Different Types of Current Distribution Display

  Different Types of Current Distribution Display

            On the MGRID, we can display 6-types of current distribution. What are the meanings of the different types? This appendix explains the differences.

            At any location, the time-harmonic current density can be described as a complex vector:

J(x,y,z) = Jx  x + Jy y + Jz z                                                                                                                           (L-1)

where x, y and z are the unit vectors.

            Jx = Jxr + j Jxi                                                                                                                                                  (L-2)

            Jy = Jyr + j Jyi                                                                                                                                                  (L-3)

            Jz = Jzr + j Jzi                                                                                                                                                   (L-4)

Then, we can get,

J(x,y,z) = ( Jxr  x + Jyr y + Jzr z ) + j ( Jxi  x + Jyi y + Jzi z )                                                                      (L-5)

J(x,y,z) = Jmr r + j Jmi i                                                                                                                                 (L-6)

where r and i are the unit vectors for the real and imaginary parts, respectively

Jmr = Ö ( Jxr² + Jyr² + Jzr² )                                                                                                                           (L-7)

Jmi = Ö ( Jxi² + Jyi² + Jzi² )                                                                                                                            (L-8)

r = ( Jxr  x + Jyr y + Jzr z ) / Jmr                                                                                                                    (L-9)

I = ( Jxi  x + Jyi y + Jzi z ) / Jmr                                                                                                                   (L-10)

At a specific time, the time-harmonic current density is,

J(x,y,z, t) = Re[ J(x,y,z) exp(jwt) ] = r Jmr cos( wt ) - i Jmi sin( wt)                                                      (L-11)

            Equation (L-11) is the actual current density at a specific location at a specific time. Clearly, both the value and direction of J(x,y,z, t) are changing with time. Table L-.1 shows the correspondence between the current display functions and the quantities.

Table L-1 Correspondence between the current display functions and the quantities

Display Type	Quantity	Display Features
Average Current Distribution	Ö ( Jmr2 + Jmi2 )	Shows the average intensity at each location
Vector Current Distribution	J(x,y,z, t)	Shows the direction and intensity at specific location and time as vectors on arrows.
Average and Vector Current Distribution	Ö ( Jmr2 + Jmi2 ) and J(x,y,z, t)	Shows the average intensity as color on polygons and direction of current density at a specific time with vectors on arrows.
Scalar Current Distribution Animation	|J(x,y,z, t)|	Shows the current density at different locations at different time.
Vector Current Distribution Animation	J(x,y,z, t)	Shows the direction and intensity at different locations at different time.
Scalar and Vector Current Distribution Animation	|J(x,y,z, t)| and J(x,y,z, t)	Shows the direction and intensity at different locations at different time.

.     Rectangular Mesh Versus Triangular Mesh

         IE3D uses a non-uniform, mixed rectangular and triangular meshing scheme. Usually, rectangular cells are efficient for regular shaped portion of a structure. Each rectangular cell is equivalent to 2 triangular cells. Triangular cells are flexible on modeling irregular shaped portion of a structure. It can fit the irregular boundary easily (see Figure M.1). Some people claimed that triangular cells could not yield accurate results because of the zigzag in the meshing. Such a claim is certainly not true, at least for IE3D. In this appendix, we will show that modeling using triangular cells  is as accurate as modeling using rectangular cells. We will also demonstrate the efficiency of rectangular cells.

Figure M.1 Comparison between triangular and rectangular cells.

           

            A simple rectangular patch antenna is used as our example. The structure using rectangular mesh is saved in c:\ie3d\samples\rcell.geo and the structure using triangular mesh is saved in c:\ie3d\samples\tcell.geo. They are identical except the difference in meshing. The meshed structure is compared in Figure M.2.

Figure M.2 Meshed structure using rectangular cells (rcell.geo) and triangular cells (tcell.geo).

            Table M.1 and Figure M.3 show the comparison between the simulation results using rectangular cells (c:\ie3d\samples\rcell.geo) and triangular cells (c:\ie3d\samples\tcell.geo). They are almost identical on the Smith Chart. In fact, there is a slightly frequency shift due to the difference in the meshing. Certainly, the triangular meshing scheme takes more memory and time to simulate the structure.

Table M.1 The comparison between rectangular cells and triangular cells.

	Number of Cells	Number of Unknowns	Memory Required	Simulation Time Per Frequency
Rectangular Mesh (c:\ie3d\samples\rcell.geo)	126	225	2 M	0.6 seconds
Triangular Mesh (c:\ie3d\samples\tcell.geo)	248	347	4 M	2 seconds

Figure M.3 The comparison between rectangular meshing and triangular meshing.

.     Uniform Grid Versus Non-Uniform Grid

         We have discussed the theoretical comparison between uniform grid and non-uniform grid in Section 2 of Chapter 1. We will provide an actual structure for comparison between using the uniform grid and non-uniform grid. The structure is derived from the geometry saved in c:\ie3d\samples\lpass.geo. The structure in lpass.geo can be best fitted into a uniform grid with grid size of 1 mil. The problem we encountered is that we cannot simulate the structure lpass.geo using a uniform grid of size 1 mil on the IE3D with even 256 M RAM without swapping the memory. What we can do is to first shorten the feed-line and then adjust the geometry to fit it into a uniform grid of 2 mils. In such a case, we can solve the problem using about 32 M RAM without swapping. We simulated the structure using the following 3 schemes:

1.     Non-Uniform Grid without Edge Cells:

                The structure is saved into c:\ie3d\samples\lpass1.geo and shown in Figure N.2a.

2.     Non-Uniform Grid with Edge Cells for accuracy enhancement:

                The structure is saved into c:\ie3d\samples\lpass2.geo and shown in Figure N.2b.

3.     Uniform Grid Structure:

                The structure is saved into c:\ie3d\samples\lpass3.geo and shown in Figure N.2c.

Comparison between the three schemes is shown in Table N.1. The simulation result is shown in Figure N.1. It is interesting to note that the results between Non-Uniform Grid with Edge Cells and Uniform Grid Structure compare very well. However, the Non-Uniform Grid with Edge Cells uses much less memory and simulation time to solve the problem. We still get close result without adding the edge cells. Certainly, adding the edge vertices to create small edge cells will improve the simulation accuracy.

         It is also interesting to note that the adjusted structure (lpass1.geo, lpass2.geo and lpass3.geo) yield quite different results from the original structure in c:\ie3d\samples\lpass.geo. Our conclusion on the uniform and non-uniform grids is:

Figure N.1 The comparison of the results from uniform grid, non-uniform grid with or without edge cells.

1.     Non-uniform grid scheme is much more efficient than uniform grid scheme.

2.     Non-uniform grid with edge cells enhancement is at of the same accuracy of the uniform grid scheme when the edge cell size of the non-uniform grid is the same as that for the uniform grid.

3.     Non-uniform grid without edge cells enhancement yields reasonably accurate result with extremely high efficiency.

4.     Fitting structure into a uniform grid may create significant error.

5.     Uniform grid is extremely low efficiency and it is not suitable for large circuit analysis.

Table N.1      Comparison between Uniform Grid and Non-Uniform

	Number of Cells	Number of Unknowns	Memory Required	Simulation Time Per Frequency
Non-Uniform without AEC (lpass1.geo)	141	180	2 M	1 seconds
Non-uniform with AEC (lpass2.geo)	467	762	5 M	9 seconds
Uniform Grid (lpass3.geo)	919	1556	25 M	60 seconds

Figure N.2 The meshed structure using uniform and non-uniform schemes.