A broadband hilbert-like antenna for partial discharge detection

Abstract

The application provides a broadband Hilbert-like antenna for partial discharge detection, which adopts a fourth-order Hilbert fractal curve as a basic framework, wherein each line segment is realized by a cosine function curve under parameterized control, so that the surface current path of the broadband Hilbert-like antenna extends in a wavy shape. The technical scheme of the application organically combines the cosine curve with fractal geometry, and solves the key problem that the broadband and omnidirectional of the partial discharge detection antenna are difficult to be considered simultaneously through accurate parameterized control. The practical application proves that the antenna has simple structure, superior performance and good engineering application prospect.

Description

Technical Field

[0001] This invention belongs to the field of high-voltage equipment and partial discharge detection technology, specifically relating to a broadband Hilbert antenna for partial discharge detection. Background Technology

[0002] With the continuous expansion of the power grid, the requirements for the reliability and safety of power equipment are also gradually increasing. Partial discharge detection is an important means of assessing the insulation status of power equipment, and the performance of the antenna, as a key component for electromagnetic signal sensing, directly affects the detection effect. Traditional partial discharge detection antennas mainly include dipole antennas, horn antennas, and fractal antennas. Among them, fractal antennas, due to their self-similarity characteristics and space-filling capabilities, can achieve multi-band or broadband operation in a compact size, and have gradually become a research hotspot.

[0003] Hilbert fractal antennas, as an important branch of fractal antennas, are widely used in various communication systems due to their excellent space utilization and self-similarity characteristics. Traditional Hilbert antennas consist of multiple straight segments, with their electrical length increased through spatial folding. However, in partial discharge detection applications, traditional Hilbert antennas have significant shortcomings: firstly, their bandwidth is relatively limited, making it difficult to cover the wide-spectrum characteristics of partial discharge signals; secondly, the radiation pattern is prone to distortion with frequency changes, affecting detection reliability; and thirdly, the antenna structure parameters have low degrees of freedom for adjustment, making it difficult to optimize for different application scenarios.

[0004] In recent years, scholars both domestically and internationally have made various attempts to improve Hilbert antennas. Some studies have adjusted performance by changing the number of fractal iterations or linewidth, but the improvement in bandwidth is limited; other studies have adopted hybrid fractal structures, but this has increased design complexity. Regarding curved antennas, sine and cosine curves, due to their smooth transition characteristics, have been used in dipole antenna design, showing potential for improving bandwidth and radiation patterns, but there are no reports of combining them with fractal structures.

[0005] The electromagnetic pulse signals generated by partial discharge have nanosecond-level rise times and wide spectral characteristics (typically covering 500MHz-2GHz), which imposes stringent requirements on the detection antenna for broadband and omnidirectional performance. Existing antennas struggle to simultaneously meet the demands for wide bandwidth, stable radiation patterns, and miniaturization, thus limiting the improvement of partial discharge detection accuracy. Therefore, developing a novel broadband omnidirectional fractal antenna has significant engineering application value. Summary of the Invention

[0006] To address the problems existing in the prior art, the present invention provides a broadband Hilbert-like antenna for partial discharge detection.

[0007] A broadband Hilbert-like antenna for partial discharge detection is provided. The broadband Hilbert-like antenna adopts a fourth-order Hilbert fractal curve as its basic framework, wherein each line segment is realized by a parameterized controlled cosine function curve, so that the surface current path of the broadband Hilbert-like antenna extends in a wavy shape.

[0008] Preferably, the broadband Hilbert antenna covers the partial discharge detection requirements of 300MHz-3GHz.

[0009] Preferably, the control parameters of the cosine function curve include amplitude and period;

[0010] The amplitude is used to control the fluctuation range of the cosine function curve, with an optimization range of 0.3-1.5.

[0011] The period is used to control the fluctuation density of the cosine function curve, with an optimization range of 0.5-2.0.

[0012] Preferably, the expression for the cosine function curve is as follows:

[0013]

[0014] Where A is the amplitude parameter and K is the period parameter. This is the phase parameter.

