The spectral domain method is a very fast and powerful technique to analyse planar microwave circuits. Available techniques for simulating the excitation of open planar microwave circuits are not very effective at relatively low frequencies. In this contribution new compensation functions are introduced which correctly model the excitation over the whole microwave frequency regionThe spectral domain method is a very fast and powerful technique to analyse planar microwave circuits. Available techniques for simulating the excitation of open planar microwave circuits are not very effective at relatively low frequencies. In this contribution new compensation functions are introduced which correctly model the excitation over the whole microwave frequency regio

Balik, HH

Railton, CJ

English

Explore Bristol Research

Balik, HH., & Railton, CJ. (1998). Efficient enhancements in spectraldomain method to speed up open planar circuit analysis. In 9thMediterranean Electrotechnical Conference, 1998 (MELECON 98),Tel Aviv (Vol. 1, pp. 459 - 463). Institute of Electrical and ElectronicsEngineers (IEEE). https://doi.org/10.1109/MELCON.1998.692468Peer reviewed versionLink to published version (if available):10.1109/MELCON.1998.692468Link to publication record on the Bristol Research PortalPDF-documentUniversity of Bristol – Bristol Research PortalGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/brp-terms/Efficient Enhancements in Spectral Dom.ain Method To Speed up Open Planar Circuit Analysis Hasan H. Balik Dept. of Elect. and Electronics Eng. Karadeniz Technical University Trabzon, TURKEY 61080 Abstract The Spectral Domain Method is a very fast and powerful technique to  analyse planar microwave cir- cuits. But available techniques f o r  simulating the ex- citation of open planar microwave circuits are not very effective at relatively low frequencies. I n  this con- tribution new compensation functions are introduced which correctly model the excitation over the whole mi- crowave frequency region. 1 Introduction For the last few decades, designers of microwave and millimetre-wave integrated circuits have come de- pend heavily on computer-aided techniques to  reduce design time and improve performance. This is because most of the practical electromagnetic problems can be solved numerically and cannot be solved analytic- ally. Recently as the complexity, component density and operating frequencies of these circuits have in- creased dramatically, the problems posed to the de- signer of CAD tools have become much more difficult. Moreover the rapid development and availability of in- creasing computing power do not solve the problem. The complexity of the circuit to be designed and com- putational overhead required for accurate modelling will outstretch any development in hardware. Most of the early analysis methods for microwave integrated circuits were based on either quasi-static or magnetic wall approximation which are only valid at low frequencies. The use of high operating frequen- cies make such approximations insufficiently accurate. At high frequencies it is necessary to use full--wave analysis to get accurate results. In this context the descriptive term full-wave is used to define the set of numerical techniques which include all field com- ponents and thus allow rigorous analysis of dispersive structures where coupling between circuits elements cannot be ignored in the microwave frequency band of interest. C. J. Railton Dept. of Elect. and Electronics Eng. University of Bristol Bristol, ENGLAND BS8 1TR The Spectral Domain Method which is based on solving coupled integral equation [l, chapter 3 and 51 is one of the widely used full-wave analysis tech- niques. It is, however, limited by the requirement for a large amount of computer resources. Notwithstanding the rapid increase in the availability of the computer power, work is still necessary to  increase the efficiency of the algorithm used. The first requirement for the analysis of open struc- tures in SlDM is the definition of the unknown current distribution on the metallisation. Therefore the first step of thle analysis is to expand the unknown sur- face current as a set of known basis functions with unknown coefficients. It has been shown and com- monly used that a rooftop function [2,3] as a current basis function allows the unknown surface current on an irregular shaped metallisation to  be defined. But this approach results in a large numbers of basis func- tions for convergence and the rooftop functions must be defined identical apart from a shift from tthe ori- gin [2,3]. The use of pre-calculated basis functions reduces the impedance matrix to be calculated, but pre-calculated basis functions must be a combination of rooftop functions [4] to exploit the FFT. Balik has defined the rooftop function as a function of its loc- ation in [5] allowing subgridding. With this defin- ition the discontinuity has modelled by fine rooftop functions whereas course rooftops are used where only slow changes occur in current distribution. The pre- calculated basis function as a linear combination of the rooftop function as well as a current wave function with a pre-calculated wave number was also defined in [5 ] .  