Miniaturization of a Koch-Type Fractal Antenna for Wi-Fi Applications

fractal and fractional

Article Miniaturization of a Koch-Type Fractal Antenna for Wi-Fi Applications

Dmitrii Tumakov 1,* , Dmitry Chikrin 2 and Petr Kokunin 2

1 Institute of Computer Mathematics and Information Technologies, Kazan Federal University; 18 Kremlevskaya Str., 420008 Kazan, Russia 2 Institute of Physics, Kazan Federal University; 18 Kremlevskaya Str., 420008 Kazan, Russia; [email protected] (D.C.); [email protected] (P.K.) * Correspondence: [email protected]

Received: 28 February 2020; Accepted: 2 June 2020; Published: 4 June 2020

Abstract: Koch-type wire dipole antennas are considered herein. In the case of a first-order prefractal, such antennas differ from a Koch-type dipole by the position of the central vertex of the dipole arm. Earlier, we investigated the dependence of the base frequency for different antenna scales for an arm in the form of a first-order prefractal. In this paper, dipoles for second-order prefractals are considered. The dependence of the base frequency and the reflection coefficient on the dipole wire length and scale is analyzed. It is shown that it is possible to distinguish a family of antennas operating at a given (identical) base frequency. The same length of a Koch-type curve can be obtained with different coordinates of the central vertex. This allows for obtaining numerous antennas with various scales and geometries of the arm. An algorithm for obtaining small antennas for Wi-Fi applications is proposed. Two antennas were obtained: an antenna with the smallest linear dimensions and a minimum antenna for a given reflection coefficient.

Keywords: Koch-type antenna; fractal antenna; antenna miniaturization; Wi-Fi applications

1. Introduction Wire antennas are widely used in modern telecommunication systems [1]. Despite the fact that the simplest wire half-wave dipole antennas are well explored [2], antennas with complex geometry are a separate subject for investigation [3]. The latter antennas are by far the most promising devices [4,5]. By complicating the geometry, it is possible both to minimize the dimensions of the antenna itself and to improve its electrodynamic characteristics [6,7]. In practice, various forms of broken balanced dipoles are used [1,8,9]. However, the most common way to minimize or improve the properties of the antennas is through their fractalization [10–16]. The most studied fractal antenna is a dipole constructed on the basis of the Koch fractal. A sufficient number of works have been devoted to the study and analysis of the base characteristics of both the Koch dipole and monopole [17,18], as well as its various modifications [19–21]. One of these modifications is the Koch-type fractal [22]. In addition, some other fractal structures are also used, including the Minkowski curve [23], Spidron fractal [24], crinkle curve [25], Peano fractal [26], and many other curves. Hybrid structures representing combinations of the above are also popular [27]. For designing antennas, in practice, both two-dimensional and three-dimensional structures are applied [28,29]. For the simplest half-wave dipole, especially in the case of the base frequency, the interconnections between its various parameters are well known [30]. The relationship between the base frequency and the reflector geometry was also investigated by the authors for Koch-type dipoles [22,31]. Such a relationship can be used for the designing of antennas [32]. We previously used this approach to model a Koch-type dipole of the first iteration for Wi-Fi applications [33].

Fractal Fract. 2020, 4, 25; doi:10.3390/fractalfract4020025 www.mdpi.com/journal/fractalfract Fractal Fract. 2020, 4, x FOR PEER REVIEW 2 of 12

[22,31]. Such a relationship can be used for the designing of antennas [32]. We previously used this approach to model a Koch-type dipole of the first iteration for Wi-Fi applications [33]. In this paper, a Koch-type miniature wire antenna is designed using regression analysis. A second-order prefractal is chosen for the arm geometry. This choice is due to the fact that, because of the small linear dimensions of the dipole, it is technically very difficult to construct antennas with a complex geometry (higher-order prefractals). It appears to be possible only by means of significant reduction in the diameter of the wire.

Fractal Fract. 2020, 4, 25 2 of 12 2. Koch-Type Prefractals A Koch-type fractal is not a classical fractal; it received its name due to its similarity to the Koch fractalIn [18]. this The paper, first aiterations Koch-type of these miniature fractals wire diffe antennar only in is the designed position usingof the regressioncentral vertices analysis. (see AFigure second-order 1). When constructing prefractal is iterations chosen for of Koch-typ the arme geometry. fractals, a fractal This choiceinterpolation is due algorithm to the fact is used that, [34,35].because A of description the small linear of this dimensions algorithm of is the given dipole, in detail it is technically below. very difficult to construct antennas withLet a complex geometry (higher-order prefractals). It appears to be possible only by means of significant reduction in the diameter of the wire. = (,,)∈0, 1 ×× | 0= < <⋯< =1 (1) 2. Koch-Type Prefractals be some interpolation points for =1... For the Koch-type fractal, the number =4 is chosen. The valuesA Koch-type =0, fractal=1/3, is not =2/3 a classical, =1 fractal; and it received = = its name= =0 due at to arbitrary its similarity and to the Koch>0 formfractal interpolation [18]. The first points iterations for a offamily these of fractals Koch-type differ curves. only in Also, the position for antenna of the applications, central vertices the

(seerestriction Figure 10<). When < constructing is imposed. iterations Note that of in Koch-type the case fractals, =1/2 a, fractal =1/ interpolation√12, the first algorithm iteration isof theused classical [34,35]. Koch A description curve is obtained. of this algorithm is given in detail below.

