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Dairesel Dalga Kılavuzlarının Modlarını Belirlemede İkiye Bölme Temelli Hızlı Bir Analizör Tasarımı

Year 2018, Volume: 6 Issue: 1, 100 - 113, 01.03.2018

Coşkun Deniz

https://doi.org/10.15317/Scitech.2018.118

Abstract

Dairesel dalga kılavuzlarının desteklediği TE ve TM modlarının belirlenmesinde, Bessel fonksiyonlarının ve türevlerinin sıfırlarının bulunması elzemdir. Ancak, bu fonksiyonlar konvansiyonel olarak sonsuz seri şeklinde tanımlandığından, sayısal değerlerinin ve sıfırlarının hızlı ve makul güvenirlikte hesaplanması, geliştirilmiş sayısal teknikleri ya da yaklaşım yapmayı gerektirir. Ayrıca, modlar genellikle insan tarafından kontrol edilerek sıralandırılır ve özellikle yüksek mod endekslerinde, doğru olarak sıralanmış modlara anında erişim önemlidir. Burada, sıralanmış modları hızlı olarak hesaplayan, sayısal yöntemlerden ikiye bölme (Bisection) temeline dayanan bir algoritma tasarımı sunulmaktadır. Önerimiz şu hususları içermektedir: i) Bessel fonksiyonlarının ve türevlerinin sıfırlarına yakın kritik noktaların, kullanıcı tarafından seçilen örnekleme genişliğine göre (tipik olarak=0.01) belirlenmesi, ii) Kullanıcı tarafından seçilen hassasiyet değeri elde edilinceye kadar, bu kritik noktaları kullanarak, kullanıcı tarafından seçilen maksimum indeks değerine kadar taranan bu fonksiyonlara ardışık olarak sayısal ikiye bölme yönteminin uygulanması, iii) Birleştirilmiş kökler matrisinin köpük sıralaması (bubble sorting), iv) köpük sıralaması yapılmış kökler matrisinin taranarak mod tipinin belirlenmesi. Neticede, tasarımımız hızlı bir hesaplamayla, ilgili modları, kesim frekansları ve ilerleyen dalga frekansları ile birlikte doğru sıralanmış olarak, kullanıcı kontrollü hesaplanabilir doküman dosyası (CDF) ortamında bulmaktadır.

Keywords

Bessel fonksiyonları Dairesel dalga kılavuzları Hesaplanabilir doküman formatı Lindirik Dalga kılavuzları Gerçek zamanlı hesaplama TE modları TM modları

