Analyze Transmission Lines With Almost No Math

7/27/2019 Analyze Transmission Lines With Almost No Math

1/5Mar ch 18, 1999 | ed n 143www.ednmag.com

designfeature By Ron Schmitt, Nortel Networks

WITH AN UNDERSTANDING OF THE UNDERLYING PHYSICS, YOU

CAN OFTEN GO A LOT FURTHER IN ANALYZING TRANSMISSION

LINES THAN YOU CAN BY MANIPULATING DOZENS OF COM-

PLEX EQUATIONS. THE BASIC INFORMATION THAT YOU NEED

TO GET STARTED IS RIGHT HERE.

As wirel ess designs become more prevalentand as digital designs reach higher and higherfrequencies, a thorough understanding of

transmission-line theory is becoming increasinglyimportant.With the aid of graphical representationsof analog and digital signals,you can gain a solid in-tuitive understanding of transmission lines. More-

over, this approach requires little background inelectromagnetic-field theory.

Unfortunately, you probably learned transmis-sion-line theory during a few lectures in an elec-tromagnetic-fields class. If so, you learned trans-mission-line theory with wave equations and a lotof difficult math. In addition, youve probablyheard that transmission-line effects become ap-parent at higher frequencies, but rarely does any-one explain why. Why are transmission-line effectsusually noticeable only at high fre-quencies? What happens at low fre-quencies? What are the definitions of high and

low? In practice, you can more easily and com-pletely grasp transmission-line theory just by un-derstanding the basic physics.

THE CIRCUIT MODEL

As the name implies, a transmission line is a setof conductors used for transmitting electrical sig-nals. In general, every connection in an electric cir-cuit is a transmission line. However, implicit in mostdiscussions of transmission-line theory is the as-sumption that the lines are uniform.

A uniform transmission line is one whose geom-etry and materials are uniform. That is, the con-

ductor shape, size, and spacing are constant,and theelectrical characteristics of the conductors and the

material between them are uniform.Some examplesof uniform transmission lines are coaxial cables,twisted-wire pairs, and parallel-wire pairs. For pcboards, the common transmission lines are strip-line and microstrip.

In a simple transmission line, a source provides asignal that must reach a load (Figure 1a). The fig-

ure shows the transmission line as a pair of parallelconductors. In basic circuit theory, you assume thatthe wires that make up the transmission line are ide-al and hence that the voltage at all points on thewires is exactly the same. In reality, this situation isnever quite true. Any real wire has series resistance(R) and inductance (L). Additionally, a capacitance(C) exists between any pair of real wires. Moreover,because all dielectrics exhibit some leakage, a small

Analyze transmission lineswith (almost) no math

SOURCE

SOURCE

SOURCE

(a)

(b)

(c)

LOAD

LOAD

LOAD

L L

1/G C

C C C

1/G C

R R

L L L

These circuits model a transmission line as an ideal wire pair (a), a lumped circuit

that accounts for the nonideal characteristics of real wires (b), and a lossless lumpedcircuit (c).

F i g u r e 1


2/5144 ed n | Mar ch 18, 1999 www.ednmag.com

conductance (G) (that is, a highshunt resistance) exists betweenthe two wires. You can model thetransmission line using a basiccircuit that consists of an infinite

series of infinitesimal R, L, and Celements (Figure 1b). Be-cause the elements are infin-itesimal, the model parameters(L, C, R, and G) are usually spec-ified in units per meter.

To simplify the discussion, youcan ignore the resistances. Figure1c shows the resulting LC circuit.A trans-mission line that is assumed to have noresistance is a losslesstransmission line.Notice several important points. First,with the LC model, points A and B may

be at different potentials. Second,a signaltransmitted from the source charges anddischarges the lines inductance and ca-pacitance. Hence, the signal does not ar-rive instantly at Point B but is delayed.

Last, the impedance at points A and Band each node in between depends notjust on the source and load resistance,butalso on the LC values of the transmissionline.

How does this circuit react at differ-ent frequencies? Recall that inductive andcapacitive reactance depend on frequen-cy. At low frequencies, the LC pairs in-troduce negligible delay and impedance,reducing the model to a simple pair ofideal wires.At higher frequencies, the LCeffects dominate the behavior, and youmust not ignore them.

THE WAVEGUIDE MODEL

An equivalent method of characteriz-ing transmission lines describes a trans-

mission line as a guide for electromag-netic waves.With this method, the sourcesends the electromagnetic signal, whichconsists of a voltage wave and a currentwave, to the load (Figure 2).

