Application of Launch Point Extrapolation Technique to Measure Characteristic Impedance of High Frequency Cables with TDR

Transcription

1 DesignCon 2009 Application of Launch Point Extrapolation Technique to Measure Characteristic Impedance of High Frequency Cables with TDR Luis Navarro, Tyco Electronics Eugene Mayevskiy, Tyco Electronics Timothy Chairet, Tyco Electronics 1

2 Abstract A new methodology for measuring cable characteristic impedance (Zo) is presented. The approach employs a least square fit to extract accurate and repeatable Zo values from the TDR measurements. The RLGC circuit modeling explores various cable losses and shows that the proposed approach agrees with the transmission line theory. A frequency domain plot of Zo measurements correlates with the new methodology. The proposed methodology equips a test engineer with a quantifiable and efficient way to evaluate the cable performance as a function of environmental and physical changes. Luis Navarro s Biography Throughout his career Luis Navarro was involved in solving complex engineering problems. His work experience includes working for a small start up, where he developed the hardware and software to extract models for transistors and op-amps, and working for Tektronix, where he led the efforts to develop the most popular digital oscilloscopes for Tektronix. The features and innovations resulted in several patents that are used in every digital oscilloscope today. Luis currently with Tyco Electronics-Precision Interconnect, where he has developed a number of automatic measurements to measure parameters to characterize high frequency transmission lines as well as medical monitoring cables. In addition, he is involved with teams solving unusually complex problems with product design and manufacturing processes using statistical methods. In his spare time, Luis and his wife Nenita like to travel and spend time with the family. Luis holds a B.S.E.E and M.S.E.E degrees from the University of Nebraska. 2

3 Eugene Mayevskiy s Biography Eugene Mayevskiy holds the B.S.E.E and M.S.E.E. degrees from Oregon State University where he has done substantial research on measurement-based modeling and analysis of passive devices fabricated for radio frequency integrated circuits. His work experience includes product development, training and problem resolution for TDR/VNA measurements, modeling and SPICE/IBIS circuit simulations. He has published a number of papers and given seminars on measurement-based modeling for high-speed designs. Eugene Mayevskiy is currently with Tyco Electronics working as an electrical test engineer. Timothy Chairet s Biography Timothy Chairet is an electrical engineering student at the Portland State University. He is currently working as a MECOP intern at Tyco Electronics, Precision Interconnect in Wilsonville, OR. His job responsibilities include working on TDR and VNA measurements and generating measurement-based SPICE models. 3

4 Introduction The time domain reflectometry oscilloscope (TDR) oscilloscope is traditionally a tool of choice to measure time domain characteristic impedance of cables. When impedance is measured with a TDR oscilloscope, the generator produces a time domain square waveform with a fast rising edge. Only a short region near the rising edge of the waveform is used, so that part of the waveform looks like a step on the oscilloscope s screen. This step is used to measure impedance because when the step propagates through a cable, it results in reflections, which are directly related to the amplitude and duration of a physical discontinuity. Even though the principal operation of the TDR instruments is the same in many platforms, there is a variety of methodologies used in industry to interpret the oscilloscope s readout. The major issue in the interpretation of the results is that the cable time domain impedance curve in general exhibits a slope due to the cable loss. For example, the beginning of the TDR data can show 48Ω impedance whereas the end of the cable can be 52Ω. The impedance can also deviate due to reflections or manufacturing process variations. Hence, the test engineer working with TDR oscilloscope has to make an intelligent decision about what impedance value to report. The industry techniques can vary from a simple oscilloscope measurement of the absolute impedance with the cursors to the sophisticated impedance measurement algorithms aimed to eliminate undesired losses and reflections in interconnect and in the measurement system. Perhaps the simplest way to measure characteristic impedance of cables is to specify the time range and find out if the measured impedance exhibits expected results. This approach is often used for compliance testing in various communication standards such as SATA, InfiniBand, or DisplayPort. Those standards specify impedance masks relative to reference time position, which is usually found by measuring open or short from the test adaptor. Design features, such as fixture footprints, mated connectors, and solder points, shown on the TDR waveform help to define a mask that is applicable for a specific application. For example in a SATA cable, the reflections from a mated connector in the cable assembly can occupy approximately ps of the TDR response. Hence, the SATA compliance test specification procedures require to measure and record maximum and minimum cable differential impedance values in the first 500ps of cable response following any vestige of the connector response [1]. By doing this, the impedance of the beginning of the cable is captured, and the loss behavior of the cable is ignored. Although, the specification limits can be very well defined, the question about the real value of the characteristic impedance can be left open. The institute for printed circuits test methods manual (IPC-TM 650) describes impedance measurements of transmission lines using a calibration standard and a defined region [2]. The method corrects for the high-frequency loss exhibited by the TDR connecting cable and allows measuring impedance almost as accurate as the standard used in calibration process. The major drawback of this method, however, is that measured impedance is an 4

