Research on the Defected Ground Structure With Von Koch Snowflake Fractals

This paper studies the electromagnetic properties of defected ground structure (DGS) with von Koch snowflake fractals. First, the snowflake fractal is employed in the dumbbell shaped DGS, and we have discovered that it has the characteristics of a low-pass filter and the out-of-band rejection increases with fractal iterations. In consideration of fabrication accuracy and performance, the third iteration has been adopted. Based on this structure, we have designed a narrow-band filter and a wide-band filter to verify our theory. From the simulated and measured results, we can see that the out-of-band rejection level has been greatly improved. The proposed structure can be applied in communication planar circuits.

plane, so that the distributed capacitance and inductance of the transmission line are changed [20]- [24]. It can be regarded as an electromagnetic bandgap (EBG) structure and its equivalent circuit can be model as a parallel resonant circuit. If it is designed properly, transmission zeroes can be created in the transition band, thus increasing the out-of-band rejection.
In the area of mathematics, fractals belong to a subset of Euclidean space that exhibit patterns with self-similarity. The topology replicates exactly itself at a smaller sale with infinity. Snowflake fractal was proposed by the Swedish mathematician Helge von Koch in 1904 [25]. It is created from a base triangle, from which sides grow smaller triangles, and so infinitum [26]. Figure 1 shows the topology of von Koch snowflake fractal at the zeroth, first, second and third iterations. We can see that several more triangles grow onto the original pattern after each iteration. The circumfluence can theoretically increase to infinity in a confined space.  It looks like a paradox. But because of this property, we have discovered the potential applications in the microwave circuit.
In our design, we apply von Koch snowflake in the DGS structure and explore its electromagnetic properties. We also have designed two microwave planar filter with excellent performance. The filters are simulated and measured, and the results are in good agreement, which proves our proposed theory.