The transmission line, which effec-tively acts as a transmission medium,guides the signal along the way. The sig-nal travels through this medium at thespeed of light within that medium. Youcan calculate the speed of light, , in atransmission line from the permittivity() and permeability () of the dielectric

between the conductors=1/ .For example, if Teflon separates the

designfeature Transmission lines

SOURCE LOAD

I

+

V

Z0,

In this waveguide model, I represents the current wave,

and V represents the voltage wave, each of which travels

along the transmission line. ZO

is the lines characteristic impedance, and

is the speed of light in the transmis-

sion lines dielectric.

F i g u r e 2

TIME t

TIME t+tX=l

LOAD

X=0

SOURCE Z0=2

I

V

This example of a signal pulse traveling along a

transmission line shows the voltage and current

waves along the line. The figure represents two snapshots in time.

F i g u r e 3

SOURCE LOAD

RS=0 RL=100

RS RL

l=1m

REFLECTIONS

OCCUR

HERE

Z0=50; =0.7c

In this transmission-line example, a source andload are connected through a 1m Teflon coaxi-

al cable with a characteristic impedance of 50. Reflections occur

at the boundary between the source and the cable because of the

difference in impedance. Reflections also occur at the boundary

between the load and the cable. (Assume that the source-to-cable

and load-to-cable wires are of negligible length.)

F i gu r e 5

PATTY 05ms5094

Z1 Z2

INCIDENT

WAVE

REFLECTED

WAVE

TRANSMITTED

WAVE

At a boundary

between two

regions of different impedance,

Z1

and Z2, some of the incident

energy passes through the

boundary, and some is reflected.

F i g u r e 4

DISTANCE

TIMESOURCE

X=0

LOAD

X=l

t=0

t=

t=2

t=3

t=4

t=5

V=0.11

V=+0.11

V=+0.15

V=+0.33

V=0.33

V=+1

V=+1

SOURCE

1= LOAD

= LOAD

=1

V=+1.33

V=0.44

+

+

=

3 5SIGNAL RECEIVED

AT LOAD

t

1

31

3

This lattice diagram shows the propagation of a 1V step wave and its subse-

quent reflections along the transmission line of Figure 5. The waves received atthe load are at the right of the diagram. The diagram shows the first five reflections.F i g u r e 6


3/5

pair of wires that makes up the trans-mission line, the wave travels at the speedof light in Teflon, which is approximate-ly 70% of the speed of light in a vacuum.

(TEFLON=0.7c, where c is the speed oflight in a vacuum.) As the signal travelsalong the transmission line, the voltagewave defines the voltage at each point,and the current wave defines the currentat each point. Figure 3 shows a pulse sig-nal traveling along a transmission line,

with the voltage and current values ateach point. Along the entire length of theline, the ratio of the voltage to the cur-rent is constant. The ratio is the char-

acteristic impedance, Z0, and is definedby the geometry of the line and the per-mittivity and permeability of the dielec-tric. The characteristic-impedance equa-tions are often complex. For example, ifyou have a wire-pair transmission line,Z

WIREPAIR= 1/ / cosh1 (s/d),where

s is the distance between the wires and dis the diameter of each wire.

RELATIONSHIP BETWEEN THE MODELS

Now you have two models for a trans-mission line: a circuit comprising infini-tesimal inductances and capacitanceswith parameters L and C and a waveguidefor signals with parameters and Z

0.The

following equations relate the parametersof the two lossless models:Z

0= LC,and

146 ed n | Mar ch 18, 1999 www.ednmag.com


In the circuit of Figure 5, when the source signal is a 1V step with varying rise times, the load receives these signals. (The diagram shows only the firstfour waves to reach the load.) In (a), t

RISE=0. In (b), t

RISE=5 nsec. In (c), t

RISE=15 nsec.

