Capacitance matrix and integral equation method

In summary, the conversation discusses the calculation of the capacitance matrix using an integral equation method for two parallel strips in vacuum. The method involves using the potential of a line charge and imposing the boundary condition that the strip surface is equipotential. The resulting matrix is then approximated to obtain the capacitance values. However, there is confusion about how to extract individual capacitance values from the matrix.
  • #1
Jack the Stri
39
0
Dear all,

I'm having some trouble calculating the capacitance matrix, as outlined below.

So first of all, the lecture notes I'm using use an integral equation method (method of moments) to determine the capacitance of two infinitely long and thin parallel strips in vacuum, at a distance d of each other. The z-axis is taken along one of both strips, and image theory teaches us that this problem is equivalent to one strip at a distance d/2 of a PEC plane.

The notes start from the potential of a line charge, and use the superposition principle to express the potential of the surface charge density, yielding

[tex]\phi(\rho) = \int^{}_{c}\rho_{l}(\rho')G(\rho,\rho')dc'[/tex]

With the position vectors in bold (rho and rho' being each other's image), [tex]\rho_{l}[/tex] the surface charge density, c the strip width and G the Green's function for a two-dimensional Laplace equation in a homogeneous half-space on top of a PEC plane.

The next step is to impose the boundary condition that the strip surface is equipotential, e.g. at 1. Hence

[tex]lim_{\rho\rightarrow c} \phi(\rho) = lim_{\rho\rightarrow c} \int^{}_{c}\rho_{l}(\rho')G(\rho,\rho')dc'=1[/tex]

This is solved by subdividing the strip into N segments of length [tex]\Delta[/tex], allowing the above expression to be rewritten as
[tex]lim_{\rho\rightarrow c}\sum^{N}_{i=1}\rho_{li}\int^{}_{c_{i}}G(\rho,\rho')dc'_{i}=1[/tex]

The text then goes on to explain that, when one chooses the centres [tex]\rho_{j}[/tex] as the limit c to approach, the sum forms a set of N linear equations with N unknown coefficients. In matrix form this is written:

[tex]I_{N}=C^{-1}Q[/tex]

With IN an Nx1 column matrix with elements 1, Q the Nx1 column matrix with the elementary line charge densities (or at least, the average of the segment, the way I see it), and C-1 the inverted capacitance matrix.

The values of Cij are then approximated to fill in the matrix.

However.

I understand the steps leading up to this, but in this particular case, the only output for the capacitance is one number, not an NxN matrix (which makes sense otherwise the matrix size would be dependent upon the accuracy of your calculation). The question then is, how do you get the capacitance C from this matrix, and, in the more general case of a capacitance matrix, how do you extract each individual Cij (since these are not the same as the Cij mentioned above)?

I can post some more elaboration on the maths followed if necessary.

Edit: LaTeX doesn't seem to be co-operating with me as far as formatting certain formula parts in bold, sorry...
 
Last edited:
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  • #2
Bumpity bump. From what I can make out of it each element i,j of the matrix represents the capacitance between two segments [tex]\Delta_{i}[/tex] and [tex]\Delta_{j}[/tex], or the self-capacitance. The capacitance value would then be the sum of the capacitances for each individual segment, i.e. [tex]C = \sum C_{i} = \sum (C_{ii} - \sum_{j(j \neq i)} |C_{ij}|)[/tex], or the diagonal minus the absolute values of the rest.

This doesn't match the textbook though. Any expertise around here?
 

What is a capacitance matrix?

A capacitance matrix is a mathematical representation of the capacitance values between multiple conductive objects in a system. It is a square matrix where each element represents the capacitance between two objects, and the diagonal elements represent the self-capacitance of each object.

How is a capacitance matrix calculated?

A capacitance matrix can be calculated using various methods, such as numerical simulations or analytical solutions. One common method is the integral equation method, which involves solving a set of equations derived from Maxwell's equations and boundary conditions.

What is the integral equation method?

The integral equation method is a mathematical technique used to solve for the capacitance matrix of a system. It involves setting up and solving a set of integral equations based on the geometry and material properties of the system. This method is often used in electromagnetic simulations and can provide accurate results for complex systems.

What are the advantages of using the integral equation method?

The integral equation method has several advantages over other methods of calculating capacitance matrices. It can handle complex geometries and material properties, it is computationally efficient, and it can provide accurate results even for high-frequency systems. Additionally, this method can be easily adapted to include other effects, such as surface roughness or non-linear behavior.

How is the capacitance matrix used in practical applications?

The capacitance matrix is a valuable tool in the design and analysis of electronic circuits, antennas, and other systems involving conductive objects. It can be used to calculate the capacitance values between components, which can then be used to determine the circuit's overall performance, such as its resonant frequency or power dissipation. Additionally, the capacitance matrix can aid in the optimization of circuit layouts and the reduction of electromagnetic interference.

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