a brief history of work in transmission lines for emc applications

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 49, NO. 2, MAY 2007 237

A Brief History of Work in TransmissionLines for EMC Applications

Clayton R. Paul, Life Fellow, IEEE

(Invited Paper)

Abstract—A brief chronology of the application of transmissionline theory to electromagnetic compatibility (EMC) applications ispresented. Transmission line studies in EMC began in the 1950sand 1960s with the frequency-domain analysis of crosstalk in ca-bles. Nuclear electromagnetic pulse (EMP) concerns in the 1970scaused an increasing emphasis on the study of incident field exci-tation of the lines. The advent of digital technology in the 1980smoved the research emphasis toward the analysis of the transmis-sion lines in the time domain. Early work concentrated on losslesslines whose solutions are very simple. After the 1980s, the impactof high-speed digital technology has driven much of the researchtoward the study of ways to incorporate line losses (particularly,frequency-dependent losses as with skin effect) into the solutions.In addition, the increasingly complex digital systems have resultedin the study of how to optimize the representation and solutionof large interconnected networks of transmission lines. This paperattempts to put the historical evolution of the study of transmissionlines in EMC in a chronological perspective.

Index Terms—Complex frequency hopping, crosstalk, guidedwaves, method of characteristics, moments, multiconductor trans-mission lines, Pade’ method, Prony’s method, recursive convolu-tion, skin effect, transverse electromagnetic (TEM) mode, vectorfitting.

I. INTRODUCTION

THE USE of parallel conductors to guide signals from onepoint to another (a transmission line) has a long history dat-

ing back to the early days of the telegraph. To begin our discus-sion, it is appropriate to review the transmission line equationsfor a multiconductor line consisting of n+ 1 conductors of to-tal length L. The general case of a multiconductor transmissionline (MTL) is shown in Fig. 1. The n + 1 conductors are parallelto each other and the z-axis. Terminal networks at the source(at z = 0) and at the load (at z = L) are illustrated as gener-alized Thevenin equivalent representations that contain lumpedexcitation sources. An incident field, perhaps from a lightningstrike or a radar pulse, is also included as a possible excita-tion source. The MTL equations are obtained by assuming thetransverse electromagnetic (TEM) mode of propagation, wherethe electric and magnetic field vectors lie in the transverse x–yplane that is perpendicular to the line’s z-axis. In other words,there are no components of the electric and magnetic fields thatare directed along the line’s z-axis. Under this assumption of aTEM field structure, the line voltages can be uniquely definedin the same fashion as for dc fields, i.e., as the line integral in the

Manuscript received December 17, 2006.C. R. Paul is with the Department of Electrical and Computer Engineering,

Mercer University, Macon, GA 31207 USA (e-mail: [email protected];[email protected]).

Digital Object Identifier 10.1109/TEMC.2007.897162

Fig. 1. MTL with terminations and an incident field.

transverse plane of the transverse electric field intensity vectorEt from one conductor to the other [1], [2] as

Vi(z, t) = −∫

ci (x,y)

Et(x, y, z, t) · dl (1)

where ci(x, y) is a contour or path in the transverse x–y planefrom the reference or zeroth conductor to the ith conductor. Theline currents can be similarly defined as for dc fields, i.e., as theline integral of the transverse magnetic field intensity vectorHt in the transverse plane around each conductor [1], [2]

Ii(z, t) =∮

c′i(x,y)

Ht(x, y, z, t) · dl ′ (2)

where c′i(x, y) is a closed contour in the transverse x–y plane

encircling the i th conductor. Other higher order non-TEM modefield structures can exist if the line cross section is electricallylarge, i.e., a significant portion of a wavelength [1]–[5]. Hence,the following transmission-line model requires the existence ofonly the TEM mode, and therefore, the cross-sectional dimen-sions of the line such as wire separations must be electricallysmall, i.e., much less than a wavelength, λ = v/f [1], [2].

Under this assumption of a TEM field structure, a ∆z lengthof the line is characterized by the per-unit-length equivalentcircuit shown in Fig. 2. The MTL equations become [1], [2]

∂

∂zV (z, t) + RI(z, t) + L

∂

∂tI(z, t) = V F (z, t) (3a)

∂

∂zI(z, t) + GV (z, t) + C

∂

∂tV (z, t) = IF (z, t). (3b)

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238 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 49, NO. 2, MAY 2007

Fig. 2. Per-unit-length equivalent circuit.

The n × 1 vectors V and I contain the n line voltages as[V (z, t)]i = Vi(z, t) and the n line currents as [I(z, t)]i =Ii(z, t), where the entry in the ith row of a vector V is de-noted as [V ]i, and t denotes time. The n line voltages are withrespect to the (arbitrarily chosen) reference conductor that isdenoted as the zeroth conductor. The entries in the per-unit-length resistance matrix R, per-unit-length conductance matrixG, per-unit-length inductance matrix L, and per-unit-length ca-pacitance matrix C are given by [1], [2]

R =

(r1 + r0) r0 · · · r0

r0 (r2 + r0) · · · r0

......

. . ....

r0 r0 · · · (rn + r0)

Ω

m(4a)

G =

∑n

k=1 g1k −g12 · · · −g1n

−g12

∑nk=1 g2k · · · −g2n

......

. . ....

−g1n −g2n · · ·∑n

k=1 gnk

S

m(4b)

L =

l11 l12 · · · l1n

l12 l22 · · · l2n...

.... . .

...l1n l2n · · · lnn

H

m(4c)

C =

∑n

k=1 c1k −c12 · · · −c1n

−c12

∑nk=1 c2k · · · −c2n

......

. . ....

−c1n −c2n · · ·∑n

k=1 cnk

F

m. (4d)

The per-unit-length resistance matrix in (4a) has a nice formbut is restricted to reference conductors that are finite-sizedconductors such as wires. In the case of a reference conductorbeing a large ground plane, each current returning to the groundplane will be concentrated beneath the “going down” conductor.These currents spread out in the ground plane, and cause theoff-diagonal terms in the per-unit-length resistance matrix to beunequal. Under the assumption of a TEM field structure, it canbe shown that the entries in G, L, and C are obtained as a static

(dc) field solution in the 2-D transverse plane [1], [2]. It can alsobe shown that all four per-unit-length matrices R, G, L, and C,are symmetric and G, L, and C are positive definite [1], [2]. Theeffects of the incident field are contained in the n × 1 vectorsV F (z, t) and IF (z, t) which are given by [1], [2], [6]–[8]

V F (z, t) =

...

− ∂∂z

∫ i

0Einc

t · dl +

Eincz (ith)

−Eincz (reference)

...

(5a)

and

IF (z, t) = −G

...∫ i

0Einc

t · dl...

− C

∂

∂t

...∫ i

0Einc

t · dl...

(5b)

where∫ i

0Einc

t · dl denotes the line integral in the transverse x–yplane between the reference conductor and the ith conductor ofthe component of the incident electric field that is transverse tothe line, and Einc

z (ith) denotes the z-directed component of theincident electric field that is along the ith conductor.

The above MTL equations are said to be in the time domain.In other words, the line excitation and response may be in theform of any general time-domain waveform. Alternatively, wemay be interested in the sinusoidal steady-state or phasor so-lution where the excitation is a single-frequency sinusoid, andhas been applied for a sufficiently long time that the line is insteady state. Hence, the excitation is in the form of a sinusoidx(t) = Xcos(ωt + θX), and the line voltage or current solu-tions will also be in the form of a sinusoid at the same frequencyy(t) = Y cos(ωt + φY ). This is said to be the frequency-domainsolution. The MTL equations in the frequency domain are

d

dzV (z) + Z(ω) I(z) = V F (z) (6a)

d

dzI(z) + Y (ω) V (z) = IF (z) (6b)

where the n × n per-unit-length impedance and admittancematrices are

Z(ω) = R + jωL (7a)

Y (ω) = G + jωC. (7b)

Here, the carat over the quantity, e.g., V and I , denotes thecomplex-valued frequency-domain phasor quantity, and ω =2πf .

