Canavero, Flavio

Grivet-Talocia, S.

English

Crossref

Weak solution of the nonuniform multiconductor transmission lines

GRIVET TALOCIA, STEFANO

CANAVERO, Flavio

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEWeak solution of the nonuniform multiconductor transmission lines / GRIVET-TALOCIA S.; F. CANAVERO. - STAMPA. -(1998), pp. 964-968. ((Intervento presentato al convegno IEEE International Symposium on ElectromagneticCompatibility tenutosi a Denver (USA) nel August 24-28, 1998.OriginalWeak solution of the nonuniform multiconductor transmission linesPublisher:PublishedDOI:10.1109/ISEMC.1998.750338Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1508195 since: 2015-07-14T12:51:46ZPiscataway, N.J. : IEEEWeak Solution of the Nonuniform Multiconductor Transmission Lines S. Grivet-Talocia Dip. Elettronica, Politecnico di Torino Corso Duca degli Abruzzi 24 1-10129, Torino, Italy Abstmct: A time domain technique for the simulation of Nonuniform Multiconductor Transmission Lines (NMTL) with external sources is presented. The technique is based on a weak formulation of the NMTL equations obtained through expansion of the voltage and current vectors, per unit length matrices, and distributed sources into a set of locally supported basis functions. The particular choice of these functions determines the approximation order of the method. Examples are shown for practical interest structures. INTRODUCTION This paper presents a new time domain simulation method for the Nonuniform Multiconductor Transmission Lines equations. This topic has been collecting a lot of interest in the recent literature, because many interconnections of practical interest are characterized by cross sections which are not translation-invariant. Examples can be impedance matching networks or cables in complex structures, like automobiles or airplanes. Therefore, it is of crucial im- portance to model the effects of the nonuniformity of the interconnections both from signal integrity and EM1 stand- points. Several techniques have been presented for the simulation of the NMTLs. These techniques can be subdivided in two main classes, performing simulation in the frequency domain or in the time domain, respectively. The former can obtain closed-form solutions [l] in some cases, but can also be used to analyze more general structures through a piecewise constant discretization of the line [2]. If the transient response is wanted, inverse FFT can be used. However, when signals with complex waveforms are ap- plied to unmatched lines and long transients are generated, the number of points for the evaluation of the FFT can be very large. In fact, as the result of inverse FFT is al- ways a periodic waveform, the period must be chosen long enough for the transients to be extinguished. This is the reason why numerical schemes performing the simulation directly in the time domain have been recently proposed. Among these we can cite the methods based on the scat- F. Canavero Dip. Elettronica, Politecnico di Torino Corso Duca degli Abruzzi 24 1-10129, Torino, Italy tering representation [5 ] ,  the method of characteristics [9], and the waveform relaxation analysis [4]. The method pre- sented here is based on a weak formulation of the NMTL equations, which leads to a class of numerical schemes of different approximation order according to the particular choice of some trial and test functions. MATHEMATICAL FORMULATION Let us consider the NMTL equations a a dZ at - I ( Z , t )  + C(z)-V(Z,t) + G(z)V(.Z,t) = IF(Z, t ) ,  with V(z, t )  and I(z, t )  indicating the voltage and current vectors at location z and time t .  The line is assumed to have P + 1 conductors, and the per-unit-length parameters L(z), C(z), R(z), and G ( z )  are P x P matrices whose en- tries are arbitrary functions of the space variable z. With- out loss of generality we will consider the length of the line to be normalized, i.e., z E [0,1]. The distributed sources VF and IF are due to external field excitation and must be known at any z and t. For simplicity, the line is supposed here to be terminated by Thhvhnin loads, V(0, t )  = Vs(t) - RsI(0, t ) ,  (1) V(1,t) = VL(t) + R~1(1, t ) .  (2) However, any other type of linear and resistive termina- tions can be used with the following formulation. The approximate voltages and currents along the line are sought for in terms of expansion coefficients into a set of trial functions (9, : n = 1, . . . , ATv}, "p V(Z,t) = (Pn(z)Vn(t), (3) I(z,t) = cpn(z)In(t). (4) n=l N ,  n= 1 The external sources axe also projected onto the same ap- 0-7803-5015-4/98/$10.00 0 1998 JEEE 964 proximation space through the expansion "p v F ( z , ~ )  = E (Pn(Z)VFn(t), (5) n=l N ,  IF(Z,t) = (Pn(Z)IFn(t). (6) n=l A similar expansion can be used, with possibly different basis functions {@A : k = 1 , .  . . , N G } ,  for the entries in the per unit length matrices k = l  and similarly for C(Z), R(z), and G(z). Projecting now the NMTL equations onto a third set of functions (71, : m = 1 , .  . . , Nv}, we get a set of ODE'S representing the spatial discretization of the original NMTL equations, (7) with k l  where Zp is the P x P identity matrix, and ~ 2 ;  = (Vn $ k , q m ) .  (9) h h Similar expressions hold for Cmn, Rmn, and Gmn. We consider now the inclusion of the loads (1) and (2). It is convenient to  choose the trial and test functions such that only one is nonzero at the boundaries, i.e., Vn(O) = 0, V n  = 2 , . . . ,N ,+ , ,  ( ~ ~ ( 1 )  = 0, V n =  1 ,..., N,+, - 1, ~ ~ ( 1 )  = 0, V m = l  , . . . , N , + , - l .  ~ ~ ( 0 )  = 0, Vm= 2,...,N,+,, (10) This is not a real restriction because whatever be the initial choice of basis functions, a change of basis can always be performed to obtain only one nonzero function at both edges. The two edge trial functions will also be normalized so that (Pl(0) = (PN, (1) = 1. Substituting now the expansions (3) and (4) into the load equations (1) and (2) and using the conditions (lo), we get. vi(t) = Vs(t) - RsIi(t) (11) VN, ( t )  = V L ( ~ )  + RLIN, ( t )  (12) These expressions for the loads can be used to eliminate the two unknowns Vl(t) and VN,(~ )  from the system (7)-(8). It should be noted, however, that the number of scalar un- knowns in the system is 2N,P, which matches the number of scalar equations. If we eliminate the voltages at the two edges, i.e., 2 P  scalar unknowns, also 2 P  equations must be suppressed in order to keep the balance even. These equations are obviously the ones involving the projection onto the border test functions 71 and q ~ ,  . As a result, the system of ODE'S to be solved reads d dt Q-x(t) + cPx(t) = rsvs(t) + r L v L ( t )  + (13) d d rSD-vS(t) dt + rLDzVL(t) + nVVF(t) + OIfF(t) where Q is nonsingular if the trial and test functions are linearly independent. The array x collects the expansion coefficients o,f voltage and current along the line, while the arrays VF, IF collect the time-varying expansion coeffi- cients of the distributed sources. The system (13) can be solved with a suitable integration method. We used here a a 5th - 6th order explicit Runge- Kutta scheme [6]. If the trial and test functions have a local support, the system matrices Q and ii! have a banded structure with few nonvanishing codiagonals. Therefore, the computation of the product *-'ax can be performed in O(N,,,) operations. The same applies to the external forcing terms Slv, 521. It should be noted that the forcing terms in the system (13) include also the time derivatives of the source vectors Vs(t) and VL(t). Therefore, singular waveforms like delta functions or step functions cannot be handled by this method. EXAMPLES The numerical scheme was applied to several examples and validated through comparison with a reference solution, obtained by applying inverse FFT to the frequency do- main solution. The weak solution for the following exam- ples was determined with piecewise polynomial (B-splines) basis functions for the three sets (Pn, Cpk, and qm. With these basis functions the approximation error is expected to decrease as N g L ,  where L - 1 is the local polynomial order of the basis functions. 