Independently of the capacitor shape the capacitance is defined as the ratio between the charge Q on the electrode and the voltage U between the electrodes.
The charge stored on electrodes of surface A separated by a dielectric material of permittivity ε , is given by the surface integral of the scalar product of the electric displacement D with the surface vector of the surface element dA
CapacitanceThe potential between the electrodes is given by the line integral of E along the arc s:
which leads to the expression of the capacitance
Capacitance determinationThere are several methods used to determine the capacitance of a system:
2)Exact analytical method
ApproximationIn most of industrial capacitor the dielectric thickness d is much smaller than the electrode surface. In this case it's sufficient to perform the calculation with the approximation of flat parallel electrodes.
In this approximation the capacitance doesn't depend on the electric field E.
Exact analytic methodFrom Laplace law
The capacitance is calculated with
Example: spherical capacitorThe potential of a point charge is given by
The capacitance is given by:
Where re is the radius of the external electrode, ri the radius of the internal electrode and ε the dielectric permittivity of the material which between the 2 spheres.
Conform transformationAn interesting method for calculating the capacitance is to use the conform transformation between complex surfaces. The 2 dimensional Laplace equation is used as transformation function
Example: cylindrical capacitor
The capacitance per unit of length is given by
Example: conductive planar bands
The thickness of the bands is supposed to be very thin. The calculation requires elliptic functions.The solution for the capacitance per unit of length is given by:
In the next geometry the general solution is the same. The approximation differs and gives the following result:
Rounding the electrode hedgesThe sharper the electrode hedges are, the larger is the electric field close to the hedge. It's often at this place that partial discharges or breakdowns are initiating.
Given an electrode thickness a, a rounding radius ρ and a distance between the electrode d, two cases must be distinguished:
Case 1: the electrode thickness 2a << d the distance between the electrodes
E is the electric field in the center of the capacitor where it is homogeneous,
E' is the electric field close to the hedge,
a is the electrode thickness,
d is the distance between the electrode,
ρ is the rounding radius.
Case 2: the inverse situation where 2a > d
Geometry: windingA winding is a spiral made of 2 electrode bands and 2 dielectric systems. Each of these dielectric system is composed of one or a plurality of different dielectric films which can be (paper, polypropylene (PP), polyethylene terephtalate (PET), ….).
S is surface of the active part of the electrode band. S = electrode length L x electrode width W.
Geometry II: Ceramic capacitor and feed through capacitance