These enhancements remedy the limitations given in [2,3] The SClM is a frequency domain technique and re- quires repeated calculation of the impedance matrix at each frequency point of interest. In [6], the adapt- ive integrakion technique which consists of an adaptive 0-7803-3879-0 I 98 I $10.00 -459 - truncation and adaptive integration step has been in- troduced to  limit the two-dimensional continuous nu- merical integration over an infinite surface because of having open structure to  finite computer resources. The location vector calculation, which reduces the number of impedance matrix elements, is introduced and used in [7] in the region where identical sub- domain basis functions used. For this region, a sub- domain basis function is divided into two parts. First one is identical for each function and second one rep- resents its location. By this enhancement, N(N+l)/2 number of impedance matrix elements calculated are reduced to 3N, where N is the size of the imped- ance matrix. In addition, the asymptotic form of the Green’s function, which is defined for large values of transform variables, is used to calculate the imped- ance matrix. The symmetry in Green function has also been exploited to  reduce the run-time required for the impedance matrix calculation. The enhancement introduced in [8] is effective on the first and second requirement in SDM analysis. It is proposed by Balik to  reduce the number of current basis function required in one dimension to speed up overall two dimensional numerical integration. This is very effective and could give up-to %99 improvement. The circuits are presented by N-port network para- meters. Scattering parameters, otherwise called S- parameters are utilised but can be transformed to other representations by using standard formulae. This paper highlights the lack of information, draw- backs of available excitation modelling techniques for the derivation of the S-parameter of the open planar circuits and introduces a novel excitation modelling for efficient, fast accurate analysis in SDM. 2 Available Excitation Modellings The details available in the literature for the extrac- tion of the S-parameters of planar circuits using the SDM or other related techniques are limited in scope as the topic is avoided or ignored in most papers. For shielded planar microwave circuits, the S-parameters at a spot frequency are derived in [9], [lo, chapter51 from the solution of a matrix equation. In these im- plementations, the tangential electric field is assumed to be identically zero because of the existence of side- walls. Since there do not exist in open structures a different method must be sought. In addition, the ef- fects of the port/feed arrangement is assumed to  be negligible. The extraction of the S-parameters from the knowledge of the surface current distribution is achieved by applying transmission line theory to the feedlines (de-embedding algorithm) [ll, 121, but this method can only be used when the line length is big- ger than a half wavelength. A method was introduced by Jackson [13] to calculate S-parameters of gap- discontinuities in 1985 for open planar circuits, but his technique, in contrast to the method described here, is not complete and efficient at relatively low frequencies. This is because the cosine portion of the travelling wave is truncated one-quarter of a guide wavelength from a zero of the sine. The length of the truncated portion is a function of the operating fre- quency. At low frequencies, its length becomes larger than the entire circuit’s dimension and a large num- ber of extra rooftop functions are required in order to avoid spurious numerical reflections. 3 Improved Technique for Excitation Modelling For the formulation described in this paper, all cir- cuits are considered to be connected to semi-infinite feedline, whose width is identical to  the joining port, at each port. The fundamental microstrip mode is assumed to propagate on the feedlines and thus the travelling current waves are chosen as current basis functions given in equation 1. Here I C ,  is the pre-calculated wavenumber of the feed- line, L is the length of the feedline and z ,  (s = i ,o)  is the offset of the port from the origin. The letters i, t ,  T indicate the incident, transmitted and reflected cur- rent waves respectively and the unknown coefficients a,, at are coefficients of the reflected and transmitted current waves which are to  be calculated. For the ex- cited port, the current is; and for the ports which are not excited, the current is; Joutput = Jt (3) The Fourier transforms of the current basis func- tions for the €eedlines as defined in equation 1 are -460 - given by; L is theoretically extends to infinity, but in practice is chosen to be an integer number of half wavelengths [3,13]. As shown in equations 2, 3 and in figure 1, the current basis functions of the feedlines have a real (cosk,~) and an imaginary (sinIcnz) part. The real part causes current discontinuities in the direction of current flow, because the triangle function as a rooftop function component is identically zero at  the interface between the port and the adjacent feedline, whereas cos(k,z) has a finite value. The problem is the same in the direction perpendicular to current flow, because in this case the step function as a rooftop function com- ponent has a finite value, whereas sin(knz) is identic- ally zero. Therefore extra basis functions are still re- quired at  each interface between the port and adjacent feedline for an accurate solution. These functions are introduced and referred to by the author as compens- ation functions which are shown in figure 2. The compensation functions are defined for the input port in the space domain by: (5) zi < z < zi + 1, otherwise and for the output port; (6) z ,  - E ,  5 z 5 z,  otherwise J,(z) = { (7) -z--I, z, - 1, 5 z 5 z,  J x ( z )  otherwise where 1, is the size of the compensation functions in the direction of propagation. J,(z)  in equation 4 and figure 2(a) transfers the effect of the cosine portion of the incident current wave in the direction of current flow, whereas J x ( z )  in equa- tion 5 and figure 2(a) transfers the effect of the sine portion of the incident current wave in the direction perpendicular to the current flow. Similarly J z ( z )  in equation 6 and figure 2(b) transfers the current wave to the output port in the direction of propagation V Compensation Functions Figure 1: Illustration of the excitation in the direction of current flow ' Z  (a) Input port . (zo4Z) (b) Output port Figure 2: Compensation functions for input and out- put ports whereas J x ( z )  in equation 7 and figure 2(b) transfers the currlent wave in the transverse direction. As illustrated in figure 2, the compensation func- tion is actually a semi-rooftop function with an offset from the origin in either -z or +z directions and the missing portion of the rooftop function has been com- pleted b:y the current waves. Fourier transforms of the half rooftop functions are given by; where Jl,ft and Jriyht indicate the Fourier transform of the left and right hand side of a rooftop function. The derivation of the compensation functions from the left and right hand side of the rooftop function for the input port are given by; -461- and for the output port, they are given by; J t ( lcz )  = J l e f t ( k z ) e j k z z o f f  (10) J , ( k , )  = J r i g h t ( k z ) e j k z ( z o f f - l . )  (11) Including all current basis functions of the entire system, the total current is expressed as: Jtotal = Jboz (k, , N x ( a p n J p o r t .  (kc, knkz) n=l acnJcomp, (kz, k z ) )  where N is the number of ports, J b o z  refers to the basis functions of the microwave circuit, JpoTt ,  is either the sum of the incident and reflected current wave for the excited port or the transmitted current wave for unexcited ports and Jcomp, is the Fourier trans- form of the compensation function. To calculate the S-parameters of the circuit, apn must be known. The Method of Moments is employed to eliminate the elec- tric field components and to find the unknown coef- ficients. A total of N weighting functions must be defined to complete the algorithm. These are chosen to be a triangle function which straddles the lines sep- arating each port and the feedline in the direction of current flow and a pre-calculated basis function in the direction perpendicular to current flow. After applic- ation of the Method of Moments, the matrix equation yields, Because two travelling waves, which are the incident current wave with unit amplitude and the reflected current wave with unknown amplitude, are used for the feedline connected to the input port of the cir- cuit, Z is not a square matrix and contains unknown coefficients, as well as the known unit amplitude. The column of the impedance matrix corresponding to the incident current wave products is moved to the right hand side of the equation and the equation 12 be- comes, where Z, consists of the elements related to unknown current basis functions and Z; is a column vector con- taining the elements related to incident current waves. Any root-finding procedure can be applied to solve equation 13 and the unknown coefficients can be found. The S-parameters of the N-port circuit are actually the coefficients of the reflected and the trans- mitted current waves. E E rs Q, t-o Figure 3: Low-pass filter detail 4 Numerical Results It must be noted that, Jackson’s excitation tech- nique is used with no extra rooftop functions to com- plete the truncated portion of the current wave. This is because of using identical number of basis functions in each model and illustrating the improvement in the accuracy for identical number of basis functions. 4.1 Simple Low-pass Filter Measurement results are available for the mi- crostrip low-pass filter [14] shown in figure 3. This filter is analysed to demonstrate the total improve- ment in this implementation and to compare the cal- culated results to available measurement data. The dimensions and parameters of the dielectric substrate are given in figure 3. It is assumed that the truncated current wave and the extra set of rooftop functions to complete this truncated portion are used to excite the open circuit of interest (Jackson’s technique). The minimum operat- ing frequency is 2 GHz and the corresponding length of the cosine is 27.42 mm, which is 45 times bigger than the size of the rooftop which is 0.60325 mm in this analysis. For the maximum operating frequency of 20 GHz, the corresponding length of truncation is 2.6925 mm, which is still 4 times bigger than the rooftop. To complete the missing portion in order to get accur- ate results, 528 extra rooftop functions are required. In contrast to this, with the use of the compensation functions, 4 functions are sufficient. The S-parameter results are plotted in figures 4 where it can be seen that the calculated results and measurements are in very good agreement. 5 Conclusion We have shown that realistically complex mi- crostrip circuits can be rigorously analysed at  relat- ively low frequencies using new compensation func- tions which are defined at the interface between a port and a feedline in the Spectral Domain Method. The compensation functions are shown to be effective to - 462- [7] H. H. Balik and C. J.  Railton, “Full-wave spectal do.main analysis of open planar circuits and an- tennas,” in 1998 National Radio Science Meeting, p. 127, January 1998. S-Parameters (dB) ‘ e ,  . [8] H. H. Balik and C. J. Railton, “Efficient numer- ical integration of the impedance matrix for the analysis of open planar circuits and antennas,” in 1998 International Symposium on Electromag- ne,tic Theory, p. accepted, May 1998. . .  .. lo l2 l4 l6 l8 2o [9] J. C. Rautio and R. F. Harrington, “An elec- tromagnetic time-harmonic analysis of shielded mi’crostrip circuits,’’ IEEE Transaction on Mi- crowave Theory and Technique, vol. 35, pp. 726- 72!3, August 1987. Frequency GHz Figure 4: Plot of S-parameters’ magnitude for the low-pass filter [lo] S. A. Meade, The Rapid and Rigorous Mathemat- icd Modelling of Passive Planar Microwave Cir- cuits. PhD thesis, University of Bristol, March 19!34. model the effect of the excitation of the circuit. The results compare well with published measurements. [ll] S. A. Meade and C. J .  Railton, “Efficient ex- traction of s-parameters from the spectral do- main method for passive planar microwave cir- cui ts,” in 4 th. European Microwave Conference, pp. 1084-1089, September 1994. References [l] T. Itoh, Numerical Techniques for Microwave and Millimeter- Wave Passive Structures. John Wil- ley and Sons, 1989. i2] s- A- Meade and c. J. Railton, “Efficient imple- [12] w, p. ~~~~k~~~~ and p. B. ~ ~ t ~ h i ,  IlCharac- terisation of microstrip discontinuities on mul- tilayer dielectric substrates including radiation losses,” IEEE Transaction on Microwave Theory and Technique, vol. 37, pp. 2058-2066, December mentation of the spectral domain method includ- ing precalculated corner basis functions,” IEEE Transaction on Microwave Theory and Tech- nique, vol. 42, pp. 1678-1684, September 1994. [3] R. W. Jackson, “Full-wave, finite element ana- lysis of irregular microstrip discontinuities,” IEEE Transaction on Microwave Theory and Technique, vol. 37, pp. 81-89, January 1989. [4] C. J. Railton and S. A. Meade, “Fast rigor- ous analysis of shielded planar filters,” IEEE Transaction on Microwave Theory and Tech- nique, vol. 40, pp. 978-985, May 1992. 19139. [13] R. W. Jackson and D. M. Pozar, “Full-wave ana- lysis of microstrip open-end and gap discontinu- ities,” IEEE Transaction on Microwave The0 y and Technique, vol. 33, pp. 1036-1042, October 1985. [14] D. M. Sheen, S. M. Ali, M. D. Ab- douzahra, and J .  A. Kong, “Application of the three-dimensional finite-difference time-domain method of the analysis of planar microstrip cir- cui.ts,” IEEE Transaction on Microwave Theory and Technique, vol. 38, pp. 849-857, July 1990. [5] H. H. Balik and C. J. Railton, “Sub-gridding in the spectral domain method for the ana- lysis of planar circuits and antennas,” in 3rd In- ternational conference on telecommunications in modern satellite, cable and broadcasting services, pp. 592-595, October 1997. [6] H. H. Balik and C. J. Railton, “Adaptive numerical integration technique for the ana- lysis of open planar circuits and antennas,” in 3rd High Frequency Postgraduate Student Col- luquium, pp. 100-105, September 1997. -463 - 

Efficient enhancements in spectral domain method to speed up open planar circuit analysis

                          Balik, H. H., & Railton, C. J. (1998). Efficient enhancements in spectraldomain method to speed up open planar circuit analysis. In 9thMediterranean Electrotechnical Conference, 1998 (MELECON 98), TelAviv. (Vol. 1, pp. 459 - 463). Institute of Electrical and ElectronicsEngineers (IEE). 10.1109/MELCON.1998.692468Link to published version (if available):10.1109/MELCON.1998.692468Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.Efficient Enhancements in Spectral Dom.ain Method To Speed up Open Planar Circuit Analysis Hasan H. Balik Dept. of Elect. and Electronics Eng. Karadeniz Technical University Trabzon, TURKEY 61080 Abstract The Spectral Domain Method is a very fast and powerful technique to  analyse planar microwave cir- cuits. But available techniques f o r  simulating the ex- citation of open planar microwave circuits are not very effective at relatively low frequencies. I n  this con- tribution new compensation functions are introduced which correctly model the excitation over the whole mi- crowave frequency region. 1 Introduction For the last few decades, designers of microwave and millimetre-wave integrated circuits have come de- pend heavily on computer-aided techniques to  reduce design time and improve performance. This is because most of the practical electromagnetic problems can be solved numerically and cannot be solved analytic- ally. Recently as the complexity, component density and operating frequencies of these circuits have in- creased dramatically, the problems posed to the de- signer of CAD tools have become much more difficult. Moreover the rapid development and availability of in- creasing computing power do not solve the problem. The complexity of the circuit to be designed and com- putational overhead required for accurate modelling will outstretch any development in hardware. Most of the early analysis methods for microwave integrated circuits were based on either quasi-static or magnetic wall approximation which are only valid at low frequencies. The use of high operating frequen- cies make such approximations insufficiently accurate. At high frequencies it is necessary to use full--wave analysis to get accurate results. In this context the descriptive term full-wave is used to define the set of numerical techniques which include all field com- ponents and thus allow rigorous analysis of dispersive structures where coupling between circuits elements cannot be ignored in the microwave frequency band of interest. C. J. Railton Dept. of Elect. and Electronics Eng. University of Bristol Bristol, ENGLAND BS8 1TR The Spectral Domain Method which is based on solving coupled integral equation [l, chapter 3 and 51 is one of the widely used full-wave analysis tech- niques. It is, however, limited by the requirement for a large amount of computer resources. Notwithstanding the rapid increase in the availability of the computer power, work is still necessary to  increase the efficiency of the algorithm used. The first requirement for the analysis of open struc- tures in SlDM is the definition of the unknown current distribution on the metallisation. Therefore the first step of thle analysis is to expand the unknown sur- face current as a set of known basis functions with unknown coefficients. It has been shown and com- monly used that a rooftop function [2,3] as a current basis function allows the unknown surface current on an irregular shaped metallisation to  be defined. But this approach results in a large numbers of basis func- tions for convergence and the rooftop functions must be defined identical apart from a shift from tthe ori- gin [2,3]. The use of pre-calculated basis functions reduces the impedance matrix to be calculated, but pre-calculated basis functions must be a combination of rooftop functions [4] to exploit the FFT. Balik has defined the rooftop function as a function of its loc- ation in [5] allowing subgridding. With this defin- ition the discontinuity has modelled by fine rooftop functions whereas course rooftops are used where only slow changes occur in current distribution. The pre- calculated basis function as a linear combination of the rooftop function as well as a current wave function with a pre-calculated wave number was also defined in [5 ] .  These enhancements remedy the limitations given in [2,3] The SClM is a frequency domain technique and re- quires repeated calculation of the impedance matrix at each frequency point of interest. In [6], the adapt- ive integrakion technique which consists of an adaptive 0-7803-3879-0 I 98 I $10.00 -459 - truncation and adaptive integration step has been in- troduced to  limit the two-dimensional continuous nu- merical integration over an infinite surface because of having open structure to  finite computer resources. The location vector calculation, which reduces the number of impedance matrix elements, is introduced and used in [7] in the region where identical sub- domain basis functions used. For this region, a sub- domain basis function is divided into two parts. First one is identical for each function and second one rep- resents its location. By this enhancement, N(N+l)/2 number of impedance matrix elements calculated are reduced to 3N, where N is the size of the imped- ance matrix. In addition, the asymptotic form of the Green’s function, which is defined for large values of transform variables, is used to calculate the imped- ance matrix. The symmetry in Green function has also been exploited to  reduce the run-time required for the impedance matrix calculation. The enhancement introduced in [8] is effective on the first and second requirement in SDM analysis. It is proposed by Balik to  reduce the number of current basis function required in one dimension to speed up overall two dimensional numerical integration. This is very effective and could give up-to %99 improvement. The circuits are presented by N-port network para- meters. Scattering parameters, otherwise called S- parameters are utilised but can be transformed to other representations by using standard formulae. This paper highlights the lack of information, draw- backs of available excitation modelling techniques for the derivation of the S-parameter of the open planar circuits and introduces a novel excitation modelling for efficient, fast accurate analysis in SDM. 2 Available Excitation Modellings The details available in the literature for the extrac- tion of the S-parameters of planar circuits using the SDM or other related techniques are limited in scope as the topic is avoided or ignored in most papers. For shielded planar microwave circuits, the S-parameters at a spot frequency are derived in [9], [lo, chapter51 from the solution of a matrix equation. In these im- plementations, the tangential electric field is assumed to be identically zero because of the existence of side- walls. Since there do not exist in open structures a different method must be sought. In addition, the ef- fects of the port/feed arrangement is assumed to  be negligible. The extraction of the S-parameters from the knowledge of the surface current distribution is achieved by applying transmission line theory to the feedlines (de-embedding algorithm) [ll, 121, but this method can only be used when the line length is big- ger than a half wavelength. A method was introduced by Jackson [13] to calculate S-parameters of gap- discontinuities in 1985 for open planar circuits, but his technique, in contrast to the method described here, is not complete and efficient at relatively low frequencies. This is because the cosine portion of the travelling wave is truncated one-quarter of a guide wavelength from a zero of the sine. The length of the truncated portion is a function of the operating fre- quency. At low frequencies, its length becomes larger than the entire circuit’s dimension and a large num- ber of extra rooftop functions are required in order to avoid spurious numerical reflections. 3 Improved Technique for Excitation Modelling For the formulation described in this paper, all cir- cuits are considered to be connected to semi-infinite feedline, whose width is identical to  the joining port, at each port. The fundamental microstrip mode is assumed to propagate on the feedlines and thus the travelling current waves are chosen as current basis functions given in equation 1. Here I C ,  is the pre-calculated wavenumber of the feed- line, L is the length of the feedline and z ,  (s = i ,o)  is the offset of the port from the origin. The letters i, t ,  T indicate the incident, transmitted and reflected cur- rent waves respectively and the unknown coefficients a,, at are coefficients of the reflected and transmitted current waves which are to  be calculated. For the ex- cited port, the current is; and for the ports which are not excited, the current is; Joutput = Jt (3) The Fourier transforms of the current basis func- tions for the €eedlines as defined in equation 1 are -460 - given by; L is theoretically extends to infinity, but in practice is chosen to be an integer number of half wavelengths [3,13]. As shown in equations 2, 3 and in figure 1, the current basis functions of the feedlines have a real (cosk,~) and an imaginary (sinIcnz) part. The real part causes current discontinuities in the direction of current flow, because the triangle function as a rooftop function component is identically zero at  the interface between the port and the adjacent feedline, whereas cos(k,z) has a finite value. The problem is the same in the direction perpendicular to current flow, because in this case the step function as a rooftop function com- ponent has a finite value, whereas sin(knz) is identic- ally zero. Therefore extra basis functions are still re- quired at  each interface between the port and adjacent feedline for an accurate solution. These functions are introduced and referred to by the author as compens- ation functions which are shown in figure 2. The compensation functions are defined for the input port in the space domain by: (5) zi < z < zi + 1, otherwise and for the output port; (6) z ,  - E ,  5 z 5 z,  otherwise J,(z) = { (7) -z--I, z, - 1, 5 z 5 z,  J x ( z )  otherwise where 1, is the size of the compensation functions in the direction of propagation. J,(z)  in equation 4 and figure 2(a) transfers the effect of the cosine portion of the incident current wave in the direction of current flow, whereas J x ( z )  in equa- tion 5 and figure 2(a) transfers the effect of the sine portion of the incident current wave in the direction perpendicular to the current flow. Similarly J z ( z )  in equation 6 and figure 2(b) transfers the current wave to the output port in the direction of propagation V Compensation Functions Figure 1: Illustration of the excitation in the direction of current flow ' Z  (a) Input port . (zo4Z) (b) Output port Figure 2: Compensation functions for input and out- put ports whereas J x ( z )  in equation 7 and figure 2(b) transfers the currlent wave in the transverse direction. As illustrated in figure 2, the compensation func- tion is actually a semi-rooftop function with an offset from the origin in either -z or +z directions and the missing portion of the rooftop function has been com- pleted b:y the current waves. Fourier transforms of the half rooftop functions are given by; where Jl,ft and Jriyht indicate the Fourier transform of the left and right hand side of a rooftop function. The derivation of the compensation functions from the left and right hand side of the rooftop function for the input port are given by; -461- and for the output port, they are given by; J t ( lcz )  = J l e f t ( k z ) e j k z z o f f  (10) J , ( k , )  = J r i g h t ( k z ) e j k z ( z o f f - l . )  (11) Including all current basis functions of the entire system, the total current is expressed as: Jtotal = Jboz (k, , N x ( a p n J p o r t .  (kc, knkz) n=l acnJcomp, (kz, k z ) )  where N is the number of ports, J b o z  refers to the basis functions of the microwave circuit, JpoTt ,  is either the sum of the incident and reflected current wave for the excited port or the transmitted current wave for unexcited ports and Jcomp, is the Fourier trans- form of the compensation function. To calculate the S-parameters of the circuit, apn must be known. The Method of Moments is employed to eliminate the elec- tric field components and to find the unknown coef- ficients. A total of N weighting functions must be defined to complete the algorithm. These are chosen to be a triangle function which straddles the lines sep- arating each port and the feedline in the direction of current flow and a pre-calculated basis function in the direction perpendicular to current flow. After applic- ation of the Method of Moments, the matrix equation yields, Because two travelling waves, which are the incident current wave with unit amplitude and the reflected current wave with unknown amplitude, are used for the feedline connected to the input port of the cir- cuit, Z is not a square matrix and contains unknown coefficients, as well as the known unit amplitude. The column of the impedance matrix corresponding to the incident current wave products is moved to the right hand side of the equation and the equation 12 be- comes, where Z, consists of the elements related to unknown current basis functions and Z; is a column vector con- taining the elements related to incident current waves. Any root-finding procedure can be applied to solve equation 13 and the unknown coefficients can be found. The S-parameters of the N-port circuit are actually the coefficients of the reflected and the trans- mitted current waves. E E rs Q, t-o Figure 3: Low-pass filter detail 4 Numerical Results It must be noted that, Jackson’s excitation tech- nique is used with no extra rooftop functions to com- plete the truncated portion of the current wave. This is because of using identical number of basis functions in each model and illustrating the improvement in the accuracy for identical number of basis functions. 4.1 Simple Low-pass Filter Measurement results are available for the mi- crostrip low-pass filter [14] shown in figure 3. This filter is analysed to demonstrate the total improve- ment in this implementation and to compare the cal- culated results to available measurement data. The dimensions and parameters of the dielectric substrate are given in figure 3. It is assumed that the truncated current wave and the extra set of rooftop functions to complete this truncated portion are used to excite the open circuit of interest (Jackson’s technique). The minimum operat- ing frequency is 2 GHz and the corresponding length of the cosine is 27.42 mm, which is 45 times bigger than the size of the rooftop which is 0.60325 mm in this analysis. For the maximum operating frequency of 20 GHz, the corresponding length of truncation is 2.6925 mm, which is still 4 times bigger than the rooftop. To complete the missing portion in order to get accur- ate results, 528 extra rooftop functions are required. In contrast to this, with the use of the compensation functions, 4 functions are sufficient. The S-parameter results are plotted in figures 4 where it can be seen that the calculated results and measurements are in very good agreement. 