965 Scalar exponential line We chose the scalar exponential line as a test case for our numerical scheme because the analytical solution in the frequency domain is well known and understood [3]. Let us consider a scalar nonuniform line ( P  = 1) of unitary length characterized by ~ ( z )  = L0e6", C(z) = Coe-6", R(z) = 0, G(z) = 0, where the parameter 6 controls the rate of taper and Lo, CO are the nominal per unit length inductance and ca- pacitance at the edge z = 0. The nominal characteristic impedance of the line is therefore where 2 0  is the nominal characteristic impedance at the edge z = 0. As the impedance level increases at rate b with z, it can be shown that the positive voltage wave increases in magnitude and the corresponding current wave decreases at rate 612. The converse holds for negative voltage and current waves. The parameters of the line that will be investigated here are normalized. More precisely, Lo = 1H/m, CO = 1F/m, 6 = log4. This corresponds to a 1:4 impedance stepping line. The waveform of the voltage source is set here to a gaussian pulse, vs(t) = Voexp { -~ '";)2}, with amplitude V, = lV ,  center T, = 2 s, and semi-width A, = 0.2 s. The line is matched at both ends. A reference solution in the time domain is obtained from the frequency domain analytical solution through inverse FFT. The total simulation time is set here to Tma, = 8 S, which means that the input signal is considered as a peri- odic pulse train with period Tma,. As the one-way delay is T = 1 s, there are no interactions between two adjacent pulses, because the transient associated to one pulse due to  the nonuniformity of the line is already extinguished when the next pulse comes through. Of course, this holds only when at least one of the two line ends is matched. The voltage at the two terminations obtained with the weak formulation by using piecewise linear functions is plotted and compared to the reference solution in Fig- ure 1. The number of basis functions in these simulations is NV = 65. The figures show clearly that the weak so- lution (thin continuous line) is undistinguishable from the reference solution (thick dashed line). I 0 2 4 6 8 -0.1 ' time [s] v ( L t )  [VI 1 I -0.2; 2 4 6 8 time [SI Figure 1: Voltage at the left (top panel) and right (bottom panel) terminations of the matched 1:4 exponential line. The convergence properties of the method as the dimension NV of the approximation spaces increases is now investi- gated. The approximation error on voltage and current is computed for each Nq according to Ev(N9) = mzm I V N ~  (z, t) - wref(Z, t)l, Ei(NV) = m ~ m a J i ~ , ( z , t )  t r  - iref(z,t)lr where Oref(z, t),  iref(Z, t) represent the reference voltage and current while v ~ ~ ( z , t ) ,  i ~ ~ ( z , t )  are the voltage and current obtained with our method. The voltage approxi- mation errors are reported in Figure 2 as functions of NV. The parameter L controls the approximation order of the 966 1 o-2, I 0 L = 2  + L = 3  L = 4  x L = 5  . L = 6  I I 1 o2 N Figure 2: Matched 1:4 exponential line with gaus- sian excitation. Maximum absolute error on volt- age as a function of Nv and of the approximation order L of the basis functions. basis functions. As expected, we have a power law decay of the type N G ~ .  Consequently, small errors can be obtained by using few high order basis functions. We repeated the same analysis by using a 50% trapezoidal pulse voltage source with a rise time T = 0.4 s. This func- tion has a singluarity in the first derivative. Therefore, the decay of the approximation error cannot be faster than N;' for any choice of basis functions. This is confirmed by Figure 3, which shows the behavior of the approximation error for voltage and current in the case of piecewise linear functions. Nonparallel wires above a ground plane This section will examine the crosstalk on a nonuniform line made of two wires above a ground plane, sketched below. The two wires are supposed to be parallel to the ground plane, but their distance increases linearly along the length l0-'r------ 10' 1 oc N Figure 3: Approximation errors on voltage (cir- cles) and current (stars) as functions of "p for the matched 1:4 exponential line with trapezoidal pulse excitation. The dash-dotted line corresponds to a slope N;'. of the line. Lines of this type have been studied in [7, 21. Let us consider two wires with radius r = 1 mm placed at h = 3 cm above a ground plane. Their separation is DO = 5 mm at z = 0 and D1 = 15 mm at z = 1. The medium is supposed to be free space. Due to these par- ticular conditions, the expressions for the per-unit-length inductance and capacitance matrices can be obtained with the wide-separation approximation [8].  