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A moment-method characterization of V-Koch fractal dipole antennas
Int. J. Electron. Commun. (AEÜ) 63 (2009) 279 – 286 www.elsevier.de/aeue
A moment-method characterization of V-Koch fractal dipole antennas Rowdra Ghataka,∗ , Dipak R. Poddarb , Rabindra K. Mishrac a Radionics Laboratory, Department of Physics, The University of Burdwan, Burdwan-713104, West Bengal, India b Electronics and Telecommunication Engineering Department, Jadavpur University, Kolkata-760032, India c Electronic Science Department, Berhampur University, Bhanja Vihar, Berhampur-761003, Orissa, India
Received 26 June 2007; accepted 22 January 2008
Abstract This article compares input impedances and radiation characteristics of half wavelength Koch fractal V–electric dipoles having included angles 60◦ , 90◦ and 120◦ . The study considers three structures. In the 1st structure the Koch arms open into the V-region, in 2nd structure they open away from the V-region and in the third structure, one arm opens into and the other away from the V-region. A first iteration, structure 1 of V-Koch electric dipole antenna with included angle of 120◦ was fabricated and the experimental return loss was in good agreement with simulation. At their first resonances the antennas’ gain and input resistance decrease with decrease in included angles, an observation synonymous to Euclidian electric dipoles. In terms of gain, the first structure is found to give better performance than the other two. For this structure, the pattern distortion at the second resonance was also less compared to the other structures. 䉷 2008 Elsevier GmbH. All rights reserved. Keywords: Wire antenna; Fractal; V-Koch electric dipole
1. Introduction The electric dipole antenna is one of the most widely studied radiating systems [1]. It has many derivatives like the V-electric dipole antenna. The analysis of such antennas use the Euclidian system. In such coordinate system the dimension is always an integer number. For analysis, the antenna is assumed to be an array of elementary electric dipole elements of infinitesimal dimension with zero spacing. The same analysis method can be universal basis for antenna analysis. With the same endpoints as that of conventional electric dipoles and the monopoles, the electrical length of an antenna increases by adopting fractal geometry [2,3]. Such ∗ Corresponding author. Tel.: +91 9434147929.
E-mail addresses: [email protected] (R. Ghatak), [email protected] (D.R. Poddar), [email protected] (R.K. Mishra). 1434-8411/$ - see front matter 䉷 2008 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2008.01.010
realization, of large electrical length, results in simultaneous operation at multiple frequencies [4] with sufficient miniaturization [5]. These fractal shaped antennas exhibit many improved features in comparison to their Euclidian counterparts. Since the electric dipole is supposed to be the most common antenna in Euclidian geometry, its counterpart in the Fractal geometry shall be the Koch electric dipole. Rigorous physics-based analysis is yet to be developed for Koch electric dipole antenna. Therefore, Koch pre-fractal electric dipole antenna has been studied using method of moment (MoM) for its resonant frequency [2,3] and other characteristics. This work considers the modification of simple Koch electric dipole into V-Koch electric dipole structures with included angles between the V-arms, as 60◦ , 90◦ and 120◦ . These antennas are simulated using a MoM software WIPL-D姠 [6]. These simulations are based on MoM,
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R. Ghatak et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 279 – 286
which considers the current distribution on the electric dipole arms. The study considers three structures. In the 1st structure the notches on the arms open into the V-region, in 2nd structure they open away from the V-region and in the third structure, they open inward on one arm and outward from the V-region on the other.
that result in (3). W (A) =
N
wn (A).
(3)
n=1
In (3) W is the Hutchinson operator. The desired iteration can be obtained by applying such repeated transformation to the geometry as
2. Koch curve
A1 = W (A0 ),
It is one of the most popular shapes in the fractal geometry. Many fractals are made up of parts that are similar to the whole shape. Koch fractal geometry is one such fractal. The self-similarity is one of the properties of fractals that can be used to define a fractal. The iterated function system [7] provides a unified approach for construction of fractals. This is illustrated in Fig. 1 for Koch fractal geometry. The general procedure for obtaining such curves is by subdividing a straight line into three equal parts and then replacing the original line by four such sub-parts. Thus a straight line (i.e. Euclidian geometry) is transformed to a Koch shape (i.e. fractal geometry). Analytically a transform such as the affine transformation (1) in the plane : R2 → R2 can give more insight into the resulting structure. a b x1 e x1 = + = Ax + t. (1) c d f x2 x2
The method of calculating the dimensions of these structures is to compute the fractal dimension of a line as a function of two measurements taken while enclosing the fractal line with a number of discrete boxes. If the number of boxes N (), of linear size , are necessary to cover a set of points distributed in a plane, then the box dimension is defined as the power D in (5)
In (1) x1 and x2 are the coordinates of the point x and the matrix A is expressed as cos − sin A= . (2) sin cos In (2), is scaling factor, which takes values in the range 0 to 1 and is the angle of rotation of the subparts to form the curve. The matrix t gives the translation motion. The parameters a, b, c and d control the rotation of coordinates as well as scaling, while the parameters e and f control linear translation. Let us now assume that w1 , w2 , w3 , . . . , wN are a set of affine transformation, and let A be the initial geometry defined by the matrix as given above. The new geometry is obtained by applying appropriate transformation to (2)
Fig. 1. Geometry of a Koch curve fractal.