[0015] Preferably, the broadband characteristics of the broadband Hilbert-like antenna are achieved through multi-resonant coupling, a wavy current path, and an impedance gradient structure; wherein,

[0016] The curvature change introduced by the cosine function curve generates coupled resonance at multiple frequency points, and the overlapping of the coupled resonance points forms a wideband response.

[0017] Preferably, the omnidirectional radiation characteristics of the broadband Hilbert antenna are achieved through symmetrical current distribution, phase center stabilization, and coordinated operation of the radiating elements;

[0018] The symmetry of the cosine function curve causes the current in each radiating unit to be symmetrically distributed.

[0019] Compared with existing technologies, the advantages of this invention are as follows: This invention effectively extends the current path on the antenna surface, increasing design freedom. Compared with traditional Hilbert antennas, it achieves a more stable omnidirectional radiation pattern and lower signal distortion over a wide frequency band. Results show that the antenna of this invention exhibits excellent broadband characteristics in the 300MHz-3GHz frequency band, with a voltage standing wave ratio (VSWR) of less than 4.0, making it particularly suitable for broadband electromagnetic signal capture in partial discharge detection of power equipment. This antenna has a compact structure and stable performance, solving the problems of narrow bandwidth and large radiation pattern distortion in traditional partial discharge detection antennas, providing a more reliable detection tool for power equipment condition monitoring. Attached Figure Description

[0020] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 This is a schematic diagram of a conventional fourth-order Hilbert fractal antenna structure according to an embodiment of the present invention;

[0022] Figure 2 The radiation pattern of a conventional fourth-order Hilbert fractal antenna at a frequency of 2.0 GHz is shown in an embodiment of the present invention.

[0023] Figure 3 The radiation pattern of a conventional fourth-order Hilbert fractal antenna at a frequency of 2.5 GHz is shown in an embodiment of the present invention.

[0024] Figure 4 This is a schematic diagram of a broadband Hilbert-like antenna structure with A=1.0 and K=1 according to an embodiment of the present invention.

[0025] Figure 5 This is a schematic diagram of a broadband Hilbert-like antenna structure with A=1.5 and K=1 according to an embodiment of the present invention.

[0026] Figure 6 This is a schematic diagram of a broadband Hilbert-like antenna structure with A=0.5 and K=2 according to an embodiment of the present invention.

[0027] Figure 7 This is a schematic diagram of the VSWR according to an embodiment of the present invention;

[0028] Figure 8 This is a three-dimensional radiation pattern of a broadband Hilbert antenna according to an embodiment of the present invention;

[0029] Figure 9 The radiation pattern of the broadband Hilbert antenna at a frequency of 0.3 GHz is shown in an embodiment of the present invention.

[0030] Figure 10 The radiation pattern of the broadband Hilbert antenna with a frequency of 0.5 GHz is shown in an embodiment of the present invention.

[0031] Figure 11 The radiation pattern of the broadband Hilbert antenna at a frequency of 0.7 GHz in an embodiment of the present invention is shown.

[0032] Figure 12 The radiation pattern of a broadband Hilbert antenna with a frequency of 1.0 GHz is shown in an embodiment of the present invention.

[0033] Figure 13 The radiation pattern of a broadband Hilbert antenna with a frequency of 1.5 GHz is shown in an embodiment of the present invention.

[0034] Figure 14 The radiation pattern of the broadband Hilbert antenna with a frequency of 2.0 GHz is shown in an embodiment of the present invention.

[0035] Figure 15 The radiation pattern of the broadband Hilbert antenna with a frequency of 2.5 GHz is shown in an embodiment of the present invention.

[0036] Figure 16 The radiation pattern of the broadband Hilbert antenna with a frequency of 3.0 GHz is shown in an embodiment of the present invention. Detailed Implementation

[0037] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0038] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0039] Example 1

[0040] like Figure 1 As shown, a broadband Hilbert-like antenna for partial discharge detection is presented. This antenna employs a fourth-order Hilbert fractal curve as its basic framework, where each line segment is implemented using a parameterized cosine function curve, causing the surface current path of the broadband Hilbert-like antenna to extend in a wavy shape. Essentially, the straight lines of a traditional Hilbert fractal antenna are replaced with cosine curve segments, while maintaining the self-similarity characteristics of the fractal structure. Specifically, the nth-order cosine Hilbert curve is shown below. Figure 1 As shown. Figure 2The radiation pattern of a conventional fourth-order Hilbert fractal antenna at a frequency of 2.0 GHz is shown in an embodiment of the present invention. Figure 3 The radiation pattern of a conventional fourth-order Hilbert fractal antenna with a frequency of 2.5 GHz is shown in an embodiment of the present invention.