References

  • Abramowitz, A., Stegun, I.A., 1965, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, 3rd Printing with Corrections, Vol. 55 of NBS Applied Mathematics Series, Superintendent of Documents, US Government Printing Office, Washington DC, pp. 355—479.
  • Abuelma’atti, M. T., 1999, “Trigonometric Approximations for Some Bessel functions”, Active and Passive Elec. Comp., Vol. 22, pp. 75-85.
  • Al-Shamali, F., 2015, “Interactive CDF Diagrams in Introductory Physics Course”, Proceedings of 20th International Conference on Multimedia in Physics Teaching and Learning, LMU Munich, p. 77.
  • Arfken, H.J., Weber, G.B., 2005, Mathematical Methods for Physicists (6th ed.), Elsevier Academic Press, pp. 675—686.
  • Arora, N., Kumar, S., Tamta, V.K., 2012, “A Novel Sorting Algorithm and Comparison with Bubble Sort and Insertion Sort”, International Journal of Computer Applications, Vol. 45(1), pp. 31-32.
  • Astrachan, O., 2003, “Bubble Sort: An Archaeological Algorithmic Analysis”, SIGCSE '03 Proceedings of the 34th SIGCSE Technical Symposium on Computer Science Education, NY, pp. 1-5.
  • Balanis, C., 1989, Advanced Engineering Electromagnetics (2nd ed.), Wiley, NY, pp. 483—500.
  • Beattie, C. L., 1958, “Table of First 700 Zeros of Bessel Functions”—Jl(x) and Jl’(x), Bell System Technical Journal, Vol. 37, pp. 689-697.
  • Beaulieu, J.R., 2012, A Dynamic, Interactive Approach to Learning Engineering and Mathematics, MSc. Thesis in Mechanical Eng., Virginia Polytechnic Institute and State University, Blacksburg, VA.
  • Bell, W.W., 1968, Special Functions for Scientists and Engineers, D. Van Nostrand Compant Ltd., London, pp. 92-110.
  • Blachman, N. M., Mousavinezhad, S. H., 1986, “Trigonometric Approximations for Bessel functions”, EEE Transactions on Aerospace and Electronic Systems, Vol. AES-22(1), pp.2-7.
  • Boas, L.M., 2006, Mathematical Methods in the Physical Sciences (3rd ed.), Wiley, NY, pp. 587—606.
  • Chapra, S., Canale, R., 2014, Numerical Methods for Engineers (7th ed.), WCB/McGraw-Hill, NY, pp. 148-154.
  • Cheng, D. K., 1989, Field and Wave Electromagnetics (2nd ed.), Addison-Wesley, London, pp., 562-572.
  • Cormen, T. H., Leiserson, C. E., Rivest, R. L., Stein, C., 2009, Introduction to Algorithms (3rd ed.), MIT Press, USA, p. 40.
  • Deniz, C., 2016, “A Secant Based Roots Finding Algorithm Design and its Applications to Circular Waveguides”, IOSR Journal of Electrical and Electronics Engineering, Vol. 11(6), Ver. III, pp. 84-91.
  • Deniz, C., 2017, “A Newton Raphson Based Roots Finding Algorithm Design and its Applications to Circular Waveguides”, El-Cezeri Journal of Science and Engineering, Vol. 4(1), pp. 32-45.
  • Deshmukh, S., 2012, Illustration of Fundamentals of Vibrations Using Computable Document Format, MSc. Thesis in Mechanical Eng., The University of Texas At Arlington.
  • Guillermo, J., León, S., 2017, Mathematica Beyond Mathematics: The Wolfram Language in the Real World, CRC press- Taylor & Francis group, NY.
  • Hamming, R. W., 1987, Numerical Methods for Scientists and Engineers (2nd Revised ed., Dover Books on Mathematics), Dover Publications, NY, pp. 68-72.
  • Harrison, J., 2009, “Fast and Accurate Bessel Function Computation”, Computer Arithmetic-proc. of 19th IEEE symp. on Computer Arithmetic, Portland-Oregon, pp. 104-113.
  • Hastings, C., Mischo, K., Morrison, M., 2016, Hands-On Start to Wolfram Mathematica: And Programming with the Wolfram Language, 2nd ed., Wolfram Media Inc., Canada.
  • Hoffman, J.D., 2001, Numerical Methods for Engineers and Scientists, (2nd ed.), Marcel Dekkel, NY-Basel, pp. 141-154.
  • Hollingsworth, M.L. and Narayanan, N.H., 2016, “Building a Better eTextbook”, Bulletin of the IEEE Technical Committee on Learning Technology, Vol. 18, No. 2/3, pp. 14-17.
  • Hong, W., 2016, Art of Mathematics, Dorrance Publishing, Pittsburg, pp. 165—166.
  • Jones, J., 2014, The Technical and Social History of Software Engineering, Pearson, NJ, pp. 201-203.
  • Kahle, D., 2014, “Animating Statistics: A New Kind of Applet for Exploring Probability Distributions”, Journal of Statistics Education, Vol. 22, No. 2, pp. 1-21.
  • Khairullah, Md., 2013, “Enhancing Worst Sorting Algorithms”, International Journal of Advanced Science and Technology, Vol. 56, pp. 13-26.
  • Korenev, B.G., 2002, Bessel Functions and Their Applications, Taylor and Francis, NY.
  • Kushwaha, R. K., Srivastava, S., Chaursiya, V., 2014, “A Novel Approach to Analyze a Circular Waveguide in Air and Dielectric Medium”, SOP Transactions on Wireless Communications, Vol. 1, No. 2, pp. 1-18.
  • Luke, Y. L., 1975, Approximation of Special Functions, Academic Press, NY.
  • Millane R. P., Eads, J. L., 2003, “Polynomial Approximations to Bessel Functions”, IEEE Transactions on Antennas and Propagation, Vol. 51(6), pp. 1398-1400.
  • Newman, J. N., 1984, “Approximations for the Bessel and Struve functions”, Mathematics of Computation, Vol. 43(168), pp. 551-556.
  • Richards, D., 2002, Advanced Mathematical Methods with Maple, Cambridge University Press, UK, pp. 325-331.
  • Rohil, H., Manisha, 2014, “Run Time Bubble Sort–An enhancement of Bubble Sort”, International Journal of Computer Trends and Technology (IJCTT), Vol. 14(1), pp. 36-38.
  • Russel, D. A., 2013, “Creating İnteractive Acoustics Animations using Mathematica's Computable Document Format”, Proceedings of Meetings on Acoustics, Vol. 19, 025006.
  • Sekeljic, N., 2010, “Asymptotic Expansion of Bessel Functions; Applications to Electromagnetics”, Dynamics at the Horsetooth, Focused Issue: Asymptotics and Perturbations, Vol. 2A, pp. 1-11.
  • Selinger, J. V., 2016, Introduction to the Theory of Soft Matter from İdeal Gases to Liquid Crystals, Springer, OH-USA.
  • Tasgal, R. S., Band, Y.B., 2015, “Sound Waves and Modulational İnstabilities on Continuous-Wave Solutions in Spinor Bose-Einstein Condensates”, Physical Review, A 91, 013615, pp. 1-15.
  • Waldron, R.A., 1981, “Formulas for Computation of Approximate Values of Some Bessel Functions”, Proceedings of the IEEE, Vol. 69, pp. 1686-1588.
  • Watson, G. N., 1995, A Treatise on the Theory of Bessel Functions, (2nd ed.), Cambridge University Press, NY.
  • Wellin, P., 2016, Essentials of Programming in Mathematica, 1st ed., Cambridge, Cambridge University Press, pp. 449-493.
  • Wolfram, S., 2003, The Mathematica Book (5th ed.), Wolfram Media Inc., 5th edition, USA, pp. 29-35 & pp. 102-110.
  • Wolfram, S., 2017a, Wolfram Documentation Center, BesselJ, http://reference.wolfram.com/mathematica/ref/BesselJ.html (accessed in 22 February 2017).
  • Wolfram, S., 2017b, Wolfram CDF player, https://www.wolfram.com/cdf-player/ (accessed in 22 February 2017).
  • Wolfram, S., 2017c, Computable Document Format, http://www.wolfram.com/events/siam-2016/files/CDF-4.pdf (accessed in 22 February 2017).
  • Wolfram Research, Inc., 2017, Mathematica, Version 11.1, Champaign, IL.

