1 Introduction

Microwave band pass filters play a significant role in wireless communication systems. Transmitted and received signals have to be filtered at a certain center frequency with a specific bandwidth. The filter design is proposed to implement using microstrip lines. In designing a micro strip filters, the first step is to carry out an approximated calculation of microstrip lines length, width and spacing between them [1, 2]. Experimental verification gives comparison, how close the theoretical results and measurements look alike.

To construct specific filters, the desired frequency characteristics are related to the parameters of the filter structure. The general synthesis of filters proceeds from tabulated low-pass prototypes. Various physical forms of filters such as parallel coupled line filters, edge coupled line filters, interdigital filters and combline filters can be realized in distributed structures.

The parallel coupled line filter consists of a cascade of pairs of parallel coupled open circuited lines. The lines are quarter wave long at the center frequency of the filter [3, 4]. There are N + 1 coupled line section including input and output transformers in an Nth degree filter is shown in Fig. 1.

Fig. 1
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Parallel coupled line filter of degree 5

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The parallel coupled line filter is often used in microwave subassemblies as it is easy to fabricate due to the absence of short circuits.

A pair of coupled lines and its equivalent circuit is shown below in Fig. 2. Here it is noted that the equivalent circuit consists of series-open circuited stubs separated by an unit element (UE) in Fig. 2. Zoe and Zoo are the even and odd mode characteristic impedances of the coupled line pair as depicted in Fig. 3. The UE may be decomposed into a pair of open circuited stubs separated by an inverter as shown in Fig. 4. Combining this with the equivalent circuit in Fig. 2, a final equivalent circuit consisting of series open circuit stubs separated by inverters are obtained as shown in Fig. 5 is obtained.

Fig. 2
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Equivalent circuit of a parallel coupled line pair

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Fig. 3
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Equivalent circuit of a unit element

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Fig. 4
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Equivalent circuit of a coupled line pair

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Fig. 5
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Equivalent circuit of a parallel

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A cascade of N + 1 coupled line pair results in a circuit consisting of N series stubs separate by inverters as in Fig. 5.

2 Procedure and design of bandpass filter

Narrowband bandpass filters can be made with cascaded coupled line sections of the form shown in Fig. 1. The design starts with the specifications mentioned as in the Table 1.

Table 1 Design specification
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The fractional bandwidth (FBW) is calculated as, [5,6,7,8]

$$ \begin{aligned} \Delta = & \frac{{f_{2} - f_{1} }}{{f_{0} }} \\ \Delta = & \frac{10.25 - 9.75}{10} = \frac{0.5}{10} = 0.05 \\ \end{aligned} $$
(1)

Using Fig. 6 the attenuation at the stop bands is estimated [9].

Fig. 6
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Attenuation versus normalized frequency for maximally flat

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To convert this frequency to the normalized low pass form

$$ \begin{aligned} \omega \leftarrow & \frac{1}{\Delta }\left( {\frac{\omega }{{\omega_{o} }} - \frac{{\omega_{o} }}{\omega }} \right)\; {\text{is}}\;{\text{used}} \\ \omega \leftarrow & \frac{1}{0.05}\left( {\frac{9.5}{10} - \frac{10}{9.5}} \right) = 20\left( {0.95 - 1.0526} \right) = - \,2.052 \\ \end{aligned} $$
(2)

The approximate order of the filter is found with the required stop band attenuation

$$ \left| {\frac{\omega }{{\omega_{c} }}} \right| - 1 $$
(3)

\( \left| {\frac{\omega }{{\omega_{c} }}} \right| - 1 = \left| { - \,2.052} \right| - 1 = 1.052 \approx 1.1 \) [10, 11]. This indicates an attenuation of 25 dB for N = 4. The low pass prototype element values are given in Table 2.

Table 2 Element values for maximally flat low pass filter prototypes
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The design equations for a bandpass filter with N + 1 coupled line sections to calculate the admittance inverter constants \( J_{n} \) are listed here [12, 13].

