**3. Results**

To verify the proposed analytical model of CGP-EBG MTM-DTLs, it is compared to full-wave simulations based on the finite element method (FEM) [18] and measurements. Comprehensive validations were performed considering various cases. Firstly, the CM characteristics of the CGP-EBG MTM-DTLs between the proposed model and full-wave simulations are compared. For the full-wave simulation model, geometrical parameters *d*H, *h*H, *d*L, *h*L, *w*, *s*, *w*G, and *s*v are set to 10, 1.0, 10, 0.08, 0.1, 0.1, 10, and 1.3 mm, respectively. The values were determined considering a commercial PCB process. FR-4 and copper (35 μm thick) were used as a dielectric material and conductor, respectively. The physical dimensions of the CGP-EBG MTM-DTL present *<sup>Z</sup>*oe,H, *<sup>Z</sup>*oo,H, *<sup>Z</sup>*oe,L, and *<sup>Z</sup>*oo,L of 218, 66.3, 66, and 52 Ω, respectively. The dimensions and parameters for this validation are listed in Table 1.

The FEM simulation model for the CGP-EBG MTM-DTLs including three unit cells and their mesh generation result are shown in Figure 5. The waveguide ports are adopted for the excitation of the CM and DM waves at ports 1, 2, 3, and 4 of the differential signal lines. The perfect magnetic conductor and radiation boundaries are assigned at the sides and top of the vacuum box, which is shown as a red box in the Figure 5a. The dielectric constant and loss tangent of FR-4 are 4.4 and 0.02, respectively. The via in the FEM model is modeled using a polyhedron with 12 segments. Its radius and height are 0.2 and 0.9 mm, respectively. The meshes are generated with the solution frequency of 10 GHz, which is the maximum frequency of interest. As can be seen in Figure 5b, most meshes are placed in differential signal lines and ground plane transition.

**Figure 5.** (**a**) Finite element method (FEM) simulation model, and (**b**) mesh generation result of CGP-EBG MTM-DTLs with three unit cells.

Figure 6 depicts the eight curves of parameter *S*cc21 from the CGP-EBG MTM-DTLs considering one to four UCs, using the proposed analytical model (solid lines) and full-wave simulations (dashed lines). The proposed model suitably agrees with the full-wave simulations in the four cases. From the proposed model, the minimum values of CM noise suppression from one to four UCs are −6.1, −15.6, −25.9, and −36.3 dB, respectively. Overall, CM noise suppression improves as the number of UCs increases.


**Table 1.** Parameters for verification of the proposed analytical model. DTL—differential transmission line; HZ—high *Z*oe; LZ—low *Z*oe; UC—unit cell; CM—common mode; DM—differential mode.

**Figure 6.** CM noise suppression with varying number of unit cells (UCs) for model verification.

Examining the suppression levels where the suppression bandwidth of 3 GHz is ensured, the values for two to four UCs are −10.1, −14.1, and −18.6 dB, respectively. The value of 3 GHz was selected because it corresponds to the suppression bandwidth predicted by the dispersion analysis based on Floquet theory. The high and low cut-off frequencies from Floquet theory were obtained as 2.7 and 5.7 GHz, respectively. The periodic analysis only estimates the cut-off frequencies, but not the suppression level. As seen in Figure 6, the suppression level corresponding to Floquet theory notably varies according to the number of UCs. In the practical use of MTM-DTLs for high-speed PCBs, a periodic condition is not commonly ensured, thus requiring the development of approaches such as the proposed model considering MTM DTLs with a finite and small number of UCs.

Remarkably, the proposed analytical model achieves a drastic reduction in computation time compared to the full-wave simulation. For instance, the time for determining *S*cc21 of the CGP-EBG MTM-DTL with four UCs using the proposed model was 0.3 s, whereas that using the full-wave simulation was 18,257 s. Hence, the proposed model substantially reduces the computation time while providing a suitable accuracy compared to the full-wave simulation. The time reduction results are listed in Table 2.


simulation

**Table 2.** Computation time for estimating CM and differential transmission characteristics of the corrugated ground-plane electromagnetic bandgap (CGP-EBG) metamaterial (MTM)-DTL.

 Computation platform: Intel Xeon processor (3.2 GHz), 512 GB RAM (E5-2667 v4 @3.20 GHz, Intel, Santa Clara, CA, USA).

