### I. Introduction

Filter designers focus on the initial design of CM filters, considering the proper magnetic core material and shape selection, calculating the number of turns, and employing proper winding schemes. Achieving sufficient insertion loss characteristics for the CM filter is the goal of such design procedures.

Circuit designers are intended to select the best filter among numerous commercially available filters for their specific applications.

*S*-parameters and modal models, have been proposed [10, 11], which adds even more complexity to the modeling process [12]. Consequently, impedance measurement-based methods were developed and are now the most frequently used techniques for modeling passive filters [13]. In this respect, a lumped-element high-frequency model for CM chokes is utilized in [13, 14]. To extract the parameters of the model, heuristic methods [15], rational function approximation [8, 16], iterative rational function [12], and fitting methods [17] are employed. Furthermore, transfer function-based modeling is also employed, in which the numerator and denominator coefficients are obtained by implementing the experimental measurements into the available system identification toolboxes [18]. To summarize these previous studies, the topology of the high-frequency model is either complex or chosen heuristically, which requires significant experience and testing. Furthermore, although the estimated models are accurate, the circuit diagram of the filter is complex. These concerns make the existing schemes complicated for modeling, especially from the circuit designer’s perspective, for which a simple but accurate modeling approach is needed.

### II. EMI Filters for Suppressing Common-Mode Noise

### III. Proposed Analytical Formulation and CM Filter Modelling

*μ*′) and imaginary (

*μ*″) permeabilities, the complex permeability of Eq. (4) is reformed to Eq. (5).

*μ*′ and

*μ*″ may be physically related to each other by a physical relation, such as the Kramers–Kronig relation for non-passive materials [29]. A more accurate expression is provided by [30], which is accurate even for passive materials. However, the physics-based modeling of a CM filter or its components necessitates extensive electromagnetic analysis, which makes it very complicated and limits the applicability to specific applications [12, 31, 32]. For this reason, measurement-based modeling approaches are widely used for EMI filters in the field of power electronics. These modeling methods are more general, direct, and accurate for describing the behavior and characteristics of the CM filters [12], and they are also used in this paper. Fig. 2 shows the permeability versus frequency characteristics of ferrite and nanocrystalline CM cores, which are the most well-defined and widely-used magnetic core materials in CM noise filtering in power electronics [33]. As can be seen from Fig. 2, there is a cut-off frequency for both cores in which the permeability curve starts to decrease from the initial value. In this frequency,

*μ*′ start to decrease, and

*μ*″ would be the dominant component. According to the conducted emission regulations, the EMI studies are performed in the frequency range of 150 kHz to 30 MHz. However, the frequency range of 10 kHz to 1 MHz is considered in this paper, where magnetic core characteristics play a dominant role rather than the parasitic capacitance of the inductor. In this respect, it should be noted that the resonance in the common-mode currents with large peak values occurs in the specified frequency range in high-power converter applications, as can be seen in [22, 23]. Modeling of the filters in this range is needed for the circuit designer to calculate and limit the peak of common-mode currents to the standard levels. Accordingly,

*μ*′ and

*μ*″ are modeled in the specified frequency band according to the experimental results obtained in Fig. 2 for ferrite and nanocrystalline CM cores.

*ω*.

*N*,

*A*,

*ω*, and

*l*specify the number of turns, cross-section area, angular frequency, and length of the flux path.

*μ*″ and

*μ*′ are frequency-dependent components, as shown in Fig. 2. Below the cut-off frequency,

*μ*′ is the dominant component, and

*μ*″ can be ignored due to its negligible value compared with

*μ*′. However, in higher frequencies, the

*μ*″ component becomes larger, introducing a resistive part. Then, the impedance can be represented by Eq. (9).

*f*≤

*f*

*.*

_{cut}### IV. Model Parameter Estimation Using Experimental Measurements

As expected from the analysis of the previous section, there is a cut-off frequency for this filter at 90 kHz (

*f*= 90 kHz)._{cut}Below this frequency

*f*, the impedance characteristic is mainly inductive and the resistive part is negligible, as expected._{cut}Above the cut-off frequency, the resistive part becomes dominant and the slope of the curve decreases noticeably, as expected.

*f*

*. To further analyze the equations of Eq. (12), the different parts are separated as follows.*

_{cut}*B*

_{1}is the unknown but constant value. Moreover,

*A*

_{2}and

*B*

_{2}are unknown values that can be calculated from the experimental results of Fig. 3.

### 1. CM Choke Impedance Interpolation

### 2. Model Parameter Estimation

*f*>

*f*

*is elaborated in the following. For*

_{cut}*f*≤

*f*

*, the EMI filter is dominantly inductive (see (10)), and the inductance value can be obtained simply through the experimental tests provided in the datasheets (normally at 10 kHz).*

_{cut}