0.30 0.25 0.20 0.15 0.10 0.05 x 0.2 0.4 0.6 0.8 1.0 Figure 1. First iteration. Black Black solid solid line for the Koch-t Koch-typeype fractal, green dashed line for the classical Koch fractal.

ForLet all , the affine transformation is introduced according to the following rule: K1 = (ti, xi, yi) [0, 1] R R 0 = t0 < t1 < . . . < tn = 1 (1) ∈ × × 00 be some interpolation points: → for i, = 1..n . For≔ the Koch-type fractal,+ the number . n = 4 is chosen.(2) The values x0 = 0, x1 = 1/3, x3 = 2/3, x4 = 1 and y0 = y1 = y3 = y4 = 0 at arbitrary x2 and y2 > 0 formHere, interpolation the matrices points for havea family the of form Koch-type curves. Also, for antenna applications, the restriction x x x = y = √ 0 < 2 < 4 is imposed. Note0.333 that in 0 the case 2 0.1671/2, 2 −0.2891/ 12, the ﬁrst iteration0.167 0.289 of the classical = = , = , = . (3) Koch curve is obtained. 0 0.333 0.289 0.167 −0.289 0.167 For all i, the aﬃne transformation is introduced according to the following rule: Next, we require that all satisfy the following conditions:         ti   ai 0 0  ti   ei  3 3        A R R A=x  = ,c D D= x +.  d  (4) i : , i  i  :  i1 i1 i2  i   i1  . (2) →        yi ci2 Di3 Di4 yi di2 Then, the following is obtained: Here, the matrices Dij have the form = −, =, (5) ! ! ! 0.333 0 0.167 0.289 0.167 0.289 D = D = , D = − , D = . (3) i1 i4 0 0.333 i2 0.289 0.167 i3 0.289 0.167 − Next, we require that all i satisfy the following conditions:          t0   ti 1   t4   t4     −      A  x  =  x  A  x  =  x  i 0   i 1 , i  4   4  . (4)    −      y0 yi 1 y4 y4 − Fractal Fract. 2020, 4, 25 3 of 12

Then, the following is obtained:

Fractal Fract. 2020, 4, x FOR PEER REVIEW 3 of 12 ai = ti ti 1, ei = ti 1, (5) − − − ! ! ! ! ! ! ! ! ci1 xi xi 1 −Di1 Di2 x4 x0− fi1 xi1 Di1 Di2 x0 = =− − , , == − − . . (6) − − − − ci2 yi yi 1 − Di3 Di4 y4 y0 fi2 yi 1 − Di3 Di4 y0 − − − − With this deﬁnitiondefinition ofof thethe operatorsoperatorsA i,, the the straight straight line line segment segment connecting connecting the the points pointsx 0 and andx4 merges merges into into a polyline, a polyline, connecting connecting the the interpolation interpolation points points in a in consecutive a consecutive manner. manner. Thus, a set of points for the second and thirdthird iterations isis obtainedobtained asas follows:follows: n n K2== Ai((K1)),,K3 = = Ai(K2)(. ) . i=1 i=1 ∪ ∪ The graph of the obtained Koch-type prefractalsprefractals isis presentedpresented inin FigureFigure2 2..

0.30 0.25 0.20 0.15 0.10 0.05 x 0.2 0.4 0.6 0.8 1.0 Figure 2. First (black(black solidsolid line), line), second second (green (green dashed), dashed), and and third third (red (red dotted) dotted) iterations iterations of the of Koch-typethe Koch- = = fractal.type fractal. Coordinates Coordinates of the of central the central vertex: vertex:x2 0.7, =0.7y2 , 0.33. =0.33.

