atlc
In order to test the accuracy of atlc
, some simple geometries were devised, for which there are known exact analytical solutions. Three cases were considered for two-wire lines - the standard coaxial cable, an off-centre or eccentric coaxial cable and a symmetrical strip transmission line. For three-wires, which can be used two make a 4-port directional coupler, the accuracy was compared using two edge-on strip lines..
In order to verify atlc is running properly on each system, extensive testing is performed on different platforms. However, after running make, it is highly reccomended to run make check, which will verify atlc is behaving as expected, and there are no issues related to the particular platform, operating system, comiler or optimisation flags used. At the bottom of the page, there is an explanation of how make check
works.
One obvious case is the round coaxial cable, which has a well known and exact formula for its impedance. In the case of an vacuum dielectric, it is simply
Zo=60.0*loge(D/d)
, where D is the inner diameter of the outer conductor and d is the outer diameter of the inner conductor.
Circular conductors can never be defined exactly using a square grid, so differences between the exact answer and atlc
's answer are due to:
atlc
uses. Filename | D | d | Er | Zo (theory) | Zo (atlc ) |
Error (%) |
coax-a | 500 | 400 | 1.0 | 13.388 | 13.3710 | -0.127 % |
coax-b | 500 | 200 | 1.0 | 54.977 | 54.9031 | -0.134% |
coax-c | 500 | 200 | 100.0 | 5.4977 | 5.49031 | -0.134% |
coax-d | 500 | 100 | 1.0 | 96.566 | 96.4277 | -0.143% |
coax-e | 500 | 50 | 1.0 | 138.155 | 137.8984 | -0.186% |
coax-f | 500 | 25 | 1.0 | 179.744 | 179.8969 | +0.085 % |
According to the book Microwave and Optical Components, Volume 1, - Microwave Passive and Antenna Components, page 7, there is an exact formula for the impedance of a coaxial line (see below). If O is the offset between the centres of the two conductors, then the impedance Zo assuming Er=1, is
60 loge(x+sqrt(x^2-1)) where x=(d2+D2-4 O2)/(2*D*d)
Filename | D | d | O | Er | Zo (theory) | Zo (atlc ) |
Error (%) |
eccen-a | 400 | 320 | 0 | 1.0 | 13.388 | 13.375 | -0.097 % |
eccen-b | 500 | 400 | 40 | 1.0 | 8.044 | 8.024 | -0.246% |
eccen-c | 400 | 160 | 0 | 1.0 | 54.977 | 54.955 | -0.040% |
eccen-d | 500 | 200 | 150 | 1.0 | 52.021 | 51.998 | -0.043% |
eccen-e | 500 | 200 | 100 | 1.0 | 41.589 | 41.562 | -0.063% |
eccen-f | 500 | 100 | 0 | 1.0 | 96.566 | 96.489 | -0..080% |
eccen-g | 500 | 100 | 50 | 1.0 | 94.008 | 93.930 | -0.083% |
eccen-h | 500 | 100 | 100 | 1.0 | 85.525 | 85.443 | -0.096% |
eccen-i | 400 | 40 | 0 | 1.0 | 138.155 | 138.052 | -0.075% |
eccen-j | 500 | 50 | 50 | 1.0 | 135.679 | 135.575 | -0.077% |
eccen-k | 500 | 50 | 100 | 1.0 | 127.556 | 127.448 | -0.085% |
eccen-l | 400 | 40 | 12 | 1.0 | 110.835 | 110.709 | -0.113% |
eccen-m | 400 | 40 | 160 | 1.0 | 73.540 | 73.237 | -0.411% |
eccen-n | 1600 | 160 | 640 | 1.0 | 73.540 | 73.238 | -0.134% |
Note, due to their large size, the eccentric coax files are not distributed. They are made automatically when you run
make check
, which should be done as part of the installation procedure (see the file INSTALL). Running
make check
calls create_bmp_for_circ_in_circ, which is used for creating a circular conductor inside another circular conductor.
create_bmp_for_circ_in_circ 500 400 0 1 eccentric-a.bmp
create_bmp_for_circ_in_circ 500 400 40 1 eccentric-b.bmp
create_bmp_for_circ_in_circ 500 200 0 1 eccentric-c.bmp
Comparisions between atlc and a symmetrical strip transmission line
Another obvious test is a symmetrical strip transmission line - see diagramme below.
