1 Introduction

Microwave and radio frequency (RF) technology is more widespread than ever before. These are used in many modern applications such as the commercial sector, including cellular phones, smartphones, wireless networking via 3G and WiFi, millimeter wave collision sensors, direct broadcast satellites for radio, television, and networking, global positioning systems, radio frequency ID tagging, ultra-wideband radios, and radars, as well as microwave environmental remote sensing systems [1,2,3,4,5,6,7]. It is expected that there will be no shortage of challenging problems in radio frequency and microwave engineering in the near future, and engineers need both the understanding of microwave engineering fundamentals and the creativity to apply that knowledge to practical problems [8]. Microwave engineers today use computer-aided design (CAD) software and network analyzers as essential tools to face these challenges.

One of the fundamental building elements of communication networks is filters. Microwave filters have diverse practical applications across various fields, including telecommunications, aerospace, medical, and defense [8, 9]. These filters are integral components in 5G and beyond wireless communication systems for frequency selection, interference suppression, and signal conditioning [10, 11]. In modern radar applications, such as automotive radar and unmanned aerial vehicle (UAV) sensing, there is a growing demand for compact, high-performance filters [12, 13]. Any microwave system must have bandpass filters (BPFs) because they allow the signals in the targeted band to pass through while blocking out signals from external systems that could interfere with and reduce the system's performance as well as signals generated by the system's active components (such as harmonics and intermodulation products) [14,15,16,17]. Additionally, they lower the system's noise, improving the system's dynamic range and performance. BPFs can be used in various technologies, such as coaxial, and planar technology, rectangular and circular waveguides, and the more current waveguide such as substrate integrated waveguide (SIW) [18]. Microwave microstrip bandpass filters find numerous practical applications across various domains due to their ability to selectively pass a certain range of frequencies while attenuating others. These filters are extensively used in wireless communication systems such as cellular networks, Wi-Fi routers, and satellite communication terminals to isolate desired frequency bands and reject out-of-band interference [1, 19, 20]. The rapid advancement of broadband wireless communication systems has sparked a lot of interest in the design of wide and ultra-wide bandpass filters over the past several years [21,22,23,24,25,26,27,28,29].

In Ishida and Araki [23], a very wide-band BPF is designed whose bandwidth was formed by adding zeroes in the sections of the transmission line (TL). It has an extremely narrow stopband and only has notches in its frequency response at the band edges. In Wang et al. [24], the microstrip-and-CPW BPF was presented for broadband applications. This design was based on the multi-mode resonator (MMR) in the form of multiples of a quarter wavelength to extend the rejection region and broaden the bandwidth. In Sheikhi et al. [25], researchers introduce a single-ended BPF with open stubs and coupled lines, which has a fractional bandwidth (FBW) of 97% and a narrow stopband region. SIRs (stepped impedance resonators) were used to achieve ultra-wideband BPFs. The coupled lines of a quarter wavelength are utilized as the inverter and the MMR of the half wavelength is also a concept in Sun and Zhu [26]. This work shows how increased coupling has caused the lower and higher stopbands to expand. In Dong et al. [27], a wideband differential BPF is designed by loading two capacitor pairs along a half-wavelength-coupled microstrip line at different positions. The capacitances and loaded positions of capacitors are combined with the even–odd-mode impedance of the coupled microstrip line. This design is compact and has low insertion loss when compared to state-of-the-art wideband differential BPFs. Full-wave simulations are used here. In Menzel et al. [28], researchers combined low- and high-pass filters as suspended stripline structures with various planes to create a composite wide-band filter. [29] presented composite microstrip filters for high-speed communication applications in which seven or eight TL sections of approximately a quarter wavelength are successively connected.