Note that the transformation (2) is an affine affine tran transformation,sformation, not a dilatation; therefore, the curves shift from their original lines (Figure2 2).). InIn otherother words,words, thethe setssets areare self-aself-affineffine sets, and the formulas are valid for the dimension S of self-similar setssets ([([36],36], p.p. 130–132): Xn S ri ==1,1, (7) i=1 = q( − ) +( − ) is not valid in this case. The values of S obtained via Equation (7) 2 2 ri = (yi yi 1) + (xi xi 1) is not valid in this case. The values of S obtained via Equation (7) can serve only− − as an upper− −estimate of the fractal dimension of self-affine curves. Nevertheless, the cantransformation serve only (2) as anclearly upper preserves estimate the of geometry the fractal of dimensionthe classical of Koch self-a fractalffine curves.(self-similar Nevertheless, set), and, thein addition, transformation satisfies (2) the clearly Equation preserves (7) for it. the geometry of the classical Koch fractal (self-similar set), and, in addition, satisfies the Equation (7) for it. 3. Problem Statement 3. Problem Statement A wire balanced dipole antenna is considered herein, the arm geometry of which is a second- A wire balanced dipole antenna is considered herein, the arm geometry of which is a second-order order Koch-type prefractal. The antenna feed point is located at (0, 0). An example of such a dipole Koch-type prefractal. The antenna feed point is located at (0, 0). An example of such a dipole antenna antenna is shown in Figure 3. is shown in Figure3. The forming prefractal (first-order prefractal) is defined on the interval [0, 1]. The arms of the The forming prefractal (first-order prefractal) is defined on the interval [0, 1]. The arms of antennas are considered, which are Koch prefractals scaled by a factor of v. That is, the actually the antennas are considered, which are Koch prefractals scaled by a factor of v. That is, the actually forming prefractal is defined on [0, v]. Obviously, the same length can be obtained at different scales forming prefractal is defined on [0, v]. Obviously, the same length can be obtained at different scales of of v and at different initial positions of the central vertex. Since the base frequency depends on the v and at different initial positions of the central vertex. Since the base frequency depends on the length length of the conductor [31], many antennas of a given type can have the same operating frequency. of the conductor [31], many antennas of a given type can have the same operating frequency. The aim of this work is to design miniature Koch-type fractal antennas operating at a given frequency (2.45 GHz). Moreover, two subtasks can be distinguished. The first subtask is to obtain the smallest antenna of this type. The second subtask is to design a minimum size antenna with a given matching. By antenna size, the minimum radius of a circle describing a dipole is implied (see Figure 3).

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Fractal Fract. 2020, 4, x FOR PEER REVIEW 4 of 12

FigureFigure 3. 3.Balanced Balanced dipole wire wire antenna antenna with withan arm an form armed by formed the Koch-type by the prefractal Koch-type of the prefractal second of the seconditeration. iteration.

The aimFor regression of this work analysis is to at designa fixed scale, miniature 440 antennas Koch-type were fractalcalculated antennas in the FEKO operating program, at a in given frequencywhich (2.45the central GHz). vertex Moreover, of the two generator subtasks curve can (p berefractal distinguished. of the first The order) first varies subtask within is to the obtain the smallestfollowing antenna limits: ofThe this x coordinate type. The changes second from subtask 0.25 isto to 0.75 design with a step minimum of 0.025, size and antenna the y coordinate with a given changes from 0.25 to 0.8, also with a step of 0.025. The diameter of the antenna wire is 0.4 mm. matching.Figure By antenna 3. Balanced size, dipole the wire minimum antenna radiuswith an ofarm a form circleed describingby the Koch-type a dipole prefractal is implied of the second (see Figure3). iteration. For4. Base regression Frequency analysis at a fixed scale, 440 antennas were calculated in the FEKO program, in which the central vertex of the generator curve (prefractal of the first order) varies within the following limits: ForThe regressiondependence analysis of the baseat a fixedfrequency scale, on 440 the antennas length of were the conductor calculated was in the analyzed. FEKO program, To perform in The x coordinate changes from 0.25 to 0.75 with a step of 0.025, and the y coordinate changes from 0.25 whichthe analysis, the central the resonances vertex of at the various generator scales curvev were (p calculated.refractal of The the results first areorder) presented varies inwithin Figure the 4. to 0.8,following also with limits: a step The of x 0.025. coordinate The diameterchanges from of the 0.25 antenna to 0.75 with wire a step is 0.4 of mm. 0.025, and the y coordinate The graphs show the dependence of on L. Values of for v = 0.012 m and v = 0.028 m are changesindicated from in black. 0.25 toFor 0.8, v =also 0.016 with m, a the step values of 0.025. are Thein blue; diameter for v =of 0.020 the antenna m, the valueswire is are 0.4 inmm. red; and 4. Base Frequency for v = 0.024 m, the values are in green. 4. Base Frequency The dependence of the base frequency on the length of the conductor was analyzed. To perform The dependencef1,MHz of the base frequency on the length of the conductor was analyzed. To perform the analysis, the resonances4000 at various scales v were calculated. The results are presented in Figure4. the analysis, the resonances at various scales v were calculated. The results are presented in Figure 4. The graphs show the dependence of f1 on L. Values of f1 for v = 0.012 m and v = 0.028 m are indicated in The graphs show the dependence of on L. Values of for v = 0.012 m and v = 0.028 m are black. For v = 0.0163500 m, the values are in blue; for v = 0.020 m, the values are in red; and for v = 0.024 m, indicated in black. For v = 0.016 m, the values are in blue; for v = 0.020 m, the values are in red; and the values are in green. 0.012 for v = 0.024 m, the values are in green. 3000

