4_Twisted_p

1Digital Kommunikasjon 2002 1

NTNUDepartment of Telecommunications

Transmission properties ofpair cables

Nils Holte, NTNU

Digital Kommunikasjon 2002 2


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 existingcable plant in an optimum way



Part IBasic design of pair cables

A single twisted pair

Binder groups

The cross-stranding principle

Building binder groups and cables ofdifferent sizes

A complete cable



Cross section of a single pair

Most common conductor diameters: 0.4 mm, 0.6 mm(0.5 mm, 0.9 mm)

Insulation

Conductor



A twisted pair



Typical pair cables of theNorwegian access network

0.4 and 0.6 mm conductor diameter

Polyethylene insulation (expanded)

Twisting periods in the interval50 - 150 mm

10 pair cross-stranded binder groups

10 - 2000 pairs in a cable



Pair positions in a 10 pair binder group

Cross-stranding [13]:The positions ofall the pairs arealternated randomlyalong the cable.Interference is thusrandomised, and allpairs will be almostuniform



Cross-stranding

Cross-strandingtechnique:Each pair runsthrough a die in thecross-strandingmatrix.The positions of thedies are set by arandom generator



Design of binder groups up to 100 pairs



Design of binder groups up to 1000 pairs



Typical cable design (Kombikabel)

This cable may be used as: 1) overhead cable2) buried cable3) in water (fresh water)



Part IIProperties 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



A single pair in uniform insulation

r

2a2r

A coarse estimate of the primary parameters R, L, C, Gmay be found assuming a >> r, uniform insulation, andno surrounding conductors



Model of a single pair in a cable

The electricalinfluence ofthe surroundingpairs in the cable may be modelledas an equivalentshield. This modelwill give accurateestimates of R, L, C and G [12]



Capacitance of a single pair

C a

r

r= 0

2ln

The capacitance per unit length of a single pair is given by:

The capacitance of cables in the access network is 45 nF/km



Inductance of a single pair

La

r=

0 2ln

The inductance per unit length of a single pair is given by:



Skin effectAt high frequencies the current will flow in the outer part of the conductors, and the skin depth is given by [12]:

=1

0f r is the conductance of the conductors

For copper the skin depth is given by:

= 2 11,FkHz

mm

=

=

2 11

0 067

.

.

mm at 1 kHz

mm at 1 MHz



Resistance of a conductorThe resistance per unit length of a conductor is for r >

10



Resistance of a pairThe resistance per unit length of a pair will be the sum of the resitances of the two conductors and is given by:

R Rr

r

rr

C= =

>>

11



Per unit length model of a pair

Lx Rx

GxCx

U+UI+I

x



Model of a pair of length l

R, L, C, G

x=0 x=l

I(x)

U(x)

12



The telegraph equation

d

dxU x R j L I x Z I x

d

dxI x G j C U x Y U x

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

= + =

= + =

d

dxU x Z Y U x

2

( ) ( )=

Combining the equations:

From the circuit diagram:



Solution of the telegraph equation

U(x) c e c e

I(x)c

Ze

c

Ze

1x

2x

1 x 2 x

= +

= +

0 0

c1 and c2 are constants is propagation constantZ0 is characteristic impedance

13



Propagation constant

is the attenuation constant in Neper/km is the phase constant in rad/km

dB = =20

108 69

ln( ).

Neper to dB:

= = + + =

+

Z Y R j L G j C

R j L j C

( ) ( )

( )

= + j



Characteristic impedance

ZZ

Y

R j L

G j C

R j L

j C0= =

++

=+( )

( )

( )

At high frequencies R

14



Attenuation constant

= + = = R CL

G L

C

R C

Lk f

2 2 2 1

At high frequencies (f > 100 kHz) R L. Hence:

= = +

= =

=

j C R jR C

R Ck f

( )12

2 2



Attenuation constant of pair cables

15



Phase constantAt high frequencies (f > 100 kHz):

