Determination of Propagation Constants of Transmission Lines using 1 -port TDR measurements

Transcription

1 Determination of Propagation Constants of Transmission Lines using 1 -port TDR measurements Woopoung Kim, Seock Hee Lee, Man Cheol Seo, Madhavan Swaminathan**, and R. R. Tummala- Packaging Research Center, School of Electrical & Computer Eng. Georgia Institute of Technology, Atlanta, GA 30332, USA Tel) , fax) ~ozisi@ieee.org, **n2an havati.swnminathan~~ece.gatec~i.en u, it tshlee@ece.gatech.edii, rao.tummala@ee.gatech.edu ABSTRACT : The propagation constants of transmission lines were measured from 1-port TDR measurements. Since the TDR measurement is a 1-port measurement, error can be smaller than 2-port measurement techniques. Moreover, the available frequency is determined by the rise time of the TDR step pulse unlike TRL methods. The propagation constant of a lossy transmission line was extracted from DC to 1OG& Simulation of the lossy transmission line using the extracted propagation constant shows good agreement with TDR measurement, demonstrating the accuracy of the TDR measurement technique. I. INTRODUCTION The GHz clock signal in digital circuits requires exact simulation of transmission limes in the time domain. Signal integrity is a major problem for the time domain response of digital circuits. The simulation of transmission lines for digital circuits, therefore, needs good models extracted from the measurement of transmission limes. In [I], the simulation of lossy transmission lines using nonphysical RLGC models bas been discussed. model requires the extraction of the characteristic impedance and propagation constant of transmission lines to construct the non-physical RLGC models. The extraction of the frequency-dependent characteristic impedance was introduced using 1-port TDR measurements in [2]. In this paper, the measurement of the propagation constant of the transmission lines using TDR measurements has been discussed. Using the measured characteristic impedance and propagation constant, the time domain wavefnrm has been simulated, which shows good agreement with measurements. Since the propagation constant is a parameter in the 6equency domain, most of the propagation constant measurements has been performed using Network Analyzer (NA) in the past [3]. Well-developed calibration methods such as Tbru-Reflect-Line (TRL,) in NA have resulted in good measurement of the propagation constant. However, since NA measurements assume that the characteristic impedance of the transmission line to be measured is 50 ohms, which is the impedance of the NA, error can be introduced in the extraction[4]. Several time domain techniques for measuring the propagation constant have been developed using TDmT measurements [5][6]. However, due to the difficulty of calibration in the lime domain, the time domain techniques have been only thought as alternatives to the expensive NA. In this paper, TDR was used for characterizing the propagation constant. While NA can measure only the steady state response of the Device Under Test(DUT), TDR can measure both the transient response and steadystate response. From the transient response, the frequencydependent characteristic impedance can be extracted [2], and 6om the steady-state response, the propagation constant can be measured as shown in this paper. The advantages of using TDR for the measurement of the propagation constant are i) TDR is a one-port measurement. The measurement error is smaller than twoport measurements. ii) There is no frequency limitation due to the length of the transmission lines in measuring the propagation constant. The available maximum fiequency is determined by the rise time of the TDR pulse. For a 36 ps rise time pulse, the maximum frequency is around 15GHz. iii) TDR can extract the propagation constant of transmission limes with arbitrary characteristic impedances. U. EXPERIMENTAL SETUP TDR was initially developed for locating faults on long electrical systems such as telephone wires and network lines. TDR represents the reflected time signamre of an incident step waveform that can be used to extract the characteristics of the Device Under Test (DUT), as shown in Fig. 1. TDR measurements display the round trip electrical delay of cables and the DUT. The size of the discontinuity that can be characterized is a fimction of the risetime of the step pulse. Commercial TDR equipment such as 60m Tektronix supports a risetime of 391s with a 250mV amplitude pulse /02/$ IEEE. 119

