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In this paper an effective size reduction technique using fractal structure is suggested. The proposed technique has been applied on a band stop and a low pass filter separately. This technique provides 42% reduction of size for the band stop filter and about 26% for the low pass counterpart. Both the designed structures are fabricated and the measured results are compared with the simulated results. The proposed technique does not require any recalculation or optimization of dimensions of the filter, and is straightforward to implement. The band stop filter is designed for the center frequency of 3 GHz where as the cut-off frequency for low pass filter is 2.5 GHz. A good agreement between the simulated and measured results is observed. A comprehensible explanation of the proposed technique is also provided.

Filters are a very important component in the microwave and RF communication systems. In the advanced communication systems, a compact filter circuit with better performance is a basic requirement these days. In this paper the band stop and low pass filters are considered to demonstrate the proposed technique. To remove the unwanted frequency bands from the microwave and radio frequency signals a band stop filter (BSF) plays a very important role in wireless communication systems. Thus the BSF have been widely used in several wireless circuits and systems. The low pass filters are the indispensable components in any microwave communication system especially at the receiving end. There are various BSF structures developed using planar substrates to achieve a compact circuit size. A wide band stop filter is proposed by using single quarter wave length resonator with one section of anti-coupled lines with shorted ant other end [

Using the microstrip line with open end stubs separated with quarter wavelength lines at the center frequency of stop band is one of the most widely used method to design a wide band stop filter [8-10]. This technique is one of the conventional and more widely used methods. These quarter wave length connecting lines are used as unit elements. Since the unit elements of this type of band stop filter are redundant, and their filtering properties are not well utilized, the resultant band stop filter is not an optimum one. For band stop filters the unit elements can be made nearly as effective as the open-circuited stubs. Therefore, by incorporating the unit elements in the design, significantly steeper attenuation characteristics can be obtained for the same number of stubs as is possible for filters designed with redundant unit elements. Also, a specified filter characteristic can be met with a more compact configuration using fewer stubs if the filter is designed by an optimum method [11,12]. This optimum method is used in this paper to design the band stop filter and then the fractal structure [

Similar technique of size reduction is applied on the low pass microstrip filter. The fractal curve has been applied on the high impedance line of the stepped impedance microstrip low pass filter. Using microstrip structure the low pass filter is designed by using either open stub or stepped impedance technique [8,9]. The stepped impedance technique is more common because of its simpler design methodology. The low pass filter design using this technique occupies larger area. To achieve the compact structure several efforts have been made. A compact low pass filter has been designed using slow wave resonators [

The basic structure of 1-D, 90˚ angles Kotch curve [

The length of the line that is the distance between the two end points p and q is considered as d which is shown in

the physical length of the line unchanged as shown in

The basic structure of the proposed band stop filter is based on the open circuited transmission line stub network as depicted in _{i} for the open-circuited stubs, and characteristic impedances Z_{i}_{,i+1} for the unit elements, as well as two terminating impedances Z_{A} and Z_{B} [8,9].

The synthesis of the above shown ladder network

shown in

where ε is the pass band ripple constant and A_{n} is the filtering function represented by the following function

where t is the Richards’ transform variable which is given by

where f_{0} is the mid-band frequency of the band-stop filter and FBW is the fractional bandwidth. and are the Chebysheve functions of the first and second kinds of order n:

The elemental values are normalized admittances, and for a reference Impedance Z_{0} = 50 ohm, the impedances are determined by the set of Equation (8).

An optimum microstrip band-stop filter with three open-circuited stubs and a fractional bandwidth FBW = 100% at a mid-band frequency is designed. By considering a pass band return loss of 20 dB, which corresponds to a ripple constant ε = 0.1005. For optimized filter the normalized element values are taken from [_{0} = 50 ohm, and from Equation (8) we determine the electrical design parameters for the desired filter network as given below:

The lengths and widths of the corresponding microstrip line open stub filter are determined. The microstrip BSF of such design with their dimensions is shown in

The fractal structure shown in

there are no change in the dimensions, only the structure of connecting lines are changed. With the application of Kotch fractal the distance between the two open end stubs becomes half. The total length reduction of L = 15.5 mm is achieved.

