Transcript Transmission lines () - Lyle School of Engineering

Transmission Line Fundamentals
Southern Methodist University
EETS8320
Fall 2005
Session 5
Slides only. (No notes.)
Rev. 2.8;Page 1
©1996-2005, R.Levine
Major Transmission Facts 1
• Electromagnetic waves flow via the nonconductive space in or around wire/cable
conductors, and not via the metal conductor
itself.
– Some of this electromagnetic power may be coupled
to/from other nearby wires, producing “crosstalk”
– Transpositions, twisted pairs, or use of co-axial cable
(having minimum external EM fields) minimize crosstalk
• Electromagnetic waves are guided by
conductors (in twisted pair, co-axial cable, or
wave guides).
• Power loss is due to:
– A. Longitudinal metallic resistance of wire/cable
– B. Radiation losses (particularly for twisted pairs)
Rev. 2.8;Page 2
©1996-2005, R.Levine
Diagram of EM Fields Around Wire Pair
Figure taken from web site http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/audio/part6/page3.html
The H field is related to the B field by the equation B=µ·H
Rev. 2.8;Page 3
©1996-2005, R.Levine
Structure and EM Fields in Co-ax
See footnote on previous page.
Rev. 2.8;Page 4
©1996-2005, R.Levine
Major Transmission Facts 2
• Analysis of transmission via examination of the EM
fields is most accurate, but also complicated
• For transverse EM waves we can determine four
“lumped” parameters that approximately describe the
properties of the section of wire/cable:
–
–
–
–
R. Longitudinal resistance (ohm/meter)
L. Loop inductance (henry/meter)
G. Parallel insulation conductance (mho/meter or 1/ohm•meter)
C. Parallel wire pair capacitance (farad/meter)
• Derived parameters:
– Fractional power loss in percent/meter or dB/meter
– Characteristic impedance or surge impedance, Z0 :
Z0  R  jL2f  G  jC2f 
– Wave speed (phase velocity)
cm  1 / R  jL2f   G  jC2f 
* Transverse EM waves have both their E and H fields in the cross-sectional plane
perpendicular to the direction of EM power flow. (H field is B/µ .) H is
measured in amp/meter, analogous to lumped element current, and more
convenient for analysis of power flow. Symbol j=-1. Mathematicians use
symbol i, but engineers use i for current.
Rev. 2.8;Page 5
©1996-2005, R.Levine
Major Transmission Facts 3
• R typically increases proportional to frequency
because the “skin” depth of EM wave penetration into
the metallic conductors is inversely proportional to
frequency.
• Aside from power loss, typical wire/cable transmission
medium has slightly different wave speeds for different
sine wave frequency components of a complicated
waveform, thus producing an altered waveform after
passing through many km of wire/cable
– Attenuation is a problem for voice and modem signals both.
– Waveform changes are a problem primarily for modems.
• When two transmission wires/cables having different
Z0 are “spliced,” EM wave power is partially reflected
and partially transmitted. This produces “echo.”
Rev. 2.8;Page 6
©1996-2005, R.Levine
Electrical Resistance
• Most metal objects have “linear” resistance properties. Ohm’s “law”
applies: v= R•i, where i is current (amps), R is resistance (ohms), and
v is voltage (volts)
area
length
• Longitudinal electric resistance of a wire is determined by:
– R =  • length/area
– where  is the material resistivity* (unit: ohm•meter), with a high value for
some materials (e.g. platinum) and low for others (e.g. silver). The unit
ohm•centimeter is also used historically.
– area is •r2 for circular wire of radius r (but note later about “skin”effect)
• Power “lost” due to electrical resistance R carrying current i, is i2•R
(also equivalent to v2/R or v•i)
– This formula describes dc (constant current) power loss accurately. Current
density is uniform throughout the area for unvarying or “direct” current.
