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Transmission Line Fundamentals Southern Methodist University EETS7320 Fall 2007 Session 5 Slides only. (No notes.)
Major Transmission Facts 1 • Electromagnetic waves flow via the non-conductive 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)
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
Structure and EM Fields in Co-ax See footnote on previous page.
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 : • Wave speed (phase velocity) * 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.
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.”
Electrical Resistance area • 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) • 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. length
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.
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] Abbreviations: AWG=American Wire Gauge, B&S=Brown & Sharpe (manufacturer of measuring equipment), = Ohms Electric power uses Electric power uses Telephone history interest Telephone use today Telephone use today
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
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
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
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 amps a V I b t t T 0 T 0
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.) volts a-b a amps V I b t t T T 0 0
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.
Illustration of Skin Effect B or H field circulates In clockwise direction. Cross-section of wire carrying current into paper. Intensity of H field (amp/m) f=0 kHz (DC) f=1 kHz External H field falls off asymptotically inversely proportional to distance from wire center. f=2 kHz Diametrical distance Inside wire (mm) 0 1 2
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 Mof resistance Note: These parameters are all dc values for 20º C temperature.
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.
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
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
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
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
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
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.
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, • 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.
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.
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.
Characteristic Impedance Zo=(L/C) • Zo is ratio of V/I in a traveling wave. V is transverse voltage (wire-to-wire), 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 orproperties 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
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 “cross-arm” 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 in-phase) is often used as the nominal surge impedance in technical specification documents, etc. • 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. 900 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.
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.
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 0.7940.7940.794 = 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.”
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!
Transmission Loss Also Depends On... • Wire diameter (gauge). At 1 kHz: • Frequency (due primarily to skin effect R) 19 ga • Temperature (due primarily to increased R) • Loss per mi (or per km) is greater at higher temperature
“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
Linear Loss Described Using Logarithmic dB T(x)=10•log(P(x))
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.
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
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
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.
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.
Loading Coils Have Mainly Historical Significance Today • Due to use of lumped inductors, loaded line has better loss only at low frequencies, and has muchworse 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.