The level of difficulty in solving the MTL equations dependsstrongly on the type of MTL under consideration, or equiva-lently, on the assumptions about the line that one makes. Thethree primary classifications of the line are: 1) a homogeneous orinhomogeneous medium surrounding the conductors; 2) a uni-form or a nonuniform line; and 3) a lossless or a lossy line. Thedielectric medium surrounding the conductors is characterizedby a permittivity ε = εrε0, where εr is the relative permittivity

PAUL: A BRIEF HISTORY OF WORK IN TRANSMISSION LINES FOR EMC APPLICATIONS 239

or dielectric constant and ε0∼= (1/36π) × 10−9 F/m is the per-

mittivity of free space, and a permeability, i.e., the permeabil-ity of free space µ = µ0 = 4π × 10−7 H/m. A homogeneousmedium is one in which the permittivity at any xy cross sectionis independent of x and y. Dielectric insulations surroundingwires are an example of an inhomogeneous medium, since theelectric fields lie partly in the dielectric insulations and partlyin the air that occupies the remainder of the space around theconductors. A printed circuit board (PCB) is also an exam-ple of an inhomogeneous medium, since the electric fields liepartly in the glass-epoxy board and partly in the air that oc-cupies the remainder of the space around the conductors. Anyinhomogeneity of the surrounding dielectric medium is con-tained in the per-unit-length parameters of G and C. The idealTEM mode of propagation cannot exist in an inhomogeneousmedium. A homogeneous medium supports a TEM wave withonly one velocity of propagation v = 1/

√εµ m/s. However,

the field structure for lines with dielectric insulations or forPCBs is approximately TEM in structure, which is referred toas the quasi-TEM mode assumption. Inhomogeneous surround-ing media cause multiple velocities of the modes. This causesdispersion resulting in distortion of pulses traveling on the line.The entries in G result from a surrounding medium that has anonzero conductivity. Losses in dielectric media are character-ized by a complex permittivity ε = ε′ − jε′′. Various handbookstabulate the effective conductivity in terms of the loss tangenttan δ of the medium at various frequencies as [1], [2], [9]

σeff =ωε tan δ. (8)

In the case of a homogeneous medium, the conductance matrixG can be obtained in terms of the capacitance matrix as [1], [2]

G =σeff

εC. (9)

In addition, for a homogeneous medium the inductance andcapacitance matrices are related as [1], [2]

LC = CL = µε1n (10a)

where 1n is the n × n identity matrix with ones on the maindiagonal and zeros elsewhere. The surrounding medium is as-sumed not to be magnetic so that

LC0 = µ0ε01n (10b)

where C0 is the per-unit-length capacitance matrix with thesurrounding dielectric removed and replaced with free space.

A uniform line is one in which the cross-sectional dimensionsof the conductors and any inhomogeneous surrounding mediumare invariant along the length of the line. In other words, the per-unit-length parameters of a nonuniform line are functions of z.The coefficients in the transmission-line equations for a nonuni-form line are, therefore, functions of the independent variable z.Those types of differential equations are very difficult to solve.It is possible to approximate nonuniform lines as a cascade ofuniform lines with different per-unit-length parameters [1], [2].Although there are some important uses of nonuniform lines inmicrowave circuits, it will be assumed in this paper that the linesunder consideration are uniform lines.

The entries in R occur as a result of imperfect line conduc-tors, i.e., conductors with finite nonzero conductivity σ. Imper-fect line conductors R = 0 invalidate the TEM field structureassumption. The reason is that there will be a component ofthe electric field directed in the z direction along the conductorsurfaces due to the voltage drop along their surfaces [1], [2].Nevertheless, it is assumed that any conductor losses will be“small,” and therefore, this perturbation of the fields from TEMstructure is small. Thus, lossy conductors are included in an ap-proximate manner, which is also referred to as the quasi-TEMassumption.

Solution of the MTL equations for a lossless line, i.e., forR = G = 0 is a trivial process [1], [2], [10]–[12]. The majordifficulty in the solution of the MTL equations occurs for lossylines. Conductor losses at the higher frequencies are functionsof the square root of frequency

√f due to skin effect [1], [2].

Frequency-dependent losses are easily represented in the fre-quency domain. The frequency-domain MTL equations in (6)become [1], [2]

d

dzV (z) + Z int(f)I(z) + jωLI(z) = V F (z) (11a)

d

dzI(z) + G(f)V (z) + jωCV (z) = IF (z). (11a)

At low frequencies, the currents are distributed uniformly acrossthe conductor cross sections leading to a dc resistance Rdc andinternal inductance Lint,dc, where Lint,dc denotes the induc-tance of the conductors due to magnetic flux internal to theconductors. As frequency increases, the currents crowd to theoutside of the conductors, and are eventually located primarilyin a thickness on the order of a skin depth

δ =1

√πµσ

1√f

. (12)

Hence, at higher frequencies, the resistance increases at a rateof

√f . In addition, the magnetic flux internal to the conduc-

tors creates an internal inductance that decreases as√

f . Thesecontribute to an internal conductor impedance of

Z int(f) = R(f) + jωLint(f). (13)

Losses in a homogeneous medium tend to increase linearlywith frequency as shown by (8). However, various dipole mo-ments cause resonances [13]. This contributes to a frequency-dependent conductance matrix G(f). The matrices L and Ctend to be relatively frequency independent at frequencies be-low the low gigahertz range.

Solution of the phasor MTL equations in (11), which in-clude frequency-dependent losses in Z int(f) and/or G(f), isvery simple. Compute the values of Z int(f) and G(f) at eachfrequency, solve the equations, recompute these at the next fre-quency, re-solve the equations at that frequency, and so on.In the time domain, solution of these MTL equations whenthey include frequency-dependent losses in Z int(f) and G(f)becomes very problematic for the following reasons. In thetime domain, the frequency-domain MTL equations in (11) are


modified as∂

∂zV (z, t)+Z int(t) ∗ I(z, t)+ L

∂

∂tI(z, t) = V F (z, t) (14a)

∂

∂zI(z, t) + G(t) ∗ V (z, t) + C

∂

∂tV (z, t) = IF (z, t) (14b)

where ∗ denotes convolution as

h(t) ∗ x(t) =∫ t

0

h(t − τ)x(τ)dτ (15)

and the Fourier (Laplace) transforms relating a frequency-domain function and its time-domain equivalent are

Z int(t) ⇔ Z int(f) = R(f) + jwLint(f) (16a)

G(t) ⇔ G(f). (16b)

Hence, losses require the time-consuming computation of con-volutions in the time domain. If the losses are frequency inde-pendent, i.e., dc, then no convolutions are required in the timedomain but the solution of the time-domain MTL equations in(3) is still very difficult.

This brief discussion provides the background for understand-ing where the work in MTLs has been concentrated over theyears. It will also facilitate our understanding of the new prob-lems created by technology developments.

