Transmission properties of pair cables

Transmission properties of pair cables Nils Holte, Digital Kommunikasjon 22 1 Overview Part I: Physical design of a pair cable Part II: Electrical properties of a single pair Part III: Interference between pairs, crosstalk Part IV: Estimates of channel capacity of pair cables. How to exploit the existing cable plant in an optimum way Digital Kommunikasjon 22 2 1

Part I Basic design of pair cables A single twisted pair Binder groups The cross-stranding principle Building binder groups and cables of different sizes A complete cable Digital Kommunikasjon 22 3 Cross section of a single pair Insulation Conductor Most common conductor diameters:.4 mm,.6 mm (.5 mm,.9 mm) Digital Kommunikasjon 22 4 2

A twisted pair Digital Kommunikasjon 22 5 Typical pair cables of the Norwegian access network.4 and.6 mm conductor diameter Polyethylene insulation (expanded) Twisting periods in the interval 5-15 mm 1 pair cross-stranded binder groups 1-2 pairs in a cable Digital Kommunikasjon 22 6 3

Pair positions in a 1 pair binder group Cross-stranding [13]: The positions of all the pairs are alternated randomly along the cable. Interference is thus randomised, and all pairs will be almost uniform Digital Kommunikasjon 22 7 Cross-stranding Cross-stranding technique: Each pair runs through a die in the cross-stranding matrix. The positions of the dies are set by a random generator Digital Kommunikasjon 22 8 4

Design of binder groups up to 1 pairs Digital Kommunikasjon 22 9 Design of binder groups up to 1 pairs Digital Kommunikasjon 22 1 5

Typical cable design ( Kombikabel ) This cable may be used as: 1) overhead cable 2) buried cable 3) in water (fresh water) Digital Kommunikasjon 22 11 Part II Properties of a single pair Equivalent model of a single pair Capacitance Inductance Skin effect Resistance Per unit length model of a pair The telegraph equation - solution Propagation constant Characteristic impedance Reflection coefficient, terminations Digital Kommunikasjon 22 12 6

A single pair in uniform insulation 2r 2a ε r A coarse estimate of the primary parameters R, L, C, G may be found assuming a >> r, uniform insulation, and no surrounding conductors Digital Kommunikasjon 22 13 Model of a single pair in a cable The electrical influence of the surrounding pairs in the cable may be modelled as an equivalent shield. This model will give accurate estimates of R, L, C and G [12] Digital Kommunikasjon 22 14 7

Capacitance of a single pair The capacitance per unit length of a single pair is given by: C = πε ε r 2 a ln r The capacitance of cables in the access network is 45 nf/km Digital Kommunikasjon 22 15 Inductance of a single pair The inductance per unit length of a single pair is given by: L = µ π ln 2 a r Digital Kommunikasjon 22 16 8

Skin effect At high frequencies the current will flow in the outer part of the conductors, and the skin depth is given by [12]: δ = 1 f r π µµσ σ is the conductance of the conductors For copper the skin depth is given by: δ = 211, F khz mm δ = 211. mm at 1 khz δ =. 67 mm at 1 MHz Digital Kommunikasjon 22 17 Resistance of a conductor The resistance per unit length of a conductor is for r << a given by: R C = 1 2 σπr 1 2σπrδ for for δ >> r δ << r (low frequencies) (high frequencies) Digital Kommunikasjon 22 18 9

Resistance of a pair The resistance per unit length of a pair will be the sum of the resitances of the two conductors and is given by: 2 2 σπr R= 2 RC = 1 σπrδ for for δ >> r δ << r (low frequencies) (high frequencies) Digital Kommunikasjon 22 19 Conductance of a pair The conductance per unit length of a pair is given by: G = δ ωc l δ l is the dielectric loss factor A typical value of the loss factor is δ l =.3. This means that the conductance is usually negligible for pair cables. Digital Kommunikasjon 22 2 1

Per unit length model of a pair L x R x I+ I U+ U C x G x x Digital Kommunikasjon 22 21 Model of a pair of length l I(x) R, L, C, G U(x) x= x=l Digital Kommunikasjon 22 22 11

