On the identification of cables for metallic access networks

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/3899322 On the identification of cables for metallic access networks Conference Paper · February 2001 DOI: 10.1109/IMTC.2001.928292 · Source: IEEE Xplore CITATIONS READS 17 31 5 authors, including: Leo P Van Biesen Tom Bostoen 137 PUBLICATIONS 1,220 CITATIONS 28 PUBLICATIONS 738 CITATIONS Vrije Universiteit Brussel SEE PROFILE Alcatel Lucent SEE PROFILE Thierry Pollet Alcatel Lucent 34 PUBLICATIONS 262 CITATIONS SEE PROFILE All content following this page was uploaded by Tom Bostoen on 30 December 2013. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. IEEE Instrumentation and Measurement Technology Conference Budapest, Hungary, May 21-23, 2001 On the Identification of Cables for Metallic Acces Networks Patrick BOETS1, Mohamed ZEKRI1, Leo VAN BIESEN1 Tom BOSTOEN2, Thierry POLLET2 1 FREE UNIVERSITY BRUSSELS, Dept. ELEC, Pleinlaan 2, B-1050 Brussel, Belgium, Email:pboets@vub.ac.be 2 ALCATEL CRC ANTWERP, F. Wellesplein 1, B-2018 Antwerpen, Belgium Email:tom.bostoen@alcatel.be Abstract: - A new physical model ‘VUB1’ is proposed to characterize a metallic parallel wire or twisted wire pair. The VUB1 model will be compared with a rational function in the sdomain and with the popular physical model ‘BT0’ of British Telecom as described in the standards for Metallic Acces Networks. The VUB model contains a limited parameter set and is linear and time invariant there were the models used in the standards use more parameters and are semi-empirical, which can lead to non-causal systems. Metallic cable characterization with physical models will be compared with rational functions in the Laplace-domain. A Maximum Likelihood Estimation framework is used in the frequency domain to identify the model parameters. The performance of the models will be compared with a simulation directly based on measurements. Keywords - Metallic cables, Identification INTRODUCTION Metallic acces networks are nowadays revalued. They are now exploited almost to the Shannon bound because there is a large demand for more bandwidth mainly by the Internet applications. The acces network contains a variety of cable types, mostly twisted pair and sometimes parallel wire (drop wire), which differ in wire diameter (gauge) and insulator material such as PolyEthylene (PE), Paper (Pulp) and PolyVinylChloride (PVC). Depending of the topology of a subscriber loop, a wire segment can even be screened or located in a cable bundle where it is surrounded by neighbouring metallic conductors that act as a screen. Digital Subscriber Line (DSL) modems, which use the old-fashioned telephony metallic acces networks as transmission medium, contain state-of-the-art analog and digital technologies in order to achieve a high throughput. Manufacturers still expend a lot of effort on research and development to improve existing products and even to create new designs. This means that cable models for twisted pair and drop wires are frequently used in order to simulate the throughput of a system, to design hybrid circuits, etc. The cable models based on the physical geometry and properties facilitate investigation of the electrical behaviour when a design parameter is modified, although they do not necessarily correspond exactly to the measured response due to the mechanical deviation (e.g. the twist, multiple insulators, neighbouring wires, screens, etc.) of the commercially available lines from the idealized geometry. The simulation of the transmission line on a computer, based on the true frequency response of a real installed line from the cable plant, is still very important. The applications areas are frequent: the design and testing of adaptive equalizers for high throughput digital communication apparatus, the investigation of hybrid couplers on