A FAST BISECTION BASED ANALYZER DESIGN FOR THE DETERMINATION OF MODES IN CIRCULAR WAVEGUIDES

Year 2018, Volume: 6 Issue: 1, 100 - 113, 01.03.2018

Coşkun Deniz

https://doi.org/10.15317/Scitech.2018.118

Abstract

Determination of zeros of Bessel functions and their derivatives are essential in the TE and TM modes supported by the circular waveguides. However, since these functions are conventionally defined as infinite series, fast calculation of their numerical values and zeros with reliable accuracy requires improved numerical techniques or approximations. Moreover, modes are usually sorted by human inspection and instant retrieval of correctly ordered modes becomes essential especially for higher mode-index values. Here, a fast-computational algorithm design based on the numerical Bisection method to determine the sorted TE and TM mode solutions of the circular waveguides is presented. Our suggestion involves: i) determination of the critical points close to the zeros of Bessel functions and their derivatives within the user selected sampling width (typically =0.01), ii) application of the numerical Bisection method to these functions one after another to scan up to the user selected maximum index number by using these critical points up to maintain the user selected sensitivity values, iii) Bubble sorting of the unified roots matrix, iv) scan the bubble sorted roots matrix to decide the mode type. As a result, our design finds the related TE and TM modes along with the cut-off and propagating wave frequencies in the correct order with a very fast calculation by the user controlled Computable Document File (CDF) environment.