The telegraph equation From the circuit diagram: d Ux ( ) = ( R+ jωlix ) ( ) = Z Ix ( ) dx d Ix ( ) = ( G+ jωcux ) ( ) = YUx ( ) dx Combining the equations: 2 d dx Ux ( ) = ZYUx ( ) Digital Kommunikasjon 22 23 Solution of the telegraph equation γ x γ x U(x) = c1 e + c2 e c I(x) Z e γ c = + Z e c 1 and c 2 are constants γ is propagation constant Z is characteristic impedance 1 x 2 γ x Digital Kommunikasjon 22 24 12

Propagation constant γ = Z Y = ( R+ jωl) ( G+ jωc) = γ = α + jβ α is the attenuation constant in Neper/km β is the phase constant in rad/km Neper to db: ( R+ jωl) jωc α = 2 db α = 869. α ln( 1) Digital Kommunikasjon 22 25 Characteristic impedance Z Z = R j L Y = ( + ω ) ( G+ jωc) = ( R+ jωl) jωc At high frequencies R << jωl : Z = L C The characteristic impedance is approximately 12 ohms at high frequencies for pair cables in the access network Digital Kommunikasjon 22 26 13

Attenuation constant At high frequencies (f > 1 khz) R <<ωl. By series expansion of γ : R C G L R C α = + = = k1 2 L 2 C 2 L f At low frequencies (f < 1 khz) R >>ωl. Hence: ω R C γ = jωc R = ( 1 + j) 2 α = β = ω R C = k f 2 2 Digital Kommunikasjon 22 27 Attenuation constant of pair cables Digital Kommunikasjon 22 28 14

Phase constant At high frequencies (f > 1 khz): β = ω L C = k f 3 Phase velocity: x wavelength 2πβ ω v = = = = = t cycle 2πω β 1 L C = c ε r Phase velocity is 2 km/s for pair cables at high frequencies (ε r = 2.3 for polyethylene) Digital Kommunikasjon 22 29 Termination of a pair I(l) R, L, C, G U(l) Z T x= x=l Digital Kommunikasjon 22 3 15

Reflection coefficient γ l x U(x) = V+ e + V e V I(x) Z e γ l x V = Z e Z T U() l = = Z V I() l V ( ) γ ( l x) + ( ) γ ( l x) + V V + + Reflection coefficient: V ρ = = Z Z V Z + T T + Z Z Solution of telegraph equation including termination imp. Z T V + is voltage of wave in positive direction at x=l V - is voltage of reflected wave in at x=l No reflections (ρ =) for Z T = Z. Ideal terminations are assumed in later crosstalk calculations Digital Kommunikasjon 22 31 Part III Crosstalk in pair cables Basic coupling mechanisms Crosstalk coupling per unit length Near end crosstalk, NEXT Far end crosstalk, FEXT Statistical crosstalk coupling Average NEXT and FEXT Crosstalk power sum - crosstalk from many pairs Statistical distributions of crosstalk Digital Kommunikasjon 22 32 16

Main contributions: Capacitive coupling Inductive coupling Crosstalk mechanisms Digital Kommunikasjon 22 33 Ideal and real twisting of a pair Ideal twisting Actual twisting of a real pair The crosstalk level observed in real cables is caused mainly by deviations from ideal twisting Digital Kommunikasjon 22 34 17

Crosstalk coupling per unit length Digital Kommunikasjon 22 35 Crosstalk coupling per unit length Normalised NEXT coupling coefficient [11]: κ Nij, 1 du C N ij, ( x) Lij, ( x) 2 1 ( x) = = + jβ U dx 2 C L 1 Normalised FEXT coupling coefficient [11]: 2F ij, ij, Fij, ( x) = = jβ U1 dx 2 C L C i,j (x) is the mutual capacitance per unit length between pair i and j L i,j (x) is the mutual inductance per unit length between pair i and j β is the lossless phase constant, β = ω L C κ 1 du 1 C ( x) L ( x) Digital Kommunikasjon 22 36 18

Near end crosstalk, NEXT Digital Kommunikasjon 22 37 Near end crosstalk, NEXT Assuming weak coupling, the near end voltage transfer function is given by [11]: H NE V2 2αx 2jβx ( f) = = jβ N ( x) e dx V κ 1 l Digital Kommunikasjon 22 38 19