5 averaged impedance value of the specified region. Given that the displayed characteristic impedance is changing due to the cable loss, such results will be highly dependent on the region used in the measurements as illustrated in Figure 1. Figure 1. Impedance of a coaxial cable measured at different regions; the reflections from the launch and the end of the cable are excluded. Note that the impedance slope due to loss mechanisms in the cable is causing three different regions to have three different characteristic impedance average values. The time span of the cable is twice the propagation delay since this is the TDR response of the cable. More rigorous techniques, such as the one described by Bechhover [3], try to tackle the major sources of the discrepancies such as reflections and attenuation due to high frequency losses by applying algorithms that minimize undesired effects, such as inverse scattering method followed by the inverse filtering. Although, this methodology can be very helpful in determining the location of the cable discontinuity or failure, the impedance value obtained is dependent on at least two assumptions that are not true in the real cables. First, impedance inverse scattering method assumes a lossless behavior of the cable measured. Second, the inverse filtering assumes that a high frequency loss in the system is uniform. However, the real cable assembly has both a non-uniform losses and significant reflections that affect the accuracy of the algorithms used. In this paper we present a new method of measuring characteristic impedance Zo, which involves capturing the waveform of impedance as a function of time in a Time Domain Reflectometer (TDR), defining the cable impedance measurement region that includes only cable impedance variations, performing a least square fit to the measured characteristic, and finally extrapolating the impedance result to the time location of the reference plane (or the launch time point) as shown in Figure 2. This point represents impedance that would be seen by the transmitter if someone could connect a cable to it without any reflections. The new methodology offers several advantages over the other methodologies used in industry such as, repeatability, independence on physical properties of the cable, and provides a quantifiable way to evaluate the cable characteristic impedance of a cable as a function of environmental changes. 5

6 Figure 2. Impedance of a coaxial cable measured using the launch point extrapolation technique; the reflections from the launch (1) and the end (3) of the cable are excluded from the measurement zone (2) and the characteristic impedance is measured from the least square fit extrapolation to the time position of the reference plane. The paper starts from the discussion about the physical effects of losses in time and in frequency domains. Then the impedance measurement methodology is explained in details. Finally, the validity of the measurement technique will be explored by analyzing the accuracy, sensitivity, and repeatability results from the measurements of the test cables. Physical Effects A lossless transmission line represents interconnecting media where the loss effects from dielectric and conductor are essentially equal to zero. The physical behavior of the lossless transmission line can be described by two major characteristics: characteristic impedance Zo in Ohms (Ω) and propagation delay in seconds per meter (Tpd). The governing equations for these characteristics can be derived from the telegrapher s equations [4] and they are listed below: L Z o = (1) C T pd = L C (2) Where L is inductance per unit length in Henries (H/m) and C is capacitance per unit length in Farads (F/m). In the presence of loss, (1) and (2) change to the following representations: Z o = R + jωl G + j ω C (3) 6

