1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML>
<HEAD>
<META name="generator" content="bluefish and vi">
<META http-equiv="CONTENT-TYPE" content="text/html; charset=iso-8859-1">
<meta name="author" content="Dr. David Kirkby">
<META name="DESCRIPTION" content="Accuracy of atlc - Arbitrary Transmission Line Calculator">
<META name="KEYWORDS" content="transmission lines, Transmission line, atlc, finite difference">
<TITLE>Accuracy of atlc - Arbitrary Transmission Line Calculator</TITLE>
</HEAD>
<BODY>
<H1>Accuracy of <CODE>atlc</code></H1>
<p>In order to test the accuracy of <CODE>atlc</code>, some simple geometries were devised, for which there are known exact closed-form analytical solutions. The following have been tested. To date, all tests have yielded acceptable accuracy. Out of 48 tests, the higest error measured is 3.05%, with However, note that in most cases where the error is over 1%, is is clearly very obvious why it is. The <a href="accuracy.html#conclusions">conclusions</a> at the end explain the reason for the higher errors observered on some simulations.
<ol>
<li>Two_conductor_uniform_dielectric<a href="accuracy.html#Two_conductor_uniform_dielectric">Two-conductor transmission lines, with a uniform dielectric</a> are discussed in <a href="accuracy.html#Two_conductor_uniform_dielectric">section 1.0</a>. Three cases were considered - the <a href="accuracy.html#Accuracy_coax">standard coaxial cable</a>, an off-centre or <a href="accuracy.html#Accuracy_eccentric_coax">eccentric coaxial cable</a> and a <a href="accuracy.html#Accuracy_symmetrical_strip">symmetrical strip transmission line</a>. These cases were consided since there are closed form exact analytical solutions on which to compare the results of atlc. Other structures will be tested at a later date.
</li>
<li><a href="accuracy.html#Two_conductors_hetrogeneous_dielectrics"> Two conductor transmission lines,
with two different dielectrics</a> are described in <a href="accuracy.html#section_2">section 2</a>.
The only structure considered has been a dual coaxial cable, which has an inner and outer conductor like normal coaxiale cable, but has two different concentric dielectrics. This has an exact closed-form analytical solution. Testing a coaxial cable with thress or more multiple concentric dielectrics will later be performed, as it should be possible to derive an analytical formula for such a strucutre, although this has not currently be done.
I'm unaware of any other structure which has an exact solution when there are two or more dielectrics. If you know of any, please <a href="jpgs/home-email.jpg">e-mail</A> the details. </li>
<li>Three conductors (<a href="accuracy.html#Accuracy_coupler">directional coupler</a>), with a single dielectric are described in <a href="accuracy.html#Section_3_Accuracy_coupler">section 3.0</a>. For three-wires, which can be used to make a 4-port <a href="accuracy.html#Accuracy_coupler">directional coupler</a>, the accuracy was compared using <a href="accuracy.html#Accuracy_coupler">two edge-on strip lines.</a>, as that is the only case I am aware of that has an exact analytical solution. However, comparisions with commercial tools such as the expensive HFSS package will be done at a later date. However, I am relieant on the help of others for this, since I don't personally have access to such expensive commerical software. </li>
</ol>
<H2><a name="Two_conductor_uniform_dielectric">Section 1. Two conductor Transmission Lines with a Uniform Dielectric</a></H2>
This section describes several tests for atlc using just two conductors and one dielectric. Several geometries have exact results, so it makes testing relatively easy.
<H3><a name="Accuracy_coax" ><strong>1.1 Comparisons between atlc and a round coaxial cabl</strong>e</a></H3>
<p>
<img src="jpgs/coax2.jpg" ALT="coaxial line" align="left">An obvious structure to test atlc with two conductor and a single dielectric is the round coaxial cable, which has an impedance:
<br>
<STRONG>
Zo=59.95849160*log<sub>e</sub>(D/d)/sqrt(Er) Ohms</STRONG>
<br>
where D is the inner diameter of the outer conductor and d is the outer diameter of the inner conductor. (The number 59.958491602 is usually seen as 60 in most books, but that is only an approximation).
<br>
Circular conductors can never be defined exactly using a square grid, so differences between the exact answer and <CODE>atlc</code>'s answer are due to:
<UL>
<LI>Errors in representing a circle on a square grid</LI>
<LI>Errors in the method <COde>atlc</code> uses. </LI>
</UL>
<br>
Seven coaxial cables were defined, which exibited a range of impedances between 5.5 and 179.6 Ohms. All eccept one used a vacuum dielectric. The table below shows the theoetical results and the results computed by <code>atlc</code>.