A2 = W (A1 ), . . . , Ak = W (Ak−1 ).
N () = −D .
(4)
(5)
Taking the logarithm of both sides it becomes log[N ()] = −D log(). Physically it implies that the dimension is the ratio of the logarithm of the number of new elements N (), formed from the parent object, to the logarithm of the inverse of the scaling factor . Analytically, it can be written as D=
log[N ()] . log 1/
(6)
As an example (refer to Fig. 1), let us take the Koch fractal, which is obtained from a line by dividing the line of length L into three equal parts and then replacing the middle portion with an equilateral triangle of side length L/3. Its dimension (i.e. (log 4)/log(1/3)) is 1.26.
3. Antenna structures and computational complexities The design starts with a half wave electric dipole of length 166.5 mm corresponding to the 900 MHz. The wire antenna radius is taken as 1.4478 mm (gauge no. 9 wire). Fig. 2 shows, from (a) to (c), the structures 1 to 3, respectively for first iteration V-Koch electric dipole antenna. Fig. 2d shows the structure 1 for a second iteration antenna. A fabricated prototype of first iteration V-Koch electric dipole antenna with included angle of 120◦ is shown in Fig. 2e. As an example of the nomenclature for the antenna used in this paper consider V60K2I2; it means a V-dipole based on second iteration Koch with included angle of 60◦ and the last number 2 depicts the second structure, which means that the notch on the V-arm open outward. Similarly in the 1st structure the Koch curve opens inwards on the V-arm and in the 3rd structure the curve on one arm opens inward while the curve on the other arm opens outward. As the notches on antenna arms may open up in any of the three ways, so a family of structures is considered for the
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the shortest wavelength that corresponds to the maximum desirable frequency in the analysis domain. A trade-off between number of segments and the highest frequency in the analysis with a given radius of the wire is desired when V-Koch fractal electric dipole antenna needs to be analysed using MoM. The starting point of a Koch curve is division of a line into three equal sub-parts (Fig. 1 of previous section). For the verification of the antenna shown in Fig. 2e) entire span was sub-divided into small frequency segments of 1.5 GHz/2 GHz intervals like 0.5–2.5, 2.5–4.5 and then 4.5 to 6.0 GHz. For the second iteration V-Koch electric dipole antenna three frequency ranges first one from 0.5 to 2.5 GHz, second from 2.5 to 4.5 GHz and third from 4.5 to 6.5 GHz with 151 frequency points in each sub-span was chosen. It is also observed that the results match at the sub-span edges. The simulated results of WIPL-D姠 were then accumulated and concatenated. Fig. 2. V-Koch fractal electric dipole antenna (a) structures 1, (b) structure 2 and (c) structure 3. (d) The structure 1 for second iteration. (e) Fabricated prototype of first iteration inward opening V-Koch electric dipole antenna.
present discussion. For the inward opening V-Koch dipole with included angle 60◦ , the arms get shorted and hence its analysis is excluded. The method discussed in previous section in conjunction with coordinate rotation algorithms calculates the coordinates for different iterations of Koch curve dipoles with included angles of 60◦ , 90◦ , 120◦ and 180◦ . The coordinate points calculated using MATLAB姠 is exported to WIPL-D姠 that computes antenna current and hence other characteristics. In MoM formulation the segment length to radius of wire ratio (/a) is critical for accuracy and validation [8]. For iterated structures like fractal geometries it gets far more important due to increasing number of inflection points. Within the realm of computational electromagnetics and specifically in MoM the manner in which the kernel is evaluated is very important. If thin wire kernel approximation is used then it has a limitation that it is applicable to problems with /a > 2. However, when the kernel is evaluated without thin wire approximation, which is used in commercial software like WIPL-D姠 [6], this limitation is warded off. Still it must be clarified that for too small a value of /a the convergence of numerical computation may be slow. This situation may arise when highly iterated fractal structures are analysed, for example a sixth-order Koch curve electric dipole antenna. It is not only computationally challenging to analyse it but also difficult to fabricate. So the benefits of fractal antenna in general are harvested till certain stages commonly known as pre-fractal geometry. MoM simulation, for 2nd and higher iteration Koch wire antenna to find resonant frequencies of upper bands, also relies on the segment length to wavelength ratio and radius to wavelength ratio. Here wavelength means
4. Resonance characteristics We want to observe the changes in the resonance behaviour of the Koch electric dipole antenna when the included angle between the arms is changed that result in a V-Koch dipole antenna. The computed input resistance of the electric dipole, Koch electric dipole, and V-Koch electric dipole for various included angles are tabulated in Table 1. These results show that the input resistance of the VKoch electric dipole antenna decreases with the included angle, in conformity with their Euclidian counterparts of the same length. In each of the iterations, the input resistance is highest for the second structure. With decrease in the included angle the arms get closer and the capacitance between them increases. This is further enhanced for the structure one, where the arms of the dipole open up inwards. For all structures with increase in number of iterations, the electrical length increases; it results in addition of equivalent inductance for the additional length of wire. Table 2 shows that the resonant frequencies drift towards lower side Table 1. Computed input resistance for Koch dipole antenna structures Antenna type KDPL1I KDPL 2I V-KOCH V120K1I V120K2I V90K1I V90K2I V60K1I V60K2I NC: not computed.