II. 1950S–1960S

Most of the early work in analyzing transmission lines con-cerned high-voltage power transmission lines [14]–[35]. Muchof this work was focused on solving the frequency-domain MTLequations without incident field excitation

d

dzV (z) + ZI(z) = 0 (17a)

d

dzI(z) + Y V (z) = 0. (17b)

In the case of high-voltage power transmission lines, the n × nper-unit-length impedance and admittance matrices are givenby

Z = (Rcond(f) + Rgnd(f) + jωL) (18a)

Y = jωC. (18b)

For power transmission lines, the effect of a lossy ground (earth)was investigated by modeling the earth loss with the Sommer-feld integral and Rgnd = rgnd(f)Un, where Un is the unitmatrix with ones in every position [31], [32]. The losses in theline conductors (all of which are identical) are included in theterm Rcond = rcond(f)1n. In all these solutions, the numberof conductors is n = 3. The MTL equations are neverthelesscoupled and must be solved simultaneously. This is one of theprimary problems that distinguishes MTLs from the more fa-miliar two-conductor lines [36]–[38]

The second-order phasor MTL equations can be obtained bydifferentiating (17a) with respect to z and substituting (17b),and vice-versa to give

d2

dz2V = ZY V (19a)

d2

dz2I = Y ZI. (19b)

The matrices Z and Y generally do not commute, so their orderof multiplication must be strictly observed.

The primary method of solving the frequency-domain MTLequations is to decouple them by using a change of vari-ables [39]. For example, define a change of variables to modequantities as

V = T V V m (20a)

I = T I Im (20b)

where T V and T I are n × n and the n × 1 vectors V m andIm are said to be the mode voltages and currents, respectively.Substituting (20) into (19) yields

d2

dz2V m = T

−1

V ZY T V V m (21a)

d2

dz2Im = T

−1

I Y ZT I Im. (21b)

If we can find a T V or a T I such that

T−1

V ZY T V = T−1

I Y ZT I = γ2 (22)

where γ2 is an n × n diagonal matrix with γ2i on the main diag-

onal and zeros elsewhere, the frequency-domain MTL equationsin terms of the mode voltages and currents in (21) are uncoupledas

d2

dz2Vm1 = γ2

1 Vm1

d2

dz2Im1 = γ2

1 Im1

...

d2

dz2Vmn = γ2

nVmn

d2

dz2Imn = γ2

nImn. (23)

These are in the form of n two-conductor lines that do notinteract. The solutions are then familiar [36]–[38]

Vm1(z) = V +m1e

−γ1z + V −m1e

γ1z

Im1(z) = I+m1e

−γ1z − I−m1eγ1z

...

Vmn(z) = V +mne−γn z + V −

mneγn z

Imn(z) = I+mne−γn z − I−mneγn z (24)

where the V +mi, V

−mi, I

+mi, I

−mi are undetermined constants. The

superscript of + in V +mi, I

+mi denote forward-traveling waves,

i.e., waves traveling in the +z direction, whereas the superscriptof – in V −

mi, I−mi denote backward-traveling waves, i.e., waves

traveling in the −z direction [36]–[38]. Substituting these into


the mode transformations in (20) gives

V (z) = T V

(e−γ zV

+

m + eγ zV−m

)(25a)

I(z) = T I

(e−γ z I

+

m − eγ z I−m

)(25b)

where

e±γ z =

e±γ1 z 0 · · · 0

0. . . 0

...... 0

. . . 00 · · · 0 e±γn z

(26)

and [V±m] i = V ±

mi and [I±m]i = I±mi. The V ±

mi can be found interms of the I±mi by substituting (25b) into (17b) to yield

V (z) = −Y−1 d

dzI(z)

= Y−1

T I γ(e−γz I

+

m + eγz I−m

)

=Y

−1T I γT

−1

I︸︷︷︸ZC

T I(eγz I+

m + eγz I−m)

(27a)

I(z) = T I

(e−γ z I

+

m − eγ z I−m

). (27b)

The n × n characteristic impedance matrix is then defined as

ZC = Y−1

T I γ T−1

I . (28)

There are numerous other equivalent definitions of this char-acteristic impedance matrix [1], [2], [39], [40]. Notice that ingeneral, the mode transformations T V and T I will be functionsof frequency, since the entries in Z and Y are in general func-tions of frequency because the conductor and medium losses arefrequency dependent. Hence, they must be recomputed at eachnew frequency. The transformations T V and T I can be chosensuch that they are related as [39]

Tt

V T I = 1n. (29)

The solutions in (27) contain 2n unknowns: n in I+

m and n

in I−m. These can be determined by substituting the terminal

constraints. One way of characterizing the terminations is as ageneralized Thevenin equivalent, as shown in Fig. 1. Then, wehave

V (0) = V S − ZS I(0) (30a)

V (L) = V L + ZLI(L). (30b)

Substituting these into (27) yield 2n equations in 2n unknowns[1], [2][

(ZC + ZS)T I (ZC − ZS)T I

(ZC − ZL)T Ie−γ L (ZC + ZL)T Ieγ L

][I

+

m

I−m

]=[

V S

V L

].

(31)This general procedure can be used for any frequency-domainsolution of the MTL equations. There are also similar formu-lations for the terminations in terms of a generalized Nortonequivalent or a mixed representation [1], [2]. These alternative

representations are useful for extreme values of the terminationimpedances. For example, a short-circuit load impedance hasa value of zero, and hence, the generalized Thevenin equiva-lent representation would be appropriate. On the other hand, anopen-circuit load impedance has an impedance of infinite butan admittance of zero. Hence, a generalized Norton equivalentrepresentation would be appropriate.

The primary assumption in solving the MTL equations forpower transmission lines is that the lines are transposed. Powerlines frequently have their positions interchanged at regular in-tervals in order to avoid interference with nearby telephone lines.This means that the lines can be assumed to occupy every po-sition at some time, and hence, the impedance and admittancematrices have a special form [1], [2], [39], [40]

Z =

Zs Zm Zm

Zm Zs Zm

Zm Zm Zs

(32a)

Y =

Ys Ym Ym

Ym Ys Ym

Ym Ym Ys

. (32b)

The mode transformations that diagonalize these as in (22) are

T V = T I = T =1√3

1 1 1

1 1 120 1 − 120

1 1 − 120 1 120

(33a)

T−1

V = T−1

I = T∗=

1√3

1 1 1

1 1 − 120 1 120

1 1 120 1 − 120

(33b)

and T∗

is the complex conjugate of T . The propagation con-stants become

γ21 =

Zs + Zm + Zm

Ys + Ym + Ym

= (Zs + 2Zm)(Ys + 2Ym) (34a)

γ22 =

Zs 0o + Zm 120o + Zm − 120o

×

Ys 0o + Ym 120o + Ym − 120o

= (Zs − Zm)(Ys − Ym) (34b)

γ23 =

Zs 0o + Zm − 120o + Zm 120o

×

Ys 0o + Ym − 120 + Ym 120o

= (Zs − Zm)(Ys − Ym) (34c)

since 1 120 + 1 − 120 = −1. Two of the propagation con-stants are equal. The first, γ1, is called the “ground mode,” andγ2 and γ3 are called the “aerial modes.” This is often referredto in the power system literature as the method of symmetricalcomponents [18]–[20]. Observe that the mode transformationsare not only independent of the entries in Z and Y but are alsoindependent of frequency even though the entries in Z and Yare functions of frequency.


