Finite Difference Method for a Numerical Solution to the Laplace Equation

To develop a numerical solution using to the the voltage and electrical field in a two-dimensional cross section of a coaxial cable of a given shape.

A numerical solution to a problem works by solving the problem over a series of discrete points rather than solving over the entire continuous domain. For a two-dimensional problem, this generally means solving over a grid. The grid must contain within it the entire problem domain, in this practical this means that the coaxial cable cross-section must be fully contained within the grid.

Additionally, the numerical solution is calculated iteratively. This means that a new estimate is made for each point based on the current estimates, and this is continuously updated. If the estimation method is accurate, the points will eventually converge towards the correct solution.

Voltage can be related to electric field, \(\textbf{E}\), in two dimensions by the following formulae:

Combining \eqref{eq:V_E} with the property of the electric field that its divergence is 0, \(\nabla\cdot\mathbf{E}=0\) [5], yields Laplace's equation:

$$\nabla^2V=0$$

where \(\nabla^2\) is the Laplacian operator [1]. Laplace's equation is a partial differential equation, and can also be given for two dimensions in Cartesian form as:

Finding a solution to Laplace's equation required knowledge of the boundary conditions, and as such it is referred to as a boundary value problem (BVP). BVPs can be solved numerically using a method known as the finide difference (FD) method [2][3].

1. Using equations \eqref{eq:laplace_FD} and \eqref{eq:E_V_FD}, the MATLAB code give in the appendix was developed. The following describes each code block:
   1. % Input Parameters
      • shieldxs [...]: Shield polygon x coordinates.
      • shieldys [...]: Shield polygon y coordinates.
      • shieldv: Shield voltage.
      • corexs [...]: Core polygon x coordinates.
      • coreys [...]: Core polygon y coordinates.
      • corev: Core voltage.
      • gridsize [x,y]: No. x and y grid points.
      • maxits: Maximum no. of iterations to perform.
      • err: Maximum allowable error before convergence.
   2. % Field (real units)

      The field is initialised to cover the coaxial cable cross section.

   3. % Grid (pixels)

      A computational grid is defined over the field.

   4. % Set grid initial value

      Initialise outer shield voltages to shieldv, inner core voltages to corev, and space voltages to \(\frac{1}{2}(\texttt{shieldv}+\texttt{corev})\)

   5. % iterations

      Iterate [eq:laplace_FD] for all points except edge points, reset core and shield voltages to initial values, and check for convergence.

   6. % Calculate E

      Using \eqref{eq:E_V_FD}, calculate the electric field. Also, ensure that the field inside the conductors is 0, preventing slight errors creeping in at the edges.

   7. % Plot V and E

      Plot the resultant voltage and electric fields.

2. The above simulation was run for a number of different cases. In each, the outer shield was an equilateral triangle with sides of 10 cm, and the core was a square with sides of 2 cm, gridsize was \(129\times129\), maxits was 10,000 and err was 0.01. Other test parameters used are given in table 1.

   Table 1: Test Parameters

   Test no.	shieldv	corev
   1	0 V	20 V
   2	20 V	0 V
   3	-10 V	10 V
   4	10 V	-10 V

3. In addition, the simulation was run using test conditions from test no. 1 for a range of grid sizes, \([2^n+1, 2^n+1]\) for \(n=3, 4, ..., 10\). Increasing the grid size allowed for smaller \(d_x\) and \(d_y\) values to be investigated, and their associated error.

Results to the four experiments outlined in 1 are given in tables 2, 3, 4, and 5.

Table 2: Test no. 1 results left) \(V\) field, right) \(\textbf{E}\) field.

Table 3: Test no. 2 results left) \(V\) field, right) \(\textbf{E}\) field.

Table 4: Test no. 3 results left) \(V\) field, right) \(\textbf{E}\) field.

Table 5: Test no. 4 results left) \(V\) field, right) \(\textbf{E}\) field.

Table 6: Volatge at the point \(x=5\) cm, \(y=6.495\) cm, for various values of \(d_x\).

The simulations appeared to be accurate and results were logical and consistent with known properties of the voltage and electric field.

Voltage was constant within the conductors, both the shield and core, in tables 2, 3, 4, and 5 left) (although this was expected, as it was strictly enforced within the code). Additionally, there were no voltage minima or maxima within the space between, the minima and maxima occurred only at the conductors. These are both qualitatively verifiable properties of a voltage field that were expected [5] and observed, supporting the accuracy of the simulation.

The electric field in tables 2, 3, 4, and 5 right) similarly demonstrated the expected properties. The field formed a smooth vector field, with no noticeable curl. The electric field also pointed from positive voltage to negative voltage, as expected. Finally, vectors intersected conductor surfaces at approximately 90. These properties were again expected [5], and supported confidence in the accuracy of the simulation.

Interestingly noted was that tables 2 and 4 were identical, with the exception of the colour scales in tables 2 and 4 left), as were tables 3 and 5. This phenomena is due to the voltage field being described by linear differential equations. Thus linearly translated boundary conditions yield correspondingly linearly scaled solutions. The electric field, however, is related by the derivative of the voltage field. Therefore, constant translations have no effect on the electric field, as demonstrated in tables 2 and 4 right), and tables 3 and 5 right). For the same reasons, the voltage fields in tables 2 and 4 left) were opposite to those in tables 3 and 5 left), and the electric fields in tables 2 and 4 right) were equal in magnitude (see the contour lines) and opposite in direction to those in tables 3 and 5 right). The boundary conditions between these two pairs were simply reversed.

Finally, table fig:v_convergence shows the calculated voltage for a given \(d_x\). Here it was noted that the variation between Voltage estimates was considerably lower for low values of \(d_x\) and increased with \(d_x\). This result was consistent with the theory outlined in equations \eqref{eq:1stO_FD_x}, \eqref{eq:1stO_FD_y}, \eqref{eq:2ndO_FD_x}, and \eqref{eq:2ndO_FD_y}, stating the associated error was in the order of the square of \(d_x\).

A numerical solution to the voltage and electrical field in a two-dimensional cross section of a coaxial cable, where the outer shield was an equilateral triangle with sides of 10 cm, and the core was a square with sides of 2 cm, was developed using methods. The accuracy of this method was verified using empirical means, based on qualitative observations of the resultant fields. These results indicated confidence in the accuracy of the method. Additional properties of the linearity of the electrical and voltage fields were noted, specifically the invariance of the electric field under boundary voltage translation, given the difference was the same.