Capacitance by solving Poisson equation

2020 2 7
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更新时间：2024 3 25
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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 ?

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回答（1 个）

Ravi Kumar
Ravi Kumar 2020-2-7

0 个投票

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

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Lalson Vincent
Lalson Vincent 2020-2-10
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
Ravi Kumar 2020-2-10
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
Lalson Vincent 2020-2-12
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
Ravi Kumar 2020-2-12
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
Lalson Vincent 2020-2-13
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 ?

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