transmission equation solutions

8/3/2019 Transmission Equation Solutions

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 44, NO. 3, AUGUST 2002 413

Solution of the Transmission-Line Equations Underthe Weak-Coupling Assumption

Clayton R. Paul , Fellow, IEEE

AbstractThe transmission-line equations for a three-con-ductor lossless line in a nonhomogeneous medium are solvedin symbolic form (as opposed to a numerical solution) in thetime domain under the assumption of weak coupling. The resultprovides insight into crosstalk between the lines. Numericalsolutions of this problem exist but do not illustrate the factorsthat influence the crosstalk waveforms. The symbolic solutiongiven in this paper makes those factors clear. The validity of theweak-coupling assumption is also investigated.

Index TermsCrosstalk, literal solution, transmission lines,weak coupling.

I. INTRODUCTION

CONSIDER the three-conductor line shown in Fig. 1. Theground symbol merely symbolizes the third or referenceconductor for the line voltages. This reference conductor may

be an infinite ground plane, a wire, a printed circuit board (PCB)

land, an overall shield, etc. The other two conductors may also

be of various types such as wires, PCB lands, etc. The line is as-

sumed to be a uniform line in that theconductor and surrounding

dielectric cross sections do not vary along the line. The line is

assumed to be lossless in that the conductors are assumed to be

perfect conductors, and the surrounding medium, which may

be inhomogeneous as in a PCB, is assumed to be lossless. One

conductor with the reference conductor is driven by a source

having open-circuit voltage and source resistance andis terminated in a resistive load . This circuit is referred to as

the Generator circuit. The other conductor with the reference

conductor is referred to as the Receptor circuit and is terminated

at the near end in a resistive load, , and at the far end in a

resistive load, . Electric andmagnetic fields areproducedby

the voltage and current of the generator circuit and interact with

the receptor circuit producing crosstalk voltages at the terminals

of that circuit, and . The objective of this paper is to

provide a symbolic solution (as opposed to a numerical one) for

these crosstalk voltages.

The voltages (with respect to the reference conductor),

and , and currents of each circuit,

and , are functions of position and time . These arerelated, for the TEM mode of propagation by the transmis-

sion-line equations [1]. For the generator circuit we have

(1a)

(1b)

Manuscript received January 14, 2002; revised March 18, 2002.The author is with the Department of Electrical and Computer Engineering,

Mercer University, Macon, GA 31207 USA.Publisher Item Identifier 10.1109/TEMC.2002.801753.

For the receptor circuit we have

(2a)

(2b)

These transmission-line equations are obtained from the per-

unit-length equivalent circuit shown in Fig. 2 [1], [2]. The quan-

tities and are the per-unit-length inductances of the re-

spective circuits, whereas the quantities and are the per-

unit-length capacitances of the respective circuits. The quanti-

ties and are the per-unit-length mutual inductance and ca-

pacitance between the two circuits, respectively. Equations (1)and (2) are coupled via the terms on the right-hand sides of the

equations. The terminal conditions must be incorporated into

the general solution of (1) and (2) and these are, from Fig. 1

(3a)

(3b)

and

(4a)

(4b)

where the line is of total length and extends from to

.The solution of these equations requires the solution of the

transmission-line equations in (1) and (2) along with the incor-

poration of the terminal loads given in (3) and (4). A number

of numerical solution techniques are available for this problem.

The finite-difference time-domain (FDTD) method is described

in [1]. A SPICE subcircuit model can also be obtained for in-

clusion in a SPICE or PSPICE solution [1], [2]. In these numer-

ical methods, the various factors that contribute to the crosstalk

waveform are not evident; only a numerical solution is obtained.

A symbolic solution of these equations along with the incorpo-

ration of the terminal loads into that solution would make the

effects of the line and load parameters on the waveform clear.

Such a solution for a three-conductor lossless line was obtainedin [1]. However, in order to facilitate a solution, it was assumed

that the surrounding medium was homogeneous. For example,

bare wires in air would correspond to this case but wires with

dielectric insulations or a PCB would not. The results of this

paper will provide a similar solution for the case of an inhomo-

geneous surrounding medium. Although the solution in [1] for a

homogeneous medium was exact, it appears impractical to pro-

vide a similar, exact solution for the case of an inhomogeneous

medium. In order to facilitate a solution for the case of an in-

homogeneous medium, it will be assumed that the two circuits

0018-9375/02$17.00 2002 IEEE
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414 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 44, NO. 3, AUGUST 2002

Fig. 1. The crosstalk circuit to be investigated.

