A simple method to compute crosstalk on printed circuit boards

Figure 3 Solid-wire skin effect resistance compared with dc value.Ž . Ž .a Over a wide frequency range. b Illustrating performance at lowfrequencies

computational resource. Although the new solution is basedon the plane analysis for curved conductors at high frequen-

w xcies, as in 3 , the new formula provides an accurate solutionfor all frequencies. Because the approach is based on an

w Ž .xequivalent current density Eq. 9 , we believe that it is notlimited to the round wire illustrated here, and we are cur-rently investigating the application of this approach to irregu-lar cross-section conductors, e.g., microwave strip lines.Moreover, the approach may be extended to determine thereactance of wires of arbitrary cross section at any frequency.

REFERENCES

1. A. W. Lotfi and F. C. Lee, ‘‘Two-Dimensional Skin Effect inPower Foils for High-Frequency Applications,’’ IEEE Trans.Magn., Vol. 31, No. 2, 1995, pp. 1003]1006.

2. L. J. Giacoletto, ‘‘Frequency- and Time-Domain Analysis of SkinEffect,’’ IEEE Trans. Magn., Vol. 32, No. 1, 1996, pp. 220]229.

3. S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Waës inCommunication Electronics, 3rd ed., John Wiley & Sons, NewYork, 1994, pp. 180]186.

4. B. Z. Wang, ‘‘Exact Expressions for the AC Resistance andInternal Inductance of a Lossy Circular 2-Wire Transmission-Line,’’ Microwa e Opt. Technol. Lett., Vol. 7, No. 10, 1994, pp.451]452.

5. L. Rade and B. Westergren, Mathematics Handbook for Scienceand Engineering, 3rd ed., studebtlitterature, 1995.

6. N. W. McLachlan, Bessel Functions for Engineers, 2nd ed., OxfordUniversity Press, Oxford, England, 1961.

7. S. A. Schelkunoff, ‘‘The Electromagnetic Theory of Coaxial Trans-mission Lines and Cylindrical Shields,’’ Bell Syst. Tech. J., Vol. 13,No. 4, 1934, pp. 532]579.

Q 1998 John Wiley & Sons, Inc.CCC 0895-2477r98

A SIMPLE METHOD TO COMPUTECROSSTALK ON PRINTEDCIRCUIT BOARDSJan Carlsson1 and Per-Simon Kildal21 Swedish National Testing and Research InstituteS-501 15 Boras, Sweden˚2 ( )Chalmers University of Technology CTHGothenburg, Sweden

Recei ed 3 March 1998

ABSTRACT: A simple method for computing the crosstalk betweenparallel conductors on printed circuit boards is presented. The crosstalkis computed by using a lumped-circuit element model for the conductors,where the per unit-length parameters are computed by a finite-differencescheme. The method is applied to analyze seëral examples showing howcrosstalk on circuit boards can be reduced. The results are ërified bycomparison with preïously published results and measurements. Q 1998John Wiley & Sons, Inc. Microwave Opt Technol Lett 19: 87]94,1998.

Key words: crosstalk; multiconductor transmission lines

1. INTRODUCTION

Ž .Many EMC problems on printed circuit boards PCBs areŽ .caused by the crosstalk mutual coupling between adjacent

traces on the PCB. Therefore, when designing a PCB, it isimportant to know how the routing should be done in orderto keep the coupling on an acceptable level. It is also impor-tant to be able to compute the coupling before the PCB ismanufactured in order to reduce the development time. Thispaper presents a simple method for computing the crosstalkthat gives results which compare well with measurements.Several examples are given in order to illustrate how thecrosstalk can be reduced by simple methods. The crosstalkcan be computed accurately by using a full-wave formulation

Ž .solved by methods such as the method of moments MoM ,Ž .the finite-difference method FDTD , or the finite-element

Ž .method FEM , but the computational expense of thesemethods is much higher than using the multiconductor trans-

Ž .mission line MTL equations. In this paper, we use the MTLequations as the basis of our formulation. For the MTLequations to be valid, a TEM mode of propagation is as-sumed. For inhomogeneous MTLs or when conductor lossesare included, a pure TEM mode is not supported, but never-theless, the MTL approach can be used as an approximate

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 19, No. 2, October 5 1998 87

solution. The MTL equations can be solved by, e.g., using aw xfinite-difference scheme; see, for instance, Roden et al. 1 .