Computed input resistance ()
Structure 1 40 32 20 19.82 NC NC
50 45.85 Structure 2 45.96 41.35 40 33 20 20
Structure 3 40 37 30 30 20 10
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Table 2. Tabulated resonant frequencies and corresponding gains of various structures of Koch and V-Koch dipole antenna Iteration 1
Resonance (GHz)
Gain (dB)
Iteration 2
Resonance (GHz)
Gain (dB)
KDPL1I V60K1I2 V60K1I3 V90K1I1 V90K1I2 V90K1I3 V120K1I1 V120K1I2 V120K1I3
0.771, 0.793, 0.814, 0.786, 0.757, 0.771, 0.757, 0.743, 0.750,
2.00, 1.47, 1.61, 1.88, 1.71, 1.78, 1.97, 1.89, 1.92,
KDPL2I V60K2I2 V60K2I3 V90K2I1 V90K2I2 V90K2I3 V120K2I1 V120K2I2 V120K2I3
0.741, 0.788, 0.816, 0.806, 0.760, 0.779, 0.779, 0.751, 0.760,
2.00, 2.82, 1.52 1.48, 2.95, 5.65 1.63, 3.07, 1.53, 1.50 1.9, 5.48, 1.30, 2.10 1.71, 1.70, 1.64 1.78, 4.11, 1.85 1.97, 6.26, 1.62, 3.50 1.87, 1.50, 1.31 1.90, 4.11, 1.43
2.279 2.193 2.243 2.121 2.271 2.221 2.157 2.257 2.200
2.82 3.23 3.24 5.76 3.30 4.53 6.30 2.50 4.64
2.162, 2.255, 2.236, 2.116, 2.236, 2.218, 2.153, 2.209, 2.199,
3.462 3.434 3.481, 4.966 3.564, 5.031 3.444 3.546 3.49, 5.022 3.462 3.518
Fig. 3. Measured and simulated return loss of the fabricated prototype of the antenna as shown in Fig. 2e.
with higher iteration. The fabricated prototype as depicted in Fig. 2e was experimented using a HP 8722C VNA. The simulated and measured return loss is plotted in Fig. 3 that is in good agreement. However, the difference in measured and simulation result is due to fabrication tolerance in the region of feed. The simulated resonance characteristics for first iteration and second iteration of Koch electric dipole as well as V-Koch electric dipole antennas, by changing the included angle, are shown in Fig. 4. The distinct observation is that for first iteration there are two distinct resonance frequencies and for second iteration there are three distinct bands. The resonance frequencies are not harmonically related rather the antenna behaves as multi-band antenna.
5. Radiation characteristics The current path in each segment of the arms of the V-Koch wire antenna, as shown in Fig. 5, helps in understanding the radiation mechanism for all three-structure with an included angle . The first resonance is due to the entire length of arms. Higher order resonance is also obtained due to the translation of the electrical behaviour of smaller segments in various iterations. The collective effect of the
Fig. 4. Return loss plot of (a) first and (b) second iteration of Koch and V-Koch dipoles.
current in each segment on the far field determines the radiation pattern of the antenna. Fig. 6a–c depicts numerically computed gain variation of each antenna with increasing included angle, except an inward opening Koch curve V-electric dipole of 60◦ . It is a fact that a half wavelength electric V-electric dipole antenna
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designed for any frequency show a maximum gain for an optimum flare angle of 180◦ [9] (i.e. on becoming a normal electric dipole). However, it is observed that for a V-Koch electric dipole, this phenomenon is true only at its first resonant frequency as seen in Table 2. At this resonance, in all three structures, gain increases for increasing included angles, an observation synonymous to Euclidian electric dipoles. In terms of gain, the first structure is found to give better performance than the other two. For this structure, the pattern
Fig. 5. Current paths in all three structures of first iteration V-Koch dipole antenna.
283
distortion at the second resonance is also less compared to the other structures. It is observed that for all structures the gain at the second resonant frequency, as tabulated in Table 2, is more than that at the first one. Gain is more for both iteration of structure 1 as compared to other structures. This may be due to the following reason. Due to the Koch curve there is elongation in over all length of the arm. It results in added inductive effect. In the notched portion, the capacitive effect is higher for inward opening than any other openings. So the inward opening, i.e. the structure 1, compensates the inductive effect better than the other structures. Thus it gives better gain. With a similar reasoning, the third structure shall give better gain than the second structure, which is also evident in the figure. The gain for both iterations of structure 2 is lowest amongst all. Thus proximity of the conductors influences gain in V-Koch antennas. Since, the gain varies with resonant frequency, so the multi-band operation of the V-Koch antenna does not make its radiation characteristics frequency independent. However, it does not mean that the radiation occurs at higher order modes, since the resonant frequencies are not harmonically related. At each resonant frequency, the radiation is in
Fig. 6. Variation in gain at the first and second resonance as per Table 2. Left hand side are for (a) V90K1I1 and V120K1I1; (b) V60K1I2, V90K1I2 and V120K1I2; (c) V60K1I3, V90K1I3 and V120K1I3. The right hand side are for (a) V90K2I1 and V120K2I1; (b) V60K2I2, V90K2I2 and V120K2I2; (c) V60K2I3, V90K2I3 and V120K2I3.
