Applied Science and Convergence Technology 2023; 32(2): 54-57
Published online March 30, 2023
Copyright © The Korean Vacuum Society.
aDepartment of Physics, Chungnam National University, Daejeon 34134, Republic of Korea
bInstitute of Quantum Systems (IQS), Chungnam National University, Daejeon 34134, Republic of Korea
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Despite the importance of self-resonance frequency (SRF) in the operation of inductors, which is only effective below the SRF, a comprehensive analysis of the SRF in different inductor structures has yet to be conducted. This work employs a three-dimensional electromagnetic wave simulation to analyze SRF in the both solenoid and planar inductor structures with various inductor structure parameters such as the number of turns, radius, and inter-coil distance. We summarize the behavior of the SRF with these parameters. This result is valuable for radio-frequency engineering applications.
Keywords: Inductor, Self-resonance frequency, Electromagnetic wave simulation
An inductor, which stores energy as a magnetic field inside it, has been used in various industrial fields such as wireless energy transfer, radio-frequency (RF) engineering, inductor communication, and plasma engineering [1–6]. Especially, an inductor plays a significant role in a plasma source , an impedance matcher , and a voltagecurrent sensor . In an inductively plasma source, which is a common source used in plasma processing, an inductor converts electric energy from a RF generator to electromagnetic energy, of which an electric field induced by a time-varying magnetic field generates and sustains plasma. In the impedance matcher, which changes the plasma source impedance to output impedance of the RF generator, several inductors are used to remove the reactance of the plasma source impedance. Furthermore, an inductor is used in a voltage-current sensor as the current sensor picking up the time-varying magnetic field induced by RF current.
In RF range, inductor design is important due to parasitic capacitors originating from capacitive coupling between the inter-coils of the inductor. It causes degradation of coupling performance. Leakage current flowing through the inter-coils as the displacement current reduces the conduction current flowing on the inductor. It results in the reduction of the magnetic field and as a result, degrades the inductive coupling.
Furthermore, the parasitic capacitor causes self-resonance of the inductor, which is the resonance between the inductor and the parasitic capacitor . The frequency at which the self-resonance forms is called the self-resonance frequency (SRF), which is dependent on the inductor geometric parameters. For a simple understanding the self-resonance, the parasitic capacitor can be assumed as the parallel connection with the inductor . In terms of that, the self-resonance is the parallel LC resonance between the inductor and parasitic capacitor. Furthermore, the self-resonance condition is that the parasitic capacitor impedance (ZC) is the same with the ideal inductor impedance (ZL) without the ZC. Provided that the ZL is
SRF is significant as it is related with the operation of the inductor, which is only effective below the SRF. Several studies have investigated the SRF based on simple analysis. Lope
In the next section, simulation details about the configuration, boundary condition, and structure parameters are described. In the third section, we summarize the comprehensive simulation results and discuss the behavior of the SRF. In the last section, we will present the conclusion of this work.
To investigate the SRF of both inductor structures, we adopted a precise electromagnetic wave simulation, called CST Microwave Studio Suite, which is is widely used in plasma engineering [15,16]. It solves Maxwell’s equations in three-dimension space with the finitedifference time-domain method. In this simulation, we used the timedomain solver which provides frequency responses of simulation domain, for instance, S-parameters.
In this simulation, we find the SRF from a reactance frequency spectrum (
where the Z0 is the characteristic impedance of the coaxial cable. Then, the
Figures 2(a) and 2(b) depict simulation configurations for solenoid and planar inductor structures. As for the planar inductor structure, the diameter decreases with increasing number of turns as shown in Fig. 2(b), but the diameter is fixed at the one-turn planar inductor case shown in Fig. 2(c). Furthermore, all cases have the same coaxial as shown in Fig. 2, of which characteristic impedance is 50 Ω, which is the common value.
Here, simulation variables are number of turns, diameter, and intercoil distance and well described in Fig. 2. As for the solenoid inductor structure, the number of turns varies from 2 to 6, the diameter from 50 to 150 mm, and inter-coil distance from 2 to 6 mm. As for the planar inductor structure, the number of turns varies from 1 to 5, the diameter from 100 to 300 mm, and the inter-coil distance from 4 to 15 mm. Those values are common values used in plasma engineering. Since copper is the common material used in plasma engineering, we used copper as the inductor materials of which electric conductivity is 401 S/m. For solenoid and planar inductor structures, material information and variables are well organized in Tables I and II.
We analyzed the impact of the variables on SRF with following orders: the number of turns, inter-coil distance, and diameter. Figure 3(a) shows reactance spectra with numbers of turns in the case of the solenoid structure. As the number of turns increases, the SRF, indicated as an arrow in this figure, shifts toward lower frequency. It results from the increase in its inductance as the number of turns rises; as mentioned in Introduction section, the SRF is inversely proportional to the square root of the product between its inductance. The inductance in solenoid inductor is
Figure 3(b) represents the reactance spectra with several inter-coil distances. The SRF slightly shifts towards higher frequency with increasing the inter-coil distance. It is due to the dominant decrease of the parasitic capacitance  rather than the decrease in its inductance. Increasing the inter-coil distance causes the slight decrease in the inductance. The slope of reactance frequency spectrum (
As the diameter increases, the SRF abruptly shifts toward lower frequency as shown in Fig. 3(c). The increase in diameter corresponds to the increase in the area of solenoid, which causes the increase in the inductance. This effect is similar to that in the number of turns; the increase in the both its inductance and parasitic capacitance with expanding its radius.
It is noted that engineers have to deliberate its use as matcher component and plasma generation antenna, regarding that the common deriving frequency used in plasma engineering ranges from 2 to 60 MHz. For insight for the design of solenoid inductors, we summarize the SRF variation for all variables in Fig. 4. Quite similar trend for the diameter and the number of turns can be observed in this figure.
As for the planar inductor structure, the similar trend with the solenoid was observed and thus, we focus on the difference between the solenoid and planar inductors. As shown in Fig. 5, it is noted that the dependence of the SRF for the number of turns in the planar inductor is stronger than that in the solenoid inductor. This difference comes from the antenna structure. Increasing the number of turns induces the decrease in the inductor area as shown in Figs. 2(b) and 2(c), since magnetic fields between the inter-coils cancel out. Furthermore, the overall SRF in the planar inductor are higher than the solenoid at similar condition, for instance, 150 mm in a diameter and 2-3 in the number of turns [Figs. 4(b) and 5(a)]. This is due to the smaller parasitic capacitance in the planar inductor than that in the solenoid.
In this work, we analyzed the SRF of solenoid and planar inductor structures through three-dimensional electromagnetic wave simulation. We varied the diameter, number of turns, and inter-coil distances, which are commonly used in plasma engineering, to determine the effect on the SRF. The results revealed a dramatic decrease in SRF with an increase in diameter and number of turns, and a slight decrease in inter-coil distance for both structures. We organized the overall behavior of the SRF in relation to these variables. Our results provide insights for the design of inductors in plasma engineering applications.
This research was supported by a research fund from Chungnam National University.
The authors declare no conflicts of interest.