Design of an optimized load-resilient conjugate T for the ICRH system in the LHD using a novel hybrid circuit/3DLHDAP code and experimental results

Maximizing antenna coupling to plasma is one of the main challenges for high-power and long-pulse operation in both tokamaks and stellarators. Many experimental studies and simulations have been dedicated to the improvement of antenna coupling and relevant impedance matching techniques [1–7]. As is well known, coupling loading R, which is defined as the equivalent resistance terminating a transmission line with characteristic impedance Z_c , is highly dependent on the edge density profile and changes rapidly (⩽ ms) under the fast-changing density resulting from phenomena such as edge localized modes (ELMs), L-H transition, and plasma start up, etc [3–5, 8, 9], which occurs too quickly to be tracked by a real-time controlled traditional matching system (e.g. a lumped capacitor and an inductance, liquid impedance matching device, whose matching time is greater than or equal to 10 ms [4]). Therefore, load resilient systems, providing an inherently voltage standing wave ratio (VSWR) less than VSWR_max (typically VSWR_max= 2:1) are hence required [3, 4].

In 2003, Monakhov proposed to use one type of conjugate-T device to improve the ion cyclotron resonance heating (ICRH) ELM tolerance on JET [10]. The conjugate-T circuit comprises two complex conjugate impedances. Simulations have demonstrated an excellent circuit tolerance to the coupling loading R perturbation. Although a low VSWR can be obtained in a large variation range, this is an ideal case that does not take into account the plasma-dependent mutual coupling between antennas. Quantitatively, coupling resistance R can only reasonably describe the coupling properties of a single antenna with plasma. A complex scattering matrix (S-matrix), in the presence of plasma, can provide a consistent description of the RF properties of a multi-strap antenna [6]. In 2018, Monakhov measured the S-matrix of 4-port ICRH antennas at JET by sweeping the relative phases of the voltages applied to straps, a comparison of TOPICA simulations of the A2 antennas with the experimental results confirms that the existing model plausibly describes the RF coupling of the antenna straps with plasma [6]. In 2019, Tierens used RAPLICASOL and TOPICA to predict the S-matrix entries and capacitor settings for the WEST ICRH antenna system [1]. Recently, the load-resilience of a conjugate-T loop consisting of two capacitors and a 3-branch node had been successfully demonstrated on plasma for WEST ICRH launchers, which agreed with simulations involving S-matrices computed on TOPICA/HFSS and RF network computations performed on SIDON [3, 5]. All these works provide the possibility to predict and verify the experimental S-matrix measurements and impedance matching setting in the presence of plasma by computer simulations.

The Large Helical Device (LHD) is a superconducting device with a R₀ (major radius) of 3.9 m, an a₀ (minor radius) of 0.6 m, an m (toroidal mode number of helical coil) of 10, an l (multipolarity) of 2 [11]. The LHD has used the poloidal array (PA), handshake form (HAS) and field-aligned-impedance-transforming (FAIT) antennas for ICRH [2, 12], and a world-record injected energy of 3.36 GJ in a long-pulse operation was achieved [12]. To realize the goal of high-power and long-pulse operation, the ICRH devices of the LHD were optimized by adopting a devised in-vessel impedance transformer (IVIT) and an ex-vessel impedance transformer (EVIT) [2]. Although triple liquid stub tuners have been employed for real-time impedance matching on the LHD, their adjustment speed is too slow to cope with the fast variations of ICRH antenna input impedance arising from fast plasma events (e.g. plasma start-up and pellet injection). To increase the ICRH injection power with fast coupling changes, the ICRH antenna system of the LHD is ready for the design of a load-resilient conjugate-T device to improve the situation. Therefore, ICRH experiments to measure S-matrices with hydrogen plasma were first conducted on the LHD recently. A program hybrid circuit/3DLHDAP was also developed, consisting of two parts: 3DLHDAP (three dimensional LHD antenna with plasma), and an RF circuit model to calculate the S parameters and optimize the load-resilient conjugate-T device for the LHD ICRH antenna system.