<TABLE border align="left">
<TR>
<TD><STRONG>Filename</STRONG></td>
<TD><STRONG>D</STRONG></TD>
<TD><STRONG>d</STRONG></TD>
<TD>Er</TD>
<TD><STRONG>Zo (theory)</STRONG></TD>
<TD><STRONG>Zo (<COde>atlc</code>)</STRONG></TD>
<TD><STRONG>Error (%)</STRONG></TD>
</TR>
<TR>
<TD>coax-500-200-Er=100.bmp</TD>
<TD>500</TD>
<TD>200</TD>
<TD>100.0</TD>
<TD>5.494</TD>
<TD>5.492</TD>
<TD>-0.036 %</TD>
<TD>0m:43s</TD>
</TR>
<TR>
<td>coax-500-400.bmp</td>
<TD>500</TD>
<TD>400</TD>
<TD>1.0</TD>
<TD>13.379</TD>
<TD>13.381</TD>
<TD>+ 0.020 %</TD>
<TD>0m:11s</TD>
</TR>
<TR>
<TD>coax-500-200.bmp</TD>
<TD>500</TD>
<TD>200</TD>
<TD>1.0</TD>
<TD>54.939</TD>
<TD>54.919</TD>
<TD>-0.036 %</TD>
<TD>0m:43s</TD>
</TR>
<TR>
<TD>coax-400-82.bmp</TD>
<TD>400</TD>
<TD>82</TD>
<TD>1.0</TD>
<TD>95.019</TD>
<TD>95.023</TD>
<TD>+0.004 %</TD>
<TD>0m:31s</TD>
</TR>
<TR>
<TD>coax-500-100.bmp</TD>
<TD>500</TD>
<TD>100</TD>
<TD>1.0</TD>
<TD>96.499</TD>
<TD>96.448</TD>
<TD>-0.053 %</TD>
<TD>1m:08s</TD>
</TR>
<TR>
<TD><a name="coax-500-50.bmp">coax-500-50.bmp</a></TD>
<TD>500</TD>
<TD>50</TD>
<TD>1.0</TD>
<TD>138.060</TD>
<TD>137.944</TD>
<TD>-0.008 %</TD>
<TD>1m:22s</TD>
</TR>
<TR>
<TD>coax-500-25.bmp</TD>
<TD>500</TD>
<TD>25</TD>
<TD>1.0</TD>
<TD>179.620</TD>
<TD>180.022</TD>
<TD>+0.244 %</TD>
<TD>1m:28s</TD>
</TR>
</TABLE><br clear="all">
<br>
<STRONG>Notes:</STRONG>
<br>
<OL>
<LI>These results were obtained with version 4.6.0 of atlc. Other versions will undoubtablly differ slightly as effort is made to improve the algorithms in atlc. </LI>
<LI>Run times quoted are for a Sun Ultra 80 with 4 x 450 MHz and 4 Gb RAM, running Solaris 9. The compiler was gcc-3.2.2 with compiler options <code>-O2 -g</code>.</LI>
<LI>The only option used on atlc was the -d option on the occasions where the permittivity was not one of atlc's known values, so it needed to be specified on the command line</LI>
<LI>
<STRONG>The largest error for the coaxial cables is only 0.244 %, the mean error is 0.017 % with the RMS error being 0.089 %. </STRONG>
</LI>
<LI>The larger error in the last result is due to the difficulty in accurately representing a small conductor on a reasonable sized grid. The ratio between the diameters of the outer and inner conductors is 20:1, making it difficult to represent both accurately without a grid become huge and so taking a lot of CPU time.
</LI>
<LI>
In some situations, accuracy can be improved at the expense of memory and CPU time, by using a finer grid and altering the cutoff parameter of atlc. </LI>
<LI>As from version 4.6.0 of atlc, a new program called coax has been provided. This allows the quick computation of the impedance of a coaxial cable. To find the impedance of a coax with an inner of 32 mm, and outer of 120 mm and a relative permittivity of 2.2, just run coax:<br><br>
<code>% coax 32 120 2.2</code>
</OL>
<!-- Section 1.2 --- Comparisons between atlc and an eccentric coaxial line<-->
<H3><STRONG><a name="Accuracy_eccentric_coax" >1.2 Comparisons between atlc and an eccentric coaxial line</A></STRONG></H3>
<img src="jpgs/eccentric_coax2.jpg" ALT="eccentric coaxial line" align="bottom"><p>According to the book <EM>Microwave and Optical Components, Volume 1, - Microwave Passive and Antenna Components</EM>, 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 given by the following equation.
<p><STRONG> 60 log<sub>e</sub>(x+sqrt(x^2-1)) </STRONG>where <STRONG>x=(d<sup>2</sup>+D<sup>2</sup>-4 O<sup>2</sup>)/(2*D*d) </STRONG>
<br>
<p>This will allow a second check of atlc's accuracy with two conductors and one dielectric. Any problems which might be masked by the symmetry of the stardard coaxial cable will be eliminated. </p>
<p>Whenever the number 60 appears in formula for transmission lines, it should in fact be replaced by the number 59.9585. 60 is just a good approximation, but since we are testing atlc, the following formula will be used:<br><br>
<STRONG> 59.9585 log<sub>e</sub>(x+sqrt(x^2-1)) /sqrt(Er) </STRONG>where <STRONG>x=(d<sup>2</sup>+D<sup>2</sup>-4 O<sup>2</sup>)/(2*D*d) </STRONG>
<br>
<br>
Of course, one could constuct the a number of such transmission lines with a graphics package using its ability to draw circles, but getting the correct diamters and offsets would be time confusming. For this reasons the program <a href="create_bmp_for_circ_in_circ.1.html">create_bmp_for_circ_in_circ</a> was used to generate a number of bitamps quickly with the following diameters and offsets.