NEXT between two pairs 1 9 pair combination pair combination pair combination average NEXT Near end crosstalk, db 8 7 6 5 4 3 1-1 1 Frequency, MHz 1 1 Digital Kommunikasjon 22 39 Stochastic model of crosstalk couplings Crosstalk coupling factors are white Gaussian stocastic processes: NEXT autocorrelation function [6]: RN( τ) = E[ κ N( x) κ N( x+ τ) ]= kn δ( τ) FEXT autocorrelation function [6]: RF( τ) = E[ κf( x) κf( x+ τ) ]= kf δ( τ) k N and k F are constants Digital Kommunikasjon 22 4 2

Average NEXT Average NEXT power transfer function between two pairs [6]: p( f) E H f E V NE ( ) V = [ ] = l 2 2 4α β k β kn e dx = 4α 2 2 = 2 4l β k ( 1 e ) 4α x N N NEXT increases 15 db/decade with frequency 1 2 = k N2 f 15. Digital Kommunikasjon 22 41 Far end crosstalk, FEXT Digital Kommunikasjon 22 42 21

Far end crosstalk, FEXT Assuming weak coupling, the far end voltage transfer function is given by [11]: H FE V2 l ( f) = = jβ F ( x) dx V κ 1l l Digital Kommunikasjon 22 43 Average FEXT Average FEXT power transfer function between two pairs [6]: q( f) E H f E V FE ( ) V 2 2l = [ ] = = 2 2 2 β kf dx = kf β l = kf2 f l l Digital Kommunikasjon 22 44 1l FEXT increases 2 db/decade with frequency FEXT increases 1 db/decade with cable length 2 22

Crosstalk from N different pairs crosstalk power sum Only crosstalk between identical systems is considered (self NEXT and self FEXT) Crosstalk from different pairs add on a power basis Effective crosstalk is given by the sum of crosstalk power transfer functions, which is denoted crosstalk power sum Digital Kommunikasjon 22 45 Crosstalk from N different pairs crosstalk power sum NEXT crosstalk power sum for pair no i: N 2 2 HNE ps( f) = [ HNE i, j( f) ] j = 1 j i FEXT crosstalk power sum for pair no i: N 2 2 HFE ps( f) = [ HFE i, j( f) ] j = 1 j i Digital Kommunikasjon 22 46 23

NEXT power sum 7 65 6 NEXT ps, cable 1 NEXT ps, cable 2 NEXT ps, cable 3 average NEXT ps NEXT power sum, db 55 5 45 4 35 3 25 1-1 1 Frequency, MHz 1 1 Digital Kommunikasjon 22 47 Probability distributions of crosstalk Crosstalk power transfer function for a single pair combination at a given frequency is gamma-distributed with probability density [6]: p () z z 1 ν ν ν 1 = z Γ( ν) a e νz a ν = 1. for NEXT ν =.5 for FEXT a is average crosstalk power Digital Kommunikasjon 22 48 24

Probability density of NEXT and FEXT for a single pair combination 2 1.8 ny =.5 (FEXT) ny = 1. (NEXT) Probability density 1.6 1.4 1.2 1.8.6.4.2.5 1 1.5 2 2.5 3 Amplitude relative to average power transfer function Digital Kommunikasjon 22 49 Probability of crosstalk for an arbitrary pair combination Different pair combinations will have different levels of crosstalk coupling (different k N and k F ). It can be shown that: The crosstalk power transfer function for a random pair combination is approximately gamma-distributed The number of degrees of freedom, ν must be found empirically. ν < 1. for NEXT and ν <.5 for FEXT Digital Kommunikasjon 22 5 25

Probability of crosstalk power sum Crosstalk from different pairs add on a power basis. Hence, crosstalk power sum is approximately gamma-distributed with probability distribution: p ps ps 1 ps ps () z = ν z e ( ps ) ν 1 a Γ ν ps ν ν a ps ps z a ps =Na, where a is the average crosstalk of one pair combination ν ps =Nν, where ν is the number of degrees of freedom for a random pair combination N is the number of disturbing pairs Digital Kommunikasjon 22 51 Probability density of crosstalk power sum Probability density 2 1.8 1.6 1.4 1.2 1.8.6 ny =.2 ny =.5 ny = 1. ny = 2. ny = 5..4.2.5 1 1.5 2 2.5 3 Amplitude relative to average power sum Digital Kommunikasjon 22 52 26