7 γ = ( R + jωl)( G + jωc) (4) Where R is the series component of the loss per unit length in Ohms (Ω/m), G is the shunt component of the loss per unit length in Siemens (S/m) and γ is the propagation function. The loss elements in (3) and (4) are frequency dependent, and that frequency dependency can be approximated as: R ( f ) R R f (5) DC + AC G( f ) G + G f (6) DC AC Where f is the frequency of operation in Hertz (Hz) and subscripts DC and AC indicate low and high frequency dependent series loss and shunt loss terms in Ohms (Ω) and Siemens (S) respectively. The loss representations (5) and (6) are often used in commercially available measurement-based model extractors such as IConnect software by Tektronix Inc. [5]. Having defined the loss model for the cables, let s look at how time and frequency impedance responses are affected by these loss contributors. Accurate RLGC models can be used to remove the various losses by setting them to zero and analyzing the cable performance both in time and in frequency domains. Time Domain Three coaxial cables (45Ω, 50Ω, and 75Ω) were first measured using a TDR oscilloscope. IConnect model extraction software was then used to generate a general measurementbased RLGC model for each cable. The resulting correlation of the models with the measurements in the time domain is shown in Figure 3. Finally, the loss parameters in (5) and (6) are used to analyze physical effects of various losses. (a) 7

8 (b) (c) Figure 3. RLGC model correlations for 45Ω (a), 50Ω (b), and 75Ω (c) coaxial cables, with the measurements in time domain. Note the transmission rise time degradation and a slight increase in reflection waveform profile due to the losses. The RLGC model parameters are listed in the legend. The model parameters can be easily adjusted to look at the behavior when one or more loss components are set to zero. This would help to analyze possible cable characteristic changes when one or more components are idealized. For example, setting the G components to zero in (6) eliminates the dielectric loss, which in real life would mean that air dielectric is used in the cable design. Whereas, setting the R components to zero in the equation (5) would eliminate the conductor loss. If both loss components are set to zero, then the resulting response will be the same as the one from the lossless transmission line. Therefore, the loss model obtained from TDR measurements is 8

9 modified to derive three more models: lossless, series loss only, and a shunt loss only. The loss parameters for all models are listed in the Table 1. Table 1. Loss parameters for extracted models shown in figure 3. Note the loss is represented by first four rows in the table; the original model is All Losses (green highlighted field). The experiment is performed by setting the conductor (series) and the dielectric (shunt) loss components shown in yellow-highlighted fields to zero. Model Parameter All Losses Lossless Series Loss Only Shunt Loss Only 45Ω 50Ω 75Ω 45Ω 50Ω 75Ω 45Ω 50Ω 75Ω 45Ω 50Ω 75Ω Rdc(Ω) Rac(mΩ/Hz 1/2) Gdc(nS) Gac(pS/Hz) L (nh) C(pF) The time domain impedance profiles of these responses for the 50Ω cable are shown in Figure 4. The impedance variations for 45Ω and 75Ω samples are shown in appendix A. A careful observation of the trends in all cases allows one to draw some interesting conclusions. Figure 4. Removal of the various loss components in the 50Ω cable case. Note that the removal of the shunt losses results in more upward slope of the data, whereas the removal of the series losses results in the downward slope. Since the removal of the shunt loss elements ( Series Loss Only case) resulted in a greater upward change in the impedance slope in all samples, the slope of the TDR impedance can be linked to the series loss elements of (5). Hence, the positive slope would be an indicator of the domination of the series loss in the cable assembly. On the other hand, the shunt conductance ( Shunt Loss Only case) results in the downward 9

10 impedance change, and if such behavior is observed it can be attributed to the high dielectric loss. Since cables normally are designed to have very low dielectric loss, the upward slope in the impedance profile is usually observed in TDR measurements. When there is no loss present ( Lossless case), the TDR impedance profile is flat. Another prominent effect is the curvature 1 of the TDR impedance waveform. The source of this curvature can be explained by the frequency dependent loss of the cable. Note that the impedance profile is curved more in the vicinity of the launch point. This happens because the TDR step has the highest frequency content at the beginning of its travel. As it propagates, the loss results in the TDR rise time degradation, and the TDR step encounters less high frequency dependent loss ( R AC f ). To illustrate this, the comparison of the impedance profiles obtained using different rise times for the TDR source is shown in Figure 5. Clearly, as the TDR step rise time (Tr) increases, the linear trend line shows a better fit (R 2 is closer to 1). Figure 5. The effects of the rise time on the impedance profile curvature. As the TDR step rise time (Tr) increases, the linear trend line shows a better fit (R 2 is closer to 1). This happens because the propagating signal encounters less high-frequency loss. To verify this observation, the model s parameters can be altered to decrease the frequency dependent resistive loss (Rac) in the series component and to increase the frequency dependent conductive loss (Gac) in the shunt component. The result of this alteration is shown in Figure 6 where the arrows indicate the direction of the curvature increase. 1 Normal loss will result in a concave curve (frown), while extremely high dielectric loss in respect with copper loss may show a convex curve (smile), this is not likely to happen. However, undesirable variations of impedance as a function of length due to center conductor or dielectric diameter as well as relative permittivity can show an unwelcome "smile". 10