*Material resistivity of copper can be increased by repeatedly bending and flexing the wire to modify the
atomic level crystal structure. Newly manufactured “soft drawn” copper wire has slightly lower resistivity
than “hard drawn” wire that was repeatedly flexed via roller machines before selling. Hard drawn wire is
mechanically stronger and can be pulled with less breakage.
Rev. 2.8;Page 7
©1996-2005, R.Levine
Power Loss
• Power really flows via an electromagnetic wave in the space
surrounding the wires (only a little electric field in the copper)
– Wave speed is affected by the insulation material (e.g.,plastics, paper pulp,
silk or other woven fibers, etc.)
– Only a surface portion of the copper carries alternating current, so-called
“skin effect,” -- to form a “boundary” for EM wave
– depth of the current “skin” is inversely proportional to square root ( of
frequency -- therefore effective resistance is higher at higher frequency due
to smaller effective current-carrying area
• Resistance of the wire causes i2•R loss, the conversion of electric
power into heat
– Silver would be slightly better, but too costly (silver coating/plating sometimes used)
– Aluminum’s low resistivity is close to Cu -- also lighter in weight!... but its surface oxide
is a poor conductor*
• Some EM Field power Radiates into Space
– Particularly for non-shielded wire, curved wires, etc.
– Even with super-conducting wires (zero resistance) there would be some
radiation losses
*Resistive surface aluminum oxide led to heating and home fires in 1960s through 1980s. Consequently
Aluminum power wiring was banned, or installed only with special coating or terminal fittings.
Rev. 2.8;Page 8
©1996-2005, R.Levine
Wire “Gauge”
• In North America, wire diameter is described by
peculiar “gauge” (ga or AWG) number
– Based on the number of times the wire is drawn through smaller
and smaller conical diamond forming dies during manufacture.
Larger ga or AWG number implies smaller diameter
• Most other countries list actual diameter (in mm)
[dc resistance stated in table]
B&S or
Diameter
AWG Copper (inches)
Wire Gauge
12
0.08
Diameter  per km (at dc, 0 Hz)
(mm)
[loop  is twice the resistance of
one wire]
2.053
10.42
Electric power uses
14
19
22
24
1.628
0.91
0.644
0.511
0.064
0.036
0.025
0.020
16.56
51.6
103.8
164.4
Electric power uses
Telephone history interest
Telephone use today
Telephone use today
Abbreviations: AWG=American Wire Gauge, B&S=Brown & Sharpe (manufacturer
of measuring equipment),  = Ohms
Rev. 2.8;Page 9
©1996-2005, R.Levine
Transmission Lines
• Electromagnetic waves propagate or flow in a direction
parallel to the wire’s axis, but power flow is mostly in
the electromagnetic field outside the metallic wires
– The wires act as a waveguide, although the name
“waveguide”sometimes describes a hollow tube
• The most accurate, but complicated, method of
analysis is to examine the electromagnetic wave
pattern in space
– Is the propagation completely parallel to the wires, or do waves
bounce around on diagonal reflected paths as in a hollow
waveguide or a multi-mode optical fiber?
• A sufficiently accurate method for many applications is
to describe the transmission line properties by
approximate “lumped” electrical parameters
Rev. 2.8;Page 10
©1996-2005, R.Levine
Free-wave Coupling
• Why don’t the EM waves just flow out into space away
from the wires?
– With certain geometrical arrangements, they do just that:
» Parallel wires separated far more than their diameters
» Wires bent to right angles from parallel (so-called dipole
antenna) like the lines above...