Taking the ground plane as the reference conductor, we have the approximate expressions L11 = L22 = log (F) , 2lr where po indicates the permeability of free space and the distance along the line is D ( z )  = Do + z(D1 - Do). As the surrounding medium is homogeneous we can easily derive the per-unit-length capacitance matrix, C ( z )  = &oPoL-l(z), where EO is the permettivity of free space. We will apply a voltage source consisting of a 1 MHz , 50% duty cycle trapezoidal pulse train with raise and fall times equal to 20 ns to one of the two wires at the edge t = 0, and calculate the near end crosstalk on the other 967 Figure 4: Near end crosstalk for the nonparallel wires above a ground plane. The thin continu- ous line represents the weak solution and the thick dashed line is the reference solution. wire. Each conductor is terminated with a 50 R resistance connected to the reference. As there is no closed form solution for lines of this type, we used an approximate method to obtain a reference solu- tion for this problem. The standard approach is to divide the line into N,  uniform subsections and to perform the analysis in the frequency domain [8, 21. Each subsection is analyzed separately by deriving its chain matrix, which can be evaluated in closed form. The chain matrix of the overall structure is then obtained by multiplying the chain matrices of each subsection of the line, and the solution for the voltages and currents at the line ends is found by in- corporating the terminal conditions. Finally, inverse FFT is applied to get the time domain waveform. This method converges to the exact solution when N,  increases. Some numerical tests on the convergence have been conducted to obtain the minimum number of subdivisions that insures a good approximation for the nonuniformity of the line. A number of Nz = 32 subdivisions resulted beyond this limit and was used in the simulations. Figure 4 shows the results of the simulations with our method (Nv  = 65) and with the approximate piecewise uniform solution. The number of points for the evaluation of the inverse FFT in the latter was set to 1024. The two curves are undistinguishable. We can conclude that the two methods give practically the same results. CONCLUSIONS A new time-domain scheme for the numerical solution of the Nonuniform Multiconductor Transmission Lines (NMTL) has been presented. The method is based on the spatial expansion of the solution, the external sources, and the per unit length matrices into some approximat- ing functions. The projection of the NMTL equations through suitable test functions leads to a discrete system of ODE’S that can be solved by an appropriate time inte- gration scheme. The approximation order of the method depends on the particular choice of trial and test functions. Numerical tests show that small approximation errors can be obtained with few high order basis functions. REFERENCES [l] C. E. Baum, J.  B. Nitsch, and R. J .  Sturm, “Analyti- cal Solution for Uniform and Nonuniform Multiconduc- tor Transmission Lines with Sources”, The Review of Radio Science 1993-96, URSI-Oxford University Press, 1996, NY. [2] P. Besnier, and P. Degauque, “Electromagnetic Topol- ogy: Investigations of Nonuniform Transmission Line Networks”, IEEE Trans. Electromagn. Compat., vol. 37, 1995, 227-233. [3] C. R. Burrows, “The exponential transmission line”, Bell System Tech. J., vol. 17, 1938, 555-573p. [4] F. Chang, “Transient Simulation of Nonuniform Cou- pled Lossy Transmission Lines Characterized with frequency-Dependent Parameters-Part I: Waveform Relaxation Analysis”, IEEE Trans. Circuits Syst. I ,  vol. 39, 1992, 585-603. [5] T. Dhaene, L. Martens, and D. De Zutter, “Transient Simulation of Arbitrary Nonuniform Interconnection Structures Characterized by Scattering Parameters”, IEEE Trans. Circuits Syst. I ,  vol. 39, 1992, 928-937. [6] IMSL, IMSL MATH/LIBRARY User’s Manual, Ver- sion 2.0, 1991, IMSL, Houston. [7] J .  Nitsch, C. E. Baum , “Splitting of degenerate natural frequencies in coupled two-conductor lines by distance variation”, Interaction Notes, Note 477, July 1989. [8] C. R. Paul, Analysis of Multiconductor Transmission Lines, John Wiley and Sons, NY, 1994. [9] V. K.  Tripathi, N. Orhanovic, “Time-Domain Charac- terization and Analysis of Dispersive Dissipative Inter- connects”, IEEE Trans. Circuits Svst. I,  vol. 39, 1992, 938-945. 968 