Keywords

Bessel functions Circular waveguides Computable document file (CDF) Cylindrical waveguides Real time computation TE modes TM modes

References

  • Abramowitz, A., Stegun, I.A., 1965, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, 3rd Printing with Corrections, Vol. 55 of NBS Applied Mathematics Series, Superintendent of Documents, US Government Printing Office, Washington DC, pp. 355—479.
  • Abuelma’atti, M. T., 1999, “Trigonometric Approximations for Some Bessel functions”, Active and Passive Elec. Comp., Vol. 22, pp. 75-85.
  • Al-Shamali, F., 2015, “Interactive CDF Diagrams in Introductory Physics Course”, Proceedings of 20th International Conference on Multimedia in Physics Teaching and Learning, LMU Munich, p. 77.
  • Arfken, H.J., Weber, G.B., 2005, Mathematical Methods for Physicists (6th ed.), Elsevier Academic Press, pp. 675—686.
  • Arora, N., Kumar, S., Tamta, V.K., 2012, “A Novel Sorting Algorithm and Comparison with Bubble Sort and Insertion Sort”, International Journal of Computer Applications, Vol. 45(1), pp. 31-32.
  • Astrachan, O., 2003, “Bubble Sort: An Archaeological Algorithmic Analysis”, SIGCSE '03 Proceedings of the 34th SIGCSE Technical Symposium on Computer Science Education, NY, pp. 1-5.
  • Balanis, C., 1989, Advanced Engineering Electromagnetics (2nd ed.), Wiley, NY, pp. 483—500.
  • Beattie, C. L., 1958, “Table of First 700 Zeros of Bessel Functions”—Jl(x) and Jl’(x), Bell System Technical Journal, Vol. 37, pp. 689-697.
  • Beaulieu, J.R., 2012, A Dynamic, Interactive Approach to Learning Engineering and Mathematics, MSc. Thesis in Mechanical Eng., Virginia Polytechnic Institute and State University, Blacksburg, VA.
  • Bell, W.W., 1968, Special Functions for Scientists and Engineers, D. Van Nostrand Compant Ltd., London, pp. 92-110.
  • Blachman, N. M., Mousavinezhad, S. H., 1986, “Trigonometric Approximations for Bessel functions”, EEE Transactions on Aerospace and Electronic Systems, Vol. AES-22(1), pp.2-7.
  • Boas, L.M., 2006, Mathematical Methods in the Physical Sciences (3rd ed.), Wiley, NY, pp. 587—606.
  • Chapra, S., Canale, R., 2014, Numerical Methods for Engineers (7th ed.), WCB/McGraw-Hill, NY, pp. 148-154.
  • Cheng, D. K., 1989, Field and Wave Electromagnetics (2nd ed.), Addison-Wesley, London, pp., 562-572.
  • Cormen, T. H., Leiserson, C. E., Rivest, R. L., Stein, C., 2009, Introduction to Algorithms (3rd ed.), MIT Press, USA, p. 40.
  • Deniz, C., 2016, “A Secant Based Roots Finding Algorithm Design and its Applications to Circular Waveguides”, IOSR Journal of Electrical and Electronics Engineering, Vol. 11(6), Ver. III, pp. 84-91.
  • Deniz, C., 2017, “A Newton Raphson Based Roots Finding Algorithm Design and its Applications to Circular Waveguides”, El-Cezeri Journal of Science and Engineering, Vol. 4(1), pp. 32-45.
  • Deshmukh, S., 2012, Illustration of Fundamentals of Vibrations Using Computable Document Format, MSc. Thesis in Mechanical Eng., The University of Texas At Arlington.
  • Guillermo, J., León, S., 2017, Mathematica Beyond Mathematics: The Wolfram Language in the Real World, CRC press- Taylor & Francis group, NY.
  • Hamming, R. W., 1987, Numerical Methods for Scientists and Engineers (2nd Revised ed., Dover Books on Mathematics), Dover Publications, NY, pp. 68-72.
  • Harrison, J., 2009, “Fast and Accurate Bessel Function Computation”, Computer Arithmetic-proc. of 19th IEEE symp. on Computer Arithmetic, Portland-Oregon, pp. 104-113.
  • Hastings, C., Mischo, K., Morrison, M., 2016, Hands-On Start to Wolfram Mathematica: And Programming with the Wolfram Language, 2nd ed., Wolfram Media Inc., Canada.
  • Hoffman, J.D., 2001, Numerical Methods for Engineers and Scientists, (2nd ed.), Marcel Dekkel, NY-Basel, pp. 141-154.
  • Hollingsworth, M.L. and Narayanan, N.H., 2016, “Building a Better eTextbook”, Bulletin of the IEEE Technical Committee on Learning Technology, Vol. 18, No. 2/3, pp. 14-17.
  • Hong, W., 2016, Art of Mathematics, Dorrance Publishing, Pittsburg, pp. 165—166.
  • Jones, J., 2014, The Technical and Social History of Software Engineering, Pearson, NJ, pp. 201-203.
  • Kahle, D., 2014, “Animating Statistics: A New Kind of Applet for Exploring Probability Distributions”, Journal of Statistics Education, Vol. 22, No. 2, pp. 1-21.
  • Khairullah, Md., 2013, “Enhancing Worst Sorting Algorithms”, International Journal of Advanced Science and Technology, Vol. 56, pp. 13-26.
  • Korenev, B.G., 2002, Bessel Functions and Their Applications, Taylor and Francis, NY.
  • Kushwaha, R. K., Srivastava, S., Chaursiya, V., 2014, “A Novel Approach to Analyze a Circular Waveguide in Air and Dielectric Medium”, SOP Transactions on Wireless Communications, Vol. 1, No. 2, pp. 1-18.
  • Luke, Y. L., 1975, Approximation of Special Functions, Academic Press, NY.
  • Millane R. P., Eads, J. L., 2003, “Polynomial Approximations to Bessel Functions”, IEEE Transactions on Antennas and Propagation, Vol. 51(6), pp. 1398-1400.
  • Newman, J. N., 1984, “Approximations for the Bessel and Struve functions”, Mathematics of Computation, Vol. 43(168), pp. 551-556.
  • Richards, D., 2002, Advanced Mathematical Methods with Maple, Cambridge University Press, UK, pp. 325-331.
  • Rohil, H., Manisha, 2014, “Run Time Bubble Sort–An enhancement of Bubble Sort”, International Journal of Computer Trends and Technology (IJCTT), Vol. 14(1), pp. 36-38.
  • Russel, D. A., 2013, “Creating İnteractive Acoustics Animations using Mathematica's Computable Document Format”, Proceedings of Meetings on Acoustics, Vol. 19, 025006.
  • Sekeljic, N., 2010, “Asymptotic Expansion of Bessel Functions; Applications to Electromagnetics”, Dynamics at the Horsetooth, Focused Issue: Asymptotics and Perturbations, Vol. 2A, pp. 1-11.
  • Selinger, J. V., 2016, Introduction to the Theory of Soft Matter from İdeal Gases to Liquid Crystals, Springer, OH-USA.
  • Tasgal, R. S., Band, Y.B., 2015, “Sound Waves and Modulational İnstabilities on Continuous-Wave Solutions in Spinor Bose-Einstein Condensates”, Physical Review, A 91, 013615, pp. 1-15.
  • Waldron, R.A., 1981, “Formulas for Computation of Approximate Values of Some Bessel Functions”, Proceedings of the IEEE, Vol. 69, pp. 1686-1588.
  • Watson, G. N., 1995, A Treatise on the Theory of Bessel Functions, (2nd ed.), Cambridge University Press, NY.
  • Wellin, P., 2016, Essentials of Programming in Mathematica, 1st ed., Cambridge, Cambridge University Press, pp. 449-493.
  • Wolfram, S., 2003, The Mathematica Book (5th ed.), Wolfram Media Inc., 5th edition, USA, pp. 29-35 & pp. 102-110.
  • Wolfram, S., 2017a, Wolfram Documentation Center, BesselJ, http://reference.wolfram.com/mathematica/ref/BesselJ.html (accessed in 22 February 2017).
  • Wolfram, S., 2017b, Wolfram CDF player, https://www.wolfram.com/cdf-player/ (accessed in 22 February 2017).
  • Wolfram, S., 2017c, Computable Document Format, http://www.wolfram.com/events/siam-2016/files/CDF-4.pdf (accessed in 22 February 2017).
  • Wolfram Research, Inc., 2017, Mathematica, Version 11.1, Champaign, IL.
There are 47 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Coşkun Deniz

