1、 Solved with COMSOL Multiphysics 4.2a.Capacitance matrixIntroductionThe 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. I
2、f 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. T
3、he 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 un
4、ique 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.Q1Q2Q3C11C12C13C21C22C23C31C32C33V1V2V3=V1, Q1V2, Q2V3, Q3Terminal 1Terminal 2Terminal 3CAPA
5、CITANCE MATRIX | 1Solved with COMSOL Multiphysics 4.2a.2 | CAPACITAIn 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 th
6、e components of the capacitance matrix is elaborated in the ACDC Module Users Guide.Model DefinitionIn 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 or
7、der 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
8、 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 t
9、o 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 capacitanc
10、e 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
11、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
12、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=NCE MATRIXSolved with COMSOL Multiphysics 4.2a.Results a
13、nd DiscussionThe 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
14、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 termi
15、nals 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.0,0214 0,0125 0,00890,0125 0,02
16、77 0,01520,0089 0,0152 0,0242CAPACITANCE MATRIX | 3Solved with COMSOL Multiphysics 4.2a.4 | CAPACITAFigure 3: Surface plot of electric potential when Terminal 1 is excited with 1 V.Model Library path: ACDC_Module/Tutorials/capacitance_matrixModeling InstructionsMODEL WIZARD1 Go to the Model Wizard w
17、indow.2 Click the 2D button.3 Click Next.4 In the Add physics tree, select AC/DCElectrostatics (es).5 Click Next.6 Find the Studies subsection. In the tree, select Preset StudiesStationary.7 Click Finish.NCE MATRIXSolved with COMSOL Multiphysics 4.2a.GEOMETRY 1Square 1 (sq1)1 In the Model Builder wi
18、ndow, right-click Model 1 (mod1)Geometry 1 and choose Square.2 Go to the Settings window for Square.3 Locate the Position section. From the Base list, choose Center.4 Click to expand the Layers section.5 In the table, enter the following settings:6 Select the Layers to the left check box.7 Select th
19、e Layers to the right check box.8 Select the Layers on top check box.9 Click the Build Selected button.Rectangle 1 (r1)1 In the Model Builder window, right-click Geometry 1 and choose Rectangle.2 Go to the Settings window for Rectangle.3 Locate the Position section. In the x edit field, type -0.3.4
20、In the y edit field, type 0.2.5 Locate the Size section. In the Width edit field, type 0.2.6 In the Height edit field, type 0.1.Rectangle 2 (r2)1 In the Model Builder window, right-click Geometry 1 and choose Rectangle.2 Go to the Settings window for Rectangle.3 Locate the Position section. In the x
21、 edit field, type -0.1.4 In the y edit field, type -0.3.5 Locate the Size section. In the Width edit field, type 0.2.6 In the Height edit field, type 0.1.Rectangle 3 (r3)1 In the Model Builder window, right-click Geometry 1 and choose Rectangle.LAYER NAMETHICKNESS (M)Layer 1 0.1CAPACITANCE MATRIX |
22、5Solved with COMSOL Multiphysics 4.2a.6 | CAPACITA2 Go to the Settings window for Rectangle.3 Locate the Position section. In the x edit field, type 0.1.4 In the y edit field, type 0.5 Locate the Size section. In the Width edit field, type 0.2.6 In the Height edit field, type 0.1.7 Click the Build A
23、ll button.MATERIALS1 In the Model Builder window, right-click Model 1 (mod1)Materials and choose Open Material Browser.2 Go to the Material Browser window.3 Locate the Materials section. In the Materials tree, select Built-InAir.4 Right-click and choose Add Material to Model from the menu.Air1 In th
24、e Model Builder window, click Model 1 (mod1)MaterialsAir.2 Select Domains 16 and 1012 only.Domains 7, 8 and 9 represent conductors and hence they need to be deselected.ELECTROSTATICS (ES)1 In the Model Builder window, click Model 1 (mod1)Electrostatics (es).2 Select Domains 16 and 1012 only.Since Do
25、mains 7, 8 and 9 are conductors, the electrostatics problem will not be solved in these domains. They must be deselected.3 Go to the Settings window for Electrostatics.4 Locate the Port Sweep Settings section. Select the Activate port sweep check box.Note that a port parameter appears by default. We