Worst case crosstalk Crosstalk dimesioning is usually based upon the 99% point of NEXT and FEXT power sum, which is given by (1% of the pairs will have crosstalk that exceeds this limit): 15. p ( f) = N E[ k ] f c ( ν ) ps99 N2 2 q ( f) = N E[ k ] f l c ( ν ) ps99 F2 99 99 ps ps for for NEXT FEXT c 99 (ν ps ) is the ratio between the 99% point and the average power sum in the gamma distribution The expectations E[k N2 ] and E[k F2 ] are taken over all pair combinations Digital Kommunikasjon 22 53 Empirical worst case NEXT model International model of 99% point of NEXT power sum based upon 5 pair binder groups: p ps 99 4 N ( F) = 1 49 6. F 15. N is the number of disturbing pairs in the cable F is the frequency in MHz This model fits well with 1% filled Nowegian cables with 1 pair binder groups Digital Kommunikasjon 22 54 27

Empirical worst case FEXT model International model of 99% point of FEXT power sum based upon 5 pair binder groups: q ps 99 4 N ( F) = 31 49 6. F 2 L N is the number of disturbing pairs in the cable F is the frequency in MHz L is the cable length in km This model fits well with 1% filled Nowegian cables with 1 pair binder groups Digital Kommunikasjon 22 55 Part IV Channel capacity estimates [1,2,8] Shannon s channel capacity formula Signal and noise models Realistic estimates of channel capacity Channel capacity per bandwidth unit One-way transmission Two-way transmission Crosstalk between different types of systems, alien NEXT, alien FEXT Digital Kommunikasjon 22 56 28

Shannon s theoretical channel capacity Maximum theoretical channel capacity in the frequency band [fl,fh]: C Sh fh S f = log + df 1 ( ) 2 N( f) bit/s fl S(f): signal power density N(f): noise power density Digital Kommunikasjon 22 57 Capacity per bandwidth unit A realistic estimate of bandwidth efficiency [2]: C S( f) η( f ) = = keff log + λ f 1 2 N( f) bit/s/hz S(f): signal power density N(f): noise power density λ 1: factor for margin (safety margin + margin for mod.meth.) k eff 1: factor for overhead (sync bits, RS-code, cyclic prefix) Digital Kommunikasjon 22 58 29

Signal transmission Attenuation proportional to f due to skin effect (f > 1 khz) Signal transfer function: α db l ( ) H( f) = 1 2 = exp kl f α db : attenuation constant in db Digital Kommunikasjon 22 59 Signal: Noise models: Signal and noise models SF ( )= e 2αL 4 15. NNEXT = 1 F NEXT 4 2 2αL NF ( ) = NFEXT = 31 F L e FEXT 8 NAWGN = 1 AWGN Digital Kommunikasjon 22 6 3

Channel capacity vs. frequency Potential bandwith efficiency, bit/s/hz 15 1 5 FEXT AWGN NEXT L=1 km 2 4 6 8 1 Frequency, MHz Digital Kommunikasjon 22 61 Channel capacity vs frequency II L=1 km L=3 km Potential bandwith efficiency, bit/s/hz 15 1 5 one-way transmission two-way transmission 2*two-way transmission Potential bandwith efficiency, bit/s/hz 15 1 5 one-way transmission two-way transmission 2*two-way transmission 2 4 6 8 1 Frequency, MHz.5 1 1.5 2 Frequency, MHz Digital Kommunikasjon 22 62 31

R R Total channel capacity One-way transmission: fh SF = keff log ( ) 2 1 + λ N + N one way Two-way transmission: fl FEXT AWGN df fh SF = keff log ( ) 2 1 + λ N + N + N two way fl NEXT FEXT AWGN df The channel capacity is somewhat greater than this expression for two-way transmission due to uncorrelated NEXT in different frequency bands [9,1] Digital Kommunikasjon 22 63 Assumptions for estimation of bitrates For one-way transmission, the total bitrate found in the calculations must be divided by downstream and upstream transmission Identical systems in all pairs of the cable (only self NEXT and self FEXT) All pairs are used, the cable is 1% filled Net bitrate is 9% of total bitrate (k eff =.9) Frequency band: f 1 khz, upper limit 11 MHz Digital Kommunikasjon 22 64 32