the performance of echocancellation based digital communication, the training of operators of cable fault localization devices, the simulation of the propagation of signals, etc.. The existing physical models as described in the standards are semi-empirical and exhibit some inconsistencies which appear as pulse-like fore-runners in the impulse response. Therefore, a new physical model is presented to improve the impulse response of the metallic cable model. CABLE MODELS Definitions The metallic cable which consist out of 2 conductors surrounded by an insulator is modelled as an homogenous transmission line. The medium between the two conductors is a linear and isotropic dielectric. The transfer function of the line is given by: H(w) = e –g(w )l (1) with g(w) the propagation function and l the length of the line. The propagation function is described by: g(w) = Z s(w)Y p(w) with Z s(w) the serial impedance per meter and Y p(w) the parallel admittance per meter. The characteristic impedance of the cable can be obtained as follows: Z c( w ) = Z s(w) -------------Y p(w) (2) The metallic reference models The cable models that are used in the standards for xDSL systems [9], [10] and [7] are based on the physical properties of the materials where the skin-effect and the dielectric losses are taken into account. To improve the 0-7803-6646-8/01/$10.00 ©2001 IEEE 1348 approximation between the models and the measurements, additional model parameters were added on empirical basis. However, this addition makes the ‘improved’ models noncausal: the real and imaginary part of Z s(w) and often also Y p(w) do not fulfil to the Hilbert Relations [8]. And the constants A n are solutions of: (8) ln Q nm = T nm – d nm -------D 2n (9) A = ( A 1, A 2 ¼ ) t (10) with A very popular model is the BT0-model from British Telecom. It is adopted by the xDSL-standards. It contains 11 parameters: 6 parameters to describe Z s(w) and 5 for Y p(w) . ( m + n – 1 )! T nm = ----------------------------------------( n – 1 )! ( m – 1 )! The VUB1-model n Jn – 1 (ka) l n = – ---------------------------Jn + 1 (ka) a (11) and d nm the Kronecker delta with D = a ¤ D , J n the Bessel function of the first kind and order n . D Fig. 1 Q A= 1 A cross-section of the parallel wire line The total series impedance Z s(w) of the per-unitlength distributed circuit consists of four contributions: Z s(w) = Z sk1(w) + Z sk2(w) + jwL + Z prox(w) (3) If the series (7) is truncated up to the first 2 terms, then it can be shown [11] that the maximum relative error is below 0.5% for D ¤ a ³ 3 in a band up to 10MHz. Solving the equation (8) for the first 2 terms, the following expression for the total series impedance of the per-unit-length distributed circuit can be found in the s -plane: Z ski is the impedance of the conductor ( i = 1, 2 ) if the return conductor is infinitely distant. The skin effect is included in this contribution, thus kaJ 0(ka) Z ski = R 0 -----------------------2J 1(ka) J Z s(s ) = A 0 s + A 1 – s -----0- + J1 3D 4 J 3 J 2 + 2J 1 J 2 + D 2 J 0 J 3 æ ö A 2 s ç ---------------------------------------------------------------------------------------------÷ 6 2 4 è D J 2 J 3 + D J 1 J 2 + 3D J 0 J 3 + J 0 J 1ø (4) with (12) with 1 R 0 = ------------- and k = psa 2 – jwms and w stands for the angular frequency, a the radius, s the conductivity, m the permeability of the conductor, R 0 the DC-resistance of one wire and Ji (ka) the Bessel function of order i . The external self-inductance L , attributed to the field between the conductors, which are separated by distance D , is given by: m D L = --- ln ---p a (6) Belevitch [3] has given a set of equations to solve the proximity effect in multiwire cables for the screened and unscreened cases. It was shown [3] that no closed form solution exists for calculating Z prox(w) . The impedance Z prox(w) , due to the proximity effect, can be found by theoretically solving an infinitely linear system: jwm ¥ Z prox = ---------- å A n p n =1 A0 = L (5) (7) A 1 = R 0 a ms and Ji = Ji (A 3 A 3= a ms (13) m A 2 = D 2 -----2p – s) the Bessel functions of order i with A extra degree of freedom is introduced by considering A 0 as an independent parameter because the screening effect of the neighbouring idle line pairs will decrease the external inductance. Moreover, the unmodelled wire twist will increase the inductance. Therefore, the series impedance becomes: J Z s(s ) = a 4 s + a 1 – s -----0- + J1 a æ 3a 22 J 3 J 2 + 2J 1 J 2 + a 2 J 0 J 3 ö a 1 a 3 -----2- s ç ----------------------------------------------------------------------------------------------÷ 2 3 2 è a 2 J 2 J 3 + a 2 J 1 J 2 + 3a 2 J 0 J 3 + J 0 J 1ø (14) The admittance of the per-unit-length distributed circuit is given by: Y p(w) = G(w) + sC(w) 1349 (15) For dielectrics that show very few losses, such as PE or XLPE, the following model is used: Y p(w) = wC tan d + sC (16) however for PVC and Pulp cables more advanced dielectric model equations should be used [1][2]. The parameter vector P = ( a 1, a 2, a 3, a 4, C, tan d ) t for the VUB1-model. functions. The instrument calibration method mixes both waves whereafter they can not be separated any more to fall in the errors-in-variables estimation framework. Therefore, the sample variance of the measured G in(w) will be used to turn errors-in-variables structure into an output-error model (Fig. 2). In this case, the Maximum Likelihood Estimator for an errors-in-variables structure is reduced to a Weighted Least Squares Estimator (WLSE). Rational functions 1 From an approximation point of view, a rational function in the Laplace-domain (s-domain) with delay was in the past tested out with success on the transfer function of coaxial cables [5]. Therefore, the transfer function of a twisted pair will be identified using the same models. This methodology will even be extended to the approximation of Z s(w) and Y p(w) directly. This approach stands closer to the physics because the rational functions represent an impedance and admittance function now. It was demonstrated that there is a direct relationship between the parameters of the rational function in the s-domain and the physical behaviour of coaxial cables [4][11]. On the same assumptions, it can be expected that a parallel wire systems also obeys the same relationship. THE ESTIMATOR The transfer function and the characteristic impedance of a cable can be obtained from 2 input reflection coefficient G in(w) measurements: G in(w) = b(w) ¤ a(w) (17) and a(w) and b(w) denote the incident and reflected voltage wave in base Z b respectively. Thus the line functions are: ( 1 – G in, s G in, o ) 2 ( 1 – G in, s G in, o ) e – 2gl = – ----------------------------------------+ æè -----------------------------------------öø – 1 (18) ( G in, s – G in, o ) ( G in, s – G in, o ) also called TF 2(w) and ( 1 + G in, o ) ( 1 – e –2gl ) Z 0(w) = Z b ------------------------------------------------------( 1 – G in, s ) ( 1 + e –2gl ) The total series impedance and parallel admittance can be obtained as follows: Y p(w) = ( – log TF(w) ) ¤ Z 0(w) G in (w ) + G in, m(w ) s G2 in (w) Fig. 2 The output-error model. The expression for the MLE cost-function [6] is turned into the WLSE cost-function: N CF = H(W k) – H m, k 2 å ----------------------------------------2 k=1 s H, k (21) with H(W k) the model, H m, k the measured Frequency Re2 sponse Function (FRF), s H the variance of the measured ,k FRF and k the spectral index. SIMULATIONS AND MEASUREMENTS All the cable models discussed are tried out on measurements done on pair 10 of a 900m long Belgacom 0.5mm 10 quad gelly filled PolyEthylene cable. An HP3577B Network Analyzer was used to measure G in(w) in a frequency band from 