# PFC Inductor Design Considering Suppression of the Negative Effects of Fringing Flux
## 1 Introduction
KoolM\(\mu\) (Sendust) material has a soft saturation curve with a B\({}_{\text{sat}}\) in the range of around 1 T. The preferred range of magnetic induction in amorphous and nanocrystalline cores is also, due to core loss effects, in the range up to 1 T [4; 15; 16; 17; 18]. When approaching the air gap, the magnetic field lines, which until that point are entirely enclosed by the core, tend to fringe out beyond the cross-sectional area of the core and enter the surrounding medium, usually copper winding [19]. In this paper, the research task is devoted to the practical verification of the above statements from the mentioned studies. One of the main aims is also to examine other phenomena related to fringing flux, which have not yet been considered in terms of the construction or operational performance of the power converter. Many studies have addressed the impact of fringing flux on the properties of a magnetic component. However, the examination of the secondary impact, i.e., to the system operation of the power converter as a whole, was no longer sufficiently investigated. The article gradually presents the influence of fringing flux from the thermal performance point of view of the magnetic component, while a comparison between a component with one air gap and with several smaller distributed gaps is also presented. Subsequently, the space is devoted to the analysis of the influence of fringing flux in terms of the design of the magnetic component, while the main indicator is the stability of the inductance value of the PFC converter coil. The final consequence of the analysis is the influence of various coil designs forcing the reduction of fringing flux, and consequent evaluation of the impact on the efficiency parameter of the PFC converter. Effect of the Air Gap in Inductor Magnetic Core--Air Gap of Inductor Reflected in the Generation of Fringing Flux  
Generally, the main function of the coil's air gap is to store the energy of the inductor. This effect can be described by Equation (1); the left part represents the operational variablesof the coil and the right part reflects inductor core dimensions for given permeability \(\mu_{r}\) and maximum saturation \(B_{m}\). \[Ll_{m}^{2}=\frac{B_{m}^{2}S_{Fe}}{\mu_{0}}\Big{(}\frac{l_{Fe}}{\mu_{r}}+l_{v} \Big{)} \tag{1}\]  
where \(L\)--inductance of the coil, \(I_{m}\)--maximum current of the inductor, \(B_{m}\)--saturation induction, \(S_{Fe}\)--core cross-section area, \(l_{Fe}\)--mean length path, \(l_{v}\)--length of the air gap, \(\mu\)--relative permeability, and \(\mu_{0}\)--permeability of vacuum. Considering the geometry and design of the inductor core, if the situation when \(l_{v}=0\), i.e., the core is without air gap, the product of values \(Ll_{m}{}^{2}\) requires the high product of the second part \(S_{Fe}\) and \(l_{Fe}\). It will then reflect into the bulky and heavy magnetic core. On the other hand, the higher the relative permeability of the magnetic material \(\mu_{r}\), the higher the product of \(S_{Fe}\) and \(l_{Fe}\) required. Moreover, the magnetic conductivity of the core is defined by the conductivity of the ferromagnetic material. Because of the dependency \(\mu_{r}=f(B)\), the magnetic conduction is also dependent on the induction, thus on the value of the coil's current. Based on this fact, the value of inductance is dependent on the value of current. As a result, a core without an air gap can be saturated relatively quickly. In the case when \(l_{v}>>l_{Fe}/\mu_{r}\), the product of \(S_{Fe}\) and \(l_{Fe}\) shall be lowered targeting the required product of \(Ll_{m}{}^{2}\), resulting in smaller dimensions of the magnetic core. Simultaneously, if magnetic resistance \(R_{v}\) is much higher than \(R_{Fe}\), then the magnetic resistance of ferromagnetic material has limited influence on magnetic conduction. The value of the inductance will then be practically independent of the properties of ferromagnetic material, thus the inductance will be independent of the value of the inductor current.