Assumptions II Multicarrier modulation [7] Adaptive modulation in each sub-band M-TCM modulation in each sub-band 4 M 16384, 1-13 bit/s/hz Distance to Shannon (λ): 9 db (6 db margin + 3 db for modulation) White noise: 8 db below output signal Cable:.4 mm, 22.5 db/km at 1 MHz Digital Kommunikasjon 22 65 Potential range for.4 mm cable 6 6 5 5 Bitrate, Bitrate, Mbit/s Mbit/s 4 4 3 3 2 2 1 1 VDSL one-way transmission one-way two-way transmission two-way transmission ADSL 1 2 3 4 55 Range, Km Digital Kommunikasjon 22 66 33

Potential range for.4 mm cable II Bitrate, Mbit/s 4 3.5 3 2.5 2 1.5 1.5 one-way transmission two-way transmission SHDSL+ 2 2.5 3 3.5 4 4.5 5 Range, Km SHDSL+ means a multicarrier system using approximately the same frequency band as SHDSL Digital Kommunikasjon 22 67 Digital Subscriber Line systems, xdsl ADSL; Asymmetric Digital Subscriber Lines [5] Asymmetric data rates, 256 kbit/s - 8 Mbit/s downstream, range up to 4-5 km SHDSL; Symmetric High-speed Digital Subscriber Lines [3] Symmetric data rates, 192 kbit/s - 2.3 Mbit/s, range up to 6-7 km VDSL; Very high-speed Digital Subscriber Lines Asymmetric or symmetric data rates (still under standardisation), up to 52 Mbit/s downstream, range typically 1 km [4] Digital Kommunikasjon 22 68 34

Upstream Frequency allocations in the access network 2 Mbit/s SHDSL SHDSL low ADSL Frequency, khz 125 211 4 Downstream 114 SHDSL low 64 kbit/s 2 Mbit/s SHDSL SHDSL low ADSL + VDSL VDSL Frequency, khz 125 276 4 114 Digital Kommunikasjon 22 69 Conclusions Frequency planning in pair cables is very important New systems should be introduced with great care in order to preserve the potential transmission capacity of the cable Full rate SHDSL systems overlaps with ADSL Digital Kommunikasjon 22 7 35

References I [1] J.-J. Werner, "The HDSL environment," IEEE Journal on Sel. Areas in Commun., August 1991, pp. 785-8 [2] T. Starr, J. M. Cioffi, P. J. Silverman, "Understanding digital subscriber line technology," Prentice Hall, Upper Saddle River, 1999 [3] ITU-T Recommendation G.991.2, " Single-Pair High-Speed Digital Subscriber Line (SHDSL) transceivers", Geneva, 21 [4] ETSI TS 11 27-1, V1.2.1, "Very high speed Digital Subscriber Line (VDSL); Part 1: Functional requirements", Sophia Antipolis, 1999 [5] ITU-T Recommendation G.992.1, "Asymmetric Digital Subscriber Line (ADSL) transceivers", Geneva, 1999 Digital Kommunikasjon 22 71 References II [6] H. Cravis, T. V. Crater, "Engineering of T1 carrier system repeatered lines," Bell System Techn. Journal, March 1963, pp. 431-486 [7] J. A. C. Bingham, "Multicarrier modulation for data transmission: An idea whose time has come," IEEE Communications Magazine, May 199, pp. 5-19 [8] N. Holte, Broadband communication in existing copper cables by means of xdsl systems - a tutorial, NORSIG, Trondheim, October 21 [9] N. Holte, A new method for calculating the channel capacity in pair cables with respect to near end crosstalk, DSLcon Europe, Munich, November 21 Digital Kommunikasjon 22 72 36

References III [1] A. J. Gibbs, R. Addie, "The covariance of near end crosstalk and its application to PCM system engineering in multipair cable," IEEE Trans. on Commun., Vol. COM-27, No. 2, Feb. 1979, pp.469-477 [11] W. Klein, "Die Theorie des Nebensprechens auf Leitungen," Springer-Verlag, Berlin, 1955 [12] H. Kaden, Wirbelströme und Schirmüng in der Nachrichtentechnik, Springer-Verlag, Berlin 1959 [13] S. Nordblad, "Multi-paired Cable of nonlayer design for low capacitance unbalance telecommunication networks," Int. Wire and cable symp., 1971, pp. 156-163 Digital Kommunikasjon 22 73 37