= = L C k f3Phase velocity:

vx

t L C

c

r

= = = = =

=

wavelength

cycle

2

2

1

Phase velocity is 200 000 km/s for pair cables at high frequencies (r = 2.3 for polyethylene)



Termination of a pair

R, L, C, G

x=0

I(l)

U(l)

ZT

x=l

16



Reflection coefficient

U(x) V e V e

I(x)V

Ze

V

Ze

ZU

IZ

V V

V V

x x

x x

T

= +

=

= =+

+

+

+

+

( ) ( )

( ) ( )

( )

( )

l l

l l

l

l

0 0

0

= = +

+

V

VZ

Z Z

Z ZT

T0

0

0

Solution of telegraph equationincluding termination imp. ZTV+ is voltage of wave in positive direction at x=lV- is voltage of reflectedwave in at x=l

Reflection coefficient:No reflections ( =0) for ZT= Z0 . Ideal terminationsare assumed in latercrosstalk calculations



Part IIICrosstalk 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

17



Crosstalk mechanismsMain contributions:Capacitive couplingInductive coupling



Ideal and real twisting of a pairIdeal twisting

Actual twisting of a real pair

The crosstalk level observed in real cables is caused mainly by deviations from ideal twisting

18



Crosstalk coupling per unit length



Crosstalk coupling per unit lengthNormalised NEXT coupling coefficient [11]:

Normalised FEXT coupling coefficient [11]:

N

Ni j

i j i jxj

dU

U dx

C x

C

L x

L,, ,( )

( ) ( )=

= +

1 1

20

2

1

Ci,j(x) is the mutual capacitance per unit length between pair i and jLi,j(x) is the mutual inductance per unit length between pair i and j0 is the lossless phase constant, 0 = L C

F

Fi j

i j i jxj

dU

U dx

C x

C

L x

L,, ,( )

( ) ( )=

=

1 1

20

2

1

19



Near end crosstalk, NEXT



Near end crosstalk, NEXT

H fV

Vj x e dxNE N

x j x( ) ( )= = 2010

02 2

0

l

Assuming weak coupling, the near end voltage transfer function is given by [11]:

20



NEXT between two pairs

10-1 100 10130

40

50

60

70

80

90

100

Frequency, MHz

Nea

r en

d cr

osst

alk,

dB

pair combinationpair combinationpair combinationaverage NEXT



Stochastic model of crosstalk couplings

Crosstalk coupling factors are white Gaussian stocastic processes:

NEXT autocorrelation function [6]:

FEXT autocorrelation function [6]:

R E x x kF F F F( ) ( ) ( ) ( ) = +[ ] =

R E x x kN N N N( ) ( ) ( ) ( ) = +[ ] =

kN and kF are constants

21



Average NEXT

p f E H f EV

V

k e dxk

ek

k f

NE

Nx N N

( ) ( )

.

= [ ] =

=

= ( ) =

2 20

10

2

02 4

0

02

4 02

21 5

41

4

ll

N

NEXT increases 15 dB/decade with frequency

Average NEXT power transfer function between two pairs [6]:



Far end crosstalk, FEXT

22



Far end crosstalk, FEXT

H fV

Vj x dxFE F( ) ( )= = 2

10

0

l

l

l

Assuming weak coupling, the far end voltage transfer function is given by [11]:



Average FEXT

q f E H f EV

V

k dx k k f

FE

F F

( ) ( )= [ ] =

=

= =

2 2

1

2

02

0

02 2

l

l

l

l l F2

FEXT increases 20 dB/decade with frequencyFEXT increases 10 dB/decade with cable length

Average FEXT power transfer function between two pairs [6]:

23



Crosstalk from N different pairscrosstalk power sum

Only crosstalk between identical systems isconsidered (self NEXT and self FEXT)

Crosstalk from different pairs add on apower basis

Effective crosstalk is given by the sum ofcrosstalk power transfer functions, which isdenoted crosstalk power sum



