Fractal antennas for television. Fractal ultra-wideband antenna based on a circular monopole. Then, a plane electromagnetic wave was sent to the designed fractal antenna, and the program calculated the propagation of the field before and after

In mathematics, fractals are sets consisting of elements similar to the set as a whole. Best example: If you look closely at the line of an ellipse, it will become straight. A fractal – no matter how close you zoom in – the picture will remain complex and similar to the general view. The elements are arranged in a bizarre way. Consequently, we consider concentric circles to be the simplest example of a fractal. No matter how close you get, new circles appear. There are many examples of fractals. For example, Wikipedia gives a drawing of Romanesco cabbage, where the head of cabbage consists of cones that exactly resemble the drawn head of cabbage. Readers now understand that making fractal antennas is not easy. But it's interesting.

Why are fractal antennas needed?

The purpose of a fractal antenna is to catch more with less. In Western videos, it is possible to find a paraboloid where a piece of fractal tape will serve as the emitter. They are already making elements of microwave devices from foil that are more efficient than ordinary ones. We'll show you how to complete a fractal antenna, and deal with the matching alone with the SWR meter. Let us mention that there is a whole website, foreign of course, where the corresponding product is promoted for commercial purposes; there are no drawings. Our homemade fractal antenna is simpler, the main advantage is that you can make the design with your own hands.

The first fractal antennas - biconical - appeared, according to a video from the website fractenna.com, in 1897 by Oliver Lodge. Don't look on Wikipedia. Compared to a conventional dipole, a pair of triangles instead of a vibrator gives a band expansion of 20%. By creating periodic repeating structures, it was possible to assemble miniature antennas no worse than their larger counterparts. You will often find a biconical antenna in the form of two frames or oddly shaped plates.

Ultimately, this will allow more television channels to be received.

If you type a request on YouTube, a video on making fractal antennas appears. You will better understand how it works if you imagine the six-pointed star of the Israeli flag, the corner of which was cut off along with the shoulders. It turned out that three corners remained, two had one side in place, the other not. The sixth corner is completely absent. Now we will place two similar stars vertically, with central angles to each other, slits to the left and right, and above them - a similar pair. The result was an antenna array - the simplest fractal antenna.

The stars are connected at the corners by a feeder. In pairs by columns. The signal is taken from the line, exactly in the middle of each wire. The structure is assembled with bolts on a dielectric (plastic) substrate of the appropriate size. The side of the star is exactly an inch, the distance between the corners of the stars vertically (the length of the feeder) is four inches, and the horizontal distance (the distance between the two wires of the feeder) is an inch. The stars have angles of 60 degrees at their vertices; now the reader will draw something similar in the form of a template, so that later he can make a fractal antenna himself. We made a working sketch, but the scale was not met. We can’t guarantee that the stars came out exactly, Microsoft Paint without much ability to produce accurate drawings. Just look at the picture for the structure of the fractal antenna to become obvious:

1. The brown rectangle shows the dielectric substrate. The fractal antenna shown in the figure has a symmetrical radiation pattern. If the emitter is protected from interference, the screen is placed on four posts behind the substrate at a distance of an inch. At frequencies there is no need to place a solid sheet of metal, a mesh with a side of a quarter of an inch will suffice, do not forget to connect the screen to the cable braid.
2. A feeder with a characteristic impedance of 75 Ohms requires coordination. Find or make a transformer that converts 300 ohms to 75 ohms. It’s better to stock up on an SWR meter and select the necessary parameters not by touch, but by using the device.
3. Four stars, bend from copper wire. We will clean the varnish insulation at the junction with the feeder (if any). The antenna's internal feed consists of two parallel pieces of wire. It is a good idea to place the antenna in a box to protect it from bad weather.