f1,MHz 4000 2500 0.016

3500 2000 0.012 0.020 3000 1500 0.024 0.028 2500 0.016 L, m 0.04 0.06 0.08 0.10 2000 Figure 4. Dependence of the base frequency on the0.020 length of the arm’s wire for various scales. The wire diameter is 0.4 mm. The blue color indicates the calculation results of 440 antennas with various central vertices1500 for v = 0.016 m, etc. The black curves show the 0.024dependences at a continuous change in scale for a specific Koch prefractal: the solid line corresponds to the antenna0.028 with a central vertex at (0.725, 0.775); the dashed line is for (0.576, 0.538); and the dotted line is for (0.475, 0.25).L, m 0.04 0.06 0.08 0.10 Figure 4 shows the points for the scales of various antennas with step ∆v = 0.002 m. Apparently, FigureFigure 4. Dependence 4. Dependence of of the the basebase frequency frequency f1 onon the the length length of the of arm’s thearm’s wire for wire various for variousscales. The scales. ∆ Thewith wire wirea decrease diameter in is is 0.4the 0.4 mm. step mm. The The v,blue theblue color points colorindicates for indicates identicalthe calculation the antennas calculation results will of results440 continuously antennas of 440 with antennas change various withtheir various central central vertices vertices for v for= 0.016 v = m,0.016 etc.m, The etc. black The curves black show curves the showdependences the dependences at a continuous at a continuouschange changein scale in scale for fora specific a speciﬁc Koch Koch prefractal: prefractal: the solid the line solid corresponds line corresponds to the antenna to the with antenna a central with vertex a central vertexat at(0.725, (0.725, 0.775); 0.775); the thedashed dashed line is line for is(0.576, for (0.576, 0.538); 0.538);and the and dotted the line dotted is for line (0.475, is for 0.25). (0.475, 0.25).

Figure 4 shows the points for the scales of various antennas with step ∆v = 0.002 m. Apparently, with a decrease in the step ∆v, the points for identical antennas will continuously change their

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Figure4 shows the points for the scales of various antennas with step ∆v = 0.002 m. Apparently, with a decrease in the step ∆v, the points for identical antennas will continuously change their coordinates from a smaller scale to a larger scale. Examples of such a change for three antennas are shown in Figure4: the antenna dipole with a central vertex (0.576, 0.538) is shown by a dashed line. It should be noted that the length of the Koch prefractal of the nth iteration is calculated as L = Sn, where S = 4/3 (the value of S is determined from Equation (7)). For the Koch prefractals, there is no explicit formula for calculating L; Equation (7) can be used only for estimating the values of S. Note that the upper threshold for computing the base frequency is 4 GHz. With a scale of v = 0.012 m, some of the antennas have base frequencies above 4 GHz, so these antennas were not included in our consideration. Thus, a total of 2041 antennas were considered: 281 antennas were obtained for v = 0.012 m, whereas for each of the remaining four scales, 440 antennas were obtained. Following [33], a regression analysis of the set of points in Figure4 was conducted separately at each scale according to the formula k2 L fˆ1 = k1 e− (8) where k1, k2 are the desired parameters of the model, which are sought by the least squares method. Here, f1 is measured in MHz, and L is measured in meters. The obtained values for the coeﬃcients of the model are given in Table1.

Table 1. Coeﬃcients of regression model (1) for the base frequency.

Scale, m k1 k2 0.012 7329.63 21.5032 − 0.016 5824.25 17.8957 − 0.020 4633.42 14.6327 − 0.024 3863.48 12.3748 − 0.028 3306.20 10.7503 −

In the next step, a regression model was built by adding a scale to its parameters. In this case, the approach proposed in [33] was used. The following formula was obtained:

1 1 fˆ = Exp L (9) 1 0.000007451 + 0.01048v· −0.00993 + 2.95v The relative error for model (2) was calculated by the following formula:

n 1 X Yi Yˆ(Xi) δ = − 100 %, (10) n Yi i=1 which is equal to δ 1.22 %. ≈ 5. Dipole Antenna Reﬂection Coeﬃcient

