Understanding Voltage Distribution in Series Capacitors

In an electrical circuit, capacitors behave in unique ways when connected in series. Unlike resistors, which share voltage, capacitors in series share charge but not current. This article delves into how the voltages across each capacitor differ based on their capacitances. Learn the fundamental principles, relationships, and an example to solidify your understanding.

Total Voltage Across Series Capacitors

When capacitors are connected in series, the total voltage V_{text{total}} across the series combination is the sum of the voltages across each individual capacitor. This relationship is summarized by the equation:

V_{text{total}} V_1 V_2 V_3 ... V_n

Charges in Series Capacitors

A key feature of series capacitors is that the charge Q on each capacitor is the same. This means that regardless of the individual capacitances, the charge through each capacitor will be identical. The formula for voltage V in terms of capacitance C and charge Q is given by:

V frac{Q}{C}

Voltage Distribution in Series Capacitors

Interestingly, the voltage across each capacitor in a series arrangement is inversely proportional to its capacitance. This means that smaller capacitors, having less capacitance, will have a higher voltage while larger capacitors, having more capacitance, will have a lower voltage. Mathematically, for capacitors with capacitances C_1, C_2, C_3, ..., C_n, the voltages across them are:

V_1 frac{Q}{C_1} quad V_2 frac{Q}{C_2} quad V_3 frac{Q}{C_3} ...

An Example of Capacitors in Series

Let's consider an example where two capacitors are connected in series with capacitances C_1 2 , mutext{F} and C_2 4 , mutext{F}. If the total voltage across this series combination is V_{text{total}} 12 , text{V}, we can calculate the voltage across each capacitor.

Equivalent Capacitance

The equivalent capacitance C_{text{eq}} for series capacitors is calculated using:

frac{1}{C_{text{eq}}} frac{1}{C_1} frac{1}{C_2} quad Rightarrow quad C_{text{eq}} frac{C_1 C_2}{C_1 C_2}

Plugging in the values, we get:

C_{text{eq}} frac{2 cdot 4}{2 4} frac{8}{6} frac{4}{3} , mutext{F}

Charge Across the Capacitors

The charge Q on the equivalent capacitor can be calculated as:

Q C_{text{eq}} cdot V_{text{total}} frac{4}{3} , mutext{F} cdot 12 , text{V} 16 , mutext{C}

Voltages Across Each Capacitor

Using the charge and the individual capacitances, we can now calculate the voltages across each capacitor:

V_1 frac{Q}{C_1} frac{16 , mutext{C}}{2 , mutext{F}} 8 , text{V}

V_2 frac{Q}{C_2} frac{16 , mutext{C}}{4 , mutext{F}} 4 , text{V}

As expected, the voltage across the 2 μF capacitor is 8 V and across the 4 μF capacitor is 4 V, summing up to the total voltage of 12 V.

Conclusion

In a series configuration, the voltage distribution across each capacitor is inversely proportional to its capacitance. This unique behavior of capacitors demonstrates a fundamental principle in electronics and further emphasizes the importance of understanding capacitors in series for design and analysis.