Assembling a fractal antenna for digital television

After reading this review to the end, anyone can make fractal antennas. We got so deep into the design that we forgot to talk about polarization. We assume it is linear and horizontal. This stems from considerations:

• The video is obviously of American origin, the conversation is about HDTV. Therefore, we can adopt the fashion of the specified country.
• As you know, few countries on the planet broadcast from satellites using circular polarization, among them the Russian Federation and the United States. Therefore, we believe that other information transmission technologies are similar. Why? There was a Cold War, we believe that both countries strategically chose what and how to transfer, other countries proceeded from purely practical considerations. Circular polarization was introduced specifically for spy satellites (moving constantly relative to the observer). Hence there is reason to believe that there are similarities in television and radio broadcasting.
• The antenna structure says it is linear. There is simply nowhere to get circular or elliptical polarization. Therefore - unless among our readers there are professionals who own MMANA - if the antenna does not catch in the accepted position, rotate 90 degrees in the plane of the emitter. The polarization will change to vertical. By the way, many will be able to catch FM if the dimensions are set 4 times larger. It is better to take a thicker wire (for example, 10 mm).

We hope we explained to readers how to use a fractal antenna. A couple of tips for easy assembly. So, try to find wire with varnished protection. Bend the shapes as shown in the picture. Then the designers diverge, we recommend doing this:

1. Strip the stars and feeder wires at the junction points. Secure the feeder wires by the ears with bolts to the backing in the middle parts. To perform the action correctly, measure an inch in advance and draw two parallel lines with a pencil. There should be wires along them.
2. Solder a single structure, carefully checking the distances. The authors of the video recommend making the emitter so that the stars lay flat on the feeders with their corners, and rest with their opposite ends on the edge of the substrate (each in two places). For an approximate star, the locations are marked in blue.
3. To fulfill the condition, tighten each star in one place with a bolt with a dielectric clamp (for example, PVA wires made of cambric and the like). In the figure, the mounting locations are shown in red for one star. The bolt is schematically drawn with a circle.

The power cable runs (optional) from reverse side. Drill holes in place. The SWR is adjusted by changing the distance between the feeder wires, but in this design this is a sadistic method. We recommend simply measuring the impedance of the antenna. Let us remind you how this is done. You will need a generator at the frequency of the program you are viewing, for example, 500 MHz, and additionally a high-frequency voltmeter that will not give up on the signal.

Then the voltage produced by the generator is measured, for which it is connected to a voltmeter (in parallel). We assemble a resistive divider from a variable resistance with an extremely low self-inductance and an antenna (we connect it in series after the generator, first the resistance, then the antenna). We measure the voltage with a voltmeter variable resistor, while simultaneously adjusting the rating until the generator readings without load (see point above) become twice as high as the current ones. This means that the value of the variable resistor has become equal to the wave impedance of the antenna at a frequency of 500 MHz.

It is now possible to manufacture the transformer as required. It’s difficult to find what you need on the Internet; for those who like to catch radio broadcasts, we found a ready-made answer http://www.cqham.ru/tr.htm. It is written and drawn on the website how to match the load with a 50 Ohm cable. Please note that the frequencies correspond to the HF range, SW fits partially here. The characteristic impedance of the antenna is maintained in the range of 50 – 200 Ohms. It’s hard to say how much the star will give. If you have a device on your farm for measuring the wave impedance of a line, let us remind you: if the length of the feeder is a multiple of a quarter of the wavelength, the antenna impedance is transmitted to the output without changes. For small and large ranges, it is impossible to provide such conditions (remember that especially fractal antennas also include an extended range), but for measurement purposes the mentioned fact is used everywhere.

Now readers know everything about these amazing transceiver devices. Such an unusual shape suggests that the diversity of the Universe does not fit into typical boundaries.