In this section, the behavior of the reflection coefficient Sˆ11 with change in the conductor length is studied. This study was also carried out for various scales. The results are shown in Figure5. It can be noted that for a fixed scale, long antennas have a matching which is worse than that of short antennas. Regression analysis of a set of points was conducted separately for each scale using the following formula:

k Sˆ = 3 , (11) 11 k L 4 − where k3, k4 are unknown parameters of the model. The parameters were also found by the least squares method. The obtained values for the coeﬃcients of the model are given in Table2. Fractal Fract. 2020, 4, 25 6 of 12 Fractal Fract. 2020, 4, x FOR PEER REVIEW 6 of 12

S11,dB L, m 0.04 0.06 0.08 0.10 0.012 10

20 0.028

30 0.024 0.020 0.016 40

FigureFigure 5.5. TheThe dependencedependence ofof thethe reflectionreflection coecoefficientfficient SS1111 on the length of thethe arm’sarm’s wirewire forfor variousvarious scales.scales. TheThe wirewire diameter diameter is is 0.4 0.4 mm. mm. The The blue blue color color indicates indicates the the calculation calculation results results of 440 of antennas 440 antennas with variouswith various central central vertices vertices for v = fo0.016r v m,= 0.016 etc. The m, blacketc. The curves black show curves the dependencesshow the dependences at a continuous at a changecontinuous in scale change for a in specific scale for Koch a specific prefractal: Koch the pr solidefractal: line correspondsthe solid line to corresponds the antenna to with the aantenna central vertexwith a atcentral (0.725, vertex 0.775); at the (0.725, dashed 0.775); line isthe for dashed (0.576, line 0.538); is for and (0.576, the dotted 0.538); line and is forthe (0.475,dotted 0.25).line is for (0.475, 0.25).

Table 2. Coeﬃcients of regression model (4) for Sˆ11 at the base resonance. Analysis of Figure 5 indicates that as L increases, antenna matching worsens. In other words, the more the wire fills the space (theScale, closer m the adjaceknt3 antenna elementsk4 are located to each other), the worse the dipole matching. In Figure0.012 5, the solid 0.162 black line corresponds 0.017 to an antenna with a long length, the dashed line corresponds0.016 to an antenna 0.227 with an 0.020 average length, and the dotted line corresponds to an antenna with a 0.020minimum arm 0.271 length. Thus, 0.024 the following can be concluded: 0.024 0.327 0.028 0.028() < 0.372(), for 0.032<. (12) Comprehension of condition (12) leads to the logical conclusion that minimizing the antenna leadsAnalysis to a deterioration of Figure 5in indicates its matching. that as L increases, antenna matching worsens. In other words, the moreThe solid the wire black ﬁlls curve the spacecorresponds (the closer to a thenormalized adjacent (at antenna v = 1) length elements of 1.64; are located varies to each from other), −7.38 theto − worse6.07 with the an dipole increase matching. in scale Infrom Figure 0.0125, to the 0.028 solid (with black an line increase corresponds in length to from an antenna 4.5 to 10.5 with cm). a long length,At a continuous the dashed change line correspondsin scale, antenna to an matching antenna withchanges an averagecontinuously. length, and the dotted line corresponds to an antenna with a minimum arm length. Thus, the following can be concluded:

Table 2. Coefficients of regression model (4) for at the base resonance. S11(L1) < S11(L2), for L1 < L2. (12) Scale, m Comprehension of condition (12) leads0.012 to the0.162 logical 0.017 conclusion that minimizing the antenna 0.016 0.227 0.020 leads to a deterioration in its matching. 0.020 0.271 0.024 The solid black curve corresponds to a normalized (at v = 1) length of 1.64; S varies from 7.38 0.024 0.327 0.028 11 − to 6.07 with an increase in scale from 0.012 to 0.028 (with an increase in length from 4.5 to 10.5 cm). − 0.028 0.372 0.032 At a continuous change in scale, antenna matching changes continuously. JustJust asas in in the the previous previous paragraph, paragraph, a regression a regression model model for any for scale any was scale constructed. was constructed. The following The regressionfollowing regression model was model obtained: was obtained: 0.012 + 12.986v Sˆ11 = 0.012 + 12.986 v , (13) =0.00495 + 0.973v L (13) 0.00495 + 0.973− v − with relative error obtained by formula (13), equal to δ 12.59 %. ≈ withNote relative that error the resultingobtained by formula formula is less(13), accurateequal to thanδ ≈ 12.59 a similar %. formula for the base frequency. This isNote also that evident the resulting from comparison formula ofis theless graphs accurate in Figuresthan a similar2 and3. formula for the base frequency. This is also evident from comparison of the graphs in Figures 2 and 3.

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6. Results Two examplesexamples are are considered considered here. here. The The ﬁrst first example example is related is related to obtaining to obtaining the smallest the antenna.smallest Theantenna. second The example second isexample related is to related designing to designing a minimum a minimum size antenna size withantenna a given with matching. a given matching.

