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 startree impedance model for the 5node 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 onsilicon 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 fivenode 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 5parameter 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 startree network Figure 1 has just four Z (or C) parameters.
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, Z_{1}Z_{4}.
Let C_{i} be a column vector of parametric capacitances for the model in Figure 1.
(1)
(2)
(3)
A matrix, for AG=b 
G_{1}  G_{2}  G_{3}  G_{4} 
b_{1}=1/C_{12}  2  0  0  0 
b_{2}=1/C_{13}  1  1  0  1 
b_{3}=1/C_{14}  1  1  0  1 
b_{4}=1/C_{15}  1  0  1  0 
b_{5}=1/C_{23}  1  1  0  1 
b_{6}=1/C_{24}  1  1  0  1 
b_{7}=1/C_{25}  1  0  1  0 
b_{8}=1/C_{34}  0  2  0  0 
b_{9}=1/C_{35}  0  1  1  1 
b_{10}=1/C_{45}  0  1  1  1 
Table 1a: 10×4 Amatrix for all combinations
In Eqns. 13, 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)
Reduced A matrix  G_{1}  G_{2}  G_{3}  G_{4} 
b_{1}=1/C_{12}  2  0  0  0 
b_{2}=1/avg(C_{13}, C_{14}, C_{23}, C_{24})  1  1  0  1 
b_{3}=1/avg(C_{15}, C_{25})  1  0  1  0 
b_{4}=1/C_{34}  0  2  0  0 
b_{5}=1/avg(C_{35}, C_{45})  0  1  1  1 
Table 1b: 5×4 reduced matrix based on Table 1a Amatrix.
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. 15 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.
Capacitances  measured caps  Averaged cap  Linear Regression  % Diff (Averaged Cap) 
C_{12}  124.89  124.89  124.71  0.1635 
C_{13}  77.95  78.58  78.66  0.1028 
C_{14}  76.55  
C_{23}  79.75  
C_{24}  80.08  
C_{15}  212.33  212.19  212.73  0.2777 
C_{25}  212.05  
C_{34}  63.02  63.02  63.02  0 
C_{35}  106.57  106.57  106.44  0.1395 
C_{45}  106.57 
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 startree 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 Z_{1}Z_{4} values, found from C_{1}C_{4} using
(6)
Our best value of propagation velocity V_{p} for the transmission lines was measured by timedomain 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 startree impedances along with the calculated impedance values Z_{i} and capacitance values C_{i} from G.
The D+/D twisted pair impedance 2Z_{2} = 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 Z_{3} and Z_{4} 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 startree 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 highspeed cables.
References
 S. Marathe, P. Wei, S. Ze, L. Guan, D. Pommerenke, “Scenarios of ESD Discharges to USB Connectors,” 2017 EOS/ESD Proceedings, 3A.4.
 M. Coenen, “Cable Discharge Event (CDE),” Interference Technology, July 31, 2019.
 https://en.wikipedia.org/wiki/Starmesh_transform
 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 subsystems 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 coauthor of the book Basic ESD and I/O Design (Wiley, 1998) and is a Fellow of the IEEE. 
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