The world is not without good people:-)
Valery UR3CAH: "Good afternoon, Egor. I think this article (namely the section "Fractal antennas: less is more") corresponds to the theme of your site and will be of interest to you:) 73!"
Yes, of course it’s interesting. We have already touched on this topic to some extent when discussing the geometry of hexabims. There, too, there was a dilemma with “packing” the electrical length into geometric dimensions :-). So thank you, Valery, very much for sending the material.
Fractal antennas: less is more
Over the past half century, life has rapidly begun to change. Most of us accept achievements modern technologies for granted. You get used to everything that makes life more comfortable very quickly. Rarely does anyone ask the questions “Where did this come from?” and “How does it work?” A microwave heats up breakfast - great, a smartphone gives you the opportunity to talk to another person - great. This seems like an obvious possibility to us.
But life could have been completely different if a person had not sought an explanation for the events taking place. Take cell phones, for example. Remember the retractable antennas on the first models? They interfered, increased the size of the device, and in the end, often broke. We believe they have sunk into oblivion forever, and part of the reason for this is... fractals.
Fractal patterns fascinate with their patterns. They definitely resemble images of cosmic objects - nebulae, galaxy clusters, and so on. It is therefore quite natural that when Mandelbrot voiced his theory of fractals, his research aroused increased interest among those who studied astronomy. One of these amateurs named Nathan Cohen, after attending a lecture by Benoit Mandelbrot in Budapest, got the idea practical application acquired knowledge. True, he did this intuitively, and chance played an important role in his discovery. As a radio amateur, Nathan sought to create an antenna with the highest possible sensitivity.
The only way to improve the parameters of the antenna, which was known at that time, consisted of increasing its geometric dimensions. However, the owner of the property in downtown Boston that Nathan rented was categorically against installing large devices on the roof. Then Nathan began experimenting with different antenna shapes, trying to get the maximum result with the minimum size. Inspired by the idea of ​​fractal forms, Cohen, as they say, randomly made one of the most famous fractals from wire - the “Koch snowflake”. Swedish mathematician Helge von Koch came up with this curve back in 1904. It is obtained by dividing a segment into three parts and replacing the middle segment with an equilateral triangle without a side coinciding with this segment. The definition is a little difficult to understand, but in the figure everything is clear and simple.
There are also other variations of the Koch curve, but the approximate shape of the curve remains similar.

When Nathan connected the antenna to the radio receiver, he was very surprised - the sensitivity increased dramatically. After a series of experiments, the future professor at Boston University realized that an antenna made according to a fractal pattern has high efficiency and covers a much wider frequency range compared to classical solutions. In addition, the shape of the antenna in the form of a fractal curve makes it possible to significantly reduce the geometric dimensions. Nathan Cohen even came up with a theorem proving that to create broadband antenna it is enough to give it the shape of a self-similar fractal curve.

The author patented his discovery and founded a company for the development and design of fractal antennas, Fractal Antenna Systems, rightly believing that in the future, thanks to his discovery, cell phones will be able to get rid of bulky antennas and become more compact. In principle, this is what happened. True, to this day Nathan is engaged in a legal battle with large corporations, who illegally use his discovery to produce compact communication devices. Some famous manufacturers mobile devices, such as Motorola, have already reached a peace agreement with the inventor of the fractal antenna. Original source

Over the past few years, I have regularly been faced with the challenges of developing UWB (ultra-wideband) microwave modules and functional units. And as sad as it is for me to say this, I get almost all the information on the topic from foreign sources. However, some time ago, in search of the information I needed, I came across one that promised a solution to all my problems. I want to talk about how the problems were not resolved.

One of the constant “headaches” in the development of UWB microwave devices is the development of UWB antennas, which must have a set of certain properties. Among these properties are the following:

1. Agreement in the operating frequency band (for example, from 1 to 4 GHz). However, it happens when it is necessary to agree in the frequency range from 0.5 GHz to 5 GHz. And here the problem arises of going below 1 GHz in frequency. I generally got the impression that the 1 GHz frequency has some kind of mystical power - you can get close to it, but it’s very difficult to overcome it, because in this case, another requirement for the antenna is violated, namely

2. Compactness. After all, it’s no secret that now few people need a waveguide horn antenna of enormous size. Everyone wants an antenna that is small, light and compact so that it can be shoved into a housing. portable device. But when compacting the antenna, it becomes very difficult to comply with paragraph 1 of the requirements for the antenna, because The minimum frequency of the operating range is closely related to the maximum size of the antenna. Someone will say that you can make an antenna on a dielectric with a high relative dielectric constant... And they will be right, but this contradicts the next item on our list, which says that

3. The antenna should be as cheap as possible and made from the most accessible and inexpensive materials (for example, FR-4). Because no one wants to pay a lot, a lot of money for an antenna, even if it’s three times brilliant. Everyone wants the cost of the antenna at the manufacturing stage printed circuit board tended to zero. Because this is our world...

