# Numerical Laplace Inversion Method for Through-Silicon Via (TSV) Noise Coupling in 3D-IC Design
## 2 Calculation of Time-Domain TSV Noise Coupling in 3D-IC Design with NILT
\begin{array}{cc}1&R_{tsv}+sL_{tsv}\\ 0&1\end{array} \right] \tag{18}\]  
\[Z_{eq}=\frac{2}{2C_{TSV-equiv}s}+\frac{R_{sub-equiv}}{1+R_{sub-equiv}C_{sub -equiv}S} \tag{19}\]  
Observing the circuit, Equations (14) and (15) are found:  
\[V_{in}(s)=Z_{1}(s)I_{1}(s)+V_{1}(s) \tag{20}\]  
\[V_{2}(s)=-Z_{4}(s)I_{2}(s) \tag{21}\]Figure 1: The through-silicon via (TSV)–TSV noise coupling structure with I/O termination. Figure 2: Lumped circuit model of TSV–TSV noise coupling. By exploiting Equations (13)-(15), the noise V\({}_{2}\) could be expressed in the frequency domain according to V\({}_{\text{in}}\), then the NILT method can be applied, by replacing F(s) by V\({}_{2}\)(s) in previous equations, to find the noise in the time domain. The voltage source V\({}_{\text{in}}\) is a periodic trapezoidal signal switching expressed by Equation (16). \[V_{in}(s)=\sum\limits_{n=0}^{\infty}\exp(-snT).E(s) \tag{22}\]  
where T is the period and E(s) represents the trapeze shape. Then, while \(\frac{1}{1-x}=\sum\limits_{n=0}^{\infty}x^{n}\), Equation (16) could be written as:  
\[V_{in}(s)=\frac{1}{1-\exp(-Ts)}E(s) \tag{23}\]  
The second analyzed structure is given in Figure 3. It represents the conceptual view of TSV-active circuit noise coupling. The equivalent circuit model of this structure is similar to that in Figure 2, except that the capacity on the right is eliminated [3]. Consequently, the calculation was also done in the same way. Because of the diversity of electronic devices, and the presence of many stacked dies in 3D technology, the second studied circuit contains two stacked dies with two interconnect lines. The concerned structure is presented in Figure 4. First, the noise coupling was calculated without taking into consideration the coupling between the two interconnect lines, only the coupling between the TSVs in each level was considered. This conceptual structure is modeled by a lumped circuit, as given in Figure 5.  
\[V_{in}(s)=Z_{1}(s)I_{1}(s)+V_{1}(s) \tag{20}\]0 The conceptual view of TSV-active circuit noise coupling. \[V_{in}(s)=Z_{1}(s)I_{1}(s)+V_{1}(s) \tag{20}\]1 The conceptual view of a TSV noise coupling structure with interconnect lines and I/O drivers. \[V_{in}(s)=Z_{1}(s)I_{1}(s)+V_{1}(s) \tag{20}\]2 The equivalent circuit model of TSV noise coupling with interconnect line. The electrical schema presented in Figure 5 is composed of a lumped circuit model of TSV-TSV noise coupling in each die, two interconnect lines to distribute signals between dies, and I/O drivers modeled by \(Z_{1}\), \(Z_{2}\), \(Z_{3}\), and \(Z_{4}\). As explained above, before applying the NILT method, the global T-matrix of the circuit must be found. The matrices \(T_{sub}\), \(T_{tsv}\), \(T_{tl}\), \(T_{3}\), and \(T_{4}\) were used. First, \(T_{1}\) and \(T_{2}\) were calculated using Equations (18) and (19), respectively, then a transformation to \(Y_{1}\) and \(Y_{2}\) of \(T_{1}\) and \(T_{2}\), respectively, was made. This transformation was performed to find the global \(Y_{g}\) of the circuit without \(Z_{1}\), \(Z_{4}\), and \(Z_{tsv}\) near \(Z_{1}\) and \(Z_{4}\). Then another transformation from \(Y_{g}\) to \(T_{g}\) was performed. When finding \(T_{g}\), it is multiplied by \(T_{tsv}\) on the left and right sides, and by using Equations (13), (14), and (21) \(V_{2}\) is found according to \(V_{in}\). \[V_{in}(s)=Z_{1}(s)I_{1}(s)+V_{1}(s) \tag{20}\]3  
\[V_{in}(s)=Z_{1}(s)I_{1}(s)+V_{1}(s) \tag{20}\]4  
\[V_{in}(s)=Z_{1}(s)I_{1}(s)+V_{1}(s) \tag{20}\]5  
\[V_{in}(s)=Z_{1}(s)I_{1}(s)+V_{1}(s) \tag{20}\]6  
where:  
\[[T_{tl}]=\left[ \begin{array}{cc}\cos(\beta l)&jZ_{0}\sin(\beta l)\\ j\sin(\beta l)/Z_{0}&\cos(\beta l)\end{array} \right] \tag{28}\]  
where \(\beta\) is the propagation constant, \(l\) and \(Z_{0}\) are the length and the characteristic impedance, respectively, of the interconnect line, and:  
\[[T_{sub}]=\left[ \begin{array}{cc}1&Zeq\\ 0&1\end{array} \right] \tag{29}\]  
\[[T_{tsv}]=\left[ \begin{array}{cc}1&Z_{tsv}\\ 0&1\end{array} \right] \tag{30}\]  
\[[T_{4}]=\left[