F i g u r e 7

INITIAL

WAVEAT LOAD

SECOND

WAVE

AT LOAD

THIRD

WAVE

AT LOAD

FOURTH

WAVE

AT LOAD

RESULTING

LOAD

SIGNAL

(a) (c)(b)

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.50 10 20 30 40 50

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.5

0 10 20 30 40 50

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.50 10 20 30 40 50

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.5

0 10 20 30 40 50

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.50 10 20 30 40 50

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.5

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.5

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.5

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.5

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.5

10 0 10 20 30 40

10 0 10 20 30 40

10 0 10 20 30 40

10 0 10 20 30 40

10 0 10 20 30 40

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.50 10 20 30 40 50

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.50 10 20 30 40 50

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.50 10 20 30 40 50

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.50 10 20 30 40 50

TIME (nSEC)

VOLTAGE

(V)

1.5

1

0.5

0

0.50 10 20 30 40 50


4/5148 ed n | Mar ch 18, 1999 www.ednmag.com

=1/ LC.Although these models are in-terchangeable, the waveguide model isusually more useful for transmission-lineanalysis; the remainder of this article ex-clusively uses the waveguide model.

REFLECTIONS

Whenever an electromagnetic waveencounters a change in impedance, someof the signal is transmitted and some ofthe signal is reflected (Figure 4). The in-terface between two regions of differentimpedances is an impedance boundary.

An analogy helps in understanding thisconcept. Imagine yourself sitting in asmall boat in a pond, looking down intothe water. A fish swimming by sees youfrom below the waters surface because

water is transparent. In addition,you faintly see your image as a re-flection upon the waters surface.Hence,some of the light from your image trav-els through the water to the fish, andsome of the light reflects back to you.This phenomenon occurs because of thedifference in optical impedance of waterand air.This same phenomenon also oc-curs at electronic-signal frequencies.Thedifference between the impedances de-termines the amplitude of the reflectedand transmitted waves. The reflection

coefficient, , for the voltage wave is=V

REFLECTED/V

INCIDENT=(Z

2Z

1)/Z

2+Z

1,

whereas the transmission coefficient is=V

TRANSMITTED/V

INCIDENT=1+.

An interesting and often underem-phasized fact is that the amount of re-flection is independent of frequency andoccurs at all frequencies. This fact seemscontrary to the common belief that re-flections are high-frequency phenome-na. The next section should clear up thisconfusion.

PUTTING IT ALL TOGETHER

In summary: A signal traveling along a transmis-

sion line has voltage and currentwaves related by the characteristicimpedance of the line.

Signal reflections occur at impedanceboundaries.

As it travels down the line, a signalhas delay associated with it.

These three elements combine to pro-duce transmission-line effects. The firsttwo items imply that a circuit has reflec-

tions unless the transmission-line,source, and load impedances are all

equal. The third item implies that re-flected waves reach the load staggered intime. An example helps to illustrate. Fig-ure 5 shows a transmission line (coaxialcable) with a source and load connected.

The reflection and transmission coeffi-cients for this circuit follow:

Let the source produce a 1V step func-tion at time t=0. This wave travels downthe cable and reaches the load at time

t=l/=1/0.7c=5 nsec. Because the cablescharacteristic impedance (50) is differ-ent from the impedance of the load(100), some of the incident wave trans-mits to the load, and some is reflected by

the load. The reflected wave travels backthrough the cable and arrives back at thesource at time 2t.Because the source im-pedance (zero) does not match the char-acteristic impedance of the cable (50),another reflection occurs. This reflectiontravels toward the load and arrives at theload at time 3t.

As with the initial wave, the load ab-sorbs some of the wave, and some is re-flected. Subsequent waves arrive at theload in this manner ad infinitum, de-


In the circuit of Figure 5, when the source signal is a 1V sine wave at different frequencies, the load

receives these signals. (The diagram shows only the first four waves to reach the load.) In (a), f=50MHz. In (b), f=1 MHz.

F i gu r e 8

INITIAL

WAVE

AT LOAD

SECOND

WAVE

AT LOAD

THIRD

WAVE

AT LOAD

FOURTH

WAVE

AT LOAD

RESULTING

LOAD

SIGNAL

TIME (10 nSEC)

TIME (10 nSEC)

TIME (10 nSEC)

TIME (10 nSEC)

TIME (10 nSEC) TIME (SEC)

TIME (SEC)

TIME (SEC)

TIME (SEC)

TIME (SEC)

VOLTAGE

(V)

VOLTAGE

(V)

VOLTAGE

(V)

VOLTAGE

(V)

VOLTAGE

(V)

VOLTAGE

(V)

VOLTAGE

(V)

VOLTAGE

(V)

VOLTAGE

(V)

VOLTAGE

(V)

(a) (b)