4. There is one more requirement that arises when solving various problems associated, for example, with short-range location, as well as with the creation of various sensors using UWB technology (here it must be clarified that we're talking about about low power applications where every dBm counts). And this requirement states that the radiation pattern (DP) of the designed antenna should be formed in only one hemisphere. What is it for? In order for the antenna to “shine” only in one direction, without dissipating precious power into the “return”. This also allows you to improve a number of indicators of the system in which such an antenna is used.

Why am I writing all this..? In order for the inquisitive reader to understand that the developer of such an antenna is faced with a lot of restrictions and prohibitions that he needs to heroically or wittily overcome.

And suddenly, like a revelation, an article appears that promises a solution to all the above problems (as well as those that were not mentioned). Reading this article evokes a slight feeling of euphoria. Although the first time you do not fully understand what is written, the magic word “fractal” sounds very promising, because Euclidean geometry has already exhausted its arguments.

We get down to business boldly and feed the structure proposed by the author of the article to the simulator. The simulator growls gutturally like a computer cooler, chewing gigabytes of numbers, and spits out the digested result... Looking at the simulation results, you feel like a little deceived boy. Tears well up in my eyes, because... again your childhood airy dreams collided with cast-iron...reality. There is no coordination in the frequency range 0.1 GHz - 24 GHz. Even in the range of 0.5 GHz - 5 GHz there is nothing similar.

There is still a timid hope that you didn’t understand something, did something wrong... The search for the switching point begins, various variations with the topology, but everything is in vain - it’s dead!

The saddest thing in this situation is that until the last moment you are looking for the reason for failure in yourself. Thanks to my fellow workers who explained that everything was correct - it shouldn’t work.

P.S. I hope my Friday post brought a smile to your face.
The moral of this presentation is this: be vigilant!
(And I really wanted to write an ANTI-article about this, because I was deceived).

The first thing I would like to write about is a little introduction to the history, theory and use of fractal antennas. Fractal antennas were recently discovered. They were first invented by Nathan Cohen in 1988, then he published his research on how to make a TV antenna from wire and patented it in 1995.

The fractal antenna has several unique characteristics, as written on Wikipedia:

“A fractal antenna is an antenna that uses a fractal, self-repeating design to maximize the length or increase the perimeter (on internal areas or external structure) of a material that can receive or transmit electromagnetic signals within a given total surface area or volume.”

What exactly does this mean? Well, you need to know what a fractal is. Also from Wikipedia:

“A fractal is typically a rough or fragmented geometric shape that can be divided into parts, each part being a smaller copy of the whole—a property called self-similarity.”

Thus, a fractal is a geometric shape that repeats itself over and over again, regardless of the size of the individual parts.

Fractal antennas have been found to be approximately 20% more efficient than conventional antennas. This can be useful especially if you want your TV antenna to receive digital or high definition video, increase cellular range, Wi-Fi range, FM or AM radio reception, etc.

In the majority cell phones There are already fractal antennas. You may have noticed this because Cell phones no longer have antennas on the outside. This is because they have fractal antennas inside them, etched into the circuit board, which allows them to receive better signal and pick up more frequencies, such as Bluetooth, cellular and Wi-Fi from one antenna.

Wikipedia:

“The fractal antenna's response is noticeably different from traditional antenna designs in that it is capable of operating with good performance at different frequencies simultaneously. The frequency of standard antennas must be cut to be able to receive only that frequency. Therefore, a fractal antenna, unlike a conventional antenna, is an excellent design for wideband and multi-band applications.”