PORTO Publications Open Repository TOrino

Weak Solution of the 
Nonuniform Multiconductor Transmission Lines 
S. Grivet-Talocia 
Dip. Elettronica, Politecnico di Torino 
Corso Duca degli Abruzzi 24 
1-10129, Torino, Italy 
Abstmct: A time domain technique for the simulation of 
Nonuniform Multiconductor Transmission Lines (NMTL) 
with external sources is presented. The technique is based 
on a weak formulation of the NMTL equations obtained 
through expansion of the voltage and current vectors, per 
unit length matrices, and distributed sources into a set of 
locally supported basis functions. The particular choice 
of these functions determines the approximation order of 
the method. Examples are shown for practical interest 
structures. 
INTRODUCTION 
This paper presents a new time domain simulation method 
for the Nonuniform Multiconductor Transmission Lines 
equations. This topic has been collecting a lot of interest 
in the recent literature, because many interconnections of 
practical interest are characterized by cross sections which 
are not translation-invariant. Examples can be impedance 
matching networks or cables in complex structures, like 
automobiles or airplanes. Therefore, it is of crucial im- 
portance to model the effects of the nonuniformity of the 
interconnections both from signal integrity and EM1 stand- 
points. 
Several techniques have been presented for the simulation 
of the NMTLs. These techniques can be subdivided in 
two main classes, performing simulation in the frequency 
domain or in the time domain, respectively. The former 
can obtain closed-form solutions [l] in some cases, but can 
also be used to analyze more general structures through 
a piecewise constant discretization of the line [2]. If the 
transient response is wanted, inverse FFT can be used. 
However, when signals with complex waveforms are ap- 
plied to unmatched lines and long transients are generated, 
the number of points for the evaluation of the FFT can be 
very large. In fact, as the result of inverse FFT is al- 
ways a periodic waveform, the period must be chosen long 
enough for the transients to be extinguished. This is the 
reason why numerical schemes performing the simulation 
directly in the time domain have been recently proposed. 
Among these we can cite the methods based on the scat- 
F. Canavero 
Dip. Elettronica, Politecnico di Torino 
Corso Duca degli Abruzzi 24 
1-10129, Torino, Italy 
tering representation [5 ] ,  the method of characteristics [9], 
and the waveform relaxation analysis [4]. The method pre- 
sented here is based on a weak formulation of the NMTL 
equations, which leads to a class of numerical schemes of 
different approximation order according to the particular 
choice of some trial and test functions. 
MATHEMATICAL FORMULATION 
Let us consider the NMTL equations 
a a 
dZ at - I ( Z , t )  + C(z)-V(Z,t) + G(z)V(.Z,t) = IF(Z, t ) ,  
with V(z, t )  and I(z, t )  indicating the voltage and current 
vectors at location z and time t .  The line is assumed to 
have P + 1 conductors, and the per-unit-length parameters 
L(z), C(z), R(z), and G ( z )  are P x P matrices whose en- 
tries are arbitrary functions of the space variable z. With- 
out loss of generality we will consider the length of the line 
to be normalized, i.e., z E [0,1]. The distributed sources 
VF and IF are due to external field excitation and must be 
known at any z and t. For simplicity, the line is supposed 
here to be terminated by Thhvhnin loads, 
V(0, t )  = Vs(t) - RsI(0, t ) ,  (1) 
V(1,t) = VL(t) + R~1(1, t ) .  (2) 
However, any other type of linear and resistive termina- 
tions can be used with the following formulation. 
The approximate voltages and currents along the line are 
sought for in terms of expansion coefficients into a set of 
trial functions (9, : n = 1, . . . , ATv}, 
"p 
V(Z,t) = (Pn(z)Vn(t), (3) 
I(z,t) = cpn(z)In(t). (4) 
n=l 
N ,  
n= 1 
The external sources axe also projected onto the same ap- 
0-7803-5015-4/98/$10.00 0 1998 JEEE 
964 
proximation space through the expansion 
"p 
v F ( z , ~ )  = E (Pn(Z)VFn(t), (5) 
n=l 
N ,  
IF(Z,t) = (Pn(Z)IFn(t). (6) 
n=l 
A similar expansion can be used, with possibly different 
basis functions {@A : k = 1 , .  . . , N G } ,  for the entries in the 
per unit length matrices 
k = l  
and similarly for C(Z), R(z), and G(z). Projecting now 
the NMTL equations onto a third set of functions (71, : 
m = 1 , .  . . , Nv}, we get a set of ODE'S representing the 