» A bend in the two parallel wires (over large distance
compared to the wavelength)
• EM waves from other sources may induce voltage or
current on wires
– One cause of “cross talk,” particularly at audio frequencies
– Called a “radio receiving antenna” when intentional
– Electromagnetic waves may cause primarily magnetic or
primarily electrostatic coupling or induction, depending on
geometrical arrangement
Rev. 2.8;Page 11
©1996-2005, R.Levine
Transmission Line Properties
• Approximate “lumped” section model of wave transmission
• Resistance per unit (loop) length, R
– unit: ohm/meter
• Inductance per unit (loop) length, L
– unit: henry/meter (where henry=volt•sec/amp)
• Leakage Conductance per unit length, G
– unit: mho/meter or 1/(ohm•meter) of conductance per unit length (“leakage”
from one wire to another)
– Conductance is 1/resistance (informal unit “mho” is ohm spelled
backwards -- official name “siemens”)*
– plastic insulation is very good so very little “mhos”
• Capacitance per unit length, C
– unit: farad/meter (where farad= amp•sec/volt)
• Following two “thought experiments” require relatively short section of
wire, so EM waves travel to far end in a very short time.
* Backward spelling is also used informally: 1/henry=yrneh (“ernie”), 1/farad=daraf
Rev. 2.8;Page 12
©1996-2005, R.Levine
Inductance/unit length
• Isolate a unit length of transmission wire pair,
“short circuit” the two wires at the far end
– Theoretically, it is desirable to chill the material to a low
(super-conducting??) temperature, so the electrical
resistance does not complicate the measurement!
– This is what scientists call a “thought experiment”
• Apply a constant voltage Va-b for T seconds.
The current i will increase “slowly” and the
magnetic field increases proportional to i.
• Compute V•T/i at the end of the time. This is
the inductance L. (Blue “area” is V•T.)
volts a-b
a
amps
V
I
b
Rev. 2.8;Page 13
t
©1996-2005, 0R.Levine T
t
0
T
Capacitance/unit length
• Isolate a unit length of transmission wire pair
• Apply a constant current I for T seconds. The
voltage Va-b will increase “slowly” as the
electric field increases. Positive electric
charge is drawn away from the lower wire and
pumped up to the upper wire. The total
amount of charge transferred in T seconds is
I •T (amp•sec or coulomb)
• Compute I•T/V at the end of the time. This is
the capacitance C. (Green “area” is I•T.)
a
volts a-b
amps
V
I
b
t
Rev. 2.8;Page 14
©1996-2005, 0R.Levine T
t
0
T
Resistance, Conductance, etc.
• The loop resistance per unit length is measured in an
experiment similar to measuring inductance. We find dc loop
resistance from the ratio V/I using constant current I.
• The conductance between the two wires is measured in an
experiment similar to measuring capacitance, except we
measure the “leakage” current I that flows from one wire to
another due to imperfect insulation.
• All of these measurements can be made in a more practical way
using sine wave test current or voltage at different frequencies.
The effects of inductance and series resistance can be
mathematically calculated using the measured ratio of voltage
to loop current. Similarly, the effect of capacitance and
conductance can be mathematically calculated.
• We find that each of these four parameter measurements give
slightly different results at different frequencies. For example,
skin effect produces higher measured effective series
resistance (ESR) at higher frequencies.
Rev. 2.8;Page 15
©1996-2005, R.Levine
Illustration of Skin Effect
B or H field
circulates
In clockwise
direction.
Cross-section
of wire carrying
current into
paper.
f=0 kHz (DC)
Intensity of
H field (amp/m)
External H field falls off
asymptotically inversely
proportional to distance
from wire center.
Rev. 2.8;Page 16
f=1 kHz
f=2 kHz
0
1
©1996-2005, R.Levine
2
Diametrical distance
Inside wire (mm)
Lumped Element Model for
Transmission Line
• This represents a 1 km loop of 19 ga copper wire, with
typical plastic insulation.
• Leakage conductance between wires is more often
described as 0.14 µmho or µsiemens of conductance,
instead of 7.14 Mof resistance
Note: These parameters are all dc values for 20º C temperature.
Rev. 2.8;Page 17
©1996-2005, R.Levine
Common (Longitudinal) Mode
• Electrical “Balance” is important in telephone transmission lines
– Electrical characteristics such as capacitance or leakage conductance from
either wire to ground should be the same (symmetrical).