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Fig. 7. Phi-cut of radiation pattern of Koch dipole and V-Koch dipole. In (a), (b) and (c) the structures 1, 2 and 3 are compared respectively for the first iteration and in (d), (e) and (f) for the second iteration. The left-hand side figures pertain to first resonance and the right-hand side figures are for second resonance.
the fundamental mode and the structure behaves as a different antenna. These arguments are further substantiated from the radiation pattern of the V-Koch antenna structures. Fig. 7 puts forward the phi cut of radiation pattern of all the structures tried out with V-Koch electric dipole antennas. The left hand figure shows the radiation pattern for first resonance and the right hand figure for that of the second resonance. For all structures it can be observed that radiation pattern at the lowest radiating frequency resembles somewhat to that of a electric dipole; but at higher radiating frequencies the
pattern considerably deviate from that of the half wavelength electric dipole which was the starting point of our design. Observing the right-hand side pattern for the structure 1 in Fig. 7a and d no deviation is found. Let us now consider the radiation characteristics for higher resonance of V60K2I2, V60K2I3, V90K2I1 and V120K2I1. A gain of 5.65 dB at 3.434 GHz for K60K2I2 as compared to a gain of 1.3 dB at nearly same frequency of 3.564 GHz for V90K2I1 is seen in Table 2 and left-hand side of Fig. 8. It is further noted that the patterns are distinct from each other as the maxima of one coincides with the minima
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Fig. 7. (continued).
Fig. 8. Comparison of radiation characteristics for higher resonance of antennas namely V60K2I2 and V90K2I1 in left-hand side and in right-hand side for antennas namely V60K2I3, V90K2I1and V120K2I1.
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of the other. For clarity, in right-hand side of Fig. 8 the maximum of V60K2I3 is in the broadside direction. The dotted arrows show the direction of maximum for V90K2I1 and the solid arrow shows that for V120K2I1.
6. Conclusion The simulated results for V-Koch antennas are presented. While exhibiting the expected Multiband characteristics, they show improved gain for the second Koch-iteration, which are more prominent in structures for which Koch arms open inwards to the V-region. Gain at the first resonance show variation similar to a Euclidian antenna but the same is not followed at higher resonance. The input resistance decreases with decreasing included angle. Both for second and third structures, the radiation patterns vary with resonant frequencies. However, there is certain degree of consistency in the radiation pattern for inward opening V-Koch electric dipole.
Acknowledgements The authors thankfully acknowledge the helps received in using WIPL-D姠 from IIT Kharagpur. The authors are grateful to the anonymous reviewers for their constructive and helpful comments and suggestions.
References [1] Balanis CA. Antenna theory: analysis and design. New Delhi: John Wiley & Sons; 1997. [2] Puente C, Romeu J, Cardama A. The Koch monopole: a small fractal antenna. IEEE Trans Antennas Propag 2000;48: 1773–81. [3] Vinoy KJ, Abraham JK, Varadan VK. On the relationship between fractal dimension and the performance of multiresonant dipole antennas using Koch curves. IEEE Trans Antennas Propag 2003;51:2296–303. [4] Douglas HW, Raj M. Frontiers in electromagnetic. New York: Wiley-IEEE Press; 1999. [5] Gianvittorio JP, Samii YR. Fractal antennas: a novel miniaturization technique, and applications. IEEE Antennas Propag Mag 2002;44:20–35. [6] Kolunzija BM, Ognjanovic JS, Sarkar TK. WIPL-D — Electromagnetic modelling of composite metallic and dielectric structures. Software and User’s Manual. Boston: Artech House; 2000.
[7] Peitgen HO, Jurgens H, Saupe D. Chaos and Fractals: new frontiers of science. New York: Springer; 1992. [8] David BD. Computational electromagnetics for RF and microwave engineering. UK: Cambridge University Press; 2005. [9] Schelkunoff SA, Friis HT. Antennas: theory and practice. New York: John Wiley and Sons, Inc.; 1966.
Rowdra Ghatak completed his M.Sc. (Physics) with specialization in Radio Physics and Electronics and M.Tech. (Microwave Engineering) from The University of Burdwan in ‘99 and ‘02, respectively. He is currently in the Physics Department, The University of Burdwan as a Scientific Officer. He is a recipient of the URSI Young Scientist Award in 2005. He has 20 publications in various National/International conferences and journals. His research interest lies in the areas of Fractal antenna, mobile radio communication and application of Genetic algorithms to electromagnetic optimization. He is a life member of ISTE and member IEEE.
Dipak Ranjan Poddar is a Professor in the Department of Electronics and TeleCommunication Engineering in Jadavpur University. He has supervised more than 15 Ph.D. theses. He has more than 90 publications in various National/International conference and journals. His areas of research include EMI/EMC, antennas and metamaterials. He is a senior member of IEEE. Rabindra K. Mishra is a Professor in Electronics Science Department of Berhampur University. He has published more than 40 research papers in International/National journals. He has received many awards including Hari Ohm Ashram Prerit Harivalabh Das Chunilal Research Endowment Award, IETE JC Bose Award, British Commonwealth Fellowship, INSA Visiting Fellowship, etc. He is a reviewer of many International Journals including IEEE Transactions on Antennas and Propagation. He is a Senior Member of IEEE and Life Member of IETE & ISTE.
