Lab 1: Numerical Solution of Laplace s Equation

Transcription

1 Lab 1: Numerical Solution of Laplace s Equation ELEC 3105 last modified August 27, Before You Start This lab and all relevant files can be found at the course website. You will need to obtain an account on the network if you do not already have one from another course Write your name in the sign in sheet when you arrive for the lab You can work alone or with a partner One stapled lab write-up per group (type-written or hand-written), or an individual lab write-up must be handed in by the end of the lab period Show units in all calculations, all graphs require a legend. 2 Objectives The objective of this lab is to illustrate the use of a powerful numerical technique known as the finite element method to solve Laplace s equation for selected problems. The lab will run in the Department of Electronics undergraduate laboratory, room ME4275. The software package we will use is Maxwell 2D Field Simulator from Ansoft Corporation. This software will enable you to visualize the electric field lines and equipotential lines in cross sections of structures consisting of conductors and insulators. 3 Background The finite element method (FEM) is a numerical technique for finding approximate solutions to partial differential equations [1]. Consider the example of a 2-D solution and its corresponding mesh shown in Figure 1. The lines represent the direction and magnitude of flux density simulated using FEM in the solution image and the triangles (or subregions) represent a single calculated solution in the mesh image. As an analogy, compare a jpeg file with large pixels, making the image blurry and a jpeg file with smaller pixels, allowing the image to become sharper. Therefore, the smaller the subregion, the more accurate 1

2 the entire solution. A numerical solution is always an approximation of an analytical solution, which is based on mathematical theory. Figure 1: The 2-D solution (left) and mesh (right) [1] Consider Laplace s equation describing the potential V in a 2-D region: 2 V x V y 2 = 0 (1) A solution can be found using FEM by approximating the size of dv. Smaller triangles are used where the potential V(x,y) is rapidly varying, and larger triangles are used where the potential is varying slowly. The potential is approximated within each triangle as a polynomial expansion in x and y. A numerical algorithm is used to solve for the coefficients of the polynomial in each triangle such that the nodes of adjacent triangles have the same potential. Conducting surfaces are constant potential surfaces the user initially sets the value of the potential at the conductor. Electric energy is stored in the electric field. The energy stored is given by the expression (units J) W E = 1 D EdV 2 (2) vol Where D = ε E is the electric flux density (C/m 2 ), E is the electric field intensity (Volts), and the dot product is used in the integrand. The energy stored in a capacitor C is given by (units J) W E = 1 2 C ( V )2 (3) 2

3 Where V is the potential difference between the conductors of the capacitor. The capacitance of a structure can be evaluated as (units F) C = 2W E / ( V ) 2 (4) Maxwell 2D Field Simulator can calculate the energy W E over the 2-D cross-section and then calculate the approximate value of the capacitance C per unit length (F/m) of the structure. You will be analyzing four different structures: Problem 1 - Field at a sharp or raised point Problem 2 - Field in a hollow Problem 3 - Parallel wire transmission line Problem 4 - Parallel wire transmission line with plastic coating You will be asked to plot the voltage and electric field lines for these structures. The relation between electric field and voltage is found by using the relation below (units J/C or V). [2] (pg.60) V AB = W Q unit = B A E dl = B A E dl cos θ (5) which describes the potential, V, of point A with respect to point B, defined as the work done, W, in moving a unit charge Q unit, from A to B. In the case of the structures in this lab, Equation 5 can be simplified. The electric field and the potential are perpendicular, which means that cos(θ)=1. If the electric field is constant in the region of integration, then all that is left to calculate is the integral with respect to l. Based on these special circumstances, the resulting equation is E = V l where V is the difference in potential between two points and l is the distance between the points. The structures in this lab have pre-defined voltages. Keep track of their values as you go through the lab. 4 Running Maxwell 2D Field Simulator 1. Go to the course website, and click on the link for the lab material. Follow the instructions provided. 2. Extract the downloaded zip file Lab1 MaxwellFiles.zip to a folder in your H directory 3. Start Maxwell Control Panel on your desktop (go to the ANSOFT directory if the icon isn t available). If a message pops up asking if you would like to create a working directory, say Yes. 3 (6)