11 Figure 6. TDR waveform curvatures as a result of the frequency dependent loss components for 50Ω cable. The arrows indicate the curvature increases as the loss is increasing. When the series frequency dependent loss (Rac) was set to zero while the constant term (Rdc) in (5) was set to 2Ω (a value corresponding to the total resistive loss at 30 MHz), the curvature disappeared, whereas when the shunt frequency depended term (Gac) was increased significantly, the impedance waveform had pronounced increased curvature. The same trends are observed in 45Ω and 75Ω samples as well as shown in Appendix A. Lastly, it can be observed that all impedance waveforms start approximately from the same impedance point, and then they deviate after being affected by the loss. Since this point corresponds to the impedance at time zero, this beginning value represents the impedance seen by the transmitter and is important for matching purposes. Moreover, since the starting point of this impedance is located at a time when the TDR step enters the cable under test and therefore has the highest frequency content, this value should relate to the high frequency value of the impedance. To visualize this effect we will look at the frequency domain in the next section of the paper. Frequency Domain To observe the frequency domain trends, (3) can be used. At the low frequencies the series loss term dominates, and (3) can be rewritten as: R + jωl Z 0 (7) jωc At the high frequencies the L and C terms become more prominent and the asymptote reaches the lossless transmission line value expressed by (1). The loss model defined in the Table 1 along with (1), (3), and (7) can be used to plot high, medium, and low 11

12 frequency trends respectively in the frequency domain as shown in Figure 7 for the 50Ω cable case. Figure 7 shows that for the cable used in the experiment, the low frequency transition happens at the frequencies below 100kHz. The cable impedance is approaching the high frequency asymptote after about 2MHz. The VNA measurement data, which starts at 40MHz, is also approaching the high frequency asymptote, and the deviations from this line can be explained by reflections that come from the cable s connectors. Since observations in time and in frequency domain indicate that the starting point of cable impedance in the TDR waveform does not have strong dependency on the cable loss and represents the high frequency asymptote, the extrapolated value of the measured impedance waveform to the launch time point can be used as a practical representation of the cable characteristic impedance. Figure 7. Frequency Domain Plot of Characteristic Impedance of the RLGC cable model asymptotes for 50Ω cable. Note the high frequency asymptote and the VNA measurements approaches the lossless representation of the measured data. HF is high frequency trend. LF is low frequency trend. MF is middle frequency trend. Methodology Description To measure the characteristic impedance with the proposed technique, the time domain reflection data has to be acquired first. This can be done with a TDR capable oscilloscope or with a Vector Network Analyzer (VNA). In case of the VNA measurement, the data 12

13 needs to be converted into time domain using the inverse of the Fast Fourier transformation (IFFT). Often the beginning and the end of the time domain reflection data from the cable measurements are affected by the connector reflections. These reflections are superimposed on the cable response and need to be excluded from the cable impedance data. Then a linear or higher order least squares fit can be performed on the data between the launch and the end exclusion zones. Finally, the plot of the least squares fit is extrapolated to time t =To where the open end of the launching fixture is. The curve fit intercept with the impedance axis at that time is then defined as the measured characteristic impedance of the cable (Zo). Figure 8 shows the results of the impedance measurements of three coaxial cables having different characteristic impedances. Note that in all cases the impedance slope linear trends are very well defined. Figure 8. Impedance measurements of three coaxial cables having different characteristic impedances. Note that the top waveform is plotted on the secondary axis on the right. Error Correction for TDR Measurements of Cables When performing the TDR impedance measurements using the launch point extrapolation (LPE) technique it is important to correctly calibrate the measurement instrument ensuring that the results are consistent between different test instruments. Errors may come from the instrument imperfections such as TDR step aberrations or a non-flatness of the baseline display response, as well as from the loss and reflections introduced by the connecting cables. These errors may substantially affect the accuracy and repeatability of the measured response. The instrument s specifications are usually within 1-2% of the accuracy. This translates to about ±.5-1Ω in cases when the measurements are performed in the vicinity of 50Ω. Practically it means that if we measure the same cable using two different instruments we might get potentially 1-2Ω 13

