# Simulation Study of the Small-Signal Characteristics of Single-Phase Common-Mode Inductors with Capacitance Cancellation

## Abstract

**:**

## 1. Introduction

_{can}between this center tap and a reference potential. Theoretically, C

_{can}neutralizes the effect of stray capacitance C

_{p}on the inductor’s frequency characteristic, making the inductor with the CC method act like an ideal inductor. Wang et al. applied CC to a single-phase CMI and experimentally showed that CC could boost CM emission attenuation by up to 20 dB at frequencies above 10 MHz [18]. Heldwein et al. used the CC technique on three-phase CMIs [19]. However, previous studies identified stray capacitances from the measured attenuation characteristics and impedances of manufactured CMIs and determined the capacitance of cancellation capacitors accordingly. To date, to the authors’ knowledge, there is a lack of circuit simulation models that can accurately simulate CMIs’ attenuation characteristics with CC applied, making it challenging to assess these characteristics during the EMI filter design phase.

## 2. Simulation Model of CMIs with the CC Method

#### 2.1. Configuration of the Proposed Model

_{c}of the magnetic core is implemented in the magnetic circuit part of the model, and the winding resistance R

_{w}and the parallel stray capacitance C

_{p}are implemented in the electric circuit part. Coupling between the magnetic and electric circuit parts is realized using a gyrator, an impedance inverting component. The gain G of the gyrator corresponds to the turn number and is set to half of the turn number (G = N/2). Furthermore, cancellation capacitors C

_{can}are connected between the connection point of two gyrators (the center tap of the winding) and the ground to simulate the CC method.

#### 2.2. Modeling the Complex Permeance of the Core

_{s}and an equivalent series resistance R

_{s}[26]. Consequently, the real and imaginary components of the complex permeability, μ

_{r}

^{′}and μ

_{r}

^{″}, can be expressed by Equations (1) and (2), respectively.

_{c}represents the magnetic core’s cross-sectional area, l

_{c}is the magnetic path length, N

_{m}refers to the inductor’s turn number under measurement, and μ

_{0}is the permeability of free space (4π × 10

^{−7}H/m).

_{s}and R

_{s}with a network analyzer. Inserting these measured frequency characteristics of L

_{s}and R

_{s}into Equations (1) and (2) allows for calculating the complex permeability’s real and imaginary parts, respectively.

_{m}, as depicted in Figure 3.

_{1}, a

_{2}, a

_{3}, b

_{0}, b

_{1}, b

_{2}, and b

_{3}, the admittance Y

_{m}can be represented as

_{1}, R

_{2}, R

_{3}, C

_{1}, C

_{2}, and C

_{3}can be calculated based on the identified fitting parameters.

#### 2.3. Calculating the Winding DC Resistance

_{T}is assumed, as shown in Figure 5. Thus, l

_{T}is obtained from the following Equation (7), where OD, ID, and HT are the outer diameter, inner diameter, and height of the toroidal core, respectively. In addition, d

_{o}is the outer diameter of the wire, including an insulation coating.

_{w,DC}, can be expressed by Equation (8). In this equation, ρ

_{Cu}denotes the resistivity of copper, d

_{w}represents the diameter of the conductor, and N refers to the number of turns.

_{Cu}(20 °C) is the resistivity of copper at 20 °C (1.724 × 10

^{−8}(Ω ∙ m)), and α

_{Cu}is the temperature coefficient of copper (0.00393 (1/°C)). In this article, the winding DC resistance of the winding was calculated, where T was set to the room temperature (25 °C).

## 3. Estimation of Stray Capacitance of Single-Phase CMIs

#### 3.1. Turn-to-Turn Capacitance

_{p}, is modeled as a ladder circuit consisting of the turn-to-turn capacitance, C

_{t−t}, and the turn-to-core capacitance, C

_{t−c}[29]. Initially, the turn-to-turn capacitance is determined using Equations (10) and (11), which are based on the assumption of an electric force line existing between adjacent windings, as depicted in Figure 7 [30].

_{w}is the total winding length, ε

_{ri}is the permittivity of the insulation coating, and p is the distance between the center of the conductors.

_{i}, p

_{o}, and p

_{l}. Based on Figure 8, p

_{i}and p

_{o}are obtained as Equations (12) and (13), respectively, as a function of the angle φ, at which the winding covers the core [20].

_{l}is obtained as the average of p

_{i}and p

_{o}(p

_{l}= (p

_{i}+ p

_{o})/2).

_{wi}, l

_{wo}, and l

_{wl}are calculated as

_{t−t,i}, C

_{t−t,o}, and C

_{t−t,l}can be calculated, respectively. The turn-to-turn capacitance C

_{t−t}is obtained as the sum of calculated results in each part of the core (C

_{t−t}= C

_{t−t,i}+ C

_{t−t,o}+ 2C

_{t−t,l}).

#### 3.2. Turn-to-Core Capacitance

_{t−c}, an assumption is made regarding the electric force line between the wire conductor and the magnetic core, as shown in Figure 9 [20].

_{t−c,i}, which is due to the winding’s insulation coating; C

_{t−c,g}, resulting from the air gap between the winding and core, and C

_{t−c,c}, originating from the core’s insulation. The calculation of the turn-to-core capacitance is, therefore, performed using Equations (16) and (17) [20].

_{c}represents the thickness of the magnetic core’s insulation, and ε

_{rc}is the relative permittivity of the core’s insulation.

#### 3.3. Total CM Capacitance

_{p}is obtained as the ladder circuit of the turn-to-turn capacitance and turn-to-core capacitance. Based on the delta–wye transformation, Equation (18) is derived [20].

_{CM}of single-phase CMIs is the parallel connected C

_{p}and is obtained as C

_{CM}= 2C

_{p}.

## 4. Evaluation of Small-Signal Characteristics of Single-Phase CMIs

#### 4.1. Configuration of the Evaluation System

_{can}(= 4C

_{p}) calculated based on the estimated parallel stray capacitance were also mounted on the printed circuit boards. The picture of the fabricated CMI is shown in Figure 10, and the specifications of the fabricated CMIs are listed in Table 1.

_{o1}when the CMI is not connected and the voltage V

_{o2}when the CMI is connected. Based on simulated V

_{o1}and V

_{o2}, the attenuation characteristic G

_{CM}of the CMI can be calculated using Equation (19).

#### 4.2. Evaluation of the Small-Signal Characteristics

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{c}is calculated using the imaginary part of the complex permeability μ

_{r}

^{″}and is obtained as

_{w}is given by

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**Figure 4.**Measured complex permeabilities: (

**a**) Nanocrystalline (Vitroperm 500F); (

**b**) MnZn ferrite (N30).

**Figure 6.**Capacitance network for calculating the total stray capacitance of the winding with N turns.

**Figure 13.**Configurations for calculating attenuation characteristics of the fabricated CMIs: (

**a**) without CMI; (

**b**) with CMI.

**Figure 14.**Comparison between the measured and simulated attenuation characteristics of CMI−1: (

**a**) without C

_{can}; (

**b**) with C

_{can}.

**Figure 15.**Comparison between the measured and simulated attenuation characteristics of CMI−2: (

**a**) without C

_{can}; (

**b**) with C

_{can}.

CMI-1 | CMI-2 | |
---|---|---|

Material | Nanocrystalline | MnZn ferrite |

Parts number | T60006-L2030-W358 | B64290L0048X830 |

OD [mm] | 32.80 | 35.50 |

ID [mm] | 17.60 | 19.20 |

HT [mm] | 12.50 | 13.60 |

A_{c} [mm^{2}] | 40.00 | 82.60 |

l_{c} [mm] | 79.00 | 82.06 |

t_{c} [mm] | 1.25 | 0.55 |

d_{o} [mm] | 0.88 | 0.88 |

d_{w} [mm] | 0.81 | 0.81 |

N | 14 | 14 |

φ | 5π/6 | 5π/6 |

Insulation | Plastic case | Coating |

ε_{ri} | 3.5 | 3.5 |

ε_{rc} | 3.0 | 3.6 |

CMI-1 | CMI-2 | |
---|---|---|

R_{1} [Ω] | 9.66 × 10^{−3} | 9.34 × 10^{−2} |

R_{2} [Ω] | 1.80 × 10^{−2} | −5.18 × 10^{−2} |

R_{3} [Ω] | 6.07 × 10^{−2} | −3.12 × 10^{−3} |

C_{1} [F] | 1.31 × 10^{−6} | −1.41 × 10^{−7} |

C_{2} [F] | 3.93 × 10^{−6} | −9.81 × 10^{−6} |

C_{3} [F] | 1.20 × 10^{−5} | 1.58 × 10^{−5} |

R_{w} [Ω] | 20.8 | 22.3 |

C_{p} [pF] | 0.61 | 1.20 |

C_{can} [pF] | 2.0 | 5.1 |

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**MDPI and ACS Style**

Takahashi, S.
Simulation Study of the Small-Signal Characteristics of Single-Phase Common-Mode Inductors with Capacitance Cancellation. *Appl. Sci.* **2024**, *14*, 1172.
https://doi.org/10.3390/app14031172

**AMA Style**

Takahashi S.
Simulation Study of the Small-Signal Characteristics of Single-Phase Common-Mode Inductors with Capacitance Cancellation. *Applied Sciences*. 2024; 14(3):1172.
https://doi.org/10.3390/app14031172

**Chicago/Turabian Style**

Takahashi, Shotaro.
2024. "Simulation Study of the Small-Signal Characteristics of Single-Phase Common-Mode Inductors with Capacitance Cancellation" *Applied Sciences* 14, no. 3: 1172.
https://doi.org/10.3390/app14031172