**4.2 Design-2 [SOMI using one open stub, one short stub and three serial lines]**

The design-2 SOMI configuration is illustrated in **Figure 5**. The optimized characteristic impedances of design-2 SOMI obtained by PSO, CSA, and GSA. The algorithm steps have discussed in Section 3 by which these characteristic impedances are attained. **Table 8** offers the optimized characteristic impedances of the line elements of Design-2 SOMI. For the frequency response analysis of the design-2 SOMI, the same parameters are selected as the design-1 SOMI as shown in **Figure 6 (a)**–**(f)**. In **Table 9**, the total magnitude and phase errors are listed. **Table 10** and **Table 11** summarize the statistical analysis and the qualitatively analyzed data of different characteristics of magnitude error of the design-2 SOMI. It is observed that GSA-based results have the least values in all aspects as compared to the PSO and CSA. The pole-zero plot of the Design-2 SOMI for stability analysis in which all the poles and zeros lie inside the unit circle is shown in **Figure 6(d)**. This plot asserts that SOMI Design-2 is stable. The convergence profile of all three algorithms in which GSA is clearly demonstrated to be faster than PSO and CSA with less execution time is shown in **Figure 6(e)**, which is also summarized in **Table 12**. The percentage improvement comparison in magnitude error for the design-2 SOMI as CSA to PSO, GSA to PSO, and GSA to CSA and the percentage improvement comparison in process error, summarized in **Table 13**, are shown in **Figure 6(f)**.

#### **Figure 5.**

*Configuration of the design-2 SOMI.*


#### **Table 8.**

*Optimized characteristic impedances of the design-2 SOMI.*

### **Figure 6.**

*Profile of the design-2 SOMI (a) Magnitude response, (b) Magnitude error response, (c) Phase response, (d) Pole-zero plot, (e) Convergence profile, and (f) Improvement bar graph.*

**Figure 7** depicts the comparison of the designed SOMIs with the existing second

comparative magnitude response and AME of the designed SOMI GSA-based optimization with an existing integrator. The corresponding qualitative and statistical magnitude error analysis of the existing SOMIs and the designed integrator is shown in **Table 14**. **Figure 7** and **Table 14** confirm that GSA-based designed SOMIs are more suitable in terms of magnitude, minimal magnitude error, and wide

**Algorithm Mean Variance Standard deviation** PSO 0.0324 <sup>5</sup>*:*<sup>6118</sup> � <sup>10</sup>�<sup>4</sup> 0.0237 CSA 0.0303 0.0013 0.0354 GSA 0.0300 0.0012 0.0343

**Algorithm Maximum Minimum Average** PSO 0.1048 <sup>2</sup>*:*<sup>59</sup> � <sup>10</sup>�<sup>4</sup> 0.0324 CSA 0.2097 <sup>9</sup>*:*<sup>35</sup> � <sup>10</sup>�<sup>4</sup> 0.0303 GSA 0.1425 <sup>3</sup>*:*<sup>67</sup> � <sup>10</sup>�<sup>4</sup> 0.0300

PSO 400 301.5411 0.5918 CSA 400 273.1862 0.5356 GSA 400 244.6284 0.4802

**Algorithm Magnitude PI (%) Phase PI (%)** PSO 5.8571 �11.22 CSA 6.7419 �7.443 GSA 0.9398 3.400

**per iteration**

**Execution time (s) per cycle**

**Table 10.**

**Table 11.**

**Table 12.**

**Table 13.**

**143**

*Convergence profile for design-2 SOMI.*

*Statistical data of magnitude error for design-2 SOMI.*

*Analysis of Wideband Second-Order Microwave Integrators*

*DOI: http://dx.doi.org/10.5772/intechopen.94843*

*Qualitatively data of magnitude error for design-2 SOMI.*

**Algorithm Iteration cycle Execution time (s)**

The results of the proposed-2 SOMI GSA-based leads in performance to the proposed-1 SOMI counterpart. All simulation results of the design-2 SOMI approximates with the ideal one and have linear phase response in its frequency range 3.0– 15 GHz. Therefore, the designed-2 SOMI is simulated on ADS by using microstrip lines. To simulate the designed-2 SOMI on ADS, RT/duroid substrate is selected with dielectric constant *ϵr*=2.2, height of substrate ð Þ¼ *H* 30 *mil* ð Þ 0*:*762 *mm* , loss tangent tan ð Þ¼ *δ* 0*:*001, and copper cladding is 35 um. The magnitude and phase response of the designed-2 SOMI GSA-based is illustrated in **Figure 8(a)** and **(b)** respectively.

order microwave integrator. **Figure 7(a)** and **(b)**, respectively, display the

bandwidth, especially for high-frequency ranges.

*Percentage improvement in magnitude and phase error for design-2 SOMI.*


#### **Table 9.**

*Comparison summary of total magnitude and phase error for design-2 SOMI.*

The Design-2 SOMI GSA-based magnitude response closely matches that of the ideal one in the 3.0–15 GHz frequency range. Based on the observations, compared to PSO and CSA, GSA has the least magnitude error and highest convergence speed.