2 r'(r)= (3) where r'(t) and p'(t) are the derivatives of the reflected and incident waveforms, respectively. Using the Z- transform of equation (3), U Fig.1. Experimental Setup. III.Conversion to ffequency domain A few methods have been developed in the past to extract the frequency response of the DUT using TDR, details of which are available in [7][5]. In [7], Fourier Transform techniques have been used to extract the impulse response of the DUT using the relation where H(o) is the impulse response of the DUT, R(w) is the Fourier transform of the reflected signal and P(w) is the Fourier transform of the incident signal. However, this method did not use calibration structures to de-embed discontinuities and parasitics, resulting in error in the extracted frequency response. In [5], a method was demonstrated for extracting the impulse response through rational functions by using thru and short calibration. In this paper, a method is discussed for extracting the impulse response kom TDR measurements by using open, short and load calibration. This method is more robust and produced better results as compared to [5] for extracting the frequency dependent characteristic impedance of transmission limes. If the DUT is a linear time-invariant system, the derivative of the incident signal p(t) can be thought of as a virtual signal generated by the sampling bead. Similarly, the derivative of r(t), the reflected waveform, can he thought of as a virtual signal reflected by the DUT. In the time domain, m = p(0 8 MO (2) where 0 is the convolution operator and h(t) is the impulse response of the DUT. Taking the derivative of equation (2) R'(Z) = P'(Z) H(Z) (4) where the Z-transform of a discrete signal, X(Z), is defmed as 01 X(Z) = Cx(n)z-' n=o In equation (5), T, is the sampling interval, x(n) are the discrete samples and w is the angular frequency. The sampling interval, Ts, determines the bandwidth that can he obtained using the method discussed in this paper. A smaller sampling interval will produce a larger frequency bandwidth. Typically, OSps sampling time was used in this paper. IV. CALIBRATION OF TDR High Frequency measurements require the specification of reference planes. A DUT is always characterized at or hetween reference planes for a I-port or 2-port measurement, respectively. Calibration structures are required to de-embed parasitics and discontinuities fiom the measurements at the reference planes. Calibration shuctures such as short, open, load, thru, reflect, line, etc are often used in measurements. De-embedding using a subset of the calibration structures have been developed by the microwave community. For example, calibration using short-open-load-thru (SOLT) and thru-reflect-he (TRL) have been developed for network analysis. In this paper we have used the Short-Open-Load calibration for 1-port TDR measurements. As mentioned earlier, the parasitics and other discontinuities affect the accuracy of the measurements. For a 1-port TDR measurement, an error model using a signal flow graph can be constructed as shown in Fig. 3 [SI. In the Figure, x, y and z are the parameters calculated from the open, short and 50 ohm load calibration measurements. SllA is the response of the DUT and SllM is the measured response which includes the parasitic effect caused by error variables, x, y and z. From Fig. 3, using the signal flow graph, the fiequency response of the DUT can be derived as 120

3 SI," = SI, --x ys,, -xy+r RF in - Fig.3. Error model for calibration. If' the open, short and load measurements are expressed as SIIMO, Slim and Slim respectively, the variables x, y and z in equation (6) can be computed as : Once x, y and z are computed, these parameters can be used to calibrate the DUT using equation (6). This is similar to the 1-port calibration method used in a Network Analyzer. An interesting point to note in equations (6) and (7) is that the incident signal, dt), need not be measured. Since S-parameters are expressed as the ratio (reflected signavmcident signal), the reference signal in equations (6) and (7) are cancelled. For example, if the reflected waveforms from the open, short, load and DUT are known, the calibrated frequency response of the DUT can be extracted without measuring the incident signal. In addition, linear operations on the four signals such as differentiation and integration do not change the results in the extraction procedure. Hence, using the method described in the previous section, the derivatives of the signals have been used to compute SIIA, using equations (6) and (7). When the TDR is calibrated with a SMA-connector calibration kit (Open-Short-Load), the reference planes of Open, Short and Load are almost at the same location. However, for micro-probes, the reference planes can be different depending on the physical structures of the probes. A delay of -3 ps between an Open-standard and a Short-standard could be observed on the TDR measurement in this paper. Therefore for the probe calibration an extra calibration structure is required to adjust the delay of the Open-standard signal. In this paper a coplanar transmission line with a short end was used for calibration. When the Short-line was measured after the Open, Short, and Load calibration, the amplitude of the S- parameter could be larger than one due to the delay between the Open and Short standards. The inaccuracy in the delay between the Open and Short standards can move the circle of the Short-lime up and down on the smith chart. This is a deviation from the S-parameter response of an ideal Short-line which is a circle with the center at the origin. Hence, using the Short-line, the optimum delay of the Open was extracted in order for the Short-line to have its ideal response. V. De-embedding of pad transitions When SMA connectors or pads for probes are used to connect transmission lines to TDR equipment, parasitic inductances and capacitances in the transitions affect the extracted characteristic impedance in the RF frequency range. Hence, de-embedding of the pad transitions is required for the extraction of the characteristic impedance. In this paper, a model based on the physical structure of the pads has been used. The advantage of this model is that the dsembeddmg problem becomes a one-variable minimization problem on the Smith Chart, which implies that the deembedding algorithm is robust and accurate if the physical model is valid [Z]. Rd d Fig. 4. Pad modeling. (a) physical structure (b) physical pad model. Fig. 4 shows the physical modeling of the pads and vias for a microstrip line. A capacitor between the two pads, a capacitor between the center pad and the ground plane, and an inductor for vias are the parasitic components in the transition. Since all these parasitic components are lumped in our model, the valid frequency range for this model is around 15GHz for typical pad dimensions. Above this frequency, the model is invalid due to the distributed transmission line effects. Since large pads and vias add large discontinuities, smaller pads and vias are recommended for obtaining a larger bandwidth in the 121

