Derivation of Equivalent Circuits for. Multilayer Printed Circuit Board. Discontinuities Using Full Wave Models. Edward Pillai, Student Member, IEEE, and Werner ...
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 9, SEPTEMBER 1994
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Derivation of Equivalent Circuits for Multilayer Printed Circuit Board Discontinuities Using Full Wave Models Edward Pillai, Student Member, IEEE, and Werner Wiesbeck, Fellow, IEEE
Abstract- Recent advances in increasing transmission rates for digital communication systems necessitates special attention during printed circuit board (PCB) design. This contribution provides a methodology for developing appropriate simple equivalent circuits to describe the discontinuities in multilayer boards. The effects of inhomogeneoussubstrate material, via holes, crossovers and coupled lines are examined. The incorporation of substrate loss in the difference equation system is additionally treated. The equivalent circuits are intended for system simulation, where minimal computational overhead is desired. The Finite Difference Frequency Domain (FDFD) and Finite Difference Time Domain (FDTD) methods provide in conjunction the basis for calculating Of the discontinuities, the scattering parmeters [SI and from which equivalent circuits are derived. Excellent s parameter agreement between field computation and equivalent circuit is achieved upto 20 GHz for all structures considered. Measurements are indicated where appropriate.
Fig. 1. spital Multilayer FCB including via holes, crossovers and coupled lines designed on low cost inhomogeneous substrate.
I. INTRODUCTION
A
S data rates in high speed digital transmission systems using light wave technology move into the ever increasing GbiVs range, it has become not only necessary to characterize the occurrence of frequency dependent discontinuities that lie in the PCB signal path, but also to design and implement them in computer aided design (CAD). A typical PCB with dielectric material removed, is shown in Fig. 1. Here, unwanted effects such as dispersion, resonances, reflections, EXCITATION PLANE A coupling and radiation can be expected. These shortcomings Fig. 2. Microstrip discretization indicating excitation planes. have taken on particular significance as multiplexor technology in demodulating light wave signals arriving with frequencies upto 100 GHz is currently in the development process. This The FDTD method [3]-[6] is a powerful tool, yet cannot calls for simple design rules with respect to discontinuity fulfill all requirements pertaining to the solution of planar geometry. Reference [11 developed an equivalent circuit for structures. One particular shortcoming is the inability to hana via hole, but as the via showed resonances and dispersion dle frequency dependent behavior of substrate material in a over the frequency range S parameter agreement for equivalent straight forward manner, especially when its behavior is the circuit and FDTD proved difficult to attain. Similarly, as dominating factor. This is seen in [l], where discrepancy bethe dimensions of the via were electromagnetically large, [2] tween FDTD computation and measurement can be accounted required a sixth order equivalent circuit to achieve agreement for by the difficulty in modeling the loss tangent of the between a FDTD pulse and an equivalent circuit pulse for a polyphenylene oxide (PPO) substrate [ 1 I] used. This is treated microstrip via. Although a higher order equivalent circuit can here by sinusoidal excitation for the FDTD and the memory be accurate, it has the additional drawback of causing overhead intensive convolution algorithm [161, incorporating frequency during its use in system simulation. dependency is avoided. Both the FDTD and FDFD methods are used in coniunction as analysis tools. The versatility of Manuscript received October 29, 1993; revised April 2, 1994. both methods lies in the number of physical features they The authors are with the Institut fuer Hoechstfrequenztechnik und Elekincorporate. The FDFD method [9], [lo] includes definition tronik, University of Karlsruhe, 76128 Karlsruhe, Germany. of nonequidistant Cartesian cells, metal thickness, ohmic loss, IEEE Log Number 9404130. 0018-9480/94$04.00 0 1994 IEEE
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PILLAI AND WESBECK DERIVATION OF EQUIVALENT CIRCUITS FOR MULTILAYER PRINTED CIRCUIT BOARD DISCONTINUITIES
Fig. 5. Crosssectional impression of (3200 substrate; Glass fibres pressed together with epoxy.
GROUND PLANESA Fig. 3. Microstrip via configuration: side view and front view respectively; PPO substrate and air as media. Taken originally from [l]. Microstripline width of 3.3 mm.