<TABLE border align="left">
<TR>
<TD><STRONG>Filename</STRONG></td>
<TD><STRONG>D</STRONG></TD>
<TD><STRONG>d</STRONG></TD>
<TD><STRONG>O</STRONG></TD>
<TD>Er</TD>
<TD><STRONG>Zo (theory)</STRONG></TD>
<TD><STRONG>Zo (<COde>atlc</code>)</STRONG></TD>
<TD><STRONG>Error (%)</STRONG></TD>
</TR>
<TR>
<td>eccentric-a.bmp</td>
<TD>500</TD>
<TD>400</TD>
<TD>40</TD>
<TD>2.15</TD>
<TD>5.482</TD>
<TD>5.487</TD>
<TD>+0.091 %</TD>
</TR>
<TR>
<td>eccentric-b.bmp</td>
<TD>400</TD>
<TD>320</TD>
<TD>0</TD>
<TD>1.0</TD>
<TD>13.379</TD>
<TD>13.389</TD>
<TD>+0.075 %</TD>
</TR>
<TR>
<td>eccentric-c.bmp</td>
<TD>500</TD>
<TD>100</TD>
<TD>100</TD>
<TD>10.0</TD>
<TD>29.707</TD>
<TD>29.713</TD>
<TD>+0.020 %</TD>
</TR>
<TR>
<td>eccentric-d.bmp</td>
<TD>500</TD>
<TD>200</TD>
<TD>100</TD>
<TD>1.0</TD>
<TD>41.560</TD>
<TD>41.587</TD>
<TD>+0.065 %</TD>
</TR>
<TR>
<td>eccentric-e.bmp</td>
<TD>500</TD>
<TD>200</TD>
<TD>10</TD>
<TD>1.0</TD>
<TD>54.825</TD>
<TD>54.862</TD>
<TD>+0.067 %</TD>
</TR>
<TR>
<td>eccentric-f.bmp</td>
<TD>400</TD>
<TD>160</TD>
<TD>0</TD>
<TD>1.0</TD>
<TD>54.939</TD>
<TD>54.976</TD>
<TD>+0.067 %</TD>
</TR>
<TR>
<td>eccentric-g.bmp</td>
<TD>400</TD>
<TD>40</TD>
<TD>12</TD>
<TD>5.0</TD>
<TD>61.644</TD>
<TD>61.676</TD>
<TD>+0.052 %</TD>
</TR>
<TR>
<td>eccentric-h.bmp</td>
<TD>400</TD>
<TD>40</TD>
<TD>160</TD>
<TD>1.0</TD>
<TD>73.489</TD>
<TD>73.330</TD>
<TD>-0.216%</TD>
</TR>
<TR>
<td>eccentric-i.bmp</td>
<TD>1600</TD>
<TD>160</TD>
<TD>640</TD>
<TD>1.0</TD>
<TD>73.489</TD>
<TD>73.330</TD>
<TD>-0.216 %</TD>
</TR>
<TR>
<td>eccentric-j.bmp</td>
<TD>500</TD>
<TD>100</TD>
<TD>50</TD>
<TD>1.0</TD>
<TD>93.943</TD>
<TD>93.961</TD>
<TD>+0.019 %</TD>
</TR>
<TR>
<td>eccentric-k.bmp</td>
<TD>500</TD>
<TD>100</TD>
<TD>0</TD>
<TD>1.0</TD>
<TD>96.499</TD>
<TD>96.524</TD>
<TD>+0.019 %</TD>
</TR>
<TR>
<td>eccentric-l.bmp</td>
<TD>500</TD>
<TD>50</TD>
<TD>100</TD>
<TD>1.0</TD>
<TD>127.467</TD>
<TD>127.524</TD>
<TD>+0.045 %</TD>
</TR>
<TR>
<td>eccentric-m.bmp</td>
<TD>500</TD>
<TD>50</TD>
<TD>50</TD>
<TD>1.0</TD>
<TD>135.586</TD>
<TD>135.654</TD>
<TD>+0.050 %</TD>
</TR>
<TR>
<td>eccentric-n.bmp</td>
<TD>400</TD>
<TD>40</TD>
<TD>20</TD>
<TD>1.0</TD>
<TD>137.451</TD>
<TD>137.519</TD>
<TD>+0.049 %</TD>
</TR>
</TABLE><p><br clear="all"><br>
<ol>
<li>These results were produced with version 4.6.0 of atlc. Results from other versions will probably differ, as efforts are made to further improve atlc.</li>
<li>Due to their large size, the eccentric-?.bmp coax files are not distributed, but you could make them and compute their impedances like this. Add the -v option to create_bmp_for_circ_in_circ (as in example eccentric-d.bmp) if you want the theoretical results printed.
