# Calculation of Capacitance Matrix in Electrical Systems

In typical electrical systems, the capacitance between multiple conductors is of primary interest. To facilitate calculations, it is possible to arrange the mutual capacitances of a system of N conductors in a matrix format. For precise calculations of capacitance matrices in electrical systems, INTEGRATED’s ELECTRO™ AND COULOMB™ programs prove invaluable. Below are a few examples:

For the computation of the capacitance matrix in a twisted multiple cable model and to speed up the solution process, the symmetric and periodicity setup feature in 3D Electric Field Solver COULOMB™ is exceptionally useful. The image below shows four conductors embedded in a dielectric material with εr = 4. Each conductor has a radius of 5mm within a cylindrical shaped cable of radius 20mm (see Image 1).

For a one-meter length of cable, the computation of the capacitance matrix may require approximately 5 to 10 minutes because of the large aspect ratio between the radii of the conductors and their length. To address this issue, we conducted an analysis using a 200mm length cable centrally located at the point (0,0,0). The number of sections used in the Linear Periodic Condition was 5. Around 10,000 2D triangular surface elements were used to solve the model. COULOMB™ efficiently solved the model in just 85 seconds utilizing 118 MB of RAM. Image 2 shows the resulting capacitance matrix for this model.

Similarly, an electrical bushing model was also solved using COULOMB™ with symmetric conditions applied along y=0 and z=0 planes which effectively simplified the model by 4 folds (as seen in Image 3). Capacitance calculations were performed between the Conductor-1 and the Ground resulting in a capacitance value of C11= 3.086301222E-11 (F).

ELECTRO™ applied Boundary Element Method to compute the capacitance matrix in an electric power lines model (as shown in Image 4). The elements in the capacitance matrix are calculated as described below:

The first column of the capacitance matrix is calculated by applying 1V to Conductor 1 and 0V to Conductor 2 and grounding. This yields C11 = Q1/1V = 1.468076963E-11 F and C21= -Q2/1V = -1.831800433E-12 F. Where C21 is called the mutual capacitance between Conductor 1 and Conductor 2. The partial capacitance of Conductor 1 with Ground is the algebraic sum of C11+C21 = 1.468076963E-11 F - 1.831800433E-12 F = 1.28489692E-11 F.

Similarly, the second column of the capacitance matrix is calculated by assigning 1V to the Conductor 2 and 0V to the Conductor 1, while maintaining the Ground at 0V. Since this model is a symmetric model, C22=C11. In this model, there are no dielectric materials. Hence, the Free-space Capacitance matrix is the same as the Capacitance matrix. If dielectric materials were present, ELECTRO™ would recompute the model by replacing the dielectric materials with free-space and calculate the free-space capacitance matrix.