Behavioral Modeling of Tranmiion Line Channel via Linear Tranformation Albert Vareljian albertv@ieeeorg Member, IEEE, Canada Abtract An approach baed on the linear tranformation of network port variable i augmented and applied to behavioral modeling of inhomogeneou tranmiion line channel with lumped and ditributed dicontinuitie Generic tool Spice-like and Simulink can be ued for channel imulation in the frequency or time domain Introduction Recent development in very high bit rate wireline data tranmiion application a 000BASE-T Gigabit Ethernet (GE) [], VDSL [] and other, have motivated continued reearcher interet in accurate imulation of data channel baed on the unhielded twited pair (UTP) medium A the ignaling bandwidth approach the order of 00 MHz, even relatively minor channel dicontinuitie uch a hort jumper, connector etc need to be accounted for in the ytem deign Such dicontinuitie caue multiple reflection in the channel, becoming an impairment for the ytem performance An approach baed on the linear tranformation of network port variable, originally introduced for ynthei of active filter [3], ha been augmented and applied to the behavioral modeling of inhomogeneou tranmiion line channel with lumped and ditributed dicontinuitie Firt, a channel with dicontinuitie i plit into a cacade connection of uniform block: two-port, repreenting lumped (dicrete) circuit element, or homogeneou ection of tranmiion line The bidirectional channel model tructure i then derived in a regular procedure, uing electively choen port variable tranformation matrice The overall model conit of a cacade interconnection of the matrix tranfer function module of the two type Module of the firt type correpond on an element-by-element bai to the channel dicrete intance, or homogeneou tranmiion line ection, wherea the econd type are the o called port-matching module The later erve a an interface between input and output port for the adjacent module of the firt type It i hown that the port-matching requirement reult from the dicontinuity boundarie in the phyical channel and yield a matrix tranfer function whoe element are in eence the reflection and refraction coefficient The overall channel model, referred to a the linear tranformation equivalent (LTE), can be readily imulated in time or frequency domain uing generic tool like Matlab/Simulink or Spice Phyical phenomena in tranmiion line uch a kin effect, dielectric lo and characteritic impedance variation can alo be modeled a hown Homogeneou tranmiion line ection are decribed in term of their correponding characteritic impedance, propagation delay and rational -domain tranfer function approximating inertion lo frequency repone The near and far end echo and tranmiion waveform are generated imultaneouly The propoed method offer certain advantage compared to the approach traditionally ued in the communication community baed on the frequency domain computation of the channel chain matrix ABCD parameter [4], [5], with ubequent frequency-to-time dicrete-domain Laplace inverion by IFFT The LTE model naturally yield a continuou-time ytem which can be imulated uing analog variable-tep tool Thi provide mean for a relatively traightforward and efficient behavioral imulation and analyi of complex mixed-ignal ytem, epecially under condition of timing reference frequency drift and/or offet, a i the cae, for example, with a pair of data communication tranceiver with timing recovery The method alo allow effective what-if, or component variation analyi Application example i given whereby GE timing recovery tet channel i modeled together with a typical tranformer line interface Linear Tranformation Approach The linear tranformation method wa originally developed by Dimopoulo and Contantinide [3] with the main aim of active filter ynthei from the doubly terminated lole LC ladder filter The philoophy of the approach i that phyical variable (current and voltage) of a network are firt linearly tranformed into a new domain where the correponding relationhip are then realized by mean of tranfer function By varying the linear tranformation parameter different tructure correponding to the ame prototype network may be obtained, poeing feature deirable for a targeted realization Due to the univeral nature of the underlaying concept, the baic approach could be adopted for application in ytem modelling, a demontrated in thi paper for the cae of a tranmiion line channel with ditributed and lumped dicontinuitie The concept of linear tranformation i briefly preented below
