Evaluation of the propagation constants of differential PCB interconnections Vittorio Ricchiuti*, Antonio Orlandio, Giulio Antoninio * TechnoLabs S.p.A., L’Aquila-Italy o UAq-EMC Lab., Dept of Electrical Engineering, Univ. of L’Aquila, L’Aquila-Italy vittorio.ricchiuti@technolabs.it, orlandi, antonini@ing.univaq.it Abstract Today the telecommunication equipments are characterized by transmission bandwidths of tents of GHz. Consequently the digital signals into play reach data rates of some Gbps with rise/fall times of few picoseconds. In this context it is important to correctly characterize the transmission lines on boards for studying the signal propagation as a function of the frequency. The more effective technique starts from measurements, connecting the studied traces to the measurement instruments by means of transition networks or fixtures. But in this case the connective errors due to the connector mismatches and discontinuities have to be removed. The paper proposes different mathematical deembedding procedures to evaluate the propagation constant of differential traces, de-embedded by the effects of the SMA’s connecting them to the VNA. Introduction Nowadays the trend in the telecommunication market is connecting digital boards by means of backplanes, providing transmission bandwidths of tents of GHz. For example, the last PICMG 3.x specifications announced ATCA (Advanced Telecom Computing Architecture) backplanes with transmission channels up to 10-12.5Gbps. The increased signal data rate and the corresponding reduced rise/fall times create higher order harmonics (> 10GHz) in the signal transmission medium. Consequently it is evident that, to study and simulate the signal propagation along the traces of a board, the basic parameters of the transmission lines as a function of the frequency have to be known. A transmission line is characterized by its propagation function and characteristic impedance. The first parameter takes into account the loss and delay during the signal propagation, while the second one considers the matching conditions along the line and then the possible reflections. Knowing the electrical characteristics of the traces on the PCB (Printed Circuit Board), it is possible to choose the proper and cheapest dielectric material supporting the considered signal data rate. The frequency dependent transmission line parameters can be obtained theoretically or experimentally. The theoretical approach is difficult, because it leads to solve complicated problems and equations, so that the experimental approach is preferable. In the case of microstrips and striplines, they cannot be connected directly to the coaxial ports of the measurement instrument, but using some kind of transition network or fixture (e.g. SMA connectors). Consequently the measured data contain systematic errors, due to connector mismatches and discontinuities, which have to be removed, using some mathematical de-embedding techniques. Obviously such mathematical techniques have to be applied to differential traces, commonly used for transmitting high speed digital signals. The following paper, starting from S-parameter measurements of only two PCB differential traces of different length, proposes three different methods to evaluate the propagation constant of these differential traces, de-embedded by the effects of the SMA connecting them to the VNA. Error-removal procedures Let’s refer to the error matrices model in Fig. 1 [2]. Fig.1: Error matrices model for the evaluation of the true propagation constant of differential PCB traces. Let L be the length of the differential trace with odd mode propagation constant γo = αo + j βo and even mode propagation constant γe = αe + j βe for the dominant quasi-TEM mode of propagation, TA, B the left and right error chain parameter matrices (4x4) of the connective discontinuities due to the SMA’s, TL the chain parameter matrix of the differential trace (4x4) and TM the measured chain parameter matrix (4x4) of the entire structure. The differential trace is a 4-port network and the relative TM matrix can be obtained from the measured S-parameter matrix using the formulas in Appendix. 