5 Conclusion We have shown that realistically complex mi- crostrip circuits can be rigorously analysed at  relat- ively low frequencies using new compensation func- tions which are defined at the interface between a port and a feedline in the Spectral Domain Method. The compensation functions are shown to be effective to - 462- [7] H. H. Balik and C. J.  Railton, “Full-wave spectal do.main analysis of open planar circuits and an- tennas,” in 1998 National Radio Science Meeting, p. 127, January 1998. S-Parameters (dB) ‘ e ,  . [8] H. H. Balik and C. J. Railton, “Efficient numer- ical integration of the impedance matrix for the analysis of open planar circuits and antennas,” in 1998 International Symposium on Electromag- ne,tic Theory, p. accepted, May 1998. . .  .. lo l2 l4 l6 l8 2o [9] J. C. Rautio and R. F. Harrington, “An elec- tromagnetic time-harmonic analysis of shielded mi’crostrip circuits,’’ IEEE Transaction on Mi- crowave Theory and Technique, vol. 35, pp. 726- 72!3, August 1987. Frequency GHz Figure 4: Plot of S-parameters’ magnitude for the low-pass filter [lo] S. A. Meade, The Rapid and Rigorous Mathemat- icd Modelling of Passive Planar Microwave Cir- cuits. PhD thesis, University of Bristol, March 19!34. model the effect of the excitation of the circuit. The results compare well with published measurements. [ll] S. A. Meade and C. J .  Railton, “Efficient ex- traction of s-parameters from the spectral do- main method for passive planar microwave cir- cui ts,” in 4 th. European Microwave Conference, pp. 1084-1089, September 1994. References [l] T. Itoh, Numerical Techniques for Microwave and Millimeter- Wave Passive Structures. John Wil- ley and Sons, 1989. i2] s- A- Meade and c. J. Railton, “Efficient imple- [12] w, p. ~~~~k~~~~ and p. B. ~ ~ t ~ h i ,  IlCharac- terisation of microstrip discontinuities on mul- tilayer dielectric substrates including radiation losses,” IEEE Transaction on Microwave Theory and Technique, vol. 37, pp. 2058-2066, December mentation of the spectral domain method includ- ing precalculated corner basis functions,” IEEE Transaction on Microwave Theory and Tech- nique, vol. 42, pp. 1678-1684, September 1994. [3] R. W. Jackson, “Full-wave, finite element ana- lysis of irregular microstrip discontinuities,” IEEE Transaction on Microwave Theory and Technique, vol. 37, pp. 81-89, January 1989. [4] C. J. Railton and S. A. Meade, “Fast rigor- ous analysis of shielded planar filters,” IEEE Transaction on Microwave Theory and Tech- nique, vol. 40, pp. 978-985, May 1992. 19139. [13] R. W. Jackson and D. M. Pozar, “Full-wave ana- lysis of microstrip open-end and gap discontinu- ities,” IEEE Transaction on Microwave The0 y and Technique, vol. 33, pp. 1036-1042, October 1985. [14] D. M. Sheen, S. M. Ali, M. D. Ab- douzahra, and J .  A. Kong, “Application of the three-dimensional finite-difference time-domain method of the analysis of planar microstrip cir- cui.ts,” IEEE Transaction on Microwave Theory and Technique, vol. 38, pp. 849-857, July 1990. [5] H. H. Balik and C. J. Railton, “Sub-gridding in the spectral domain method for the ana- lysis of planar circuits and antennas,” in 3rd In- ternational conference on telecommunications in modern satellite, cable and broadcasting services, pp. 592-595, October 1997. [6] H. H. Balik and C. J. Railton, “Adaptive numerical integration technique for the ana- lysis of open planar circuits and antennas,” in 3rd High Frequency Postgraduate Student Col- luquium, pp. 100-105, September 1997. -463 - 

Adaptive numerical integration technique for the analysis of open planar circuits and antennas,”

Application of the three-dimensional finite-difference time-domain method of the analysis of planar microstrip circui.ts,”

Efficient imple- [12] w, p. ~~~~k~~~~

Fast rigorous analysis of shielded planar filters,”

Full-wave analysis of microstrip open-end and gap discontinuities,”

Full-wave, finite element analysis of irregular microstrip discontinuities,”

Numerical Techniques for Microwave and Millimeter- Wave Passive Structures. John Willey and Sons,

Sub-gridding in the spectral domain method for the analysis of planar circuits and antennas,”

https://research-information.bris.ac.uk/ws/files/3010814/balik_MELECON_1998.pdf

Efficient enhancements in spectral domain method to speed up open planar circuit analysis

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