spatial discretization of the original NMTL equations, 
(7) 
with 
k l  
where Zp is the P x P identity matrix, and 
~ 2 ;  = (Vn $ k , q m ) .  (9) 
h h 
Similar expressions hold for Cmn, Rmn, and Gmn. 
We consider now the inclusion of the loads (1) and (2). It 
is convenient to  choose the trial and test functions such 
that only one is nonzero at the boundaries, i.e., 
Vn(O) = 0, V n  = 2 , . . . ,N ,+ , ,  
( ~ ~ ( 1 )  = 0, V n =  1 ,..., N,+, - 1, 
~ ~ ( 1 )  = 0, V m = l  , . . . , N , + , - l .  
~ ~ ( 0 )  = 0, Vm= 2,...,N,+,, (10) 
This is not a real restriction because whatever be the initial 
choice of basis functions, a change of basis can always be 
performed to obtain only one nonzero function at both 
edges. The two edge trial functions will also be normalized 
so that 
(Pl(0) = (PN, (1) = 1. 
Substituting now the expansions (3) and (4) into the load 
equations (1) and (2) and using the conditions (lo), we get. 
vi(t) = Vs(t) - RsIi(t) (11) 
VN, ( t )  = V L ( ~ )  + RLIN, ( t )  (12) 
These expressions for the loads can be used to eliminate the 
two unknowns Vl(t) and VN,(~ )  from the system (7)-(8). 
It should be noted, however, that the number of scalar un- 
knowns in the system is 2N,P, which matches the number 
of scalar equations. If we eliminate the voltages at the two 
edges, i.e., 2 P  scalar unknowns, also 2 P  equations must 
be suppressed in order to keep the balance even. These 
equations are obviously the ones involving the projection 
onto the border test functions 71 and q ~ ,  . As a result, the 
system of ODE'S to be solved reads 
d 
dt 
Q-x(t) + cPx(t) = 
rsvs(t) + r L v L ( t )  + 
(13) 
d d 
rSD-vS(t) dt + rLDzVL(t) + 
nVVF(t) + OIfF(t) 
where Q is nonsingular if the trial and test functions are 
linearly independent. The array x collects the expansion 
coefficients o,f voltage and current along the line, while the 
arrays VF, IF collect the time-varying expansion coeffi- 
cients of the distributed sources. 
The system (13) can be solved with a suitable integration 
method. We used here a a 5th - 6th order explicit Runge- 
Kutta scheme [6]. If the trial and test functions have a 
local support, the system matrices Q and ii! have a banded 
structure with few nonvanishing codiagonals. Therefore, 
the computation of the product *-'ax can be performed 
in O(N,,,) operations. The same applies to the external 
forcing terms Slv, 521. It should be noted that the forcing 
terms in the system (13) include also the time derivatives 
of the source vectors Vs(t) and VL(t). Therefore, singular 
waveforms like delta functions or step functions cannot be 
handled by this method. 
EXAMPLES 
The numerical scheme was applied to several examples and 
validated through comparison with a reference solution, 
obtained by applying inverse FFT to the frequency do- 
main solution. The weak solution for the following exam- 
ples was determined with piecewise polynomial (B-splines) 
basis functions for the three sets (Pn, Cpk, and qm. With 
these basis functions the approximation error is expected 
to decrease as N g L ,  where L - 1 is the local polynomial 
order of the basis functions. 
965 
Scalar exponential line 
We chose the scalar exponential line as a test case for our 
numerical scheme because the analytical solution in the 
frequency domain is well known and understood [3]. 
Let us consider a scalar nonuniform line ( P  = 1) of unitary 
length characterized by 
~ ( z )  = L0e6", 
C(z) = Coe-6", 
R(z) = 0, 
G(z) = 0, 
where the parameter 6 controls the rate of taper and Lo, 
CO are the nominal per unit length inductance and ca- 
pacitance at the edge z = 0. The nominal characteristic 
impedance of the line is therefore 
where 2 0  is the nominal characteristic impedance at the 
edge z = 0. As the impedance level increases at rate b 
with z, it can be shown that the positive voltage wave 
increases in magnitude and the corresponding current wave 
decreases at rate 612. The converse holds for negative 
voltage and current waves. 
The parameters of the line that will be investigated here 
are normalized. More precisely, 
Lo = 1H/m, CO = 1F/m, 6 = log4. 
This corresponds to a 1:4 impedance stepping line. The 
waveform of the voltage source is set here to a gaussian 
pulse, 
vs(t) = Voexp { -~ '";)2}, 
with amplitude V, = lV ,  center T, = 2 s, and semi-width 
A, = 0.2 s. The line is matched at both ends. 
A reference solution in the time domain is obtained from 
the frequency domain analytical solution through inverse 