<pre><CODE>
create_bmp_for_circ_in_circ 500 400 40 2.2 eccentric-a.bmp && atlc -d caff00=2.15 eccentric-a.bmp
create_bmp_for_circ_in_circ 400 320 0 1 eccentric-b.bmp && atlc eccentric-b.bmp
create_bmp_for_circ_in_circ 500 100 100 10 eccentric-c.bmp && atlc -d caff00=10 eccentric-c.bmp
create_bmp_for_circ_in_circ -v 500 200 100 1 eccentric-d.bmp && atlc
create_bmp_for_circ_in_circ 500 200 10 1 eccentric-e.bmp && atlc
create_bmp_for_circ_in_circ 400 160 0 1 eccentric-f.bmp && atlc
create_bmp_for_circ_in_circ 400 40 12 5 eccentric-g.bmp && atlc -d caff00=5 eccentric-g.bmp
create_bmp_for_circ_in_circ 400 400 160 1 eccentric-h.bmp && atlc
etc. </CODE>
</pre>
</li>
<li> If you add the <code>-v</code> option to <code>create_bmp_for_circ_in_circ</code>, it will print the <strong>exact</strong> theoretical values for you too.
</li>
<LI>If you only want to compute the impedance of an offset coax, without actually creating the bitmap, the program coax can be used, if you supply the <code>-o offset</code> option. For example:
<pre>
coax -o 40 400 500 1
Zo = 8.038255 Ohms
</pre>
<LI>Since there is an exact answer to this geometry, even when the inner is offset, there is not a lot of point in spending seconds or minutes running <code>atlc</code> to come an approximate numerical answer, when you can compute an exact one in a small fraction of a second. However, using atlc to compute a few of these gives you confidence atlc is working properly.</LI>
</ol>
<p>
<!-- Section 1.3 *** Comparisions between atlc and a symmetrical strip transmission line -->
<H3><a name="Accuracy_symmetrical_strip" ><strong>1.3 Comparisions between atlc and a symmetrical strip transmission line</strong></A></H3>
Another obvious test to determine the performance of atlc with two conductors and one dielectric is a symmetrical strip transmission line - see diagramme below. <p>
<img src="jpgs/symmetrical_strip.jpg" ALT="Symmetrical Strip Transmission Line"><p>
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 <CODE>atlc.</CODE> </p>
<p>
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.</p>
The <code>-i</code> option to <CODE>create_bmp_for_symmetrical_stripline</code>,
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 <code>-i</code> option is used, a valid test of <code>atlc</code>'s
accuracy can be made<br>
Without the <code>-i</code> 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 that is cleraly visable.
<br>
<br>
<CODE>create_bmp_for_symmetrical_stripline -vv -i 0 201 290 50-201.bmp<br>
<br>
For this to be a valid test of atlc, the width should be<br>
infinite. Since you used the -i option (indicationg you<br>
want the width W to effectively infinite, W must exceed w + 4xH.<br>
Therefore W has been is set to 1134<br>
w=290 H=201 w/H=1.442786 xo=23.7538<br>
Zo is theoretically 49.989477 Ohms (assuming W is infinite)<br>
<br>
</code>
<p>
This structure, which has a w/H value of 1.442786, has a theoretical impedance close to 50 Ohms (49.989477 to be precise). Version 4.6.0 of <code>atlc</code> calculates this to be 49.899 Ohms, an error of -0.181%, when using a grid 1134x201.</p>
<TABLE border align="left">
<TR>
<TD><STRONG>Filename</STRONG></TD>
<TD><STRONG>W</STRONG></TD>
<TD><STRONG>H</STRONG></TD>
<TD><STRONG>w</STRONG></TD>
<TD><STRONG>w/H</STRONG></TD>
<TD><STRONG>Zo<sup>exact</sup></STRONG></TD>
<TD><STRONG>Zo<sup>atlc</sup></STRONG></TD>
<TD><STRONG>Error</STRONG></TD>
<TD><STRONG>Time</STRONG></TD>
</TR>
<TR>
<TD>25ohm-201h.bmp</TD>
<TD>1512</TD>
<TD>201</TD>
<TD>668</TD>
<TD>3.3234</TD>
<TD>25.018</TD>