Another place where this type of symmetry guarantees a sim-ple choice for the mode transformation matrices is in the mi-crowave community. Many microwave devices consist of twolands on a substrate with a ground plane on the other side. Thelands are identical in cross section and at the same heights abovethe ground plane giving rise to per-unit-length impedance andadmittance matrices of the form

Z =[

Zs Zm

Zm Zs

](35a)

Y =[

Ys Ym

Ym Ys

]. (35b)

In this case of symmetry, the mode transformations simplify to

T V = T I = T =1√2

[1 11 −1

]. (36)

This is referred to in the microwave literature as the even-mode,odd-mode transformation [39]. Observe that these mode trans-formations are independent of not only the entries in Z and Ybut are also independent of frequency, and need to be computedonly once.

In the 1950s–1960s, the interest in the electromagnetic com-patibility (EMC) community was focused on the effects ofthese lines in generating electromagnetic interference in thefrequency domain as with power transmission lines. This inter-est in MTLs and their interference potential began to accelerateafter World War II. Electronic devices were fairly simple bytoday’s standards, and were interconnected by cables consist-ing of a relatively small number of wires (circular-cylindricalconductors with dielectric insulations). Digital devices had notbeen developed, and hence, the need for time-domain analysiswas minimal. The unintended coupling of the electromagneticfields surrounding the conductors, from one conductor to an-other, caused some crosstalk to occur at the line terminations.Generally, this was a simple matter to fix; wires were reroutedinto other bundles, were twisted together, or shielded. As theelectronics became more sophisticated, the numbers of wires inthe cables were increasing, which made the analysis of theseMTLs more difficult.

III. 1970S–1980S

In the 1970s, there arose considerable interest in “harden-ing” communication and control facilities to the effects of anelectromagnetic pulse (EMP) from a nuclear detonation. Thisled to considerable interest in solving the MTL equations fortheir response to incident fields [6]–[8], [40]–[51]. Generally,this concerned the frequency-domain solution to (6), where theincident field illumination was in the form of a uniform planewave [1], [2], [6]. The solution to these equations was facilitatedby recognizing that the frequency-domain MTL equations in (6)can be placed in the form of state-variable equations, which arethe focus of work in automatic control and general linear sys-tems [52], [53]. The frequency-domain MTL equations in (6)

can be put in matrix form as in state-variable equations as

d

dzX(z) = AX(z) + U(z) (37a)

where

X(z) =[

V (z)I(z)

](37b)

A =[

0 −Z−Y 0

](37c)

U =[

V F (z)IF (z)

]. (37d)

The solution to these equations is well known and is givenby [52], [53]

X(z) = Φ(z − z0)X(z0) +∫ z

z0

Φ(z − τ)[

V F (τ)IF (τ)

]dτ

(38)and the 2n × 2n state-transition matrix is [52], [53]

Φ(z) = eAz

= 1n +z

1!A +

z2

2!A

2+ · · · . (39)

The state-transition matrix relates the line voltages and cur-rents at one end of the line z to those at the other end z0. Inparticular, we can specialize these to the two endpoints of theline z = L and z0 = 0, giving the 2n × 2n chain parametermatrix Φ(L), sometimes referred to as the ABCD matrix, as[

V (L)I(L)

]= Φ(L)

[V (0)I(0)

]+∫ L

0

Φ(L − τ)[

V F (τ)IF (τ)

]dτ .

(40)A convenient form of the 2n × 2n chain parameter matrix interms of the mode transformation T I that diagonalizes the prod-

uct Y Z as T−1

I Y ZT I = γ2 is [1], [2], [39], [40]

Φ(L) =[Φ11(L) Φ12(L)Φ21(L) Φ22(L)

](41)

where the n × n submatrices are given by

Φ11(L) =12Y

−1T I

(eγ L + e−γ L

)T

−1

I Y (42a)

Φ12(L) = −12Y

−1T I γ

(eγ L − e−γ L

)T

−1

I (42b)

Φ21(L) = −12T I

(eγ L − e−γ L

)γ−1 T

−1

I Y (42c)

Φ22(L) =12T I

(eγ L + e−γ L

)T

−1

I . (42d)

Numerous other forms of these chain parameter submatricescan be obtained [1], [2], [39], [40]. Hence, the solution to thefrequency-domain MTL equations for incident field illumina-tion involves the convolution of the chain parameter matrix andthe vector of sources due to the incident field. Carrying out theconvolution in (40) gives[

V (L)I(L)

]= Φ(L)

[V (0)I(0)

]+[

V FT (L)IFT (L)

](43)


Fig. 3. Equivalent circuit for incident field illumination.

where [V FT (L)IFT (L)

]=∫ L

0

Φ (L − τ)[

V F (τ)IF (τ)

]dτ (44)

or equivalently

V FT (L)=∫ L

0

Φ11(L−τ)V F (τ)dτ +∫ L

0

Φ12(L−τ)IF (τ)dτ

(45a)

IFT (L)=∫ L

0

Φ21(L−τ)V F (τ)dτ +∫ L

0

Φ22(L−τ)IF (τ)dτ .

(45b)

This gives the equivalent circuit shown in Fig. 3. Numerouscomputer programs are available that give this solution for anincident field that is in the form of a uniform plane wave [1],[2], [50].

The terminal constraints can also be incorporated into this so-lution in a rather straightforward way. Rewriting (43) by movingthe incident field sources to the left-hand side gives[

V ′(L)I ′(L)

]=[

V (L) − V FT (L)I(L) − IFT (L)

]= Φ(L)

[V (0)I(0)

](46)

where V′(L) and I

′(L) are terminal voltages and currents at

the right end of the unexcited line as shown in Fig. 3. Hence, wemay modify the generalized Thevenin equivalent representationof the terminations without incident field illumination given in(30) by modifying (30b) as

[V′(L) + V FT (L)] = V L + ZL[I

′(L) + IFT (L)]. (47)

Hence, we only need to replace in (31)

V L → V L − V FT (L) + ZLIFT (L). (48)

Solving (31) again but with the replacement in (48) gives

the I±m which, when substituted into (27) and evaluated at

z = L, gives the solutions for V′(L) and I

′(L). The ac-

tual terminal voltages and currents can then be obtained fromV (L) = V

′(L) + V FT (L) and I(L) = I

′(L) + IFT (L).

The spectral content of signals carried by transmission lineswas increasing at a rapid rate in the 1970s, and interference inthe form of crosstalk was becoming more of a concern. This led

to a renewed interest in the use of shielded wires and twistedpairs of wires to reduce the crosstalk. In addition, ribbon cablesand flatpack cables were being used more frequently insteadof random cable harnesses to automate the wiring installation.Hence, a renewed interest in the frequency-domain solution ofthe MTL equations for these types of wiring configurationsdeveloped [54]–[63].

Toward the end of the 1970s, personal computers (PCs) werebeginning to appear. These were very slow compared to today’sstandards with clock speeds of the order of 10 MHz and digitalpulse rise-/falltimes of the order of 20 ns. Hence, the spectralcontent or bandwidth of these digital pulses extended into thelow megahertz range. A measure of the spectral content of atrapezoidal pulse train is the inverse of the pulse rise time [37].Hence, a digital clock or data pulse train having a repetitionfrequency of 10 MHz and a rise-/falltime of 20 ns has a band-width of around 50 MHz. In this frequency range, the skin-effectconductor losses and the losses in the medium were negligibleso that it could be assumed that R = G = 0. In addition, theemphasis for digital signals shifted from the frequency domainto the time domain. Hence, the analysis concentrated on thetime-domain solution of the lossless MTL equations

∂

∂zV (z, t) = −L

∂

∂tI(z, t) (49a)

∂

∂zI(z, t) = −C

∂

∂tV (z, t). (49b)

Again, the primary method of solution was to use a change ofvariables to the mode quantities as [1], [2], [10], [12], [37], [39]

V = T V V m (50a)

I = T IIm. (50b)

Substituting (50) into (49) yields

∂

∂zV m(z, t) = −T−1

V LT I∂

∂tIm(z, t) (51a)

∂

∂zIm(z, t) = −T−1

I CT V∂

∂tV (z, t). (51b)

If these n × n real mode transformations can be found such that

Lm = T−1V LT I

=

lm1 · · · 0

.... . .

