- Capacitor Definition: A capacitor is a device that stores energy in an electric field, created by two metal plates separated by a dielectric material.
- Series Capacitance: In a series connection, capacitors decrease the total capacitance, which can be calculated using the formula 1/C = 1/C1 + 1/C2 + … + 1/Cn.
- Parallel Capacitance: In a parallel connection, capacitors increase the total capacitance, calculated by adding their individual capacitances, C = C1 + C2 + … + Cn.
- Charge and Voltage in Series and Parallel: In series, the charge across each capacitor is the same, while in parallel, the voltage across each capacitor is the same.
- Applications of Capacitors: Series and parallel capacitor connections are crucial for achieving specific capacitance values needed in different electronic devices and power systems.
Capacitors in Series
Consider several capacitors in series across an applied voltage of V volts.
Let the capacitance values be C1, C2, C3 through Cn, and let C represent the equivalent capacitance. The individual voltage drops across the capacitors are V1, V2, V3 through Vn.
For an ideal, initially uncharged series string with no leakage or other connection at its intermediate nodes, each capacitor develops the same charge magnitude Q.
The total applied voltage is the sum of the capacitor voltages. Because each voltage is Q divided by its capacitance, a smaller capacitor takes a larger share of the voltage.
Substituting these values into equation (i) gives the reciprocal rule for the series combination.
Capacitors in Parallel
A capacitor stores electrostatic energy in its electric field. At a fixed voltage, increasing capacitance increases the stored energy because the energy is one-half of capacitance multiplied by voltage squared.
A capacitor has two conducting plates separated by an insulating dielectric such as glass, mica or ceramic. The charge accumulates on the plates; the dielectric separates them and polarises in the electric field. When a voltage source is connected across the plates, equal and opposite charges develop. The magnitude of charge q is proportional to voltage V:
Here C is the proportionality constant called capacitance. For an ideal parallel-plate capacitor, it depends on the plate geometry and dielectric permittivity.
Here ε is the permittivity, A is the effective plate area and d is the plate separation.
In a circuit, two or more capacitors are connected in parallel across the same pair of nodes. This electrical connection differs from the parallel plate geometry inside one capacitor. Every branch has the same voltage, so Veq = Va = Vb. The total current ieq divides into branch currents ia and ib. For an ideal capacitor,
Substituting q from equation (1) gives:
When capacitance is constant, its time derivative is zero, so:
Apply Kirchhoff’s Current Law at the incoming node:
The result is:
For n capacitors in parallel, the equivalent capacitance is the sum of the individual capacitances. This rule has the same form as the equivalent resistance of resistors in series.
Deriving the Equivalent Capacitance of Parallel Capacitors
Connect n capacitors in parallel across a voltage source of V volts.
Let the individual capacitances be C1, C2, C3 through Cn, and let C be the equivalent capacitance. Because the capacitors are connected in parallel, they have the same applied voltage. Their charges are not generally equal: each charge is its capacitance multiplied by V, and the total charge is their sum.
Here Q1, Q2, Q3 through Qn are the charges on capacitors C1, C2, C3 through Cn, respectively.
Substituting into equation (2) gives:
