2011

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

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.

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

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.

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