## 4 Design and Measurement  
To demonstrate the proposed design method, two Wilkinson power dividers were designed at the resonant frequencies of 2 and 3 GHz by following the algorithm below, which is illustrated in the flowchart of Figure 5(a). 1. Draw the proposed WPD circuit and label its essential parameters. 2. Apply circuit analysis and conjugate matching theory to extract the required parameters. 3. Utilize Microsoft Excel or MATLAB to compose a spreadsheet program or script, in order to calculate the unknown parameters, i.e., \(\theta_{1}\), \(\theta_{2}\) and isolation resistance R.
4. Perform circuit simulation and three-dimensional field analysis using ANSYS Electronics Desktop. Figure 6: Design flow of the proposed circuit. * Insert the DGS section into the ground plane utilizing the physical dimensions \(a,b\) and \(g\), which are acquired by following the procedure illustrated in Figure 6b, and perform EM simulation. * If the achieved results are desirable, then fabricate the proposed circuit to check the validity of the circuit. As the proposed circuit should be viable and compact for fabrication and measurement in order to achieve preferable results, a compromise was made between the availability of the substrate and the compactness of the device as well as the constraints of the fabrication facility. Hence, the impedances of both pairs of transmission lines of the divider, i.e., \(Z_{1}\) and \(Z_{2}\), are fixed at \(60\,\Omega\) to obtain improved results with the best substrate at hand. Thus, by following the design method given in Figure 6a, the electrical lengths of both transmission lines with impedances \(Z_{1}\) and \(Z_{2}\) are calculated. Other parameters are obtained by using ANSYS HFSS via circuit and EM Simulation. The proposed circuit is simulated with different substrates (available) to examine the performances of substrates and choose the best accessible, available and reliable one which can accommodate the preferred outcome of the proposed WPD. Figure 7 shows that all substrates under test have the capacity to achieve good results with minor variations in return loss and isolation, as shown in Table 1. However, according to the data shown in Table 1, Taconic substrate is well suited for the proposed WPD as it yields better isolation, i.e., \(S_{32}=-55.84\) dB, and return loss, i.e., \(S_{11}=-55.02\) dB, as compared to other substrates. Furthermore, in the case of FR4 and Rogers, due to low build quality, noise creates signal integrity problems during measurement. Hence, these factors can affect the efficiency and overall performance of the WPD as compared to WPD powered with Taconic substrate. Therefore, the proposed circuits were simulated and fabricated on a Taconic substrate with a dielectric constant of \(\varepsilon_{r}=2.97\), loss tangent of \(\tan\delta=0.0012\) and thickness of H = 0.76 mm. Table 2 shows the calculated optimized values and dimensions of both WPDs. Therefore, Equations (9), (11) and (14) were plugged into Microsoft Excel or MATLAB by fixing \(Y_{1}\) and \(Y_{2}\), while changing the value of R from 1 to nR, i.e., 1 to 114 \(\Omega\). Hence, applying this method, the solution of Equation (14) approached zero when R approached 36 \(\Omega\). The physical dimensions of the DGS cell, i.e., \(a\), \(b\) and \(g\), were achieved by following the steps illustrated in Figure 6b:  
1. Guess the dimensions of the DGS section (trial-and-error method) by taking an approximation from its calculated lumped elements i.e., capacitance and inductance given in Equations (19) and (20). 2. Select dielectric material and thickness. 3. 3D simulate the DGS section separately by taking the TL2 line and fabricate the DGS section below the TL2 transmission line. 4. Perform full-wave analysis and extract the S-parameters versus frequency, as shown in Figure 8.
5. Change the dimensions of the DGS iteratively, until the frequency response meets the required acceptable result. 6.