The demand for these types of research to design and optimization of microwave components is required. One must often work with Maxwell’s equations and their solutions in RF and microwave engineering [1, 8], which is a traditional approach to solving microwave problems. In terms of device design, there are a number of traditional methods (electromagnetic theory-based) that can be applied, but they are not very straightforward. To use any of these methods, a designer must have a solid understanding of microwave technology, electromagnetic theory, antenna theory, wave propagation, etc. In light of this, the design of microwave devices is considered to be a challenging task due to various factors such as complexity of optimization [8, 30], miniaturization and size constraints [31,32,33], manufacturability and fabrication challenges [1, 8, 34]. Since conventional techniques are hindered by all of these difficulties, unconventional techniques must be explored. So, the need for soft computing techniques arises. Soft-computing techniques provide numerous advantages in engineering optimization because of its flexibility, robustness, large search capability and ease of implementation and integration [35,36,37,38,39]. Genetic algorithm (GA)-based optimization has become popular for building antennas and microwave filters that are more efficient and perform better [40, 41]. GA has been used in the design of antennas [40,41,42,43,44] and microwave circuits [45,46,47,48] for the optimization of high-frequency circuit components. A way to specify the topology and physical or geometrical parameters of a circuit made up of microstrip-line segments was presented in Nishino and Itoh [47], and this method was subsequently combined with conventional GA. In Sanada et al. [49], it is explained how to utilize GA to build transmission line filters with matching circuits that feature nonuniform transmission lines that constantly change and shunt-connected open circuit stubs. For the manufacture of microwave filters applying a certain circuit architecture, researchers in refs. [50,51,52] focused on obtaining the most suitable electrical parameters.

In this paper, we design microstrip filters using GAs with transmission line models (TLs). The genetic algorithm (GA) and transmission line model (TLM) are well-suited for microstrip filter design due to their complementary strengths [53, 54]. The GA offers a robust optimization framework capable of exploring the complex and nonlinear solution space inherent in microstrip filter design, facilitating both global exploration of diverse solutions and exploitation of promising regions [42, 48, 50]. Meanwhile, the TLM accurately represents the physical behavior of microstrip structures, providing fast and efficient simulations that yield insights into the impact of design parameters on filter performance [1, 8, 55]. By integrating the GA’s optimization capabilities with the TLM’s accurate modeling, designers can efficiently navigate the design space, achieving high-performance microstrip filters tailored to specific application requirements. In this approach first fixed topology of the filter is considered to reduce the search space. After that, GA operators are applied to obtain possible feasible solutions. The transmission line model is used to compute fitness value in GA in order to make the proposed procedure fast and effective. Each individual in the fitness function is assigned a fitness score according to its closeness to the desired response. This methodology is demonstrated by optimizing a second-order wideband microstrip bandpass filter centered at 2.3 GHz and validating the optimized design using full-wave methods (MoM and EM simulator CST). This paper is structured as follows, Sect. 2 describes the design problem of the second-order microstrip bandpass filter, Sect. 3 gives a detailed explanation about the methodology used in the proposed work, Sect. 4 presents the results and discussion, which is followed by Sect. 5 conclusion.

2 Problem design

The paper proposes a more straightforward method for creating a single bandpass filter. In Fig. 1, the wideband second-order bandpass filter with center frequency of 2.3 GHz is examined with a pair of short-circuited stubs and a pair of open-circuited stubs.

Fig. 1
figure 1

Second-order bandpass filter

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Zα and Zβ stand for the stubs and the transmission line's respective characteristic impedances of the second-order bandpass filter. The electrical length of two short-circuited stubs in a second-order bandpass filter is θα and that of open-circuited stubs is θγ. The combined transmission line can be equivalent to a parallel resonant circuit when the combined electrical length of the open-circuited stub, the short-circuited stub, and the connecting line between them is a quarter wavelength, or θα + θβ + θγ = 90° in Fig. 1.

Additionally, the connecting line can be thought of as an admittance inverter when the electrical length of the connecting line between the open-circuited stubs is a quarter wavelength, i.e., θδ = 90°. The quarter wavelength resonator is fed using a tapped-line coupling arrangement, as seen in Fig. 1. The placement of the feeding point affects the coupling strength. The equivalent circuit of the second-order bandpass filter is illustrated in Fig. 2.

Fig. 2
figure 2

Equivalent circuit of the second-order bandpass filter

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In accordance with the information presented in Hong and Lancaster [8], the connection that exists between the values of the lumped elements, the values of the admittance inverters, and the characteristic impedance of the stubs and connecting line can be described as follows:

$${C}_{1}= \frac{\pi }{4 {\omega }_{0}{Z}_{\alpha }} , {L}_{1}= \frac{1}{{\omega }_{0}^{2} {C}_{1}}$$
(1)
$${J}_{0, 1}= \sqrt{\frac{\pi\, FBW}{{4\text{ g}}_{0}{\text{g}}_{1}{Z}_{0}{Z}_{\alpha }}}$$
(2)
$${J}_{\text{1,2}}= \frac{\pi\, FBW}{4{ Z}_{\alpha }\sqrt{{\text{g}}_{0}{\text{g}}_{1}}}= \frac{1}{{Z}_{\beta }}$$
(3)
$${J}_{2, 3}= \sqrt{\frac{\pi\, FBW}{{4\text{ g}}_{2}{\text{g}}_{3}{Z}_{0}{Z}_{\alpha }}}$$
(4)

