# Numerical Laplace Inversion Method for Through-Silicon Via (TSV) Noise Coupling in 3D-IC Design
## 3 Results and Discussions
### Time-Domain Analysis of the Coupling Noise with I/O Drivers Load  
In Figures 9-11, the TSV coupling noise was computed based on the assumption that all TSVs are terminated with 50 \(\Omega\). However, TSVs are usually terminated with I/O drivers; therefore, the TSV I/O terminations must be considered as mentioned before. For the analysis, \(Z_{2}\) and \(Z_{4}\) were replaced by a capacitance of 10 fF. Figure 11 depicts the TSV-TSV noise coupling for a trapezoidal signal switching  
Figure 11: The proposed method and measured coupling noise of the TSV-active circuit (the input clock frequency at port 1 is 1 GHz). Figure 10: The proposed method and measured coupling of the TSV–TSV test vehicle (the input clock frequency at port 1 is 1 GHz). from 0 to 1 V and from 0 to 1.8 V. The results show that the coupling noise increases when Z\({}_{2}\) and Z\({}_{4}\) are replaced by the capacitances. The peak-to-peak coupling noise increases from 80 mV (Figure 10) to 170 mV (Figure 12). The peak-to-peak coupling noise increases from 170 mV to 310 mV when the source changes from 1 V to 1.8 V. These results imply that the type of termination and the source significantly affects the coupling noise. The TSV I/O buffer size also influences TSV noise coupling and must be considered. The RDL redistributes the signals to connect I/Os or power/ground when two different dies with via-last processed TSVs are integrated vertically. Therefore, for advanced 3D-IC design, analyzing TSV noise coupling with RDLs is very important. The results found for the circuit presented in Figure 5 are illustrated in Figures 13-16 separately for \(l_{\text{RDL}}=200~{}\upmu\)m and \(l_{\text{RDL}}=500~{}\upmu\)m. These results present the TSV noise coupling without the coupling among the RDLs. A trapezoidal signal switching from 0 to 1.8 V with a rising/falling time of 10 ps and a source resistance of 50 \(\Omega\) at frequency 1 GHz was used, Z\({}_{1}\) and Z\({}_{3}\) were replaced by resistances of 50 \(\Omega\), and Z\({}_{2}\) and Z\({}_{4}\) were replaced by capacitances of 10 fF. Figure 12: The proposed method and Pspice simulation of the coupling noise of TSV–TSV (V\({}_{\text{in}}=1\) V and 1.8 V). Figure 14: The proposed method and Pspice simulation of the TSV–TSV coupling noise with uncoupled RDLs (\(l_{\mathrm{RDL}}=500\) μm) at port 4. Figure 13: The proposed method and Pspice simulation of the TSV–TSV coupling noise with uncoupled RDLs (\(l_{\mathrm{RDL}}=200\) μm) at port 4. It is observed that the coupling noise spreads on the stacked dies through used interconnections. The peak-to-peak coupling noise increases from 50 mV to 80 mV when the length of the interconnect line (RDL) changes. It is also observed that both ports 3 and 4, which represent, respectively, the input and the output drivers, are affected by the coupling noise. By analyzing the obtained results, the presence of horizontal interconnections can add the coupling noise. In high frequencies, coupling among the horizontal interconnections cannot be neglected. Indeed, a study including the coupling between the RDLs was done. The obtained results based on Figure 6 are depicted in Figures 17-19. Figure 16: The proposed method and Pspice simulation of the TSV–TSV coupling noise with uncoupled RDLs (\(l_{\text{RDL}}=500\) μm) at port 3. Figure 15: The proposed method and Pspice simulation of the TSV–TSV coupling noise with uncoupled RDLs (\(l_{\text{RDL}}=200\) μm) at port 3. Figure 17: The proposed method and Pspice simulation of the TSV–TSV coupling noise with coupled RDLs (\(l_{\rm RDL}=200\) μm and \(t_{\rm r}\)= 10 ps) at port 4. Figure 18: The proposed method and Pspice simulation of the TSV–TSV coupling noise with coupled RDLs (\(l_{\rm RDL}=500\) μm and \(t_{\rm r}=10\) ps) at port 4. The simulations were done for different RDL lengths and several rise/fall time values. The noise was studied only at port 4. Observing Figures 13 and 17, the peak-to-peak coupling noise increases when the coupling between RDLs is added.