14 difference in the results. It is important to make sure that the deviation between two measurements is minimized. There are different techniques used in industry that help to minimize the errors, and perhaps the most suitable for the long (greater than 1ft) cable measurements is the error waveform subtraction. The error waveform can be obtained from the TDR measurements of the known termination resistor. This approach is illustrated in Figure 9. Figure 9. Error correction with a known calibration termination. The DC resistance of the termination can be measured with DMM. The difference of the DMM value and the settled response approximates the measurement error in the measurement region. The reflections can be windowed out by using a 500ps long test fixture. The known resistor calibration technique involves calibration with a known resistor, which value is independently measured. The resistance can be accurately measured with a calibrated digital multi-meter (DMM), and this value can be used for the TDR correction. The DMM measures DC resistance of the termination, and the TDR measurement may have high frequency variations, therefore a settled response of the resistor s TDR profile will be used when calibrating the measurement. Practically, it means that the resistor is connected to the reference plane that is earlier in time than the measured impedance region. The time position can be defined by the length of the unsettled region of the resistor s TDR response. Typically, the termination resistor can exhibit about 500ps of reflections before its response settles out as shown in Figure 9. In this case a fixture length of at least 500ps should be used to calibrate the response as shown in Figure 9. Any impedance difference of this settled response from the known resistor value determined from DMM represents the measurement error and will be subtracted from the measurements. 14

15 Perhaps, the greatest benefit of having this calibration is that any variations between test setups can be effectively minimized. The calibration standard does not have to be perfect, but it is desirable to have one standard for different test instruments. The golden device under test (DUT) can be used to verify the accuracy of the measurements. Properly calibrated equipment should generate the same results when such a DUT is measured. (a) (b) Figure 10. Golden DUT measured with (a) Tektronix ten years ago and (b) DSA

16 Figure 10 shows measurements of the golden DUT that was performed ten years apart with different TDR instruments (Tektronix and DSA8200), but with the same calibration procedure. The results indicate excellent correlation in impedance values. Methodology Validation Initial measurements performed indicated that the measurements were repeatable enough to perform a Design of Experiments 2 to characterize the sensitivity of the measurements. Later on, additional measurements were made to quantify the repeatability. Finally, the accuracy was verified using independent measurements of capacitance. These results are shown below. Sensitivity The first investigation was to study the sensitivity of the impedance measurement to choices on how to make the measurements. The variables were as follows: #1 the test method (TDR or TDT 3 ), #2 whether the far end sample was open or terminated into 50Ω and #3 the variation introduced by the equipment setup. This was accomplished by measuring two coax transmission lines, 50Ω and 75Ω nominally and designing a full factorial 16 trial experiment design. Since there are four variables (A=Coax impedance, B=TDR or TDT method, C=Open or 50Ω and D=setup 1 and 2) and two values each, it takes 16 trials (2 4 ) to measure the effect of those variables and all possible interactions. Figure 11 shows the sensitivity (Effects) of the measurement contributed by the different variables, A the coax type (50Ω or 75Ω), B test method (TDR or TDT), C termination (open or 50Ω). The tests were repeated twice, so the last variable D represented setup. Statistics teaches us to determine a criterion using a Student-t distribution 4 to determine if the effect is significant based on the correct sample size (16), the standard deviation of the data (0.3Ω) and the significant economic resolution we desire for this experiment (0.7Ω). If any variable is above the criterion we can state that we have 95% confidence that it is significant (5% Alpha error) and 99% that we did not miss a significant effect (1% Beta error). 5 The calculated criterion is 0.263Ω. 2 Design of Experiments (DOE) is a tool that is used to measure the effect (response) to a set of variables (inputs) when changed at the same time. The advantage over the classical approach of hanging one variable at a time while leaving others fixed is that it will measure interaction (product) of two variables. 3 Typically, the TDR measurement is made with the far end of the transmission line open. At times, the far end may be terminated in its characteristic impedance to measure the transmitted waveform. This is known as TDT (Time Domain Transmission). Some TDR measurements are made with the far end shorted. These are conditions that may affect the quality of the measurement. 4 The Student-t distribution is used instead of the Normal distributions to estimate probabilities when the sample size is less than There are four possibilities when measuring the response of our system. 1- We measure the response as "Important" and the truth is that it is Important. 2- We measure the response as "Not Important" and the truth is that it is Not Important. 3- We measure the response as "Important" and the truth is that it is Not Important. This is an alpha error. 4- We measure the response as "Not Important" and the truth is that it is Important. This is a beta error. Note that a beta error represents a missed opportunity, while an alpha error represents a false error. 16