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Fig. 6. Material discretization patterns; Microstripline (white square), epoxy (grey), glass (black). (a) Microstripline (width = 0.1833 mm, sub. height = 0.36 mm) placed directly above glass fibre bundle, (b) Microstripline placed in between two glass fibre bundles, (c) Discretization to model glass fibres at 45 degrees to microstripline, (d) Pattern indicating the relative Cartesian coordinate positions of the electric field components directly below the microstripline.
(b) Fig. 4. Magnitude 5’11 and ,912 for the via in Fig. 3 compared with values from the literature. (a) Magnitude S11 (b) Magnitude Slz.
conductivity ((T), dielectric constant ( E ~ )and loss tangent (tans). Off grid diagonal material variation for (T, E , t a n s can also be defined using weighted field components [17]. The FDTD method is similarly adapted to handle most of the features listed above, including, additionally the presence of a Mur’s first order Absorbing Boundary Condition (ABC) [15]. In both cases the staircase approximation is used, where sufficient discretization causes minimal discrepancy [6]. The popular method of validating computational results is to compare with measurement. The dimensional limits placed on the PCB structures here however, produce only parasitic variation in electromagnetic behavior and cannot be measured with the accuracy offered by conventional Thru
Reflect Line (TRL) [13] calibration techniques. Hence an analytical approach is better suited to calculate the incremental impedances [8] of these triplate structures, as an interference free environment is guaranteed. Comparison of the FDFD and FDTD methods with each other is provided for verification of results, as is comparison to measurement of structures with larger dimensions [ 141. Hence the development of equivalent circuits for PCB discontinuities with dimensional limits must be based on computed S parameters. The FDFD method requires the solution first of an exact 2D eigenvalue problem for the two excitation planes. Then the 3D equation system is solved iteratively. Although the FDTD has a predefinable stability criterium and guarantees convergence, it requires a longer computation time with resonant structures, as oscillations from the pulse excitation need to die down. The geometries of the PCB structures considered here exclude such
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 9, SEPTEMBER 1994
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(6) (b) Fig. 7. The magnitude of the electric field computed using the FDFD. (a) corresponds to Fig. 6(a) pattern. (b) corresponds to Fig. 6(b) pattern.
Fig. 8. Response to Gaussian excitation (FDTD)as registered in time steps for the positions indicated in Fig. 6(d). (a) Inhomogeneous substrate case as depicted in Fig. 6(d) (glass = 5 . 5 , epoxy = 3.6). (b) Homogeneous case for comparison cr = 3.9.
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TABLE I SEVENMICROSTRPLINES MEASURED FOR EACH OF THE CASESGIVEN IN ORDERTO OBTAIN ADDITIONAL POSITIONAL VAFUATION INFORMATION
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distortion and deliver healthy computation times. For these structures with approximately 150000 cells, the FDFD method requires approximately 5 minutes per frequency point ( 2 minutes vector time) on a Fujitsu vectorized 5 Gflop machine and the FDTD about 45 minutes per complete simulation (no vector time). In the sinusoidal excitation mode, where a fewer number of time steps are required until steady state is achieved, the computation lasts approximately 15 minutes. 11. DESCRIPTION OF THE STRUCTURES AND
DISCONTINUITIES INVESTIGATED
In order to make the PCB's cost effective, low cost materials are sought. FR4 and (3200 are substrates made up of glass and epoxy having differing material properties. The geometrical
Fig. 9.
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Magnitude Slz for the structures of Fig. 6(a)-(c).
variation of the glass layers in the substrate may affect the transmission characteristics. This is tested by calculation and measurement of propagation characteristics along a transmission line, with respect to its position and orientation on the surface of the substrate. The FDFD algorithm is used to calculate the scattering parameters and fields for various cases of material distribution. The dimensions of these geometrical material distributions are accurately modeled, based on scaled cross sectional microscopic photographs of the substrate. The scattering parameter results show limited variation with position in the frequency range of interest.
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PILLAI AND WIESBECK: DERIVATION OF EQUIVALENT CIRCUITS FOR MULTILAYER PRINTED CIRCUIT BOARD DISCONTINUITIES
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Fig. 11. Triplate via configuration, side view and front view respectively. G200 substrate used. Stripline width = 112 mm, thickness = 50 mm.Reference plane for S parameters taken at via edge.
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Fig. 12. Magnitude of electric from the FDFD.
Fig. 13. Equivalent circuit used to fully describe the via in Fig. 11.
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(d) Fig. 10. Measurement of 15 cm lines with orientation variation. (a) line width = 0.3833 mm, G200 substrate. (b) line width = 0.1833 nun, (3200 substrate. (c) line width = 0.3833 mm, FR4 substrate. (d) line width = 0.1833 mm, FR4 substrate.
Further accurate measurements of the scattering parameters categorically show no significant variation. This allows the use of quasi-TEM analysis to determine an equivalent dielectric constant E , and loss tangent t a n 6 for the combined glassepoxy material in both the case of FR4 and G200 substrates valid for transmission rates with bandwidths upto 20 GHz. Based on the material parameters ascertained for the G200 substrate the next step is to characterize the triplate via discontinuity. By interpretation of the electric (capacitive) and magnetic (inductive) field in the vicinity of the via attained using the FDFD,it is possible to derive an equivalent circuit.
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The via is varied in height (H),rod diameter ( d ) and platform diameter (D), and the S parameters calculated using the FDFD. The S parameters of the equivalent circuit are fitted to those of the FDFD. The circuit component values derived during the fitting procedure are listed in a table, except for the resistance due to substrate loss, which is frequency dependent and displayed graphically. In a similar manner the FDTD is used with sinusoidal excitation to calculate coupling between conductors in a crossover configuration. The use of a sinusoidal excitation allows for the modeling of high substrate loss ( t a n s ) encountered in
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 9, SEPTEMBER 1994
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Fig. 14. S parameter agreement between FDFD and equivalent circuit with variation in via diameter (6).
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the G200 substrate using an equivalent conductivity (c)and circumvents the need to use the tedious convolution algorithm [16] for frequency dependent behavior in the time domain. When no adverse electromagnetic effects are expected from a discontinuity a fewer number of frequency points need to be calculated, hence the sinusoidal excitation option is a worthy alternative. Further the case of coupled triplate lines is investigated. Here too, consideration of the field distributions leads to the derivation of an equivalent circuit, whose 5' parameters are fitted to obtain circuit component values. The values are computed with respect to variation of the separation (s) between the lines. The length (1) of the coupled lines for which the equivalent circuit holds, is additionally tested.
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The FDTD [4] and FDFD [9] are well documented and will not be dealt with in detail. The emphasis is placed here on adapting the algorithms in such a way as to reflect the dominant physical characteristics encountered in the discontinuities. The use of the G200 substrate means that substrate loss overrides other forms of loss and must be dealt with in computation specifically. Initially, a structure editor is used to define the discontinuities, which can be read in by both the FDTD and FDTD algorithms. The editor provides definition of E ~ p, r , tanS, (T and metal, including a non-equidistant grid for each Cartesian cell shown in Fig. 2. The FDTD requires additionally the definition of time constants so that the computation is stable and the optimum pulse response is achieved. The FDFD algorithm employs the integral form of the time harmonic equations for a source free structure.
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111. ADAPTATION OF THE FDFD AND FDTD METHODSFOR SUBSTRATE LOSS
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Fig. 15. (a) ,511, phase comparison between the FDFD and equivalent circuit for the via case H =270 jtm, h = 280 jtm, d = 120 jtm (Ref. plane taken at via edge), (b) Frequency dependence of resistance R with respect to variation in via height (H), (c) Frequency dependence of resistance R with respect to variation in via diameter (6), (d) Frequency dependence of resistance R with respect to variation in via platform diameter ( D ) .
where
PILLAI AND WIESBECK: DERIVATION OF EQUIVALENT CIRCUITS FOR MULTnAYER PRINTED CIRCUIT BOARD DISCONTINUITIES
5 13.9 w = 112pm h=280pn s-270pm
the gaussian pulse [5], except inside the computation domain, so that the excitation plane A shown in Fig. 2 is at all times an ABC wall. The substrate loss is then taken into account using an effective conductivity (0)valid for that frequency. This formulation of the FDTD is particularly suited to dealing with coupled line structures with substrate loss, as the S parameters for the four ports can be registered in the time evolution of signals, avoiding the use of odd and even mode excitation used in frequency domain methods. The central difference time domain equation system is well documented [4]. The introduction of substrate loss only requires modification of the electric field equations. The corresponding Maxwell equation is +
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The discretization of (3) for the Electric Field E, component using the Yee cell [3] with (i,j,k ) dependence yields
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Fig. 16. Triplate crossover configuration, (a) The geometrical data pertaining to the computation, (b) The magnitude of the electric field indicating the coupling effect as captured using the FDFD.
Ell and E, have similar forms. By considering the real and imaginary parts of the time harmonic case of (3) an effective conductivity can be derived. u = tan ~ W E O E , .
Equation (2) takes into account the substrate loss (tans) of G200. The discretization and term relocation of (1) according to a Yee cell [3] using a lowest order integration formula applied to the excitation planes A and B of Fig. 2, produces a 2D eigenvalue problem. The eigenvalues computed are the complex propagation constants, which are in turn used with gaussian elimination to derive the 2D fields. With the remaining surfaces of the structure modeled as electric walls, an equation system for a closed 3D structure is derived. An iterative equation solver is used to converge on the 3D fields. Convergence of the algorithm for material with high substrate loss housing the discontinuities proves to be good. The S parameters are gained by orthogonal decomposition of the fields. The treatment of substrate loss poses difficulty in the time domain, as it is frequency dependent. This can be treated with a convolution [16], but the computation overhead is high. Indeed, when the structure is electromagnetically well behaved, interpolation from a few frequency points is sufficient to observe S parameter variation. The time harmonic use of the FDTD is achieved by using a monofrequent (sinusoidal) excitation. This is applied in the traditional manner, similar to
(5)
Equation (5) placed in (4) can now treat substrate loss. For verification purposes of the FDFD and FDTD methods outlined hitherto, the S parameters of a via structure with electromagnetically large dimensions connecting through a ground plane, taken from [l] shown in Fig. 3 is calculated and compared to the traditional FDTD approach used by them. The transmission parameter IS121 in Fig. 4(b) from the traditional method shows some discrepancy with respect to the measurement from [7]. The treatment of substrate loss clearly provides better agreement with measurement.
Iv.
INHOMOGENEOUS SUBSTRATE ANALYSIS AND
RESULTS
The accurate analysis of FR4 and G200 substrates is based on material dimensions attained by way of microscopic photographs, Fig. 5. From the photographs a repetitive pattern is recognizable. The bundles of glass fibres are seen to approximate an elliptic structure. Hence the cross sectional areas of the bundles are approximated by an ellipse. The two extreme cases measured are calculated with the FDFD algorithm. The case of Fig. 6(a) demonstrates the microstrip line directly above the bundle and the case of Fig. 6(b) shows the microstrip line placed in between two bundles. Initial reference values for the dielectric constants and loss tangents of glass and epoxy are taken from [12] (glass E, = 5.5 and tan6 = 10.10-3, epoxy E, = 3.6 and tan 6 = 20.10W3).
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 9, SEPTEMBER 1994
0.00 -0.02 -0.04
. 8
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Fig. 17. S parameter agreement for a one element equivalent circuit with C = 3 5 fF and s = 2 7 0 pm. (a) Magnitude (c) Magnitude 513.
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Fig. 18. Triplate coupled lines configuration, (a) The geometrical data pertaining to the computation, (b) Equivalent circuit to fully describe the coupled lines of Fig. 18(a).
Fig. 7 shows the difference in the Electric Field magnitude with position. From the intensity of the field just below the microstrip line in Fig. 7(a), it is clear that in the case of Fig. 6(a) the field is better conserved than that of Fig. 6(b). This is due to the higher dielectric constant of glass, which supports higher field intensity in the substrate and less radiation with quasi-TEM mode propagation. Another interesting effect seen, is the way in which the electric field in the glass below the microstrip in Fig. 7(a) having a higher dielectric constant couples better to the epoxy below it, which has a lower dielectric constant. The reverse is true in Fig. 7(b). To test the orientation dependence, an additional discretization with glass fibres at 45 degrees to the direction of transmission is conducted as shown in Fig. 6(c). Here an impression of a cross section of the substrate as seen from above indicates how the pattern in Fig. 6(a) is made to follow a diagonal path instead of parallel to the direction of propagation. On the adjacent plane above or below that shown in Fig. 6(c),
Sii, (b) Magnitude Siz,
the next group of glass fibre bundles is also placed diagonally except to follow the opposite direction (90 degrees to Fig. 5 pattern). The FDTD is used on this pattern and the gaussian pulse (100 GHz bandwidth) variation is registered for the E, component directly under the transmission line (TIL)for the Cartesian positions shown in Fig. 6(d). Fig. 8(a) shows how an interference pattern is set up for the pulse passing through the different points. Fig. 8(b) contrasts the homogeneous case. However, to see the precise variation in transmission for a given frequency range the S parameters need to be calculated for all cases considered. The corresponding ISlz( result is shown in Fig. 9, where only minor variations are noted. In order to add greater validity to the results, measurements are made. Seven lines per orientation are measured corresponding to Table I. By measuring upto 7 lines, it is expected in addition to seeing orientation variation, to discover the influence of positional variation with respect to the nearest bundle of glass fibre. Hence conductor width is also varied for two impedances. Substrate orientation is then chosen for 0 degrees and 45 degrees. For both the FR4 and the G200 materials, this produces 8 x 7 measurements as can be interpreted from Table I. The measurements with the HP85 10B vector network analyzer are TRL calibrated for each material and orientation with the microstrip line lengths set at 15 cm. The measurements show minimal variations. Figures l o a 4 show results of lSl2l measurements for the cases of orientation variation referenced in Table I. Positional variation is unnoticeable. As expected the reference values for the dielectric constants and loss tangents used for glass and epoxy in modeling the substrates, caused the measurement results to be marginally different to the FDFD curves of Fig. 9. However a linear factorial relationship is seen. As the detailed investigation into substrate effects in the frequency range upto 18 GHz shows no significant variation, a quasi-TEM based analysis [ 121 is used to optimize equivalent dielectric constants ( E ~ )and equivalent loss tangents (tan 6) for the combined gZass/epoxy mixture for both G200 and FR4 materials. The measurements produce S l 2 magnitude and effective dielectric constant values. The magnitude of S12 represents the losses in the substrate and conductor. Several attempts are then made using quasi-TEM equations to reproduce the measurement results. For example, FR4's substrate thickness h = 0.36 mm, line width w = 0.3833 mm and line thickness t = 0.035 mm produces the airhubstrate
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PILLAI AND WIESBECK DERIVATION OF EQUIVALENT CIRCUITS FOR MULTILAYER PRINTED CIRCUIT BOARD DISCONTINUITIES
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Fig. 19. S parameter agreement for the triplate coupled lines with respect to line separation (s), (a) Magnitude S ~ Z(b) , Magnitude S14.
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Fig. 20. Frequency dependence of conductance Gk.
effective dielectric constant ~ , t f = 3.2. This enables value assignment of E , = 4.5 and tan 6= 19.2-1(Y3for FR4. Similarly G200 is assigned E , = 3.9 and tan S = 14.10-3 according to measurements.
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(b) Fig. 21. Validity of equivalent circuit with respect to the length of line (1) over the frequency range. (a) m mutual inductance variation, (b) Ck coupling capacitance variation.
structure becomes a larger fraction of the guide wavelength. By interpretation of the electric (capacitive energy) field shown in Fig. 12 attained using the FDFD, the equivalent circuit of Fig. 13 is derived. The equivalent circuit is then fitted for H , d, D variation and the excellent agreement achieved is shown once again in Fig. 14 for the ( d ) variation case. Phase agreement for the case H=270 pm, d=120 pm, D=200 pm is shown in Fig. 15(a) to indicate the extremely accurate modeling. The component values derived using the fitting procedure are shown in Table I1 except for the resistance, which is frequency dependent and shown in Fig. 15(b)-(d).The resistance corresponds to the substrate loss, which increases linearly upto 11 GHz and nonlinearly thereafter due to the presence of higher order modes. This may nevertheless be treated in a circuit simulation program using a simple formula.
AND RESULTS v . VIA ANALYSIS
AND RESULTS OF VI. ANALYSIS CROSSOVERS AND COUPLED LINES
The Triplate via is used in PCBs to connect between striplines forming a crossover in different planes. The triplate structure allows pure TEM propagation and by constraining the largest dimension of the via to one tenth of the guide wavelength at 20 GHz, it is possible to avoid signal distortion and describe the via with a simple equivalent circuit. The via structure shown in Fig. 11 placed in the G200 substrate is calculated with the staircase approximation using the FDFD. Variation of height ( H ) , diameter (d), and platform diameter ( D ) is undertaken and the corresponding S parameters for the case of (d) variation is shown in Fig. 14. In general, increase of any one of the dimensions causes poorer transmission and larger reflection. This effect is further exaggerated with increase of frequency as the
For the calculation of S parameters for structures in this section, the FDTD with sinusoidal excitation including substrate loss is used. As an introductory task the geometry shown in Fig. 16(a) (G200 substrate) with the parameters given is calculated. Fig. 16(b) shows the electric field distribution between the lines as captured by the FDFD method. By noting the capacitive coupling of the electric field, a single element capacitance as an equivalent circuit is used for optimization. The S parameters attained with the FDTD are shown in Fig. 17(a)-(c). For the equivalent circuit with G = 35 fF the corresponding 5' parameters are also shown in Fig. 17(a)-(c). The agreement is extremely good considering the fact that substrate loss is not accounted for in the equivalent circuit. Yet this example demonstrates that, when the dimensions of
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TABLE 111 CAPACITANCE C , , C k AND ~NDUCTANCE L , m WITH VARIATION OF SEPARATION (s).
geometry, linear interpolation of the component values for dimensions in between the steps may be taken.
REFERENCES
0.060
‘
the crossover are in the order given here, firstly coupling is negligible and secondly agreement with an one element equivalent circuit is attainable. Similarly for s = 210 pm and 240 pm, capacitance values of 39 fF and 37 fF are attained respectively. By considering the coupling between parallel lines for the configuration shown in Fig. 18(a), the equivalent circuit of Fig. 18(b) is derived. The conductance G,, capacitance C,, self inductance L and resistance R represent the transmission line equivalent circuit of a single line and the mutual and conductance Gk represent inductance m, capacitance coupling elements between the lines. Again using the FDTD the S parameters of Fig. 19(a) and (b) are computed and the corresponding fitted curves are shown. The equivalent circuit values with respect to the separation s attained during the fitting procedure are given in Table 111, except for G,, R and Gk, which are frequency dependent and Gk is shown in Fig. 20. In Table 111, the self impedance values C,, and L of the single line show little or no variation as expected. In order to test the length of coupled line for which the equivalent circuit holds, the behavior of the coupling elements m and ck are displayed over the frequency range in Fig. 21(a) and (b) respectively. It can be seen that the equivalent circuit is only valid upto 5 GHz when a 3 mm length is considered, but at lmm validity upto 20 GHz is guaranteed. In a circuit simulation program several of these equivalent circuits maybe placed in series to model longer lengths of line. VII. CONCLUSION This contribution addresses the issue of developing equivalent circuits for triplate PCB discontinuities in a comprehensive manner. Firstly, the inclusion of substrate loss for the FDFD and FDTD computational techniques leads to accurate modeling of discontinuities in the G200 substrate. Secondly, triplate discontinuity configurations with dimensional limits are proposed, which avoid S parameter magnitude and phase distortion upto 20 GHz. This enables the derivation of extremely accurate simple equivalent circuits to describe the via in a system design and simulation program. The equivalent circuit itself can be used generally for any triplate discontinuity in that the capacitance and inductance account for phase lag and advance respectively, while resistance compensates for losses. The shortest possible guide wavelength in transmission should not exceed one tenth of the discontinuity’s dimensions. The equivalent circuit component values provided here are with respect to discrete dimensional steps. Yet with a triplate
W. D. Becker, P. H. Harms, and R. Mittra, “Time-domain electromagnetic analysis of interconnects in a computer chip package,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 2155-2163, Dec. 1992. P. Harms, J. Lee, and R. Mittra, “Characterizing the cylindrical via dispp. 153-156, continuity,” IEEE Trans. Microwave Theory Tech., vol. 41, .. Aug. 1992. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Tram. Antennas Propagat., vol. AP-14, pp. 302-307, May 1966. X. Zhang and K. K. Mei, “Time-domain finite difference approach to the calculation of the frequency-dependent characteristics of microstrip dicontinuities,” IEEE Trans. Microwave Theory Tech., vol. Mll-36, pp. 1775-1787, Dec. 1988. D. M. Sheen, S. M. Ali, M. D. Abouzahra, and J. A. Kong, “Application of the three-dimensional finite-difference time-domain method to the analysis of planar microstrip circuits,’’ IEEE Trans. Microwave Theory Tech., vol. 38, pp. 849-857, July 1990. A. Taflove, “Review of the formulation and applications of the finitedifference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion, vol. 10, pp. 547-582, Dec. 1988. S. Maeda, T. Kashiwa, and I. Fukai, “Full wave analysis of propagation characteristics of a through hole using the finite-difference timedomain method,” IEEE Trans. Microwave Theory Tech., vol. 39, pp. 2154-2159, Dec. 1992. P. Harms and R. Mittra, “Equivalent circuits for multiconductor microstrip bend discontinuities,” IEEE Trans. Microwave Theory Tech., Voi. 4i, pp. 6 2 4 9 , an. 1993. S. Haffa, D. Hollmann, and W. Wiesbeck, “The finite difference method for s-parameter calculation of arbitrary three-dimensional structures,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 1602-1610, Aug. 1992. A. Christ and H. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microwave Theory Tech., vol. MTT-35, pp. 688-696, Aug. 1987. W. V. Miinch, “Werkstoffe der elektrotechnik,” Eubner Studienskripten, Stuttgart, 1989. G. Meinke, Taschenbuch der Hochfrequenztechnik, Vol 1. Heidelberg, Germany: Springer-Verlag, 1986. G. F. Engen and C. A. Hoer, “Thru-reflect-line: an improved technique for calibrating the dual six-port automatic network Analyzer,” IEEE Trans. Microwave Theory Tech., vol. MlT-27, pp. 987-993, Dec. 1979. E. Pillai, F. Rostan, and W. Wiesbeck, “Derivation of equivalent circuits for via holes from full wave models,” IEE Electron. Lett., vol. 29, no. 11, pp. 1026-1028, May 27 1993. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromag. Compat., vol. EMC-23, pp. 377-382, Nov. 1981. D. M. Sullivan, “Frequency-dependent fdtd methods using 2 transforms,” IEEE Trans. Antennas Propagat., vol. 40, pp. 1223-1230, Oct. 1992. T. Weiland, “Verlustbehaftete wellenleiter mit beliebiger randkontur und materialbelegung,” Archiv Elektr. Uberrr., vol. 33, no. 4, pp. 170-174, 1979.
Edward Pillai (S’90) received the B.Sc. degree from the University of Wales, Swansea, Great Britain and the M.Eng. degree from McMaster University, Hamilton, Canada, in 1987 and 1991, respectively. He is currently pursuing the Dr.-Ing. degree at the Institut fuer Hoechstfrequenztechnik und Elektronik, University of Karlsruhe, Germany From 1987 to 1989, he worked on telecommunication systems at Tie Communications Inc. and on flight simulators at Atlantis Flight Research, both in Toronto, Canada. His research interests include microstrip patch antennas, optical communications, microwave devices, and microwave circuit design.
PILLAI AND WIESBECK DERIVATION OF EQUIVALENTCIRCUITS FOR MULTILAYER PRINTED CIRCUITBOARD DISCONTINUITIES
Werner Wiesbeck (SM87-F'94) was bom near Munich, Germany in 1942. He received the Dip1.Ing. and the Dr.-Ing. degrees from the Technical University Munich in 1969 and 1972, respectively. From 1972 to 1983, he was with AEG-Telefunken in various positions including head of the Research and Development of the Microwave Division in Flensburg, Germany and Marketing Director Receiver and Direction Finder Division, Ulm, Germany. He had product responsibility for mm-wave radars, receivers, direction finders and electronic warfare systems. Since 1983, he has been Director of the Institut fuer Hoechstfrequenztechnik und Elektronik, University of Karlsruhe, Germany. His research interests include radar, remote sensing, wave propagation and antennas. Dr. Wiesbeck is a member of the IEEE GRS AdCom, Chainnan of the GRS Awards Committee and Treasurer of the IEEE German Section. He has been General Chairman of the '93 Conference on Microwaves and Optics and he is a member of scientific committees of many conferences. For the Carl Cranz Series for scientific education, he is a Lecturer for radar system engineering and wave propagation. He is a member of an Advisory Committee of the EU-Joint Research Centre.
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