6.1. Maximum Antenna Miniaturization 6.1. Maximum Antenna Miniaturization ˆ Modeling the antenna of minimum radius is considered next. The isoline f1 = 2450 MHz Modeling the antenna of minimum radius is considered next. The isoline = 2450 MHz was was constructed (in Figure4, it is shown by the curved line). Around the contour, a region constructed (in Figure 4, it is shown by the curved line). Around the contour, a region ∈ f (2.4 GHz, 2.5 GHz) is highlighted. The range of allowable values for L is shown in yellow. (2.41 ∈ GHz, 2.5 GHz) is highlighted. The range of allowable values for L is shown in yellow. Thus, the Thus, the allowable length of the arm length varies from L = v to the maximum possible wire length at allowable length of the arm length varies from L = v to the maximum possible wire length at a given a given scale (L = 3.64 v). The maximum value of L is determined from the condition that the antenna is scale (L = 3.64v). The maximum value of L is determined from the condition that the antenna is located located inside a circle of radius v. inside a circle of radius v. From analysis of Figure6, it follows that the curve f1 = 2450 MHz crosses the straight line L = 3.64v From analysis of Figure 6, it follows that the curve = 2450 MHz crosses the straight line L = at3.64v v = at0.0138 v = 0.0138 m. Thism. This scale scale will will correspond correspond to to the the minimum minimum antenna antenna at at which which thethe base frequency ˆf L == 24502450 MHz MHz is is reached. reached. The point of intersectionintersection forfor Lis is around around 0.05 0.05 m. m.

Figure 6. Wi-FiWi-Fi frequency isolines and allowable areas for L and v.

The antennaantenna at at the the found found scale scale is now is no considered,w considered, with with length length equal toequalL = 0.0497to L = m.0.0497 Such m. an Such antenna an canantenna be obtained can be obtained with a forming with a prefractalforming prefractal having a central having vertex a central at (0.675 vertex at0.0138 (0.675 m, × 0.750.01380.0138 m, 0.75 m). × × × This0.0138 antenna m). This reaches antenna resonance reaches at resonance the frequency at the of frequencyˆf = 2.47 GHz of with = 2.47 reflection GHz with coefficient reflectionSˆ =coefficient7.2 dB. 11 − = It−7.2 can dB. be noted that the resulting antenna matching is not the most optimal. One possible further improvementIt can be noted may bethat rotation the resulting of a dipole antenna arm matching [37,38]. However, is not the most simulating optimal. an One antenna possible with further lower reflectanceimprovement with may no rotationbe rotation of the of arma dipole was attempted.arm [37,38]. To However, do this, the simulating dependence an antenna of Sˆ11 on with the lengthlower Lreflectanceis considered. with no rotation of the arm was attempted. To do this, the dependence of on the length L is considered. 6.2. Designing a Minimum Antenna with a Specified Reflection Coefficient 6.2. Designing a Minimum Antenna with a Specified Reflection Coefficient Now the antenna is modelled, taking into account formula (5). The values of f1 = 2450 MHz and Sˆ = 15 dB were set. The system of Equations (2) and (5) was solved with respect to the base Now11 −the antenna is modelled, taking into account formula (5). The values of = 2450 MHz and frequency and the scale. The scale was rounded to v = 0.02 m to obtain L = 0.043 m. In this case, = −15 dB were set. The system of Equations (2) and (5) was solved with respect to the base frequency and the scale. The scale was rounded to v = 0.02 m to obtain L = 0.043 m. In this case, the

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Fractal Fract. 2020, 4, x FOR PEER REVIEW 8 of 12 the coordinates of the central vertex of the forming prefractal are (0.625 0.02 m, 0.375 0.02 m). coordinates of the central vertex of the forming prefractal are (0.625 × 0.02 m,× 0.375 × 0.02 m). Based× Based on the simulated parameters, an antenna was fabricated and is shown in Figure7. on the simulated parameters, an antenna was fabricated and is shown in Figure 7.

FigureFigure 7. The 7. The fabricated fabricated antenna antenna with with centralcentral vertex vertex (0.625, (0.625, 0.375) 0.375) for for v = v0.02= 0.02 m. m.

