The compact size and suppression of unwanted frequency components with excellent passband characteristic is the major concern of the microwave low pass filters design. To achieve the compact size printedcircuit technology is generally preferred to design the planar microwave filters. This technique also provides easy fabrication, low cost as well as easy integration with other microwave circuits. Conventional design of microstrip low pass filters basically involves either use of shunt stubs or using the stepped impedance network that is high-low impedance transmission line [1-2]. Using the above methods, a large number of inductive and capacitive elements are required to achieve shaper roll-off characteristic thus the resultant filters occupy larger area. There are several methods to reduce the size of microstrip low pass filters which have been reported [3-9]. One of the effective methods of size reduction is to introduce slow wave structure either in the main line [4-5] or on the ground plane [6-9]. In [4] a spiral resonator is used to replace the stepped impedance ladder network in the conventional design. The unsymmetrical stubs have also been used in [5] to design a compact filter. These methods increase the circuit complexity. The slow wave effect on the ground plane can be introduced by using the defective ground structures (DGS). The overall inductance and capacitance of the resonators can be increased using DGS and thus the compact circuit size can be achieved. The defects on the ground plane may cause unwanted radiation and it is difficult to integrate with other microwave components.

In this paper, a compact microstrip low pass filter has been proposed with sharper roll-off characteristics. An empirical expression has been derived to provide the direct calculation of the lengths of the resonators. The basic concept used for the proposed technique has also been discussed critically. The designed filter has been fabricated and tested.

To design passive low pass prototype filter, amplitudesquared transfer function may be used [1]. In the proposed design the insertion loss technique with Chebyshev approximation has been applied. Chebyshev approximation of a two port filter network is a mathematical description of the filter response which results equal-ripple in pass band and maximally flat stop band. In microwave filters it defines the transmission parameter S21. The Chebyshev low pass prototype filter is a ladder network with series inductors and shunt capacitors, for five poles it has been shown in Figure 1. The inductance and capacitance of various components can be calculated from the normalized values of g_k (where k = 1, 2, 3, 4, 5) as given in Equations (1) and (2) respectively.

where L and C are the inductances and capacitances of the resonators. Z₀ is the source and load impedances, f_c is the cut-off frequency in hertz. The electrical lengths for the inductors and capacitors for microstrip structure are proportional to the respective inductances and capacitances [1].

      for inductor

    for capacitor where β is the phase constants, l is physical lengths of resonators, Z_high and Z_low are the impedance of inductive and capacitive lines respectively.

The constant g0 and g6 shown in Figure 1 are the normalized values of source and load impedances.

To calculate the lengths of the resonators for the proposed method, the value of f(x) is determined for desired cut-off frequency x (in GHz) using Equation (3) which is a polynomial of the fourth order that has been empirically derived. This equation has been derived by using curve fitting of simulated results for different cut off frequencies. From the Equations (4) and (5) it is clear that the lengths of inductive components are inversely proportional to the corresponding impedance, whereas the lengths of capacitive elements are proportional to the corresponding impedances. Since as the cut-off frequency decreases the length of the resonators increases. This makes the length of resonators for L-band application so large. The value of f(x) has been used to calculate the lengths of inductive and capacitive lines by using the Equations (4) and (5) respectively. The layout of the filter is shown in Figure 2. From this figure it can be visualized that this structure is high-low impedance ladder

network in which the high impedance lines are placed at one end of the low impedance stubs. It has been studied that as the position of high impedance line shifted towards the end from the central position the values of overall capacitance increases and hence the cut-off frequency reduces.

where and are the physical lengths and and effective dielectric constant of inductive (smaller width) and capacitive lines (wider width).

For the proposed design, a fifth order Chebyshev response for 0.1 dB ripple has been considered. The cut off frequency is considered as 1.7 GHz. The presented filter has been fabricated using the substrate FR4 with dielectric constant 4.5 and height 1.5 mm. By using the design equations given in the previous section the lengths of the inductive and capacitive line sections have been calculated for the frequency x = 1.7 GHz. The dimensions are calculated as length of low impedance resonators L₁ = 1.41 mm, length of central stub is L₃ = 2.44 mm, lengths of the inductive lines are L₂ = 2.64 mm width of inductive lines are W₂ = 0.1 mm and width of the capacitive lines are W₃ = 20 mm and the width of the 50 ohm line is W₁ = 2.81 mm. The designed structure is simulated using MoM based full wave electromagnetic simulation software IE3D [9] and fabricated using photolithographic technique. The fabricated structure is shown in Figure 3. There is a good agreement between the simulated and measured results. The measured and simulated S₁₁ and S₂₁ parameters are shown in the Figures 4 and 5 respectively. In the pass band the maximum return loss in 22.9 dB. The

insertion loss reaches more than 25 dB at about 2.2 GHz.

In this work a simple design method has been proposed to design a compact low pass filter. The size of the proposed design occupies 2.34 times lesser area than the conventionally designed low pass filters. That is using the proposed technique about 57% size reduction has been realized. This filter shows more than 22 dB insertion loss within 0.5 GHz.