[0041] By replacing straight segments with cosine curves, the surface current path is effectively extended, achieving a wider impedance bandwidth. A further implementation allows for broadband Hilbert-like antennas to cover partial discharge detection requirements from 300MHz to 3GHz. The smoothness of the cosine curve improves current distribution and reduces phase differences in the high-frequency band, resulting in a more stable omnidirectional radiation pattern. The novel curved fractal structure, while maintaining miniaturization, optimizes the current path and improves radiation efficiency through curve curvature adjustment.

[0042] A further implementation involves using control parameters for the cosine function curve, including amplitude and period, providing greater design freedom and allowing for optimization of antenna performance based on specific applications. The parameter range is primarily optimized through multi-step processes, i.e., correlation. The value of A has a greater impact on the radiation pattern over a wide bandwidth, while the value of the feed location has a greater impact on VSWR.

[0043] Amplitude is used to control the fluctuation range of the cosine function curve, directly affecting the antenna's equivalent electrical length and radiation resistance. A larger value of A increases the current path and lowers the resonant frequency, but it also increases the overall antenna size. The optimization range is 0.3-1.5; specifically, due to the optimized design of A, a larger value of A increases the current path and lowers the resonant frequency, but it also increases the overall antenna size. The value of A is between 0.3 and 1.5. This stage mainly ensures minimal pattern distortion and no splitting over a wide frequency band.

[0044] The period is used to control the fluctuation density of the cosine function curve, affecting the high-frequency characteristics of the antenna. A smaller K value (more fluctuation) improves the high-frequency response but increases conductor loss; the optimization range is 0.5-2.0. Specifically, the period parameter K is designed according to the antenna frequency band requirements. A smaller K value (more fluctuation) improves the high-frequency response but increases conductor loss. The optimization range of K is 0.5-2.0, with larger values ​​used for low-frequency applications and smaller values ​​used for broadband applications. Since partial discharge mainly operates in the ultra-high frequency and long-wave bands, K is set to 1.

[0045] In this embodiment, the power supply location is optimized, and this stage mainly ensures the optimization of the VSWR value.

[0046] Hilbert antennas, due to their fixed fractal rules, are difficult to optimize and exhibit pattern distortion at high frequencies. This can even lead to pattern splitting, affecting the reliability of partial discharge. Traditional Hilbert antennas use a "gate"-shaped structure composed of straight line segments; this invention replaces this with a cosine curve. Let the side length of a standard Hilbert element be L, the expression for the cosine function curve is as follows:

[0047]

[0048] Where A is the amplitude parameter and K is the period parameter. This is the phase parameter (usually taken as 0 or π / 2).

[0049] Figure 4 A schematic diagram of a broadband Hilbert antenna structure with A=1.0 and K=1;

[0050] Figure 5 A schematic diagram of a broadband Hilbert antenna structure with A=1.5 and K=1;

[0051] Figure 6 A schematic diagram of a broadband Hilbert antenna structure with A=0.5 and K=2.

[0052] Specifically, in the parameter control and optimization process of this invention, a multi-stage screening mechanism is first used to initially set reasonable value ranges for key geometric parameters A and K, avoiding getting trapped in local optima due to low global search efficiency. Subsequently, a multi-objective, multi-parameter collaborative optimization mechanism is introduced in the simulation design stage. Taking into account electromagnetic performance, structural continuity, and engineering feasibility, the following three targeted multi-objective tuning strategies are adopted to effectively achieve the refined design of A and K:

[0053] 1. Equivalent electrical length coupling design

[0054] This strategy focuses on the excitation capability of multiple resonant modes in an antenna. By jointly adjusting parameters A and K, it achieves coordinated control of the equivalent electrical length of multiple "quasi-resonant units" in the antenna structure. This mechanism can enhance the energy coupling between resonant frequencies, improve the overall spectral response density, and broaden the usable bandwidth.