where gn (n = 0, 1, 2, 3) represents the element value of the lowpass filter prototype of the second order and Z0 represents the impedance at the terminal. The notation ω0 and FBW is used to refer to the bandpass filter’s center angular frequency and its fractional bandwidth, respectively. The admittance inverters are analogous to the coupling element that is used between the resonators, and the values of the admittance inverters may be derived from Eqs. (2), (3) and (4) using the given information.

3 Genetic algorithm

Chromosomes are parameters that GA uses to represent optimization variables. Initially, microstrip circuits are represented as data structures. Each data structure in the set provides a simple two-port network together with the appropriate connecting technique and electrical specifications. By using a fixed topology for the filter, the search space (variables) can be drastically reduced in comparison to using a genetic algorithm. An arbitrary two-port microstrip circuit [1, 56] can be broken down into its fundamental circuit elements, as shown in Fig. 3. The description of such a fundamental circuit element is then applied using the data structure depicted in Fig. 3. There are two parts to it. The first component represents the number of fundamental elements in the filter topology. The second section is made up of numerous real values that describe the electrical parameters of elements. Table 1 shows the details of the basic circuit elements [1].

Fig. 3
figure 3

Representation scheme in the genetic algorithm: a decomposition of the circuit of second-order bandpass filter into basic circuit elements, b chromosome of the circuit in (a)

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Table 1 Basic elements in the modified genetic algorithm
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A chromosome can be viewed as a set of fundamental circuit components in the GA [57, 58]. Chromosomes are a set of parameters that GA defines to represent the optimization variables. Thus, a chromosome can serve as a representation for any two-port circuit. It is a set of data consisting of the number of elements, electrical lengths, and widths. As shown in Fig. 3b, chromosome contains electrical lengths (Z1, Z2, Z3, Z4, Z5, Z6, and Z7) and widths (θ1, θ2, θ3, θ4, θ5, θ6, and θ7) corresponding to the seven elements of the circuit topology. In this work, a second-order microstrip bandpass filter with better design efficiency is designed using genetic algorithm optimization.

In order to achieve more effectiveness, a GA is used in this study to build a second-order bandpass microstrip filter based on loaded open-/short-circuited stubs. The filter design is implemented on a substrate with a thickness h of 0.8 mm, a loss tangent δ = 0.0029, and a dielectric constant εr = 4.3.

The following steps are taken to finalize the filter design. To begin with, the fixed topology of the filter base design is used in order to allow the genetic algorithm to generate the initial random population of electrical parameters (Z0i, θ0i) by simply altering the electrical lengths and widths of the stubs. Here the filter parameters corresponding to the given substrate with thickness h and dielectric constant εr for the required resonant frequency fr with the specified mode numbers n and m must be found. In the proposed method, the specified parameters for the design, i.e., substrate thickness h, dielectric constant εr, required resonant frequency fr, and the mode numbers n and m, are the inputs to generate the filter’s electrical parameters such as Zα, Zβ, θβ, and θγ for second order bandpass filter. In the genetic algorithm, the initial random population (potential population set) is generated for the filter’s electrical parameters. The fitness function is computed using the response of the potential solution, which is acquired by the transmission line theory model to make this process time-efficient. The ABCD parameters of the transmission line model are transformed into S-parameters. The computed S-parameters are compared to the desired response or S-parameters in the fitness function. The solution that has a greater fitness value is more likely to produce the desired results or responses. The final electrical length (θ0final) and impedance (Z0final) are then transformed into the physical length (Li) and width (Wi), respectively. Finally, the full-wave technique (MoM and EM simulator CST) is used to validate the genetic algorithm's optimal solution. After validation, and finalizing the filter design.