The trick is to design your fractal antenna to resonate at the specific center frequency you want. This means that the antenna will look different depending on what you want to achieve. To do this you need to use mathematics (or an online calculator).

In my example I'm going to do simple antenna, but you can make it more complex. The more complex the better. I'll use a coil of 18-strand solid core wire to make the antenna, but you can customize your own circuit boards to suit your aesthetic, make it smaller or more complex with greater resolution and resonance.

I'm going to make a TV antenna to receive digital TV or TV high resolution. These frequencies are easier to work with and range in length from about 15 cm to 150 cm for half wavelength. For simplicity and low cost of parts, I'm going to place it on a common dipole antenna, it will catch waves in the 136-174 MHz range (VHF).

To receive UHF waves (400-512 MHz), you can add a director or reflector, but this will make the reception more dependent on the direction of the antenna. VHF is also directional, but instead of pointing directly at the TV station in a UHF installation, you will need to mount the VHF ears perpendicular to the TV station. This is where you'll need to put in a little more effort. I want to make the design as simple as possible, because this is already quite a complex thing.

Main components:

• Mounting surface, such as a plastic housing (20 cm x 15 cm x 8 cm)
• 6 screws. I used steel sheet metal screws
• Transformer with resistance from 300 Ohm to 75 Ohm.
• 18 AWG (0.8 mm) Mounting Wire
• RG-6 coaxial cable with terminators (and with a rubber sheath if installation will be outdoors)
• Aluminum when using a reflector. There was one in the attachment above.
• Fine marker
• Two pairs of small pliers
• The ruler is no shorter than 20 cm.
• Conveyor for angle measurement
• Two drill bits, one slightly smaller in diameter than your screws
• Small wire cutter
• Screwdriver or screwdriver

Note: The bottom of the aluminum wire antenna is on the right side of the picture where the transformer is sticking out.

Step 1: Adding a Reflector

Assemble the housing with the reflector under the plastic cover

Step 2: Drilling Holes and Installing Mounting Points

Drill small outlet holes on the opposite side of the reflector in these positions and place a conductive screw.

Step 3: Measure, Cut and Strip Wires

Cut four 20cm pieces of wire and place them on the body.

Step 4: Measuring and Marking Wires

Using a marker, mark every 2.5 cm on the wire (there will be bends at these points)

Step 5: Creating Fractals

This step must be repeated for each piece of wire. Each bend should be exactly 60 degrees, since we will be making equilateral triangles for the fractal. I used two pairs of pliers and a protractor. Each bend is made on a mark. Before making folds, visualize the direction of each of them. Please use the attached diagram for this.

Step 6: Creating Dipoles

Cut two more pieces of wire that are at least 6 inches long. Wrap these wires around the top and bottom screws along the long side, and then wrap them around the center screws. Then trim off the excess length.

Step 7: Installation of dipoles and installation of transformer

Secure each of the fractals onto the corner screws.

Attach a transformer of the appropriate impedance to the two center screws and tighten them.

Assembly complete! Check it out and enjoy!

Step 8: More Iterations/Experiments

I made some new elements using a paper template from GIMP. I used a small solid telephone wire. It was small, strong and pliable enough to bend into the complex shapes required for the center frequency (554 MHz). This is the average digital signal UHF for channels terrestrial television in my area.

Photo attached. It may be difficult to see the copper wires in low light against the cardboard and tape on top, but you get the idea.

At this size, the elements are quite fragile, so they need to be handled carefully.

I have also added a template in png format. To print the size you want, you'll need to open it in a photo editor like GIMP. The template is not perfect because I made it by hand using a mouse, but it is comfortable enough for human hands.