The electricalThe electrical characteristics characteristics of of the the resulting resulting antennaantenna were were calculated. calculated. The The resulting resulting antenna antenna has has reflection coefficient = −16 dB at the base frequency = 2430 MHz. The results of the calculation reflection coefficient Sˆ11 = 16 dB at the base frequency f1 = 2430 MHz. The results of the calculation of the reflection coefficient− of this antenna and of the antenna obtained earlier are shown in Figure 8. of the reflection coefficient of this antenna and of the antenna obtained earlier are shown in Figure8. An improvement in antenna matching with increasing scale can be observed. By increasing the scale An improvementfrom v = 0.0138 in antennam to v = 0.02 matching m, the withantenna increasing matching scale can canbe improved. be observed. By increasing the scale from v = 0.0138 m to v = 0.02 m, the antenna matching can be improved. In addition,S11 in Figure8, the red dotted line shows the reflection coe fficient for the prototype of the second “optimal” antenna for v 0.02 m. f1,GHz 2.3 = 2.4 2.5 2.6 Images of the arms of the obtained antennas are presented in Figure9. Also, to compare the sizes of the antennas, their radii are depicted. The black color indicates the antenna with scale v = 0.02 m, and the blue color indicates an antenna with v = 0.0138 m. The parameters5 of a wire dipole based on the Koch curve (a special case of a Koch-type dipole) are presented; like the antennas obtained above, it has a wire diameter of 0.4 mm. The required frequency of 2.44 GHz for this dipole is achieved at a radius of 2.25 cm. This value is greater by 8.7 mm than the radius of the first antenna and greater by 2.5 mm than the radius of the second antenna. The10 matching of the Koch dipole is better (S = 21 dB) in comparison with the matching of 11 − the obtained optimal antennas. It can also be noted that the matching of the obtained “minimized” antennas can be somewhat improved by applying iterative methods [39].

Figure 8. The dependence of the reflection coefficient S11 on the frequency of the two antennas. The blue graph corresponds to the smallest antenna, the green graph corresponds to the simulated second antenna, and the red graph corresponds to the fabricated second antenna.

In addition, in Figure 8, the red dotted line shows the reflection coefficient for the prototype of the second “optimal” antenna for v = 0.02 m.

Fractal Fract. 2020, 4, x FOR PEER REVIEW 8 of 12 coordinates of the central vertex of the forming prefractal are (0.625 × 0.02 m, 0.375 × 0.02 m). Based on the simulated parameters, an antenna was fabricated and is shown in Figure 7.

Figure 7. The fabricated antenna with central vertex (0.625, 0.375) for v = 0.02 m.

The electrical characteristics of the resulting antenna were calculated. The resulting antenna has reflection coefficient = −16 dB at the base frequency = 2430 MHz. The results of the calculation of the reflection coefficient of this antenna and of the antenna obtained earlier are shown in Figure 8. AnFractal improvement Fract. 2020, 4, 25 in antenna matching with increasing scale can be observed. By increasing the 9scale of 12 from v = 0.0138 m to v = 0.02 m, the antenna matching can be improved.

S11 f ,GHz 2.3 2.4 2.5 2.6 1

Fractal Fract. 2020, 4, x FOR PEER REVIEW 9 of 12 Figure 8. The dependence of the reflection coefficient S11 on the frequency of the two antennas. The Figure 8. The dependence of the reﬂection coeﬃcient S11 on the frequency of the two antennas. The blue blueImages graph of correspondsthe arms of tothe the obtained smallest antennasantenna, th aree green presented graph corresponds in Figure 9. to Al theso, simulated to compare second the sizes graph corresponds to the smallest antenna, the green graph corresponds to the simulated second antenna, and the red graph corresponds to the fabricated second antenna. of theantenna, antennas, and their the red radii graph are corresponds depicted. The to the black fabricated color secondindicates antenna. the antenna with scale v = 0.02 m, and the blue color indicates an antenna with v = 0.0138 m. In addition, in Figure 8, the red dotted line shows the reflection coefficient for the prototype of the second “optimal” antenna for v = 0.02 m.

Figure 9. The arms and envelopes of the designed antennas. The blue color corresponds to the smallest antenna with S 11 = −77 dB dB (radius (radius is is 14.1 14.1 mm), mm), while while the black color corresponds to the miniatureminiature 11 − antenna with S 11 == −1515 dB dB (radius (radius is is 20 20 mm). mm). 11 − ItThe can parameters be noted of that a wire the dipole antennas based obtained on the atKoch diﬀ curveerent (a scales special of vcase have of a linear Koch-type polarization dipole) andare presented; practically thelike samethe antennas radiation obtained pattern and above, power it has gain. a Figurewire diameter 10 shows of the 0.4 gain mm. for The the obtainedrequired optimalfrequency antennas of 2.44 atGHz frequency for this equaldipole to is 2430 achieved MHz. at a radius of 2.25 cm. This value is greater by 8.7 mm than the radius of the first antenna and greater by 2.5 mm than the radius of the second antenna.

The matching of the Koch dipole is better ( = −21 dB) in comparison with the matching of the obtained optimal antennas. It can also be noted that the matching of the obtained “minimized” antennas can be somewhat improved by applying iterative methods [39]. It can be noted that the antennas obtained at different scales of v have linear polarization and practically the same radiation pattern and power gain. Figure 10 shows the gain for the obtained optimal antennas at frequency equal to 2430 MHz.

