In general, a chamber wall is grounded for safety when experimenting with radio frequency (RF) power [1–4]. However, although the exterior wall of the chamber is grounded, the RF current does not flow along the ground line. This is because the area of the current loop formed between the coaxial cable and the chamber is smaller than that made with the ground line [5]. Thus, the current flows toward a smaller impedance, and the smaller the area, the smaller the impedance of the formed closed loop. Consequently, the current does not flow to the ground line but returns through the coaxial line shield. Therefore, grounding the chamber wall is necessary only for safety reasons.

However, if a magnetic gradient field is generated around the chamber, a current induced by picking up the magnetic field flux can flow through a current loop formed with a ground line [6]. This induced current can flow to the exterior wall of the chamber and cause safety accidents. In addition, the magnetic gradient field can affect experimental results or equipment. Typically, several apparatuses are present in a laboratory generating magnetic gradient flux, such as inductively coupled plasma (ICP). Thus, understanding how changes in the surrounding magnetic field affect experimental results or equipment is necessary. However, the effect of grounding the exterior wall of the chamber under a peripheral magnetic gradient field should be investigated. In this study, the RF current path flow was confirmed through simulation by grounding or opening the exterior wall of the chamber. Moreover, the effect of grounding the chamber exterior wall on safety was discussed for the case when the solenoid shape was designed around the chamber and a coaxial line to generate a magnetic gradient flux.

A high-frequency time-domain solver provided by CST Microwave Suite (Dassault Systèmes, France) was used to analyze the effect of the grounding chamber wall. The software can calculate simulation models, including surface current and electromagnetic field, in the range of 0 to several GHz by solving the Maxwell equation in a threedimensional space. In this simulation, the profile of the surface current and electromagnetic field and the voltage values for each position of the coaxial cable shield at 13.56 MHz were obtained.

Figure 1(a) shows the chamber and solenoid configurations in our simulation, where the diameter and height of the chamber are 40 mm and 19 mm, respectively. Moreover, the diameter and number of turns of the solenoid are 30 mm and 5, respectively. The distance between the electrodes inside the chamber is 20 mm, and plasma with a sheath thickness of 0.234 mm is between the electrodes. The plasma is assumed to be a dispersive dielectric material in the Drude model [7]. In the simulations, the density of the plasma was 10¹¹ cm⁻³, and the pressure was 10 Torr. Figure 1(b) shows the boundary conditions for the simulation. In particular, the simulation condition was divided into two types: one, where the bottom of the coaxial cable under the chamber was grounded, and in the other, it was opened. Thus, the boundary conditions of the simulation were set to the ground only on the top and bottom surfaces, and the other surfaces were set to open space.

Figure 1. (a) Configuration of the simulation structure and y-plane cut. (b) Boundary conditions in the simulation.

The simulation was conducted considering two situations: (i) a simulation where RF power was applied to Port 1 to check the current flowing in the loop formed between the coaxial cable and the chamber, depending on whether the chamber is grounded. (ii) a simulation where RF power is applied to Port 2 to generate magnetic gradient fields in the solenoid and confirm the result of the chamber ground. All the materials used in the simulation were set to a perfect electric conductor, and all dielectrics were set to polytetrafluoroethylene (PTFE).

3.1. Coaxial cable signal applied

In a high-frequency circuit, the current flows along a path with low impedance (Z), particularly reactance (X). This is because the higher the frequency, the greater the effect of reactance. The impedance and reactance equation is as follows:

Where R is the resistance, L is the inductance, and C is the capac- itance, respectively. The second term on the right side represents a reactance equation. The equation shows that the reactance increases as L increases and C decreases. Thus, when the other riables are constant, the current flows toward a smaller L.

As shown in Fig. 2, when a signal is applied to Port 1, two current loops are present: (i) The path where the return current flows from the coaxial cable core through the chamber and inside the shield. (ii) The path where the current returns from the coaxial cable core through the chamber to the ground line. The impedance of these two paths can be confirmed using the loop inductance (L_LOOP) equation. Assuming a square, as shown in Fig. 3, the loop inductance expression is as follows [5]:

Figure 2. Current path configuration that can flow.

Figure 3. Rectangular ring of lead wire with radius r.

where Φ_total is the total magnetic flux, I is the current on closed loop. As shown in Eq. (2), the loop inductance decreases when either a or b decreases. That is, when the area of the loop is reduced, inductance is also reduced. Thus, current flows through path (i), where the loop inductance is small, regardless of the presence or absence of the chamber ground.

