CDE Modeling Using Star-Tree Impedance Networks for USB2 Cable

Previous studies of cable discharge events (CDE) have often used oversimplified models of the cable, such as a single 50Ω transmission line. This is not bad for an initial investigation, but the next level of detail is not difficult to capture for some familiar data cables. This work focuses on a star-tree impedance model for the 5-node USB2 cable and outlines a methodology for treating other cables, such as USB3 and HDMI.

The Cable Discharge Event (CDE) is an important ESD topic of continuing interest [1,2]. But to quantify CDE and observations, a simple and accurate electrical model of the cable itself is needed for future studies of on-silicon ESD protection optimizing for cost performance and improved reliability. Industry specifications leave much latitude at the expense of clarity. After considerable study and to promote understanding, we devised simple, lucid models for USB2, USB3, and HDMI cables based on star and tree networks [3]. These utilize measurements of capacitance and propagation velocity (and therefore inductance and impedance) that give models with reduced parameter count and agree well with the experiment. In this brief article, we model the five-node USB2 cable (Figure 1) and plan to cover similar models for USB3 and HDMI in the 2024
EOS/ESD Symposium.

Methodology & Experiments

We began, using methods to be described below, by noting that a 5-parameter star network with a single center point [3] fits the 10 (i.e., C(5,2)) pairwise capacitance measurements for USB2 reasonably well. But then we found that adding an extra element from the center point to each pair’s D+/D- and V_BUS/GND was substantially better, forming trees. One of those elements, from the center to the V_BUS/GND tree, turned out to be nearly zero impedance and thus is shorted in Figure 1. Finally, the elements of the two pairs D+/D- and V_BUS/GND are so similar that using a single impedance/capacitance for each pair and symmetrizing the associated measurements was done. The resulting star-tree network Figure 1 has just four Z (or C) parameters.

Figure 1: Star-tree capacitance (impedance) network for USB2 cable including VBUS, GND, D+, and D- signal lines and the shield, plus nodal naming 1-5.

The final model will be transmission line impedance Z that captures both capacitance and velocity measurements, as Z=L/C and inductance L are derived from velocity v=1/LC, as discussed below. (C and L quantities are per unit length.) Let conductance G=1/C here, as it is the capacitive reactance with j/w normalized out as unity. Then, we can easily formulate a matrix describing the summed G elements corresponding to the ten pairwise (reciprocal) capacitance measurements between the numbered nodes in Figure 1. Linear regression using matrices, as below, solve for the four parametric values of (reciprocal) capacitance, whereupon we use measured velocity v to solve for each Z, Z₁-Z₄.

Let C_i be a column vector of parametric capacitances for the model in Figure 1.

   (1)

   (2)

   (3)

Table 1a: 10×4 A-matrix for all combinations

In Eqns. 1-3, b is a 10×1 measured vector of 1/C_ij, A is a 10×4 matrix of 1s and 0s (see Table 1a), considering the duplicates, Table 1a reduces to Table 1b, and G is a 4×1 vector (Eq. 2) of parametric reciprocal C values. Multiplying both sides of Eq. 3 by the transpose of the A matrix A^T (4 x 10) results in

   (4)

Table 1b: 5×4 reduced matrix based on Table 1a A-matrix.

Solving for G, where [C] = [1/G],

  (5)

Table 2 compares the various capacitances, as measured (by a capacitance meter) and as predicted by solving for G and applying A. The best solution comes from averaging the measured capacitances that should all be the same, as indicated by Figure 1 (Averaged cap column, Table 2). G comes from linear regression, using linear algebra as in Eqs. 1-5 and employing the 5×4 A matrix of Table 1b and the 5×1 b vector of averaged (reciprocal) measured capacitances. As shown in Table 2, this gave agreement with the experiment to <0.3% worst case.

Table 2: Measured capacitances/meter vs. and averaged values for the network in Figure 1. Right: 5×4 matrix solution for predicted measurement and percent error.

Table 2 shows the ten combinations of capacitance/length between the five nodes, beginning with measured values from the lab, with averaged and linear regression AG capacitance. Note the small difference between predicted and measured results. We achieved this with an averaged star-tree network of only four C (or Z) values, fitting five averaged values from 10 measurements as shown.

The four C values C_i, derived from the G solution, are converted to transmission line impedances (Z_i) once we have a propagation velocity V_p. Figure 2 shows Z₁Z₄ values, found from C₁C₄ using

   (6)

Figure 2: USB2 star-tree impedances as derived from the capacitance solution of Figure 1.

Our best value of propagation velocity V_p for the transmission lines was measured by time-domain reflectometry on the cable to be 0.68c, c the speed of light. Values for different wire pairs were found to be very close, thereby simplifying the result. Figure 2, very similar to Figure 1, shows the star-tree impedances along with the calculated impedance values Z_i and capacitance values C_i from G.

The D+/D- twisted pair impedance 2Z₂ = 77.84 ohms is within the USB2 spec limit of 90 ohms ±15%. For CDE problems, the capacitive DC limit, Figure 1, is used to describe initial charge storage, and the full impedance model of Figure 2 plus line terminations and switching can be used to determine the sequence and timing of CDE pulses. The Figure 2 network is comprehensive enough to describe line coupling in terms of even and odd mode impedances that can be written down by inspection, with Z₃ and Z₄ playing major roles. [4] Note that the twisted pair lines are more strongly coupled than the power lines, as desired. This comprehensive USB2 cable model is, of course, applicable beyond CDE and, for example could allow a quick grasp of USB2 signal integrity issues.

Simple star-tree networks, as shown here for USB2, should have wide applicability for cables carrying fast signals. The authors have extracted similar networks for USB3 and HDMI cables and plan to follow the present work by publishing solutions for those more complicated high-speed cables.

References

1. S. Marathe, P. Wei, S. Ze, L. Guan, D. Pommerenke, “Scenarios of ESD Discharges to USB Connectors,” 2017 EOS/ESD Proceedings, 3A.4.
2. M. Coenen, “Cable Discharge Event (CDE),” Interference Technology, July 31, 2019.
   
3. https://en.wikipedia.org/wiki/Star-mesh_transform
4. See Maloney and Poon, 2004, https://bit.ly/3z7BTVe, and references therein.

	Peyman Ensaf is currently a Quality & Reliability Research and Development Engineer focused on ESD/RFI with Intel. While a student he completed summer internships at the Phillips Laboratory U.S. Air Force focusing on numerical electromagnetics, studying electromagnetic field behaviors within complex cavities of satellite sub-systems using numerical tools.
	Timothy J. Maloney was a Senior Principal Engineer at Intel before retirement in June of 2016. He received the Intel Achievement Award for his patented ESD protection devices, which have achieved breakthrough ESD performance enhancements for a wide variety of Intel products. He is co-author of the book Basic ESD and I/O Design (Wiley, 1998) and is a Fellow of the IEEE.