4 6equency response. It is important to note that the extracted pad parasitics L, C1, and C2 in Fig. 4(b) are frequency dependent parameters. the simulated measurement data generated 60m the extracted values. This produced the same values as the original results. The maximum frequency at which this physical de-embedding technique is valid depends on the physical dimensions of the pad transitions. As a rule of thumb, the maximum 6equency for 5 mm length pad transitions is -3GHz. For 1 mm length pad transitions the maximum frequency is -7 GHz and is >loghz for 300 um length pad transitions W. COPLNAR LINES as > ~. ' ,' Fig. 5. Pad transition on Smith Chart To extract the frequency dependent values for the pad parasitics, the behavior of the pad transition model on the Smith Chart is first investigated at a frequency. The characteristic impedance of the transmission lme is plotted as (Pl) on the Smith Chart in Fig. 5. The capacitance, C2 rotates the point (Pl) downward to (P2) along the circle determined by the characteristic impedance. Inductance L rotates the point (P2) to (P3) along the circle shown in Fig. 5. Finally, capacitance C1 rotates the point (P3) to the measurement point (F'4) along the circle determined by the measurement impedance at the given frequency. On the Smith chart, the only unknown variable is the circle related to the inductance L that needs to he determined Since this is a onevariable minimization problem, the results are repeatable. Since the characteristic impedance varies with frequency, the characteristic impedance at a frequency, for example IO=, can be assumed the same as the characteristic impedance at DC, which is displayed on the TDR instrument. In addition, the parasitics, C1, L and C2 at low frequency also do not vary rapidly since the values are based on physical dimensions. From the characteristic impedance at lomhz (Pl) and the measurement at lomhz (P4), the parasitics, C1, L, and C2 can he determined by " izmg the norm of the error between the measured and simulated values at the next frequency. The simulated values are based on the C1, L, C2, and characteristic impedance values calculated at the previous frequency. The algorithm is based on a leapfrog scheme which leads to good results. To ensure that the extracted characteristic impedance and parasitics, C1, L, and C2 are repeatable, the same procedure was repeated again with The coplanar line in Fig. 6 was measured using TDR for extracting the characteristic impedance and propagation constant. The width and thickness of the center conductor is 5 mils and 1 mil, respectively. The gaps between the center conductor and the ground conductors are 3 mils. There is a metal plane with 10 um thickness on the bottom layer with a dielectric of Er = 3.8 and loss tangent = 0.02 above the bottom metal layer. The metal pattern in Fig. 6 was deposited on the dielectric layer. Using the deembedding algorithm and TDR measurements[2], the frequency-dependent characteristic impedance of the coplanar lme was extracted as shown in Fig ,.;.....,. Fig. 6. A coplanar line to he characterized. n m.6 m 26.6 I U L 8 T 8 9 IO FWWW fie extracted frequency-dependent characteristic impedance of a coplanar lme using the deembedding algorithm[2]. 122

