Design of dual-band Wilkinson power divider based on novel stubs using PSO algorithm

Abstract

In this paper, a new dual-band Wilkinson power divider (WPD) is designed and fabricated using novel low and high impedance stubs instead of quarter-wavelength transmission lines. The proposed circuit was analyzed using odd and even mode analysis, and the optimal values of design parameters were obtained using the particle swarm optimization algorithm. The designed power divider has input reflection coefficients (S₁₁) of −22.1 and −17 dB at the first operating frequency of 2.2 GHz and the second operating frequency of 14.2 GHz, respectively. It also improves stop-band and fractional bandwidth (FBW) while maintaining a simple topology. The proposed WPD suppresses undesired harmonics from the second to the fifth with an attenuation level of less than −20 dB in the first band and generates a broad stop-band (4.4–11.5 GHz). In the first band, the FBW is 54.5%, and in the second band, it is 20.1%.

Introduction

Microwave circuits are now a common component of wireless systems, and three-port power dividers (PDs) are a common component of microwave circuits. Conventional PDs can be used in microwave circuits, but their operational bandwidth is often limited. As a result, attaining an appropriate bandwidth is one of their design problems [1, 2].

The Wilkinson power divider (WPD) divides the input power evenly between the two output ports, ideally without losses, and can also function as a power coupler. The characteristics of a suitable WPD are the matching of ports, small dimensions, wide bandwidth, and isolated output ports [3].

A band-pass WPD is provided in [4], where the quarter-wavelength (λ/4) transmission lines are substituted by coupling lines. However, this method increases the circuit size compared to the conventional WPD. A WPD with open-circuit stubs at each port is used in [5] to eliminate harmonics with a high attenuation level. But, due to the high-frequency noise effect, the measured S parameters for the suppressed fifth harmonic have a fuzzy shift. Two triangular-shaped resonators, two open-circuit stubs, and a trapezoidal-shaped suppressor with a 68% attenuation and small dimensions are used in [6] to provide an ultra-wide PD. This WPD has a small size but is single-band and has limited applications.

Various techniques, such as rectangular-shaped resonators [7], circular-shaped resonators [8, 9], open stubs [10, 11], taper transmission lines [12], and coupled lines [13], have been used to design a PD by attenuating undesired signals and small dimensions. These structures are inefficient and complex, despite their excellent features.

On the other hand, the Gysel structure is one way to design a PD. Despite their excellent isolation and high thermal conductivity, Gysel structures have large circuit dimensions that limit their applications and suffer from complex architectures [14–16].

A two-way WPD with equal power division was proposed in [17] and featured a wide cutoff band and low insertion loss. Two impedance-matching lines and a two-sided composite transmission line is used in the design of this circuit. The proposed dual-band PD has two central frequencies at 0.90 and 2.45 GHz, making it suitable for RF/microwave applications. In [18], a dual-band WPD is developed based on an isolation network with additional lumped elements. This structure has six transmission zeros but is large and has an unsuitable in-band input reflection coefficient. Another compact dual-band coplanar WPD is presented in [19] which modified coplanar waveguides separated by a series stub. It is used to achieve two adjustable frequency bands. However, this circuit does not have a wide stop band. In [20], a compact structure is presented, which includes a short stub, three transmission lines, and lumped components (RLC). This PD operates at frequencies of 1 and 2.5 GHz, and due to the RLC circuit, it has high output matching and perfect isolation. But it does not reject spurious harmonics and has a fluctuated passband.

A series-in-phase 3-way dual-band PD is proposed in [21]. This divider is intended and implemented to operate at 0.915 and 2.440 GHz. Its design employs the composite right- and left-handed transmission lines technique. It has a low input reflection coefficient and an equal power split to the output ports in two bands. However, its bandwidth is limited. Additionally, there is no harmonic suppression.

