Capacitance

Capacitance is the measure of an object’s ability to store electric charge.

Capacitance is the measure of an object’s ability to store electric charge. Any body capable of being charged in any way has a value of capacitance.

The unit of capacitance is known as the Farad (F), which can be adjusted into subunits (the millifarad (mF), for example) for ease of working in practical orders of magnitude. The Farad can be equated to many quotients of units, including JV^-2, WsV^-2, CV^-1, and C2J^-1.

The most common capacitor is known as a parallel-plate capacitor which involves two separate conductor plates separated from one another by a dielectric. Capacitance (C) can be calculated as a function of charge an object can store (q) and potential difference (V) between the two plates:

[latex]\text{C}=\frac {\text{q}}{\text{V}}[/latex]

Ultimately, in such a capacitor, q depends on the surface area (A) of the conductor plates, while V depends on the distance (d) between the plates and the permittivity (ε_r) of the dielectric between them. For a parallel-plate capacitor, this equation can be used to calculate capacitance:

[latex]\text{C}=\epsilon_\text{r} \epsilon_0 \frac {\text{A}}{\text{d}}[/latex]

Where ε₀ is the electric constant. The product of length and height of the plates can be substituted in place of A.

In storing charge, capacitors also store potential energy, which is equal to the work (W) required to charge them. For a capacitor with plates holding charges of +q and -q, this can be calculated:

[latex]\text{W}_{\text{charging}}=\int_0^\text{Q} \! \frac {\text{q}}{\text{C}} \mathrm{\text{d}} \text{q}=\frac {\text{CV}^2}{2}=\text{W}_{\text{stored}}[/latex]

Thus, either through calculus or algebraically (if C and V are known), stored energy (W_stored) can be calculated. In a parallel-plate capacitor, this can be simplified to:

[latex]\text{W}_{\text{stored}}= \frac {\epsilon_\text{r} \epsilon_0 \text{AV}^2}{2\text{d}}[/latex]

Parallel-Plate Capacitor

The parallel-plate capacitor is one that includes two conductor plates, each connected to wires, separated from one another by a thin space.

One of the most commonly used capacitors in industry and in the academic setting is the parallel-plate capacitor. This is a capacitor that includes two conductor plates, each connected to wires, separated from one another by a thin space. Between them can be a vacuum or a dielectric material, but not a conductor.

The purpose of a capacitor is to store charge, and in a parallel-plate capacitor one plate will take on an excess of positive charge while the other becomes more negative.

Assuming the plates extend uniformly over an area of A and hold ± Q charge, their charge density is ±, where ρ=Q/A. Assuming that the dimensions of length and width for the plates are significantly greater than the distance (d) between them, the electric field (E) near the center of the plates can be calculated by:

[latex]\text{E}=\frac {\rho}{\epsilon}[/latex]

Potential (V) between the plates can be calculated from the line integral of the electric field (E):

[latex]\text{V}=\int_0^\text{d} \! \text{E} \mathrm{\text{d}}\text{z}[/latex]

where z is the axis perpendicular to both plates. Through simplification and substitution, this integral can be changed to:

[latex]\text{V}=\frac {\rho \text{d}}{\epsilon}=\frac {\text{Qd}}{\epsilon \text{A}}[/latex]

Given that capacitance is the quotient of charge and potential:

[latex]\text{C}=\frac {\epsilon \text{A}}{\text{d}}[/latex]

Accordingly, capacitance is greatest in devices with high permittivity, large plate area, and minimal separation between the plates.

The maximum energy (U) a capacitor can store can be calculated as a function of U_d, the dielectric strength per distance, as well as capacitor’s voltage (V) at its breakdown limit (the maximum voltage before the dielectric ionizes and no longer operates as an insulator):

[latex]\text{U}=\frac {\text{CV}^2}{2}=\frac {\epsilon \text{A}(\text{U}_\text{dd})^2}{2\text{d}}=\frac {\epsilon \text{AdU}_\text{d}^2}{2}[/latex]

Combinations of Capacitors: Series and Parallel

Like any other form of electrical circuitry device, capacitors can be used in series and/or in parallel within circuits.

Like any other form of electrical circuitry device, capacitors can be used in combination in circuits. These combinations can be in series (in which multiple capacitors can be found along the same path of wire) and in parallel (in which multiple capacitors can be found along different paths of wire).

Capacitors in Series

Like in the case of resistors in parallel, the reciprocal of the circuit’s total capacitance is equal to the sum of the reciprocals of the capacitance of each individual capacitor:

[latex]\frac {1}{\text{C}_{\text{total}}}=\frac {1}{\text{C}_1}+\frac {1}{\text{C}_2}+...+\frac {1}{\text{C}_\text{n}}[/latex]

This can also be expressed as:

[latex]\text{C}_{\text{total}}=\frac{1}{\frac {1}{\text{C}_1}+\frac {1}{\text{C}_2}+...+\frac {1}{\text{C}_\text{n}}}[/latex]

Parallel Capacitors

Total capacitance for a circuit involving several capacitors in parallel (and none in series) can be found by simply summing the individual capacitances of each individual capacitor.

[latex]\text{C}_{\text{total}}=\text{C}_1+\text{C}_2+...+\text{C}_\text{n}[/latex]

Capacitors in Series and in Parallel

It is possible for a circuit to contain capacitors that are both in series and in parallel. To find total capacitance of the circuit, simply break it into segments and solve piecewise.

The circuit shown in (a) contains C₁ and C₂ in series. However, these are both in parallel with C₃. If we find the capacitance for the series including C₁ and C₂, we can treat that total as that from a single capacitor (b). This value can be calculated as approximately equal to 0.83 μF.

With effectively two capacitors left in parallel, we can add their respective capacitances (c) to find the total capacitance for the circuit. This sum is approximately 8.83 μF.

Dieletrics and their Breakdown

Dielectric breakdown is the phenomenon in which a dielectric loses its ability to insulate, and instead becomes a conductor.

Dielectric breakdown (illustrated in ) is the phenomenon in which a dielectric loses its ability to insulate, and instead becomes a conductor. Dielectrics are commonly used either to isolate conductors from a variable external environment (e.g., as coating for electrical wires) or to isolate conductors from one another (e.g., between plates of a parallel-plate capacitor). In all applications, they are selected for their ability to act as insulators. By definition, an insulator is unable to conduct electricity. Under certain conditions, however, a material that is an insulator can become a conductor.

Eventually, exposing any insulator to increasing voltage will result in the insulator becoming conductive. This point (the minimum voltage for the insulator to become a conductor) is known as the breakdown voltage. Breakdown is more of a rough concept than an exact science. A material’s breakdown voltage cannot be precisely defined. As a failure, there is a probabilistic element and thus a dielectric may experience a breakdown at any of a range of voltages. Additionally, the nature of the voltage used to induce breakdown must be considered. Short pulses can be used in stress testing to resemble lightning strikes, as could a continuous applied voltage.

However, for the case of a gas being used as a dielectric, the following equation has been proven to be rather reliable in predicting breakdown voltage (Vb):
[latex]\text{V}_\text{b}=\frac {\text{Bpd}}{\ln {\text{Apd}} - \ln {(\ln {(1+\frac {1}{\gamma_{\text{se}}})})}}[/latex]

where A and B are constants that depend on the surrounding gas, p is the pressure of the surrounding gas, d is distance between the electrodes (in cm) and γ_se is the secondary electron emission coefficient. Gaseous dielectrics commonly experience breakdown in nature (the phenomenon of lightning is the most common example).