For simplicity, we have used another approach, which alsogives us the possibility of using a commercial circuit analysis

w xprogram, such as, e.g., SPICE 2 , in predicting the crosstalk.In this approach, we model the MTL as a number of cas-caded short sections of lumped-circuit elements. Anothermethod to solve the MTL equations would be to use thetransmission line model available in the SPICE program, as

w xhas been done by Paul 3 . However, this model is valid forlossless two-conductor lines, which means that we have touncouple the MTL equations in order to use this approachfor crosstalk analysis. It is also difficult to generalize thismethod to include conductor losses which are important forlow frequencies where common impedance coupling domi-nates. The use of a circuit analysis program also means thatwe easily can compute the responses either in the time or inthe frequency domain. Another advantage of using this sim-ple model is that the coupling phenomena can easily beunderstood, and thereby methods for reducing the couplingcan be generated more easily.

The main difficulty in using the MTL approach is that wehave to determine the per-unit-length parameters. For homo-geneous MTLs with a simple cross section, these can becomputed with exact or approximate formulas. For MTLswhich are inhomogeneous or have a complicated cross sec-tion, numerical methods must be used. Several differentmethods for computing the per-unit-length parameters have

w xbeen published; see, for instance, 4]7 . In this paper, we useŽ . w xthe finite-difference method FDM in 8, Sect. 3 , and extend

this to calculate the per-unit-length parameters. One advan-tage of the FDM formulation is that it is straightforward andcan easily be implemented with a computer program.

The described approach for computing the crosstalk canbe used for frequencies where the transmission line theory isvalid, i.e., the TEM mode of propagation is assumed. Themethod can be used for computing the crosstalk between anarbitrary number of parallel lines, and the surrounding mate-rial can either be free space or dielectric. Even a combinationof free-space and dielectric material can be treated, as is thecase for, e.g., a microstrip line. For the case of nonparallel

lines, we can approximate the lines by a number of parallellines placed in a staircase configuration.

2. THEORY

The multiconductor transmission line configuration treated inthis paper is shown in Figure 1. We will consider the config-

w xuration shown in Figure 1 for an N-line 9 . The N-line isdefined as a multiconductor line with N q 1 conductors,including the reference conductor. The reason for consider-ing the configuration in Figure 1 for an N-line is that, for thisconfiguration, we can define N unique voltages and N uniquecurrents. We choose to define the voltages as the potentialdifferences between the conductors and the reference con-ductor, conductor 0. All currents are assumed to be returningto the sources through the reference conductor. The matricesw x w xZ and Z represent the source and load impedances,S Lrespectively. The transmission line equations for an MTL canbe obtained by simply replacing the scalar quantities in theordinary transmission line equations for a two-conductor linewith the corresponding vector quantities. Also, the per-unit-length parameters R, L, C, and G have to be replaced by thecorresponding matrix expressions. The MTL equations in thefrequency domain can thus be written as

d¡ w x w x w xV s y Z I w x w x w xZ s R q jv Ldz~ Ž .where 1½ w x w x w xd Y s G q jv C .w x w x w xI s y Y V¢

dz

The MTL equations can be solved exactly in a similar fashionas the ordinary transmission line equations; see, for instance,w x w x3 and 9 . However, the solution process involves quitecomplicated matrix manipulations, and we have thereforechosen a simpler approach, which is to model the MTL usinga lumped-circuit representation. One advantage of this sim-ple model is that the physics of the problem is preserved, andwe can more easily understand the coupling phenomena. Yetanother advantage is that we easily can implement the modelfor our N-line in an ordinary circuit simulator, such as, e.g.,SPICE, and obtain the desired responses either in the time or

Ž .Figure 1 Multiconductor transmission line N-line excited at z s 0

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 19, No. 2, October 5 199888

in the frequency domain. For our model, which is an approxi-mate model, the line is divided into a number of smallsections. Each section is characterized by the per-unit-lengthparameters. Several different models for the small sectionsare possible; the one that we have chosen to use is known as

Ž .the p-section model Fig. 2 . By comparing the MTL equa-Ž .tions 1 and the equations for the lumped-circuit model

obtained by using Kirchhoffs laws, the following relations areobtained:

R s R , i / j L s M , i / ji j 0 i j i j, ,½ ½R s R q R L s Lii i 0 i i i

Ž .and 2

C s yC , i / j¡ i j mi j~C s C q CÝii i mi j¢j/ i

where the left-hand sides represent the entries in the per-unit-length matrices, the right-hand sides represent the com-ponents in the lumped-circuit model, and M s M , C si j ji mi jC . The dimension of the matrices will be N = N for anm jiN-line. For lossless lines, the resistance R and the conduc-tance G matrices are both equal to zero. The assumption of alossless line is often a very good approximation for real cases,and is consequently often used. It is, however, possible totake conductor losses into account by assuming a finite seriesresistance and keeping the zero shunt conductance. Conduc-tor losses are only important when occurring in the common

Ž .reference conductor ground , and they cause coupling due tothe voltage drop over this ground. This effect dominates theinductive and capacitive coupling at low frequencies. Fortraces on a PCB of normal size, this is usually the case forfrequencies below approximately 100 kHz. The capacitancematrix is a symmetric matrix, i.e., C s C , and the elementsi j jisatisfy the following relations:

C G 0¡ ii~C F 0 for i / ji j¢C s C .i j ji

The ij-element in the capacitance matrix can be determinedby letting the potential on all conductors except the jth beequal to zero and evaluating the charge on the ith conductor,

w x Ž . <i.e., 4 , C s Q rV . Thus, in order to determineV s0, m / ji j i j m

the capacitance matrix for an N-line, we have to solve theLaplace equation for the configuration N times with differ-

ent boundary conditions. The solution of the Laplace equa-tion gives the potential distribution in the region, and byapplying Gauss’ law, we can determine the charge per unitlength on conductor i as Q s yH «=V ? n dl , where l is aî l i ii

closed line around conductor i, n is an outward directed unitˆvector, and V is the potential distribution. The inductancematrix can be computed by the knowledge of the capacitancematrix for the case when all material in the cross section is

w x w x w xy1 w xfree space, i.e., 5 , L s m « C , where C is the0 0 0 0capacitance matrix when all dielectric material in the crosssection is replaced by free space. The solution of the Laplaceequation is obtained numerically by using a finite-differenceapproach. Starting with Maxwell’s equation for the two-di-mensional electrostatic case,

D s « EŽ .3E s y=V½

= ? D s 0

and insertion of the first and second relations in the lastŽ .relation in 3 gives for the homogenous case,

2 2 V V V VŽ .= ? «=V s = ? x« q y« s « q s 0.ˆ ˆ 2 2ž / x y x y

Ž .4

Ž .In order to discretize Eq. 4 , we approximate the derivativesw xwith the following central differences 10 :

2 V q V y 2V V iq1, j iy1, j i , jf ,2 2 x D xŽ .5

2 V q V y 2V V i , jq1 i , jy1 i , jf2 2 y D y

where the indexes i and j are defined as shown in Figure 3.Ž . Ž .Equation 4 , together with the approximations 5 , stipulates

that the potential at node i, j can be determined by theknowledge of the potentials at the surrounding nodes. Thus,by defining a number of nodes in the region that we areinterested in and stepping through the nodes, we can deter-mine the potential at all nodes. In order to do this, we needto have a finite region with a closed boundary. We also needto define the potentials at the boundary nodes, i.e., we putthem to zero. If we have an infinite problem region, such as,e.g., an unshielded microstrip line, we have to truncate theregion. The truncation of an infinite region will, in practice,

Ž .Figure 2 Lumped-circuit section p-section for a 2-line


Figure 3 Part of the finite-difference mesh

not cause us any problems since we can make the finiteregion sufficiently large so that the boundary will practicallyhave no influence on the results. By inserting the approxima-

Ž . Ž .tions 5 into 4 and rearranging, we are able to express thepotential at node i, j by the potentials at the four surround-ing nodes as

Ž .V s C V q C V q C V q C V 6i , j y iy1, j y iq1, j x i , jy1 x i , jq1

where the constants C and C are given byx y

D x2 D y2

C s , C s .x y2 2 2 2Ž . Ž .2 D x q D y 2 D x q D y

Now, if we give all nodes in the region an initial estimate V 0 ,i, jwe can compute the potential at all nodes by an iterativeprocedure. By scanning through the nodes in the grid fromleft to right starting with the bottom row, i.e., increasing ibefore j, the following iterative formulation is obtained:

nq1 nq1 n nq1 n Ž .V s C V q C V q C V q C V 7i , j y iy1, j y iq1, j x i , jy1 x i , jq1

where the superscript n stands for iteration number n. Equa-Ž .tion 7 is commonly referred to as a Gauss]Seidel iteration

w x8, Sect. 3.3.2 . When we have one or several subregions withdifferent permittivities within the problem space, special caremust be taken. The constraint we must put on the solution isthat the normal component of the electric flux density D mustbe continuous across the boundary. For the case of a horizon-tal boundary between two subregions with dielectric con-