• Telephone lines run parallel to electric power wires for miles, on
telephone poles or in underground conduits
– Power wires are furthest from the street level for safety of telephone repair
crews
• Longitudinal voltage can be magnetically coupled to both
telephone wires
– “Common Mode” voltage appears on both wires with respect to ground/earth
– A device that senses the “differential mode” (voltage difference between the
two wires) will not respond to a common mode voltage. Example: telephone
set
• Longitudinal voltage produces significant ac power frequency
“hum” if telephone line is “unbalanced”
– Example: unbalance occurs when one wire has lower resistance than other
wire vis-à-vis “ground/earth,” due to damaged or wet insulation.
Rev. 2.8;Page 18
©1996-2005, R.Levine
Unbalanced Model
• Real transmission lines must have well
balanced electrical characteristics to prevent
longitudinal or common mode induced
voltages from appearing at the ends
• However, for many theoretical purposes, an
“unbalanced” model with the same total loop
parameter values is simpler for analysis
Rev. 2.8;Page 19
©1996-2005, R.Levine
dc or Resistive Model
• A model which ignores L and C is only useful for the
single special purpose of computing dc loop current
• Omitting inductance and capacitance theoretically
removes time delay and waveform distortions. Power
loss still occurs.
– Note for dc that L becomes a zero ohm resistance or a short
circuit, while C becomes an open circuit
or
Rev. 2.8;Page 20
©1996-2005, R.Levine
Wire Resistance R Depends On...
• Material resistivity (copper, aluminum, etc.)
• Resistivity partly depends on metallic atomic arrangement
– Hard drawn (“work hardened”) copper has small irregular metal
crystals, higher resistance, but it is less damaged by handling or
installation.
– Soft drawn copper has large regular crystals of metal, lower resistance
• Temperature: resistance of metal increases about 1% for
each higher degree Celsius
– Standard room temperature is 20º C (=68º F)
• Wire Diameter (more generally, current carrying cross
sectional area). Larger diameter implies lower resistance.
• Signal frequency: due to frequency-dependent skin effect
– Higher equivalent resistance for higher frequency
– Because current-carrying area is smaller at high frequency
Rev. 2.8;Page 21
©1996-2005, R.Levine
Inductance L Depends On...
• Inductance is the ratio of the total “flux linkage” to the
current. Flux linkaage is measured in volt•sec, and is
found by integrating the magnetic field intensity over a
suitable surface between the two conductors
• In general, L depends on geometric shape and
separation of conductors. Major types are:
– Parallel round/cylinder wires (usually “twisted pair”)
– Co-axial cable (outer and inner cylindrical conductors)
• Use of magnetic materials
– Magnetic materials in the field region can affect L, but usually
non-magnetic materials (µ/µo=1) are used
– Some older cables were made with a magnetic alloy
(e.g.,“permalloy”) built in between the current carrying wires.
• L is very slightly dependent on frequency, indirectly
due to skin effect
Rev. 2.8;Page 22
©1996-2005, R.Levine
Conductance G Depends on...
• “Leakage” conductance is ratio of wire-to-wire leakage
current, divided by voltage. It is determined by….
• Intrinsic resistance of insulation material
• Thickness of the insulation. Thicker insulation gives
lower G value.
• Conductive impurities such as water (particularly with
dissolved ions) which can permeate through the
plastic under some conditions
– Much more serious problem with older porous pulp or fiber
(cotton or silk) insulation
• “Wet” cable can be dried out by use of dry nitrogen
(N2) gas under continuous pressure from an
evaporating tank of liquid nitrogen
• Slightly temperature dependent
Rev. 2.8;Page 23
©1996-2005, R.Levine
Capacitance C Depends On...