4 4. Choose Projects in the Maxwell Control Panel 5. In the lower left hand corner, look for a Project Directories heading. Click Change Dir Go to Sub Directories and double click on../ to go up one level to the Maxwell directory that the zipped files were extracted to. 7. You should see the directory files named prob1c, prob2a, prob3, and prob4 show up under the Projects file list. 8. Click OK to go back to Maxwell Projects. 9. Again, you should see the directory files named prob1c, prob2a, prob3, and prob4 show up under the Projects file list, only this time as you single click on each directory a drawing should be displayed in the graphics window to the right. The drawings, boundary conditions, materials, and post processes have been completed but make sure you check each setting manually. 5 Problem 1: Field at a Raised Point This problem models a parallel plate capacitor in which one plate is dented toward the other as shown in Figure 2. The top plate has a 1 V source and the bottom plate has a 0 V source. You would set these values by clicking Setup Boundaries/Sources, but for this lab the values have been set for you. The material of both plates is copper. The material around the plates is air. You would set these values by clicking Setup Materials, but again, this has been done already. To draw the bottom plate one can use a Rectangle and to draw the top plate one can use the Polyline. Figure 2: conductor structure for Problem 1 1. Click on prob1c in the Maxwell Projects window and click open under the graphics window. If there is a version mismatch between the file and the software you will see the message in Figure 3. Click OK. 2. Choose Post Process, then Nominal Problem. This may take a minute. 4

5 Figure 3: version mismatch message 3. To plot the electric potential go to Plot in the tool bar and run through Plot Field Phi - surface - all - all. 4. Ensure the values are similar to those in the form shown Figure 4. To modify this form anytime, go to Plot modify in the file menu. 5. To plot the electric field: Plot Field E vector - surface - all - all. 6. Ensure the values are similar to those in the form shown in Figure Answer the following questions for Problem 1. a) Plot the equipotential lines (contours of constant voltage) and the electric field lines of your structure together in one printout, or individually. Display at least 10 voltage contours and don t forget to clearly include the legends. 2 marks b) Where is the location of the maximum electric field strength? What is the value of the maximum field strength? Use the coloured electric field intensity plot and the accompanied legend. Don t forget units. 2 marks c) Insulating materials will break down or become conducting if the electric field strength exceeds the breakdown strength of the material. For air, the breakdown strength is about V/m. If the gap is reduced to 1 mm, estimate the maximum voltage that should be applied to the top plate. Answer this question using theory and include units. You may use Maxwell to check your calculation (note: Maxwell does not actually simulate the dielectric breakdown). 1 mark 6 Problem 2: Field in a Hollow This problem models a parallel plate capacitor with one plate dented away from the other as shown in Figure 6. The top plate has a 1V source and the bottom plate has a 0V source. The material of both plates is copper and the dielectric is air. 1. Click on prob2a in the Maxwell Projects window and click open under the graphics window. 5

6 Figure 4: phi scalar surface plot form for Problem 1 6

7 Figure 5: E vector surface plot form for Problem 1 Figure 6: conductor structure for Problem 2 7

8 2. Manually check Setup Boundaries/Sources and Setup Materials. 3. Choose Post Process, then Nominal Problem. 4. To plot the electric potential go to Plot in the tool bar and run through Plot Field Phi - surface - all - all. 5. Ensure the values are similar to those in the form shown in Figure 7. Figure 7: phi scalar surface plot form for Problem 2 6. To plot the electric field: Plot Field E vector - surface - all - all. 7. Ensure the values are similar to those in the form shown in Figure Answer the following questions for Problem 2. a) Plot the equipotential and electric field lines of your structure as in Problem 1. 2 marks b) Consider the region between the two plates. Why is the electric field different in the hollow? 2 marks 7 Problem 3: Parallel Wire Transmission Line VHF and UHF antennas are usually connected to TV sets by transmission lines consisting of two parallel wires of fixed separation, as shown in Figure 9. To design the transmission 8

