Capacitance is the ability of an electrical component, a capacitor, to store an electrical charge. This charge is held by an electric field between two conductive plates separated by an insulating material. The standard unit for measuring capacitance is the Farad (F).
Because the Farad is a very large unit, electronic components typically use smaller subunits. Common units include the microfarad (\(\mu\text{F}\)), which is one-millionth of a Farad, and the picofarad (\(\text{pF}\)). Calculating capacitance requires different formulas depending on whether you are determining the inherent capacity of a single component or the combined capacity of multiple components in a circuit.
Calculating Capacitance for Parallel Plates
The inherent capacitance of a component is determined by its physical design and materials used in its construction. For a parallel-plate capacitor, the capacity is calculated using the formula \(C = \frac{\epsilon A}{d}\).
In this equation, \(A\) represents the surface area of one conductive plate, and \(d\) is the distance separating the two plates. Capacitance increases directly with plate area (\(A\)) but decreases as the distance (\(d\)) increases.
The term \(\epsilon\) (epsilon) is the permittivity of the insulating material, or dielectric, between the plates. Permittivity measures a material’s ability to transmit an electric field. This value is the product of the permittivity of free space (\(\epsilon_0\)), approximately \(8.85 \times 10^{-12}\) Farads per meter (\(\text{F/m}\)), and the dielectric constant (\(\kappa\)) of the specific material.
Calculating Total Capacitance in Series Circuits
When multiple capacitors are connected in a series circuit, the total capacitance of the combination is calculated using a reciprocal relationship. This arrangement increases the effective separation distance between the outermost plates, which reduces the overall capacity of the system. The charge stored on each capacitor in a series connection is identical.
The source voltage is divided among the capacitors in the series, with a smaller capacitor experiencing a larger voltage drop. To find the total capacitance (\(C_{\text{total}}\)), you must first sum the reciprocals of all the individual capacitances (\(C_1, C_2, C_3, \dots\)). The formula for this sum is \(\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots\).
After summing the reciprocals, take the reciprocal of that result to determine \(C_{\text{total}}\). For example, if you have a \(10 \, \mu\text{F}\) capacitor and a \(20 \, \mu\text{F}\) capacitor in series, the reciprocal sum is \(\frac{3}{20}\). Taking the reciprocal gives \(6.67 \, \mu\text{F}\). The total capacitance in a series circuit will always be less than the value of the smallest individual capacitor in the group.
Calculating Total Capacitance in Parallel Circuits
Connecting capacitors in a parallel circuit is a simpler calculation. This configuration is equivalent to creating a single, larger capacitor by adding the plate areas of all individual components together. Since capacitance increases directly with plate area, the total capacity of the system is the straightforward sum of the individual values.
The formula for the total capacitance (\(C_{\text{total}}\)) in a parallel circuit is \(C_{\text{total}} = C_1 + C_2 + C_3 + \dots\). This additive nature means that the resulting capacitance is always greater than any single capacitor in the circuit.
For instance, if a \(10 \, \mu\text{F}\) capacitor and a \(20 \, \mu\text{F}\) capacitor are connected in parallel, the total capacitance is \(30 \, \mu\text{F}\). Every capacitor shares the same voltage across its plates, which is equal to the source voltage. This ability to increase capacitance makes the parallel configuration widely used in many practical applications.
Calculating Capacitance Using Stored Energy
Capacitance can be determined indirectly when the amount of energy stored within the component is known. This method is used when the physical geometry of the capacitor is unknown or when measuring the stored energy is more practical than measuring charge or plate dimensions. The energy stored in a capacitor is a form of electric potential energy, measured in Joules (\(J\)).
The relationship between stored energy (\(E\)), capacitance (\(C\)), and the voltage (\(V\)) across the capacitor is given by the formula \(E = \frac{1}{2} C V^2\). This equation shows that the stored energy is proportional to the capacitance and the square of the voltage.
To calculate capacitance, the formula must be algebraically rearranged to solve for \(C\): \(C = \frac{2E}{V^2}\). For example, if a capacitor stores \(0.06\) Joules of energy when a \(20\) Volt potential difference is applied, the calculation yields a capacitance of \(0.0003\) Farads, or \(300\) microfarads (\(\mu\text{F}\)).