UDC 621.396

fractal ultra-wideband antenna based on a circular monopole

G.I. Abdrakhmanova

Ufa State Aviation Technical University,

Universita degli studi di Trento

Annotation.The article discusses the problem of designing an ultra-wideband antenna based on fractal technology. The results of studies of changes in radiation characteristics depending on the scale factor are presented.and iteration level. Parametric optimization of the antenna geometry was carried out to meet the requirements of the reflection coefficient. The dimensions of the developed antenna are 34 × 28 mm 2, and the operating frequency range is 3.09 ÷ 15 GHz.

Keywords:ultra-wideband radio communications, fractal technology, antennas, reflectivity.

Abstract:The development of a new ultra-wideband antenna on the basis of fractal technology is described in the paper. The research results on radiation characteristics changes depending on the value of scale factor and iteration level are presented. The parametric optimization of the antenna geometry for satisfying the reflection coefficient requirements was applied. The developed antenna size is 28 × 34 mm 2 , and the bandwidth is 3.09 ÷ 15 GHz.

Key words:ultra-wideband radio communication, fractal technology, antennas, reflection coefficient.

1. Introduction

Today, ultra-wideband (UWB) communication systems are of great interest to developers and manufacturers of telecommunications equipment, since they make it possible to transmit huge data streams at high speeds in an ultra-wide frequency band on a license-free basis. The peculiarities of the transmitted signals imply the absence of powerful amplifiers and complex signal processing components as part of the transceiver complexes, but they limit the range (5-10 m).

The lack of an appropriate element base capable of effectively working with ultrashort pulses is holding back the mass adoption of UWB technology.

Transceiver antennas are one of the key elements that influence the quality of signal transmission/reception. The main direction of patents and research in the field of designing antenna technology for UWB devices is miniaturization and reduction of production costs while ensuring the required frequency and energy characteristics, as well as the use of new forms and structures.

Thus, the antenna geometry is built on the basis of a spline with a rectangular U-shaped slot in the center, which allows it to operate in the UWB band with a blocking function WLAN -band, antenna dimensions - 45.6 × 29 mm 2. An asymmetric E-shaped figure measuring 28×10 mm 2, located at a height of 7 mm relative to the conducting plane (50×50 mm 2) was chosen as the radiating element in. A planar monopole antenna (22x22mm2) designed based on a rectangular radiating element and a ladder resonant structure on the reverse side is presented.

2 Statement of the problem

Due to the fact that circular structures can provide a fairly wide bandwidth, simplified design, small size and reduced production costs, this paper proposes to develop a UWB antenna based on a circular monopole. Required operating frequency range – 3.1 ÷ 10.6 GHz at a level of -10 dB reflection coefficient S 11, (Fig. 1).

Rice. 1. Required mask for reflectance S 11

For the purpose of miniaturization, the geometry of the antenna will be modernized through the use of fractal technology, which will also make it possible to study the dependence of the radiation characteristics on the value of the scale factor δ and the level of fractal iteration.

Next, we set the task of optimizing the developed fractal antenna in order to expand the operating range by changing the following parameters: the length of the central conductor (CP) of the coplanar waveguide (HF), the length of the ground plane (GP) of the HF, the distance “CP HF - radiating element (E)”.

Antenna modeling and numerical experiments are carried out in the " CST Microwave Studio".

3 Selecting antenna geometry

A circular monopole was chosen as the basic element, the dimensions of which are a quarter of the wavelength of the required range:

Where L ar– length of the radiating element of the antenna without taking into account the CPU;f L– lower limit frequency,f L = f min uwb = 3.1·10 9 Hz; With– speed of light, With = 3·10 8 m/s 2 .

We get L ar= 24.19 mm ≈ 24 mm. Considering that a circle with a radius ofr = L ar / 2 = 12 mm, and taking the original CPU lengthLf also equal r, we get the zero iteration (Fig. 2).

Rice. 2. Zero iteration of the antenna

Dielectric substrate thicknessT sand with parameter valuesεs = 3.38, tg δ = 0.0025 is used as a base on the front side of which IE, CPU and PZ . At the same time, the distances " PZ-CP" Zv and "PZ-IE" Zh taken equal to 0.76 mm. The values ​​of other parameters used in the modeling process are presented in Table 1.