In recent years, various methods have been used to improve the performance of PDs. In [22], the particle swarm optimization (PSO) algorithm and finite difference time domain methods are combined to achieve an optimal PD design for a suitable specification. But the PD is single-band, and its insertion loss level is about −4 dB, so it is not very useful. In [23], four isolation resistors and four sections of microstrip lines with various characteristic impedances and lengths make up the proposed PD. All PD variables are calculated using a genetic algorithm. In [24], to design a WPD, a new neuro-based approach based on a feed-forward neural network is proposed. In the typical configuration, quarter-wave transmission lines are replaced by symmetrical modified T-shaped resonators in the proposed PD.

Although these designs are suitable in isolation and matching, they often have limited bandwidth. They are large, and it is still a great challenge for many designers and engineers to design and fabricate a small dual-band PD with broad bandwidth.

The primary goal of this paper is to present a dual-band PD with small dimensions and suitable bandwidth, which uses a new low and high impedance stub to realize it. The design has a simplified and symmetrical architecture that can be easily implemented in microstrip technology. At working frequencies of 2.2 and 14.2 GHz, the proposed PD has more than 17 dB input reflection coefficients on all ports. It also has small dimensions and high fractional bandwidth (FBW). The simulated and measured results are compared and found to be highly correlated. To our knowledge, a dual-band PD in the S and Ku bands has never been designed before, and for the first time using the PSO algorithm, a new high and low impedance stub has been considered for use in a dual-band PD. Therefore, it can be used in radars, satellites, and many wireless applications.

Design procedure

The following is a breakdown of the steps involved in designing a dual-band WPD with a compact size and high bandwidth:

1. (1) Design a modified circuit for WPD: The first step is to consider the new arrangement of transmission lines to attain a new WPD.

2. (2) Calculation of line impedances and electrical lengths: The second step is performed using even and odd mode analysis and PSO algorithm to accurately calculate the values.

3. (3) WPD implementation: This is done using microstrip technology because it is less expensive and faster.

4. (4) Design of the WPD layout: In the structure of the modified WPD, low and high impedance stubs and symmetrical rectangular-shaped resonators are used instead of transmission lines.

Figure 1 depicts the conventional WPD, including 1/4 wavelength transmission lines and isolation resistance. Transmission lines 1/4 of the wavelength have an impedance of 70.7 Ω, an electrical length of 90°, and an isolation resistance of 100 Ω. Furthermore, all ports are matched with a 50 ohm resistance [3].

Fig. 1. The conventional WPD [3].

The conventional design has drawbacks, such as low S ₁₁ and small bandwidth. It is a single band that ignores any extra harmonics.

In this paper, instead of the transmission lines of each branch of the conventional design, a modified circuit, shown in Fig. 2, is used to create a dual-band structure and the appropriate bandwidth.

Fig. 2. The proposed modified circuit.

Nine transmission lines are used in the modified circuit, and its equivalent impedance is the same as the original structure's impedance.

One of the methods of analyzing telecommunication circuits, especially the PD, is the even and odd mode analysis method. The even mode will be modeled as an open circuit, while the odd mode will be modeled as a short circuit, and this method will be used to calculate line impedances and electrical lengths.

Odd mode analysis

The voltage in the middle of the circuit is zero in the excitation of the odd mode. As a result, the circuit can be divided into two halves by connecting the middle plate to the ground.

The equivalent circuit of the suggested PD in odd mode is shown in Fig. 3. The input and output ports' impedance are indicated by Z ₀, and R is the isolation resistance between the output ports.

Fig. 3. The schematic of the equivalent odd-mode half circuit.

The following formulas for equivalent impedance under odd mode excitation can be derived using the high-frequency transmission line model:

(1)$$z_{o7} = jz_7\tan \theta _7$$

(2)$$z_{o6} = {-}jz_6\cot \theta _6$$

The characteristic impedance of the closed-end transmission line is calculated as equation (1), and to obtain the characteristic impedance of the open-end transmission line, its relation is as equation (2) [1].