1. Similar matrices procedure (smp) When the coupling errors can be ignored, considering the chain parameter matrix of the structure in Fig. 1, it can be write [1, 3]: 202 T M = T A ⋅T L ⋅T B (1) where SPI2007 eγ e L 0 TL = 0 0 eγ o L 0 e−γ e L 0 0 0 0 0 e −γ o L 0 0 0 With a little bit of mathematics, solving with respect to γο, it can be write: B + −1 = 0 8 For two differential traces of lengths L1 and L2 with the same cross-sectional dimensions, supposing identical the SMA’s at their ends, it is: T1M = T A ⋅ T L1 ⋅ T B (3) T2M = T A ⋅ T L2 ⋅ T B (4). ( ) −1 = T A ⋅T L1, 2 ( ) ⋅ TA −1 (5) where T L1 , 2 e γ e ( L1 − L 2 ) 0 = 0 0 e 0 The two matrices 0 0 0 e − γ e ( L1 − L2 ) 0 0 0 0 0 e − γ o ( L1 − L 2 ) γ o ( L1 − L 2 ) T1,M2 and T L1, 2 ( ) (T ) T1,Modd = T1Modd T2Modd 2 T1,Meven = T1Meven 2 e ( L1 − L2 ) ) −1 (11) Meven −1 2 (12) The matrix T1,2Modd and T1,2Meven have respectively the following eigenvalues: are similar, then they have (λ1 , λ2 ) = (eγ the same eigenvalues λ1, λ2, λ3, λ4: (λ1 , λ2 , λ3 , λ4 ) = (eγ (10) Known γο from (10), it is possible to calculate γe using (8) or (9). 3. Cascaded networks procedure (cnp) A reciprocal and balanced four-port network is fully determined by its odd and even mode subnetworks via the mixed-mode transformation [4], [5]. From [4], the odd/even mode of two cascaded four-port networks equals the scattering matrix of the cascade of the implicit odd/even mode two-port networks. Consequently, for the structure in Fig. 1, we can extract the odd/even equivalent models and then we can apply the similar matrices procedure to the two-port even/mode cascaded networks. By letting TiModd(even) be the measured odd (even) chain parameter matrix (2x2) of the differential line Li, obtained starting from the measured mixed-mode S parameters, it is: Then T1,M2 = T1M T2M A cosh{γ o ( L1 − L2 )} + 8 cos 2 h{γ o ( L1 − L2 )} + (2). , eγ o ( L1 − L2 ) , e −γ e ( L1 − L2 ) , e − γ o ( L1 − L2 ) Consequently, evaluating the eigenvalues of the matrix T1,M2 obtained from measurements, it is possible to determine the propagation constants γο, γe of the PCB differential trace, de-embedding the effects of the connectors used for connecting the measurement instrument. 2. Determinant procedure (dp) Starting from eqs. (3), (4), it can be written: T1M − T2M = T A (T L1 − T L2 )T B (6) T1M + T2M = T A (T L1 + T L2 )T B (7). Let’s define the two parameters A, B as follows: A= det(T1M − T2M ) det(T1M + T2M ) − det(T1M ) det(T1M ) (8) B= det(T1M − T2M ) det(T1M + T2M ) + det(T1M ) det(T1M ) (9). (λ , λ ) = (e * 1 ) * 2 ) ) ( L1− L 2 ) ) , e −γ o ( L1− L 2 ) γ e ( L1− L 2 ) ) − γ e ( L1− L 2 ) o ,e (13) (14) Consequently, evaluating the eigenvalues, it is possible to determine the true propagation constant γο, γe of the PCB differential trace. Test board The test board is a 12 layers board (Fig. 2) in FR4, where three different High Frequency (HF) chambers (HF1, HF2, HF3) in IS620 dielectric material have been laid out. To check the previous error removal techniques, two differential striplines of length L1 = 100cm and L2 = 20cm are laid out on both the layers HF2 and HF3. Each differential stripline consists of two traces of width W = 180µm, separated by S = 200µm. The traces are connected to the four-ports VNA for Sparameter measurements by means of SMA connectors at their ends. A standard full ports calibration procedure (SOLT calibration) has been made at the end of the test probes for calibrating the network analyzer with high quality calibration standards. The considered measurement frequency range is 40MHz – 18GHz, where the quasi-TEM mode of propagation is dominant. 