<TD>24.932</TD>
<TD>-0.344 %</TD>
<TD>0h:00m:46s</TD>
</TR>
<TR>
<TD>25ohm-401h.bmp</TD>
<TD>2978</TD>
<TD>401</TD>
<TD>1334</TD>
<TD>3.3267</TD>
<TD>24.996</TD>
<TD>24.940</TD>
<TD>-0.224%</TD>
<TD>0h:08m:52s</TD>
</TR>
<TR>
<TD>25ohm-801h.bmp</TD>
<TD>6000</TD>
<TD>801</TD>
<TD>2664</TD>
<TD>3.3267</TD>
<TD>25.001</TD>
<TD>24.935</TD>
<TD>-0.264%</TD>
<TD>1h:49m:46s</TD>
</TR>
<TR>
<td>50ohm-201h.bmp</td>
<TD>1134</TD>
<TD>201</TD>
<TD>290</TD>
<TD>1.42786</TD>
<TD>49.989</TD>
<TD>49.899</TD>
<TD>-0.180%</TD>
<TD>0h:00m:37s</TD>
</TR>
<TR>
<TD>50ohm-401h.bmp</TD>
<TD>2222</TD>
<TD>401</TD>
<TD>578</TD>
<TD>1.4419</TD>
<TD>50.026</TD>
<TD>49.944</TD>
<TD>-0.164%</TD>
<TD>0h:07m:16s</TD>
</TR>
<TR>
<TD>50ohm-801h.bmp</TD>
<TD>4399</TD>
<TD>801</TD>
<TD>1155</TD>
<TD>1.4419</TD>
<TD>50.012</TD>
<TD>49.878</TD>
<TD>-0.268%</TD>
<TD>1h:46m:31</TD>
</TR>
<TR>
<TD>100ohm-201h.bmp</TD>
<TD>945</TD>
<TD>201</TD>
<TD>101</TD>
<TD>0.5025</TD>
<TD>100.161</TD>
<TD>100.319</TD>
<TD>+0.158%</TD>
<TD>0h:00m:34s</TD>
</TR>
<TR>
<TD>100ohm-401h.bmp</TD>
<TD>1846</TD>
<TD>401</TD>
<TD>202</TD>
<TD>0.5037</TD>
<TD>100.02</TD>
<TD>99.998</TD>
<TD>-0.022%</TD>
<TD>0h:06m:42s</TD>
</TR>
<TR>
<TD>100ohm-801h.bmp</TD>
<TD>3647</TD>
<TD>801</TD>
<TD>403</TD>
<TD>0.5037</TD>
<TD>100.09</TD>
<TD>99.857</TD>
<TD>-0.233%</TD>
<TD>1h:29m:17s</TD>
</TR>
<TR>
<TD>200ohm-201h.bmp</TD>
<TD>862</TD>
<TD>201</TD>
<TD>18</TD>
<TD>0.0896</TD>
<TD>200.81</TD>
<TD>204.210</TD>
<TD>+1.693%</TD>
<TD>0h:0m:31s</TD>
</TR>
<TR>
<TD>200ohm-401h.bmp</TD>
<TD>1680</TD>
<TD>401</TD>
<TD>36</TD>
<TD>0.08978</TD>
<TD>200.669</TD>
<TD>201.844</TD>
<TD>+0.586%</TD>
<TD>0h:06m:22s</TD>
</TR>
<TR>
<TD>200ohm-801h</TD>
<TD>3317</TD>
<TD>801</TD>
<TD>73</TD>
<TD>0.09114</TD>
<TD>199.771</TD>
<TD>199.734</TD>
<TD>-0.019%</TD>
<TD>1h:23m:08s</TD>
</TR>
<TR>
<TD>400ohm-1551h</TD>
<TD>6439</TD>
<TD>1551</TD>
<TD>5</TD>
<TD>0.00322</TD>
<TD>400.040</TD>
<TD>417.700</TD>
<TD>+4.415%</TD>
<TD>12h:20m:50s</TD>
</TR>
<TR>
<TD>400ohm-76610h</TD>
<TD>31109</TD>
<TD>7750</TD>
<TD>25</TD>
<TD>0.00323</TD>
<TD>400.085</TD>
<TD></TD>
<TD>%</TD>
<TD></TD>
</TR>
</TABLE>
<p><br clear="all">
Notes:
<br>
<OL>
<li>These results were produced with version 4.6.0 of atlc. Results from other versions will probably differ, as efforts are made to further improve atlc.</li>
<LI>For the same sepparation between the two outer conductors h, the width of the inner conductor w decreases as the impedance of the line is increased. When this width w is too small, accuracy suffers. In order to get reasonable accuracy it is essential to use sufficient pixels for the width of the conductor w.</LI>
<LI>Run times are given when compiled single-threaded, with <code>gcc-3.2.2</code> on a Sun Ultra 80 workstation with 4x450 MHz CPUs and 4 GB of RAM. Compiler options of <code>-O2 -g</code> were used.</LI> <LI>Only 1 CPU in the Ultra 80 would have been used, since <code>atlc</code>
was compiled single-threaded.</LI>
<LI>
The -s and -S options were given to <code>atlc</code> 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. </LI>
<LI>Compiled as a multi-threaded application to use all 4 CPUs in a Sun
Ultra 80, the run times reduce by a factor of approximately 3.5.</LI>
<LI>The result for the first 400 Ohm transmission line analysed (400ohm-1551h.bmp) is very poor (3.5% error)
since only 5 pixels could be used for the width of the inner conductor.
The results from this test are not included when calculating the overall
accuracy of atlc, since using only 5 pixels is not a fair test.