(a) I I V N V (a) I I V N V V I N I V y T () T () y (b) y y y y (b) () T () N e J N e x x (c) y y Fig (a) The paive two-port prototype (b) The SFG of the correponding LTE (c) LTE block diagram For an arbitrary linear paive network (ee Fig (a)) contained between the port and with aociated voltage-current variable vector [V I ] T and [V I ] T, one canaignatwo-portn characterized in term of the modified chain matrix ABCD a follow: The port variable vector [V I ] T and [V I ] T are linearly tranformed by a pair of noningular matrice Λ and Λ into the new variable [ y ] T and [x y ] T uch that: N e V A B V T V I C D I I α β V V Λ y γ δ I I x () Fig (a) The cacade connection of two-port (b) The interconnection of the correponding LTE block Combining () and () yield: where Λ TΛ x ------ y y d γ ( β A α B) + δ ( β C α D) A the chain matrix generally repreent invere (noncaual) input-output relation, it i deirable to group the ytem variable uch that a caual and, therefore, realizable relationhip reult If, for example, vector [y y ] T i expreed in term of [ x ] T, the correponding (realizable) relationhip can be found from (3) for any paive N, auming the baic paive network theory identity det T a follow: y y a c b d a α ( δ A γ B) + β ( δ C γ D) b α ( β A α B) + β ( β C α D) c γ ( δ A γ B) + δ ( δ C γ D) d b b x b a b x x y ( ) T ( ) T ( ) T ( ) x (3) x α β V V Λ y γ δ I where detλ 0,, Due to the choice for the port current direction flowing into the network, quantitie B and D appear in the modified form with a negative ign in contrat to the conventional notation I () (4) Equation (4) repreent a matrix tranfer function whoe SFG i hown in Fig (b) A two-input-two-output nonreciprocal (directional) network N e hown in Fig (c) will in general be required for the matrix tranfer function realization in (4) The minu ign appear a a reult of the choen direction for I
For the purpoe of thi work the SFG tructure of Fig (b) and the realizing network N e of Fig (c) are referred to a the linear tranformation equivalent (LTE) of the correponding paive network N The directional nature of the LTE i reflected in Fig (c) whereby the ignal y and y are the deignated output (denoted by triangle) generated in repone to the excitation and x (denoted by mall circle) that are enforced at the input Cacade connection of the two-port N, N hown in Fig (a) decribed in the V-I domain by V 0 I 0 V yield together with () for the correponding vector variable of the adjacent LTE the following relationhip: α δ + γ β ------------------------------------ γ δ + γ δ ---------------------------------- γ δ + γ δ where the econd ubcript denote the index of the repective block The interconnecting network J in Fig (b) i defined for the adjacent LTE a a block, realizing the matrix tranfer function in (6): The later combine ignal y and y to generate and It can be oberved from the above conideration that by varying the tranformation matrice in (), different LTE repreentation in (4) can be obtained for a given prototype Throughout the ret of thi paper tranformation matrice of the form will be employed, reulting in the wave-flow [6], repreentation in the LTE With the above aumption, variable x and y have a meaning of incident and reflected voltage wave repectively Uing (7) and auming only ymmetrical network for N (A D), the LTE tranfer function in (4) can be expreed a follow: I J J y J J ---------------------------------- γ δ + γ δ α δ + γ β ------------------------------------ γ δ + γ δ y Λ Λ β β ( B β + βc) T ------------------------------------- A B β + βc T T ------------------------------------- A B β + βc y y (5) (6) (7) (8) With (7) the matrix tranfer function (6) will be implified: β β ---------------- β + β β ---------------- β + β β ---------------- β + β β β ---------------- β + β where the ingle ubcript correpond to the network index Homogeneou Tranmiion Line Section Homogeneou ection of a tranmiion line can be decribed a a two-port with the modified chain matrix of the form T cohθl ---- inhθl Z 0 Z 0 inhθl cohθl (9) (0) where l i the length of the ection, θ ( R + L) ( G + C) and Z 0 ( R + L) ( G + C) are the propagation contant and characteritic impedance of the line with primary contant R, L, G, C per unit length In practice, the line primary contant are actually frequency dependent due to the kin effect in conductor and dielectric lo in the inulator material To reflect thi fact we adopt the following approximation [7], [8] for the erie reitance R and hunt conductance G: R r + k () G g + g d Y d ( ) () where r, g are the dc reitance and leakage repectively, k i the kin effect contant, g d i a contant, and Y d ( ) -------------- + (3) repreent the dielectric lo frequency repone with a pole at angular frequency ω d To model a ection of homogeneou tranmiion line we chooe in (7) β Z 0 Thu, from (8) and (0) we can find for the correponding LTE: T 0,and (4) The interpretation of the above tranfer equation i that there are no reflection within the line ection, and the line tranfer function i of the ame form when terminated at both end by it characteritic impedance Z 0 When two line ection with different characteritic β Z 0 and β Z 0 are connected, the LTE interconnection relationhip can be found form (9): ω d T T e θl y y
Z 0 Z --------------------- 0 Z 0 + Z 0 Z 0 --------------------- Z 0 + Z 0 Z 0 --------------------- Z 0 + Z 0 Z 0 Z --------------------- 0 Z 0 + Z 0 y y (5) y y The element of the above matrix can be readily recognized a reflection (on the diagonal) and refraction coefficient at the dicontinuity boundary for the left and right going voltage wave With thi, the LTE interconnection network function ha a meaning of a port-matching block Naturally, for the adjacent line ection with identical characteritic impedance Z 0 Z 0 the port-matching block i not needed (no dicontinuity i preent), o can be directly connected to y and to y in a cro-connect fahion hown in Fig 3 Thi follow from (5), which with Z 0 Z 0 become: 0 y (6) 0 y We will now conider tranmiion line tranfer function approximation in (4) The olution to the approximation problem i ought in the form of a cacade of two tranfer function: (7) where the exponential term on RHS of (7) repreent the wave front propagation delay τ d l LC, ( ), and H() i a rational minimal-phae function, approximating the line inertion lo frequency repone exp( θ l ) For ufficiently high frequencie (above 00-00 khz) θ can be expreed a where e θl e τd θ e θ l e τd H ( ) k 05 g d R 0 Y d ( ) + ----- r + ----- R 0 lim Z 0 (8) (9) i a purely reitive value oberved for the line characteritic impedance at high frequencie Thi aumption i well jutified for the frequency band utilized in very high bit rate application For a homogeneou ection of tranmiion line with a given length l the inertion lo frequency repone can be computed uing (8) and fitted in the frequency domain by a rational function R 0 L C R 0 Fig 3 The LTE cro-connect method The frequency fit can be performed with the help of Matlab Signal Proceing Toolbox function invfreq [9] Reult of uch approximation are hown in Fig 4 for the cae of meaured repone of a 00 m category 5 (CAT5) UTP cable ued in GE The following line electric parameter: r 938mΩ/ m, L 063 µh/m, C 56pF/m,g 0pS/m,g d 0mS/ m, ω d π90e6 rad/, and k 09mΩ/m/ / for the kin effect contant have been choen uch that a good match wa oberved for the meaured and computed baed on (8) repone Fitting wa then performed with m 4andn 6 for the computed data with frequency point normalized to 5 MHz to improve accuracy Correponding normalized polynomial coefficient for H() in (0) are given in Table I The UTP cable characteritic hown in Fig 4 wa captured with 00Ω termination at both end A it can be een from the graph, the approximation matche very well with the meaured data TABLE I NORMALIZED COEFFICIENTS FOR CAT5 (00m) CABLE RESPONSE APPROXIMATION H() i a i b i 0 654 6539 360 76703 3786 8738 3 704 3634 4 566 05559 5 33585 - H ( ) b m m + b m m + + b + b --------------------------------------------------------------------------------- 0 n + a n n + + a + a 0 (0) 6 - where m n Baud rate adopted in GE over copper
Magnitude, db 0-5 -0-5 -0 Mea Approx nel reflection, or echo With the lat two-port N n terminated into a reitor R 0, the correponding LTE output y n with x n 0 (grounded) would be equal the load voltage in the prototype An open, or hort circuit at the output of the lat two-port N n in the prototype would reult in the correponding LTE in full ignal poitive x n y n, or negative x n y n reflection repectively After all elementary two-port and tranmiion line ection in the channel are tranformed, the overall LTE model i obtained by interconnection of the adjacent LTE module baed on (9) or (6) depending on whether or not a dicontinuity i preent at the port boundary In cae of the dicontinuity a port-matching module J would be required between the adjacent LTE module, otherwie a imple cro-connect hown in Fig 3 i ued To illutrate the propoed method an application example i conidered in the next ection -5 0 4 6 8 0 Frequency, Hz x0 7 Fig 4 Approximation of CAT5 cable frequency repone Modeling of Lumped Element The modeling of channel lumped (dicrete) element; inductor, tranformer, reitor and capacitor that appear in the line interface i conidered next Similar to the approach ued in ynthei of active filter, the network i firt plit into a cacade of elementary two-port coniting olely of erie impedance Z, or hunt admittance Y (To achieve thi partitioning, all tranformer in the circuit need to be replaced with their ladder equivalent) Each elementary two-port i linearly tranformed uing matrice (7), where β i aigned a real impedance value, typically choen from practical conideration For convenience β R 0 i employed here, with R 0 correponding to the nominal characteritic impedance of the tranmiion line The elementary two-port LTE tranfer function can be obtained form (8), taking in account that the modified chain matrix of any erie or hunt arm i of the following form T i B i C i () where B i Z i, C i 0 for any erie arm Z i, and B i 0, C i Y i for any hunt arm Y i, i A all dicrete element are tranformed uing the ame β, no port-matching i needed between the correponding adjacent LTE module, o imple cro-connect a hown in Fig 3 i employed We will now conider the termination requirement for the firt and lat LTE block A prototype ignal ource E with reitance R 0 i tranformed into a direct connection between the ource and LTE input: E If no ource i active at the far end, output y would repreent the chan- Application Example An application example i given in thi ection whereby GE timing recovery tet channel [, p 69] i modeled together with a typical tranformer line interface The channel conit of the relatively long (00 m) 00Ω main UTP ection appended at each end by a hort jumper ection with a characteritic impedance of 0Ω The dicontinuitie were purpoely introduced in [] uch that the tranceiver loop-timed ynchronization ytem could be teted under condition of poor ignal to echo ratio The tet channel implified circuit diagram (referred to a the prototype) i hown in Fig 5(a) An exemplary tranformer line interface i conidered at each end of the channel with ome paraitic: the tranformer leakage inductance L,8 00 nh and a hunt capacitance C 3,7 5 pf Four dicontinuity boundarie are introduced in the channel by the two hort line egment with 0Ω characteritic impedance The prototype i firt plit into 9 ection, a indicated in Fig 5(a) by the dahed line, compriing of 3 homogeneou tranmiion line egment and 6 elementary two-port, repreenting dicrete intance The aociated port impedance value that are ued in the repective tranformation matrice (7) are hown on the top, wherea the line egment length figure are given at the bottom Correponding LTE tranfer function derived form (8) are preented in Fig 5(b) below each prototype ection Whenever applicable, the tranfer coefficient of port-matching block are alo given Relatively hort egment of line TL 4 and TL 6 have been aumed lole and, hence, are modeled in the correponding LTE a a pure delay with τ 4 076UI and τ 6 094UI, where UI 8 n correpond to the Baud unit interval adopted in GE ytem The main 00 m line egment introduce a delay of τ 5 70UI and i aumed here to be homogeneou for implicity
00Ω 0Ω 00Ω 0Ω 00Ω R00 L00 nh L800 nh R00 E L350 µh TL4 TL5 TL6 C35 pf C75 pf L9350 µh E m 00m 48m LTE No 3 J 4 J 5 J3 6 J4 7 8 9 LTE MATRIX TRANSFER FUNCTION R T T L ------------------------- + R L T T ------------------------- + R L T T ------------------------- + R L R L T T ------------------------- + R L T -------------------------- + RC 3 -------------------------- + RC 3 RC 3 T T J 009 J 009 J 09 J 0909 T T e 4 T T e τ 5 H ( ) T T e τ 6 T -------------------------- + RC 7 RC 7 T T -------------------------- + RC 7 T -------------------------- + R L 8 R L 8 T T -------------------------- + R L 8 R T T L 9 ------------------------- + R L 9 T T ------------------------- + R L 9 (a) (b) J 009 J 009 J 0909 J 09 J 009 J 009 J 09 J 0909 J 009 J 009 J 0909 J 09 (c) V V 3 J 4 J 5 J3 6 J4 7 8 9 E E Fig 5 (a) The GE timing recovery tet channel circuit diagram the prototype (b) The two-port correponding LTE tranfer function (c) Overall LTE model of the GE timing recovery tet channel Recent tudie indicate [0] that dicontinuitie due to the tructural diorder in relatively long line hould be taken into account at high operating frequencie Thi can be accomplihed by further ubdiviion of large tranmiion line egment into a relatively mall ection, which can be aumed homogeneou within their repective length The overall LTE model with properly connected LTE and port-matching block i hown in Fig 5(c) Jut like it twowire prototype the model i bi-directional in nature Thi allow for full duplex tranmiion imulation in the channel when the ource E and E are both activated In thi cae the model output V and V would repreent a uperpoition of the echo and receive ignal It hould be noted that component value variation in the prototype would reult in the tranfer function coefficient variation in the correponding LTE A hown in [] the LTE model enitivity to the coefficient variation can be made exactly the ame a the repective component enitivity in the prototype, ubject to LTE realization condition Thi important property allow realitic what-if analyi in the LTE model The diagram in Fig 5(c) ha been realized uing enitivity adequate building block hown in Fig 6 (LTE No, 3, 7, 9) and Fig 7 (LTE No, 8) The model in Fig 5(c) wa imulated in Spice and Matlab/ Simulink environment For both Spice and Simulink the imulation reult were identical within the computational preciion Fig 8 how imulated tet channel echo impule repone compared to the properly terminated homogeneou line channel: reflection wave due to characteritic impedance dicontinuity can be clearly oberved on the graph Correponding frequency repone are hown in Fig 9 A degradation of approximately 0 db in echo frequency repone in the tet v homogeneou channel can alo be oberved around 0 MHz frequency range Concluion A method baed on linear tranformation of network port variable originally introduced for ynthei of active filter ha been augmented and applied to behavioral modeling of tranmiion line channel with ditributed and lumped dicontinuitie Generic imulation tool like Simulink or Spice can be ued baed on their tandard library primitive to mo-
+ + y y 05 04 03 - T - 0 Volt 0 x + 0 Fig 6 LTE realization diagram for block No, 3, 7, 9-0 -0 Homogeneou Tet Channel + + y + y + -03 5 0 5 30 35 40 45 Time, n Fig 8 Tet v homogeneou channel echo impule repone -5 T -0-5 x Fig 7 LTE realization diagram for block No, 8 Magnitude, db -0-5 -30 Homogeneou Tet Channel del phyical phenomena in tranmiion line uch a kin effect, dielectric lo and characteritic impedance variation along with dicrete baic circuit element Reulting continuou-time model provide for imulation of high peed data channel in mixed-ignal ytem, epecially under condition of ampling frequency drift and/or offet Some additional work i need to extend the method for modeling of tranmiion line channel with bridged tap Reference [] Phyical Layer Parameter and Specification for 000 Mb/ Operation Over 4-Pair of Category 5 Balanced Copper Cabling, Type 000BASE-T, IEEE Std 803ab, 999 [] Tranmiion and Multiplexing (TM); Acce Tranmiion on Metallic Acce Cable; Very high peed Digital Subcriber Line (VDSL), ETSI Standard TS 0 70-, 00, http://wwwetiorg [3] H G Dimopoulo and A G Contantinide, Linear Tranformation Active Filter, IEEE Tran Circuit Syt, vol CAS-5, pp 845-85, 978 [4] W Y Chen, J L Dixon, D L Waring, High Bit Rate Digital Subcriber Line Echo Cancellation, IEEE J Select Area Commun, vol 9, no 6, pp 848-860, 99-35 -40-45 0 50 00 50 Frequency, MHz Fig 9 Tet v homogeneou channel echo frequency repone [5] JACBingham, ADSL, VDSL, and Multicarrier Modulation, New York,Wiley,000 [6] H Wupper and K Meerkötter, New Active Filter Synthei Baed on Scattering Parameter, IEEE Tran Circuit Syt, vol CAS-, pp 594-60, 975 [7] N S Nahman and D R Holt, Tranient Analyi of Coaxial Cable Uing the Skin Effect Approximation A+ B, IEEE Tran on CT, vol 9, pp 443-45, 97 [8] Q Yu, O W Wing, Computational Model of Tranmiion Line with Skin Effect and Dielectric Lo, IEEE Tran Circuit Syt, I, vol 4, pp 07-9, 994 [9] Signal Proceing Toolbox Uer Guide, The MathWork, 996 [0]F Wenger, T Gutafon, L J Svenon, Perturbation Theory for Inhomogeneou Tranmiion Line, IEEE Tran Circuit Syt, I, vol 49, pp 89-97, 00 [] A Vareljian, Development of Synthei Method for the High Order Active RC-filter, (in Ruian), PhD diertation, MEI, Mocow, 989