Transmission properties of pair cables Nils Holte, NTNU NTNU - PDF document

Transmission properties of pair cables Nils Holte, NTNU NTNU Department of Telecommunications Digital Kommunikasjon 2002 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 NTNU Department of Telecommunications Digital Kommunikasjon 2002 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 NTNU Department of Telecommunications Digital Kommunikasjon 2002 3 Cross section of a single pair Insulation Conductor Most common conductor diameters: 0.4 mm, 0.6 mm (0.5 mm, 0.9 mm) NTNU Department of Telecommunications Digital Kommunikasjon 2002 4 2

A twisted pair NTNU Department of Telecommunications Digital Kommunikasjon 2002 5 Typical pair cables of the Norwegian access network • 0.4 and 0.6 mm conductor diameter • Polyethylene insulation (expanded) • Twisting periods in the interval 50 - 150 mm • 10 pair cross-stranded binder groups • 10 - 2000 pairs in a cable NTNU Department of Telecommunications Digital Kommunikasjon 2002 6 3

Pair positions in a 10 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 NTNU Department of Telecommunications Digital Kommunikasjon 2002 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 NTNU Department of Telecommunications Digital Kommunikasjon 2002 8 4

Design of binder groups up to 100 pairs NTNU Department of Telecommunications Digital Kommunikasjon 2002 9 Design of binder groups up to 1000 pairs NTNU Department of Telecommunications Digital Kommunikasjon 2002 10 5

Typical cable design (”Kombikabel”) This cable may be used as: 1) overhead cable 2) buried cable 3) in water (fresh water) NTNU Department of Telecommunications Digital Kommunikasjon 2002 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 NTNU Department of Telecommunications Digital Kommunikasjon 2002 12 6

A single pair in uniform insulation 2a 2r 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 NTNU Department of Telecommunications Digital Kommunikasjon 2002 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] NTNU Department of Telecommunications Digital Kommunikasjon 2002 14 7

Capacitance of a single pair The capacitance per unit length of a single pair is given by: = πε ε 0 r C 2 a ln r The capacitance of cables in the access network is 45 nF/km NTNU Department of Telecommunications Digital Kommunikasjon 2002 15 Inductance of a single pair The inductance per unit length of a single pair is given by: 2 a = µ 0 L ln r π NTNU Department of Telecommunications Digital Kommunikasjon 2002 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]: 1 δ = f π µ µ σ r 0 σ is the conductance of the conductors For copper the skin depth is given by: δ = 2 11 , δ = 2 11 . mm at 1 kHz mm F 0 067 . mm at 1 MHz δ = kHz NTNU Department of Telecommunications Digital Kommunikasjon 2002 17 Resistance of a conductor The resistance per unit length of a conductor is for r << a given by: 1  for δ >> r (low frequencies)  2 r σπ  R C =  1  for δ << r (high frequencies)  2 r σπ δ  NTNU Department of Telecommunications Digital Kommunikasjon 2002 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  for δ >> r (low frequencies)  2 r σπ  R 2 R = ⋅ =  C 1  for δ << r (high frequencies)  r σπ δ  NTNU Department of Telecommunications Digital Kommunikasjon 2002 19 Conductance of a pair The conductance per unit length of a pair is given by: G C = δ ⋅ ω l δ l is the dielectric loss factor A typical value of the loss factor is δ l = 0.0003. This means that the conductance is usually negligible for pair cables. NTNU Department of Telecommunications Digital Kommunikasjon 2002 20 10

Per unit length model of a pair L ∆ x R ∆ x U+ ∆ U I + ∆ I C ∆ x G ∆ x ∆ x NTNU Department of Telecommunications Digital Kommunikasjon 2002 21 Model of a pair of length l I (x) U(x) R, L, C, G x=0 x= l NTNU Department of Telecommunications Digital Kommunikasjon 2002 22 11