2

1

0

1

2

2

1

0

1

2

2

1

0

1

2

2

1

0

1

2

2

1

0

1

2

2

1

0

1

2

2

1

0

1

2

2

1

0

1

2

2

1

0

1

2

2

1

0

1

2

0 1 2 3

0 1 2 3

0 1 2 3

0 1 2 3

0 1 2 30 2 4 6

0 2 4 6

0 2 4 6

0 2 4 6

0 2 4 6

,1ZR

ZR

0S

0SSOURCE =

+

=r

,3

1

ZR

ZR

0L

0LLOAD =

+

=r

.3

11)1( LOADLOAD =r+=G


5/5

creasing in amplitude after each roundtrip.The net effect is that the load receivesa voltage signal that is the superpositionof the initial incident wave and all of thesubsequent waves (Figure 6). As this ex-ample demonstrates, the resulting signal

at the load can look much different fromthe original source signal. A simple stepsignal at the source ends up producing astep wave followed by a series of oscilla-tions at the load.These oscillations even-tually settle to the value you would expectif you were to ignore the transmissionline.

In this example, the step signal is anideal digital wave; that is, a signal with

zero rise time. Of course, real-world stepsignals have rise times greater than zero.Changing the rise time of the step signalchanges the shape of the signal that ap-pears at the load. Figure 7 illustrates theload waveforms for the same transmis-sion line using various rise times. Whenthe rise time becomes much longer thanthe transmission-line delay (t), the re-flections get lost in the transition re-gion.The effect of the reflections then be-comes negligible. It is important to notethat, regardless of the rise time, the am-

plitude of the reflections is the same.Therise time affects only the superposition ofthe reflections.

The transmission-line effects (over-shoot and oscillation) become apparentwhen the rise time, t

RISE, is short com-

pared with the transmission-line delay, t.Such signals are therefore in the domainof high-frequency design. When t

RISEis

long compared with t, the transmission-line effects are negligible; these signals arein the domain of low-frequency design.For most applications, you can consider

tRISE>6t to be in the low-frequency do-main.

You can use the same analysis for ana-log signals. Using the circuit ofFigure 5,let the source be a 1V sine wave with a fre-quency of 1 MHz (period T=1sec). Thissignal undergoes the same reflections asthe step signal. The amplitude of the

waves incident on the load is also thesame, namely +1.33V, 0.44, +0.15,0.051/4. The sum of the series of waves isa 1V sine wave (Figure 8b). In this exam-ple, no transmission-line effects are no-ticeable because the delay of the trans-mission line is negligible. In fact, theround-trip cable delay is 1/100 of the signalperiod.

However,if you increase the source fre-quency to 50 MHz (period T=20 nsec),the resulting load signal is quite different,even though the incident waves are of the

same amplitude. The delay through thetransmission line is no longer negligibleat this higher frequency (Figure 8a). Infact, the round-trip delay of the trans-mission line causes the incident waves toshift 180 . With these phase shifts,the in-cident waves produce a load signal thatgradually builds to a steady-state 2V sinewave. In addition,the load signal shifts inphase from the source signal. Hence, forthis circuit,50 MHz falls in the domain ofhigh-frequency design, and 1 MHz falls inthe domain of low-frequency design. For

most applications, you can consider asine-wave period of T=20t (or =20l)the boundary between the high- and low-frequency domains.P

Re fe r e n c e s

1. Paul, Clayton, and Syed Naser, Intro-duction to Electromagnetic Fields, SecondEdition, McGraw-Hill, 1987.

2. Rosenstark, Sol, Transmission Linesin Computer Engineering, McGraw-Hill,1994.

3. Johnson, Howard, and Martin Gra-

ham, High-Speed Digital Design: A Hand-book of Black Magic, Prentice-Hall, 1993.

Au t h o r s b io g r a p h y

Ron Schmitt holds a BS in electronic engi-neering from Cornell University (Ithaca,NY) and an MSEE from University ofPennsylvania (Philadelphia). During sev-en years in the industry, he has worked onembedded-system design, signal processing,digital and RF communications systems,and analog test equipment. He has onepatent pending and has co-authored a con-

ference paper on antenna-signal processing.


150 ed n | Mar ch 18, 1999 www.ednmag.com

THE TRANSMISSION-LINE

EFFECTS (OVERSHOOT AND

OSCILLATION) BECOME

APPARENT WHEN THE RISE

TIME, tRISE

, IS SHORT COM-

PARED WITH THE TRANSMIS-

SION-LINE DELAY, t.

Analyze Transmission Lines With Almost No Math

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