• Capacitance is the ratio of the electric charge (on the surface
of one conductor) to the voltage between the two conductors
• In general, C depends on geometric shape and separation
distance of conductors
• Dielectric permittivity “epsilon” r of the insulation. Most
plastic insulation materials have relative r =e/eo)(“dielectric
constant”) value in range 3 to 8, compared to air.
• Significantly depends on temperature.
• Slight increase if water molecules permeate the insulation
• Frequency dependence due to skin effect and material
properties. See Feynman Lectures on Physics, Vol.II, chapters
10, 11 and 32, for a more fundamental physical description of
why dielectric properties depend on frequency.
Rev. 2.8;Page 24
©1996-2005, R.Levine
Wave Speed cm= 1/(LC)
• The wave speed depends on electrical parameters of the insulation
for most practical wires and cables
• Regardless of shape, for a transverse electromagnetic wave
(propagation parallel to the wires) in a lossless (non-resistive,
perfectly insulated) line, c  1 / e
m
• The wave speed described here is the “phase velocity” of a test sine
wave -- not the velocity of a general waveform
– If the phase velocity is the same for all frequency components, then the
velocity of any arbitrary waveform is the same. If the phase velocity of
different frequencies is different, then the waveform of a traveling wave
will be modified after traveling different distances!
• For lossy lines, or lines with other components inserted
periodically*, the phase velocity varies greatly at different test
frequencies
– Therefore, a non-sinusoidal waveform can have its different frequency
components arrive with different delays, thus changing the received
waveform. (an effect called “dispersion”)
* For example, when loading coils (inductors) are connected in series in the wires at
intervals of 6000 ft, the wave speed is lower than for non-loaded wires.
Rev. 2.8;Page 25
©1996-2005, R.Levine
Data Transmission “Speed”
• The wave speed or time delay depends on physical
parameters of the transmission system
• The data rate (data bits per second) affects the time required
to transmit a fixed amount of data. A channel which can
transmit more bits/second can transmit the same data file in a
shorter time. We loosely call this higher “data speed” although
the term “data rate” or “bit rate” is more accurate and
appropriate
• The bit rate capacity of a channel is sometimes called its
“bandwidth” although the term “bit rate” is more accurate and
appropriate
• When all other factors (type of modulation, etc.) are
unchanged, a higher data rate does correspond to a waveform
with a higher bandwidth. However, by changing the type of
modulation (e.g., from two level to 4 level coding) one can
change the bandwidth of a signal without changing its digital
bit rate.
Rev. 2.8;Page 26
©1996-2005, R.Levine
Lossy Distortionless Line
• A transmission line having the following ratio of parameters:
R/L=G/C, has the same loss and wave speed (phase shift or time
delay) at all frequencies. It is therefore distortionless (no change in
waveform shape), since all frequency components are reduced
proportionately in amplitude and have the same time delay. They stay
in phase with each other. The signal is reduced in amplitude as it
travels along the wires, but the waveform is otherwise unchanged.
• G/C is normally a much, much smaller ratio than R/L. The simplest
modification to achieve the same ratio with R/L is to use low
resistance insulation between the wires (to increase G), but then the
overall power loss is too much to be economically interesting, even
with amplifiers.
• A more practical method to improve transmission line loss is to
artificially increase L by installing “loading coils” described later.
• More practical method to combat dispersion, for modems and other
waveform sensitive devices, has been to use adaptive equalizers.
Equalizers combine various internally delayed copies of the received
waveform to compensate for dispersion.
Rev. 2.8;Page 27
©1996-2005, R.Levine
Characteristic Impedance Zo=(L/C)
•
Zo is ratio of V/I in a traveling wave. V is transverse voltage (wire-towire), I is longitudinal current. In contrast, ohmic series (loop)
resistance R is ratio of longitudinal voltage drop to longitudinal current
• Zo depends on geometry
– When two conductors are far separated in comparison to their diameter or
width, Zo is larger
•
For a transverse electromagnetic wave (propagation parallel to the
wires) in a lossless (non-resistive, perfectly insulated) “square” parallel
plate transmission line, Zo= (µ/) = 377 approximately
– That is an approximation assuming all significant electric and magnetic field is
almost completely confined in the space between the two parallel plates
•
•
•
Geometry with increased distance between conductors has higher Zo
value.