17 The results of the experiment are shown in Figure 11. A total of 16 numbers are obtained as a result of processing the measurements. The first one is the average impedance of 63.13Ω. Rows 1 through 15 are the contributions of the variables (A, B, C, D) to the different values used (75Ω and 50Ω coax, TDR or TDT, open end or terminated, setup) and their interactions (AB, AC, AD ABC, ABD ABCD) The Effects column is the difference between the two values of the variables. Thus the measured difference of 25.56Ω between the nominal 75Ω and 50Ω coax is close to the expected value. Variable C (open or terminated end) is the next most significant value of the effect, but below the criterion threshold at 0.18Ω or 0.7% of the coax difference. Variable B (Test method TDR or TDT) and variable D (setup) were 0.63% and 0.47% of the coax difference. This gave a solid indication that the method is quite insensitive to the chosen variables. Figure 11. Experiment design results. Top table is the numerical values of the effects for different variation and the bottom plot shows their comparison on the log scale. 17

18 Repeatability The repeatability of the measurement is demonstrated by measuring 36 coaxes using two different Tektronix oscilloscopes ( Production TDS8200 and Engineering TDS8200. Both instruments were calibrated using a 50Ω standard. Each coaxial cable was serialized and measured at random. Thus we have three sets of 36 points: two sets (A#1 and A#2) measured with Engineering TDS8200 and a third set (B#1) measured with a Production TDS8200. The sets are ordered according to their serial number and plotted against each other in a scatter plot. Figure 12 shows the plot of set A#1 as the x axis and set A#2 as the y axis. Figure 13 shows the plot of set A#1 as the x axis and set B#1 as the y axis. Clearly, if the correlation was perfect, all 36 points will lie on the y=x line. The deviation from the straight line is the measurement error/noise. We can measure the imperfection by drawing a least square fit line based on the 36 points and drawing a sausage around the line encompassing all the points except the one deviating farthest from the line. The width of the sausage is an estimate of sigma or standard deviation. The slope of the line should be close to 1 and the y intercept (or offset) measures the bias from one instrument to the other. Figure 12. Same instrument scattering plot. The ratio of 21:1 is consistent with a correlation coefficient of 99.8%. The estimate of sigma is 0.024Ω. 18

19 Another significant measure of the repeatability is the ratio of the length of the sausage to its width. This is the ability of the system to discriminate two measurements that are close together. It can be shown that a discrimination of 6:1 correspond to a correlation coefficient of 98% or R² of.96. This is the minimum desirable discrimination 6 for a suitable measurement system. Note the two populations between 49.5Ω and 51.5Ω. The ratio of 21:1 is consistent with a correlation coefficient of 99.8%. The estimate of sigma is 0.024Ω. Figure 13. Different instrument scatter plot. The discrimination is 16:1 or a correlation coefficient of 99.7%. The estimate of sigma is 0.03Ω. In this case the discrimination is 16:1 or a correlation coefficient of 99.7%. The estimate of sigma is 0.03Ω. The measurements in the Engineering instrument are an average of 0.036Ω below the measurements recorded in the Production instrument over a range of 2Ω (49.5Ω to 51.5Ω). 6 Discrimination means the ability to resolve differences. A discrimination of 6 means we can resolve six distinct "bubbles" within the length of the "sausage" This means that the measurements can be considered a variable and not an attribute (Yes-no) therefore the sample size is much smaller. A discrimination of 6 also corresponds to a correlation coefficient of 98%. 19