II. THEORY, ANALYSIS AND DESIGN PROCESS
The topology of the dumbbell DGS with a snowflake fractal of the third iteration is shown in figure 2. The yellow part is the signal line, which is on the top layer on the substrate. The red part is the metallic ground, which is on the bottom layer. The white part is the etched pattern. The snowflake fractal is on the one end of the long etched rectangle line. The fractal structure theoretically can achieve long circumference in a limited area, therefore it can effectively increase the distributed inductance, and can achieve low-pass filtering characteristics. Table 1 gives the triangles and area added after each iteration.
Let's suppose the area and circumference of S(0) are A(0) and C(0). According to geometry, A(n) can be calculated as Therefore, Then, we can see the area of the snowflake fractal is 8/5 times the initial after an infinite number of iterations. The area is  finite. C(n) can also be calculated in a similar way as Therefore, The circumference of the snowflake fractal is infinite after an infinite number of iterations. Then we can see that if n approaches infinity, the circumference also reaches infinity while the area is a finite value. This property can show that the lowpass stopband can theoretically reach infinity if it is not limited by current PCB fabrication precision. To create parallel resonance, a slot line is connected to the fractal, and the topology is shown in figure 3. Figure 4 gives the simulated electromagnetic field distribution at the passband and stopband. When the frequency is low, the electromagnetic wave travels through the snowflake fractal DGS with almost no reflection. With the increase of frequency, the electromagnetic wave is reflected and radiated by the snowflake fractal DGS, and a standing wave is formed on the input port of the line.
Considering the fabrication accuracy and our calculation, the maximum iteration used in our simulation is the third one. Figure 5 shows the simulated S 21 of the snowflake DGS with the zeroth, first, second and third iterations. First, we can VOLUME 8, 2020  see that the out-of-band rejection level increases from around 20 dB at zeroth iteration to over 40 dB at the third iteration. The steepness of the slope also increases with iteration number. At the zeroth iteration, the slope is −6.6 dB/GHz, while at the third iteration, the slope is −13.5 dB/GHz.
The lowpass characteristics is due to the etched ground pattern, and its equivalent circuit model can be given in figure 6. L g and C g are created by the slot line on the ground, and they can be calculated as [27] C g = 1 where g 1 , ω r , ω c and Z 0 are the lowpass prototype value, resonant and cutoff angular frequencies of the transmission zero and characteristic impedance, respectively. The stage number increases with the increase of iteration number. For the snowflake fractal DGS, based on our simulation, the capacitance and inductance are mainly determined by the area and circumference, respectively. Then C sn and L sn can be estimated as where d is the thickness of the substrate; v is the guided wave velocity and t is the thickness of the ground plane. In our design, quarter-wavelength resonators are employed as the resonant elements. This resonator has one open end and one shorted end. Therefore, at the resonant condition, the magnitude of voltage reaches the highest and the where λ 0 is the longest resonant wavelength. Therefore, the resonant frequencies are Therefore, the first parasitic passband appears at 3f 0 . And in order to increase the selectivity of the designed filter, this passband should be suppressed. In our paper, periodic fan-shaped capacitors are utilized. Figure 7 shows the topology and equivalent circuit of the modified quarter-wavelength resonator with one fan-shaped capacitor loaded at the open end. L r is the inductor of the line connected to the capacitor; C r is the self-capacitance of the fan-shaped capacitor, and L p is the inductance of the short-ended stub.
For fan-shaped capacitors, the capacitance and input impedance can be calculated as [29] C r = θ r r 2 0 ε eff 240πdc (11) where ε eff , d, c, β, θ r , r 0 are the effective permittivity, substrate thickness, phase velocity of electromagnetic waves in vacuum, phase constant,fan angle and radius, respectively. For planar circuit, the effective permittivity can be given as [28] where w is the line width, and ε r is the relative substrate permittivity. The equivalent circuit can generate one series resonance and one parallel resonance. The series resonant frequency f z and parallel frequency f 0 can be calculated as  The parallel resonant frequency f 0 creates the passband, while the series resonant frequency f z creates one transmission zero in the network. If f z is designed at 3f 0 , then the parasitic passband can be suppressed. The connection lines between the resonators can be treated as J -inverters with the normalized admittance of unit. The general circuit model is shown in figure 8. Based on this circuit model, the center frequency f 0 is at the position where the admittance of the resonator is zero. The relative bandwidth can be controlled by the parallel stub of the resonator. Therefore, the component values L i and C i can be calculated based on the center frequency, relative bandwidth and element values in the low pass prototype, which is [30] where g i is the element values of the low pass prototype; RBW is the relative bandwidth of the passband and ω 0 is the center angular frequency.
Since there are three elements in the equivalent circuits, then it can be obtained from previous equations that Therefore, all the component values can be calculated from g i , RBW and f 0 . The physical size of each section in the filter can be obtained from L p , L r and C r . The length of the short-ended stub and capacitor connection line can be calculated as where λ 0 is the guided wavelength of the electromagnetic wave at the center frequency. Z p and Z r are the impedance of the short-ended stub and capacitor connection line, respectively. The line width can be calculated using [28] w d =  where A = Z 60 The design specification requires the center frequency to be 3 GHz with relative bandwidth of 30%, therefore three stage Chebyshev response is chosen. The low pass element values are obtained as g 1 = g 3 = 1.367, g 2 = 2.723. Then the component values in the equivalent circuit can be calculated as L p1 = L p3 = 1.33 nH, L p2 = 0.67 nH, C r1 = C r3 = 1.38 pF, C r2 = 2.75 pF, L r1 = L r3 = 0.71 nH, L r2 = 0.35 nH. Therefore, the first resonator and the third one are of the same size, as shown in figure 9. The initial size of each section can be calculated using the component values based on the relative permittivity, loss tangent and thickness of the substrate. Figure 10 gives the simulated results with and without the snowflake fractal DGS. We can see that the second and third harmonic passbands have been suppressed greatly. Also, the out-of-band rejection is over 35 dB in the stopband.
The substrate used in the design is Rogers 5880 with the relative permittivity ε r = 2.2, loss tangent tan δ = 0.0009, and thickness h = 0.508 mm. The filter is simulated using HFSS and the photograph of the proposed filter is shown in figure 11. After optimization, the physical size of each section can be given as (length in mm): w 1 = 0.5, w 2 = 1.5, w 3 = 0.6, w 4 = 0.6, l 1 = 6.0, l 2 = 13.6, l 3 = 13.0, l 4 = 10.0, l 5 = 1.5, l 6 = 3.4, r 1 = 4.0, r 2 = 3.5, d = 0.5, and r g = 4.7. VOLUME 8, 2020   The simulated and measured results are shown in figure 12. The filter is measured using Keysight PNA-X network analyzer N5247B. The measured center frequency is 3.0 GHz with the relative bandwidth of 29%. The in-band insertion loss is less than 1.0 dB and the return loss better than 19 dB. The suppression of the parasitic passband at the third harmonic is over 40 dB. The simulated and measured results are in good agreement and the selectivity of the filter has been improved.
Also the proposed structure can be employed in the wideband band-pass filters to improve out-of-band rejection. Figure 13 shows the topology and equivalent circuit of the transverse line loaded with periodic fan-shaped capacitors.
Therefore, the low-pass cutoff frequency can be calculated as where L and C are the distributed inductance and capacitance respectively. C r is the capacitance of the fan-shaped capacitor. The characteristic impedances of the loaded line and unloaded line can be denoted as And the phase shift of the loaded line can be given as When the design specifications are given, L, C and C r are unknown variables.
In order for the stopband range to exceed over 20 GHz, and also based on the cutoff frequency of the stopband, the characteristic impedance of the line is calculated to be 50 with the phase shift of 45 • , then L, C and C r can be denoted based on the impedance, phase shift and cutoff frequency of the unit cell. The formulas can be derived as C, L and C r in our case can be calculated to be 0.18 pF, 3.5 nH and 1.4 pF, respectively. Therefore, Z 1 and f B can be calculated to be 125 and 6.3 GHz. In our design, three segments of short-ended lines are employed as the resonant units, and they produce the passband of the filter. The width of the center transverse line is thinner to create high impedance. Two fan-shaped capacitors are added between every two resonators to suppress harmonic passbands. The topology of the proposed filter is shown in figure 14. Figure 15 shows the simulated S 21 of the filters with or without the fan-shaped capacitors. The width and length of these two filters are modified due to the changes in distributed capacitance and inductance after loading the fan-shaped capacitors. It can be seen that the original filter without the capacitors have parasitic passbands at the third and fifth harmonics. After loading the capacitors, the harmonics have been suppressed without affecting the performance of the   passband. From 8 GHz to 20 GHz, the rejection level is over 40 dB and the selectivity of the filter has been greatly improved. Figure 16 shows the photograph of the proposed filter. The Rogers 5880 substrate is employed, with the relative permittivity ε r = 2.2, dissipation factor, tan δ = 0.0009 and thickness h = 0.787 mm. The geometric parameters are given as (length in mm): w 0 = 2.4, w 1 = 0.4, w 2 = 0.4, w 3 = 0.6, w 4 = 0.8, l 1 = 23.4, l 2 = 16.9, l 3 = 10.3, r = 4.4, d = 0.5, and r g = 6.9. Figure 17 shows the simulated and measured S 21 of the proposed filter. From the measured results, the selectivity of the filter has been improved. From 6 GHz to 20 GHz, the out-of-band rejection is higher than 40 dB. The center frequency of the passband is 3.4 GHz, with the relative bandwidth of 143%. The in-band insertion loss is lower than 1.2 dB, and the return loss is better than 18 dB. The measured results are in good agreement with the simulated results. Performances comparison of some other filters in terms of center frequency (f 0 ), in-band insertion loss (IB-IL), in-band return loss (IB-RL), relative bandwidth (RBW), insertion loss from the second to sixth harmonics, and maximum out-ofband rejection in the references are listed in table 2. It can be seen that comparable performances have been achieved.