Fig. 2. The per-unit-length equivalent circuit.

are weakly coupled. The accuracy of this assumption will also

be investigated.

II. THE WEAK-COUPLING ASSUMPTION

The weak-coupling assumption is reasonable for many

cases of practical interest and means the following. The

voltage and current of the generator circuit produce electric

and magnetic fields that interact with the receptor circuit and

induce voltages and currents in that circuit. This inducement

of voltages and currents in the receptor circuit are through

the terms and

on the right-hand side of (2). These induced voltages and

currents, in turn, produce electric and magnetic fields that

provide a back-interaction or second-order effect by inducing

voltages and currents in the generator circuit. This back-in-

teraction is symbolized by the terms

and on the right-hand side of (1). Theweak-coupling assumption assumes that this back interaction is

negligble so that (1) approximates to

(5a)

(5b)

The question of how weakly coupled must the lines be in

order for this to be valid can be partially answered from the

exact solution for lines in a homogeneous medium given in [1].

In that solution, it was found that the characteristic impedance

of the generator circuit could be written as

where the coupling coefficient is

(6)

This is very similar to the coupling coefficient between two

turns of a transformer. Hence, we would expect that weak cou-

pling would be a reasonable assumption so long as

(7)

As an example of a numerical solution which will be shown,

consider the PCB shown in Fig. 3(a). This is an example of a

coupled microstrip which resembles a PCB. Two 5-mil lands are

on one side of a glass-epoxy board of thickness 47

mils. The lands are 5 mils in width and 1.4-mil thickness. Theyare separated edge-to-edge by 50 mils. A ground plane (the ref-

erence conductor) is on the other side of the board. A numerical

program, MSTRPGAL.FOR described in [1], was used to de-

termine the per-unit-length parameters giving

H

mH

mpF

mpF

m
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PAUL: TRANSMISSION LINE-EQUATIONS UNDER THE WEAK-COUPLING ASSUMPTION 415

Fig. 3. The configuration investigated. (a) Cross-sectional dimensions. (b)Waveform of the source.

Using theseparameters, we calculate and

. Hence, we would expect that, for this separation, the

weak-coupling assumption would be valid.

III. THE JARVIS MODEL

There is a popular solution for the general case of a three-con-

ductor lossless line in an inhomogeneous medium that is often

used for the prediction of time-domain crosstalk in digital cir-

cuits [3][6]. This solution was apparently first published by

Jarvis [7] and will be referred to as the Jarvis Model. The deriva-

tion of the solution was facilitated by making three importantsimplifying assumptions.

Assumption #1: The Lines Are Weakly Coupled:

Assumption #2: The Line Is Symmetrical: Symmetry means

that the generator and receptor conductors are identical in cross

section, and they are equally spaced from the reference con-

ductor in a symmetrical fashion. The PCB in Fig. 3(a) is sym-

metrical although we will not make that assumption in our gen-

eral solution. This symmetry provides that the self inductances

are equal as are the self capacitances:

(8a)

(8b)

Under the weak-coupling assumption, we may logically define

the characteristic impedance ofboth circuits as

(9)

and the velocity of propagation on each line as (there are in

fact two modes of propagation on the line whose velocities of

propagation are different for an inhomogeneous medium [1],

[2])

(10)

so that the one-way time delay along the line is

(11)

Assumption #3: The Lines Are Matched at All Ends:

(12)

This last assumption is very restrictive since it is rarely achieved

in practice. Designers attempt to match with various schemes,

but a complete match is rarely achieved at all frequency com-

ponents of the signal on the line. It is this last assumption that

causes the actual crosstalk to, in many cases, differ drastically

from the predictions of this model. The derivation we will sub-

sequently provide does not assume completely matched lines

and/or symmetry. Why are these three assumptions made in the

derivation of the Jarvis model? The answer is evidently to sim-

plify the solution of the transmission-line equations.

Nevertheless, the so-called Jarvis model detailed in [3][7]

gives the predictions of the near-end and far-end crosstalk volt-

ages as

(13a)

(13b)

where the backward and forward coupling coefficients are de-fined as

(14a)

(14b)

where the total mutual inductance and capacitance are the

product of the per-unit-length values and the total line length

and are denoted as and . The results

for the near- and far-end crosstalk voltages given in (13) are

the commonly-used prediction model based on the above

three assumptions [3][7]. The backward and forward

coefficients, and , in (14) are defined slightly different

than in [3][7] but the result is equivalent.