[0055] 2. Directional pattern equalization control mechanism

[0056] Based on system simulation analysis of the impact of A and K modulation on antenna current distribution and spatial radiation characteristics, this strategy introduces a quantitative evaluation model of the influence of spatial curvature on radiation pattern distortion. By selecting an AK combination that makes the current path distribution tend to be symmetrical, abnormal changes such as main lobe tilt and sidelobe enhancement in the radiation pattern are effectively suppressed, achieving stability and uniformity control of the radiation pattern within the operating frequency band.

[0057] 3. VSWR Targeted Tuning Mechanism

[0058] In antenna matching characteristic optimization, this strategy links the feed point location with the AK structural parameters for control, forming a targeted VSWR matching region construction method. A three-parameter response matrix of "feed-AK" is established using high-density simulation samples, avoiding high-reflection regions within the bandwidth and locking in a stable operating range of VSWR < 4.0. This method balances wideband stability of antenna input impedance with manufacturing constraints, ensuring good system matching performance while maintaining design compactness.

[0059] Higher-order fractals are formed by rotating, scaling, and connecting lower-order curves. Similar to traditional Hilbert curves, this invention maintains space-filling properties, but achieves optimized extension of the surface current path through curvature variations in the cosine curve.

[0060] A further implementation method involves achieving the broadband characteristics of the broadband Hilbert-like antenna through multi-resonant coupling, a wavy current path, and an impedance gradient structure; wherein,

[0061] Multi-resonant coupling: The curvature variation introduced by the cosine function curve generates coupled resonances at multiple frequency points, and the overlapping of these coupled resonance points forms a wideband response. Compared with traditional linear fractal antennas, the curved structure generates richer current modes in higher-order fractals.

[0062] Current path optimization: The cosine curve extends the surface current path in a wavy shape, effectively increasing the equivalent electrical length compared to a straight structure. Experimental results show that, for the same external dimensions, the current path length of the antenna of this invention increases by 30-50%, which lowers the low-frequency cutoff frequency. Simultaneously, the smooth transition of the curve reduces current abrupt changes in the high-frequency band, improving the high-frequency response.

[0063] Impedance gradient characteristics: The continuous curvature change of the cosine curve forms a natural impedance gradient structure, achieving better impedance matching broadband characteristics. By adjusting the A and K parameters, this gradient characteristic can be controlled to achieve impedance matching with VSWR < 4.0 in the 300MHz-3GHz range.

[0064] Partial discharge antennas are classified into external and internal types based on their installation method and location. This invention primarily addresses internal installation, typically placed inside transformers or GIS equipment. Since the GIS or transformer itself has a metal cavity structure, it is close to the discharge source, resulting in strong signals and high sensitivity. Furthermore, due to the shielding of the transformer casing, external interference signals are less likely to affect the internal antenna, thus providing good anti-interference capabilities. Currently, the industry requires modifications to the transformer casing, including pre-drilling windows filled with insulating media to ensure the casing's airtightness, thereby allowing the signal to be extracted.

[0065] This invention also has advantages in planar and miniaturized structure. The spiral antenna is also a type of partial discharge antenna, but it is difficult to embed a three-dimensional antenna.

[0066] This invention not only features a planar structure but also employs a curved wire design, resulting in a smaller size and easier integration.

[0067] A further implementation method is that the omnidirectional radiation characteristics of the broadband Hilbert antenna are achieved through symmetrical current distribution, phase center stabilization, and the coordinated operation of the radiating elements.

[0068] Symmetrical current distribution: The symmetry of the cosine function curve makes the current of each radiating unit symmetrically distributed, which can maintain good omnidirectional characteristics even in the high frequency band.

[0069] Phase center stability: The smooth transition of the curved structure reduces phase center drift in the high-frequency band, which is key to pattern stability. Experimental results show that at 3 GHz, the phase center offset of the antenna of this invention is only 1 / 3 of that of conventional designs.

[0070] Radiation unit coordination: The cosine curve units in the multi-level fractal structure work together in different frequency bands. Low frequency is mainly radiated by large-size units, while high frequency is dominated by small-size units. This natural frequency division characteristic ensures radiation stability across the entire frequency band.