FFT. The total simulation time is set here to Tma, = 8 S, 
which means that the input signal is considered as a peri- 
odic pulse train with period Tma,. As the one-way delay 
is T = 1 s, there are no interactions between two adjacent 
pulses, because the transient associated to one pulse due to  
the nonuniformity of the line is already extinguished when 
the next pulse comes through. Of course, this holds only 
when at least one of the two line ends is matched. 
The voltage at the two terminations obtained with the 
weak formulation by using piecewise linear functions is 
plotted and compared to the reference solution in Fig- 
ure 1. The number of basis functions in these simulations 
is NV = 65. The figures show clearly that the weak so- 
lution (thin continuous line) is undistinguishable from the 
reference solution (thick dashed line). 
I 
0 2 4 6 8 -0.1 ' 
time [s] 
v ( L t )  [VI 
1 
I 
-0.2; 2 4 6 8 
time [SI 
Figure 1: Voltage at the left (top panel) and right 
(bottom panel) terminations of the matched 1:4 
exponential line. 
The convergence properties of the method as the dimension 
NV of the approximation spaces increases is now investi- 
gated. The approximation error on voltage and current is 
computed for each Nq according to 
Ev(N9) = mzm I V N ~  (z, t) - wref(Z, t)l, 
Ei(NV) = m ~ m a J i ~ , ( z , t )  t r  - iref(z,t)lr 
where Oref(z, t),  iref(Z, t) represent the reference voltage 
and current while v ~ ~ ( z , t ) ,  i ~ ~ ( z , t )  are the voltage and 
current obtained with our method. The voltage approxi- 
mation errors are reported in Figure 2 as functions of NV. 
The parameter L controls the approximation order of the 
966 
1 o-2, I 
0 L = 2  
+ L = 3  
L = 4  
x L = 5  
. L = 6  
I I 
1 o2 
N 
Figure 2: Matched 1:4 exponential line with gaus- 
sian excitation. Maximum absolute error on volt- 
age as a function of Nv and of the approximation 
order L of the basis functions. 
basis functions. As expected, we have a power law decay of 
the type N G ~ .  Consequently, small errors can be obtained 
by using few high order basis functions. 
We repeated the same analysis by using a 50% trapezoidal 
pulse voltage source with a rise time T = 0.4 s. This func- 
tion has a singluarity in the first derivative. Therefore, 
the decay of the approximation error cannot be faster than 
N;' for any choice of basis functions. This is confirmed by 
Figure 3, which shows the behavior of the approximation 
error for voltage and current in the case of piecewise linear 
functions. 
Nonparallel wires above a ground plane 
This section will examine the crosstalk on a nonuniform 
line made of two wires above a ground plane, sketched 
below. 
The two wires are supposed to be parallel to the ground 
plane, but their distance increases linearly along the length 
l0-'r------ 
10' 1 oc 
N 
Figure 3: Approximation errors on voltage (cir- 
cles) and current (stars) as functions of "p for the 
matched 1:4 exponential line with trapezoidal pulse 
excitation. The dash-dotted line corresponds to a 
slope N;'. 
of the line. Lines of this type have been studied in [7, 21. 
Let us consider two wires with radius r = 1 mm placed 
at h = 3 cm above a ground plane. Their separation is 
DO = 5 mm at z = 0 and D1 = 15 mm at z = 1. The 
medium is supposed to be free space. Due to these par- 
ticular conditions, the expressions for the per-unit-length 
inductance and capacitance matrices can be obtained with 
the wide-separation approximation [8].  Taking the ground 
plane as the reference conductor, we have the approximate 
expressions 
L11 = L22 = log (F) , 2lr 
where po indicates the permeability of free space and the 
distance along the line is 
D ( z )  = Do + z(D1 - Do). 
As the surrounding medium is homogeneous we can easily 
derive the per-unit-length capacitance matrix, 
C ( z )  = &oPoL-l(z), 
where EO is the permettivity of free space. 
We will apply a voltage source consisting of a 1 MHz , 
50% duty cycle trapezoidal pulse train with raise and fall 
times equal to 20 ns to one of the two wires at the edge 
t = 0, and calculate the near end crosstalk on the other 
967 
Figure 4: Near end crosstalk for the nonparallel 
wires above a ground plane. The thin continu- 
ous line represents the weak solution and the thick 
dashed line is the reference solution. 
wire. Each conductor is terminated with a 50 R resistance 
connected to the reference. 
As there is no closed form solution for lines of this type, 