A moment-method characterization of V-Koch fractal dipole antennas Rowdra Ghataka,∗ , Dipak R. Poddarb , Rabindra K. Mishrac a Radionics Laboratory, Department of Physics, The University of Burdwan, Burdwan-713104, West Bengal, India b Electronics and Telecommunication Engineering Department, Jadavpur University, Kolkata-760032, India c Electronic Science Department, Berhampur University, Bhanja Vihar, Berhampur-761003, Orissa, India
Received 26 June 2007; accepted 22 January 2008
Abstract This article compares input impedances and radiation characteristics of half wavelength Koch fractal V–electric dipoles having included angles 60◦ , 90◦ and 120◦ . The study considers three structures. In the 1st structure the Koch arms open into the V-region, in 2nd structure they open away from the V-region and in the third structure, one arm opens into and the other away from the V-region. A first iteration, structure 1 of V-Koch electric dipole antenna with included angle of 120◦ was fabricated and the experimental return loss was in good agreement with simulation. At their first resonances the antennas’ gain and input resistance decrease with decrease in included angles, an observation synonymous to Euclidian electric dipoles. In terms of gain, the first structure is found to give better performance than the other two. For this structure, the pattern distortion at the second resonance was also less compared to the other structures. 䉷 2008 Elsevier GmbH. All rights reserved. Keywords: Wire antenna; Fractal; V-Koch electric dipole
1. Introduction The electric dipole antenna is one of the most widely studied radiating systems [1]. It has many derivatives like the V-electric dipole antenna. The analysis of such antennas use the Euclidian system. In such coordinate system the dimension is always an integer number. For analysis, the antenna is assumed to be an array of elementary electric dipole elements of infinitesimal dimension with zero spacing. The same analysis method can be universal basis for antenna analysis. With the same endpoints as that of conventional electric dipoles and the monopoles, the electrical length of an antenna increases by adopting fractal geometry [2,3]. Such ∗ Corresponding author. Tel.: +91 9434147929.
E-mail addresses: [email protected] (R. Ghatak), [email protected] (D.R. Poddar), [email protected] (R.K. Mishra). 1434-8411/$ - see front matter 䉷 2008 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2008.01.010
realization, of large electrical length, results in simultaneous operation at multiple frequencies [4] with sufficient miniaturization [5]. These fractal shaped antennas exhibit many improved features in comparison to their Euclidian counterparts. Since the electric dipole is supposed to be the most common antenna in Euclidian geometry, its counterpart in the Fractal geometry shall be the Koch electric dipole. Rigorous physics-based analysis is yet to be developed for Koch electric dipole antenna. Therefore, Koch pre-fractal electric dipole antenna has been studied using method of moment (MoM) for its resonant frequency [2,3] and other characteristics. This work considers the modification of simple Koch electric dipole into V-Koch electric dipole structures with included angles between the V-arms, as 60◦ , 90◦ and 120◦ . These antennas are simulated using a MoM software WIPL-D姠 [6]. These simulations are based on MoM,
280
R. Ghatak et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 279 – 286
which considers the current distribution on the electric dipole arms. The study considers three structures. In the 1st structure the notches on the arms open into the V-region, in 2nd structure they open away from the V-region and in the third structure, they open inward on one arm and outward from the V-region on the other.
that result in (3). W (A) =
N
wn (A).
(3)
n=1
In (3) W is the Hutchinson operator. The desired iteration can be obtained by applying such repeated transformation to the geometry as
2. Koch curve
A1 = W (A0 ),
It is one of the most popular shapes in the fractal geometry. Many fractals are made up of parts that are similar to the whole shape. Koch fractal geometry is one such fractal. The self-similarity is one of the properties of fractals that can be used to define a fractal. The iterated function system [7] provides a unified approach for construction of fractals. This is illustrated in Fig. 1 for Koch fractal geometry. The general procedure for obtaining such curves is by subdividing a straight line into three equal parts and then replacing the original line by four such sub-parts. Thus a straight line (i.e. Euclidian geometry) is transformed to a Koch shape (i.e. fractal geometry). Analytically a transform such as the affine transformation (1) in the plane : R2 → R2 can give more insight into the resulting structure. a b x1 e x1 = + = Ax + t. (1) c d f x2 x2
The method of calculating the dimensions of these structures is to compute the fractal dimension of a line as a function of two measurements taken while enclosing the fractal line with a number of discrete boxes. If the number of boxes N (), of linear size , are necessary to cover a set of points distributed in a plane, then the box dimension is defined as the power D in (5)
In (1) x1 and x2 are the coordinates of the point x and the matrix A is expressed as cos − sin A= . (2) sin cos In (2), is scaling factor, which takes values in the range 0 to 1 and is the angle of rotation of the subparts to form the curve. The matrix t gives the translation motion. The parameters a, b, c and d control the rotation of coordinates as well as scaling, while the parameters e and f control linear translation. Let us now assume that w1 , w2 , w3 , . . . , wN are a set of affine transformation, and let A be the initial geometry defined by the matrix as given above. The new geometry is obtained by applying appropriate transformation to (2)
Fig. 1. Geometry of a Koch curve fractal.
