Chapter 6 Fractal antennas,分形天線

1、Chapter 6 Fractal Antenna6.1 Fractal Theory The term fractal, which means broken or irregular fragments, was originally coined by Mandelbrot in 1973. Possess an inherent self-similarity or self-affinity in their geometrical structure. Fractal geometry is combined with electromagnetic theory for the

2、purpose of investigating a new class of radiation, propagation, and scattering problems. One of the most promising areas of fractal-electrodynamicsresearch is in its application to antenna theory and design.The Koch Fractal34Length1Length342LengthFirst iterationAfter2 iterations6.2 Fractal geometryA

3、fter 3 iterations343LengthAfter n iterations34nLength34LengthAfter iterations(work with me here, people)The Koch snowflake is six of these put together to form . . . . . well, a snowflake.Notice that the perimeter of the Koch snowflake is infinite . . . . . but that the area it bounds is finite (ind

4、eed, it iscontained in the white square).The Koch snowflake has even been used in technology:Boston - Mar 13, 2002Fractal Antenna Systems, Inc. today disclosed that it hasfiled for patent protection on a new class of antenna arraysthat use close-packed arrangements of fractal elements toget superior

5、 performance characteristics.Fractal Tiling Arrays - Firm Reports Breakthrough in Array AntennasCan you name the movie?Each of the six sides of the Koch snowflake isself-similar: If you take a small copy of it . . . . . then dilate by a factor of 3 . . . . . you get four copies of the original.But s

6、elf-similarity is not what makes the Koch snowflakea fractal! (Contrary to a common misconception.)After all, many common geometric objects exhibitself-similarity. Consider, for example, the humblesquare.If you take a small square . . . . . and dilate by a factor of 2 . . . . . then you get 4 copies

7、 of the original.A square is self-similar, but it most certainly is not a fractal.If you take a small square . . . . . and dilate by a factor of 3 . . . . . then you get 9 copies of the original.Let k be the scale factor.Let N be the number of copies of the original that you get.Note that for the sq

8、uare, we have that:2logNkNk2Or in other words, we have:Lets computeNklogfor some other shapes.Line segmentOriginalDilatedk = scale factor = 2N = number of copies of original = 21logNkTriangleOriginalDilatedk = scale factor = 2N = number of copies of original = 42logNkCubeOriginalDilatedk = scale fac

9、tor = 2N = number of copies of original = 83logNkShapeSquare2Line segment1Triangle2Cube3NklogWhat doesNklogtell us about a shape?Thats right:Nklogtells us the dimension of the shape.(Note that for this to make sense, the shape has to beself-similar.)So for a self-similar shape, we can takeNklogto be

10、 the definition of its dimension.(It turns out that this definition coincides with a much moregeneral definition of dimension called the fractal dimension.)Now lets recall what k and N were for one side of theKoch snowflake:k = scale factor = 3N = number of copies of original = 4.261. 14loglog3NkSo

11、each side of the Koch snowflake is approximately1.261-dimensional.Thats what makes the Koch snowflake a fractal the fact that its dimension is not an integer.Start with a square of side length 3, with a square of sidelength 1 removed from its center.perimeter = 4(3) + 4(1)area = 1322The Sierpinski F

12、ractalThe Sierpinski FractalThink of this shape as consisting of eight small squares, eachof side length 1.area = 31813222From each small square, remove its central square.perimeter =31481434perimeter =31483148143422area = 31831813222222Iterate.31831831813323222222area = 31483148314814343322perimete

13、r =Iterate.The Sierpinski carpet is whats left afteryoure finished removing everything.In other words, its the intersection of all the previous sets.31483148314814343322perimeter =038434nn.31831831813323222222area = .98989813322981312= 0So the Sierpinski carpet has an infinite perimeter butit bounds a region with an area of zero!WeirdYour turn: compute the fractal dimension of the Sierpinski carpet.89. 18log3The Minkowski FractalThe Triangle Sierpinski FractalThe Hilbert Fractal6.3 Fractal antennasThe Koch Fractal AntennaThe Triangle Sierpinski Fra