The commercial software COMSOL is capable of solving Maxwell's equations using finite elements, which makes it relatively straightforward to impose fully non-uniform plasma parameters, including arbitrary three-dimensional (3D) profiles and background magnetic fields. Although the validity and correctness of ICRH simulations based on COMSOL have been proven by some codes [1, 13–15], but few codes consider both Ohmic losses and the curvature of the 3D model to describe the interaction of the ICRH antenna and plasma. The hybrid circuit/3DLHDAP is a program based on COMSOL and RF design theory that builds the model according to the actual experimental conditions while taking into account the 3D magnetic fields, the 3D shape of the LHD device and PA, FAIT, and HAS antennas, and their Ohmic losses or power losses. Experimental measurements of the ICRH antennas' S-matrices with real plasma on the LHD provides a good opportunity to demonstrate the validity of the code.

This paper is organized as follows: in section 2, we introduce the method of ICRH antenna S-matrix measurement with real plasma on the LHD. In section 3, we introduce the models and equations of the hybrid circuit/3DLHDAP code for S-matrix simulation, and compare the theoretical calculations with experimental ones. According to the experimental measurement and simulation of hybrid circuit/3DLHDAP, section 4 shows the steps to optimize the load-resilient conjugate-T circuit for the LHD ICRH antenna system. Section 5 presents the conclusions of the study.

To design a conjugate-T circuit with S-matrices considering mutual coupling between antennas and change of the effective antenna position, experiments to measure the relationships between S-matrices and line-averaged electron density, magnetic field, coupling distance, namely, the distance between the plasma confined by the last closed flux surface (LCFS) and the Faraday screen, for ICRH are firstly conducted in the 24th LHD experiment campaign in 2022. ICRH power modulation with rectangular waveform (figures 1(c) and 2(a), (b), (e)) and phase scan (figures 3(c) and (f)) experiments are performed, where ICRH, ECH, and NBI are used to heat the plasma (figures 1(a)–(c)). The experiments are performed at 38.47 MHz with hydrogen, where line-averaged electron density is measured by the far infrared laser interferometer and scanned in wide range from 0.37 × 10¹⁹m⁻³ to approximately 6.27 × 10¹⁹ m⁻³ . L_vac and L are the distance between the input port of reducer attached to the directional coupler (DC) and the position of minimum voltage with vacuum injection (ICRH pulses are injected into the vacuum vessel without plasma), and the distance between the input port of reducer of DC and the position of minimum voltage during experiments. The DC has been well calibrated with vector network analyzer, thus, its error of loading resistance is within 0.1 Ω, and its error of phase is within 0.1°.

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The magnetic field on the magnetic axis ${R_{{\text{ax}}}} = 3.6\,{\text{m}}$ is B_ax , whose direction changes from clockwise (CW) to counterclockwise (CCW). The coupling distances of the FAIT and HAS antennas are ${\Delta _{{\text{FAIT}}}}$ and ${\Delta _{{\text{HAS}}}}$, respectively. The key parameters of plasma pulses are summarized in table 1.

Table 1. Key parameters of LHD pulses with ICRH antenna power modulation and phase scan.

Pulse number	Bax	Polarity	${\Delta _{{\text{FAIT}}}}$	${\Delta _{{\text{HAS}}}}$
180275–180286	1 T	CCW	6 cm	8 cm
180287–180303	2.75 T	CCW	6 cm	8 cm
180308–180329	1 T	CW	6 cm	8 cm
180330	1 T	CW	8 cm	9 cm
180333	1 T	CW	10 cm	10 cm
180409–180424	2.75 T	CW	6 cm	8 cm
180425	2.75 T	CW	8 cm	9 cm
180427	2.75 T	CW	10 cm	10 cm

The DC measured the upper antenna's forward power ${P_f}\left( i \right)$ and reflection power ${P_r}\left( i \right)$, as well as the lower antenna's forward power $P_f^{^{\prime}}\left( i \right)$ and reflection power ${P}_r^{^{\prime}}\left( i \right)$ at different times t_i (figures 2 and 3). Analyzing the datum at two moments t_i and t_{i+ 1} with equations (1) and (2), we get the S-matrix S_{a_DC} and impedance Z_{a_DC} at the position of the DC