(3)$$z_{o5} = z_5\displaystyle{{z_{o6} + jz_5\tan \theta _5} \over {z_5 + jz_{o6}\tan \theta _5}}$$

(4)$$z_{o4} = z_4\displaystyle{{z_{o5} + jz_4\tan \theta _4} \over {z_4 + jz_{o5}\tan \theta _4}}$$

Equation (5) shows the equivalent impedance of two parallel transmission lines.

(5)$$z_{o3^{\prime}} = \displaystyle{{z_{o4} \times z_{o7}} \over {z_{o4} + z_{o7}}}$$

(6)$$z_{o3} = z_3\displaystyle{{z_{o{3}^{\prime}} + jz_3\tan \theta _3} \over {z_3 + jz_{o3^{\prime}}\tan \theta _3}}$$

Based on the equations of the high-frequency transmission line, the equivalent impedances of each section are calculated from left to right.

(7)$$z_{o1} = {-}jz_1\cot \theta _1$$

(8)$$z_{o2} = z_2\displaystyle{{z_{o1} + jz_2\tan \theta _2} \over {z_2 + jz_{o1}\tan \theta _2}}$$

(9)$$z_{{o}^{\prime}} = \displaystyle{{Z_{o3} \times Z_{o2}} \over {Z_{o3} + Z_{o2}}}$$

(10)$$z_{eq-odd} = \displaystyle{{z_{{o}^{\prime}} \times \displaystyle{R \over 2}} \over {z_{{o}^{\prime}} + \displaystyle{R \over 2}}}$$

In equation (10), z _eq−odd is the equivalent impedance of the odd mode on the right side of Fig. 3. Therefore, the derived reflection coefficient ${\rm \Gamma }_{out}^{odd}$ at the output ports is expressed as follows:

(11)$${\rm \Gamma }_{out}^{odd} = \displaystyle{{z_{eq-odd}-Z_0} \over {z_{eq-odd} + Z_0}}$$

The matching impedance is:

(12)$$Z_0 = \displaystyle{R \over 2}$$

Even mode analysis

In the excitation of the even mode, no current flows through the isolation resistor. As shown in Fig. 4, the whole structure can be bisected in half symmetrically in the horizontal direction. In this mode, the input port is 100 ohms, and the isolation resistor does not affect transmission performance. Figure 4 depicts the proposed PD's equivalent circuit in the even mode.

Fig. 4. The schematic of the equivalent even-mode half circuit.

The method of calculating the equivalent impedance of the even mode is precisely the same as the odd mode and is based on the high-frequency transmission line equations. In the even mode analysis, the following equations represent the circuit equivalent impedance:

(13)$$z_{e9} = {-}jz_9\cot \theta _9$$

(14)$$z_{e8} = z_8\displaystyle{{z_{e9} + jz_8\tan \theta _8} \over {z_8 + jz_{e9}\tan \theta _8}}$$

(15)$$z_{e7^{\prime}} = \displaystyle{{z_{e8} \times 2z_o} \over {z_{e8} + 2z_o}}$$

(16)$$z_{e7} = z_7\displaystyle{{z_{e7^{\prime}} + jz_7\tan \theta _7} \over {z_7 + jz_{e7^{\prime}}\tan \theta _7}}$$

(17)$$z_{e6} = {-}jz_6\cot \theta _6$$

(18)$$z_{e5} = z_5\displaystyle{{z_{e6} + jz_5\tan \theta _5} \over {z_5 + jz_{e6}\tan \theta _5}}$$

(19)$$z_{e4} = z_4\displaystyle{{z_{e5} + jz_4\tan \theta _4} \over {z_4 + jz_{e5}\tan \theta _4}}$$

(20)$$z_{e3^{\prime}} = \displaystyle{{z_{e4} \times z_{e7}} \over {z_{e4} + z_{e7}}}$$

