## 2-D Quasistatic Field Solvers

I love signal-integrity simulations. Unfortunately, they don't always produce the right answers. For example, most signal-integrity-software packages calculate the impedance and loss of transmission lines using a 2-D, quasistatic, discrete field solver. The field solver depends on six crucial assumptions. If your system violates any of these assumptions, the simulator produces wrong answers (references 1 to 3).

### The fringing-field assumption

Two-dimensional field solvers do not calculate the effects of fringing fields at the ends of the conductors. This omission seems reasonable as long as the main effects in the middle of the line vastly exceed the fringing-field effects at the ends. To satisfy this requirement, the transmission-line length must vastly exceed the separation between its conductors.

For typical pc-board-trace dimensions, the fringing-field assumption holds. For example, a 1-in. transmission line that is 0.005 in. above the reference plane has a length-to-separation ratio of 200 to 1. Under these conditions, the end effects probably have a less-than-1% overall effect on the behavior of the line.

### The assumption of uniformity

If, in addition to being long, the transmission line possesses a uniform cross section, you may assume by symmetry that the per-unit-length values of *r*, *l*, *g* and *c* are the same at every point along the line. The software then performs its analysis only once for a single 2-D cross section of the line.

The assumption of uniformity reduces the complexity of the simulation problem from a full 3-D simulation problem to a problem involving only one cross section (a 2-D problem).

Any percentage imperfections or wobbles in the width and height of the line directly impact the model's accuracy. For example, if your trace width is specified as 3-mil (±1), the modeling error could be as great as ±33

### The quasistatic assumption

When your simulator calculates the distribution of current (or charge) across the face of a particular 2-D slice of a transmission line, it ignores the phase. The quasistatic assumption ensures the phase will be uniform everywhere within the slice. The quasistatic assumption works only when the transmission-line waves propagate in a TEM (transverse electromagnetic) mode, which requires that the signal wavelength greatly exceed the conductor separation.

Typical pc-board traces at frequencies less than 10 GHz comply with the quasistatic assumption. For example, an FR-4 stripline placed 0.005-in above the nearest reference plane has a wavelength-to-separation ratio at 10 GHz of better than 100 to 1. Under these conditions, any quasistatic behavior probably has a less-than-one-percent overall effect on the behavior of the line.

### The small-skin-depth assumption

The inductance of a transmission line changes slightly at frequencies near the onset of the skin effect. To avoid having to contemplate frequency-varying values for inductance, most programs assume that your design operates at a frequency far above the onset of the skin effect so that changes in inductance become insignificant.

At such a high frequency, the skin depth is small. Current flows in a shallow band just beneath the surface of each conductor but not in the middle of the conductor. The importance of the small-skin-depth assumption is that the software needs to calculate values of the current distribution only around the (1-D) perimeter of each conductor cross section, instead of throughout the entire (2-D) body of each conductor. This assumption reduces the complexity of the simulation from a 2-D problem to a 1-D problem.

For rectangular traces at pc-board dimensions of w=0.008 in. t=0.00065 in. (½-oz copper), the skin-effect onset happens at the following frequency:

At frequencies well above the skin-depth onset, the 1-D calculations yield the correct answer. Furthermore, standard assumptions about how the skin effect works reasonably extrapolate the changes in inductance at lower frequencies. If, however, you are simulating conductors at frequencies near the skin-effect-onset frequency, a more comprehensive 2-D simulation of the current distribution may be necessary.

### The discrete assumption

All field solvers represent the conductors as a discrete collection of current (or charge) sources and flux windows. A 2-D quasistatic field solver represents the perimeter of a 2-D conductor cross section as an array of short line segments. It represents the current density in each segment as a 1-D vector, with each point of the vector specifying the current density in one line segment. Different simulators make various assumptions about the interpolation of current values as you move from segment to segment.

The simulator evaluates Maxwell's field equations at a finite number of field-constraint windows. In the final solution, the total flux integrated over each window must lie either tangent or perpendicular to the surface of the conductor, according to whether the simulator is working on the magnetic or electric-field part of the problem, respectively. Obviously, this discrete approach to the problem works only when the size of the discrete segments is small compared with the curvature of the conductors. Commercial simulators rarely describe in a forthcoming manner the degree of imperfection that their discrete approximations introduce.

### The round-corner assumption

Field simulators generate slightly erroneous results at corners. Most generate better-looking results (with less spurious peaking at the corners) if you round off the corners of your conductors before doing the computations, because doing so reduces the curvature of the simulated structure. However, if your corners aren't rounded in the real world, you may wonder what effect the artificial rounding has on the accuracy of the results. I do.

### References

**[1]** Archambeault, Bruce, "EMI/EMC Computational Modeling Handbook," Kluwer Academic Publishers, 1997, ISBN: 0-412-12541-2.

**[2]** Bossavit, Alain, "Computational Electromagnetism," Academic Press 1998, ISBN 0-12-118710-1.

**[3]** Balanis, Constantine, "Advanced Engineering Electromagnetics,” John Wiley, 1989,

ISBN 0-471-62194-3.