For the first section J inverter

$$ Z_{0} J_{1} = \sqrt {\frac{\pi \Delta }{{2g_{1} }}} $$
(4)

For the intermediate sections the J inverters are

$$ Z_{0} J_{n} = \frac{\pi \Delta }{{2\sqrt {g_{n - 1} g_{n} } }}\;{\text{for}}\;{\text{n}} = 2\;{\text{to}}\;{\text{N}} $$
(5)

and for the last section the J inverter is

$$ Z_{0} J_{N + 1} = \sqrt {\frac{\pi \Delta }{{2g_{N} g_{N + 1} }}} $$
(6)

where N is the order of the filter and g0,…gN+1 are the ladder type low pass filter elements. Finally the even and odd mode characteristic impedances are found using, [14,15,16]

$$ Z_{oe} = Z_{0} \left[ {1 + JZ_{0} + \left( {JZ_{0} } \right)^{2} } \right] $$
(7)
$$ Z_{O0} = Z_{0} \left[ {1 - JZ_{0} + \left( {JZ_{0} } \right)^{2} } \right] $$
(8)

The calculated values of the J inverters and their Odd and Even impedances are given in Table 3.

Table 3 Values of the J inverters, odd and even impedances
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From Table 3 it is obvious that the filter sections are symmetric about the midpoint. The physical dimensions, width (W) and length (L) of the coupled line sections and space (S) between them are calculated with the help a transmission line calculator available in NI/AWR Microwave Office Tool [17]. With respect to the electrical length \( \beta l = \frac{\lambda }{4} \) at the center resonance frequency fO, the length of all sections are calculated for substrate \( \varepsilon_{r} = 3.66 \), h = 0.508 mm and copper trace thickness t = 0.035 mm. The results are presented in Table 4 below with the corresponding space between them.

Table 4 Width, length and space of the coupling sections
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The filter structure with the calculated width, space and length delivers approximate results only as it is. After the manual tuning and optimization, the physical dimensions slightly get modified and give good results.

3 Simulation and results

An analytical approach for the design of narrowband BPF is presented based on the conventional 4th order parallel coupled line structure. The measured results agree with the simulated results with negligible deviations. The filter is capable to pass signals in the frequencies between 9.75 and 10.25 GHz. The filter response is linear and flat throughout the pass band. The performance parameters are tabulated in the Table 4.

Figure 7 depicts the two dimensional actual filter layout. At ports 1 and 2, SMA connectors are connected for practical measurements with the network analyzer. The Insertion loss, Return loss, VSWR and group delay of the filters are shown through Figs. 8, 9 and 10.

Fig. 7
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Two dimensional view of the band pass filter

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Fig. 8
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Insertion loss and return loss

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Fig. 9
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Return loss and VSWR

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Fig. 10
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Group delay

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Periodic structures generally exhibit pass band and stop band characteristics in various bands of wave number determined by the nature of the structure [18]. This was originally studied in the case of waves in crystalline lattice structures, but the results are more general. The presence or absence of propagating wave can be controlled by these periodic structures. Figure 7 shows the 2D view of the proposed BPF.

4 Conclusion

The filter is designed to give smooth cutoff at both lower and upper ends. The insertion loss, return loss, VSWR and group delay can be further reduced by incorporating either PBG/EBG or DGS structures on the signal path of the filter [18]. These periodic structures offer a promising value of these parameters. Periodic structures generally exhibit pass band and stop band characteristics in various bands of wave number determined by the nature of the structure. This was originally studied in the case of waves in crystalline lattice structures, but the results are more general. The presence or absence of propagating wave can be controlled by these periodic structures. The filter could also be designed along with the Antenna and LNA for better performance. In such case the terminal impedances at both ends may be matched with the output impedance of the antenna and the input impedance of the LNA rather than 50 Ω characteristic impedance. This will allow the reduction of the group delay (Table 5).

Table 5 Filter performance parameters
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