Crosstalk from N different pairscrosstalk power sum

H f H fNE ps NE i jjj i

N

( ) ( ),2 2

1

= [ ]=

NEXT crosstalk power sum for pair no i:

H f H fFE ps FE i jjj i

N

( ) ( ),2 2

1

= [ ]=

FEXT crosstalk power sum for pair no i:

24



NEXT power sum

10-1 100 10125

30

35

40

45

50

55

60

65

70

Frequency, MHz

NE

XT

pow

er s

um,

dBNEXT ps, cable 1NEXT ps, cable 2NEXT ps, cable 3average NEXT ps



Probability distributions of crosstalkCrosstalk power transfer function for a single pair combination at a given frequency is gamma-distributedwith probability density [6]:

= 1.0 for NEXT = 0.5 for FEXTa is average crosstalk power

p za

z ez

z

a( )( )

=

1 1

25



Probability density of NEXT andFEXT for a single pair combination

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Amplitude relative to average power transfer function

Pro

babi

lity

dens

ity

ny = 0.5 (FEXT)ny = 1.0 (NEXT)



Probability of crosstalk for anarbitrary pair combination

Different pair combinations will have different levels of crosstalk coupling (different kN and kF).

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, must be found empirically. < 1.0 for NEXT and < 0.5 for FEXT

26



Probability of crosstalk power sumCrosstalk from different pairs add on a power basis. Hence,crosstalk power sum is approximately gamma-distributedwith probability distribution:

p za

z epsps

ps

ps

z

aps

ps

ps

ps( )( )

=

1 1

aps=Na, where a is the average crosstalk of one pair combinationps=N , where is the number of degrees of freedom

for a random pair combinationN is the number of disturbing pairs



Probability density ofcrosstalk power sum

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Amplitude relative to average power sum

Pro

babi

lity

dens

ity

ny = 0.2ny = 0.5ny = 1.0ny = 2.0ny = 5.0

27



Worst case crosstalkCrosstalk 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):

p f N E k f c

q f N E k f c

ps ps

ps ps

99 21 5

99

99 22

99

( ) ( )

( ) ( )

.= [ ] = [ ]

N

F

for NEXT

for FEXT

l

c99(ps) is the ratio between the 99% point and the average power sum in the gamma distributionThe expectations E[kN2] and E[kF2] are taken over all pair combinations



Empirical worst case NEXT model

p FN

Fps 994

0 61 510

49( )

..=

N is the number of disturbing pairs in the cableF is the frequency in MHz

International model of 99% point of NEXT power sumbased upon 50 pair binder groups:

This model fits well with 100% filled Nowegian cables with 10 pair binder groups

28



Empirical worst case FEXT model

q FN

F Lps 994

0 623 10

49( )

.

=

N is the number of disturbing pairs in the cableF is the frequency in MHzL is the cable length in km

International model of 99% point of FEXT power sumbased upon 50 pair binder groups:

This model fits well with 100% filled Nowegian cables with 10 pair binder groups



Part IVChannel capacity estimates [1,2,8]

Shannons 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

29



Shannons theoretical channel capacity

Maximum theoretical channel capacity in the frequency band [fl,fh]:

CS f

N fdfSh

fl

fh

= +

log

( )( )2

1 bit/s

S(f): signal power densityN(f): noise power density



Capacity per bandwidth unitA realistic estimate of bandwidth efficiency [2]:

( ) log ( )( )

fC

fk

S f

N feff= = +

2

1 bit/s/Hz

S(f): signal power densityN(f): noise power density 1: factor for margin (safety margin + margin for mod.meth.)keff 1: factor for overhead (sync bits, RS-code, cyclic prefix)

30



Signal transmission

Attenuation proportional todue to skin effect (f > 100 kHz)

Signal transfer function:

f

H f k fdB

( ) exp= = ( )10 20 l

l

dB: attenuation constant in dB



Signal and noise models

Signal:

Noise models:

N F

N F NEXT

N F L e FEXT

N AWGN

NEXT

FEXTL

AWGN

( )

.

