# Signal integrity applications of Mathcad, Part 2: Mathcad can solve complicated SI expressions including those with complex/ imaginary terms, logarithmic and exponential functions.

MATHCAD OFFERS EFFICIENCY and flexibility for solving mathematical equations.

It can treat analytical expressions having scalar (a single number), vector (column of numbers) and matrix (a rectangular array of number). An array is a general term for vector or matrix.

The Mathcad logarithmic features include In(z) which returns the natural log of z (z [not equal to] 0) and log(z,b) which outputs the base b logarithm of z (z or b [not equal to] 0). If b is omitted, then log(z,b) equals the base 10 logarithm. Hence, log(z) returns the common logarithm of nonzero z.

Let us analyze, applying Mathcad, the concept of coupling even and odd mode impedances.

The electric and magnetic fields associated with coupled traces (i.e., traces located in close proximity) will interact in manners that depend on each line's signal patterns. The effective characteristic impedance and velocity of the transmission will be altered due to such interactions. FIGURE 5 displays the computed (using HyperLynx from Mentor Graphics) odd and even electric (in blue) and magnetic (in red) field distributions for a coupled microstrip.

Two coupled traces excited by signals of equal magnitude but opposite polarity (180 degrees phase difference) will produce the odd-mode (also called differential). When these traces are driven by identically phased signals, then the even (or common) mode will result. The relationship between the characteristic impedance of each line Z0, the odd mode impedance Z0o and the even mode impedance Z0e is governed (11) by:

Z0 = (Z0o * Z0e)^0.5 EQUATION 1

[FIGURE 5 OMITTED]

For any geometry, Z0o and Z0e depend on coupling equivalent magnitude of coupled energy depends on the odd and even mode impedances. FIGURE 6 presents impedance formulae for quarter wavelength matched lines solved with Mathcad.

As coupling intensifies, even mode impedance is likely to be high (11) since the inductance increases and capacitance diminishes (because field lines are more concentrated between coupled traces and not at the common ground/return).

In a Z0 = 60 ohms system for weak coupling (such as k = 0.03) yields Z0e = 61.83 ohms and Z0o = 58.23 ohms. For a strongly coupled case (k = 0.7) with Z0 = 60 ohms outputs Z0e = 142.83 ohms and ZOo = 25.21 ohms. Thus when coupling is strong, Z0e and Z0o can significantly differ from Z0. Frequently, Z0o and Z0e for a coupled pair are plotted against trace separation (12).

Such graphs reveal that ZOo and Z0e are close to Z0 when trace separation is large (which corresponds to weak coupling). However, when trace separation is small (resulting in strong coupling) there is a significant impedance difference.

When ascertaining the correct (12) termination for a topology (in order to minimize reflections and maintain signal integrity) it is critical to take into consideration the even and odd mode impedance values.

As another example, let us apply Mathcad to a case of coplanar structure, which demonstrates this software's capability to treat hyperbolic trigonometric such as Tanh() and special functions (such as ratios of complete elliptic integrals).

Coplanar waveguide (CPW) is a transmission line geometry that includes a central current-carrying trace on top of a dielectric substrate, with side grounds extending beyond a symmetric gap to either side of trace. There are several different types of CPW transmission lines including grounded CPW (13) that has an additional ground below the substrate and ungrounded CPW (for which the side grounds coplanar to the signal trace provide the only return path).

[FIGURE 6 OMITTED]

CPW can have a lower loss tangent than microstrip (signals coupled mostly through air) and higher skin effect losses (fields concentrated on the edges of trace and grounds). CPW is generally defined by center strip width w, gap width g, substrate height h, and substrate dielectric material. Metal thickness t can be also important, especially when t [greater than or equal to] 0.1w or t [greater than or equal to] 0.1g.

The characteristic impedance Z0 of CPW can be controlled (14) by varying the trace width, the spacing-to-ground, and the dielectric thickness or material.

Coplanar transmission line structures offer similar advantages to a microstrip in that signal is carried on an exposed surface trace where a surface mount component can be attached. However, unlike a microstrip, the CPW (at least in an ungrounded form) can have low parasitic losses between surface mounted components and an underlying ground plane. It is also possible to narrow the traces (13) to match component pad widths while maintaining constant impedance.

CPW can also provide good inherent crosstalk immunity for two layer boards, because it is less sensitive to the presence or absence of a backside of board ground plane.

The primary disadvantage of CPW is that it is more difficult to design compared to microstrip or stripline. If the CPW's aspect ratio (the ratio of gap to trace width) becomes too high or too low, parasitic modes can replace the desired CPW mode resulting in poor performance. Grounded coplanar geometry is depicted in FIGURE 7. A Mathcad script for computing the impedance of this type of coplanar structure is illustrated by FIGURE 8.

In FIGURES 7 and 8, a denotes trace width and b denotes the sum of trace width and separation, which is the same notation used in Wadell (13). However, in many other publications the trace width is represented by w, and spacing to adjacent ground traces is given by s (or gap width 9). In FIGURE 8, the expressions for Ratio1 and Ratio2 are called ratios of complete elliptic (13) integrals of the first kind.

In signal integrity there arise certain quantities containing complex (real and imaginary) terms. For instance, permittivity of dielectric materials (15) includes a real part (dielectric constant) and an imaginary part (loss). Another example of this is found in the impedance through a bypass capacitor (16) that has a real part, the equivalent series resistance ESR, and an imaginary term, capacitive/inductive reactance. Determining these quantities can necessitate the use of complex algebra or calculus.

[FIGURE 7 OMITTED]

Mathcad accepts complex numbers (17) of the form Re + (lm)i, where (Re) and (lm) represent ordinary numbers. Imaginary numbers may be followed by either i or j; however, Mathcad normally displays them followed by i where i equals the square root of negative 1. When typing complex numbers into a Mathcad formula, it is important to note that i or j alone cannot be used to represent the imaginary unit. Instead, it is necessary to type 1i or 1j, otherwise i or j will be interpreted as a variable. When the cursor is outside an equation having 1i or 1j, Mathcad will hide the superfluous 1. In Part 3 of this article, Mathcad applications to single-ended and differential signal propagation will be discussed.

[FIGURE 8 OMITTED]

ACKNOWLEDGEMENTS

REFERENCES

(11.) "Even and odd mode impedances" Microwave Encyclopedia, 2008.

(12.) "Even mode impedance--an introduction" Polar Instruments Ltd., Application Note AP157.

(13.) Brian C. Wadell, "Transmission Line Design Handbook" 1991, Artech House, PR 73-89, PP. 464-467.

(14.) Rick Hartley, "RF / Microwave PC Board Design and Layout" L-3 Avionics Systems.

(15.) A. Kumar and S. Sharma, "Measurement of Dielectric Constant and Loss Factor of the Dielectric Material at Microwave Frequencies", Progress in Electromagnetic Research, PIER 69, 47-54, 2007.

(16.) Douglas G. Brooks," ESR and Bypass Capacitor Self Resonant Behavior How to Select Bypass Caps" UltraCAD Design Inc., 2000.

(17.) Alan Felzer, "Introduction to Mathcad" August 2003.

DR. ABE (ABBAS) RIAZl is a senior staff electronic design scientist with Broadcom Corporation in Irvine, CA and can be reached at ariazi@broadcom.com.
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