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

• Series capacitors are connected end-to-end, with only one path for current to flow through all capacitors
• Equivalent capacitance of series-connected capacitors is always less than the smallest individual capacitance value in the series
  • Calculated using: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$
• 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: $C_{eq} = C_1 + C_2 + ... + C_n$

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 capacitor voltages can be calculated using voltage division formula: $V_{C_i} = V_{total} \times \frac{C_{eq}}{C_i}$
  • Where $V_{C_i}$ is the voltage across the $i$-th capacitor, $V_{total}$ is the total voltage applied to the series, and $C_{eq}$ is the equivalent capacitance of the series
• Sum of individual capacitor voltages in a series always equals the total applied voltage (Kirchhoff's Voltage Law)

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
• Equivalent inductance of series-connected inductors is the sum of all individual inductance values
  • Calculated using: $L_{eq} = L_1 + L_2 + ... + L_n$
• 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: $\frac{1}{L_{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + ... + \frac{1}{L_n}$

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 inductor currents can be calculated using current division formula: $I_{L_i} = I_{total} \times \frac{L_{eq}}{L_i}$
  • Where $I_{L_i}$ is the current through the $i$-th inductor, $I_{total}$ is the total current entering the parallel, and $L_{eq}$ is the equivalent inductance of the parallel
• Sum of individual inductor currents in a parallel always equals the total current entering the parallel (Kirchhoff's Current Law)