Mathematical tools such as Mathcad can solve a wide range of SI problems.

Mathcad is software with powerful capabilities to manipulate equations, numbers, text and graphs. The mathematical formulae and numerical values can be presented in an easy to read fashion. Mathcad allows graphics and text to be incorporated throughout the document, and if desired, the resultant files can be saved in HTML format for posting to a website.

Mathcad has proven invaluable for a wide range of engineering designs and can be used in some interesting signal integrity applications^1,2,3. For example, Johnson et al^1,4, presents Mathcad computations for DC copper resistance, capacitance of parallel plates, inductance of circular loops, coaxial cables, microstrip lines, and forward and inverse Fourier Transform (i.e., FFT and IFFT). Use of this software for Fourier transformations and square wave decomposition is also discussed by Brooks², and application of Mathcad for simulating single-ended and differential pair high-speed topologies are demonstrated by Norte³.

As an example, let us apply Mathcad to analyze the asymmetric stripling configuration depicted by Figure 1. For asymmetric stripline (also referred to as offset stripline) the signal layer is sandwiched between two plane layers, but it is closer to one plane than the other (i.e., H1 not equal H). Setting H1=H leads to case of centered (or balanced) stripline.

Figure 2 illustrates Mathcad calculations for characteristics impedance Z0 and propagation delay (Tpd). The parameters Er, H, H1 and W are defined at the top of Figure 2. The formulae for Z0 and Tpd are Equations 1 and 2, respectively.

It illustrates that for Er = 4.2, H = 5 mils, H1 = 8mils, W = 6 mils and T = 0.7 mils, Z0 = 43.06 Ohms and Tpd = 2.084 ns/ft. In Mathcad files, parameter values can be easily altered to examine what happens to all variables and the graphs that depend on them. For example, setting Er = 4.0, H = 6 mil, H1 = 10 mil W = 5 mil T = 1.4 yields Z0 at 52.7 Ohms, and Tpd at 2.034 ns/ft.

The analytical technique for determining impedance is less accurate than using a field solver⁵. Yet an analytical approach provides an efficient means for Z0 computations and can furnish useful insight regarding effects of the various parameters that affect impedance.

Mathcad allows the incorporation of text, formulae, pictures, equations and graphs in the same document. This feature is demonstrated in the next example involving via inductance.

Every via has a parasitic capacitance and inductance. The via inductance is often more critical (to digital design) than its capacitance. The parasitic inductance associated with vias can degrade signal integrity⁶ and reduce effectiveness1 of the power supply bypass capacitors. Via inductance is also important in the analyses of power distribution network (PDN) since it increases the impedance between source and load⁷.

An example of calculating via (partial) inductance using Mathcad is illustrated below:

Equation for L(h) in Figure 3 is an accurate expression for an isolated via (distanced away from other vias).

When there are other vias nearby, the mutual inductance must also be taken into account⁷.

Generally, Mathcad evaluates each statement in the program in sequence⁸, however, there are times when it is desirable to evaluate statements only when a particular condition is satisfied. This can be achieved by incorporating “if” statements.

As an example, the Serial Attached SCSI (SAS) channel insertion loss formula⁹ will be evaluated utilizing this Mathcad feature. Transmission line losses can affect the amplitude of high-speed signals and cause ISI^5,10. Hence, when designing or testing a high-speed channel, it is crucial to consider the loss specifications.

For instance, the loss budget for PCI Express Gen1 is specified as 13.2 dB at a frequency of 1.25 GHz.
For SAS, the magnitude of insertion loss (SDD21) is given by a set of equations that can be analyzed using Mathcad as shown by Figure 4.

Above examples demonstrate that Mathcad offers powerful capabilities for analyzing and solving signal integrity problems. Additional mathematical SI applications will be explored in Part 2 of this article. PCD&F

Dr. Abe (Abbas) Riazi is a senior staff electronic design scientist with Broadcom Corp. in Irvine, CA and can be reached at ariazi@broadcom.com.

REFERENCES

1. Howard Johnson and Martin Graham, “High-Speed Digital Design: A Handbook of Black Magic”, Prentice Hall, 1993, PP. 258-260, PP. 409-439.
2. Douglas Brooks, “Signal Integrity Issues and Printed Circuit Board Design”, Prentice Hall, 2003, PP. 18-21.
3. David Norte, “ Learn Signal Integrity Design Principles With Mathcad”, The EMC, Signal and Power Integrity Institute, 2005, PP. 5-29, PP. 41-48.
4. Howard Johnson and Martin Graham, “ High-Speed Signal Propagation: Advanced Black Magic”, Prentice Hall, 2003, PP. 249-250.
5. Eric Bogatin, “Signal-Integrity Simplified” Prentice Hall, 2004, P. 257-262, PP. 334-335.
6. Stephen H. Hall, Garrett W. Hall, James A. McCall, “High-Speed Digital System Design A Handbook of Interconnect Theory and Design Practices”, John Wiley & Sons Inc., 2000, PP. 102-104.
7. Istvan Novak and Jason R. Miller, “ Frequency Domain Characterization of Power Distribution Networks”, Artech House Inc., 2007, PP. 43-54.
8. “Mathcad 2001 User’s Guide”, Math Soft Inc. 2001, P. 286
9. “Information technology – Serial Attached SCSI – 1.1 (SAS-1.1)”, Working Draft American National Standard, Project T10/1601-D, Revision 10, September 21, 2005, PP. 136-146.
10. Abe Riazi, “Timing Analysis Techniques for Digital PCBs,” Printed Circuit Design & Fab, February 2008, PP. 17-19.