Publication Date March 1, 2018
Published in Issue Year 2018 Volume: 6 Issue: 1

Cite

APA Deniz, C. (2018). A FAST BISECTION BASED ANALYZER DESIGN FOR THE DETERMINATION OF MODES IN CIRCULAR WAVEGUIDES. Selçuk Üniversitesi Mühendislik, Bilim Ve Teknoloji Dergisi, 6(1), 100-113. https://doi.org/10.15317/Scitech.2018.118
AMA Deniz C. A FAST BISECTION BASED ANALYZER DESIGN FOR THE DETERMINATION OF MODES IN CIRCULAR WAVEGUIDES. sujest. March 2018;6(1):100-113. doi:10.15317/Scitech.2018.118
Chicago Deniz, Coşkun. “A FAST BISECTION BASED ANALYZER DESIGN FOR THE DETERMINATION OF MODES IN CIRCULAR WAVEGUIDES”. Selçuk Üniversitesi Mühendislik, Bilim Ve Teknoloji Dergisi 6, no. 1 (March 2018): 100-113. https://doi.org/10.15317/Scitech.2018.118.
EndNote Deniz C (March 1, 2018) A FAST BISECTION BASED ANALYZER DESIGN FOR THE DETERMINATION OF MODES IN CIRCULAR WAVEGUIDES. Selçuk Üniversitesi Mühendislik, Bilim Ve Teknoloji Dergisi 6 1 100–113.
IEEE C. Deniz, “A FAST BISECTION BASED ANALYZER DESIGN FOR THE DETERMINATION OF MODES IN CIRCULAR WAVEGUIDES”, sujest, vol. 6, no. 1, pp. 100–113, 2018, doi: 10.15317/Scitech.2018.118.
ISNAD Deniz, Coşkun. “A FAST BISECTION BASED ANALYZER DESIGN FOR THE DETERMINATION OF MODES IN CIRCULAR WAVEGUIDES”. Selçuk Üniversitesi Mühendislik, Bilim Ve Teknoloji Dergisi 6/1 (March 2018), 100-113. https://doi.org/10.15317/Scitech.2018.118.
JAMA Deniz C. A FAST BISECTION BASED ANALYZER DESIGN FOR THE DETERMINATION OF MODES IN CIRCULAR WAVEGUIDES. sujest. 2018;6:100–113.
MLA Deniz, Coşkun. “A FAST BISECTION BASED ANALYZER DESIGN FOR THE DETERMINATION OF MODES IN CIRCULAR WAVEGUIDES”. Selçuk Üniversitesi Mühendislik, Bilim Ve Teknoloji Dergisi, vol. 6, no. 1, 2018, pp. 100-13, doi:10.15317/Scitech.2018.118.
Vancouver Deniz C. A FAST BISECTION BASED ANALYZER DESIGN FOR THE DETERMINATION OF MODES IN CIRCULAR WAVEGUIDES. sujest. 2018;6(1):100-13.

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