The telegraph equation From the circuit diagram: d dx U x ( ) = − ( R + j L I x ω ) ( ) = − Z I x ⋅ ( ) d dx I x ( ) ( G j C U x ) ( ) Y U x ( ) = − + ω = − ⋅ Combining the equations: 2 d dx U x ( ) Z Y U x ( ) = ⋅ ⋅ NTNU Department of Telecommunications Digital Kommunikasjon 2002 23 Solution of the telegraph equation x x γ ⋅ − ⋅ γ U(x) c e c e = ⋅ + ⋅ 2 1 c c γ ⋅ x − ⋅ γ x 1 2 I(x) e e = − ⋅ + ⋅ Z Z 0 0 c 1 and c 2 are constants γ is propagation constant Z 0 is characteristic impedance NTNU Department of Telecommunications Digital Kommunikasjon 2002 24 12

Propagation constant Z Y ( R j L ) ( G j C ) γ = ⋅ = + ω ⋅ + ω = ( R j L ) j C + ω ⋅ ω + j γ = α β α is the attenuation constant in Neper/km β is the phase constant in rad/km Neper to dB: 20 α dB = α = 8 69 . α ln( 10 ) NTNU Department of Telecommunications Digital Kommunikasjon 2002 25 Characteristic impedance Z ( R + j L ω ) ( R + j L ω ) Z 0 = = = Y ( G j C ) j C + ω ω At high frequencies R << j ω L : L Z 0 = C The characteristic impedance is approximately 120 ohms at high frequencies for pair cables in the access network NTNU Department of Telecommunications Digital Kommunikasjon 2002 26 13

Attenuation constant At high frequencies ( f > 100 kHz) R << ω L. By series expansion of γ : R C G L R C k f α = + = = ⋅ 1 2 L 2 C 2 L At low frequencies ( f < 10 kHz) R >> ω L. Hence: ω ⋅ R C ⋅ γ = j C R ω ⋅ = ( 1 + j ) 2 ω ⋅ R C ⋅ α = β = = k f 2 2 NTNU Department of Telecommunications Digital Kommunikasjon 2002 27 Attenuation constant of pair cables NTNU Department of Telecommunications Digital Kommunikasjon 2002 28 14

Phase constant At high frequencies ( f > 100 kHz): L C k f β = ω ⋅ = ⋅ 3 Phase velocity: ∆ x wavelength 2 π β ω 1 c v = = = = = = ∆ t cycle 2 π ω β L C ⋅ ε r Phase velocity is 200 000 km/s for pair cables at high frequencies ( ε r = 2.3 for polyethylene) NTNU Department of Telecommunications Digital Kommunikasjon 2002 29 Termination of a pair I ( l ) U( l ) R, L, C, G Z T x=0 x= l NTNU Department of Telecommunications Digital Kommunikasjon 2002 30 15

Reflection coefficient l l γ ⋅ ( − x ) − ⋅ γ ( − x ) U(x) V e V e Solution of telegraph equation = ⋅ + ⋅ + − including termination imp. Z T V V l l γ ⋅ ( − x ) − ⋅ γ ( − x ) I(x) + e − e V + is voltage of wave in = ⋅ − ⋅ Z Z positive direction at x=l 0 0 V - is voltage of reflected l U ( ) Z V V + + − Z = = wave in at x=l T 0 l I ( ) V − V + − Reflection coefficient: No reflections ( ρ =0) for V Z Z − Z T 0 − Z T = Z 0 . Ideal terminations ρ = = 0 V Z Z + are assumed in later + T 0 crosstalk calculations NTNU Department of Telecommunications Digital Kommunikasjon 2002 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 NTNU Department of Telecommunications Digital Kommunikasjon 2002 32 16

Crosstalk mechanisms Main contributions: Capacitive coupling Inductive coupling NTNU Department of Telecommunications Digital Kommunikasjon 2002 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 NTNU Department of Telecommunications Digital Kommunikasjon 2002 34 17