=

= = =

10

3 10

10

4 1 5

4 2 2

8

S F e L( ) = 2

31



Channel capacity vs. frequency

0 2 4 6 8 100

5

10

15

Frequency, MHz

Pot

entia

l ban

dwith

effi

cien

cy,

bit/s

/Hz FEXT

AWGNNEXT

L=1 km



Channel capacity vs frequency II

0 2 4 6 8 100

5

10

15

Frequency, MHz

Pot

entia

l ban

dwith

effi

cien

cy,

bit/s

/Hz one-way transmission

two-way transmission2*two-way transmission

0 0.5 1 1.5 20

5

10

15

Frequency, MHz

Pot

entia

l ban

dwith

effi

cien

cy,

bit/s

/Hz one-way transmission

two-way transmission2*two-way transmission

L=1 km L=3 km

32



Total channel capacity

R kS F

N Ndfone way eff

FEXT AWGNfl

fh

= + +

log

( )2 1

R kS F

N N Ndftwo way eff

NEXT FEXT AWGNfl

fh

= + + +

log

( )2 1

One-way transmission:

Two-way transmission:

The channel capacity is somewhat greater than this expression for two-way transmission due to uncorrelated NEXT in different frequency bands [9,10]



Assumptions for estimation of bitrates For one-way transmission, the total bitrate found

in the calculations must be divided by downstreamand upstream transmission

Identical systems in all pairs of the cable (only selfNEXT and self FEXT)

All pairs are used, the cable is 100% filled

Net bitrate is 90% of total bitrate (keff=0.90)

Frequency band: f 100 kHz,upper limit 11 MHz

33



Assumptions II

Multicarrier modulation [7]

Adaptive modulation in each sub-band

M-TCM modulation in each sub-band4 M 16384, 1 - 13 bit/s/Hz

Distance to Shannon (): 9 dB(6 dB margin + 3 dB for modulation)

White noise: 80 dB below output signal

Cable: 0.4 mm, 22.5 dB/km at 1 MHz



Potential range for .4 mm cable

0 1 2 3 4 50

10

20

30

40

50

60

Range, Km

Bitr

ate,

Mbi

t/s

one-way transmissiontwo-way transmission

0 1 2 3 4 50

10

20

30

40

50

60

Range, Km

Bitr

ate,

Mbi

t/s


VDSL

ADSL

34



Potential range for .4 mm cable II

2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

Range, Km

Bitr

ate,

Mbi

t/s


SHDSL+

SHDSL+means a multicarriersystem usingapproximatelythe same frequencyband asSHDSL



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 DigitalSubscriber 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]

35



Frequency allocations in theaccess network

Upstream

Downstream

ADSL

SHDSL low ADSL + VDSL

Frequency, kHz

Frequency, kHz

2 Mbit/s SHDSL

SHDSL low

1104

2 Mbit/s SHDSL

0 125 211 400

11040 125 276 400

VDSL

SHDSL low 640 kbit/s



Conclusions

Frequency planning in pair cables is veryimportant

New systems should be introduced withgreat care in order to preserve the potentialtransmission capacity of the cable

Full rate SHDSL systems overlaps withADSL

36



References I[1] J.-J. Werner, "The HDSL environment," IEEE Journal on Sel.

Areas in Commun., August 1991, pp. 785-800[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, 2001[4] ETSI TS 101 270-1, 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 SubscriberLine (ADSL) transceivers", Geneva, 1999



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 1990, pp. 5-19

[8] N. Holte, Broadband communication in existing copper cablesby means of xDSL systems - a tutorial, NORSIG, Trondheim,October 2001

[9] N. Holte, A new method for calculating the channel capacity inpair cables with respect to near end crosstalk, DSLcon Europe,Munich, November 2001

37



References III[10] 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, Wirbelstrme und Schirmng in der Nachrichten-technik, 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

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