3.1 Initialization

Initialization is a crucial step in a genetic algorithm (GA) as it sets the initial population from which the algorithm will evolve solutions. The goal of initialization is to create a diverse set of potential solutions that can be further improved through the evolutionary process. Here, every chromosome is randomly made to initialize within the specific electrical parameters range of every basic element for better convergence. The upper and lower limits must also be assigned based on various engineering specifications and designs. To maintain the tolerance of the fabrication, the minimum width is limited to 0.1 mm, corresponding to a microstrip line with a characteristic impedance of 148 Ω. A maximum line width of 2 mm is chosen to reduce the junction discontinuity effects, which corresponds to a microstrip line with a characteristic impedance of 42 Ω.

3.2 Fitness evaluation

Genetic algorithms (GAs) depend heavily on fitness evaluation to determine how well each individual solves a given problem. In the fitness function, each individual is assigned a numerical value (fitness score) based on its closeness to the optimal solution. Individuals with higher fitness scores are more likely to be selected for reproduction in the next generation. The fitness function examines the frequency response of a chromosome through the calculation of scattering parameters (S-Parameters) using transmission-line models. The method will increase the efficiency of algorithms and the speed of calculation of procedures. Figure 4 shows the desired response of the second-order bandpass filter, and the fitness value is defined in Eq. (5).

$$F= \sum_{i=1}^{N}{w}_{i}{f}_{i}= \sum_{i=1}^{N}{w}_{i}{\left({S}_{12Di}- {S}_{12TLMi}\right)}^{2}$$
(5)

where wi represents the weighting value at the ith sampling point, fi represents the square deviation between the calculated S-parameter (S12TLMi) and the desired S-parameter (S12Di) at the ith sampling point, and N is the number of sampling points. Modified genetic algorithms terminate when the evolution count reaches the maximum or one of the fitness values of a chromosome in a solution pool reaches its target fitness value. Here, the target fitness value is specified as zero.

Fig. 4
figure 4

Desired response of the second-order BPF centered at 2.3 GHz

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3.3 Genetic operators

GAs use three primary genetic operators to simulate the process of natural selection and genetic recombination: selection, crossover, and mutation.

To produce offspring for the next generation, the selection operator chooses individuals from the current population as parents. Selection methods include roulette wheel selection, tournament selection, and rank-based selection. The objective is to converge the selection toward individuals with higher fitness scores. Tournament selection chooses each parent by choosing size players at random and then choosing the best individual out of that set to be a parent. size must be at least 2.

The crossover operator simulates the recombination of genetic material from two parent individuals to create one or more offspring. This helps propagate favorable traits from parents to offspring. The speed and ability of finding the optimal solution are determined by it. Here, crossover scattered is used as a crossover function. This creates a random binary vector and selects the genes where the vector is a 1 from the first parent, and the genes where the vector is a 0 from the second parent, and combines the genes to form the child as shown in Fig. 5.

Fig. 5
figure 5

Schematic of the scattered crossover operator in the modified GA

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In the mutation operator, small random changes are introduced into individuals, which maintains genetic diversity in the population. This mutation prevents the algorithm from getting stuck in local optima and allows it to explore the solution space. Mutation can be applied to a single gene or multiple genes in an individual. Here, Gaussian mutation is used as a mutation function that adds a random value drawn from a Gaussian (normal) distribution. This is often used when a more controlled and continuous change is desired.

4 Results and discussion

In this section, the results obtained from the proposed methodology to optimized the filter design of a second-order bandpass microstrip filter with center frequency at 2.3 GHz have been discussed and compared with conventional full-wave methods to prove the goodness of the method. The wideband second-order bandpass filter with a pair of short-circuited stubs and a pair of open-circuited stubs is examined. The chromosome for the design has three transmission lines, two short-circuited stubs, and two open-circuited stubs in it. Table 2 displays the parameter settings for the second-order bandpass filter.

Table 2 Parameters settings for the second-order bandpass filter
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Figure 6 shows the simulated S-parameters of the second-order bandpass filter centered at 2.3 GHz obtained after optimizing using a genetic algorithm. The − 3 dB cutoff frequencies are at 1.8 GHz and 2.8 GHz respectively. The simulated results indicate that the fractional bandwidth of the second-order bandpass filter is 43.48% at 2.3 GHz. In the passband, the return loss is better than − 15 dB and the insertion loss is less than − 0.1 dB. This design of a second-order bandpass filter is also verified using MoM in Matlab and CST. The electrical and physical parameters of the second-order bandpass filter are listed in Table 3. The performance of the proposed method is optimized in 98 generations as shown in Fig. 7. The normalized error in each generation is shown in Fig. 7.