For lossy lines, or lines with material µ or properties dependent on
frequency or temperature, the Zo will be different if these parameters
change
When two line sections with different Zo values (due to change in wire
diameter, insulation type, etc.) connect, some of the wave power will
be reflected and some will continue into the next section of transmission
line
Rev. 2.8;Page 28
©1996-2005, R.Levine
Nominal Zo for Subscriber Loop
• In the early days of the telephone, the two telephone
wires of a loop were installed far apart on a “crossarm” of a telephone pole. Wire centers were separated
by 20 or more times the diameter of the wires.
• Via theoretical calculations of surge impedance, we
see that wires with centers separated in air by about 5
times the wire radius will have approx Zo=600  surge
impedance
– The measured surge impedance varies slightly with frequency
due to changes in skin depth with frequency, etc.
– Despite many variations when comparing different types of wire
and cable, Zo=600 purely resistive (current and voltage inphase) is often used as the nominal surge impedance in
technical specification documents, etc.
900
• In modern telephone cables, wires are typically
separated by about 3 wire diameters, and each wire is
coated with plastic insulation. Theoretical surge
impedance of this pair is about 300 .
• Resistor-capacitor circuit model often used to better
represent an average length subscriber loop
terminated in a central office subscriber card.
Rev. 2.8;Page 29
©1996-2005, R.Levine
1.2µF
Transpositions and Helices
• A second wire pair installed on the same cross arms produced strong
magnetic field coupling (cross-talk) due to the magnetic field from
both loops sharing the space in between the wires.
• Coupling can be neutralized by “transposing” the second pair of
wires at the midpoint of installed length
• Third pair of wires can be transposed in four sections. Similarly more
transpositions can be used for the 4th, 5th, and succeeding pairs (4th,
5th not shown).
• Twisting each individual pair in a cable into a helix with different pitch
(length of one turn of the helix) helps minimize induction cross talk.
– Twisted pairs also hold the two wires comprising the same loop in close
proximity, thus reducing the area susceptible to magnetic induction. Also,
allows the installation technician to separate individual subscriber loop
pairs more easily for installation purposes.
Rev. 2.8;Page 30
©1996-2005, R.Levine
Wave Reflections
• When two transmission lines having different values of Zo are
joined, and an electromagnetic wave arrives at the joint from
one side
– Part of the power will travel through the joint into the second transmission
line
– Part of the power will be reflected back towards the source
• If the reflected wave occurs in a purely unidirectional wire pair,
this may not be a problem
– Example: one unidirectional pair of a two-pair (4 wire) system
• If the reflected wave occurs in a bi-directional wire pair, or can
get into the “return” unidirectional wire pair via a 2-to-4 wire
conversion point (a “hybrid” or directional coupler), the
participants may perceive an echo.
• We try to prevent echo, but when it occurs the best present
remedy is an echo canceller.
– The echo canceller determines the time delay, amplitude and polarity (+ or
-) of the echo waveforms, and generates a canceling signal by means of
digital signal processing (DSP).
– In dialed call service, the echo canceller must adaptively re-adjust its
parameters (time delay, etc.) for each new telephone call.
Rev. 2.8;Page 31
©1996-2005, R.Levine
Proportional Decrease In Power
• Wire to wire (transverse) voltage decreases exponentially*
with distance. There is a uniform percentage power loss
per unit length.