...0 · · · lmn

(52a)

Cm = T−1I CT V

=

cm1 · · · 0

.... . .

...0 · · · cmn

(52b)

are diagonal, then the mode MTL equations in (51) areuncoupled and consist of n uncoupled two-conductor lineshaving characteristic impedances ZCi =

√lmi/cmi and veloc-

ities of propagation vmi = 1/√

lmicmi. The solutions of these


two-conductor lines in the time domain is well known [36], [38]

Vm1(z, t) = V +m1 (t − z/vm1) + V −

m1 (t + z/vm1)

Im1(z, t) =V +

m1 (t − z/vm1)ZC1

− V −m1 (t + z/vm1)

ZC1

...

Vmn(z, t) = V +mn (t − z/vmn) + V −

mn (t + z/vmn)

Imn(z, t) =V +

mn (t − z/vmn)ZCn

− V −mn (t + z/vmn)

ZCn. (53)

The functions V +mi(t − z/vmi) and V −

mi(t + z/vmi) are as yetunknown functions but will be determined by the terminationsat the ends of the line. They are, nevertheless, functions of t andz only as (t ± z/vmi).

The remaining task is to incorporate the terminal constraintsat the ends of the line z = 0 and z = L. A very effective meansof solving these time-domain equations and incorporating theterminal characteristics for a lossless line is referred to as themethod of characteristics [1], [2], [64]. The method of char-acteristics seeks to transform the partial differential equationsof a two-conductor line into ordinary differential equations. Todo that, we define the characteristic curves in the z, t planeas

dz

dt=

1√lc

(54a)

and

dz

dt= − 1√

lc. (54b)

The differential changes in the line voltage and current are

dV (z, t) =∂V(z, t)

∂zdz +

∂V(z, t)∂t

dt (55a)

dI(z, t) =∂I(z, t)

∂zdz +

∂I(z, t)∂t

dt. (55b)

Substituting the transmission-line equations into these gives

dV (z, t) =(−l

∂I(z, t)∂t

)dz +

∂V(z, t)∂t

dt (56a)

dI(z, t) =(−c

∂V(z, t)∂t

)dz +

∂I(z, t)∂t

dt. (56b)

Along the forward characteristic defined by (54a), dz =(1/

√lc)dt, these become

dV (z, t) =(−ZC

∂I(z, t)∂t

+∂V(z, t)

∂t

)dt (57a)

dI(z, t) =(− 1

ZC

∂V(z, t)∂t

+∂I(z, t)

∂t

)dt. (57b)

Similarly, along the backward characteristic defined by (54b),dz = −(1/

√lc)dt, these become

dV (z, t) =(

ZC∂I(z, t)

∂t+

∂V(z, t)∂t

)dt (58a)

dI(z, t) =(

1ZC

∂V (z, t)∂t

+∂I(z, t)

∂t

)dt. (58b)

Multiplying (57b) by ZC and adding to (57a), and similarly,multiplying (58b) by ZC and subtracting from (58a), gives

dV (z, t) + ZCdI(z, t) = 0 (59a)

dV (z, t) − ZCdI(z, t) = 0. (59b)

Equation (59a) holds along the forward characteristic with thephase velocity v = 1/

√lc, whereas (59b) holds along the back-

ward characteristic with the phase velocity v = −1/√

lc. Theseare directly integrable to yield the important result

[V (L, t) − V (0, t − TD)] = −ZC [I(L, t) − I(0, t − TD)]

(60a)

[V (0, t) − V (L, t − TD)] = +ZC [I(0, t) − I(L, t − TD)]

(60b)

where the line one-way time delay is

TD =L

v. (61)

Equations (60) can be written in a very useful form as

V (0, t) = ZCI(0, t) + [V (L, t − TD) − ZCI(L, t − TD)]

(62a)

V (L, t) = −ZCI(L, t) + [V (0, t − TD) + ZCI(0, t − TD)].

(62b)

Equations (62) are an extraordinary result. They show that onecan compute the terminal voltage and current at one end of theline at some time in terms of the terminal voltage and current atthe other end of the line one time-delay prior to this time. TheSPICE (or PSPICE) electric circuit analysis program containsthis exact model of a two-conductor lossless line [1], [2], [37],[38], [64], [65]. This is implemented in SPICE with time-delaycontrolled sources as shown in Fig. 4. The model can be calledwith the statement.

TXXX N1 N2 N3 N4 Z0 = ZC TD = TD = L/v.

Hence, the time-domain solution of lossless transmission lines isa trivial task. Use of the SPICE model for solving transmissionlines allows the solution for very complicated and nonlinearterminations since SPICE already contains models of nonlinearelements such as diodes and transistors that can be called up.The method of characteristics can be extended to multiconductorlines, and losses can be included in that solution.

The terminations must be included in the decoupled resultsin (53). There are numerous ways of doing this [1], [2]. How-ever, the simplest and most useful way is to develop a sub-circuit model for use in the SPICE (PSPICE) circuit analysisprogram [1], [2], [37], [66], [67]. The basic idea is to use thetwo-conductor model in SPICE to model the two-conductormode lines in (53). Then, the various two-conductor solutionsin (53) are recombined by using controlled voltage and cur-rent sources to implement the linear combinations of the mode


Fig. 4. Exact SPICE model for a lossless two-conductor line.

Fig. 5. Implementing the mode transformations V = TV V m and I =T I Im with controlled sources.

variables using the mode transformations for T V and T I in(50) [1], [2], [37], [67]. This scheme is shown in Fig. 5. Expandthe mode transformations in (50) as

Ei = [T V ] i1Vmi + · · · + [T V ] inVmn (63a)

Fi = [T−1I ]i1I1 + · · · + [T−1

I ]inIn. (63b)

Hence, we sum the mode voltages using voltage-controlled volt-age sources and the currents using current-controlled currentsources. A FORTRAN computer program that produces thisSPICE subcircuit model is described in [1] and [2]. Hence, thesolution of the MTL equations and incorporation of the termi-nal conditions for a lossless MTL is a trivial task . There areno remaining problems in the solution of the MTL equationsand the incorporation of the terminal constraints for a losslessMTL.

IV. 1990S–2000S

The advent of the 1990s saw digital processing speeds ac-celerating at an astonishing rate. In the late 1990s, PCs wereavailable with clock speeds above 100 MHz and rise-/falltimesof the order of nanoseconds. Hence, the spectral content of thosesignals was approaching the gigahertz range. At present, the

digital signals in PCs and many high-end servers have spectralcontent approaching 20 GHz. The conductor losses that increaseas

√f are no longer negligible. In many cases, the losses in the

dielectric PCB boards are also becoming significant. Thus, thesolution of the MTL equations for lossless lines is no longersufficient. In addition, high-density interconnects are becominga standard requirement in the digital industry in order to fit thetechnology into a standard PCB. The number of interconnectsin a typical system is increasing at an astronomical rate, andthe traditional use of SPICE for modeling these interconnectsis rapidly becoming inefficient. The analysis methods are beingconcentrated in the reduction of the complexity and size of theinterconnect simulation models through the use of macromodelsof those dense interconnect lines.