20 Accuracy Two independent ways were used to check the accuracy of the measurements. The first one was to measure the impedance of a precision termination resistor at the end of a semi-rigid transmission line and compare it with the reading measured using a 5 digit digital multi-meter (DMM). The results are as follows: TDR method: Within ± 0.014Ω of 50.50Ω TDT method: Within ± 0.002Ω of 50.50Ω The second method was to compare the capacitance measured with a capacitor meter on a semi-rigid line 7 with the capacitance as calculated by measuring the impedance and the time delay using the following equation derived from (1) and (2): T C = d Z 0 (8) where Td is the time delay in picoseconds, and Zo the characteristic impedance in Ohms measured using the launch point extrapolation technique. The results are as follows: Calculated capacitance: 286pF Measured capacitance: 290pF Difference: -1.4% Previous measurements made using standard techniques of impedance zone averaging generated calculated capacitance values consistently 20 to 30% lower than what was measured using a capacitance meter. Since the time delay measurements were in agreement with the dielectric constant of the measured transmission lines, the characteristic impedance had to be too high. In addition, there was a variation depending on the zone where the average was measured due to the DC resistance of the sample. Conclusions In this paper the time and frequency domain cable loss effects were analyzed and the launch point extrapolation methodology to measure the cable characteristic impedance was presented. The methodology provides an effective way to perform accurate and repeatable measurements of the cable characteristic impedance. The methodology can be applied to measure the cables of various length and physical properties. It produces repeatable results regardless of the variations of impedance as a function of cable length. It can also benefit an engineer by providing a quantifiable way to evaluate cable changes as a function of temperature, humidity, stress and other environmental forces that will affect cable performance. Acknowledgments The authors thank Paul Hamilton and Tatyana Gushtyuk from Tyco Electronics Wilsonville, OR, who provided generous help with various measurements and paper revision. 7 A semi-rigid line is in general more stable than the flexible coax. 20

21 Appendix A. Loss Model Analysis Including 45Ω and 75Ω Transmission Lines. (a) (b) Figure A-1. Removal of the various loss components in the 45Ω (a) and the 75Ω (b) cables. The removal of the shunt losses results in more upward slope of the data, whereas the removal of the series losses results in the downward slope. 21

22 (a) (b) Figure A-2. TDR waveform curvatures as a result of the frequency dependent loss components for 45Ω and 75Ω cables. Notice the curvature increases as the loss is increasing. Table A-1. Loss parameters for cable models shown in Fig. A-2. * The series frequency dependent loss (Rac) was set to zero while the constant term (Rdc) for 45Ω cable was set to 12.3Ω (a value corresponding to the total resistive loss at 30 MHz) and for 75Ω cable was set to 6.39Ω (a value corresponding to the total resistive loss at 10MHz). Model Parameter Rdc Loss * High Gac Loss 45Ω 75Ω 45Ω 75Ω Rdc(Ω) Rac(mΩ/Hz 1/2) Gdc(nS) Gac(pS/Hz) L (nh) C(pF)

23 (a) (b) Figure A-3. Frequency Domain Plot of Characteristic Impedance of the RLGC cable model asymptotes for the 45Ω and 75Ω cables. Note the high frequency asymptote approaches the lossless representation of the measured data. HF is high frequency trend. LF is low frequency trend. MF is middle frequency trend. 23

24 References 1] Serial ATA International Organization, Serial ATA International Organization: Serial ATA Revision 2.6, Section , February [2] TDR Test Method Task Group (D-24a), Characteristic Impedance of Lines on Printed Boards by TDR, IPC-TM-650 TEST METHODS MANUAL, , March [3] E. Bechhoefer and Jun Yu, Algorithm Development to Ascertain the true Characteristic Impedance of a Wire for Wire Diagnostics, IEEEAC paper #1104, January 4, [4] D. Derickson and M. Müller et al. Digital Communications Test and Measurement. Prentice Hall, 2008, pp [5] Tektronix, Inc. Beaverton, OR 97077, USA; 24