Suppose that the open-circuit voltage is a ramp function il-lustrated in Fig. 3(b). Equation (13a) indicates that the near-end

crosstalk is a pulse of length twice the one-way line delay, and

(13b) indicates that the far-end crosstalk is also a pulse but has

width equal to the rise time of the source since it is propor-tional to the derivative of the source voltage. These are illus-

trated in Fig. 4(a). If the source rise time is much greater than

the one-way delay, i.e., , the crosstalk waveforms sim-

plify to those shown in Fig. 4(b). These are identical to the in-

ductive/capacitive coupling model in [1], [2] which are valid for

.

In order to illustrate the accuracy of the Jarvis model, we will

show computed crosstalk results for the PCB shown in Fig. 3(a)

having an open circuit voltage in the form of a ramp rising to 1 V

with 500 ps rise time as shown in Fig. 3(b). The total line length

is 10 inches or 0.254 m. From the above stated per-unit-length

parameters we compute , ns, and
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(a)

(b)

Fig. 4. The near-end and far-end crosstalk waveforms of the Jarvis model. (a) < T . (b) T .

and . The terminal con-

figuration is as shown in Fig. 1 with

. The results of the Jarvis model in (13)

are compared to the exact results computed with a SPICE sub-

circuit model generated by SPICEMTL.FOR described in [1].

Fig. 5 shows the near-end and far-end results for this matched

line. The Jarvis model provides reasonable prediction accuracy

although there is some residual ringing indicating that the lines

are not truly matched. In [1] it was shown that it is impossible to

completely match a multiconductor line using only single resis-

tors between the end of each line and the reference conductor. In

order to match a multiconductor line, it is required to terminate

the near and far ends with a resistive network wherein there are

not only resistors between each line and the reference conductor

but also between each line.

In order to investigate the adequacy of the weak-coupling as-

sumption, we compare the predictions of the Jarvis model to the

exact results for an edge-to-edge separation of 10 mils. For this


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(a)

(b)

Fig. 5. The crosstalk waveforms predicted by the Jarvis model compared to the exact results for 50-mil separation between the two circuits and R = R =R = R = Z . (a) Near-end crosstalk. (b) Far-end crosstalk.

separation, we compute the per-unit-length parameters as

H

mH

m pF

mpF

m

Using these parameters, we calculate and

. The total line length is again 10 inches

or 0.254 m. From the above stated per-unit-length pa-

rameters we compute ,

ns, and and . The

terminal configuration is again shown in Fig. 1 with

. The

results of the Jarvis model in (13) are compared to the exact

results computed with a SPICE subcircuit model generated by

SPICEMTL.FOR described in [1]. Fig. 6 shows the near-end

and far-end results for this matched line. Although the general

waveshapes are similar, there is inaccuracy in amplitude and in

time phase. Although , it is evident that for

this close 10-mil separation, the weak-coupling assumption is

not adequate.

IV. SOLUTION OF THE TRANSMISSION-LINE EQUATIONS

UNDER THE WEAK-COUPLING ASSUMPTION

We now present the solution of the transmission-line equa-

tions under the weak-coupling assumption. Essentially we want

to obtain the solution to (2) and (5) with the terminations given

in (3) and (4) incorporated into that solution. We will not make

the final two assumptions of the Jarvis model, i.e., in this solu-

tion, the lines need not be symmetrical or matched.
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(a)

(b)

Fig. 6. The crosstalk waveforms predicted by the Jarvis model compared to the exact results for 10-mil separation between the two circuits and R = R =R = R = Z . (a) Near-end crosstalk. (b) Far-end crosstalk.

A. The Frequency-Domain Solution

Observe that(5) are the transmission-line equationsof thegen-

eratorlineintheabsenceofthereceptorline(thisistheessenceof

the meaning of weaklycoupled). The solution of these equations

can be facilitated by taking the Laplace transform of both sides

where denotes the Laplace transform variable

(15a)

(15b)

The exact solution for this (essentially two-conductor) line with

the terminal conditions incorporated is [1], [2]

(16a)

(16b)

where the characteristic impedance, velocity of propagation,

source and load reflection coefficients, and one-way delay ofthe generator line are defined as

(17a)

(17b)

(17c)

(17d)

(17e)


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Now, we turn to the solution of the receptor circuit equations

given in (2). Laplace transforming these give

(18a)

(18b)

