Capacitor: A capacitor is an electronic component that stores electrical energy in an electric field, created by a pair of conductive plates separated by an insulating material known as a dielectric. Capacitors play a vital role in electrical circuits by influencing charge, current, voltage, and power dynamics, as well as energy storage and transfer, making them essential for various applications like filtering and timing.
Current Division: Current division is a principle in electrical engineering that describes how current splits across multiple parallel branches in a circuit. This concept is critical when analyzing circuits with components arranged in parallel, allowing for the calculation of current through each branch based on its resistance values. Understanding current division helps engineers design circuits more effectively by predicting how electrical energy will distribute among components.
Equivalent Capacitance: Equivalent capacitance refers to a single capacitance value that can replace a combination of capacitors in a circuit without changing the overall effect on voltage and charge. This concept is crucial for simplifying complex circuits into more manageable forms, allowing for easier calculations of total stored energy, voltage drops, and current distribution. The way capacitors are connected—either in series or parallel—affects how the equivalent capacitance is calculated, impacting circuit behavior and performance.
Equivalent Inductance: Equivalent inductance is a single inductance value that represents the combined effect of multiple inductors connected in a circuit, either in series or parallel. It simplifies circuit analysis by allowing you to replace complex combinations of inductors with a single, equivalent inductor. Understanding how to calculate equivalent inductance is essential for analyzing circuits and predicting how they will behave under different conditions.
Inductor: An inductor is a passive electrical component that stores energy in a magnetic field when electric current flows through it. It plays a crucial role in various electrical circuits by influencing the behavior of current and voltage, especially in response to changes over time, making it essential in understanding circuits that involve both AC and DC signals.
Kirchhoff's Current Law: Kirchhoff's Current Law (KCL) states that the total current entering a junction in an electrical circuit must equal the total current leaving that junction. This principle reflects the conservation of electric charge, ensuring that charge does not accumulate at any point within the circuit. KCL is fundamental in analyzing circuits, as it allows for the determination of unknown currents and plays a crucial role in various methods of circuit analysis.
Kirchhoff's Voltage Law: Kirchhoff's Voltage Law (KVL) states that the sum of the electrical potential differences (voltages) around any closed loop in a circuit is equal to zero. This principle is essential for analyzing circuits, as it helps determine voltage drops across components, ensuring energy conservation within electrical systems.
Parallel Capacitor: A parallel capacitor refers to a configuration where two or more capacitors are connected in parallel within a circuit, allowing them to share the same voltage across their terminals. This arrangement results in an increased overall capacitance, as the total capacitance is the sum of the individual capacitances. When capacitors are connected in parallel, they can store more charge together, which is useful in many electronic applications where energy storage and smoothing of voltage fluctuations are important.
Parallel Inductor: A parallel inductor is a configuration where two or more inductors are connected across the same voltage source, sharing the same voltage while allowing for a combined effect on inductance. This setup influences the total inductance of the circuit, as the inductors work together to store energy in a magnetic field when current flows through them. Understanding how parallel inductors behave is crucial for analyzing circuits that involve multiple inductive components and their interactions.
Series Capacitor: A series capacitor is a capacitor that is connected in a circuit such that the current flows through it in series with other components. This arrangement affects the overall impedance of the circuit and can influence the voltage across individual capacitors as well as the total capacitance of the combination. When capacitors are connected in series, the total capacitance decreases, which is essential in circuits where specific capacitance values are needed for filtering or tuning.
Series Inductor: A series inductor is an electrical component that stores energy in a magnetic field when electrical current flows through it, and is connected in series with other components in a circuit. This configuration influences the overall inductance, total impedance, and voltage across the components, which plays a critical role in how circuits behave under alternating current (AC) conditions and affects the performance of inductive devices in various applications.
Voltage Division: Voltage division is a fundamental principle in electrical engineering that explains how voltage is distributed across components in a series circuit. When resistors are connected in series, the total voltage across the circuit is divided among them proportional to their resistance values. This concept is crucial for analyzing circuits, especially when determining the voltage across individual components.
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