# How EM Works

## Introduction

Sonnet® analyzes high frequency planar circuits based on electromagnetics (EM). For example, the circuit in Figure 1 is an air bridge crossover. Two of the three dielectric layers are air. Two of four ports are shown. Vias allow the line connected to port 1 to go up and over the port 2 line. The circuit is contained inside a conducting box, the top cover and ground plane can have any value of resistivity. The lossless sidewalls form a perfect ground reference for each port, allowing an extremely precise analysis.

Sonnet is based directly on Maxwell’s equations. Starting in 1983, Sonnet has been designed to provide the most robust and most accurate EM results in the entire industry. In this paper, we describe, in simple terms, how Sonnet works and how Sonnet achieves its industry standard accuracy.

## Sonnet Theory

Sonnet uses the Method of Moments applied directly to Maxwell’s Equations to solve planar problems. For a detailed mathematical description of the Method of Moments, see [1]. For a detailed description of the theory used by Sonnet, see [2]. Following is a non-mathematical description of Sonnet’s implementation of the Method of Moments:

Sonnet first subdivides, or meshes, the metal of a circuit into small subsections. It then takes one of these subsections and, ignoring all other subsections, calculates the voltage everywhere due to current on that one subsection. Sonnet repeats this process for each subsection in turn. Then Sonnet places current on all subsections simultaneously and adjusts those currents such that the total voltage (due to current on all subsections) is zero everywhere that there is conductor. You can not have a voltage across a conductor. This is called a “boundary condition”. The current that gives us zero voltage across all conductors is the solution to the problem (see below for a modification to this when loss is included).

The FFT (Fast Fourier Transform) is familiar to many people from signal processing. Sonnet’s high accuracy and robustness is due to the use of the FFT. If you have also heard of “matrix inversion”, the following additional points are useful:

The calculation of the voltage everywhere due to current on a given subsection is done by a 2-D FFT. The size of the 2-D FFT is equal to the size of the substrate in terms of cells. To reduce memory requirements, you should specify a cell size as large as possible. This makes the FFT as small as possible.

The current-to-voltage coupling for every possible pair of subsections is stored in an NxN matrix, where N is the number of subsections. The “adjustment” of the current on each subsection is done by a matrix inversion. Memory required for the matrix increases with N squared and the time required to invert the matrix increases with N cubed. If you want a fast analysis, you must keep N as small as possible. To do this, keep the area of the metal to be analyzed (in terms of cell size) as small as possible. Free SonnetLite can handle N up to about 1,400 for lossless circuits and 1,000 if loss is included. The professional Sonnet Suite is only limited by the amount of memory available. Present-day PCs can handle up to about 30,000 subsections, depending on the type of circuit and the amount of available RAM.

Advanced readers may wish to know several other points, summarized as follows:

Not quite all subsection voltages are set to zero. If this were true, then all currents would be zero and we would be done. There must be at least one voltage source, or excitation, to make current flow. A voltage source is created by setting the voltage on one or more subsections to 1.0 Volts. Such a subsection is called a ‘port’ and can be viewed as an input and/or output as desired.

Metal loss is included by setting the voltage on each subsection (except ports), not to zero, but to a value proportional to the current flowing through it. Sound familiar? This is called Ohm’s Law. For more information on conductor loss, see An Investigation of Microstrip Conductor Loss, which can be downloaded on the Sonnet publications web page.

When an FFT is used in digital signal processing, the signal must first be sampled uniformly in the time domain. Sonnet uses a 2-D FFT for the coupling calculation. The two dimensions are the surface of the substrate and the substrate must be uniformly sampled in the space (x-y) domain. Thus, Sonnet has a uniform underlying FFT mesh that the user specifies. Alternative techniques use a 4-D numerical integration to calculate coupling. Such techniques have the advantage of not requiring an FFT mesh, however, they perform poorly in limit situations, for example, with small subsection size.

In order to reduce N, the number of subsections, Sonnet automatically groups individual FFT cells together into larger subsections. To keep analysis error low, cells are grouped into narrow subsections at the edge of conductors (where current varies rapidly) and long, wide subsections on the interior of lines (where current changes slowly).

If you are familiar with rectangular waveguide modes, you can delve deeper into how Sonnet works:

To calculate the coupling between subsections, the fields due to a patch of current must be calculated. Remember that the circuit is contained in a box with perfectly conducting sidewalls. These sidewalls can be viewed as a rectangular waveguide, with the wave propagating in the vertical direction. To calculate the coupling, we must determine the fields due to current on a given subsection. To do this, these fields are expanded as a sum (over m and n) of all TEmn and TMmn waveguide modes, the waveguide modes forming a complete orthogonal basis for the representation of any source free field. There is one sum for fields above the subsection, and there is a second sum for the fields below the subsection. The current on the subsection represents a discontinuity between the fields above and below. To determine how much each mode is excited, we invoke boundary conditions at the discontinuity. When you do all the mathematics, this becomes a 2-D sum of cosines and sines. A 2-D sum of cosines and sines is quickly evaluated with a 2-D FFT.

## Sonnet’s Accuracy and Robustness

The basic point for understanding Sonnet’s accuracy and robustness is that the perfectly conducting sidewalls allow Sonnet to use the FFT. Thus, all subsection-to-subsection couplings are calculated to full numerical precision. Unshielded EM analyses must use a 4-dimensional numerical integration to calculate these same couplings. Numerical integration introduces errors. Sometimes the additional error does not matter. Sometimes it does. This directly impacts robustness.

For example, benchmark tests have shown that the dynamic range for numerical integration techniques varies from 40 to 80 dB, depending on the specific circuit. Sonnet’s dynamic range is 100 to 180 dB, again depending on the specific circuit. If a given circuit is analyzed with an analysis dynamic range of 40 dB, then results below –20 dB start to show error. Results below –40 dB are useless. For some circuits, this is fine. For other circuits, this is not acceptable.

Accuracy in Sonnet is determined solely by subsection size. A smaller subsection size means a slower, but more accurate analysis. For a given circuit at a given level of accuracy, Sonnet’s use of the FFT provides results often as much as one to two orders of magnitude faster. A given circuit analysis time can range from a few seconds to hours depending on the subsection size selected. Thus, when comparing EM analyses, it is critically important to always compare based on the same accuracy/subsection size.

## Conclusion

Sonnet is a Method of Moments analysis of high frequency planar circuits. Sonnet’s technique is described here in non-mathematical terms and additional detail is provided which requires only minimal mathematical background. Finally, Sonnet’s industry standard accuracy and robustness is explained in terms of Sonnet’s use of the FFT which allows exceptionally efficient full precision calculation of the critical subsection-to-subsection coupling.

## References

[1] R. F. Harrington, Field Computation by Moment Methods, reprinted by IEEE Press, 1993.

[2] J. C. Rautio and R. F. Harrington, “An Electromagnetic Time-Harmonic Analysis of Shielded Microstrip Circuits,” IEEE Trans. Microwave Theory Tech., Vol. MTT-35, pp. 726-730, Aug. 1987. [download PDF]