(21)$$z_{e3} = z_3\displaystyle{{z_{e3^{\prime}} + jz_3\tan \theta _3} \over {z_3 + jz_{e3^{\prime}}\tan \theta _3}}$$

(22)$$z_{e1} = {-}jz_1\cot \theta _1$$

(23)$$z_{e2} = z_2\displaystyle{{z_{e1} + jz_2\tan \theta _2} \over {z_2 + jz_{e1}\tan \theta _2}}$$

(24)$$z_{eq-even} = \displaystyle{{z_{e2} \times z_{e3}} \over {z_{e2} + z_{e3}}}$$

Based on equation (24), z _eq−even is the equivalent impedance of the even mode to the right side of Fig. 4. So, the derived reflection coefficient at the input port or the S ₁₁ parameter of the proposed PD can be calculated as follows:

(25)$$S_{11} = {\rm \Gamma }_{in}^{even} = \displaystyle{{z_{eq-even}-2z_o} \over {z_{eq-even} + 2z_o}}$$

Next, to have a suitable WPD, the isolation between the output ports of the circuit must be zero (S ₂₃ = 0). Furthermore, for simplicity of design, the PD is assumed to be symmetric, so:

(26)$$z_1 = z_9\;.\;\theta _1 = \theta _9$$

(27)$$z_2 = z_8\;.\;\theta _2 = \theta _8$$

(28)$$z_3 = z_7\;.\;\theta _3 = \theta _7$$

The traditional PD structure has a characteristic impedance of $\sqrt 2 z_o$ and an electrical length of 90°. On the other hand, isolation I and the return loss of RL can be obtained as [25]:

(29)$$I\;( {{\rm dB}} ) = {-}20.\log \vert {S_{23}} \vert $$

(30)$$RL( {{\rm dB}} ) = {-}20.\log \left\vert {\displaystyle{{P_{in}} \over {\,p_{ref}}}} \right\vert $$

Return loss measures the efficiency with which power is delivered from a transmission line to a load. In equation (30), the power in the PD under test is P _in, and the reflected power to the source is p _ref.

For the real and imaginary parts of the S ₁₁, a value close to zero (−0.17) is chosen to give a wide bandwidth. As a result, RL is greater than 17 dB, as expected.

Now, to have suitable matching and insertion losses at working frequencies (f ₁,f ₂), we must calculate the optimal values for the parameters z ₁₋₉ and θ ₁₋₉. But equations (11) and (25) are very complicated and the number of problem variables is large, so it is impossible to solve them manually.

Optimization and calculation of design parameters

Due to its effectiveness in designing microwave circuits, the PSO method was utilized to find the best design parameters [26–28]. The objective function for optimal design is obtained as equation (31).

(31)$$\eqalign{& F = \min \Bigg[{\mathop \sum \limits_{i = 1}^2 \displaystyle{{{\rm \Gamma }_{in( {11} ) }.\;f( i ) } \over n} + \mathop \sum \limits_{i = 1}^2 \displaystyle{{{\rm \Gamma }_{in( {23} ) }.f( i ) } \over n} + \mathop \sum \limits_{i = 1}^2 \displaystyle{{{\rm \Gamma }_{in( {22} ) }.\;f( i ) } \over n} }\cr & + \mathop \sum \limits_{i = 1}^2 \displaystyle{{{\rm \Gamma }_{in( {33} ) }.\;f( i ) } \over n} \Bigg] + {\rm max}\Bigg[{\mathop \sum \limits_{i = 1}^2 \displaystyle{{{\rm \Gamma }_{in( {12} ) }.f( i ) } \over n} + \mathop \sum \limits_{i = 1}^2 \displaystyle{{{\rm \Gamma }_{in( {13} ) }.f( i ) } \over n}} \Bigg]} $$

Since our goal is to design a dual-band WPD, equation (31) is written in a two-part form. The input/output return coefficient and isolation parameters should be minimized in this objective function at the operating frequencies of the WPD, and insertion losses should be at maximum values. Therefore, the results obtained by applying these conditions are the optimal parameters of the WPD.

