## 3 Validation and Discussion  
In this section, the inductances obtained from the proposed model, the traditional circular coil method (CCM), quasi-static electromagnetic simulation with Q3D, are compared [26]. Q3D is a fast-quasi-static electromagnetic field simulator and is good at RLCG parameter extraction. For the proposed power inductor with the inductance of -1 uH, its switching frequency would not exceed 100 MHz. Therefore, Q3D can provide an accurate inductance extraction for the proposed power inductor. In the simulation environment, the two terminals of the inductor are defined as a source and sink, respectively, and the AC RL solver are adopted for inductance extraction. In the circular coil method [10,11], each loop is approximated as a circle and the radius of the \(i\)th turn is treated as \(R_{0}\) + (\(i\)-1)\(p\). The base parameters are \(w\) = 40 um, \(t\) = 20 um, \(s\) = 20 um, \(R_{0}\) = 20 um and \(n\) = 26. As shown in Figure 4, the inductances with the proposed model agree well with the results of 3D simulators over various design parameters: the error with respect to the EM results is less than 9.5%, while the error of the traditional circular coil method is 40.2%. The comparison indicates that the proposed model can be used for inductance extraction of the fully integrated power inductor. Furthermore, the results in Figure 4a,b demonstrate that \(L_{\text{T}}\) increases with the track width \(w\) and track spacing \(s\), due to the increased partial self-inductance and mutual inductance according to Equations (2) and (8). Since the variation in partial self-inductance and mutual inductance with track thickness \(t\) is little, it can be seen in Figure 4c that the impact of the parameter on \(L_{\text{T}}\) can be negligible. Figure 4: Impact of (**a**) track width, (**b**) track spacing, (**c**) track thickness and (**d**) inner radius on the total inductance. Moreover, the results in Figure 4d show the overall inductance would increase with the inner radius of the inductor. Table 2 investigates the computational efficiency of the model by comparing the simulation time of the proposed analytical model and 3D EM simulator using the parameters listed. The simulation was performed on an Intel i5 core with a processor speed of 3.0 GHz and 8 GB of RAM. It was found that lots of computational time can be saved using the proposed model. To verify the accuracy of the analytical expressions, the proposed model was applied to the fabricated power inductors. Taking into account the additional process steps for the 3D integration and the weak impact of the interposer materials on the inductor's performance, the proposed power inductor was fabricated with 2-layer Print Circuit Board (PCB) technology. In the emulated 3D integration environment shown in Figure 5a, the inductor is realized on the backside planar track. The vias in the PCB technology imitate the TSVs to connect the inductor to the frontside power circuits. Table 3 compares the inductances obtained from this model, the circular coil method (CCM) [10; 11] and electrical measurement with a Hioko IM3570 Impedance Analyzer. By connecting the two terminals of the inductor to the two probes of the impedance analyzer, capable of measurement frequencies of 4 Hz to 5 MHz, the inductance at 1 MHz can be captured. It is observed that the proposed model shows a higher accuracy in comparison to CCM. The maximum error and average error of the proposed model are 7.91% and 3.72%, respectively, which indicates the feasibility of the proposed model for inductance prediction. \begin{table}
\begin{tabular}{c c c} \hline
**Structure** & **This Model** & **3D EM Simulator** \\ \hline \(r_{1}=50\)\(\mu\)m, \(n=10\) & 0.03 s & 69.7 s \\ \hline \(r_{1}=50\)\(\mu\)m, \(n=15\) & 0.07 s & 76.0 s \\ \hline \(r_{1}=100\)\(\mu\)m, \(n=15\) & 0.08 s & 73.0 s \\ \hline \(r_{1}=100\)\(\mu\)m, \(n=26\) & 0.29 s & 86.6 s \\ \hline \(r_{1}=100\)\(\mu\)m, \(n=37\) & 0.41 s & 117.0 s \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of computational efficiency.