Fractal Fract. 2020, 4, 25 10 of 12 Fractal Fract. 2020, 4, x FOR PEER REVIEW 10 of 12

FigureFigure 10. 10.Radiation Radiation patterns patterns ofof thethe obtainedobtained antennas at at frequency frequency equal equal to to 2430 2430 MHz. MHz. Left Left picture: picture: gaingain for for the the antenna antenna with with radius radius equalequal toto 14.114.1 mm.mm. Right picture: picture: gain gain for for the the antenna antenna with with radius radius equalequal to to 20 20 mm. mm.

7.7. Conclusions Conclusions WireWire dipole dipole antennas antennas withwith armarm havinghaving a Koch-typeKoch-type geometry geometry of of the the second second iteration iteration were were consideredconsidered herein. herein. Two-parameter Two-parameter regression regression models models were were constructed constructed for the for base the frequency base frequency and andthe the reflection reflection coefficient coefficient at the at thebase base frequency frequency for the for dipole. the dipole. A single-band A single-band antenna antenna of minimum of minimum size size(radius (radius of of14.1 14.1 mm) mm) was was designed. designed. The Thereflection reflection coefficient coefficient of this of antenna this antenna was equal was equalto −7.2 to dB.7.2 dB. − InIn addition, addition, a a second second antenna antenna withwith aa radiusradius of 2 2 cm cm and and a a reflection reflection coefficient coefficient of of −16 16dB dB was was − designed.designed. Both Both of theof the proposed proposed antennas antennas were were modeled modeled using using algorithms algorithms based based on regression on regression models. Themodels. first algorithm The first algorithm allows for allows designing for designing the minimum the minimum size antenna size antenna from a from given a given family. family. The second The algorithmsecond algorithm gives the gives minimum the minimum size antenna size antenna with a givenwith a level given of level matching. of matching. It canIt can be be noted noted that that if if the the goal goal isis toto simulatesimulate a a broadband broadband antenna, antenna, then then in in the the present present study study it is it is necessarynecessary to to add add a a model model for for the the bandwidthbandwidth [[40].40]. It It is is also also possible possible to to use use all all models models for for designing designing antennasantennas with with specified specified characteristics. characteristics. Author Contributions: Conceptualization, D.T.; Investigation, D.T. and P.K.; Methodology and Writing AuthorManuscript, Contributions: D.T. and Conceptualization,D.C.; Supervision, D.C. D.T.; All Investigation, authors have D.T. read and and P.K.;Methodology agreed to the published and Writing version Manuscript, of the D.T.manuscript. and D.C.; Supervision, D.C. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Funding: This research received no external funding. Acknowledgments: The work was performed according to the Russian Government Program of Competitive GrowthAcknowledgments: of Kazan Federal TheUniversity. work was performed according to the Russian Government Program of Competitive Growth of Kazan Federal University. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest. References References 1. Balanis, C.A. Antenna Theory: Analysis and Design; John Wiley & Sons: New Delhi, India, 1997. 1. Balanis, C.A. Antenna Theory: Analysis and Design; John Wiley & Sons: New Delhi, India, 1997. 2. Poole, I.; Telenius-Lowe, S. Successful Wire Antennas; Radio Society of Great Britain: Bedford, UK, 2011. 2. Poole, I.; Telenius-Lowe, S. Successful Wire Antennas; Radio Society of Great Britain: Bedford, UK, 2011. 3. Guariglia, E. Entropy and fractal antennas. Entropy 2016, 18, 84. [CrossRef] 3. Guariglia, E. Entropy and fractal antennas. Entropy 2016, 18, 84. 4. Yao, C.; Chen, P.; Huang, C.; Chen, Y.; Qiao, P. Study on the application of an ultra-high-frequency fractal 4. Yao, C.; Chen, P.; Huang, C.; Chen, Y.; Qiao, P. Study on the application of an ultra-high-frequency fractal antennaantenna to to partial partial discharge discharge detection detection inin switchgears.switchgears. SensorsSensors 20132013, ,1313, 17362–17378., 17362–17378. [CrossRef][PubMed] 5. 5. Yang,Yang, C.-L.; C.-L.; Zheng, Zheng, G.-T. G.-T. Wireless Wireless low-power low-power integratedintegrated basal-body-temperature basal-body-temperature detection detection systems systems using using teethteeth antennas antennas in in the the MedRadio MedRadio band. band. Sensors 2015,, 1515,, 29467–29477. 29467–29477. [CrossRef][PubMed] 6. 6. Singh,Singh, K.; K.; Grewal, Grewal, V.;V.; Saxena,Saxena, R.R. FractalFractal antennas:antennas: A A novel novel miniaturization miniaturization technique technique for for wireless wireless communications.communications.Int. Int. J. J. Recent Recent TrendsTrends Eng.Eng. 2009,, 22,, 172–176. 172–176.

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