A2 = W (A1 ), . . . , Ak = W (Ak−1 ).
N () = −D .
(4)
(5)
Taking the logarithm of both sides it becomes log[N ()] = −D log(). Physically it implies that the dimension is the ratio of the logarithm of the number of new elements N (), formed from the parent object, to the logarithm of the inverse of the scaling factor . Analytically, it can be written as D=
log[N ()] . log 1/
(6)
As an example (refer to Fig. 1), let us take the Koch fractal, which is obtained from a line by dividing the line of length L into three equal parts and then replacing the middle portion with an equilateral triangle of side length L/3. Its dimension (i.e. (log 4)/log(1/3)) is 1.26.
3. Antenna structures and computational complexities The design starts with a half wave electric dipole of length 166.5 mm corresponding to the 900 MHz. The wire antenna radius is taken as 1.4478 mm (gauge no. 9 wire). Fig. 2 shows, from (a) to (c), the structures 1 to 3, respectively for first iteration V-Koch electric dipole antenna. Fig. 2d shows the structure 1 for a second iteration antenna. A fabricated prototype of first iteration V-Koch electric dipole antenna with included angle of 120◦ is shown in Fig. 2e. As an example of the nomenclature for the antenna used in this paper consider V60K2I2; it means a V-dipole based on second iteration Koch with included angle of 60◦ and the last number 2 depicts the second structure, which means that the notch on the V-arm open outward. Similarly in the 1st structure the Koch curve opens inwards on the V-arm and in the 3rd structure the curve on one arm opens inward while the curve on the other arm opens outward. As the notches on antenna arms may open up in any of the three ways, so a family of structures is considered for the
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the shortest wavelength that corresponds to the maximum desirable frequency in the analysis domain. A trade-off between number of segments and the highest frequency in the analysis with a given radius of the wire is desired when V-Koch fractal electric dipole antenna needs to be analysed using MoM. The starting point of a Koch curve is division of a line into three equal sub-parts (Fig. 1 of previous section). For the verification of the antenna shown in Fig. 2e) entire span was sub-divided into small frequency segments of 1.5 GHz/2 GHz intervals like 0.5–2.5, 2.5–4.5 and then 4.5 to 6.0 GHz. For the second iteration V-Koch electric dipole antenna three frequency ranges first one from 0.5 to 2.5 GHz, second from 2.5 to 4.5 GHz and third from 4.5 to 6.5 GHz with 151 frequency points in each sub-span was chosen. It is also observed that the results match at the sub-span edges. The simulated results of WIPL-D姠 were then accumulated and concatenated. Fig. 2. V-Koch fractal electric dipole antenna (a) structures 1, (b) structure 2 and (c) structure 3. (d) The structure 1 for second iteration. (e) Fabricated prototype of first iteration inward opening V-Koch electric dipole antenna.
present discussion. For the inward opening V-Koch dipole with included angle 60◦ , the arms get shorted and hence its analysis is excluded. The method discussed in previous section in conjunction with coordinate rotation algorithms calculates the coordinates for different iterations of Koch curve dipoles with included angles of 60◦ , 90◦ , 120◦ and 180◦ . The coordinate points calculated using MATLAB姠 is exported to WIPL-D姠 that computes antenna current and hence other characteristics. In MoM formulation the segment length to radius of wire ratio (/a) is critical for accuracy and validation [8]. For iterated structures like fractal geometries it gets far more important due to increasing number of inflection points. Within the realm of computational electromagnetics and specifically in MoM the manner in which the kernel is evaluated is very important. If thin wire kernel approximation is used then it has a limitation that it is applicable to problems with /a > 2. However, when the kernel is evaluated without thin wire approximation, which is used in commercial software like WIPL-D姠 [6], this limitation is warded off. Still it must be clarified that for too small a value of /a the convergence of numerical computation may be slow. This situation may arise when highly iterated fractal structures are analysed, for example a sixth-order Koch curve electric dipole antenna. It is not only computationally challenging to analyse it but also difficult to fabricate. So the benefits of fractal antenna in general are harvested till certain stages commonly known as pre-fractal geometry. MoM simulation, for 2nd and higher iteration Koch wire antenna to find resonant frequencies of upper bands, also relies on the segment length to wavelength ratio and radius to wavelength ratio. Here wavelength means
4. Resonance characteristics We want to observe the changes in the resonance behaviour of the Koch electric dipole antenna when the included angle between the arms is changed that result in a V-Koch dipole antenna. The computed input resistance of the electric dipole, Koch electric dipole, and V-Koch electric dipole for various included angles are tabulated in Table 1. These results show that the input resistance of the VKoch electric dipole antenna decreases with the included angle, in conformity with their Euclidian counterparts of the same length. In each of the iterations, the input resistance is highest for the second structure. With decrease in the included angle the arms get closer and the capacitance between them increases. This is further enhanced for the structure one, where the arms of the dipole open up inwards. For all structures with increase in number of iterations, the electrical length increases; it results in addition of equivalent inductance for the additional length of wire. Table 2 shows that the resonant frequencies drift towards lower side Table 1. Computed input resistance for Koch dipole antenna structures Antenna type KDPL1I KDPL 2I V-KOCH V120K1I V120K2I V90K1I V90K2I V60K1I V60K2I NC: not computed.
