Yıldırım yakalama çubuğunun elektromanyetik alan bakımından koruma bölgesinin moment yöntemi ile hesabı
Computation of protection zone in terms of electromagnetic field of a lightning rod using the method of moments
Yazar: AYBİKE EKMEKÇİ
Danışman: PROF. DR. ÖZCAN KALENDERLİ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Elektrik Mühendisliği Ana Bilim Dalı / Elektrik Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2018
Konu: Elektrik ve Elektronik Mühendisliği = Electrical and Electronics Engineering

Özet:
Bir iletkenin moment yöntemi ile elektromanyetik alan analizi dielektrik ortamda uzun paralel iletim hatlarının ve paralel düzlemlerde bulunan üç boyutlu iletkenleri içeren sistemler gibi problemler için geniş kullanım alanına sahiptir.
Bu tezde, yıldırım yakalama çubuğu çevresindeki elektromanyetik alan bakımından koruma bölgesinin belirlenmesi için Moment Yöntemi (MoM) kullanılmıştır. Çubuk üzerindeki yük dağılımı ve çevresindeki alanlar nokta yerleştirme (point-matching) yöntemi ile hesaplanmıştır. Temel işlev olarak basamak işlevi, sınama fonksiyonu olarak Dirac-Delta temel işlevi seçilmiştir. Moment yöntemi kullanırken ortaya çıkan matrislerdeki teklik problemi yaklaşık integral denklem çözümleri kullanılarak giderilmiştir. İletken düzlemlerin modellenmesi için kullanılan görüntü yöntemi, yer düzlemini temsil etmesi için kullanılarak serbest uzay Green fonksiyonları görüntü yöntemine göre değiştirilmiştir.
Bu çalışma, yıldırımın oluşumu ile ilgili temel bilgiler vererek başlamaktadır. Genel olarak elektro geometrik model ile açıklanan yıldırım yakalama çubuğu koruma bölgesi hesabı ile ilgili temel bilgiler verilerek, yuvarlanan küre yöntemi ve koruma açısı kavramlarına açıklık getirilmektedir.
Daha sonra statik elektromanyetiğin temellerini gözden geçirerek devam etmektedir. Ardından farklı yöntemler, Sonlu Elemanlar Yöntemi (SEY), Sonlu Farklar Yöntemi (SFY) vb. gibi literatürde çokça kullanılan yöntemlerin temellerini karşılaştırmalı olarak açıklayarak literatürde kısıtlı sayıda bilgi ve uygulamaya sahip Moment Yöntemi (MoM) nispeten basit bir fiziksel problemin çözümü üzerinden anlatılmaktadır.
Problemi tanımlamak için Moment Yöntemine ek yüzey tabanlı integral denklem metodolojisi kullanılmaktadır. Volumetrik yöntemlerle karşılaştırıldığında, daha küçük boyutlu ve iyi koşullandırılmış sistemle problemi çözmektedir. Bu tez çalışmasında, farklı uzunluklarda, farklı çaplarda ve farklı koordinatlara yerleştirilen üç boyutlu yıldırım yakalama çubuğu üzerindeki yükler Matlab yazılımı kullanılarak MoM yöntemi ile çözülmüştür. Elde edilen yük dağılımına göre yakalama çubuğu çevresindeki elektrik alan dağılımları yine Matlab yazılımı kullanılarak hesaplanmıştır. Çubuk çevresinde belirlenen koordinatlarda elektrik alan değerlerine ek olarak manyetik alan değerleri Matlab yazılımı kullanılarak hesaplanmıştır. 
Elde edilen sonuçlara göre yakalama çubuğu çevresinde oluşacak elektromanyetik alan bakımından koruma bölgeleri belirlenmiş, çubuk boyutlarının ve koordinatlarının koruma bölgesine etkisi tartışılmış ve geleneksel yöntemler kullanılarak bulunan yıldırımdan koruma bölgeleri ile karşılaştırılmıştır.

Abstract:
The numerical solutions of electromagnetic field problems require the solution of Maxwell equations, which's boundary conditions are determined by the system size, the geometrical structure, and the environmental conditions. The Method of Moments (MoM) solves the Maxwell equations used in the solution of electromagnetic field problems by converting them into matrix equations. Moment Method is widely used for both solution of propagation and scattering problems and can be applied to an unlimited number of electromagnetic field problems.
Electromagnetic field analysis by the Method of Moments of a conductor has a wide range of applications for problems such as long parallel transmission lines in dielectric media and systems containing three-dimensional conductors in parallel planes. In this study the lightning rod is chosen as a thin conductor to demonstrate the Moment Method technique under a lightning strike. The aim of the application of the Method of Moments for electrostatic problems is to cast the solution for the unknown charge density, which is part of the integrand on the surface of the conductor, once the electrical potential is specified.
The lightning rod, also known as Franklin rod after American physicist Benjamin Franklin (1706-1790) conducted the first studies on the lightning rod, still a widely research area. As well as the recent studies regarding protection zone of a lightning rod by using different computation techniques and experiments under investigation of the lightning phenomenon, two traditional methods, known as rolling sphere method and protection angle method are used to evaluate protection zone of an either single lightning rod or lightning protection systems which might have also included horizontal conductors and metal meshes. The protection zone of a lightning rod is defined as the region around the rod where the lightning strike does not penetrate which means the safe region in terms of lightning protection. In this study, the basic calculations of the protection zone are explained by electro-geometric models and it also clarifies the concepts of rolling sphere method and protection angle method. The protection zone of a lightning rod is calculated by traditional methods before the lightning strike attaches the lightning rod. In this study additionally, the results obtained after applying the Method of Moments, obtaining the protection zone of a lightning rod in terms of electromagnetic field at the moment of the impact.