• For a 1 kHz test signal, 19 ga wire looses approx 20% of
section input power (leaving 79.4% output) for each 1 mi
(1.6 km) section (this corresponds to ~1 dB/mi)
• 3 mi of wire delivers 0.7940.7940.794 = 0.50056, or about
1/2 of original power
• Engineers don’t like to do tedious repeated multiplication,
so they use logarithms: loss of 1 dB per mi, added 3 times
for 3 miles, yields a total loss of 3 dB (corresponding to
about 1/2 of original input power)
*The word “exponentially” is a jargon term implying a change of a
fixed percentage for each km of wire. It does not merely mean
“large change.”
Rev. 2.8;Page 32
©1996-2005, R.Levine
Transmission Loss
• “Loss” is usually expressed in dB for convenience in adding
total logarithmic loss for a chain of devices
– simpler than multiplying the numerical input/output ratios for a chain of
sections
• For a length of wire or cable, transmission* “gain” in dB is:
10•log10 (output power/input power)
• With output lower than input power, this “gain” will be a
negative number (that is, a “loss” of power)
• For 1 mi of 19 ga wire loop using 1 kHz test signal, input to
output power ratio is 1.26/1 = 1/0.794)
• Corresponds to -1 dB/mi (-0.6 dB/km) gain (+1 dB/mi loss)
• Also corresponds to input-output voltage ratio 1.122/1 (or
1/0.89) for a mile of 19 ga wire
* Be careful about often careless and confusing usage of minus sign. Strictly speaking,
negative loss is “gain” or amplification. Transmission gain could also theoretically be
produced by wire with negative resistance!
Rev. 2.8;Page 33
©1996-2005, R.Levine
Transmission Loss Also Depends On...
• Wire diameter (gauge). At 1 kHz:
AWG gauge
19
22
24
Loss (dB/mi)
1
1.79
2.2
• Frequency (due primarily to skin effect R) 19 ga
Frequency (kHz)
1
10
100
Loss (dB/mi)
1
3.2
6.1
• Temperature (due primarily to increased R)
– Loss per mi (or per km) is greater at higher temperature
Rev. 2.8;Page 34
©1996-2005, R.Levine
“Insertion” Loss
• Conceptually think of “breaking” the chain of
equipment and inserting another device of interest
(more wire, an amplifier, etc.)
• Additional loss due to this insertion of another device
is the so-called insertion loss
• Insertion loss and transmission loss are the same in a
chain of devices with the same surge impedance – that
is, the same ratio of V/I at all connection points
– That is, uniform “characteristic impedance”or “surge
impedance” at all points in the transmission chain
– Not accurate throughout the audio frequency range, but
telephone systems often approximate the surge impedance Zo of
wire pair by using 600  (resistive) as a nominal approximate
value for certain test purposes
Rev. 2.8;Page 35
©1996-2005, R.Levine
Exponential Losses in Transmission Line
1
0.9
0.8
0.7
0.6
P( x ) 0.5
0.4
0.3
0.2
0.1
0
0
1
2
x
Rev. 2.8;Page 36
©1996-2005, R.Levine
3
4
Linear Loss Described Using Logarithmic dB
1
T(x)=10•log(P(x))
0
1
T( x ) 2
3
4
5
0
1
2
x
Rev. 2.8;Page 37
©1996-2005, R.Levine
3
4
Loop Length
• Subscriber Loop length is usually limited by dc loop current
(so-called “resistance limit”)
– At least 5 to 10 mA needed to properly operate microphone and tone dial
in a telephone set. 20 mA or more is desirable.
• In contrast, trunk length is usually limited by signal loss
– This can be corrected by amplification, so there is no theoretical
physical limit from this cause (of course, signal power falls to near noise
power level, etc.)
– Longer trunks require more amplifiers
» Trunk wire with high loss requires amplifiers (one type of repeater)
with higher gain and/or closer spacing
– When dc current is used in trunks to power repeaters, the overall design
of the equipment is normally done so that dc current is not the limiting
factor.
– In some cases, the signal delay is limited due to call processing signal
requirements (even while speech delay is not yet a problem) to 1.5 or 2
milliseconds for some switching devices such as remote line modules or
concentrators used with telecom switches.