First, we will discuss the incorporation of frequency-dependent losses of the conductors into the time-domain so-lution of the MTL equations. One of the major problems pre-sented by incorporating losses is that the conductor losses at thehigh frequencies that are present in today’s high-speed digitalsignals depend on frequency as the square root of frequency,i.e.,

√f . The time-domain form of the MTL equations with

frequency-dependent losses requires: 1) the time-domain formof the frequency-domain conductor impedances are obtained asin (16a); and 2) a convolution be performed as in (14a). Hence,we are required to obtain the inverse Laplace (or Fourier) trans-forms of a

√f frequency dependence. There are few such inverse

transforms. One that has extensive use is [68], [69]

1√s⇔ 1√

π

1√t

(64)

where s is the Laplace transform variable. In order to utilizethis result, we write the frequency-domain representation of theconductor losses in terms of the square root of jω as√

jω =√

ω√2

(1 + j)

=√

π√

f(1 + j). (65)

We can then substitute jω ⇔ s, and can utilize the inverse trans-form in (64). A particularly useful representation is [70]

A + B√

s = A + B1√ss ⇔ A +

B√π

1√t

∂

∂t. (66)

Hence, the required convolution in (14a) is

Zint(t) ∗ I(z, t) = AI(z, t) +B√π

∫ t

0

1√τ

∂I(t − τ)∂(t − τ)

dτ.

(67)We can write the internal impedance of the conductors in anapproximate asymptotic form as [1], [2]

Zint(f) = rdc + rdc

√f

f0+ jωlint,hf(f)

= rdc + rdc

√ω√ω0

(1 + j) (68)

and we identify A = rdc and B = rdc1√

π√

f0. The frequency

f0 is the frequency where the resistance transitions from its dc


value to its√

f dependence. For isolated wires, this is where thewire radius is equal to two skin depths, i.e., rw = 2δ [1], [2].For wires, the high-frequency resistance and the high-frequencyinternal inductive reactance are equal, i.e., rhf = jωlint,hf . ForPCB lands, these are not equal but are approximately so for landswith high aspect ratios [1], [2]. Other representations that yieldan inverse Laplace transform are given in [71]. The numericalinversion of the Laplace transform (NILT) is also used to obtainthe time-domain result [72], [73].

The next problem we encounter is the performance of theconvolution integral. In simulators, the differential equations ofthe line are discretized in time into increments ∆t. A popularmethod of solution is the finite-difference time-domain (FDTD)method [1], [2], [74], [75]. Divide the time axis into ∆t seg-ments. The convolution in (67) can be approximated in thefollowing manner, where F (t) is the time derivative of the linecurrent, i.e., F (t) = ∂

∂tI(z, t), which may be approximated asbeing constant over the ∆t segments [1], [2]

∫ t

0

1√τ

F (t − τ)dτ ∼=∫ (n+1)∆t

0

1√τ

F ((n + 1)∆t − τ)dτ

∼=n∑

m=0

Fn+1−m

∫ (m+1)∆t

m∆t

1√τ

dτ

=√

∆t

n∑m=0

Fn+1−mZ0(m) (69a)

where

Z0(m) =∫ (m+1)

m

1√ςdς

= 2[√

m + 1 −√

m]. (69b)

We have made a change of variables in the integral in (69a) ofτ = ∆tς in order to make this integral independent of the timediscretization chosen for a specific problem.

Unfortunately, the convolution in (69) requires storage of allpast values of the currents, which presents a significant demandon storage requirements. A solution to this problem approxi-mates Z0(m) as the sum of N exponential functions via Prony’smethod [75]

Z0(m) =∫ (m+1)

m

1√ςdς

∼=N∑

j=1

ajembj . (70)

Prony’s method approximates some general function ofm, f(m) at 2N equally spaced sample points m = 0, T, 2T,. . . , (2N − 1)T over the complete interval of interest, in termsof the sampling interval T [76], [77]. These are denoted as

f(m = iT ) ≡ fi, for i = 0, 1, 2, . . . , 2N − 1 (71a)

and the first sample is at m = 0. Hence, the 2N equally spacedsamples are denoted as

f0 ≡ f(0)

f1 ≡ f(t)

f2 ≡ f(2T )

...

f2N−1 ≡ f((2N − 1)T ). (71b)

Prony’s method represents the function at the 2N sample pointsas a sum of exponentials as

fi∼=

N∑j=1

ajeibj T

= a1eib1T + a2e

ib2T + · · · + aNeibN T

= a1zi1 + a2z

i2 + · · · + aNzi

N (72a)

where we denote

zj = ebj T , for j = 1, . . . , N. (72b)

Hence, we can write the Prony approximation as

fi∼=

N∑j=1

ajzij , for i = 0, 1, 2, . . . , 2N − 1. (73)

If we determine the zj , we can recover the desired bj from (72b)as

bj =1T

ln(zj). (74)

Prony’s method provides an algorithm for determining the 2Nunknowns in the expansion, a1, a2, . . . , aN and b1, b2, . . . , bN

by matching (72) at the 2N equally spaced sample points [76],[77]. An alternative method of determining the coefficients inthe exponential expansion is the matrix pencil method [78].

Next, we turn to the reason for representing the function in theconvolution as a series of exponentials. The method of recursiveconvolution cleverly avoids the requirement to store all the pastvalues of the line currents [79], [80]. It relies on the property ofthe exponential e(a+b) = eaeb. For example, suppose we wishto represent a function of t at discrete points ti that are sep-arated by ∆t = ti − ti−1. The convolution y(t) = h(t) ∗ x(t)becomes

y(ti) =∫ ti

0

aebτ︸︷︷︸h(τ)

x(ti − τ)dτ

=∫ ti−ti−1

0

aebτx(ti − τ)dτ +∫ ti

ti−ti−1

aebτx(ti − τ)dτ

=∫ ∆t

0

aebτx(ti − τ)dτ + eb∆t

∫ ti−1

0

aebςx(ti−1 − ς)dς

=∫ ∆t

0

aebτx(ti − τ)dτ + eb∆ty(ti−1) (75)


and we have used a change of variables ς = τ − (ti − ti−1) inthe second integral. Hence, we accumulate the values of theconvolution at the previous time points as we proceed.

Another way to represent frequency-dependent losses of theconductors is to obtain a lumped circuit that simulates the skineffect process [81]. This is shown in Fig. 6(b). In a wire, wecan break the cross section into a sequence of concentric cylin-ders that are effectively in parallel. Each cylinder, by virtue ofits size, can be viewed as a resistance in series with an induc-tance. Hence, we obtain the equivalent circuit in Fig. 6(b). Asfrequency increases, the parallel RL segments open up, therebysimulating what happens in a conductor as the current crowdscloser to the conductor surface. An alternative circuit is shown inFig. 6(c) [1], [2], [82]. The conductor (in this case, a conductorof rectangular cross section block representing a PCB land) isrepresented in terms of partial inductance. The conductor crosssection is divided into rectangular subconductors, and each isrepresented with its resistance and partial inductance. Mutualpartial inductance between the subconductors causes the cur-rent to migrate toward the outer surface of the conductor, whichis what happens when skin effect is developed.