Fig. 6
figure 6

S-parameter of the second-order bandpass filter optimized by genetic algorithm

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Table 3 The electrical and physical parameters of the second-order bandpass filter
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Fig. 7
figure 7

Normalized value of fitness function in each generation of second-order bandpass filter

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The physical layout of the second-order bandpass filter is depicted in Fig. 8. The connecting wire and longer open-circuited stubs are meandering for miniaturization. A circle with a diameter of D represents the via hole, which joins the ground plane to the microstrip line. The filter has the following dimensions: W1 = 1.56 mm, W2 = 1.007 mm, W3 = 0.36 mm, and D = 0.25 mm. L1 = 7.55 mm, L2 = 2.67 mm, L3 = 4.42 mm, L4 = 7.446 mm, L5 = 3.214 mm, L6 = 4.54 mm, and L7 = 7.542 mm. Its overall dimensions are 37.46 mm × 15 mm, or 0.52 λg × 0.20 λg, where λg is the guide wavelength of a 50Ω microstrip line at 2.3 GHz, or the passband's central frequency.

Fig. 8
figure 8

Physical layout of the second-order bandpass filter

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In Fig. 9, the simulated responses of the second-order bandpass filter using MoM are shown. The pass band's respective − 3 dB cutoff frequencies using MoM are 2.09 GHz and 3.43 GHz. The filter’s fractional bandwidth is 48.55% at 2.76 GHz according to the measured data. After scaling the physical parameters of the second-order bandpass filter, the − 3 dB cutoff frequencies were obtained at 1.76 GHz and 2.84 GHz respectively with fractional bandwidth of 46.95% at 2.3 GHz, also shown in Fig. 9. The results show a low insertion loss of 0.1 dB better return loss than 30 dB. In microstrip filters, adjusting the impedance of the transmission line directly influences the coupling between the input and output ports, as well as the resonant frequencies of the filter's components. Thus, by scaling the transmission line's impedance, designers can effectively tune the bandwidth of the bandpass filter to suit specific application requirements [1, 8, 55]. The frequency response of the designed second-order bandpass filter is also verified using CST and its frequency response in comparison with the MoM method is shown in Fig. 10. In CST, the − 3 dB cutoff frequencies of the second-order bandpass filter are 1.6 GHz and 2.7 GHz. The centered frequency is 2.15 GHz with a fractional bandwidth of 51.16%. The results from Matlab and CST software both show a slight change in the resonant frequency. This small difference is acceptable and expected because each software uses a different way to calculate things [59]. Matlab uses the method of moments (MoM), while CST uses the finite element approach. The important aspect is that both resonant frequencies fall within the same frequency band. The results obtained using the genetic algorithm, CST and the MoM simulations are in good agreement.

Fig. 9
figure 9

Comparison of simulated S-parameters of the second-order bandpass filter obtained using MoM before and after scaling

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Fig. 10
figure 10

Comparison of simulated S-parameters of the second-order bandpass filter obtained using CST with MoM

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Finally, the results were compared to those from relevant recently published research, and the findings are summarized in Table 4. In comparison to the existing state of the art, the suggested wideband band-pass filter is compact with a broader − 3 dB fractional bandwidth and improved rejection level.

Table 4 Performance comparison between the proposed filter and several other BPFs
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5 Conclusion

A fast and efficient method of developing a second-order wideband loaded open-/short-circuited stubs bandpass filter is proposed. Here we have successfully demonstrated the application of modified GA with the use of the transmission-line model for efficient computation of microstrip structure. The s-parameters of the second-order bandpass filter have been confirmed by comparing the results of the modified GA, MoM method, and EM simulator (CST), which are all in good agreement. The second-order bandpass filter is small, has wide bandwidths, and operates at 2.3 GHz. It has a passband’s return loss better than − 20 dB and an insertion loss smaller than − 0.1 dB. According to the genetic algorithm’s measured data, the filter's fractional bandwidth is 43.48% at 2.3 GHz. The filter's fractional bandwidth, as determined using the full-wave technique simulations (MoM), is 46.95% at 2.3 GHz. The difference in approaches is what caused the small shift in the FBW. The results from the full-wave technique are accurate, while the genetic algorithm model is based on approximation. The design filter is suitable for the wireless communication system, such as WLAN. This proposed approach is not limited to filters. It can be applied to other microwave devices.