[0071] Existing Hilbert antennas typically use straight line segments to construct two-dimensional recursive curves to extend the current path and reduce the antenna's footprint. However, due to the large number of right-angle inflection points in the structure, the current undergoes sharp changes between the straight line segments, which easily leads to the formation of standing wave superposition at these inflection points. This results in uneven current distribution, limiting bandwidth performance and causing current hotspots.

[0072] In contrast, this invention introduces a micro-path control mechanism on the basis of the macroscopic path of the Hilbert fractal structure: by replacing each recursive line segment with a parameterized cosine function curve, the current transitions smoothly along the path, thereby effectively suppressing standing wave reflection, improving the uniformity of current distribution, significantly expanding the working bandwidth, and maintaining good omnidirectional radiation characteristics.

[0073] Unlike existing methods that control the current path by loading gaps, stubs, and patches, this scheme directly introduces a continuous function perturbation into the main path without the need for additional loading structures. The design is simpler, the principle is clearer, and the form is more regular.

[0074] Furthermore, the introduction of the cosine path effectively increases the equivalent length of the antenna current path, which is equivalent to embedding more resonant paths in a limited space, thereby increasing the number and distribution density of resonant points. By adjusting the period parameter K, the frequency distribution of the resonant points can be precisely controlled, achieving a uniform response from low to high frequencies and improving the antenna's broadband performance and spectral adaptability.

[0075] In terms of manufacturing, cosine curves have good regularity and continuity. Compared with irregular loading structures, they are more suitable for standardized processing technology and are easy to implement on flexible materials, thus possessing good practical manufacturability and engineering adaptability.

[0076] In summary, this invention, while maintaining the miniaturization advantages of traditional fractal antennas, solves the design limitation of needing to increase the order to expand the bandwidth, and significantly improves problems such as standing wave concentration, current hotspots, and radiation distortion caused by structural abrupt changes (such as sharp corners and inflection points). It is a broadband omnidirectional antenna solution with strong structural innovation, high electrical performance adjustability, and good manufacturing friendliness.

[0077] Example 2

[0078] This embodiment uses a practically designed partial discharge detection antenna as an example:

[0079] Substrate material: FR-4(ε r =4.4, thickness 1.5mm);

[0080] Fractal order: 4;

[0081] External dimensions: 100mm × 100mm;

[0082] Cosine parameters: A = 0.5, K = 1, line width 0.6mm, copper thickness 0.05mm;

[0083] Power supply method: 50Ω coaxial power supply;

[0084] Impedance bandwidth: 300MHz-3.0GHz (VSWR<4.0); e.g. Figure 7 As shown.

[0085] Gain: -32dBi@0.5GHz;

[0086] Figure 8 This is a three-dimensional radiation pattern of a broadband Hilbert antenna.

[0087] like Figures 9-16 As shown, the radiation pattern stability of similar fractal antennas is poor with frequency changes, especially in the high-frequency band where splitting and distortion are prone to occur, affecting detection reliability. The smoothing characteristics of the cosine curve in this invention improve current distribution and reduce phase differences in the high-frequency band, thereby obtaining a more stable omnidirectional radiation pattern.

[0088] The technical solution of this invention can also be extended to the following modified designs:

[0089] (1) Dual-polarization version: Dual-polarization reception is achieved by orthogonally arranging two cosine Hilbert antennas, which improves the reliability of detection.

[0090] (2) Array configuration: Multiple units are arranged into an array to improve gain while maintaining broadband characteristics, which is suitable for monitoring large equipment.

[0091] (3) Flexible version: It adopts a flexible substrate and special curve parameters to make a bonding antenna for partial discharge detection on complex surfaces.

[0092] According to the technical solution of the present invention, the following technical problems are effectively solved:

[0093] 1. Bandwidth limitation: The effective bandwidth of traditional Hilbert antennas and most fractal antennas is usually no more than 30% of the center frequency, which makes it difficult to cover the wide spectrum range of partial discharge signals, resulting in incomplete signal acquisition.

