Solved with COMSOL Multiphysics 4.2a. C A P A C I T A N C E M A T R I X | 1 Capacitance matrix Introduction The capacitance matrix of an electrical system allows us to evaluate cross talk between excitation ports. For example, in Figure 1 we see a three-terminal system in which we can excite one terminal and set the other two to ground. If we repeat this method of exciting one terminal at a time, since there are three terminals, we can evaluate nine possible values of capacitance. The capacitance component C11 is the capacitance evaluated between the grounded terminals and Terminal 1. This can be calculated by exciting Terminal 1. The capacitance between Terminals 1 and 2 would be C21. This can be calculated once we have information about C11 and C22. This means we would need to solve the model once again by exciting Terminal 2. By definition, C21 and C12 would be equal. This means that a three-terminal system will have six unique values of capacitance. The capacitance values, terminal charges and terminal voltages are linked by the following matrix relation: Figure 1: Pictorial representation of a multi-terminal electrical system. Q1 Q2 Q3 C11C12C13 C21C22C23 C31C32C33 V1 V2 V3 = V1, Q1 V2, Q2 V3, Q3 Terminal 1 Terminal 2 Terminal 3 Solved with COMSOL Multiphysics 4.2a. 2 | C A P A C I T A N C E M A T R I X In this tutorial we will find out how to find the capacitance matrix of a three-terminal system. The same idea can be extended to as many terminals as required. This model requires the ACDC Module. The methodology used to evaluate the components of the capacitance matrix is elaborated in the ACDC Module Users Guide. Model Definition In this tutorial we will model a 2D region of air surrounding three metallic terminals. This tutorial will use COMSOLs Electrostatics interface to solve the Poissons equation shown in Equation 1 in order to find the spatial distribution of electric potential in the modeling region. The only material property required to solve this model is the relative permittivity of air. Introduction of additional dielectric materials will automatically affect the capacitance values. (1) We will create geometric layers around the air domain. These layers will be assigned to the Infinite Elements feature. This feature implements the presence of an infinitely extended region hence the model would yield more accurate values of capacitance. The boundary condition for the outer edges of the layers will be set to the default Zero Charge condition which will ensure that the displacement current does not diverge. For detailed information on Infinite Elements, please refer to the ACDC Module Users Guide. In this tutorial you will use the Terminal boundary condition which automatically calculates the capacitance between ground and the excited Terminal. You will also learn to use the Port Sweep functionality which will allow you to sweep the excitation over different terminals, one at a time, in a multi-terminal system. Note that in this tutorial we do not assign a fixed ground in the geometry. The ground automatically floats between the three terminals as a result of the port sweep. However, for most cases it is in general a good practice to assign the electrical ground to appropriate boundaries which represent zero electric potential. Note that in order to calculate the capacitance, it is necessary to know the out-of-plane thickness in a 2D model. In this tutorial we will use the default value of unit thickness (i.e. 1 m). The value of out-of-plane thickness can be altered in the settings of the Electrostatics interface in COMSOL. 0rV()0= Solved with COMSOL Multiphysics 4.2a. C A P A C I T A N C E M A T R I X | 3 Results and Discussion The default plot obtained after solving the model is shown in Figure 2. This plot shows the distribution of electric potential in the modeling region. The default plot represents the case when Terminal 3 is excited. Note how the Infinite Elements feature stretch the solution to what it should be at an infinite distance within the thickness of the geometric layer. Figure 3 shows the case when Terminal 1 is excited. Similarly you can also inspect the voltage distribution when Terminal 2 is excited. Since we only model the region of air around the terminals and assumed that the terminals are at isopotential condition, the solution precludes the isopotential regions inside the terminals. The capacitance matrix evaluated (in nF) for this tutorial problem is shown below. Figure 2: Surface plot of electric potential when Terminal 3 is excited with 1 V. 0,02140,01250,0089 0,01250,02770,0152 0,00890,01520,0242 Solved with COMSOL Multiphysics 4.2a. 4 | C A P A C I T A N C E M A T R I X Figure 3: Surface plot of electric potential when Terminal 1 is excited with 1 V. Model Library path: ACDC_Module/Tutorials/capacitance_matrix Modeling Instructions M O DE L W I Z A RD 1Go to the Mo