1

Lecture 2 Transmission Line Characteristics

1. Introduction

2. Properties of Coaxial Cable

3. Telegraph Equations

4. Characteristic Impedance of Coaxial Cable

5. Reflection and Termination

6. Transfer Functions of a Transmission Line

7. Coaxial Cable Without Frequency Distortion

8. Bode Plots

9. Properties of Twisted Pair Cable

10. Impedance Matching

11. Conclusion

2

Introduction • A communication network is a collection of network elements integrated and managed to support the transfer of information. Data Transmission: links and switches. • Optical fiber• Copper coaxial cable• Unshielded twisted wire pair• Microwave or radio “wireless” links. Optical fiber and copper links are usually point-to-point links, whereas radio links are usually broadcast links.   Guided media:twisted pair (unshielded and shielded)coaxial cableoptical fiber

Unguided media:radio wavesmicrowaves (high frequency radio waves)

3

Center Wire

Braided Outer Conductor

(b) Structure of coaxial cable

Outer Insulation

Polyethylene Insulation

(a) Structure of typical UTP cable

24 gauge copper wire with insulation

Outer Insulation

Four twisted pairs

4

Coaxial cable with length l. It is composed of a number of intermediate segments with properties: R, L, G and C of a unit length of cable

x

l

S ( t )out S ( t )in e ( x )

Inner Wire

Outer Conductor

x+x

e ( x+x )

meter);per (Henrysln1021

27

m

H

R

RL

R1: radius of inner wire

R2: radius of braided outer conductor

2·10-7: derived from properties of dielectric material

5

Telegraph Equations

x

GΔx CΔx e ( x ) e ( x+x )

LΔx RΔx

x+x

t

ixLixRxexxe

)()()()(

ti

xLixRexexxe

)()()()(

x 0:

t

iLRi

x

e

t

xxexCxxexGxixxi

)(

)()()()()(

t

xxexCxxexGxixxi

)(

)()()()()(

te

CGexi

(1) (2)

Equations (1) and (2) are called the telegraph equations.

6

• They determine e( x, t ) and i( x, t ) from the initial and the

boundary conditions.

• (If we set R and L to zero in these equations (we assume no series impedance), the simplified equations are telephonic equations).

ILjRILjIRdx

Ud)(

UCjGUCjUGdxId

)(

(3)

(4)

UCjdt

deCFxjUtxeF

;),(),(

ILjdt

diLFxjItxiF

;),(),(

• Solution of equations (3,4) by differentiating of equation (3) with respect to x, and combine with equation (4):

7

UCjGLjRdx

IdLjR

dx

Ud)()()(

2

2

Define the value as 2 )()( CjGLjR

Udx

Ud 22

2 xx AAU ee 21

A1 and A2 are independent of x, and can be calculated from the boundary

conditions at x = 0 and x = l. The value of is derived from the properties of the coaxial cable and the frequency :

)()()()( jCjGLjR (7)

(6)

() is the attenuation coefficient. () is the phase shift coefficient.

(5)

8

Characteristic Impedance of Coaxial Cable

For the Fourier-transformed current insert (6) into equation (3)

)ee(

ee)(

1

)(

1

21

21

xx

xx

AALjR

AALjRdx

Ud

LjRI

from equation (7) above:

)ee(

)(

)(

1

)ee()(

)()(

21

21

xx

xx

AA

CjG

LjR

AALjR

CjGLjRI

)ee(1

;)(

)(21

xx AAZ

ICjG

LjRZ

9

Reflection and Terminationreflected voltage /

incident voltage; • To express this

signal in the time domain, we must divide (6) into its magnitude and phase components.