Computed input resistance ()
Structure 1 40 32 20 19.82 NC NC
50 45.85 Structure 2 45.96 41.35 40 33 20 20
Structure 3 40 37 30 30 20 10
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Table 2. Tabulated resonant frequencies and corresponding gains of various structures of Koch and V-Koch dipole antenna Iteration 1
Resonance (GHz)
Gain (dB)
Iteration 2
Resonance (GHz)
Gain (dB)
KDPL1I V60K1I2 V60K1I3 V90K1I1 V90K1I2 V90K1I3 V120K1I1 V120K1I2 V120K1I3
0.771, 0.793, 0.814, 0.786, 0.757, 0.771, 0.757, 0.743, 0.750,
2.00, 1.47, 1.61, 1.88, 1.71, 1.78, 1.97, 1.89, 1.92,
KDPL2I V60K2I2 V60K2I3 V90K2I1 V90K2I2 V90K2I3 V120K2I1 V120K2I2 V120K2I3
0.741, 0.788, 0.816, 0.806, 0.760, 0.779, 0.779, 0.751, 0.760,
2.00, 2.82, 1.52 1.48, 2.95, 5.65 1.63, 3.07, 1.53, 1.50 1.9, 5.48, 1.30, 2.10 1.71, 1.70, 1.64 1.78, 4.11, 1.85 1.97, 6.26, 1.62, 3.50 1.87, 1.50, 1.31 1.90, 4.11, 1.43
2.279 2.193 2.243 2.121 2.271 2.221 2.157 2.257 2.200
2.82 3.23 3.24 5.76 3.30 4.53 6.30 2.50 4.64
2.162, 2.255, 2.236, 2.116, 2.236, 2.218, 2.153, 2.209, 2.199,
3.462 3.434 3.481, 4.966 3.564, 5.031 3.444 3.546 3.49, 5.022 3.462 3.518
Fig. 3. Measured and simulated return loss of the fabricated prototype of the antenna as shown in Fig. 2e.
with higher iteration. The fabricated prototype as depicted in Fig. 2e was experimented using a HP 8722C VNA. The simulated and measured return loss is plotted in Fig. 3 that is in good agreement. However, the difference in measured and simulation result is due to fabrication tolerance in the region of feed. The simulated resonance characteristics for first iteration and second iteration of Koch electric dipole as well as V-Koch electric dipole antennas, by changing the included angle, are shown in Fig. 4. The distinct observation is that for first iteration there are two distinct resonance frequencies and for second iteration there are three distinct bands. The resonance frequencies are not harmonically related rather the antenna behaves as multi-band antenna.
5. Radiation characteristics The current path in each segment of the arms of the V-Koch wire antenna, as shown in Fig. 5, helps in understanding the radiation mechanism for all three-structure with an included angle . The first resonance is due to the entire length of arms. Higher order resonance is also obtained due to the translation of the electrical behaviour of smaller segments in various iterations. The collective effect of the
Fig. 4. Return loss plot of (a) first and (b) second iteration of Koch and V-Koch dipoles.
current in each segment on the far field determines the radiation pattern of the antenna. Fig. 6a–c depicts numerically computed gain variation of each antenna with increasing included angle, except an inward opening Koch curve V-electric dipole of 60◦ . It is a fact that a half wavelength electric V-electric dipole antenna
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designed for any frequency show a maximum gain for an optimum flare angle of 180◦ [9] (i.e. on becoming a normal electric dipole). However, it is observed that for a V-Koch electric dipole, this phenomenon is true only at its first resonant frequency as seen in Table 2. At this resonance, in all three structures, gain increases for increasing included angles, an observation synonymous to Euclidian electric dipoles. In terms of gain, the first structure is found to give better performance than the other two. For this structure, the pattern
Fig. 5. Current paths in all three structures of first iteration V-Koch dipole antenna.
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distortion at the second resonance is also less compared to the other structures. It is observed that for all structures the gain at the second resonant frequency, as tabulated in Table 2, is more than that at the first one. Gain is more for both iteration of structure 1 as compared to other structures. This may be due to the following reason. Due to the Koch curve there is elongation in over all length of the arm. It results in added inductive effect. In the notched portion, the capacitive effect is higher for inward opening than any other openings. So the inward opening, i.e. the structure 1, compensates the inductive effect better than the other structures. Thus it gives better gain. With a similar reasoning, the third structure shall give better gain than the second structure, which is also evident in the figure. The gain for both iterations of structure 2 is lowest amongst all. Thus proximity of the conductors influences gain in V-Koch antennas. Since, the gain varies with resonant frequency, so the multi-band operation of the V-Koch antenna does not make its radiation characteristics frequency independent. However, it does not mean that the radiation occurs at higher order modes, since the resonant frequencies are not harmonically related. At each resonant frequency, the radiation is in
Fig. 6. Variation in gain at the first and second resonance as per Table 2. Left hand side are for (a) V90K1I1 and V120K1I1; (b) V60K1I2, V90K1I2 and V120K1I2; (c) V60K1I3, V90K1I3 and V120K1I3. The right hand side are for (a) V90K2I1 and V120K2I1; (b) V60K2I2, V90K2I2 and V120K2I2; (c) V60K2I3, V90K2I3 and V120K2I3.