An attempt at using 25 pixels for the inner conductor's width, where accuracy should
have been better, created a 689 MB bitmap file, which could not be analysed, as
the RAM in the computer (2 GB) was insufficient, although this might be tried later since the computer has since been upgraded.</LI>
</OL>
<H2><a name="section_2">Section 2</a>. Two-conductor Transmission Lines with a non-uniform dielectric</h2>
Destermining altc's accurarcy with multiple-dielectrics is not easy, as there are few analytical methods. The only one known in dual dielectric coax. At a later date some comparisions will be made to commerical software if this is possible.
<!-- Section 2.1 **** Comparision of atlc and a dual dielectric coaxial cable -->
<H3>2.1 Comparision of atlc and a dual dielectric coaxial cable</H3>
A coaxial cable with two concentric dielectrics like that below<br>
<img src="jpgs/dual-dielectric-coax.jpg" ALT="dual dielectric coaxial line" align="left">
<br clear="all">
<p>has an exact analytical solution. The red is the inner conductor, the green forms the outer conductor. The light blue and orange regions are both dielectrics, neither of which are one of atlc's <a href="colours.html">predefined colours</a>, so the dielectric constant of both must be set by issuing the -d option to atlc. (The light blue in this image, is not to be confused with the light blue that is pre-defined for PTFE with a dielectric constant of 2.1).</p>
<p>A small program called <code>dualcoax</code> can be used to compute the impedance of a dual coaxial cable. If the diameter of the inner conductor is 135, the inner dielectric 337, the internal diameter of the outer conductor is 401, the permittivity of the inner dielectric 2.0 and the outer dielectric 3.0, then:
<br><br>
<code>$ dualcoax 135 337 401 2.0 3.0</code>
<br><br>
will compute the impedance, which is 44.912 Ohms.
<br clear="all">
The following table shows the impedances for various values of permittivity of both the inner and outer dielectrics. Note that changing the relative permittivity of the outer conductor has little effect, as it is quite thin, whereas the outer dielectric is much thicker and so has more effect on the impedance.
<table border align="left">
<tr>
<td>Filename</td>
<td>D1</td>
<td>D2</td>
<td>D3</td>
<td><strong><font color="#fd8a11">Er<sub>inner</sub></font></strong></td>
<td><strong><font color="#8b8dff">Er<sub>outer</sub></font></strong></td>
<td>Zo(theory)</td>
<td>Zo(atlc)</td>
<td>Error</td>
<td>Time</td>
</tr>
<tr>
<td>dual-dielectric-coax.bmp</td>
<td>156</td>
<td>400</td>
<td>500</td>
<td>1.0</td>
<td>1.0</td>
<td>69.837</td>
<td>69.848</td>
<td>+0.017%</td>
<td>0h:00m:59s</td>
</tr>
<tr>
<td>dual-dielectric-coax.bmp</td>
<td>156</td>
<td>400</td>
<td>500</td>
<td>3.0</td>
<td>1.0</td>
<td>47.420</td>
<td>46.681</td>
<td>-1.559%</td>
<td>0h:04m:35s</td>
</tr>
<tr>
<td>dual-dielectric-coax.bmp</td>
<td>156</td>
<td>400</td>
<td>500</td>
<td>10.0</td>
<td>1.0</td>
<td>36.451</td>
<td>35.839</td>
<td>-1.679 %</td>
<td>0h:10m:17s</td>
</tr>
<tr>
<td>dual-dielectric-coax.bmp</td>
<td>156</td>
<td>400</td>
<td>500</td>
<td>30.0</td>
<td>1.0</td>
<td>32.647</td>
<td>32.314</td>
<td>-1.020%</td>
<td>0h:17m:53s</td>
</tr>
<tr>
<td>dual-dielectric-coax.bmp</td>
<td>156</td>
<td>400</td>
<td>500</td>
<td>1000000.0</td>
<td>1.0</td>
<td>30.568</td>
<td>30.330</td>
<td>-0.779 %</td>
<td>1h:18m:17s</td>
</tr>
<tr>
<td>dual-dielectric-coax.bmp</td>
<td>156</td>
<td>400</td>
<td>500</td>
<td>1.0</td>
<td>2.0</td>
<td>66.408</td>
<td>65.974</td>
<td>+0.658%</td>
<td>0h:02m:01s</td>
</tr>
<tr>
<td>dual-dielectric-coax.bmp</td>
<td>156</td>
<td>400</td>
<td>500</td>
<td>1.0</td>
<td>1000000.0</td>
<td>62.792</td>
<td>62.727</td>
<td>-0.014%</td>
<td>0h:10m:02s</td>
</tr>
<tr>
<td>dual-dielectric-coax.bmp</td>
<td>156</td>
<td>400</td>
<td>500</td>
<td>2.5</td>
<td>3.5</td>
<td>42.943</td>
<td>42.858</td>
<td>-0.198 %</td>
<td>0h:01m:55s</td>
</tr>
</table>
<br clear="all">
Notes:
<OL>
<LI>These results were produced with version 4.6.0 of <code>atlc</code>.</LI>
<LI>
To compute these results, one must run atlc with the -d option to define what the relative dielectric constant for each colour, as described in the section on <a href="colours.html">producing suitable bitmaps</a>. The light blue colour has a hex representation of 0x8b8dff and the orange is 0xfd8a11. So for the last entry in the table, one would run<br><br>
<code>$ atlc -d fd8a11=2.5 -d 8b8dff=3.5 dual-dielectric-coax.bmp</code>
</LI>
</OL>
<br clear="all">
Ckearly the accuracy of atlc with multiple dielectrics is lower than that with a single dielectric, where typical errors are around 0.1%. This is believed to be due to the fact the equations used when there are multiple dielectrics are not precise and in a later version it is hoped to refine the equations, so accuracy improves.