Table 1. Antenna parameters ( δ = 2)

Name	Description	Formula	Meaning
L a	Antenna length	2 ∙ r + Lf	36 mm
W a	Antenna width	2 ∙ r	24 mm
Lf	CPU length	r + 0,1	12.1 mm
Wf	CPU width		1.66 mm
Lg	PZ length	r – T s	11.24 mm
L s	Substrate length	L a + G s	37 mm
W s	Substrate width	W a+ 2 ∙ G s	26 mm
G s 1	Vertical substrate gap		1 mm
G s 2	Horizontal substrate gap		1 mm
Tm	Metal thickness		0.035 mm
T s	Substrate thickness		0.76 mm
r	Radius of the circle of the 0th iteration		12 mm
r 1	Radius of the circle of the 1st iteration	r /2	6 mm
r 2	Radius of the circle of the 2nd iteration	r 1 /2	3 mm
r 3	Circle radius 3 iterations	r 2 /2	1.5 mm
εs	The dielectric constant		3,38

The antenna is powered by a coplanar waveguide consisting of a central conductor and a ground plane, SMA -connector and a coplanar waveguide port (CWP) located perpendicular to it (Fig. 3).

Where εeff – effective dielectric constant:

K – complete elliptic integral of the first kind;

	(5)

Fractality when constructing an antenna lies in a special way of packing elements: subsequent iterations of the antenna are formed by placing circles of smaller radius in the elements of the previous iteration. In this case, the scale factor δ determines how many times the sizes of neighboring iterations will differ. This process for the occasion δ = 2 is shown in Fig. 4.

Rice. 4. First, second and third iterations of the antenna ( δ = 2)

Thus, the first iteration was obtained by subtracting two circles with a radiusr 1 from the original element. The second iteration is formed by placing metal circles halved in radiusr 2 in each circle of the first iteration. The third iteration is similar to the first, but the radius isr 3 . The work examines the vertical and horizontal arrangement of circles.

3.1 Horizontal arrangement of elements

The dynamics of changes in the reflection coefficient depending on the iteration level are presented in Fig. 5 for δ = 2 and in Fig. 6 for δ = 3. Each new order corresponds to one additional resonant frequency. Thus, the zero iteration in the considered range 0 ÷ 15 GHz corresponds to 4 resonances, the first iteration – 5, etc. Moreover, starting from the second iteration, changes in the behavior of the characteristics become less noticeable.

Rice. 5. Dependence of the reflection coefficient on the iteration order ( δ = 2)

The essence of modeling is that at each stage, from the characteristics under consideration, the one that is determined to be the most promising is selected. In this regard, the following rule has been introduced:

If the excess (difference) in the range where the shelf is above -10 dB is small, then you should choose the characteristic that has a lower shelf in the operating range (below -10 dB), since as a result of optimization the first will be eliminated, and the second dropped even lower.

Rice. 6. Dependence of the reflection coefficient on the iteration order ( δ = 3)

Based on the data received and in accordance with this rule for δ = 2 the curve corresponding to the first iteration is selected for δ = 3 – second iteration.

Next, it is proposed to study the dependence of the reflection coefficient on the value of the scale factor. Consider the change δ in the range 2 ÷ 6 with step 1 within the first and second iterations (Fig. 7, 8).

An interesting behavior of the graphs is that, starting from δ = 3, the characteristics become flatter and smoother, the number of resonances remains constant, and the growth δ accompanied by an increase in the level S 11 in even ranges and a decrease in odd ones.

Rice. 7. Dependence of the reflection coefficient on the scale factor for the first iteration ( δ = 2; 3; 4; 5; 6)

In this case, the value chosen for both iterations is δ = 6.

Rice. 8. Dependence of the reflection coefficient on the scale factor for the second iteration ( δ = 2; 3; 4; 5; 6)

δ = 6, since it is characterized by the lowest shelves and deep resonances (Fig. 9).