Also, the presented objective function contains six goals for two frequencies simultaneously. The search space of this optimization is large, the range of variables is extensive, and it will be complicated to find the most optimal values; thus, calculating them individually is impossible.

PSO algorithm

The PSO optimization algorithm is one of the most significant algorithms in the swarm intelligence field [29]. Due to its simplicity and effectiveness, the PSO algorithm has recently become very popular in designing high-frequency circuits and electromagnetic systems [30, 31]. The technique is inspired by the social behaviors of living animals, such as fish and birds, which dwell in small and large groups. In this algorithm, all population members connect with each other and solve the problem by exchanging information. Each member of the population is called a particle, and these particles are spread throughout the search space of the function that is being optimized. The position of each particle is checked by calculating the objective function. Then, a direction to move is chosen by utilizing the information from its present location and the best position it has ever been in, as well as the information from one or more of the best particles in the collection. After all the particles have updated their position, one step of the algorithm ends. These steps are repeated several times until the desired answer is obtained. A collection of particles seeking the most optimal value of a function is like a group of birds looking for food.

This algorithm's foundation may be summed up as follows: each particle modifies its position in the search space at each instant by the best place it has experienced thus far and the best place among its neighbors. Like other evolutionary computations, the PSO method begins with generating a random initial population. The initial population consists of N particles that are randomly initialized. Each particle has two position and velocity values, represented by position and velocity vectors. By determining the value of the objective function, these particles begin to move in the problem space and look for better positions. To search, each particle requires two memories. The best location of each particle in the past is stored in one memory, and the best place of all particles is stored in one memory. The particles decide how to move in the following step using information from these memories. All particles adjust their velocity and location in each iteration based on the best absolute and local solutions [32]. According to equation (32), the location of each particle in the population is computed by adding the velocity of the same particle to its present position.

(32)$$X_k( i ) = X_k( {i-1} ) + V_k( i ) $$

In equation (32), X is the position of particle number k, and V represents its speed. The number of repetitions is also indicated by i. The speed parameter advanced the optimization process, representing the particle's experimental knowledge and social information exchange with its neighbors. Equation (33) is used to compute speed.

(33)$$\eqalign{V_k( i ) =& \theta ( i ).V_k( {i-1} ) + c_1r_1[ {P_{best, k}-X_k( {i-1} ) } ] \cr & + c_2r_2[ {G_{best}-X_k( {i-1} ) } ]} $$

V _k(i) is the i ^th component of the k-th particle velocity in equation (33). r ₁ and r ₂ are two random numbers with uniform distribution in the interval (0, 1).

The parameters c ₁ and c ₂ are individual and group learning factors, which usually choose c ₁ = c ₂ = 2 based on experimental results [32]. The best local place that the particle has attained so far is represented by P _best,k, and the best global place that all particles have achieved thus far is indicated by G _best.

According to equation (34), θ(i) or inertia weight is employed to control the speed of particles during test repetitions [33].

(34)$$\theta ( i ) = \theta _{\max }-\left({\displaystyle{{\theta_{\max }-\theta_{\min }} \over {i_{\max }}}} \right)i$$

θ _min or θ _max are the initial and final values of the inertia weight, respectively. The algorithm's maximum iteration number is indicated by i _max. It has been demonstrated via experimentation that for θ _min and θ _max, the optimal solution to the optimization issue will often be found if the values are θ _min = 0.4 and θ _max = 0.7 [34].

Optimization of design parameters

The PSO algorithm's specified parameters, which are listed in Table 1, have been used to optimize the design parameters for this article.

Table 1. Elective parameters of PSO for optimizing this article

Based on the relations and the objective function introduced in equation (31), the results of the PSO algorithm for electrical impedances and lengths are presented in Table 2.