The solutions forthe generator line voltage and current aregiven

in (16) and hence the right-hand sides of (18) are known and act

like sources much in the same fashion as an incident electroma-

gentic wave excites a two-conductor line. The solution can be

obtained by writing (18) in the form of state-variable equations

as [1]

(19)

The solution to these types of equations is well known to be [ 1]

(20)

where (21), shown at the bottom of the page, holds and the char-

acteristic impedance and velocity of propagation of the receptor

circuit are

(22a)

(22b)

Substituting (16) and (21) into (20) and carrying out the inte-

gration yields

(23a)

(23b)

where the one-way time delay of the receptor circuit is

(24a)

and the constants and are

(24b)

(24c)

(24d)

Incorporating the terminal conditions given in (4)

(25a)

(25b)

gives the final form of the solution as

(26a)

(26b)

where

(27a)

(27b)

(27c)

(27d)

and the receptor circuit reflection coefficients are given by

(27e)

(27f)

(21)


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B. The Time-Domain Solution

The solution of (26) in thetime domain relieson theimportant

inverse Laplace transform property

(28)

In other words, multiplying a Laplace transformed function by

results in simply delaying that function by . Before weproceed, there are two terms in the denominators of (26) which

are troublesome. These can be expanded into infinite series as

(29a)

(29b)

For a moderately mismatched line, only the first few terms of

these expansions are significant since the reflection coefficients

are less than or equal to unity. In order to illustrate this, consider

a line having terminal impedances that are twice the character-

istic impedance of the appropriate circuit. Hence, the reflection

coefficients are . Their

products are and . Consider

a source voltage , that is in the form of a periodic trape-zoidal pulse train having equal rise and fall times, . The

primary spectral content of this signal is contained below ap-

proximately [1], [2]. For this form of the signal,

we would like to have the expressions in (29) valid for all fre-

quencies up to this maximum. Substituting

into (29) gives, for example

(30a)

In the case where the rise/fall time is equal to the one-way delay,

, (30a) becomes

(30b)

and . The exact result is . Using

the first two terms gives , using the first three

terms gives, , and using the first four terms gives

. Hence, for this moderately mismatched line,

we can truncate the series after four terms and obtain an ade-

quate approximation.

The solutions in the frequency domain given by (26) can be

placed in the time domain using the fundamental result in (28).

Although these are complicated, the scheme to solve them is to

sequentially invert the components of (26) using (28). We first

invert the terms

(31a)

(31b)

using the operator in (28) that simply scales and time shifts

. Once this is done, again time shift and scale this to ob-

tain, according to (29), to yield

(32a)

(32b)

so that

(33a)

(33b)

To obtain this waveform, we simply scale and time shift the

waveforms in (31) and add the component waveforms in time.

Again, for a moderately mismatched generator line only the firstfew terms of (33) are significant. Finally, we apply

(34a)

(34b)

to yield the resulting crosstalk waveforms

(35a)

(35b)

In the case of a completely matched line,

and , all four reflection coefficients are

zero, , and the result in (26)

reduces to

(36a)

(36b)


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(a)

(b)

Fig. 7. The crosstalk waveforms predicted by the results of this paper compared to the exact results for a 50-mil separation between the two circuits and R =R = Z and R = R = 1 (open circuit). (a) Near-end crosstalk. (b) Far-end crosstalk.

with the time-domain result

(37a)

(37b)

In the case of a symmetric line, , which is also

matched, (36) reduces to

(38a)

(38b)

and we have used the following result in (36b):

(39)

In the time domain, (38) yields

(40a)

(40b)

For the same conditions as the Jarvis model (matched and sym-

metrical line) the exact result in (40) is identical to the Jarvis

model given in (13) with the forward and backward coefficients

given in (14).

V. AN IMPORTANT SPECIAL CASE

In t he c ase o f a s ymmetric l ine w herein and

, both one-way delays are the same, . This

is a frequent situation on PCBs wherein lands on one level are

above an innerplane such as is shown in Fig. 3(a). Secondly, de-

signers frequently use series matching to make .

Hence, a useful special case is where (1) the line is symmetrical


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(a)

(b)

Fig. 8. The crosstalk waveforms predicted by the results of this paper compared to the exact results for 10 mils separation between the two circuits and R =R = Z and R = R = 1 (open circuit). (a) Near-end crosstalk. (b) Far-end crosstalk.

and (2) each line is matched at at least one end. When each line

ismatchedat atleastoneend, either or are zeroandeither

or are zero. In this case, the terms resulting in infinite

series expansions in (29) are zero so that ,

thereby simplifying the general result in (26).