The frequency-dependent losses in the surrounding mediumare primarily due to the inability of the bound charge dipolesin the dielectric to align with the time-changing electric fieldas the frequency increases [1], [2], [13]. This phenomenon canbe modeled with a complex permittivity of the dielectric ε =ε′ − jε′′. The imaginary part −ε′′ is caused by this polarizationloss. The complex permittivity can be modeled with the Debyemodel as [13], [94]

ε = ε′ − jε′′

= ε0 +N∑

i=1

Ki

1 + jωτi

= ε0 +N∑

i=1

Ki

1 + ω2τ2i︸︷︷︸

ε′

−j

N∑i=1

Ki

1 + ω2τ2i

ωτi︸︷︷︸ε′′

(76)

where the τi are said to be the relaxation time constants. Theterms Ki and τi can be determined from measured data [94]. Fora homogeneous surrounding medium, as in a coupled stripline,the per-unit-length capacitance can be written as the product ofthe permittivity and a constant that depends only on the cross-sectional dimensions as c = εM [1], [2]. Substituting (76) givesthe per-unit-length admittance as

y(ω) = jωc(ω)

= jω

[ε0 +

N∑i=1

Ki

1 + jωτi

]M

= jωc0 +

[N∑

i=1

Ki

1 + jωτi

]jωM (77)

where the per-unit-length capacitance with the dielectric re-moved and replaced by free space is denoted as c0 = ε0M .

Fig. 6. Modeling frequency-dependent conductor losses with lumped circuits.

Converting to the Laplace transform jω ⇒ s, yields

y(s) =

[N∑

i=1

Ki

1 + sτi

]sM + sc0. (78)

This has the simple inverse transform [53], [65]

y(t) =

[N∑

i=1

Ki

τie−t/τi

]M

∂

∂t+ c0

∂

∂t. (79)


Hence, the convolution in the second transmission-line equationy(t) ∗ V (t) becomes

y(t)∗V (t)=N∑

i=1

[M

Ki

τi

∫ t

0

e−λ/τi∂V (t − λ)∂(t − λ)

dλ

]+c0

∂V (t)∂t

.

(80)

This can be readily evaluated using the recursive convolutionmethod, since the integrand is expressed as exponentials.

Toward the end of the 1990s, work continued on simulatinglosses in PCBs [83]–[91]. The method of characteristics wasextended to the case of lossy lines as the generalized methodof characteristics [83]. The asymptotic waveform evaluation(AWE) method was being used [84], [85]. AWE is a momentmatching method. Expanding a transfer function H(s) abouts = 0 using a Taylor series gives the Maclaurin series

H(s) ∼= H(0)+s

1!H(0)(1)+

s2

2!H(0)(2)+ · · · + sN

N !H(0)(N)

(81)

where H(0)(i) is the ith derivative of H(s) evaluated at s = 0.Expanding the Laplace transform of the time-domain responsegives

H(s) =∫ ∞

0

h(t)e−stdt

=∫ ∞

0

h(t)[1 − st

1!+

s2t2

2!− · · ·

]dt

=∫ ∞

0

h(t)dt − s

∫ ∞

0

th(t)dt + s2

∫ ∞

0

t2

2!h(t)dt−· · · .

(82)

Hence, the coefficients of the Taylor’s series expansion abouts = 0 are said to be the moments of the time-domain response[92].

At the beginning of 2000, the requirement to pack more high-speed digital systems in about the same space or into smallerspaces (as with cell phones) caused a renewed interest in makingthe simulation of high-density high-speed digital interconnectsmore efficient [92]. Losses were beginning to dramatically af-fect the performance of those devices because of the increasingspectral content of the digital signals. In addition, the pin countsof packages were increasing at a dramatic rate. More processingpower and function were being put into chips. Ball-grid arrayswith over 200 pins were becoming common. This meant thatnot only the PCB lands must be simulated but also the inter-nal bonding wires of the packages. Simulating each individualinterconnect with SPICE was becoming a losing battle. Hence,there arose the need to simulate entire blocks of interconnectswith macromodels. Measured data were being used to synthesizetransfer functions having a minimum number of poles, therebyreducing the computation time in the simulation. This is re-ferred to as model order reduction (MOR). The Pade’ methodtries to match the measured frequency-domain data to a transfer

function

H(s) =a0 + a1s + a2s

2 + · · · + aMsM

1 + b1s + b2s2 + · · · + bNsN

= c0 +N∑

i=1

ci

s − pi. (83)

The intent is, once again, to obtain the representation of theimpulse response of the system as a sequence of exponentials tobe used with the recursive convolution method to generate thetime-domain response

h(t) = c0 +N∑

i=1

ciepi t. (84)

The moments of H(s) about s = 0 in (81) are compared to (83)as

H(s) = m0 + m1s + m2s2 + · · · + mM+N+1s

M+N+1

=a0 + a1s + a2s

2 + · · · + aMsM

1 + b1s + b2s2 + · · · + bNsN. (85)

Both sides of (85) are multiplied by the denominator ofthe transfer function to give M + N + 1 equations to besolved for the M + N + 1 unknowns in the transfer functiona0, a1, . . . , aM , b1, . . . , bN [77], [91], [92]. There is an impor-tant problem with this technique. It may give poles that havepositive real parts, and hence, h(t) is unstable. Increasing theorder of representation can alleviate this to some extent. It turnsout that one can also show that the coefficients of the exponen-tial representation obtained with the Prony method in (72), i.e.,ai and bi, are the same as those in the Pade’ method-obtainedexponential representation in (84), ci and pi [77].

The Pade’ method generates a transfer function from whichwe easily obtain the time-domain representation as a sequenceof exponentials. The more negative the real part of a pole ofH(s), the less influence it has on the response. For example, apole having a value p = −1000 + jω will have a time-domainresponse of e−1000tejωt, which decays rapidly to zero. On theother hand, a pole having p = −1 + jω will have a time-domainresponse of e−tejωt and does not decay nearly as fast. Hence, thecloser the pole to the jω axis, the more effect that the pole willhave on the time-domain response. Of the many thousand polesof the actual H(s), we look for poles closest to the jω axis. Animportant problem with the Pade’ method is the expansion ofH(s) about s = 0 to generate its moments. The accuracy of theobtained H(s) representation decreases as we go away furtherfrom s = 0 in the s = σ + jω 2-D plane. To obtain other polesaway from s = 0 (and along the jω axis), we move the radiusof convergence away from s = 0 along the jω axis. This is theso-called complex frequency hopping (CFH) method [92], [93].The dominant set of poles of H(s) are obtained near the jω axisbut away from the origin, i.e., s = 0. A binary search routine isused to extract these using the Pade’ method for other expansionpoints.

And finally, we review the various methods of obtaining othermacromodels of the interconnect lines. The term macromodelrefers to a reduced-order 2n-port representation of the line.


The method of characteristics converted the partial differentialequations of a lossless line into ordinary differential equationsthat are directly integrable giving the terminal representationin (62). This gives the macromodel of a two-conductor lineshown in Fig. 4 that is used in SPICE to give an exact simu-lation of a lossless line. The extension of this macromodel tolossy lines is referred to as the generalized method of charac-teristics [91]. For lossy lines, we can put the solution of theMTL equations in a similar form. For example, the Laplace-transformed transmission-line equations for n = 1 are [1], [2]

d2

dz2V (z, s) − γ(s)2V (z, s) = 0 (86a)

d2

dz2I(z, s) − γ(s)2I(z, s) = 0 (86b)

where the complex propagation constant is written in terms ofthe Laplace transform variable s as

γ(s) =√

(Zint(s) + sL)(G(s) + sC). (87)

The solution of these is simple [see (42)] and given as

V (L, s) = cosh(γL)V (0, s) − ZC sinh(γL)I(0, s) (88a)

I(L, s) =−(1/ZC) sinh(γL)V (0, s)+ cosh(γL)I(0, s)

(88b)

where the Laplace-transformed characteristic impedance is

ZC(s) =√

(Zint(s) + sL)/(G(s) + sC). (89)

Substituting the relations for cosh and sinh, and adding andsubtracting gives [1], [2]

V (0, s)−ZC(s)I(0, s)=e−γ(s)L[V (L, s)−ZC(s)I(L, s)]

(90a)

V (L, s)+ZC(s)I(L, s)=e−γ(s)L[V (0, s)+ZC(s)I(0, s)].