<H2><a name="Section_3_Accuracy_coupler">Section 3. Accuracy of <CODE>atlc</CODE> with coupled lines</a></H2>
Testing the accuracy of <CODE>atlc</CODE> 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)
<p>
<img src="jpgs/coupler.jpg" ALT="coupled lines" width="400"><p>
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,
<pre>
----------^--------------------------------------------------------------
|
| <---w---><-----s----><---w-->
H --------- --------
| Er
|
----------v--------------------------------------------------------------
</pre>
then, according to the book by Matthaei, Young and Jones called <EM>Microwave Filters, Impedance Matching Networks and Coupling Structures</EM>, Artech House, Dedham, MA., 1980. the impedances are given by <p>
Zeven=(30*pi/sqrt(er))*(K(ke')/K(ke))<br>
Zodd=(30*pi/sqrt(er))*(K(ko')/K(ko))<p>
K(kx)=complete elliptic integral of the first kind. <p>
ke=(tanh((pi/2)*(w/H)))*tanh((pi/2)*(w+s)/H)<br>
ko=(tanh((pi/2)*(w/H)))*coth((pi/2)*(w+s)/H)<p>
ke'=sqrt(1-(ke^2))<br>
ko'=sqrt(1-(ko^2))<p>
Again, I suspect 30 is just an approximation, like 60 is used in the impedance for coax, and so the values should be:<br><br>
Zeven=(29.97924580*pi/sqrt(er))*(K(ke')/K(ke))<br>
Zodd=(29.97924580*pi/sqrt(er))*(K(ko')/K(ko))<p>
<br><br>
I'm very grateful to Paul Gili AA1LL / KB1CZP <a href="mailto:aa1ll@email.com">aa1l@email.com</a> for providing me with these equations, references and nomographs.
<p>
A programme <CODE>create_bmp_for_stripline_coupler</CODE> 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 <EM>about right</EM>, but I've no proof it is optimal). As well as producing a bitmap, <CODE>create_bmp_for_stripline_coupler</CODE>also calculates the theoretical values of impedance. </p>
The above were created using the following set of commands
<pre>
<code>
$ create_bmp_for_stripline_coupler -v 1 1 1 1 coupler1.bmp
$ create_bmp_for_stripline_coupler -v 1.991 1 1 1 coupler2.bmp
$ create_bmp_for_stripline_coupler -v 3 1 1 1 coupler3.bmp
$ create_bmp_for_stripline_coupler -v 5 1 1 1 coupler4.bmp
$ create_bmp_for_stripline_coupler -v 1 1 0.5 1 coupler5.bmp
$ create_bmp_for_stripline_coupler -v 1 1 0.099 1 coupler6.bmp
$ create_bmp_for_stripline_coupler -v 0.25 1.19 1.34 2.2 coupler7.bmp
</code>
</pre>
<TABLE border align="left">
<TR>
<TD><STRONG>Filename</STRONG></TD>
<TD><STRONG>H</STRONG></TD>
<TD><STRONG>w</STRONG></TD>
<TD><STRONG>s</STRONG></TD>
<TD><STRONG>Er</STRONG></TD>
<TD><STRONG>Zodd<sup>theory</sup></STRONG></TD>
<TD><STRONG>Zodd<sup>atlc</sup></STRONG></TD>
<TD><STRONG>Error<sup>odd</sup></STRONG></TD>
<TD><STRONG>Zeven<sup>theory</sup></STRONG></TD>
<TD><STRONG>Zeven<sup>atlc</sup></STRONG></TD>
<TD><STRONG>Error<sup>even</sup></STRONG></TD>
</TR>
<TR>
<TD>coupler1.bmp</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>64.723</TD>
<TD>64.308</TD>
<TD>-0.641%</TD>
<TD>65.969</TD>
<TD>65.540</TD>
<TD>-0.300%</TD>
</TR>
<TR>
<TD>coupler2.bmp</TD>
<TD>1.991</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>93.056</TD>
<TD>92.711</TD>
<TD>-0.371%</TD>
<TD>106.830</TD>
<TD>106.437</TD>
<TD>-0.368%</TD>
</TR>
<TR>
<TD>coupler3.bmp</TD>
<TD>3.0</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>105.409</TD>
<TD>105.072</TD>
<TD>-0.320%</TD>
<TD>139.670</TD>
<TD>139.091</TD>
<TD>-0.415%</TD>
</TR>
<TR>
<TD>coupler4.bmp</TD>
<TD>5.0</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>114.237</TD>
<TD>114.217</TD>
<TD>-0.018%</TD>
<TD>189.135</TD>
<TD>188.629</TD>
<TD>-0.268%</TD>
</TR>
<TR>
<TD>coupler5.bmp</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>0.5</TD>
<TD>1.0</TD>
<TD>62.133</TD>
<TD>61.887</TD>
<TD>-0.396%</TD>
<TD>68.215</TD>
<TD>67.941</TD>
<TD>-0.402%</TD>
</TR>
<TR>
<TD>coupler6.bmp</TD>
<TD>1.0</TD>
<TD>1.0</TD>
<TD>.099</TD>
<TD>1.0</TD>
<TD>50.614</TD>
<TD>50.546</TD>
<TD>-0.134%</TD>
<TD>74.377</TD>
<TD>73.883</TD>
<TD>-0.664%</TD>
</TR>
<TR>
<TD>coupler7.bmp</TD>
<TD>0.25</TD>
<TD>1.19</TD>