We will examine a case which meets these conditions. Theline is symmetric. The terminal conditions are

, and

so that and . The model in (26)

simplifies, in the time domain, to

(41a)

(41b)

The predictions are compared to the exact results for a 50-mil

separation in Fig. 7 and for a 10-mil separation in Fig. 8. The

predictions for the 50-mil separation in Fig. 7 are quite accurate,

whereas the predictions for the 10-mil separation in Fig. 8 again

show inaccuracy in both amplitude and time phase. This again

shows that the weak-coupling assumption is not adequate forthe 10-mil separation.

It is worth noting that if the one-way time delay for the 10-mil

separation is changed from its logically correct value of

ns to ns, the results of (41) match the exact

results very closely in time phase. Hence, it appears that one

consequence of strong coupling between the lines is to reduce

the net one-way time delay from its value with the lines isolated

from each other. However, reducing the one-way delay used in

(41) from its logically correct value is somewhat arbitrary and

cannot be relied on for other cases.

These final results show that the Jarvis model (which

assumes a symmetric line and all terminations completely


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matched) cannot be used for a line that does not satisfy the

restrictions of that model since the waveforms for the Jarvis

model in Figs. 5 and 6 do not resemble those of Figs. 7 and

8. The Jarvis model in (13) is frequently mistakenly used for

predictions wherein all terminations are not matched. This

results in grossly erroneous predictions with that model.

VI. SUMMARY AND CONCLUSIONS

The transmission-line equations were solved in symbolic

form for a three-conductor line under the weak-coupling as-

sumption. The results are applicable to inhomogeneous media

as in a PCB. The symbolic solution illustrates the factors af-

fecting the crosstalk, whereas a numerical solution does not. It

was found that so long as the separations between the generator

and receptor lines was sufficient to support the weak-coupling

assumption, the predictions of the model are very accurate.

Although the results of this paper were derived for the case of

a three-conductor line, they should also be applicable, under the

weak-coupling assumption, for -conductor lines where

. This assumes that one circuit (the generator circuit here)is driven by a source (the source of the crosstalk) and the other

lines are passive andalso weaklycoupled to the generator circuit

and each other.

REFERENCES

[1] C.R. Paul,Analysis of Multiconductor Transmission Lines. New York:Wiley Interscience, 1994.

[2] , Introduction to Electromagnetic Compatibility. New York:Wiley Interscience, 1992.

[3] MECL System Design Handbook, 4th ed., Motorola SemiconductorProducts, Phoenix, AZ, 1988.

[4] A. Feller, H. R. Kaupp, and J. J. Digiacome, Crosstalk and reflectionsin high-speed digital systems, in Proc. Fall Joint Comp.Conf., 1965,pp. 511525.

[5] J. A. DeFalco, Predicting crosstalk in digital systems, Comp. Des., no.6, pp. 6975, 1973.

[6] S. Rosenstark, Transmission Lines in Computer Engineering, NY: Mc-Graw-Hill, 1994.

[7] D. B. Jarvis, The effects of interconnections on high-speed logic cir-cuits, IEEE Trans. Electron. Comput., vol. EC-12, pp. 476487, Oct.1963.

C layt on R . Paul (S61M70SM79F87)was born in Macon, GA, on September 6, 1941.He received the B.S. degree, from The Citadel,Charleston, SC, in 1963, the M.S. degree, fromGeorgia Institute of Technology, Atlanta, in 1964,and the Ph.D. degree, from Purdue University,Lafayette, IN, in 1970, all in electrical engineering.

He is Emeritus Professor of electrical engineeringat the University of Kentucky, Lexington, where hewas a member of the faculty in the Department ofElectrical Engineering for 27 years. He is currently

the Sam Nunn Eminent Professor of AerospaceSystems Engineering and Professor of Electrical and Computer Engineering inthe Department of Electrical and Computer Engineering at Mercer University,Macon, GA. He is the author of 12 textbooks on electrical engineering subjects,and has published numerous technical papers, the majority of which are in hisprimary research area of electromagnetic compatibility (EMC) of electronicsystems. From 1970 to 1984, he conducted extensive research for the UnitedStates Air Force in modeling crosstalk in multiconductor transmission linesand printed circuit boards. From 1984 to 1990, he served as a consultant to theIBM corporation, in the area of product EMC design.

Dr. Paul is a life member of the IEEE EMC Society, and a member of TauBeta Pi and Eta Kappa Nu.

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