(90b)

This can be written as

V (0, s) − ZC(s)I(0, s) = E0(s) (91a)

V (L, s) + ZC(s)I(L, s) = EL(s) (91b)

where

E0(s) = e−γ(s)L[V (L, s) − ZC(s)I(L, s)] (92a)

EL(s) = e−γ(s)L[V (0, s) + ZC(s)I(0, s)]. (92b)

This gives the equivalent circuit shown in Fig. 7. The functionsθ(s) = e−γ(s)L and ZC(s) can be represented in the time do-main as θ(t) and ZC(t), as a sequence of exponentials using thePade’ method [91]. Alternatively, the ZC(s) can be synthesizedas a lumped circuit [87]. The time-domain forms become

V (0, t)−ZC(t) ∗ I(0, t)=θ(t) ∗ [V (L, t)−ZC(t) ∗ I(L, t)]

(93a)

Fig. 7. Equivalent circuit for the generalized method of characteristics.

V (L, t)+ZC(t) ∗ I(L, t)=θ(t) ∗ [V (0, t)+ZC(t) ∗ I(0, t)]

(93b)

where ∗ again denotes convolution. Because the time-domainfunctions can be represented as exponentials using the Pade’method, recursive convolution can be used to perform the con-volutions. This is referred to as the generalized method ofcharacteristics, and can be extended to MTLs [83], [86], [87],[89]–[91].

An alternative macromodel representation is in terms of theadmittance or Y parameters. For example, the Y parameterrepresentation for a lossy two-conductor line is [1], [2][

I(0, s)−I(L, s)

]=[

Ys −Ym

−Ym Ys

]︸︷︷︸

Y

[V (0, s)V (L, s)

](94)

where

Ys(s) =1

ZC(s) tanh(γ(s)L)(95a)

Ym(s) =1

ZC(s) sinh(γ(s)L). (95b)

Using the Pade’ method or other MOR techniques, thetime-domain admittance parameters in terms of sequences ofexponentials can be obtained so that the time-domain resultcan be written in terms of convolutions using the recursiveconvolution method as

I(0, t) = Ys(t) ∗ V (0, t) − Ym(t) ∗ V (L, t) (96a)

−I(L, t) = −Ym(t) ∗ V (0, t) + Ys(t) ∗ V (L, t) (96b)

where ∗ again denotes convolution. This gives the equivalenttime-domain circuit shown in Fig. 8 in terms of controlledcurrent sources. This can also be extended to MTLs. Analternative equivalent circuit in terms of admittance parametersis given in [94].

An alternative to the Pade’ method of generating time-domainequivalences is the use of SPICE-like equivalent circuits in termsof lumped parameters and controlled sources [95]. This usesthe vector fitting procedure to obtain the dominant poles [96],[97].


Fig. 8. Time-domain equivalent circuit in terms of the admittance parameters.

V. SUMMARY

A brief chronology of the application of MTL theory to thesolution of EMC problems is given. Rapidly changing technol-ogy (and in particular, the emergence of digital technology) hascaused a renewed interest in the analysis of transmission linesfor modern electronic system design.

In the 1950s and 1960s, transmission lines had a minor buteasily analyzed effect on the transmission of signals from onepoint to another; analysis of crosstalk and radiated emissionsfrom these lines dominated the research. In the 1970s, the abilityof cables to pick up and conduct EMP signals into communi-cation and control facilities prompted an increased interest inincident field coupling to transmission lines. With the emergenceof digital signals and technology in the 1980s, the spectral con-tent of the signals carried by these “transmission lines” beganto increase. The spectral content was confined to below the highmegahertz frequency range until the 1990s. Prior to this, lossesin the line conductors and the surrounding dielectric were notsignificant. As digital clock and data speeds entered the highmegahertz range in the 1990s and the low gigahertz range in the2000s, losses became important and no longer could be ignored.Incorporating losses into the solution of the MTL equations isone of the primary research areas today. High-density integra-tion of digital electronics in the 2000s has created the secondimportant problem, i.e., how to simulate a rapidly growing num-ber of interconnect lines.

It is important to remember that in all these, electrical dimen-sions are the important parameters that determine whether wemust treat a pair of conductors as a transmission line or sim-ply as a lumped circuit. It is particularly illuminating to pointout that the size of PCBs for PCs and embedded processorshas not changed since the early 1980s. (This size will continueto be driven by customer preferences.) However, the spectralcontent of digital signals has changed drastically. Therefore,physical line lengths have remained unchanged, but their elec-trical lengths have become a significant portion of a wavelength.Conductor lengths, some 25 years ago, were electrically shortso that the lines were inconsequential from a transmission linestandpoint. Today, these very same line lengths must be treatedas distributed-parameter circuits or transmission lines. In orderto cope with future digital designs we must be adept at analyzingtransmission lines.

It is very distressing to realize that almost all electrical engi-neering programs in the U.S. no longer have a required under-graduate course covering transmission lines even for the simpletwo-conductor line. Very few schools offer a senior technical

elective in transmission lines. Therefore, today’s electrical en-gineering graduates have little or no knowledge of transmis-sion lines and related important topics such as mismatch andreflections, time delay, etc. At the author’s institution, Mer-cer University, there is a required undergraduate course cover-ing two-conductor transmission lines as well as a senior tech-nical elective covering high-speed digital transmission lines.These graduates will be well prepared for future technologydesign. The author is, therefore, appealing to educators to re-consider their curriculum and to incorporate a required under-graduate course in, at least, the rudiments of two-conductortransmission lines if it is not already present. We must prepareour graduates to cope with the challenges of high-speed digitaldesign. As educators we owe our students nothing less.

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Clayton R. Paul (S’61–M’70–SM’79–F’87–LF’06)was born in Macon, GA, on September 6, 1941. Hereceived the B.S. degree from the Citadel, Charleston,SC, the M.S. degree from Georgia Institute of Tech-nology, Atlanta, and the Ph.D. degree from PurdueUniversity, Lafayette, IN, in 1963, 1964, and 1970,respectively, all in electrical engineering.

He is currently the Sam Nunn Eminent Professorof aerospace systems engineering and Professor ofelectrical and computer engineering at Mercer Uni-versity, Macon, GA. He is an Emeritus Professor at

the University of Kentucky, Lexington, where he was a member of the faculty inthe department of electrical engineering for 27 years. He is the author of numer-ous textbooks on electrical engineering subjects, and has published numerouspapers, the majority of which are in his primary research area of electromagneticcompatibility (EMC) of electronic systems. From 1970 to 1984, he conductedextensive research for the U.S. Air Force in modeling crosstalk in multiconduc-tor transmission lines and printed circuit boards. From 1984 to 1990, he servedas a Consultant to the IBM Corporation in the area of product EMC design.

Dr. Paul is an Honorary Life Member of the IEEE EMC Society, and amember of Tau Beta Pi and Eta Kappa Nu. He is the recipient of the 2005 IEEEElectromagnetics Award and the 2007 IEEE Undergraduate Teaching Award.

a brief history of work in transmission lines for emc applications

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