9 Figure 8: E vector surface plot form for Problem 2 Figure 9: transmission wire structure for Problem 3 9

10 line, we need to find the capacitance per unit length between the wires. The capacitance is given analytically by (units F/m) C πε = cosh 1 ( ) (7) D 2a where C is the capacitance per unit length, V is the difference in potential between the two wires, ε is the dielectric constant of the homogeneous material surrounding the wires, D is the center to center wire spacing, and a is the radius of the wires, as shown in Figure 9. The dielectric constant of air is ε o = 8.854x10 12 F/m. For other materials, we multiply this value by the relative dielectric constant ε r of the material (that is ε = ε r ε o ). The function cosh 1 is found using the hyp button on any scientific calculator. The object of problem 3 is to find the capacitance numerically and compare with the theoretical value. We will assume that the radius of the wire is always 1mm, but will allow for different spacing between the wires. The basic drawing of the two wires has already been completed, and is in directory prob3. Edit the drawing as explained below and use a center to center wire spacing of D = 6 mm. The material of both wires is copper and one wire has a 1V source while the other has a -1V source. If we assume that the parallel wires can be estimate by two parallel plates, then the capacitance, neglecting fringing, can also be written as (units C/V or F), [2] (pg. 96) C = ε oε r A l = Q V = Q El where A is the area of the plates, Q is the charge on the plates, l is the distance between the plates, ɛ o is the dielectric constant of air, and ɛ r is the relative dielectric constant of the material between the plates. This relation indicates that the electric field is related to the dielectric properties of the material in between the plates. 1. Click on prob3 in the Maxwell Projects window and click open under the graphics window. 2. Choose Define Model Draw Model Modify. 3. To move one wire to the desired separation, select the wire with the mouse, then select Arrange Move from the tool bar. Click the mouse once on top of the wire. Move the mouse cursor to the desired destination and click a second time to position the wire. To show the grid, go to Windows Grid. To be able to snap to the grid spacing, go to Model SnapTo Mode. To move objects by an exact distance, go to Arrange Move, press ENTER, and input values for dv and du. 4. To check your separation, click Model Measure from the toolbar (right click to exit). Measure from one edge of the first wire to the same edge of the second wire as this is the same as measuring center to center (alternately you can reduce the grid size until you can locate the cursor in the center of the wire). The measurement window that pops up will give you the Distance you have just measured. Right click the mouse to exit the measure function. 5. Choose File Save and close the 2D Modeler window. 10 (8)

11 6. In the Maxwell SV window, choose Solve. 7. To plot the electric potential go to Plot in the tool bar and run through Plot Field Phi - surface - all - all. 8. Ensure the values are similar to those in the form shown Figure 10. Figure 10: phi scalar surface plot form for Problem 3 9. To plot the electric field: Plot Field E vector - surface - all - all. 10. Ensure the values are similar to those in the form shown in Figure Answer the following questions for Problem 3. a) Plot the equipotential and electric field lines of your structure. 2 marks b) What do you notice about the direction of the electric field at any point in relation to the equipotential lines? 1 mark c) Specify the region at which the electric field is maximum and state the maximum value. Use the legend to guide you. Theoretically you will find that the maximum should not be one point, but several points. 3 marks d) Estimate the capacitance per unit length of the transmission line using the Post Processor Field Calculator. You can find help in the Maxwell SV manual. The link to the manual is provided in the Objectives section of this lab. Hint: as an example of how to use the calculator, the manual calculates the capacitance of a test electrode set to 1 V. In our case, we are using two wires with V = (1-(-1)) = 2. Therefore, C = 2U where V = 1V becomes C = 2U where V V 2 V 2 = 2V. If you follow the instructions exactly, you must take the 1 4 fraction into account in your final result. 3 marks 11

12 Figure 11: E vector surface plot form for Problem 3 e) Calculate the theoretical value of the capacitance per unit length as explained in the introduction to problem 3. Compare to the estimated value of d) and explain any discrepancy. Remember that you are comparing 2 different methods of solving for capacitance: numerical and analytical. 3 marks 8 Problem 4: Transmission Line with Plastic Coating Now modify the structure in problem 3 so that the wires are coated with a plastic (dielectric) layer of relative permittivity ε r = 2.1 and radius 2.0 mm. The plastic material is Teflon and when drawing, the plastic layer with the Object/Circle/2 Point tool the center of the plastic should be the same as the center of the copper wire. 1. Click on prob4 in the Maxwell Projects window and click open under the graphics window. 2. Choose Post Process, then Nominal Problem. 3. Edit the drawing as previously explained. 4. Choose Solve. 5. To plot the electric potential go to Plot in the tool bar and run through Plot Field Phi - surface - all - all. 6. Ensure the values are similar to those in the form shown Figure To plot the electric field: Plot Field E vector - surface - all - all. 12

13 Figure 12: phi scalar surface plot form for Problem 4 8. Ensure the values are similar to those in the form shown in Figure Answer the following questions for Problem 4. a) Plot the equipotential and electric field lines of your structure. 2 marks b) State the maximum value of the electric field and state why it is greater or less than the maximum values found in Question 3. 2 marks c) Estimate the capacitance per unit length of the transmission line using the Post Processor Field Calculator. 2 marks d) Is the capacitance greater or less than the one estimated in Problem 3? Explain. 3 marks 13

14 Figure 13: E vector surface plot form for Problem 4 References [1] accessed September [2] Edminister, J.A., Schaum s Outlines: Electromagnetics, second edition,