Capacitance by solving Poisson equation

Hi ,

I want to find the capacitance by solving the Poisson equation. I solved the voltage potential and was able to plot the voltage potential. I want to find the electric potential and energy from from the voltage

First i have to find electric potential which is gradient of voltage.

Then i have to find the energy Energy = integral(1/2 * Epsilon * E^2) over volume.

Can you help me find the gradient of voltage and then the energy ?

Answer 1

[Ravi Kumar]

If you setup the problem using PDE Toolbox, look here for example, then you get the gradients of the solution in the results.

Regards,

Ravi

[Lalson Vincent]

Thanks Ravi,

I am able to solve the poisson equation and find the voltage field. But i am not able to find the gradient(electric potential) and then calculated the energy from it.

[Ravi Kumar]

I might be missing something. Here is an example. If you solve a poissoin equation using:

model = createpde();
geometryFromEdges(model,@lshapeg);
applyBoundaryCondition(model,'dirichlet','Edge',1:model.Geometry.NumEdges,'u',0);
specifyCoefficients(model,'m',0,'d',0,'c',1,'a',0,'f',1);
generateMesh(model,'Hmax',0.25);
results = solvepde(model)

You get the output as:

results = 

  StationaryResults with properties:

    NodalSolution: [273×1 double]
       XGradients: [273×1 double]
       YGradients: [273×1 double]
       ZGradients: []
             Mesh: [1×1 FEMesh]
             

As you can see 'results' contains and XGradients and YGradients, which are gradients of NodalSolution.

Regards,

Ravi

[Lalson Vincent]

Thanks Again.

I am now able to find the electric field now.

Now i need to integrate the squre of electric field over volume to find the energy stored. I have used trapz to integrate it. But its not giving the required solution.

Energy = integral (1/2 * Epsilon * E^2) dV.

[Ravi Kumar]

I don't have good suggestion without knowing what is the required solution. My guess is that you might be encountering under resolved results, try increasing mesh density.

Regards,

Ravi

[Lalson Vincent]

Required solution is the capacitance of the system with the applied voltage.

Energy = integral (1/2 * Epsilon * E^2) dV = 1/2 * C * V^2

Capacitance = Energy / (1/2 * V^2).

Tried with finer mesh. Looks like, trapz is not the correct method. Is there any way to integrate this ?