Ž .stants « and « , Eq. 7 has to be modified as explained inA Bw x8, Sect. 3.4.3 which we here briefly repeat. Referring toFigure 3, the electrostatic potential in the region above theinterface between the two dielectric materials satisfies theLaplace equation =2V B s 0. And similarly for the electro-static potential in the region below the interface, =2V A s 0,where subscripts A and B denote the respective dielectricregion. Treating both regions as if they both are homogenousand filled with material with dielectric constants « and « ,A B

Ž . Žrespectively, Eq. 7 can be written as with the iteration

.number omitted

A , B A , B A , B A , B A , B Ž .V s C V q C V q C V q C V . 8i , j y iy1, j y iq1, j x i , jy1 x i , jq1

Ž . B AIn Eq. 8 , V and V are fictitious, and can be elimi-i, jy1 i, jq1nated by the following relations:

¡ A BV s V s Vi , j i , j i , j

A BV s V s Viy1, j iy1, j iy1, j~A BV s V s Viq1, j iq1, j iq1, j

A A B B¢« V y V s « V y VŽ . Ž .A i , jq1 i , jy1 B i , jq1 i , jy1

where the last relation represents the requirement of acontinuous normal component of the electric flux density

Ž .across the boundary. Finally, by multiplying Eq. 8 withŽ .« q « and using the above boundary conditions, we ob-A Btain

V nq1 s C V nq1 q C V ni , j y iy1, j y iq1, j

2« 2«A Bnq1 n Ž .q C V q C V . 9x i , jy1 x i , jq1« q « « q «A B A B

Ž .It should be noted that Eq. 9 is valid also for homogenousregions, i.e., when « s « . Vertical boundaries can beA Btreated in an analogous way. The Gauss]Seidel iteration

Ž . Ž .procedure in 7 and 9 has the disadvantage of a quite slowconvergence rate. In order to increase the convergence rate,

w xwe use an overrelaxation method 8 . For this, we let the newpotential value be equal to the old value plus a factor timesthe residual. Thus, V nq1 s V n q aR where the residual isi, j i, j i, jdefined as R s V nq1 y V n . The relaxation factor a shouldi, j i, j i, jbe greater or equal to unity, and must be smaller than 2 forthe iteration to converge. Unfortunately, the optimum valuefor the relaxation factor is strongly problem dependent, andthere is no general method to find its value. However, inorder to get an estimate of the optimum value for the


w xrelaxation factor, we use the following formula 8 :

1 1Ž .a s 2 y p 2 q 100 2 2( ž /N Nx y

where a rectangular finite-difference mesh is assumed, andŽ . Ž .N q 1 , N q 1 are the number of nodes in the x- andx yy-directions, respectively. When we know the capacitance andinductance matrices for our N-line, we can easily set up thelumped-circuit model and implement the circuit in a SPICEprogram. If we need high accuracy or need to compute theresponses for high frequencies, we might need to put severallumped circuits in cascade. Since all lumped-circuit sectionslook the same, it is easy to write a computer program thatgenerates the circuit file for an arbitrary number of sections.

3. TEST CASES

As a test of the accuracy of the FDM method for computingthe per-unit-length matrices for open regions, we have com-pared the results for several configurations with computationsperformed with other methods. The first comparison is for

w xthe coupled microstrip configuration in 5, Fig. 4 . The resultsin Table 1 show that our results are in good agreement with

w x w xresults previously reported in 5 and 6 . The difference

w xbetween our results and those of 5 , which are obtained by amoment method approach, is less than 4%. The difference

w xbetween our results and those of 6 is less than 3.5%. Ourcomputation was done with a mesh size of 601 = 281 nodesin the x- and y-directions, respectively. The step size was 0.05

Ž .in both directions D x s D y s 0.05 . The difference shouldbecome smaller if we increase the number of nodes anddecrease the step size.