10kHz up to 10MHz. A logarithmic frequency grid was selected to keep the model errors small at low frequencies because this is often the most important frequency region: e.g. channel capacity computations for xDSL modems. Due to the noise and limited resolution of the Network Analyzer, the frequency region above 2MHz was not used because the phase unwrapping in (20) became impossible. The data of this copper cable are shown in Fig. 3. (19) with G in, o and G in, o the input reflection coefficient of a line with the line end opened and shorted respectively. Z s(w) = ( – log TF(w) ) Z 0(w) G in(w ) Three models were used to approximate the measured data. In general, the stochastic errors are completely negligible with respect to the modelling errors. The observed twisted pair does not act as a nice homogeneous system and is even nonreciprocal! Due to these model errors, the cost-function will be much higher than the expected value of the cost-function. (20) Next, the serial impedance and parallel admittance will be approximated with a model using an estimator. The system G in(w) will be excited by an input signal a(w) and the system response gives b(w) , the true output signal. The excitation signal and the response are not measured separately, only their ratios are used to determine the cable 1350 The Transfer Function The Transfer Function 0 Magnitude Error 0 -5 Phase Error 0.3 -10 1 0.2 0.5 -20 0.1 -30 0 0 -40 (deg) -15 (dB) (dB ) (deg) -10 -50 -0.1 -60 -0.2 -0.5 -20 -25 4 10 5 10 Freq (Hz) 10 -70 4 10 6 The Characteristic Impedance (Real) -1 5 10 Freq (Hz) 10 -0.3 6 -0.4 4 10 The Characteristic Impedance (Imag) 200 5 10 Frequency (Hz) 10 -1.5 4 10 6 5 10 Frequency (Hz) 10 6 0 Fig. 6 180 The model errors of the transfer function of the BT0-model. -50 (9 ) (9 ) 160 Real Part Relative Error Imag Part Relative Error 5 50 140 -100 40 4 120 30 3 20 10 6 -150 4 10 5 10 Freq (Hz) 10 2 6 10 (%) 5 10 Freq (Hz) (%) 100 4 10 0 1 -10 0 -20 Fig. 3 The transfer function and the characteristic impedance of a 900m long 0.5mm Polyethylene insulated cable from Belgacom. -1 -2 4 10 Fig. 7 model. 0 3 1 10 5 0 0 -0.5 -5 -15 -1 -20 -15 -2 -20 -2.5 0 -0.5 4 10 Fig. 4 5 10 Frequency (Hz) 10 -25 4 10 6 5 10 Frequency (Hz) 10 6 The model errors of the transfer function of the VUB1-model. Real Part Relative Error -3 4 10 -25 5 10 Frequency (Hz) 10 -30 4 10 6 Real Part Relative Error 6 Imag Part Relative Error 7 20 60 6 15 50 3 5 40 4 20 3 10 0 2 0 0 1 -1 -5 -10 -2 5 (%) 1 10 30 (%) (%) 2 (%) 10 Fig. 8 The model errors of the transfer function of the rational function in s-domain. 70 4 0 -20 -10 -1 Fig. 5 model. 5 10 Frequency (Hz) Imag Part Relative Error 5 -3 4 10 6 -10 -1.5 0.5 10 Phase Error 0.5 (dB) (deg) (dB) 1 -10 5 10 Frequency (Hz) The model errors of the characteristic impedance of the BT0- 15 -5 2 -40 4 10 1.5 3.5 1.5 10 Magnitude Error Phase Error 5 2.5 10 Frequency (Hz) 6 (deg) Magnitude Error 4 -30 5 5 10 Frequency (Hz) 10 6 -30 4 10 5 10 Frequency (Hz) 10 6 The model errors of the characteristic impedance of the VUB1- -2 4 10 5 10 Frequency (Hz) 10 6 -15 4 10 5 10 Frequency (Hz) 10 6 Fig. 9 The model errors of the characteristic impedance of the rational function in the s-domain. 1351 All 3 models approximate both Z s(w) and Y p(w) whereafter the transfer function and the characteristic impedance can be calculated. 7 The approximation with the VUB1-model was of average quality (Fig. 4 and Fig. 5). Only 6 parameters describe the complete frequency behaviour. 6 The BT0-model performs better in the frequency domain than the VUB1-model (Fig. 6 and Fig. 7). The BT0 model has 11 parameters and therefore 5 extra degrees of freedom compared with the VUB1-model. 