[0094] 2. Pattern distortion: With frequency changes, the pattern stability of existing fractal antennas is poor, especially in the high-frequency band where splitting and distortion are prone to occur, affecting the reliability of detection.

[0095] 3. Low design freedom: The structural parameters (such as the number of iterations and linewidth) of traditional fractal antennas have limited impact on performance and are difficult to optimize precisely for different application scenarios.

[0096] 4. Fixed current path: The optimization space for the surface current path of traditional fractal antennas composed of straight segments is limited, which restricts the improvement of radiation efficiency.

[0097] In summary, the innovation of this invention lies in the organic combination of cosine curves and fractal geometry. Through precise parametric control, it solves the key problem of simultaneously achieving broadband and omnidirectional performance in partial discharge detection antennas. Practical applications have proven that this antenna has a simple structure, superior performance, and promising prospects for engineering applications.

[0098] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A broadband Hilbert-like antenna for partial discharge detection, characterized in that, The broadband Hilbert antenna uses a fourth-order Hilbert fractal curve as its basic framework. Each line segment is realized by a parameterized cosine function curve, which makes the surface current path of the broadband Hilbert antenna extend in a wavy shape. The broadband Hilbert antenna covers the partial discharge detection requirements of 300MHz-3GHz; The control parameters of the cosine function curve include amplitude and period; The amplitude is used to control the fluctuation range of the cosine function curve, with an optimization range of 0.3-1.5; The period is used to control the fluctuation density of the cosine function curve, with an optimization range of 0.5-2.0; The expression for the cosine function curve is as follows: y(x) = A·cos(2πxK + φ), A is the amplitude parameter, K is the period parameter, and φ is the phase parameter; The broadband characteristics of the broadband Hilbert antenna are achieved through multi-resonant point coupling, a current path extended by a wavy line, and an impedance gradient structure; the curvature change introduced by the cosine function curve generates coupled resonance at multiple frequency points, and the overlapping of the coupled resonance points forms a broadband response. The omnidirectional radiation characteristics of the broadband Hilbert antenna are achieved through symmetrical current distribution, phase center stabilization, and coordinated operation of the radiating elements. The symmetry of the cosine function curve causes the current in each radiating unit to be symmetrically distributed. In the parameter control and optimization process, a multi-stage screening mechanism is first used to initially set reasonable value ranges for key geometric parameters A and K, avoiding getting trapped in local optima due to low global search efficiency. Subsequently, a multi-objective, multi-parameter collaborative optimization mechanism is introduced in the simulation design stage. Taking into account electromagnetic performance, structural continuity, and engineering feasibility, the following three targeted multi-objective tuning strategies are adopted to effectively achieve the fine design of A and K: Equivalent electrical length coupling design: This strategy focuses on the excitation capability of multiple resonant modes of the antenna. By jointly adjusting the A and K parameters, the equivalent electrical length of multiple "resonant units" in the antenna structure is coordinated and controlled, which enhances the energy coupling between resonant frequencies, improves the overall spectral response density, and widens the available bandwidth. Pattern equalization control mechanism: Based on the system simulation analysis of the influence of A and K modulation on antenna current distribution and spatial radiation characteristics, this strategy introduces a quantitative evaluation model of the influence of spatial curvature on pattern distortion; by selecting the AK combination that makes the current path distribution tend to be symmetrical, the abnormal changes of main lobe tilt and side lobe enhancement in the pattern are effectively suppressed, and the stability and equalization control of the radiation pattern in the operating frequency band are achieved. VSWR Targeted Tuning Mechanism: In antenna matching characteristic optimization, this strategy links the feed point location with the AK structural parameter design to form a targeted VSWR matching region construction method; by establishing a "feed-AK" three-parameter response matrix through high-density simulation samples, high reflection areas are avoided within the bandwidth, locking in a stable operating range of VSWR < 4.0; the VSWR matching region construction method takes into account both the wideband stability of the antenna input impedance and the manufacturing constraints of the structure, ensuring good system matching performance while maintaining design compactness; Fourth-order Hilbert fractals are formed by rotating, scaling, and connecting lower-order curves. Similar to traditional Hilbert curves, broadband Hilbert antennas maintain space-filling characteristics, but optimize and extend the surface current path through curvature changes in cosine curves.

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