When we use the lumped-section model for an MTL, wehave to know how many lumped-circuit sections we need touse per wavelength for obtaining reliable results. In order toinvestigate this, we have compared results obtained with anexact solution of the MTL equations with results obtainedwith the lumped-circuit approach. The test was done for the

Ž .lossless homogenous 2-line shown in Figure 4 a . The lengthof the MTL was chosen to be 0.2 m, and with the given

TABLE 1 Comparison of Results for the Coupled Microstrip[ ]in 5, Fig. 4 ; Units are pF ///// m

w x w xOur Results Ref. 5 Ref. 6

C 95.31 91.65 92.2411C y8.25 y8.22 y8.5012C y8.25 y8.22 y8.5021C 95.31 91.65 92.2422

Ž . Ž .Figure 4 a Geometry for 2-line used for comparison between exact solution and lumped-circuit approach. b Computed near-endŽ .crosstalk coefficient for the 2-line in a


parameters, the velocity of propagation on the MTL is that offree space. The computed responses were the near-end andfar-end crosstalk coefficients, which are defined as CT sNEŽ . Ž .V rV , CT s V rV . Two dif-Victim < zs0 Source FE Victim < zsL Source

Ž .ferent load cases were considered; referring to Figure 4 a ,

1: R s R s R s 50;S11 L11 L22

2: R s R s R s 1000.S11 L11 L22

Load case 1 represents the case when inductive couplingbetween the lines dominates, and load case 2 represents thecase when capacitive coupling dominates. The computed re-

Ž .sults for the near end are shown in Figure 4 b ; the resultsfor the far end are similar. From the results, it can be seenthat the agreement between the exact solution and thelumped-circuit approach is good for frequencies up to wherethe length of the lumped sections is approximately one tenthof a wavelength. Thus, as a rule of thumb, at least ten lumpedsections per wavelength should be used in order to predictthe end responses accurately.

As another test case, the crosstalk between two parallellines on a PCB was computed and compared with measure-

Ž .ments. The PCB was a glass-epoxy board « s 4.7 with arcopper plane on one side and two parallel lines on the otherside. Both lines were 2.2 mm wide, and the center-to-centerdistance was 4 mm. One of the lines was excited at one end,and terminated with a 50 V termination at the other end.The second line was terminated with 50 V terminations inboth ends. The length of the lines was 160 mm. Measure-ments were performed with a network analyzer, and for thecomputation, the MTL was modeled as seven cascadedlumped-circuit sections. The per-unit-length parameters forthe MTL were computed with the FDM with a mesh size of361 = 161 nodes. The step size was 0.05 in both directionsŽ .D x s D y s 0.05 . The results are shown in Figure 5, wherean excellent agreement between measurements and computa-tions can be observed for frequencies up to the first reso-nance. For higher frequencies, the computed near-end re-

sponse starts to deviate from measurements, while the far-endresponse is still acceptable.

4. METHODS FOR REDUCING THE CROSSTALK ON PCBS

Now, when we know that the method presented in the previ-ous sections can be used for predicting the crosstalk betweenlines in an MTL, we proceed with investigating how we canminimize the crosstalk. A well-known and simple method toreduce the crosstalk is to place a shield trace between the

w Ž .xsource and the victim traces Fig. 6 a . From the results forthe far-end crosstalk coefficient for the case of dominatinginductive coupling, it can be seen that the shield trace has tobe grounded at both ends to be effective for reducing the

w Ž .xcrosstalk Fig. 6 b . This fact can be understood if oneconsiders the mechanism behind the inductive coupling, whichis the magnetic field produced by the current in the sourcetrace. The magnetic flux penetrating the loop formed by thevictim trace and the reference conductor will induce a cur-rent in the victim trace, and thereby a voltage over theterminating impedances. If we place a shield trace which isgrounded at both ends between the source and victim traces,a current also will be induced in the shield trace. This currentwill have a direction that will cause a magnetic field in theopposite direction. Therefore, the net magnetic flux penetrat-ing the loop formed by the victim and reference conductorwill be lower, and the result is lower crosstalk. Accordingly, itis important that a current can flow in the shield trace for itto be effective in reducing crosstalk by inductive coupling. Onthe other hand, when capacitive coupling dominates, it issufficient to ground the shield trace at only one end, as can

Ž .be seen from Figure 6 c . For this case, we can view theshield trace as a decoupling capacitor that is added to thecircuit.