This has an exact analytical solution, dependent on the ratio of the
width of the inner conductor w, to the distance between the two outer
conductors H. This assumes that the outer conductors extend to plus
and minus infinity and the inner conductor is infinitely thin. This
structure has the advantage of requiring no curves, so can be
represented accurately with the square grid used in atlc.
However, its impossible to have an inner conductor that is less than 1 pixel high and it is impossible to make the dimension W infinitely wide as it was take an infinite amount of disk space, RAM and CPU time. However, if the width W is made at least 4xH+w, then making it any larger does not seem to have much affect on the result.
The-i
option to create_bmp_for_symmetrical_stripline
,
forces the width
W to be equal to 4 times the internal height plus the inner conductor's
width w (unless the user specified a larger value of W). Hence, when
the -i
option is used, a valid test of atlc
's
accuracy can be made-i
option, you can made the width W and height
H any value
you want above >=5 pixels, although H must be odd, for the inner conductor
to fit equally between the two outer confuctors. As always, the bitmaps created
are 10 pixels higher and 10 pixels wider, to enforce a green metallic boundary.
create_bmp_for_symmetrical_stripline -vv -i 0 201 290 50-201.bmp
For this to be a valid test of atlc, the width should be
infinite. Since you used the -i option (indicationg you
want the width W to effectively infinite, W must exceed w + 4xH.
Therefore W has been is set to 1134
w=290 H=201 w/H=1.442786 xo=23.7538
Zo is theoretically 49.989477 Ohms (assuming W is infinite)
This structure, which has a w/H value of 1.442786, has a theoretical impedance close to 50 Ohms (49.989477 to be precise). atlc calculates this to be 49.8457 Ohms, an error of only 0.087%, when using a grid 1134x201. Increasing the number of pixels to 2222x401, the results are surprisingly somewhat worst with an error of 0.191%. I have no explanation for this. but increasing the pixels further to 4399x801, results in a larger error still, as atlc reports this as 49.9127.
Filename | W | H | w | w/H | Zoexact | Zoatlc | Error | Time |
25ohm-201h.bmp | 1512 | 201 | 668 | 3.3234 | 25.018 | 24.919 | -0.392% | 0h:00m:20s |
25ohm-401h.bmp | 2978 | 401 | 1334 | 3.3267 | 24.996 | 24.938 | -0.409% | 0h:04m:10s |
25ohm-801h.bmp | 6000 | 801 | 2664 | 3.3267 | 25.001 | 24.938 | -0.409% | 0h:48m:09s |
50ohm-201h.bmp | 1134 | 201 | 290 | 1.42786 | 49.989 | 49.846 | +0.087% | 0h:00m:15s |
50ohm-401h.bmp | 2222 | 401 | 578 | 1.4419 | 50.026 | 49.931 | -0.191% | 0h:03m:38s |
50ohm-801h.bmp | 4399 | 801 | 1155 | 1.4419 | 50.012 | 49.91 | -0.20% | 0h:47m:08s |
100ohm-201h.bmp | 945 | 201 | 101 | 0.5025 | 100.161 | 100.093 | -0.068% | 0h:00m:14s |
100ohm-401h.bmp | 1846 | 401 | 202 | 0.5037 | 100.02 | 99.911 | -0.329% | 0h:03m:16s |
100ohm-801h.bmp | 3647 | 801 | 403 | 0.5037 | 100.09 | 99.90 | -0.19% | 0h:37m:25s |
200ohm-201h.bmp | 862 | 201 | 18 | 0.0896 | 200.81 | 202.716 | +0.945% | 0h:00m:13s |
200ohm-401h.bmp | 1680 | 401 | 36 | 0.08978 | 200.669 | 201.182 | +0.255% | 0h:03m:16s |
200ohm-801h | 3317 | 801 | 73 | 0.09114 | 199.771 | 199.634 | -0.068% | 0h:37m:13s |
400ohm-1551h | 6439 | 1551 | 5 | 0.00322 | 400.040 | 412.26 | +3.05% | 6h:11m:55s |
400ohm-76610h | 31109 | 7750 | 25 | 0.00323 | 400.085 | out of RAM |
Notes:
atlc
was compiled single-threaded.atlc
so that it did
not create bitmap files. Without these options, run times would be a
little longer, due to the time to write the files to disk. atlc
with coupled linesatlc
with coupled lines is more difficult that with single isolated lines, since there is to my knowledge only one structure for which exact analytical results exist. For two infinitely thin conductors halfway between two infinitely wide groundplanes (see below)
the odd and even mode impedances can be calculated analytically. If the spacing between the two groundplanes is H, the width of the conductors w, the spacing between the conductors s, and the permittivity of the medium Er,
----------^-------------------------------------------------------------- | | <---w---><-----s----><---w--> H --------- -------- | Er | ----------v--------------------------------------------------------------then the impedances are given by.
