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) [Reference Pozar1].

(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 [Reference Bird25]:

(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.

PSO algorithm

The PSO optimization algorithm is one of the most significant algorithms in the swarm intelligence field [Reference Kennedy and Eberhart29]. Due to its simplicity and effectiveness, the PSO algorithm has recently become very popular in designing high-frequency circuits and electromagnetic systems [Reference Luo, Yang and Qian30, Reference Verma and Srivastava31]. 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 [Reference Shi and Eberhart32]. 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 [Reference Shi and Eberhart32]. 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 [Reference Shi33].

(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 [Reference Eberhart and Shi34].

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: $^\circ$)

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