Table 2. Calculated values for z and θ of the proposed WPD based on the PSO algorithm (units: z: Ω; θ: $^\circ$)

Implementation

One of the methods of implementing the proposed circuit is using microstrip lines in which the conductor is separated from the ground by a dielectric layer called the substrate. This technology is used to transmit RF frequency signals. So, in this article, instead of transmission lines, two high and low impedance stubs and symmetrical resonators have been used to improve and upgrade the PD compared to the conventional type.

Design of high and low impedance stub

Instead of wavelength 1/4 transmission lines in the conventional WPD, the proposed structure uses a combination of two symmetrically compressed microstrip resonators and a novel high and low impedance stub. Figure 5 shows the high and low impedance stubs that have replaced the impedances z ₄, z ₅, and z ₆. A high and low impedance stub is employed to produce frequency responses with a broad stop-band bandwidth.

Fig. 5. The proposed high and low impedance stub.

In the following, the dimensions of the presented stub are changed, and their effect on the simulation results is investigated.

Figures 6(a) and 6(b) depict insertion loss (S ₁₂) for various H2/H1 and L2/L1 values, respectively.

Fig. 6. The simulated S ₁₂ of the presented stub as a function of (a) H2/H1, (b) L2/L1.

The proposed stub produces two transmission zeros in the S ₁₂ result, as shown in Figs 6(a) and 6(b), and changing the stub dimensions shifts the location of the transmission zeros. Because of this change, the bigger the high impedance stub, the smaller the resonant frequency, and vice versa. Therefore, the designed structure has high flexibility, and it is easy to control the transmission zeros created by it.

Since this stub alone cannot create broad or multiple rejection bands, as shown in Fig. 2, other transmission lines must be added next to it.

Design of symmetrical resonators

Figure 7 shows results that illustrate how symmetrical resonators can produce a dual-band circuit by filtering out undesired signals. Compact microstrip cells are employed as the resonators, and in place of impedances, z ₁₋₂ and z ₈₋₉ are located. These resonators are rectangular-shaped and have inductive properties, resulting in transmission zeros in the frequency response and improved circuit performance.

Fig. 7. The layout of rectangular-shaped resonators.

The transmission zero created by the resonators is shifted by changing the length and width of the resonator bases, and the transmission zero reduces as the length of the resonator bases increases. Figures 8(a) and 8(b) show the resonator simulation results. As can be seen, when W1/W2 = 1.25 and L4 = 3.7 mm, the optimal conditions for these resonators are obtained, and a wide stop-band bandwidth is created in the S ₁₂ result.

Fig. 8. The simulated S ₁₂ of the presented rectangular-shaped resonator as a function of (a) W1/W2, (b) L4.

Based on the simulation results, a dual-band circuit can be created by combining the symmetrical resonators with the high and low impedance stub. The resonators produce the first pass band, and the stub produces the second pass band. A broader stop-band is also produced by transmission zeros

Final structure

The final PD layout is depicted in Fig. 9. This structure combines high and low impedance stubs and rectangular-shaped resonators compactly positioned to prevent dimension increase. It will replace transmission lines in the modified WPD structure. The optimal dimensions for the high and low impedance stub to create a dual-band WPD with central frequencies of 2.2 and 14.2 GHz are as follows (all in mm):

$$L_1 = 5.3, \;\,L_2 = 5.8, \;\,L_3 = 6.8, \;\,H_1 = 0.6, \;\,{\rm and}\,H_2 = 0.7$$

Fig. 9. The proposed power divider's layout.

The substrate intended for the layout design is RT Duroid 5880 with a thickness of 20 mil, ε_r = 2.2, and the dielectric loss tangent is 0.0009. The fabricated circuit is shown in Fig. 10, and its total size is 7.4 mm × 8.1 mm (0.072 λ _g × 0.079 λ _g).

Fig. 10. Photograph of the fabricated dual-band WPD.