26、 will create a parameter by the same name.GLOBAL DEFINITIONSParameters1 In the Model Builder window, right-click Global Definitions and choose Parameters.2 Go to the Settings window for Parameters.NCE MATRIXSolved with COMSOL Multiphysics 4.2a.3 Locate the Parameters section. In the Parameters table
27、, enter the following settings:ELECTROSTATICS (ES)Terminal 11 In the Model Builder window, right-click Model 1 (mod1)Electrostatics (es) and choose Terminal.2 Click the Select Box button on the Graphics toolbar.3 Select Boundaries 1517 and 21 only.4 Go to the Settings window for Terminal.5 Locate th
28、e Terminal section. From the Terminal type list, choose Voltage.Terminal 21 In the Model Builder window, right-click Electrostatics (es) and choose Terminal.2 Click the Select Box button on the Graphics toolbar.3 Select Boundaries 2326 only.4 Go to the Settings window for Terminal.5 Locate the Termi
29、nal section. From the Terminal type list, choose Voltage.6 In the V0edit field, type 0.Terminal 31 In the Model Builder window, right-click Electrostatics (es) and choose Terminal.2 Click the Select Box button on the Graphics toolbar.3 Select Boundaries 1820 and 22 only.4 Go to the Settings window f
30、or Terminal.NAME EXPRESSIONPortName 1CAPACITANCE MATRIX | 7Solved with COMSOL Multiphysics 4.2a.8 | CAPACITA5 Locate the Terminal section. From the Terminal type list, choose Voltage.6 In the V0edit field, type 0.Infinite Elements 11 In the Model Builder window, right-click Electrostatics (es) and c
31、hoose Infinite Elements.2 Select Domains 14, 6, and 1012 only.These are the outer layers of the air domain. They need to be meshed using a mapped mesh technique.MESH 1Mapped 11 In the Model Builder window, right-click Model 1 (mod1)Mesh 1 and choose Mapped.2 Go to the Settings window for Mapped.3 Lo
32、cate the Domain Selection section. From the Geometric entity level list, choose Domain.4 Select Domains 14, 6, and 1012 only.Distribution 11 Right-click Model 1 (mod1)Mesh 1Mapped 1 and choose Distribution.2 Select Boundaries 1, 2, 5, 7, 28, 33, 34, and 36 only.3 Go to the Settings window for Distri
33、bution.4 Locate the Distribution section. In the Number of elements edit field, type 2.Distribution 21 In the Model Builder window, right-click Model 1 (mod1)Mesh 1Mapped 1 and choose Distribution.2 Select Boundaries 3, 9, 14, and 35 only.3 Go to the Settings window for Distribution.4 Locate the Dis
34、tribution section. In the Number of elements edit field, type 10.Free Triangular 11 In the Model Builder window, right-click Mesh 1 and choose Free Triangular.2 In the Settings window, click Build All.NCE MATRIXSolved with COMSOL Multiphysics 4.2a.STUDY 1Parametric Sweep1 In the Model Builder window
35、, right-click Study 1 and choose Parametric Sweep.2 Go to the Settings window for Parametric Sweep.3 Locate the Study Settings section. Under Parameter names, click Add.4 Go to the Add dialog box.5 In the Parameter names list, select PortName.6 Click the OK button.7 Go to the Settings window for Par
36、ametric Sweep.8 Locate the Study Settings section. In the Parameter values edit field, type 1 2 3.The number corresponding to the value of PortName is checked against the number corresponding to the Terminal name specified in the settings of the Terminal boundary condition. In this way, the port swe
37、ep instructs the model to be solved in a loop where the excitation terminal is changed in each pass of this loop depending on the value of PortName.CAPACITANCE MATRIX | 9Solved with COMSOL Multiphysics 4.2a.10 | CAPACITAN9 In the Model Builder window, right-click Study 1 and choose Compute.RESULTSDe
38、rived Values1 In the Model Builder window, right-click ResultsDerived Values and choose Global Matrix Evaluation.2 Go to the Settings window for Global Matrix Evaluation.3 Locate the Data section. From the Data set list, choose Solution 2.4 In the upper-right corner of the Expression section, click Replace Expression.5 From the menu, choose Capacitance (es.C).6 Locate the Expression section. From the Unit list, choose nF.7 Click the Evaluate button.CE MATRIX
