A key parameter from an electrostatic solution is capacitance. For multiple conductor systems, this involves extracting self and mutual capacitance terms so that equivalent circuit lumped capacitors can be defined for use in circuit simulators. The CMATRIX command macro has been developed to extract self and mutual capacitance terms for multiple conductor systems. See the Mechanical APDL Theory Reference for more details.
Finite element simulation can readily compute and extract a "Ground" capacitance matrix of capacitance values that relate the charge on one conductor with the conductor's voltage drop (to ground). Figure 13.2: Three Conductor System illustrates a three-conductor system (one conductor is ground). The following two equations relate charges on electrodes 1 and 2, Q1 and Q2, with the voltage drops for the electrodes, U1 and U2:
Q1 = (Cg)11 (U1) + (Cg)12 (U2)
Q2 = (Cg)12 (U1) + (Cg)22 (U2)
where Cg represents a matrix of capacitances referred to as "ground capacitances". These ground capacitances do not represent lumped capacitances typically used in a circuit simulator because they do not relate the capacitances between conductors. However, the CMATRIX command macro can convert the ground capacitance matrix to a lumped capacitor matrix which is suitable for use in circuit simulators. Figure 13.3: Lumped Capacitor Equivalence of Three Conductor System illustrates the lumped capacitances between the conductors. The following two equations then relate the charges with the voltage drops:
Q1 = (Cl)11 (U1) + (Cl)12 (U1 - U2)
Q2 = (Cl)12 (U1 - U2) + (Cl)22 (U2)
where Cl represents a matrix of capacitances referred to as "lumped capacitances".
The CMATRIX command macro will perform multiple simulations and extract both the ground capacitance matrix values and the lumped capacitance matrix values. To prepare for CMATRIX, you must group the conductor nodes into node components. Do not apply any loads to the model (voltages, charge, charge density, etc). The component name applied to the conductor nodes must contain a common prefix, followed by a numerical suffix progressing from 1 to the highest numbered conductor in the system. The last numbered conductor in the system must be the ground conductor (the conductor whose potential is assumed to be zero). The procedure for using CMATRIX is as follows:
Build and mesh the solid model with electrostatic elements. Conductors are assumed to be perfect conductors and hence do not require a finite element mesh within the conductor domain. Only the surrounding dielectric regions and air regions require a mesh. The resulting nodes on the boundary of the conductors represent the nodes that will be grouped into node components.
Select the nodes on the surface of the each conductor and group them into node components.
Command(s): CMGUI:Share a command prefix for the component names, and use a numerical value sequencing from 1 to the highest numbered conductor. For example, in Figure 13.3: Lumped Capacitor Equivalence of Three Conductor System, three node components would be defined for each set of conductor nodes. Using a prefix "cond", the node component names would be "cond1", "cond2", and "cond3". The last component, "cond3" would be the nodes representing the ground.
Enter the SOLUTION processor, using either of the following:
Command(s): SOLUGUI:Select an equation solver (JCG recommended), using either of the following:
Command(s): EQSLVGUI:Invoke the CMATRIX macro, using one of the following:
Command(s): CMATRIXGUI:The CMATRIX command macro requires the following input:
A symmetry factor (
SYMFAC). If there is no symmetry in the model, the symmetry factor is 1 (default). If you wish to model only a portion of the model by taking advantage of symmetry, use the symmetry factor as a multiplier to obtain the correct capacitance.The node component prefix name (
Condname). This is the prefix of the node component name's used to define the conductor node components. In the above example, the prefix name is "cond". The command macro requires that you put single quotes around the prefix name when entering the character string. Thus, the input for this example would be 'cond'. In the GUI, the single quotes are automatically handled by the program.The number of conductor node components (
NUMCOND). Insert the total number of conductor node components. In the above example you would use "3".Enter the Ground Key option (
GRNDKEY). If your model does not contain an open boundary then the highest numbered node component represents ground. In this case, no special treatment is needed and you would set the ground key to zero (default). If your model contains an open boundary (modeled with infinite elements) and the far-field is not considered as a conductor, then you would set the ground key to zero (default). In some situations it is necessary to consider the far-field (infinity) as the ground conductor (for example, a single charged sphere in air requires the infinity location as ground in order to preserve a charge balance). When using the INFIN111 element to represent a far-field ground, set the ground key to "1".Enter a name for the stored matrix of capacitance values (
Capname). The command macro stores the computed ground and lumped matrix values in a 3D array vector where the "i" and "j" columns represent the conductor indices, the "k" column indicates ground (k = 1) or lumped (k = 2) terms. The default name is CMATRIX. For example, the command macro stores the ground terms in CMATRIX(i,j,1) and the lumped terms in CMATRIX(i,j,2). The command macro also creates a text file containing the matrix values and stores it in a file with the stored matrix name and a .TXT extension.
Do not apply inhomogeneous loads before using the CMATRIX command. Inhomogeneous loads are those created by:
CMATRIX executes a series of solutions to compute self and mutual capacitance between conductors. The solutions, which are stored in the results file, are available for postprocessing, if desired. At the end of the execution, the command macro presents a summary table.
If infinite elements (INFIN110 and INFIN111) share a common boundary with a conductor (such as a ground plane), you can consider the ground plane and infinite boundary as a single conductor (group only the ground plane nodes into a component).
Figure 13.4: Modeling Scenarios illustrates several modeling scenarios for
open and closed domain models along with the appropriate settings for
NUMCOND and
GRNDKEY.
See Example: Capacitance Calculation of this manual for an example problem using CMATRIX.