We express =F(,):

xjxxxjxx eee;eee

21 e;e 2211 jj CACA

reflectedincident U

xjx

U

xjx CCU )(2

)(1

21 eeee

reflectedincident e

x

e

x xtCxtCtxe )sin(e)sin(e),( 2211

xx AAU ee 21

10

Urec

Irec

Transmitter 50 ohm

Ztr

Receiver

Utr

l

Zrec

Coaxial Cable total resistance Rtot

characteristic impedance Z

recrecout Z IU We can find the relationship between the magnitudes of A1 and A2 if

we know that Zrec is pure resistance (a real value of impedance) and

we know the value of Z:

xxxx AAIAAU eeZ

1ZZee 21recrecrec21rec

11

xxxx AAAA eZeZeZeZ 2rec1rec21

xx AA e)ZZ(e)ZZ( 1rec2rec

ZZ

ZZ

rec

rec

rec,

rec,

incident

reflected

e U

Uq

ZZ

ZZ

rec

rec

rec,

rec,

incident

reflectedi

I

Iq

rec

rec

ZlR

ZK

rectot

rec

ZR

Z

•Z = 50, Rtot= 50, and = 2 V

•Zrec = 0,

•Zrec = 50

•Zrec =

12

Ethernet Technologies: 10Base2• 10: 10Mbps; 2: under 200 meters max cable length• thin coaxial cable in a bus topology

• repeaters used to connect up to multiple segments• repeater repeats bits it hears on one interface to its other interfaces: physical layer device only!

13

Multiport Repeater

10BASE-2 cable

Workstations with 10BASE-2

Ethernet Network Interface Cards

50 Terminator

T-connector

50 Terminators

MultiportRepeater

MultiportRepeater

10BASE-T cable

14

Ethernet Cabling

The most common kinds of Ethernet cabling.

15

ljlj

ljl

eeke

eek

U

UjH )()(

1

)()(1

input

output

1

1

C

C

lekjH )( ljH )(

Transfer Functions of Coaxial Cable

)()()CG()LR()( jjj CG

LR)(

j

jZ

RL j 0G,GC j

CZ;2Z

R l

k

RZ

ZRZ

Z

totrec

rec

L

R1LCCL1

L

jjjj

j

R

C

L

C1C

G

L1L

R

Z

j

j

For f<100 kHz

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()

-1.09 dB

() = – · τ -2 radians

(a) Amplitude Bode plot (in decibels)

(b) Phase Bode plot (in radians)

= 12.56×106 radians/s

20·log10 | H( j ) |

17

Bode Plots R

Cein eout

i

outout

in

outoutin

edt

dee

dt

deidtiedtiie

RC

;C;C

1;

C

1R

jUjUjjU outoutin RC

jUteF inin jUjdt

deFjUteF out

outoutout RCRC;

;1RC

1

)

jjU

jUjH

in

out

18

;1RC

1

)

jjU

jUjH

in

out

RCtan

;CR1

1

1

222

jH

)CR1(log10

CR1log20

CR1

1log20log20

22210

22210

2221010

jH

RC

1.1

01log10log20 10 jH

RC

1.2

RClog20CRlog10log20 10222

10 jH

11

.3 RCc

dBjH 01.32log10)11(log10log20 1010

19

()

(a) Amplitude Bode plot (in decibels)

(b) Phase Bode plot (in radians)

constant time delay

RC

20·log10 | H( j ) |

plot of -10·log10 ( RC )

c = 1

RC

c = 1

RC

- π 2

- π 4

-3.01 dB

20

Appendix

• Error analysis

• The goal of the lab experiment is to determination the transmission line bandwidth of a coaxial cable. We will use the experimental data to construct an amplitude Bode plot, and find the frequency for which the signal is attenuated by less than ‑3 dB.

• This is the bandwidth for which half or more of the input signal power is delivered to the output.

• the absolute error in 20·log10 | H( j ) | due to the errors Uout and Uin :

dB

U

UjH

in

out 3log20log20 1010

outout

in

out

inin

in

out

VV

V

V

VV

V

V

jH

1010

10

log

20

log

20log20

21

transfer from decimal to natural logarithms with the correction factor 0.434 [ log10(e) ]:

outout

in

out

inin

in

out

VV

V

V

VV

V

V

jH

lnln

434.020log20 10

inoutin

in

out

outout

out

in

out

inin

in

out

VVV

V

V

VV

V

V

V

VV

V

V

lnln

inout VVjH 434.020log20 10

The worst case will occur when Vout = -Vin . If we set Vout = 0.025:

dB434.0050.0434.020log20 10 jH

22

Amplitude bode plot with error bars on amplitude axis.

-3.01 dB

+0.434 dB

-0.434 dB

minimum bandwidth

maximum bandwidth in

out

V

V10log20