Posted: Thu Oct 18, 2012 1:29 pm
Hello,
Yes, I have been working on a model in Matlab and I have gotten really far in developing it. Here are the steps that I followed:
1. I used the basic biot-savart law to estimate the field strength at 4 points in the geometry:
A. Ring Center (which technically should be zero)
B. The joint. (where the rings are closest, ~296 Gauss)
C. The Corner (the geometric corner between 3 rings, ~193 Gauss)
D. The Axis (The center of the ring ~197 Gauss)
This is a really simple, first pass estimate but I can use it to check my work as the model gets more complex. I must state that Bussard predicted 70 to 100 Gauss at the corner, but I don't know how he estimated this. Check it the simple math here: http://hyperphysics.phy-astr.gsu.edu/hb ... oo.html#c3
=======
BTW... the rings are SPACED so that the corner and the axis field ARE EQUAL.
This makes electron confinement more uniform. Bussard stated it in his paper. THIS IS A BIG DEAL FOR RING DESIGN.
=======
I pulled two formulas from the 2011 University of Sydney paper and developed 2 more models:
2. The first is in EXCEL. The magnetic field generated by a single ring of current, in polar coordinates. They reference, and I went and found a solution in the book an "Introduction to Electrodynamics" by David Griffiths. This solution used the Complete elliptic integral of the first kind. Excel Has NO Formula for this; I had to go and find a paper on that. I found the paper: "Approximation for Elliptic Integrals" by Yudell L Luke which used the Taylor expansion to estimate it and I had to put that into excel. Here is the formula:
K(M) = 0.125 * PI*[ (1/(sqrt(1-msin^2(0.435*PI())) + (1/(sqrt(1-msin^2(0.3125*PI())) + (1/(sqrt(1-msin^2(0.1875*PI())) + (1/(sqrt(1-msin^2(0.06255*PI()))]
E(M) = 0.125*PI*[sqrt(1-m*sin^2(0.435*pi) + sqrt(1-m*sin^2(0.3125*pi) + sqrt(1-m*sin^2(0.1875*pi) + sqrt(1-m*sin^2(0.0625*pi)]
This will work, unless M gets close to 1. So, the next model I had would estimate the magnetic field at a point relative to a single current loop.
3. The second model uses superposition to add these 6 polar models together to come with a X,Y,Z coordinate model for the magnetic field by the rings. This solution amounted to adding the polar fields that align along the same axis (i.e. X, Y or Z) together. You have to add a small term, when you are converting from a polar system to a Cartesian system, but that is explained. It sort of feels like accounting, adding the correct numbers in the correct columns together. I did this in excel, but it was a big pain. Fortunately, I had my simple math estimate, to keep checking my work. That is critical. You add complexity, but you start with the simplest model first. I completed this in excel.
So now, I have jumped to Matlab and I have made significant progress along those lines. I can get contour plots and vector fields. Except that the direction vectors are not pointed correctly. Here is what I calculated (everything is in Gauss):
Simple Model:
Center: 269.0 **
Corner: 193.8
Joint: 296
Axis: 193.8
Polar Model (Excel):
Center: 269.1
Corner: -625
Joint: -1312
Axis: 197.9
Polar Model (Excel with Taylor Expansion):
Center: 269.1
Corner: -616
Joint: -1198
Axis: 198
Cartesian Model (Excel no Taylor Expansion):
Center: 90x +90y + 90z = 269
Corner: 230x + 230y + 230z = 690
Joint: 399x + 10y + 409z = 819
Axis: 207x + 22y + 22z = 257
Cartesian Model (Excel with Taylor Expansion):
Center: 90x +90y + 90z = 269
Corner: 223x + 223y + 223z = 668
Joint: 322x + 10y + 331z = 663
Axis: 207x + 8y + 25z = 240
** Technical the center field is zero but you can do the field strength from one ring, and then multiply by 6.
I had to leave this problem for the moment to work on other things. Here is what I want to do next:
1. Upload my excel file.
2. Upload a WORKING matlab file.
3. Write up a detailed explanation of all of this (with pictures) so the reader can follow everything.
Also, I would like to do this research professionally.
