Sonnet User's Guide - Sonnet Software

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Sonnet User’s Guide For evaluation on a computer, a polynomial for K(k) is available in Abramowitz and Stegun, Handbook of Mathematical Functions, pp. 590 - 592. (Be sure to note the errata, m1 = 1-m2, not 1-m.): b w k η 0 Z 0 ε r = ----- Kk' ------------ ( ) 4 Kk ( ) π = tanh -- w ⎝ ⎛ 2 --- b⎠ ⎞ k' = 1 – k 2 η 0 = 376.7303136 The expression for K(k) cited above provides an accuracy of about 1 x 10 -8 . When programmed on a computer, the following values are obtained for three different transmission line impedances (unity dielectric constant): Table 1 Stripline Benchmark Dimensions Z 0 (ohms) w/b 25.0 3.3260319 50.0 1.4423896 100.0 0.50396767 For a length of stripline, there are two parameters of interest: characteristic impedance and propagation velocity. With the w/b given above, we know the exact answer (to within 1 x 10 -8 ) for Z 0 . With a dielectric constant of 1.0, we also know the exact answer for the propagation velocity. It is the speed of light, known to about 1 x 10 -9 . Any difference from these values is error, or, DMAC. 340
Chapter 23 Accuracy Benchmarking Each of the above three benchmarks is available in the Sonnet examples. To get the 50 ohm line, get the example S50. The other benchmark circuits are in “S25 and S100. For directions on obtaining a Sonnet example, select Help ⇒ Examples from the menu of any Sonnet program, then click on the Instructions button. The “b” dimension is exactly 1.0 mm, the “w” dimension is given by the above table and the length of each line is 4.99654097 mm with a dielectric constant of 1.0. Each of these lines is precisely 0.25 wavelengths long at 15.0 GHz. The geometry projects have the subsectioning set so the lines are 16 cells wide and 128 cells long. To evaluate DMAC, do an analysis of the line at 15 GHz, with de-embedding enabled. For the error in characteristic impedance, take the percent difference between the calculated value and the exact value, above. For the error in propagation velocity, take the percent difference between the calculated S 21 phase and -90 degrees. Total error, in percent, is the sum of the two errors. Some types of analyses do not calculate characteristic impedance. A detailed error analysis shows that, to first order for a 1/4 wavelength long 50 ohm line, the value of |S 11 | is equal to the error in characteristic impedance. For example, an |S 11 | = 0.02 means that there is about 2% error in characteristic impedance. To use this approximation for, say, a 25 ohm line, the S-parameters must be converted to 25 ohm S-parameters. This may be done by adding transformers in a circuit theory program. Residual Error Evaluation We have performed a detailed analysis of the relationship between subsectioning and residual error (DMAC). The simplest way to subsection a line is to use subsections the same width as the line. In Sonnet, and in many other analyses, this results in a uniform current distribution across the width of the line. In reality, the current distribution is singular at the edges of the line. Since the current distribution is symmetrical about the center line, using either one or two subsections across the width of the line gives the same amount of error. We find that a one or two subsection wide line gives 5% to 6% error. If there is not much stray coupling, circuit theory can often give a better result. When the line is 16 cells wide, we see about 1% error, much more reasonable. We have found (and you can verify) that convergence is very strong: Double the number of cells per line width and the error is cut in half. 341
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