Zeven=(30*pi/sqrt(er))*(K(ke')/K(ke))
Zodd=(30*pi/sqrt(er))*(K(ko')/K(ko))
K(kx)=complete elliptic integral of the first kind.
ke=(tanh((pi/2)*(w/H)))*tanh((pi/2)*(w+s)/H)
ko=(tanh((pi/2)*(w/H)))*coth((pi/2)*(w+s)/H)
ke'=sqrt(1-(ke^2))
ko'=sqrt(1-(ko^2))
Those equations are taken from Matthaei, Young and Jones, Microwave Filters, Impedance Matching Networks and Coupling Structures, Artech House, Dedham, MA., 1980. I'm very grateful to Paul Gili AA1LL / KB1CZP aa1l@email.com for providing me with these equations, references and nomographs.
A programme create_bmp_for_coupler
was written to automatically generate bitmaps given the height H between the groundplanes, the conductor widths w and spacing s. Ideally this needs simulating from -infinity to +infinity, but that is not practical. It was assumed that if the complete structure width W was equal to 2*w+s+8*H that would be adequate (this seemed about right, but I've no proof it is optimal). As well as producing a bitmap, create_bmp_for_coupler
also calculate the theoretical values of impedance. In order to perform calculations of the theoetical impedance, create_bmp_for_coupler
uses the GNU scientific library gsl for the elliptic integral calculations. Hence to obtain the theoretical values, gsl, available at http://sources.redhat.com/gsl, must be installed. The bitmaps will still be produced without the gsl library.
Filename | W | H | w | s | Er | Zoddtheory | Zoddatlc | Errorodd/sup> | Zeventheory | Zevenatlc | Erroreven |
coupler1 | 11 | 1.0 | 1.0 | 1.0 | 1.0 | 64.768 | 64.375 | -0.61% | 66.015 | 65.621 | -0.56% |
coupler2 | 19 | 1.991 | 1.0 | 1.0 | 1.0 | 93.12 | 92.418 | -0.32% | 106.90 | 106.36 | -0.53% |
coupler3 | 27 | 3.0 | 1.0 | 1.0 | 1.0 | 105.48 | 104.71 | -0.73% | 139.77 | 138.93 | -0.60% |
coupler5 | 43 | 5.0 | 1.0 | 1.0 | 1.0 | 114.32 | 113.33 | -0.86% | 189.27 | 188.13 | -0.60% |
coupler6 | 10.0 | 1.0 | 1.0 | 0.5 | 1.0 | 62.200 | 61.792 | -0.66% | 68.243 | 67.869 | -0.55% |
coupler7 | 10.0 | 1.0 | 1.0 | .099 | 1.0 | 50.723 | 50.322 | -0.79% | 74.403 | 74.000 | -0.54% |
coupler8 | 5 | 0.25 | 1.0 | 1.0 | 1.0 | 21.221 | 21.069 | -0.72% | 21.221 | 21.069 | -0.71% |
Note, because of the need to simulate a large width W, these calculations are much slower than those above. The fact the simulation has to be done twice also adds to the time taken. A typical run, using the default bitmap size (18, which gives bitmap of 128 to 512kb), the time taken is approximately 2 minutes on a dual 300 MHz Sun Ultra 60.
atlc is written and supported by Dr. David Kirkby (G8WRB) It it issued under the GNU General Public License