1.7 Capacitance
When a charge is moved to a new position in an electric field, the work that is done is stored in the electric field. If the electric field is confined to a region between two conductors, the conductor geometry is called a capacitor. The capacitance of this geometry is the ratio of stored charge to potential difference.
A volume in space with an electric field stores field energy. This volume has capacitance, although it may not be in a capacitor. It is important to appreciate that nearly all field energy is stored in space. (Remember that in most practical circuits it takes very little E field in the conductors to cause current to flow. This means that there is very little field energy stored in the conductors.) Our hope is that we can confine this energy to very small regions under and between traces and in capacitors. This limits the amount of “loose” energy involved. This is how we control radiation. Conductor geometry defines the shape and extent of the field. If it is confined by conductors, it cannot radiate.
The intensity of the electric field is reduced by the presence of a dielectric. If the charge in a capacitor is held constant and a dielectric is inserted, the voltage must drop. The ratio of charge to voltage (capacitance) thus increases by a factor equal to the relative dielectric constant. The capacitance of two parallel plates with an area A and a spacing h is equal to
where ε 0 is the permittivity of free space equal to 8.85 × 10−12 F/m, ε R is the relative dielectric constant, A is the plate area, and h is the spacing. The unit of capacitance is the farad. This equation assumes that there is no fringing of the field at the edges of the plates. This is a good assumption when the spacing between conductors is very small.
Equation 1.2 states that the ratio of stored charge to voltage is a measure of capacitance.
If the stored charge is one coulomb for a voltage of 1 V then the capacitance is 1 F. In most circuit applications a farad is a very large unit. Typical decoupling capacitors used on a circuit board range from 1000 to 10,000 pF (pF stands for picofarad or 10−12 F).
A fundamental relation exists between current flow, voltage, and capacitance. If the voltage rises at 1 V/s, the current flow in a 1-F capacitor is 1 A. For a capacitor of 1 μF, if the voltage rises at 1 V/s, the current flow is 1 μA. This fact is expressed in Equation 1.5 in terms of the derivative of voltage with respect to time.
1.3 