# Board stackup's important tool: for a target differential impedance, a 2D field solver, used accurately, can be a time-saver.

THE MOST IMPORTANT tool for designing a circuit board stackup is an accurate 2D field solver. This tool will help design the optimum line widths and dielectric thickness, given the stackup and dielectric constants, for a target characteristic impedance.

Many board designs call for a target differential impedance. Many of the newest 2D field solvers also calculate differential impedance for a variety of geometries, with the assumption that each line in the pair is identical and perfectly symmetrical.

Any two uniform and symmetrical transmission lines can be considered as either one differential pair or as two single-ended transmission lines with some coupling. As a differential pair, there are two sets of equivalent terms that describe its electrical properties: differential impedance and common impedance, or odd-mode impedance and even-mode impedance. In addition to differential impedance, many 2D field solvers calculate common impedance, odd-mode and even-mode impedance.

As two coupled lines, the pair is described with a single-ended characteristic impedance of each line and a near-end (or backward) crosstalk coefficient, [k.sub.b]. Although most 2D field solvers will only calculate the various impedances for a pair of traces, by using a very simple relationship the same field solver can calculate near-end crosstalk for all geometries.

For two coupled single-ended lines, the difference between odd-mode and even-mode impedance is related to the coupling: the larger the coupling, the bigger the difference. When there is no coupling, i.e., the traces are far apart, the odd-mode and even-mode impedances are of the same value.

The relationship between the odd-mode impedance, Zodd, and even-mode impedance, Zeven, and the near-end crosstalk coefficient, [k.sub.b], is:

[k.sub.b] = -- x (Zeven - Zodd)/(Zeven + Zodd)

This is a fundamental relationship between any two lines in a differential pair and it applies to absolutely all geometries. If the 2D field solver can calculate the even-mode and odd-mode impedance of a differential pair, we can also use the tool to calculate the near-end crosstalk coefficient for the same two single-ended lines.

For example, field solvers such as Polar Instruments' SI8000 2D will calculate the differential impedance, common impedance, odd-mode and even-mode impedance for a differential pair in a variety of cross-sections. The setup and results are presented in a spreadsheet. By adding one line to the spreadsheet, we can also use it to calculate near-end crosstalk.

FIGURE 1 is an example of the geometry setup for an edge-coupled microstrip with a 0.001"-thick solder mask. TABLE 1 shows geometry features and odd- and even-mode impedances as extracted with the field solver. Added to the end of the spreadsheet is the calculated near-end crosstalk coefficient. FIGURE 2 shows the plot of the near-end crosstalk coefficient as the spacing changes.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

DR. ERIC BOGATIN is vice president and CTO of GigaTest Labs (gigatest.com). Details of the principles described in this column can be found in his new book Signal Integrity--Simplified. He can be reached at eric@gigatest.com.
```TABLE 1. Spreadsheet of all geometry features with calculated
odd-more and even-mode impedance and near-end crosstalk

H1 Er1 W1 W2 S1 T1 C1 C2 C3

6 4.2 7 6 5 1.2 1 1 1
6 4.2 7 6 10 1.2 1 1 1
6 4.2 7 6 15 1.2 1 1 1
6 4.2 7 6 20 1.2 1 1 1
6 4.2 7 6 25 1.2 1 1 1
6 4.2 7 6 30 1.2 1 1 1
6 4.2 7 6 35 1.2 1 1 1
6 4.2 7 6 40 1.2 1 1 1
6 4.2 7 6 45 1.2 1 1 1
6 4.2 7 6 50 1.2 1 1 1

H1 CEr CALC TYPE Zodd NEXT

6 4.2 Zodd 45.8 10.9%
6 4.2 Zodd 53.0 5.4%
6 4.2 Zodd 55.9 3.1%
6 4.2 Zodd 57.2 2.0%
6 4.2 Zodd 58.0 1.4%
6 4.2 Zodd 58.4 1.0%
6 4.2 Zodd 58.7 0.8%
6 4.2 Zodd 58.9 0.6%
6 4.2 Zodd 59.0 0.5%
6 4.2 Zodd 59.1 0.4%

H1 Er1 W1 W2 S1 T1 C1 C2 C3

6 4.2 7 6 5 1.2 1 1 1
6 4.2 7 6 10 1.2 1 1 1
6 4.2 7 6 15 1.2 1 1 1
6 4.2 7 6 20 1.2 1 1 1
6 4.2 7 6 25 1.2 1 1 1
6 4.2 7 6 30 1.2 1 1 1
6 4.2 7 6 35 1.2 1 1 1
6 4.2 7 6 40 1.2 1 1 1
6 4.2 7 6 45 1.2 1 1 1
6 4.2 7 6 50 1.2 1 1 1

H1 Cer CALC TYPE Zeven

6 4.2 Zeven 71.4
6 4.2 Zeven 65.9
6 4.2 Zeven 63.3
6 4.2 Zeven 62.0
6 4.2 Zeven 61.3
6 4.2 Zeven 60.9
6 4.2 Zeven 60.6
6 4.2 Zeven 60.4
6 4.2 Zeven 60.3
6 4.2 Zeven 60.2
```
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