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Fig. 7. Phi-cut of radiation pattern of Koch dipole and V-Koch dipole. In (a), (b) and (c) the structures 1, 2 and 3 are compared respectively for the first iteration and in (d), (e) and (f) for the second iteration. The left-hand side figures pertain to first resonance and the right-hand side figures are for second resonance.
the fundamental mode and the structure behaves as a different antenna. These arguments are further substantiated from the radiation pattern of the V-Koch antenna structures. Fig. 7 puts forward the phi cut of radiation pattern of all the structures tried out with V-Koch electric dipole antennas. The left hand figure shows the radiation pattern for first resonance and the right hand figure for that of the second resonance. For all structures it can be observed that radiation pattern at the lowest radiating frequency resembles somewhat to that of a electric dipole; but at higher radiating frequencies the
pattern considerably deviate from that of the half wavelength electric dipole which was the starting point of our design. Observing the right-hand side pattern for the structure 1 in Fig. 7a and d no deviation is found. Let us now consider the radiation characteristics for higher resonance of V60K2I2, V60K2I3, V90K2I1 and V120K2I1. A gain of 5.65 dB at 3.434 GHz for K60K2I2 as compared to a gain of 1.3 dB at nearly same frequency of 3.564 GHz for V90K2I1 is seen in Table 2 and left-hand side of Fig. 8. It is further noted that the patterns are distinct from each other as the maxima of one coincides with the minima
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Fig. 7. (continued).
Fig. 8. Comparison of radiation characteristics for higher resonance of antennas namely V60K2I2 and V90K2I1 in left-hand side and in right-hand side for antennas namely V60K2I3, V90K2I1and V120K2I1.
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of the other. For clarity, in right-hand side of Fig. 8 the maximum of V60K2I3 is in the broadside direction. The dotted arrows show the direction of maximum for V90K2I1 and the solid arrow shows that for V120K2I1.
6. Conclusion The simulated results for V-Koch antennas are presented. While exhibiting the expected Multiband characteristics, they show improved gain for the second Koch-iteration, which are more prominent in structures for which Koch arms open inwards to the V-region. Gain at the first resonance show variation similar to a Euclidian antenna but the same is not followed at higher resonance. The input resistance decreases with decreasing included angle. Both for second and third structures, the radiation patterns vary with resonant frequencies. However, there is certain degree of consistency in the radiation pattern for inward opening V-Koch electric dipole.
Acknowledgements The authors thankfully acknowledge the helps received in using WIPL-D姠 from IIT Kharagpur. The authors are grateful to the anonymous reviewers for their constructive and helpful comments and suggestions.
References [1] Balanis CA. Antenna theory: analysis and design. New Delhi: John Wiley & Sons; 1997. [2] Puente C, Romeu J, Cardama A. The Koch monopole: a small fractal antenna. IEEE Trans Antennas Propag 2000;48: 1773–81. [3] Vinoy KJ, Abraham JK, Varadan VK. On the relationship between fractal dimension and the performance of multiresonant dipole antennas using Koch curves. IEEE Trans Antennas Propag 2003;51:2296–303. [4] Douglas HW, Raj M. Frontiers in electromagnetic. New York: Wiley-IEEE Press; 1999. [5] Gianvittorio JP, Samii YR. Fractal antennas: a novel miniaturization technique, and applications. IEEE Antennas Propag Mag 2002;44:20–35. [6] Kolunzija BM, Ognjanovic JS, Sarkar TK. WIPL-D — Electromagnetic modelling of composite metallic and dielectric structures. Software and User’s Manual. Boston: Artech House; 2000.
[7] Peitgen HO, Jurgens H, Saupe D. Chaos and Fractals: new frontiers of science. New York: Springer; 1992. [8] David BD. Computational electromagnetics for RF and microwave engineering. UK: Cambridge University Press; 2005. [9] Schelkunoff SA, Friis HT. Antennas: theory and practice. New York: John Wiley and Sons, Inc.; 1966.
Rowdra Ghatak completed his M.Sc. (Physics) with specialization in Radio Physics and Electronics and M.Tech. (Microwave Engineering) from The University of Burdwan in ‘99 and ‘02, respectively. He is currently in the Physics Department, The University of Burdwan as a Scientific Officer. He is a recipient of the URSI Young Scientist Award in 2005. He has 20 publications in various National/International conferences and journals. His research interest lies in the areas of Fractal antenna, mobile radio communication and application of Genetic algorithms to electromagnetic optimization. He is a life member of ISTE and member IEEE.
Dipak Ranjan Poddar is a Professor in the Department of Electronics and TeleCommunication Engineering in Jadavpur University. He has supervised more than 15 Ph.D. theses. He has more than 90 publications in various National/International conference and journals. His areas of research include EMI/EMC, antennas and metamaterials. He is a senior member of IEEE. Rabindra K. Mishra is a Professor in Electronics Science Department of Berhampur University. He has published more than 40 research papers in International/National journals. He has received many awards including Hari Ohm Ashram Prerit Harivalabh Das Chunilal Research Endowment Award, IETE JC Bose Award, British Commonwealth Fellowship, INSA Visiting Fellowship, etc. He is a reviewer of many International Journals including IEEE Transactions on Antennas and Propagation. He is a Senior Member of IEEE and Life Member of IETE & ISTE.