203 SPI2007 differences at low frequencies with respect to the other ones: around 200MHz in the case of the odd-mode attenuation constant and below 100MHz in the case of the even-mode attenuation constant. Fig.4: De-embedded odd mode phase constant Fig.2: Test board. Experimental results The measurements show that the studied differential lines are reciprocal and balanced up to 6GHz. Consequently the Figs. 3-6 compare the different proposed de-embedding techniques up to 6GHz for the odd and even modes of propagation. Fig.5: De-embedded even mode attenuation constant Fig.3: De-embedded odd mode attenuation constant As it is evident, the differential striplines HF2 and HF3 have the same attenuation and phase constants. In fact they have the same cross sections even if they are on different layers in the board stack-up. The propagation constants obtained by the determinant procedure present some relevant 204 Fig.6: De-embedded even mode phase constant SPI2007 However, in spite of the fact that the propagation constants evaluated in different ways have some differences at some frequencies, the corresponding evaluated insertion losses coincide (Fig. 7). Fig. 7 compares the measured odd mode insertion loss of the HF2 differential stripline 20cm long with those obtained using the similar matrices and determinant error removal schemes. The measured |SD2D1| shows a larger loss than the error-removed insertion losses and a marked dip around 9Ghz, due to the stub offered by the via around the central pin of the SMAs. The error removal schemes are not effective in removing completely the effect of the stubs, probably due to the formation of coupling errors at high frequencies which have to take into account. By definition of the scattering parameter matrix for the 4port network in Fig.8, the outgoing wave parameters bi’s are related to the incoming wave parameters aj’s by b1 S11 b2 S = 21 b3 S31 b4 S41 S12 S22 S32 S42 S13 S 23 S33 S 43 S14 a1 S24 a2 . S34 a3 S44 a4 (15) Let’s define the sub-matrices [A], [B], [C], [D] as follows [6]: [A] = S11 S 21 S12 S 22 [B] = S13 S23 S14 S 24 [C ] = S31 S 41 S32 S 42 [D] = S33 S43 S34 S 44 The chain parameter matrix is given by: [T ] = Fig.7: Comparison among measured and extracted insertion losses for HF2 differential stripline. Conclusions The paper presents different mathematical techniques for extracting the odd and even true propagation constants of differential striplines, starting from four-port S-parameter measurements. The proposed error-removal schemes use two different length transmission lines and are effective up to frequencies where the coupling error matrices can be ignored. In this case, evaluating the modal impedances of the traces, the presented de-embedding techniques can be used for characterizing completely the transmission medium as a function of the frequency. Appendix Fig.8: Four port network with a’s and b’s defined. C −1 AC −1 − C −1D B − AC −1D References [1] V. Ricchiuti, A. Orlandi, G. Antonimi, “De-embedding methods for characterizing PCB interconnections,” IEEE workshop on Signal Propagation on interconnects, May 10-13, 2005, Garmish-Partenkirchen, Germany [2] K. Narita, T. Kushta, “An Accurate Experimental Method for Characterizing Transmission Lines Embedded in Multilayer Printed Circuit Boards,” IEEE Trans. On Advanced Packaging, Vol. 29, no. 1, February 2006, pp. 114-121. [3] T. M. Winkel, L.S. Dutta, H. Grabinski, E. Groteluschen, “Determination of the Propagation Constant of Coupled Lines on Chip Based on High Frequency Measurements”, IEEE Multi Chip Module Conference 1996 MCMC’96, , February 1996, pp. 99-104. [4] H. Shi, W. T. Beyene, J. Feng, B. Chia, X. Yuan, “Properties of Mixed-Mode Parameters of Cascaded Balanced Networks and Their applications in Modeling of Differential Interconnects”, IEEE Trans on Microwave Theory Tech, Vol. 54, no. 1, january 2006, pp. 360-372. [5] D. A. Frickey, “Conversions Between S, Z, Y, h, ABCD, and T Parameters which are Valid for Complex Source and Load Impedances”, IEEE Trans on Microwave Theory Tech, Vol. 42, no. 2, february 1994, pp. 205-211. [6] J. Fan et al., “Characterizing Multi-Port Cascaded Networks”, United States Patent, no. US 6,785,625 B1, august 31, 2004 205 SPI2007 206 SPI2007