Rice. 9. Comparison of S 11

3.2 Vertical arrangement of elements

The dynamics of changes in the reflection coefficient depending on the iteration level for the case of vertical arrangement of circles is presented in Fig. 10 for δ = 2 and in Fig. 11 for δ = 3.

Rice. 10. Dependence of the reflection coefficient on the iteration order ( δ = 2)

Based on the data obtained and in accordance with the rule for δ = 2 and δ = 3 the curve corresponding to the third iteration is selected.

Rice. 11. Dependence of the reflection coefficient on the iteration order ( δ = 3)

Consideration of the dependence of the reflection coefficient on the value of the scale factor within the first and second iterations (Fig. 12, 13) reveals the optimal value δ = 6, as in the case of horizontal arrangement.

Rice. 12. Dependence of the reflection coefficient on the scale factor for the first iteration ( δ = 2; 3; 4; 5; 6)

In this case, the value chosen for both iterations is δ = 6, which also representsn-multiple fractal, which means it may have to combine features δ = 2 and δ = 3.

Rice. 13. Dependence of the reflection coefficient on the scale factor for the second iteration ( δ = 2; 3; 4; 5; 6)

Thus, from the four compared options, the curve corresponding to the second iteration was selected, δ = 6, as in the previous case (Fig. 14).

Rice. 14. Comparison S 11 for the four antenna geometries considered

3.3 Comparison

Considering the best options for vertical and horizontal geometries obtained in the two previous subsections, the choice is made on the first (Fig. 15), although in this case the difference between these options is not so great. Operating frequency ranges: 3.825÷4.242 GHz and 6.969÷13.2 GHz. Next, the design will be modernized in order to develop an antenna operating in the entire UWB range.

Rice. 15. Comparison S 11 to select the final option

4 Optimization

This section discusses antenna optimization based on the second iteration of the fractal with the coefficient value δ = 6. Variable parameters are presented in , and the ranges of their changes are in Table 2.

Rice. 20. Appearance of the antenna: a) front side; b) reverse side

In Fig. 20 shows the characteristics reflecting the dynamics of change S 11 step by step and proving the validity of each subsequent action. Table 4 shows the resonant and cutoff frequencies used further to calculate surface currents and radiation patterns.

Table 3. Calculated antenna parameters

Name	Initial value, mm	Final value, mm
∆ Lf
Zh	Table 13,133208
6,195	27,910472
8,85	21,613615
10,6	12,503542
12,87	47,745235

The distribution of antenna surface currents at the resonant and boundary frequencies of the UWB range is shown in Fig. 21, and the radiation patterns are in Fig. 22.

a) 3.09 GHz b) 3.6 GHz

c) 6.195 GHz d) 8.85 GHz

e) 10.6 GHz f) 12.87 GHz

Rice. 21. Distribution of surface currents

A) F(φ ), θ = 0° b) F(φ ), θ = 90°

V) F(θ ), φ = 0° g) F(θ ), φ = 90°

Rice. 22. Radiation patterns in the polar coordinate system

5 Conclusion

This paper presents a new method for designing UWB antennas based on the use of fractal technology. This process involves two stages. Initially, the antenna geometry is determined by selecting the appropriate scale factor and fractal iteration level. Next, parametric optimization is applied to the resulting form based on studying the influence of the sizes of key antenna components on the radiation characteristics.

It has been established that as the iteration order increases, the number of resonant frequencies increases, and the increase in the scale factor within one iteration is characterized by a flatter behavior S 11 and constancy of resonances (starting from δ = 3).

The developed antenna provides high-quality reception of signals in the frequency band 3.09 ÷ 15 GHz in terms of level S 11 < -10 дБ. Помимо этого антенна характеризуется малыми размерами 34×28 мм 2 , а следовательно может быть успешно применена в СШП приложениях.

6 Acknowledgments

The study was supported by a grant from the European Union " Erasmus Mundus Action 2", also A.G.I. thanks professor Paolo Rocca for useful discussion.

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