III. CONCLUSION
In this paper, we have investigated and analyzed the electromagnetic property of the DGS with von Koch fractals and its potential applications in microwave planar filters. The fractal structure theoretically has an infinite rejection band if it is not limited by current precision level of PCB fabrication technology. Two band-pass filters have been designed based on the proposed structure and wide out-of-band rejection has been achieved. The proposed structure can be applied in modern planar circuits to improve performance.

APPENDIX
The fractal curve can be created using rotation matrices with building matrices. For each iteration, the fractal line is generated on the islands of the last iteration. For von Koch snowflake fractals, building functions are given as YUYU ZHU received the B.S. and M.S. degrees from the Southwest University of Science and Technology. He is currently pursuing the Ph.D. degree in non-destructive testing with the University of Electronic Science and Technology. He has long been involved in the research of power electronics applications and electromagnetic compatibility, non-destructive testing, and measurement, and control systems.
ZUXUE XIA received the Ph.D. degree in electromagnetic field and microwave from the University of Science and Technology of China. He has been involved in microwave passive components, especially microwave filter design, power divider, coupler design, and antenna design.
QIANGMING CAI received the Ph.D. degree in electromagnetic field and microwave technology from the University of Electronic Science and Technology. He has long been involved in computational electromagnetics and its engineering applications, electromagnetic dispersion and radiation, antenna principles and technologies, and electromagnetic compatibility. VOLUME 8, 2020