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©1996-2005, R.Levine
Conflicting Objectives
• Amplifiers are used in analog transmission systems to
compensate for power loss in transmission wire, cable
– In digital transmission systems, dispersion and other waveform changes
must also be compensated by repeaters. The example here considers
only amplification.
• One very high gain amplifier could, in theory, compensate for
the loss of any length of line
• But if the signal gets too small before further amplification,
the effects of thermal “noise” and interference will be severe
• If the signal is amplified too high before transmission, the
voltage will be huge (and possibly even dangerous!)
• The cost of a very high gain amplifier is also much greater
than a low gain amplifier
• The optimum engineering-economic arrangement is to use a
number of amplifiers of moderate gain, inserted at equal
distances into the transmission wires
Rev. 2.8;Page 39
©1996-2005, R.Levine
Optimum Number of Amplifiers is Set by
Economics, as well as Technology
• A definite cost model is required
– changing technology may change the cost model
• Usual practice is to place amplifiers (repeaters)
periodically at fixed distance intervals so:
– Required amplifier gain is moderate, so unit cost is moderate
– Input signal is never too low compared to noise & interference
– If the first or last section of line is not the standard interval
length, a Line Build-Out (LBO) network is connected into the
end. LBO can be made using inductors, resistors and
capacitors, or sometimes by merely using a spool of wire or
cable of the correct length.
– An example showing economic optimization of repeater spacing
will be given on the practice quiz
Rev. 2.8;Page 40
©1996-2005, R.Levine
Approximate Loss Formula
• For (G/C)<<(R/L), which is the typical transmission line case,
dB loss per km is approximately proportional to
[(R/2)• (C/L)] + [(G/2)• (L/C)] + other smaller terms.
• The first term is biggest. If we could increase L (or decrease
C or R), loss would decrease. Increasing L is the most
practical alternative.
• Both Pupin and Campbell (and others) recognized this about
1900, and added lumped inductive “loading coils” in series
with the telephone wires, thus decreasing the first loss term
(R/2)•(C/L).
• Loading Coils are passive, reliable devices, used widely until
the 1960s. Loading coils have mostly been removed since
then, but are still occasionally found in place on old outside
plant wiring.
Rev. 2.8;Page 41
©1996-2005, R.Levine
Pupin Loading Coils
• Practical approximation to increased L uses “lumped”
series inductors
– Most widely used spacing interval is 6000 feet (1.848 km)
» European systems use 2 km spacing
– Most widely used inductor is 88 mH, toroidal shape
» There is some added resistance due to thin wire in the
loading coil, but overall transmission loss is improved
• Used historically for baseband transmission on both
subscriber loops and trunks
• The 6000 ft spacing of loading coils led directly to the
same spacing later for T-1 digital carrier repeater units,
since access and enclosures were already available at
these locations.
Rev. 2.8;Page 42
©1996-2005, R.Levine
Loading Coils Have Mainly Historical Significance Today
• Due to use of lumped inductors, loaded line has better loss only at low
frequencies, and has much worse loss at high frequencies (above 4 kHz)
– Acts like a type of “low pass filter”
– Designed to pass up to 3.5 kHz audio for desired speech quality
• Loading coil toroidal cores are also used to wind transformers for radio
and other applications
– Available at low cost on the used equipment market. Used by radio “hams”
and experimenters
• In some cases where two pairs split off from one pair (a “bridged tap”), a
coil is wired in series with each pair to increase the Zo and reduce
reflected power
– This is called a “bridge lifter”
• Loading coils and bridge lifters must be removed to install any
transmission system which utilizes frequencies above about 4 kHz, such
as:
– All types of digital systems (T-1, ISDN, etc.)
– Data above voice (several proprietary systems)
– ADSL, HDSL, etc.
Rev. 2.8;Page 43
©1996-2005, R.Levine