4.4:
Equivalent Capacitance
Consider a complex capacitor network with three groups of capacitors connected in parallel, and these groups are then connected in series with each other. Determine the equivalent capacitance between the terminals of this intricate circuit.
To solve this, the group at the bottom of the circuit, consisting of four capacitors connected in parallel, is analyzed first. The sum of their capacitance gives an equivalent capacitance, represented by a single capacitor C3.
Next, the middle group, having three capacitors in parallel, is considered. Again, adding up their capacitances gives the equivalent capacitance C2.
Finally, consider the top group, which has two capacitors in parallel. The equivalent capacitance is calculated and represented with one capacitor, C1.
Now, the complex circuit has been simplified into three capacitors connected in series.
To determine the equivalent capacitance for this series combination, the sum of the reciprocals of each capacitance is taken, resulting in the inverse of the equivalent capacitance.
Upon rearranging the terms, the equivalent capacitance of the entire circuit between the terminals is obtained.
4.4:
Equivalent Capacitance
From the study of resistive circuits, it is understood that employing a series-parallel combination serves as an effective strategy for simplifying circuits. Capacitors can be arranged within a circuit in one of two ways: a series configuration or a parallel configuration. The way these capacitors are connected to a battery will influence both the potential drop across each individual capacitor and the size of the charge that each capacitor can store. This is determined by the specific type of connection in place. To simplify this scenario, the combination of capacitors can be substituted with a single equivalent capacitor. This equivalent capacitor is able to store an identical amount of charge as the original combination when subjected to the same potential difference.
When there is a parallel connection of N capacitors, they all share the same voltage across them. The equivalent capacitance for such a configuration is given by
It is crucial to note that the equivalent capacitance of N capacitors connected in parallel equals the total of their individual capacitances.
Transitioning now to the scenario where N capacitors are interconnected in a series, it is observed that the same current i, and consequently the same charge, flows through all the capacitors. The equivalent capacitance of this setup is represented as
In contrast to the parallel configuration, the equivalent capacitance of capacitors connected in series is calculated as the reciprocal of the sum of the reciprocals of each capacitor's individual capacitance.