## 2 Simulation Model of CMIs with the CC Method
### Modeling the Complex Permeance of the Core  
In the proposed model, a three-stage RC circuit in Figure 3 replicates the magnetic core's complex permeance. Since modeling accuracy did not change significantly when higher-order ladder circuits were used, the three-stage RC ladder circuit is selected in this article. The complex permeance implemented in the magnetic circuit part is derived from the core's dimensions and the magnetic material's complex permeability and is incorporated into the magnetic circuit portion of the model. Generally, manufacturers' datasheets do not offer comprehensive broadband frequency characteristics of complex permeability. Additionally, the complex permeability of MnZn ferrites is influenced by the core's dimensions and significantly diminishes at high fre  
Figure 3: RC circuit with three stages designed to emulate the magnetic core’s complex permeance. quencies. To minimize the impact of core size, manufacturers typically provide data only for cores with smaller dimensions. Therefore, measuring the frequency characteristics of complex permeability before constructing the simulation model is advisable. Within the frequency band where the effect of stray capacitance is minor and can be overlooked, it is possible to depict the inductor as an assembly of an equivalent series inductance \(L_{\text{s}}\) and an equivalent series resistance \(R_{\text{s}}\)[26]. Consequently, the real and imaginary components of the complex permeability, \(\mu_{\text{r}}^{\prime}\) and \(\mu_{\text{r}}^{\prime\prime}\), can be expressed by Equations (1) and (2), respectively. \[\mu_{\text{r}}^{\prime}(f)=L_{\text{s}}(f)\cdot\frac{l_{\text{c}}}{\mu_{0}A_{ \text{c}}N_{\text{m}}^{2}} \tag{1}\]  
\[\mu_{\text{r}}^{\prime\prime}(f)=R_{\text{s}}(f)\cdot\frac{l_{\text{c}}}{2\pi f \mu_{0}A_{\text{c}}N_{\text{m}}^{2}} \tag{2}\]  
\(A_{\text{c}}\) represents the magnetic core's cross-sectional area, \(l_{\text{c}}\) is the magnetic path length, \(N_{\text{m}}\) refers to the inductor's turn number under measurement, and \(\mu_{0}\) is the permeability of free space (4\(\pi\times 10^{-7}\) H/m). One can directly measure \(L_{\text{s}}\) and \(R_{\text{s}}\) with a network analyzer. Inserting these measured frequency characteristics of \(L_{\text{s}}\) and \(R_{\text{s}}\) into Equations (1) and (2) allows for calculating the complex permeability's real and imaginary parts, respectively. Figure 4 illustrates the measured complex permeabilities of a nanocrystalline core (Vitroperm 500F, model number: T6006-L2030-W358) and a MnZn ferrite core (N30, model number: B64290L0048X830). These measurements were conducted with a network analyzer (Bode 100, Omicron Lab, Hong Kong, China) and an impedance measurement fixture (B-WIC) across frequencies ranging from 1 kHz to 30 MHz. Figure 4a shows that the real part of the nanocrystalline core's complex permeability starts to decline from about 20 kHz. Beyond 300 kHz, the imaginary part surpasses the real part. Conversely, Figure 4b shows that the real part of the MnZn ferrite's complex permeability rapidly decreases, starting around 1 MHz, with the imaginary part displaying a resonance peak. Additionally, the real part exhibits negative polarity from approximately 1.5 MHz and gradually approaches zero. This significant alteration in the MnZn ferrite's complex permeability substantially affects the high-frequency characteristics of filter inductors [23]. Figure 4: Measured complex permeabilities: (**a**) Nanocrystalline (Vitroperm 500F); (**b**) MnZn ferrite (N30).