Ž .From Figure 6 d , it can be seen that it is advantageous tolocate the shield trace close to the source trace, or evenbetter, close to the victim trace, rather than placing it in themiddle. However, we have to place the shield trace very closeto either trace in order to obtain a significant difference. It is

Figure 5 Measured and computed crosstalk coefficients for a PCB with two parallel lines


Ž . Ž .Figure 6 a PCB with a shield trace between the source and the victim traces. b Computed far-end crosstalk coefficient for theŽ . Ž .PCB in a . RL s RL s RS s 50 V, W s W s 1 mm, d s d s 1 mm. c Computed far-end crosstalk coefficient for the PCB11 22 22 S 1 2

Ž . Ž . Ž .in a . RL s RL s RS s 1 kV, W s W s 1 mm, d s d s 1 mm. d Computed far-end crosstalk coefficient for the PCB in a .11 22 22 S 1 2RL s RL s RS s 50 V, W s W s 1 mm, d s 0.1 mm, d s 1.9 mm11 22 22 S 1, 2 2, 1


Figure 6 Continued.

Ž .also evident from Figure 6 d that we can gain a significantreduction in crosstalk by using a shield trace that is as wideas possible.

5. CONCLUSION

A simple and effective method for computing crosstalk be-tween parallel lines has been presented. The method can beused for computing crosstalk between traces on a printedcircuit board as well as, e.g., between different pins in aconnector. The main advantage of the method is that theresponses can be computed with an ordinary circuit simulatorsuch as, e.g., the SPICE program either in the time or thefrequency domain. The method for computing the per-unit-length parameters has been tested against published resultsand measurements. The agreement has been found to begood. We have also shown how the crosstalk on a PCB can bereduced by placing a shield trace between the source andvictim traces.

REFERENCES

1. J. A. Roden, C. R. Paul, W. T. Smith, and S. D. Gedney,‘‘Finite-Difference, Time-Domain Analysis of Lossy Transmis-sion Lines,’’ IEEE Trans. Electromag. Compat., Vol. 38, Feb.1996, pp. 15]24.

2. L. W. Nagel, ‘‘SPICE2: A Computer Program to Simulate Semi-conductor Circuits,’’ Memorandum M520, May 1975.

3. C. R. Paul, Introduction to Electromagnetic Compatibility, JohnWiley & Sons, New York, 1992.

4. R. Laroussi and G. I. Costache, ‘‘Finite-Element Method Ap-plied to EMC Problems,’’ IEEE Trans. Electromag. Compat., Vol.35, May 1993, pp. 178]184.

5. C. Wei, R. F. Harrington, J. R. Mautz, and T. K. Sarkar,‘‘Multiconductor Transmission Lines in Multilayered DielectricMedia,’’ IEEE Trans. Microwa e Theory Tech., Vol. MTT-32,Apr. 1984, pp. 439]450.

6. W. T. Weeks, ‘‘Calculation of Coefficients of Capacitance ofMulticonductor Transmission Line in the Presence of a Dielec-tric Interface,’’ IEEE Trans. Microwa e Theory Tech., Vol. MTT-18, Jan. 1970, pp. 35]43.

w x7. G. Plaza, F. Mesa, and M. Horno, ‘‘Quick Computation of C ,w x w x w xL , G and R Matrices of Multiconductor and MultilayeredTransmission Systems,’’ IEEE Trans. Microwa e Theory Tech.,Vol. 43, July 1995, pp. 1623]1626.

8. P.-b. Zhou, Numerical Analysis of Electromagnetic Fields,Springer-Verlag, Berlin, 1993.

9. K. S. H. Lee, Ed., ‘‘EMP Interaction: Principles, Techniques andReference Data,’’ Air Force Weapons Lab., Kirtland AFB, NM,Dec. 1980.

10. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathemati-cal Functions, Dover, New York, 1970.

Q 1998 John Wiley & Sons, Inc.CCC 0895-2477r98

SYNTHETIC BANDWIDTHCORRELATION RADIOMETER FOR THERESOLUTION IMPROVEMENT OF ATHERMAL MICROSENSORD. Allal,1 B. Bocquet,1 and Y. Leroy11 Institut d’Electronique et de Microelectronique du Nord´Departement Hyperfrequences et Semiconducteurs´ Ú.M.R. CNRS 9929Universite des Sciences et Technologies de Lille´59655 Villeneuve d’Ascq Cedex, France

Recei ed 8 April 1998

ABSTRACT: We haë concei ed a distributed thermal sensor based onthe processing of the correlation function of the thermal noise emitted bythe two ports of a small-size lossy transmission line in the microwa e

( )domain. The transducer is a lossy coplanar waëguide LCPW wherelosses are created by a thinning down of the central conductor. The twoports of the LCPW are connected to a microwa e correlation radiometer.The knowledge of electrical characteristics of the LCPW and an originalapplication of the Kalman filtering to the signals recorded as a function


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A simple method to compute crosstalk on printed circuit boards