<TD>1.34</TD>
<TD>2.2</TD>
<TD>12.208</TD>
<TD>12.062</TD>
<TD>-1.196%</TD>
<TD>12.208</TD>
<TD>12.062</TD>
<TD>-1.196%</TD>
</TR>
</TABLE>
<p><br clear="all">
Note:
<ol>
<LI>The data was collected with version 4.6.0 of atlc. As always, the data may change with later versions of atlc, as the code is further improved. </LI>
<LI>The theoretical impedance quoted are for the dimensions in the table. The actual bitmap produced will frequently not be the same as it's impossible to represent perfectly an arbitrary grid on a grid with finite resolution. The program <code>create_bmp_for_stripline_coupler</code> also computes the theoretical valus for the actual grid generated, but these have not been used.</LI>
<LI>There seems to be some systematic error for this coupler, as the impedances determined by atlc are always below the theoretical values. The source of this error will be investigated. It is clear that the error decreases as the height h is increased.</LI>
<LI>Run times quoted are for a Sun Ultra 80 with 4 x 450 MHz and 4 Gb RAM, running Solaris 9. The compiler was gcc-3.2.2 with compiler options <code>-O2 -g</code>.</LI>
</ol>
<H2>Section 4. Conclusions about the accuracy of atlc</H2>
<p>Looking at the above data it is clear that on some structures (such as a standard coaxial cable or an eccentric cable, the accuracy of atlc is excellent. Of the 21 tests for these structures, 18 had errors of below 0.1% and the other three had errors below 0.25%. Each structure is round, with a minimum diamater of 25 pixels. The outside edge has around Pi*25=79 pixels to reprsent it, so errors due to quantising the electric field are quite small. </p>
<p>Of the three results for coaxial and eccentric cables that show errors over 0.1%, the reasons are not had to understand. File <a href="accuracy.html#coax-500-25.bmp">coax-500-25.bmp</a> has the smallest number of pixels for the centre conductor in any of the standard coaxial cables. With the smallest number of pixels (a diameter of 25 and a circumference of 78 pixels), the errors can be expected to be highest. It is likely the errors could be reduced by making conductors larger, but there is no point, as the error (just 0.244%) are negligable.
You can't expect to accurately model any structure where one of the critical dimensions is represented by too few pixels. </p>
<p>Clearly if any dimention needs to be reprsented by 5.4 pizels, the nearest number is 5 pixels, so immediately an error of 8% has been introduced. But since the electric field (which varies continuously) is only computed at 5 places, the true varition can't be known accurately. </p>
<p> <strong>As a rule of thumb, try to keep any critital dimension to at least 25 pixels. </strong></p>
<a href="http://atlc.sourceforge.net">Return to the atlc homepage</a>
<br><br>
atlc is written and supported by <a href="jpgs/home-email.jpg">Dr. David Kirkby (G8WRB)</A> It it issued under the <a href="http://www.gnu.org/copyleft/gpl.html">GNU General Public License</A>
<br>
<BR>
<BR>
<A href="http://sourceforge.net"> SourceForge.net</A>
<a href="http://validator.w3.org/check/referer"><img border="0"
src="valid-html40.gif"
alt="Valid HTML 4.01!" height="31" width="88"></a>
<a href="http://bluefish.openoffice.nl/"><img SRC="jpgs/bluefish.jpg" WIDTH="88" HEIGHT="31" ALT="Bluefish"></a>
<p>The following is a trap for smammers, so they can gather loads of ficticious email address, so don't click<a href="http://homepage.ntlworld.com/drkirkby/list1.html"> anywhere</a>
<a href="http://homepage.ntlworld.com/drkirkby/list2.html"> o</a>
<a href="http://homepage.ntlworld.com/drkirkby/list3.html">n</a>
<a href="http://homepage.ntlworld.com/drkirkby/list4.html"> this</a>
<a href="http://homepage.ntlworld.com/drkirkby/list5.html"> line</a>
<a href="http://homepage.ntlworld.com/drkirkby/list6.html"> th</a>
<a href="http://homepage.ntlworld.com/drkirkby/list7.html"> anks.</a>
</body>
</html>
|