Figures 11(a) and 11(b) show a comparison of circuit measurement and electromagnetic (EM) simulation. The simulation results were obtained using ADS software, and the measurement was performed with Agilent network analyzer N5230A.

Fig. 11. Simulated and measurement results of designed WPD (a) S ₁₁ and S ₁₂, (b) S ₂₂ and S ₃₂.

The central frequencies of the two wide passbands in the proposed circuit are 2.2 and 14.2 GHz. The S ₁₁ is −22.1 and −17 dB in these operating frequencies, respectively. As is observed, for the proposed WPD, an appropriate reject band is obtained, 4.4–11.5 GHz, which can suppress 2nd up to 5th unwanted harmonics (−25, −28, −27, −33, and −18 dB). In addition, insertion losses (S ₂₁) are −3.07 and −3.1 dB in the operating frequencies. Isolation (S ₂₃) and reflection coefficient in output ports (S ₂₂ or S ₃₃) simulation results are shown in Fig. 11(b). In the operating frequencies, isolation between output ports is better than 20 dB, and the reflection coefficient of the output port is more than 17 dB. The FBW obtained by considering −15 dB as the attenuation level is 54.5% for the first band and 20.1% for the second band. So, there is a wide stop-band and good sharpness in the proposed WPD output results.

In the second working frequency, there are more losses due to the change in the loss tangent of the substrate. Moreover, the simulated and measured results are highly consistent, and their minor differences are due to measurement errors.

One of the significant advantages of the designed WPD is the high distance between the operating frequency bands, which makes it possible to use this circuit in various applications.

The phase and amplitude differences of the two isolated ports are shown in Figs 12(a) and 12(b), respectively. The phase difference around the operating frequency is less than 2 degrees, and the amplitude difference between output ports is less than 0.1 dB, indicating that the proposed circuit splits power evenly between the output ports and is entirely symmetrical.

Fig. 12. Simulated and measured (a) phase difference and (b) amplitude difference between output ports.

In the final fabricated circuit, the two output 50 ohm transmission lines have different lengths from the input 50 ohm transmission line. So, to check the performance of the microstrip lines and their effects, the phase difference between the input and output ports around the central frequency is shown in Fig. 13.

Fig. 13. Simulation results of phase difference between input and output ports.

The current distribution profile of the proposed WPD at 2.2 and 10 GHz is shown in Figs 14(a) and 14(b), respectively, which depict how resonators cause resonance at desired frequencies. According to Fig. 14(a), there is a high electrical current distribution at the output ports that is evenly led between them. But at 10 GHz, electromagnetic waves enter the PD from port 1 and are blocked by resonators when passing through transmission lines. These simulations also demonstrate that the designed PD is symmetric and can be well used at operating frequencies.

Fig. 14. The current distribution profile of the proposed WPD at (a) 2.2 GHz and (b) 10 GHz.

Table 3 represents a comparison between the proposed WPD with the previous works. It confirms that the proposed WPD is one of the best structures presented to date because it has the smallest dimensions, suppresses five undesired harmonics, and has a high FBW. It is pertinent to note that our design has the best insertion loss among other recent state-of-the-art works.

Table 3. Comparison with previous studies

^a Single-band.

The proposed WPD operates at 2.2 and 14.2 GHz frequencies and covers two frequency bands. As a result, it applies to all S-band and Ku-band devices; for example, medical sensors, smartphone processors, radars for airport security, weather radars, surface ship radars, communications satellites, etc.

Conclusion

This paper proposed a dual-band WPD with 2.2 and 14.2 GHz central frequencies. This PD comprises two similarly rectangular-shaped resonators and new high and low impedance stubs, which minimizes the size while simultaneously creating a wide cut-off bandwidth. In addition, the proposed circuit's characteristic impedances and electrical lengths were analyzed using odd and even mode analysis, and the optimal values were obtained using the PSO algorithm. This proposed PD can divide signals into various microwave amplifiers, antenna arrays, and mixers.