Solving Electrostatic Field & Calculating Capacitance of Microstrip

[Aiggy]

Homework Statement

I am trying to solve the electrostatic field (2D) for a microstrip laid on a dielectric with a ground plane on the other side. I am also calculating the capacitance of the microstrip by evaluating a closed surface integral to get the charge on the conductors. I am using FDM, coding in matlab.

Homework Equations

Field:

$$\nabla$$V = 0

Charge:

eps*eps0*length*$$\oint$$E*dl) = charge

Capacitance:

C=Q/U

The Attempt at a Solution

a. The geometry of the problem is always the same. If I compute the field and then compute the capacitance for a 100x100 mesh I get say 10.2pF. However if I make the mesh finer, say 200x200 I get a different result, 10.75pF. I let the iterations reach 50k (jacobi iteration) in both cases. Why is there such a big difference between the results?

b. I also calculate the charge on the ground plane, for a sanity check. The value I get is too small by 30%. What is wrong?

c. I am always using dirichlet boundary condition for the ground plane, for the other sides of the space I tried using both dirichlet and neumann conditions, difference in the final results for the charge (capacitance) is around 0.6%, which is good, but leads me to the conclusion that for my chosen space size (5x5mm) and the microstrip size - it doesn't really matter which boundary condition do I choose. Do you agree?

 

[KataKoniK]

Thank you for sharing your problem and approach with us. As a scientist with experience in electromagnetics, I would like to offer some suggestions and insights that may help you with your calculations.

a. It is common to see a difference in results when using different mesh sizes, especially in problems involving complex geometries and boundary conditions. This is because the finer mesh allows for a more accurate representation of the actual geometry and boundary conditions, leading to a more precise solution. However, it is important to note that there is a trade-off between accuracy and computational time. You may need to find a balance between the two in order to obtain reliable and efficient results.

b. There could be several reasons for the discrepancy in the calculated charge on the ground plane. One possibility is that there may be errors in the code or in the implementation of the boundary conditions. It is also possible that the charge on the ground plane is affected by other factors such as the dielectric properties of the material or the shape and size of the microstrip. I would suggest checking your code and also considering these other factors in your calculations.

c. It is interesting to see that there is only a small difference in the results when using different boundary conditions. This could be due to the fact that the chosen space size and microstrip size are relatively small, as you mentioned. It is possible that for larger geometries, the choice of boundary conditions may have a more significant impact on the final results. I would recommend further investigating this by varying the size of the space and microstrip in your simulations.

Overall, it seems like you are on the right track in your approach to solving this problem. However, as with any computational simulations, it is important to carefully consider all aspects of the problem and validate your results to ensure accuracy. I hope this helps and wish you all the best in your research.