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 [7], an impedance matcher [8], and a voltagecurrent sensor [9]. 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 [10]. 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 [11]. 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 (Z_C) is the same with the ideal inductor impedance (Z_L) without the Z_C. Provided that the Z_L is $i\text{ω}{L}_{\text{inc}}$ and the Z_C is $-i/\left(\text{ω}{C}_{\text{para}}\right)$, then, the SRF is ${\left(2\pi \sqrt{L_{\text{inc}}{C}_{\text{para}}}\right)}^{-1}$, where i is the complex number, ω is angular deriving frequency, $L_{\text{inc}}$ is inductance of the inductor, and $C_{\text{para}}$ is capacitance of the parasitic capacitor.

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 et al. [10] analyzed the first SRF of planar inductors with a simple circuit model. Findley et al. [11] developed the differential inductor having high SRF based on a simple SRF analysis. Yue et al. [12] proposed a physical model for planar spiral inductors on silicon. Most studies have focused on inductor design for the specific structure and its development for the use of an inductor [10–14]. Comprehensive analysis of the SRF in both solenoid and planar inductor structures, which are widely used in plasma engineering in the RF range, has yet to be conducted. Hence, this work investigated SRFs in those structures with various inductor structure parameters through a three-dimensional electromagnetic wave simulation.

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 (X_ind), which is derived from reflection frequency spectrum (S₁₁). Figure 1 shows schematic diagram of simulation domain. One end of the inductor is connected with the coaxial cable and the other is simulation domain boundary (grounded). The Gaussian pulse wave [Pin (t)] including frequency from 0 to 500 MHz enters on the wave port. After launching of the pulse wave, the reflected signals [Pref (t)] on the wave port are measured over time. Then, S₁₁ is defined as 10 log10 (P_ref (f)/P_in (f)), where P_ref (f ) and P_in (f) are derived from fast Fourier transform of the P_ref (t) and the P_in (t), respectively. Since this system in Fig. 1(a) is equivalent to the transmission line model shown in Fig. 1(b) [17], the inductor impedance (Z_ind (f)) can be derived from transmission line theory asfollows:

Figure 1. Schematic diagram of (a) simulation domain and (b) the equivalent transmission line model.

where the Z₀ is the characteristic impedance of the coaxial cable. Then, the X_ind can be derived from the imaginary part of the Z_ind. In the X_ind spectrum, the SRF is defined as the frequency where the X_ind sign abruptly changes from the positive to the negative sign, as shown in Fig. 1(b).

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.

Figure 2. Simulation configurations for (a) multi-turn solenoid inductor, (b) multi-turn planar inductor, and (c) one-turn planar inductor.

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.

Table 1 . Material and geometric parameters of the solenoid inductor simulation..

Parameter	Constant
Inductor material	Copper
Inductor electric conductivity	401 S/m
Coil thickness	4 mm
Teflon thickness	13.391
Teflon dielectric constant	2.1
The characteristic impedance of coaxial cable	50 Ω
Parameter	Variable
Inductor diameter	50, 75, 100, 125, and 150 mm
Number of Inductor Turns	2, 3, 4, 5, and 6
Inter-coil distance	2, 4, and 8 mm

Table 2 . Material and geometric parameters of the planar inductor simulation..

Parameter	Constant
Inductor material	Copper
Inductor electric conductivity	401 S/m
Coil thickness	4 mm
Teflon thickness	13.391 mm
Teflon dielectric constant	2.1
The characteristic impedance of coaxial cable	50 Ω
Parameter	Variable
Inductor diameter	100, 150, 200, 250, and 300 mm
Number of inductor turns	1, 2, 3, 4, and 5
Gap between inductor and inductor	4, 6, 10, and 15 mm

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 ${\mu }_0{N}^{2}A/l$, where ${\mu }_0$ is the permeability in vacuum, N is the number of turns, A is the area of the solenoid, and l is the length of the solenoid. Thus, the inductance is strongly dependent on the number of turns. Grandi et al. [18] proposed the distributed stray capacitance of solenoid inductor and based on their model, the parasitic capacitance decreases with increasing the N, since the length of the solenoid coil increases. It can have influence on the decrease in the SRF. Hence, increase in both its inductance and parasitic capacitance causes the decrease in the SRF with increasing the number of turns.

Figure 3. Reactance spectrum over frequency with (a) various numbers of turns at the diameter of 100 mm and the inter-coil distance of 4 mm, (b) several inter-coil distances at the diameter of 100 mm and 4 turns, and (c) various diameters at the inter-coil distance of 4 mm and 4 turns. Here, the SRFs are marked as arrows.

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 [18] 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 ($X_{\text{L}}$) in low frequency region where the parasitic capacitance is negligible corresponds to the inductance ($X_{\text{L}}=\text{ω}{L}_{\text{inc}}$). Figure 3(b) shows that the slope maintains in constants with increasing the inter-coil distance. Thus, the decrease in its inductance has no significant impact on the SRF and the dominant factor for the SRF is the decrease in the parasitic capacitance

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.

Figure 4. Contour plot for the SRF in the solenoid inductor with various number of turns and diameters at the inter-coil distances of (a) 2, (b) 4, and (c) 8 mm.

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.

Figure 5. Contour plot for the SRF in the planar inductor with various number of turns and diameters at the inter-coil distances of (a) 4, (b) 6, (c) 10, and (d) 15 mm. The unrealistic conditions as marked in this figure are excluded, for instance, conditions with diameter of 100 mm and the number of turns 4 in (c).

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.