## 4 Evaluation of Small-Signal Characteristics of Single-Phase CMIs
### Evaluation of the Small-Signal Characteristics  
Figure 14 compares the measured and simulated small-signal characteristics of a nanocrystalline-based CMI (CMI-1) with and without the implementation of a cancellation capacitor. In Figure 14a, the measured data without the cancellation capacitor indicate a gradual reduction in the attenuation characteristic's slope, starting from around 100 kHz. This observation is attributed to the significant decrease in the real component of the nanocrystalline's complex permeability from around 100 kHz and the imaginary component exceeding the real component after 300 kHz, as demonstrated in Figure 4a. Additionally, self-resonance is observed at around 10 MHz due to the stray capacitance, and CM attenuation decreases from the self-resonance frequency. The simulation reproduces \begin{table}
\begin{tabular}{c c c} \hline \hline  & **CMI-1** & **CMI-2** \\ \hline \(R_{1}\) [\(\Omega\)] & \(9.66\times 10^{-3}\) & \(9.34\times 10^{-2}\) \\ \(R_{2}\) [\(\Omega\)] & \(1.80\times 10^{-2}\) & \(-5.18\times 10^{-2}\) \\ \(R_{3}\) [\(\Omega\)] & \(6.07\times 10^{-2}\) & \(-3.12\times 10^{-3}\) \\ \(C_{1}\) [F] & \(1.31\times 10^{-6}\) & \(-1.41\times 10^{-7}\) \\ \(C_{2}\) [F] & \(3.93\times 10^{-6}\) & \(-9.81\times 10^{-6}\) \\ \(C_{3}\) [F] & \(1.20\times 10^{-5}\) & \(1.58\times 10^{-5}\) \\ \(R_{\text{w}}\) [G] & \(20.8\) & \(22.3\) \\ \(C_{\text{p}}\) [PF] & \(0.61\) & \(1.20\) \\ \(C_{\text{can}}\) [PF] & \(2.0\) & \(5.1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Model parameters. Figure 12: Flow chart of the building procedure of the proposed simulation model. the small-signal characteristic of the nanocrystalline-based CMI over a wideband frequency range, incorporating the core's frequency-responsive complex permeance and the estimated stray capacitance into the model. A minor deviation between 10-30 MHz may arise from variances between the actual and calculated stray capacitance values. Disparities in the core's actual dimensions versus datasheet specifications and inaccuracies in the winding length per turn in the lab-created CMI could contribute to errors in stray capacitance estimation. Furthermore, the high-frequency mutual coupling effect between the CMI's windings and the printed circuit board's traces might negatively impact the CMI's attenuation performance [31]. The mutual coupling effect is particularly influential in multistage EMI filters [19; 32]. However, this effect is not factored into the simulation model in this article, as it focuses solely on the attenuation characteristics of a single magnetic component. Figure 14(b) compares the measured and simulated CM attenuation for the nanocrystalline-based CMI with cancellation capacitors. The results confirm that incorporating cancellation capacitors mitigates the effects of stray capacitance, allowing the CMI to maintain robust CM attenuation even beyond the 10 MHz frequency mark. The model proficiently predicts the improved attenuation provided by the CC method at frequencies over 10 MHz. A slight  
Figure 14: Comparison between the measured and simulated attenuation characteristics of CMI\(-\)1: (**a**) without \(C_{\text{can}}\); (**b**) with \(C_{\text{can}}\). Figure 13: Configurations for calculating attenuation characteristics of the fabricated CMIs: (**a**) without CMI; (**b**) with CMI. variation around 20-30 MHz may be due to the difference between the chosen value of the off-the-shelf cancellation capacitor and the ideally required capacitance value. The data from Figure 14a,b validate the reliability of the proposed simulation model. Figure 15a demonstrates that the proposed model accurately replicates the CM attenuation characteristic of the MnZn ferrite-based CMI (CMI-2) over a wideband frequency when the CC method is not employed. Additionally, the comparison in Figure 15b confirms the model's capability to simulate the CM attenuation characteristic of CMI-2 with cancellation capacitors.