we used an approximate method to obtain a reference solu- 
tion for this problem. The standard approach is to divide 
the line into N,  uniform subsections and to perform the 
analysis in the frequency domain [8, 21. Each subsection 
is analyzed separately by deriving its chain matrix, which 
can be evaluated in closed form. The chain matrix of the 
overall structure is then obtained by multiplying the chain 
matrices of each subsection of the line, and the solution for 
the voltages and currents at the line ends is found by in- 
corporating the terminal conditions. Finally, inverse FFT 
is applied to get the time domain waveform. This method 
converges to the exact solution when N,  increases. Some 
numerical tests on the convergence have been conducted to 
obtain the minimum number of subdivisions that insures 
a good approximation for the nonuniformity of the line. A 
number of Nz = 32 subdivisions resulted beyond this limit 
and was used in the simulations. 
Figure 4 shows the results of the simulations with our 
method (Nv  = 65) and with the approximate piecewise 
uniform solution. The number of points for the evaluation 
of the inverse FFT in the latter was set to 1024. The two 
curves are undistinguishable. We can conclude that the 
two methods give practically the same results. 
CONCLUSIONS 
A new time-domain scheme for the numerical solution 
of the Nonuniform Multiconductor Transmission Lines 
(NMTL) has been presented. The method is based on 
the spatial expansion of the solution, the external sources, 
and the per unit length matrices into some approximat- 
ing functions. The projection of the NMTL equations 
through suitable test functions leads to a discrete system 
of ODE’S that can be solved by an appropriate time inte- 
gration scheme. The approximation order of the method 
depends on the particular choice of trial and test functions. 
Numerical tests show that small approximation errors can 
be obtained with few high order basis functions. 
REFERENCES 
[l] C. E. Baum, J.  B. Nitsch, and R. J .  Sturm, “Analyti- 
cal Solution for Uniform and Nonuniform Multiconduc- 
tor Transmission Lines with Sources”, The Review of 
Radio Science 1993-96, URSI-Oxford University Press, 
1996, NY. 
[2] P. Besnier, and P. Degauque, “Electromagnetic Topol- 
ogy: Investigations of Nonuniform Transmission 
Line Networks”, IEEE Trans. Electromagn. Compat., 
vol. 37, 1995, 227-233. 
[3] C. R. Burrows, “The exponential transmission line”, 
Bell System Tech. J., vol. 17, 1938, 555-573p. 
[4] F. Chang, “Transient Simulation of Nonuniform Cou- 
pled Lossy Transmission Lines Characterized with 
frequency-Dependent Parameters-Part I: Waveform 
Relaxation Analysis”, IEEE Trans. Circuits Syst. I ,  
vol. 39, 1992, 585-603. 
[5] T. Dhaene, L. Martens, and D. De Zutter, “Transient 
Simulation of Arbitrary Nonuniform Interconnection 
Structures Characterized by Scattering Parameters”, 
IEEE Trans. Circuits Syst. I ,  vol. 39, 1992, 928-937. 
[6] IMSL, IMSL MATH/LIBRARY User’s Manual, Ver- 
sion 2.0, 1991, IMSL, Houston. 
[7] J .  Nitsch, C. E. Baum , “Splitting of degenerate natural 
frequencies in coupled two-conductor lines by distance 
variation”, Interaction Notes, Note 477, July 1989. 
[8] C. R. Paul, Analysis of Multiconductor Transmission 
Lines, John Wiley and Sons, NY, 1994. 
[9] V. K.  Tripathi, N. Orhanovic, “Time-Domain Charac- 
terization and Analysis of Dispersive Dissipative Inter- 
connects”, IEEE Trans. Circuits Svst. I,  vol. 39, 1992, 
938-945. 
968 


Analysis of Multiconductor Transmission Lines,

Analytical Solution for Uniform and Nonuniform Multiconductor Transmission Lines with Sources”, The Review of Radio Science 1993-96, URSI-Oxford

Electromagnetic Topology: Investigations of Nonuniform Transmission Line Networks”,

Splitting of degenerate natural frequencies in coupled two-conductor lines by distance variation”,

The exponential transmission line”,

Time-Domain Characterization and Analysis of Dispersive Dissipative Interconnects”,

Transient Simulation of Arbitrary Nonuniform Interconnection Structures Characterized by Scattering Parameters”,

Transient Simulation of Nonuniform Coupled Lossy Transmission Lines Characterized with frequency-Dependent Parameters-Part I: Waveform Relaxation Analysis”,

https://iris.polito.it/retrieve/handle/11583/1508195/47207/cnf-1998-emcieee-weak-NMTL-IEEE.pdf

Weak solution of the nonuniform multiconductor transmission lines

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