This study begins with the basic information about lightning formation the explanation of fundamentals of lightning protection within the frame of a lightning rod. It continues reviewing the fundamentals of electrostatics and compares different methods such as Finite Element Method (FEM) and Finite Difference Method (FDM) with the Method of Moments. There are very few studies in literature and therefore limited knowledge about the application of the Method of Moments (MoM). The Method of Moments is explained through the solution of a relatively simple physical problem in this thesis.
In this thesis, Method of Moments is used to determine the protection zone in terms of the electromagnetic field around the lightning rod. Moment Method is chosen by the fact that is more practical for the small-scale problems including thin wires and plates in the open geometries, compared to differential equation methods as Finite Element Method (FEM), Finite Difference Method (FDM). The Method of Moments (MoM) as an integral equation method divides the surfaces of the conductors and the interfaces of the dielectric layers into small pieces and provides the solution of relatively small but complete matrices. 
The charge distribution on the rod and the areas around it were calculated by point-matching method whose solutions satisfy the electromagnetic boundary conditions only at discrete points. The main advantage of using point matching numerical technique no integral is required over the range of the testing function while evaluating the matrix elements. The pulse function is chosen as the basis function which is very important in numerical solution has the ability to accurately represent the anticipated unknown function while minimizing the computational effort. Dirac-Delta function is chosen as the testing function, meaning boundary conditions are matched only at discrete locations over the solution domain which satisfies an appropriate solution for a lightning rod and for several of the two-dimensional problems as well. The singularity problem in the resulting matrices when applying the Moment Method is solved by using the integral equation solutions which reduce the processing time and the processing steps. The image method used to model the conductive planes was modified to represent the ground plane and the free space Green's functions were changed according to the image method.
In order to define the problem, additional surface based integral equation methodology is used. Compared to volumetric methods, it solves the problem with a smaller and well-conditioned system. In this thesis, the charges on the three-dimensional lightning rod in different lengths, different diameters, and different coordinates were solved by MoM using Matlab software. Electric field distributions around the lightning rod were calculated using Matlab software. Magnetic field distributions around the lightning rod were calculated using Matlab software as well. According to the results obtained, the protection zones in terms of the electromagnetic field to be formed around the lightning rod were determined and the effect of the rod dimensions and coordinates on the protection zone were discussed and compared with the protection zones in terms of lightning protection found by using traditional methods.
In this thesis, short stroke charge QT which is transferred from the cloud to the earth according to the return stroke peak current IP is calculated according to Berger's equation which approximates the relation between total charge in the leader and return stroke peak current. When the return stroke peak current is increased, it is found that the short stroke charge increases. In other words, the larger the return stroke current, the higher the value of the charge transferred to the ground and the higher the amount of the charge in the thundercloud statements are confirmed. 
The copper lightning rod which has 0 = 17,8  10-9 m resistivity value is used. Since the resistivity value changes, the potential in the conductor chances accordingly. The usage of different materials will change the amount of charge on the conductor and will affect the protection zone in terms of electromagnetic field vicinity of the lightning rod. As the usage of high-resistive materials will increase the charge density on the rod, the electromagnetic field values to be generated around it at the same distances will be higher than the case of usage of low-resistivity materials.
In this study, firstly L = 5 m length and a = 10 mm radius lightning rod placed at the ground is examined by applying specified 3 kA, 5 kA, 10 kA and 15 kA return stroke peak current respectively. According to the specified return stroke peak current, the current density and the electrical field on the lightning rod are found. The charge density on the rod is found at specified lightning current by using Method of Moments and the electrical and magnetic field around the rod is calculated subsequently. The coordinate of the points which has the same electrical field value around the lightning rod creates a convex curve on z-y or z-x plane where starts on the tip of the rod and ending on the ground when the lightning rod along on the z-axis. Outside of this convex curve is said to be the safe region of the lightning rod in terms of electromagnetic field. In the results, it is found that as the value of the lightning current passing through the rod increases, the electric field value around the rod increases. With the increasing electric field, the protection zone in terms of the electromagnetic field is narrowed around the rod. 
Secondly, Method of Moments is applied to lightning rods has different radius a = 10 mm, a = 20 mm, a = 30 mm, a = 40 mm and a = 50 mm respectively by keeping the rod length constant (L = 5 m) such that the return current value is 3 kA. It is found that the charge density at the tip of the rod and the total charge on the rod increases when the radius of the lightning rod is increased by keeping the rod length constant such that the return current value is the same.
Then, L = 5 m length and a = 10 mm radius lightning rod placed at different heights from ground respectively at 5 m high and 15 m high while 3 kA lightning peak current passing through and the protection zone in terms of electromagnetic field is calculated. Then later, L = 20 m (a = 20 mm), L = 30 m (a = 30 mm), L = 45 m (a = 45 mm) and L = 60 m (a = 60 mm) lightning rods are examined under 3 kA, 5 kA, 10 kA and 15 kA lightning peak current and the protection zone in terms of electromagnetic field is calculated. 
The protection zone in terms of the electromagnetic field is calculated around the aforementioned lightning rods while the lightning current passing through the rod by Method of Moments. The results compared with the protection zones in terms of lightning protection found by using traditional methods. As a result, the most reliable regions are found for both against transient electromagnetic field effects of lightning and lightning itself.
This thesis can be developed for the design of lightning protection systems which include either single lightning rod or a lightning protection system which might have also include horizontal conductors and metal meshes. Also, this thesis can provide a preliminary study for the systems which must be protected against transient electromagnetic field effects of lightning.