Yes, I have been working on a model in Matlab and I have gotten really far in developing it. Here are the steps that I followed:
1. I used the basic biot-savart law to estimate the field strength at 4 points in the geometry:
A. Ring Center (which technically should be zero)
B. The joint. (where the rings are closest, ~296 Gauss)
C. The Corner (the geometric corner between 3 rings, ~193 Gauss)
D. The Axis (The center of the ring ~197 Gauss)
This is a really simple, first pass estimate but I can use it to check my work as the model gets more complex. I must state that Bussard predicted 70 to 100 Gauss at the corner, but I don't know how he estimated this. Check it the simple math here: http://hyperphysics.phy-astr.gsu.edu/hb ... oo.html#c3
=======
BTW... the rings are SPACED so that the corner and the axis field ARE EQUAL.
This makes electron confinement more uniform. Bussard stated it in his paper. THIS IS A BIG DEAL FOR RING DESIGN.
=======
I pulled two formulas from the 2011 University of Sydney paper and developed 2 more models:
2. The first is in EXCEL. The magnetic field generated by a single ring of current, in polar coordinates. They reference, and I went and found a solution in the book an "Introduction to Electrodynamics" by David Griffiths. This solution used the Complete elliptic integral of the first kind. Excel Has NO Formula for this; I had to go and find a paper on that. I found the paper: "Approximation for Elliptic Integrals" by Yudell L Luke which used the Taylor expansion to estimate it and I had to put that into excel. Here is the formula:
K(M) = 0.125 * PI*[ (1/(sqrt(1-msin^2(0.435*PI())) + (1/(sqrt(1-msin^2(0.3125*PI())) + (1/(sqrt(1-msin^2(0.1875*PI())) + (1/(sqrt(1-msin^2(0.06255*PI()))]
E(M) = 0.125*PI*[sqrt(1-m*sin^2(0.435*pi) + sqrt(1-m*sin^2(0.3125*pi) + sqrt(1-m*sin^2(0.1875*pi) + sqrt(1-m*sin^2(0.0625*pi)]
This will work, unless M gets close to 1. So, the next model I had would estimate the magnetic field at a point relative to a single current loop.
3. The second model uses superposition to add these 6 polar models together to come with a X,Y,Z coordinate model for the magnetic field by the rings. This solution amounted to adding the polar fields that align along the same axis (i.e. X, Y or Z) together. You have to add a small term, when you are converting from a polar system to a Cartesian system, but that is explained. It sort of feels like accounting, adding the correct numbers in the correct columns together. I did this in excel, but it was a big pain. Fortunately, I had my simple math estimate, to keep checking my work. That is critical. You add complexity, but you start with the simplest model first. I completed this in excel.
So now, I have jumped to Matlab and I have made significant progress along those lines. I can get contour plots and vector fields. Except that the direction vectors are not pointed correctly. Here is what I calculated (everything is in Gauss):
Simple Model:
Center: 269.0 **
Corner: 193.8
Joint: 296
Axis: 193.8
Polar Model (Excel):
Center: 269.1
Corner: -625
Joint: -1312
Axis: 197.9
Polar Model (Excel with Taylor Expansion):
Center: 269.1
Corner: -616
Joint: -1198
Axis: 198
Cartesian Model (Excel no Taylor Expansion):
Center: 90x +90y + 90z = 269
Corner: 230x + 230y + 230z = 690
Joint: 399x + 10y + 409z = 819
Axis: 207x + 22y + 22z = 257
Cartesian Model (Excel with Taylor Expansion):
Center: 90x +90y + 90z = 269
Corner: 223x + 223y + 223z = 668
Joint: 322x + 10y + 331z = 663
Axis: 207x + 8y + 25z = 240
** Technical the center field is zero but you can do the field strength from one ring, and then multiply by 6.
I had to leave this problem for the moment to work on other things. Here is what I want to do next:
1. Upload my excel file.
2. Upload a WORKING matlab file.
3. Write up a detailed explanation of all of this (with pictures) so the reader can follow everything.
Also, I would like to do this research professionally.