5 ~~~~~~ w. mopagatlon CONSTANT For measuring the propagation constant of the coplanar line, two coplanar lines with different lengths and same cross-section were measured. The ends of the two lines were terminated to grounds. Fig. 8 shows the TDR measurements of the two limes. The length of the solid line is 1938 mils, and the length of the dotted line 2242 mils. 0 E 4.05 P f Using (9), the propagation constant of the transmission lime can be extracted as shown in Fig. 9 without consideration on the load of the transmission line. Although the result is not related to the load short termination is preferable to open termination. To verify the propagation constant in Fig. 9 and the characteristic impedance in Fig. 7, a transmission line with the same cross-section was simulated using a nonphysical RLGC model [ 11 to compare with the extracted data fiom the TDR measurement. The length of the transmission line was 5cm and the end of the transmission line was connected to ground. Fig. 10 shows the simulation waveform and TDR measurement. The two waveforms show good agreement verifyiig the accuracy of the propagation constant and characteristic impedance Timernsl.~~~. Fig. 8. TDR measurements of two coplanar lines. The lengths of the solid lime and dotted line are 1938 mils and 2242 mils, respectively. :LZ?rl To extract the propagation constant, the total reflection coefficients of the two lines seen fiom the ends on the source side, need to he determined using the calibration procedure discussed in section IV. The total input reflection coefficients can be comuuted using - the frequency-dependent characteristic impedance in Fig. 7 after de-embedding - the Dad oarasitics for the two lines. 1. The total input reflection coeficients can he expressed as: U ' Fr~lIGtkl ( FreqIGW Fig, 9, Propagation constant of the coplanar lime, where rland Tz are the total input reflection coefficients of the short and long lines, respectively. rl is the reflection coefficient of the transmission lime and the load. y is the propagation constant and II, /z are the lengths of the transmission limes. Dividing the two input reflection coefficient results in the propagation constant: ( nm[nri -w* 1 r2 Y= In(-) -4) r, (9) Fig. 10. Comparison with simulation using the extracted propagation constant and characteristic impedance using TDR measurement. 123

6 VIII. CONCLUSION The propagation constant was extracted using 1-port TDR measurements. This method has several advantages. Since TDR is a 1-port measurement, the induced measurement error can be smaller than two-port measurements. The available frequency of the extracted propagation constant is determined by the rise-time of the TDR step pulse. However, there is no error in the extracted propagation constant due to the characteristic impedance of the transmission line. REFERENCES [l] W. Kim and M. Swaminathan, Validity of Non-Physical RLGC models for Simulating Lossy Transmission lines, accepted by AP-S [2] W. Kim, S. H. Lee, M. Swaminathan, and R. R. Tummala, Robust Extraction of the Frequency-Dependent Characteristic lmpedance of Transmission Lines using Oneport TDR Measurements, IEEE I d Topical Meeting on Eleenical Performance of Elecwonic Packaging, pp , Oct R.B. Marks, A multiline method of network analyzer calibration, EEE Trans. Microwave Theory and Tech., v. 39,pp , July D.C. DeGroot, D.K. Walker, and R.B. Marks, ImDedance mismatch effects on propagation constant measurwents, 5lh EPEP Conference, pp , Oct , M. Swaminathan, S. Pannala, and T. Roy, Extraction of frequency dependent transmission line parameters using TDR/TDT measurements, 18* Instrumentation and Measurement Tech. Cod. (IMTC 2001), vol. 3, pp ,2001. R B. Marks, L. A. Hayden, J. A. Jargon, and F. Williams, Time Domain network analysis using the multiline TRL calibration, 44* ARFTG Conference Digest, pp , Dec. 1-2,1994. R Nozaki and T.K.Bose, Measurement of the dielectric properties of materials by using Time Domain Reflectrometry, IEEE IMTC-90, pp , Feb [8] Hewlett-Pach4 Automating the HP 8410B microwave network Analyzer, Application Note 221A. June