Figures 4(a) and 4(b) show the surface current path when a signal is applied to Port 1. The surface current flows along the coaxial cable to the upper electrode of the chamber. Then, it flows under the electrode in the form of a displacement current, flowing into the coaxial cable core below the chamber. We created a coaxial cable shape without directly grounding the exterior wall of the chamber. This was performed to maximize the effect of the grounding of the exterior wall by sending the current flowing through the lower electrode to the ground without loss. If the exterior wall of the chamber is grounded, the current flowing to the coaxial cable core can flow through two paths: the ground line and the coaxial cable shield. As shown in the simulation results, the current returned to Port 1 along the inner wall of the coaxial shield. This is because the loop inductance of the path was small, and thus, the current path did not change regardless of whether the exterior wall of the chamber was grounded.

Figure 4. Current path when the power is applied to Port 1. (a) Ground condition. (b) Open condition.

3.2. Solenoid signal applied

Figure 5(a) shows the surface current path of the chamber ground condition when a signal is applied to Port 2. In the ground condition, when the current flowing through the solenoid increased, the induced current flowed counterclockwise on the shield of the coaxial cable. This result suggests that an induced current is generated because a closed loop formed with the ground line picks up the magnetic gradient flux generated by a solenoid. Because the ground line was included in the boundary surface, the current flowing along the ground line did not appear in the simulation. Figure 5(b) shows the magnetic field vector in the same phase; a magnetic field is formed in the entering direction inside the solenoid, and a magnetic field rotating in the counterclockwise direction by the induced current is formed around the coaxial cable. This indicates that the current flowing along the solenoid generates a magnetic field inside it.

Figure 5. Simulation results when power is applied to Port 2 in ground condition. (a) Current path. (b) Magnetic field vector.

Figures 6(a) and 6(b) show the surface current and magnetic field vector of the open condition, respectively. In the open condition, a current path appeared in the direction opposite to that of the ground condition. This is because the circuit is disconnected in the open condition, unlike the ground condition, where a coaxial line forms a closed loop that picks up the magnetic gradient flux with a ground line. In a disconnected circuit, an electromotive force is induced to interrupt the increasing magnetic flux inside the solenoid, and a current path is generated by the voltage difference caused by the electromotive force [8– 10]. We obtained a voltage waveform for each location of the coaxial cable shield using the voltage monitoring function to confirm whether the current flowing along the coaxial cable was due to the electromotive force. Figure 7(a) shows one period of the voltage waveform of the voltage monitors shown in Fig. 1(b). As shown in Fig. 7(a), the amplitude of the voltage increases as the distance from the ground increases. This can be observed as the voltage swinging up and down along the coaxial cable with respect to the ground, as shown in Fig. 7(b). This result shows that when the bottom of the chamber is opened, a voltage difference occurs from the port to the bottom due to the electromotive force. In the open condition, the result differs from the current direction of the short because the current flowing due to this voltage difference appears.

Figure 6. Simulation results when power is applied to Port 2 in open condition. (a) Current path. (b) Magnetic field vector.

Figure 7. (a) Results of the voltage monitoring. (b) Voltage graph by the position of the coaxial cable shield.

In this study, simulations showed that the current did not flow along the ground line of the chamber exterior wall when power was applied. This is because the impedance of the closed loop formed with the ground line was larger than that along the inner wall of the coaxial cable. Therefore, this result suggests that, in a general case, grounding the exterior wall of the chamber can be performed only for safety reasons. However, when magnetic gradient fields exist around the chamber, grounding the exterior wall of the chamber might not necessarily be safe. From the simulation results, we confirmed that the induced current flowed in the closed loop formed between the coaxial cable and the exterior wall of the chamber under magnetic gradient fields. This current is caused by the closed loop picking up the magnetic gradient flux. We performed the same simulation without grounding the exterior wall of the chamber. Consequently, the electromotive force was induced by the magnetic gradient fields in the opening path formed between the coaxial cable and the chamber, and the current flowed in the opposite direction. Therefore, grounding the chamber exterior wall may be dangerous because the current is induced in the chamber wall when peripheral magnetic gradient fields are generated. In our simulation, the solenoid was shaped similarly to a structure generating a magnetic field in the surrounding equipment. In the real industry, some structures can generate a magnetic field, such as ICP, around the equipment used, and the simulation results will help interpret the unanalyzed results that may occur in such environments. Moreover, even with frequencies other than 13.56 MHz, the wavelength of 13.56 MHz is approximately 22 m, much longer than our simulation dimension. Therefore, the simulation results are unlikely to change under similar frequency conditions.

This research was supported by a research fund from Chungnam National University in 2022.

The authors declare no conflicts of interest.