The effect of *Z*oe on parameter *S*cc21 is further examined by comparing the proposed analytical model and full-wave simulations, as shown in Figure 7. For the CGP-EBG MTM-DTL with four UCs, *<sup>Z</sup>*oe,H changes to 113, 171, and 218 Ω by adjusting *h*H to 0.2, 0.5, and 1.0 mm, respectively. The corresponding changes in the ratio of *<sup>Z</sup>*oe,H to *<sup>Z</sup>*oe,L are approximately 1.7, 2.6, and 3.3. The low and high cut-off frequencies with a suppression level of −10 dB were also investigated. The corresponding (high, low) cut-off frequencies for *<sup>Z</sup>*oe,H of 113, 171, and 218 Ω are (3.56, 4.85 GHz), (2.94, 5.72 GHz), and (2.65, 6.09 GHz), respectively. Hence, the suppression bandwidth and level improve as the ratio of *<sup>Z</sup>*oe,H to *<sup>Z</sup>*oe,L increases. Again, the results of the proposed analytical model are consistent with those of the full-wave simulation.

**Figure 7.** Effect of even-mode characteristic impedance (*Z*oe) on CM noise suppression for model verification.

In addition to CM noise suppression, the differential transmission characteristics of the CGP-EBG MTM-DTLs were investigated. Figure 8 shows parameter *S*dd21 for differential transmission using the proposed model and full-wave simulations. The design parameters and dimensions are those listed in Table 1. The values of *<sup>Z</sup>*oo,L and *<sup>Z</sup>*oo,H associated with differential transmission are 52 and 66.3 Ω, respectively. For ideal differential characteristics, the same values of *Z*oo between the HZand LZ-DTLs are preferred. However, this condition is commonly limited by the design rules of commercial PCB processes, thus making it difficult to avoid different *Z*oo values for HZ- and LZ-DTLs in practical high-speed PCBs. The effect of the *Z*oo difference on the differential characteristics is shown in Figure 8. The small difference in the *Z*oo values between HZ- and LZ-DTLs degrades differential transmission characteristics. This effect increases with the number of UCs. This phenomenon can be inferred considering the theory of a stepped impedance resonator. However, it is important to obtain the exact degradation of parameter *S*dd21 for measuring and quantitating its impact on differential signal transmission.

**Figure 8.** Differential signal transmission characteristics with varying number of UCs for model verification.

The accuracy and efficiency of the proposed analytical model for CGP-EBG MTM-DTLs are verified from comparisons to full-wave simulations based on FEM. The CM noise and differential signal transmission results of the proposed model suitably agree with those of full-wave simulations, but the computation time for obtaining the characteristics of the nonperiodic array of MTM-DTLs was substantially reduced using the proposed four-port analytical model.

To further validate the proposed model of the CGP-EBG MTM-DTL, the correlations between the proposed model, full-wave simulations, and measurements are examined using the fabricated PCB pattern of the CGP-EBG MTM-DTL. The fabricated PCB pattern and the measurement set-up for Scc21 and *S*dd21 are shown in Figure 9. The low-cost PCB process employs the FR-4 dielectric and copper metal layers. The dielectric constant and loss tangent of the FR-4 substrate are 4.4 and 0.02, respectively. The PCB pattern contains four HZ-DTLs and three LZ-DTLs. The geometric dimensions are shown in Figure 9a. To obtain the *S*cc21 and *S*dd21, four-port S-parameters of the PCB pattern are measured using a vector network analyzer and high-frequency microprobes. Figure 10 depicts the comparison of *S*cc21 and *S*dd21 between the proposed model, full-wave simulations, and measurements. The proposed model agrees well with the measurements. As can be seen in the results, CM noise is successfully suppressed, while good differential data transmission is achieved. In the *S*cc21 of the measurements, discrepancy is observed around the frequency of 5 GHz. This defect is caused by PCB manufacturing tolerances because it is not observed in the full-wave simulations.

**Figure 9.** (**a**) Fabricated CGP-EBG MTM-DTL, and (**b**) measurement set-up for S-parameters.

**Figure 10.** Comparison of *S*cc21 and *S*dd21 between proposed model, full-wave simulations, and measurements.