3 A rational function in the s-domain approximated both Z s(w) and Y p(w) . This model is very flexible because any model order can be chosen. It is not the intention to approximate the data until the model errors are below the stochastic errors. This would require very high orders and its predictive behaviour would not be better than low order models to approximate the TF(w) and Z 0(w) for another wire pair with exactly the same length and from the same type. The selected rational function order for Z s(w) amounted to 8 coefficients in the numerator and 7 coefficients in the denominator. For Y p(w) the order was limited to 3 by 2. The results are shown in Fig. 8 and Fig. 9. -1 x 10 -4 The Impulse Response (z oomed) 5 4 2 1 0 -2 -3 2 Fig. 11 3 4 5 6 Time (m s) 7 8 9 10 The impulse response of the BT0 model. To demonstrate the effect of non-causal ANSI_TP1 ADSL cable model [9] as used in the standards, the response on a bipolar excitation signal on an open ended cable of 2km long was simulated and compared with the VUB1-model. The result is shown in Fig. 12. It is clearly seen that the ADSL cable model starts to early in contrast with the VUB1-model. The impulse response of the VUB1-model is shown in Fig. 10. The pulse shape corresponds with the reflection response of an impulse excitation coming from an open ended line of 450m long which is perfectly matched with a reflectometer. The impulse response of the BT0-model contains a ‘dip’ before the actual pulse start. This effect is unrealistic and is explained by the non-causal BT0-model [8] for the series impedance Z s(w) . -4 x 10 20 15 7 -4 (V) 10 x 10 The Impulse Response (zoom ed) 5 ANSI-ADSL 6 0 5 VUB -5 4 18 3 2 22 24 26 28 Tim e (m s) 30 32 34 36 Fig. 12 The time response of a 2km long line on a bipolar pulse excitation signal. The response calculated with the ANSI-ADSL model shows an unrealistic rise of the reflection, where on the other hand the VUB model is in accordance with the physical observations. 1 0 -1 2 Fig. 10 20 3 4 5 6 Time (ms) 7 8 The impulse response of the VUB1 model. 9 10 CONCLUSIONS A new closed form expression in the Laplace-domain of a physical parallel-wire model is given.The model is causal and the number of parameters small in contrast to the models used in the standards. Twisted pairs can be modelled with the same expression if an extra degree of freedom is build-in. The impulse response of the cable model is consistent with the physical observations, which is very important for simulations in the telecommunication world. 1352 ACKNOWLEDGEMENT This work was supported by the Flemisch Community IWT. REFERENCES [1] [2] [3] [4] [5] [6] Y. Ishida, "Studies on Dielectric Behaviour of High Polymers", Kolloid-Zeitschrift, Band 168, Heft 1. C.J.F. Bötttcher and P. Bordewijk, "Theory of Electric Polarization", Vol. 2, Dielectrics in time-dependent fields, 2nd ed., Elsevier, 1978. V. Belevitch, “Theory of the Proximity Effect in Multiwire Cables, Part I”, Philips Research Reports, Vol. 32, No. 1, pp 16-43, September 1977 S. S. Yen, Z. Fazarinc and R. L. Wheeler, “Time-Domain Skin-Effect Model for Transient Analysis of Lossy Transmission Lines”, Proceedings of the IEEE, Vol. 70, No. 7, pp. 750-757, July 1982. R. Pintelon and L. Van Biesen, “Identification of Transfer Functions with Time Delay and its Application to Cable Fault Location”, IEEE Transactions on Instrumentation and Measurement, Vol. IM-39, No. 3, pp. 479–484, June, 1990. R. Pintelon, P. Guillaume, Y. Rolain, J. Schoukens, H. Van Hamme, “Parametric Identification of Transfer Functions in the Frequency 1353 Domain-A Survey”, IEEE Transactions on Automatic Control, Vol. 39, No. 11, November 1994, pp. 2245-2260 [7] R.F.M. Van Den Brink, “Cable Reference Models for Simulating Metallic Acces Networks”, ETSI STC Permanent Document TM6(97) 02, June 1998 [8] J. Musson, “Maximum Likelihood Estimation of the Primary Parameters of Twisted Pair Cables”, ETSI STC contribution TD8, TM6 meeting Madrid Spain. 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