The simulated results of the modified BSF with fractal structure are compared with the filter without modification. _{11} parameters of the filter structures shown in _{21} parameters are also compared for the filters with and without the fractal implementation which is shown in _{11} and S_{21} parameters of the fabricated structure are measured using vector network analyzer. Figures 8 and 9 compare the measured values of S_{11} and S_{21} respectively of the fabricated structure with the simulated parameters of the structure shown in

Thus the band stop filter using open stubs can be designed in a compact size with no change in the design equations given in [

In this section a compact microstrip low pass filter is designed using the fractal structure discussed in Section 2. To design a microstrip low pass filter stepped impedance technique is used in which higher and lower impedances are used to get the effects of inductive and capacitive impedances respectively from the distributive structures. In a microstrip as the impedance of the line increases the width of the line become thinner and vice-versa. So to get the appropriate values of the inductances and capacitances in microstrip

the ratio of higher to lower impedance value of lines should be as high as possible (where Z_{high} and Z_{low} are the higher and lower impedances of the microstrip lines), but limited by the practical values that can be fabricated on a printed circuit board because very thin line is impractical to fabricate. The typical values are Z_{high} = 100 W to 150 W and Z_{low} = 15 W to 25 W. Since a typical low-pass filter consists of alternating series inductors and shunt capacitors in a ladder configuration, we can implement the filter on a printed circuit board by using alternating high and low characteristic impedance section transmission lines. In this design we have selected Z_{low} = 24 Ω and Z_{high} = 100 Ω. For the filter specification given above following calculations are made. For these low and high impedances the width of the microstrip line is determined as 6.35 mm and 0.48 mm respectively. Also for Z_{0} = 50 Ω, the width of the transmission line obtained is w = 1.82 mm. The relationship of inductance and capacitance to the transmission line length at the cut-off frequency wc are:

where and _{ }are the physical lengths. Using the design Equations (9) and (10) and the lumped parameter values obtained above, the lengths of the inductive and capacitive lines are determined as below which have been depicted in

There are stray capacitive and inductive effects produced by high impedance and low impedance lines respectively. By minimizing these effects the circuit size can be made compact [

where Cp and Ls are the stray capacitances and inductances, β is the phase constant, and λ_{g} is the wave length of the respective lines. Minimizing the stray capacitance and inductance effects requires a lengthy calculation since it requires number of mathematical iterations and then the lengths of the resonators are needed to be optimized by using the simulation software.

In the proposed method, to reduce the complexity of the design process fractal structure has been applied to the inductive lines of the stepped impedance structure. It has been investigated that by using first iteration and 1/8th order of Kotch curve, the size of filter reduces by more than 26%. By using the fractal shape there is no need to recalculate the dimensions of high and low impedance lines. Using this method there is no need to go for any consideration of end effect or T-junction effects. The designed low pass filter using fractal lines reduces the length of the overall structure by l_{2} = l_{4}. The proposed structure is shown in

The simulated values of S_{11} and S_{21} of the low pass filter with and without fractal structure are shown in Figures 13 and 14. From these S_{11} and S_{21} parameters it can be clearly observed that there is no considerable deviation between the values with and without fractal structures. Moreover the pass band characteristics using the fractal structure are slightly improved. The fabricated structure of the proposed design is shown in _{11} and S_{21} parameters are shown in Figures 17 and 18 respectively. From these results it has been observed that there is a good match between simulated and measured values and a slight mismatch is due to the imperfection in fabrication.

The proposed method is an effective and efficient size reduction technique with simplicity. This technique of size reduction may be applied on most of the line based microstrip circuits. In the presented work more than 46.5% reduction of the size has been observed without change in the response over the conventional optimized band stop filter. In the case of low pass filter the proposed technique is effective to achieve the compact size with the same response as the conventional low pass filter. The size reduction in the case of low pass filter is more than 26%. This method does not require lengthy calculations or optimization. The proposed technique is

verified by comparing the measured results with the simulated results giving good agreement between the two.