$$\begin{align}\left( {\begin{array}{*{20}{c}} {{V_r}\left( i \right)} \\ {{V_r}^{^{\prime}}\left( i \right)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{S_{a\_DC\_11}}}&{{S_{a\_DC\_12}}} \\ {{S_{a\_DC\_21}}}&{{S_{a\_DC\_22}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{V_f}\left( i \right)} \\ {{V_f}^{^{\prime}}\left( i \right)} \end{array}} \right)\end{align} \tag{ 1 }$$

$$\begin{align}{Z_{a\_DC}} = \sqrt {{Z_c}} {\left( {I - {S_{a\_DC}}} \right)^{ - 1}}\left( {I + {S_{a\_DC}}} \right)\sqrt {{Z_c}} \end{align} \tag{ 2 }$$

where Z_c = 50 Ω is the characteristic impedance of the coaxial line; and V_f (i), V_r(i), V'_f (i), V'_r(i) are the incident and reflected voltages of the upper and lower antennas, respectively. The ABCD transmission matrix ${F_{a\_DC}}$ at the DC can be obtained by equation (3)

$$\begin{align}{F_{a\_DC}} = \left( {\begin{array}{*{20}{c}} {\frac{{{Z_{a\_DC\_11}}}}{{{Z_{a\_DC\_21}}}}}&{\frac{{\left( {{Z_{a\_DC\_11}} \cdot {Z_{a\_DC\_22}} - {Z_{a\_DC\_21}} \cdot {Z_{a\_DC\_12}}} \right)}}{{{Z_{a\_DC\_21}}}}} \\[3pt] {\frac{1}{{{Z_{a\_DC\_21}}}}}&{\frac{{{Z_{a\_DC\_22}}}}{{{Z_{a\_DC\_21}}}}} \end{array}} \right).\end{align} \tag{ 3 }$$

The ABCD transfer matrix ${F_{a\_minV}}$ at the position of minimum voltage with vacuum injection meets

$$\begin{align}{F_{a\_minV}} = F_{L1\_DC}^{ - 1} \cdot {F_{a\_DC}} \cdot F_{L2\_DC}^{ - 1}\end{align} \tag{ 4 }$$

with

$$\begin{align}&F_{Lm\_DC}^{ - 1}\nonumber\\ & \quad = \left( {\begin{array}{*{20}{c}} {\cos \left( {k{L_{m\_DC}}} \right)}&{ - j{Z_c}\sin \left( {k{L_{m\_DC}}} \right)} \\ {\frac{{ - j}}{{{Z_c}}}\sin \left( {k{L_{m\_DC}}} \right)}&{\cos \left( {k{L_{m\_DC}}} \right)} \end{array}} \right)\begin{array}{*{20}{c}} {}&{\left( {m = 1,2} \right)} \end{array}\end{align} \tag{ 5 }$$

where L_{1_DC} = 35.176 m and L_{2_DC} = 29.733 m are the distances between the DC and the minimum voltage position with vacuum injection for the upper antenna and the lower antenna, respectively. The impedance matrix Z_{a_minV} at the minimum voltage position with vacuum injection is

$$\begin{align}{Z_{a\_minV}} = \left( {\begin{array}{*{20}{c}} {\frac{{{F_{a\_minV\_11}}}}{{{F_{a\_minV\_21}}}}}&{\frac{{\left( {{F_{a\_minV\_11}} \cdot {F_{a\_minV\_22}} - {F_{a\_minV\_21}} \cdot {F_{a\_minV\_12}}} \right)}}{{{F_{a\_minV\_21}}}}} \\[6pt] {\frac{1}{{{F_{a\_minV\_21}}}}}&{\frac{{{F_{a\_minV\_22}}}}{{{F_{a\_minV\_21}}}}} \end{array}} \right).\end{align} \tag{ 6 }$$

Finally, S-matrix S_{a_minV} at the minimum voltage position with vacuum injection can be obtained

$$\begin{align}{S_{a\_minV}} = \left( {\frac{{{Z_{a\_minV}}}}{{{Z_c}}}} - I \right){\left( {\frac{{{Z_{a\_minV}}}}{{{Z_c}}}} + I \right)^{ - 1}}.\end{align} \tag{ 7 }$$

Figures 4 and 5 show the influence of density and magnetic field on measured S-matrices at the minimum voltage position with vacuum injection for the HAS and FAIT antennas, respectively, which are similar to the Smith chart. If the angle between S_{a_minV_ 11} (or S_{a_minV_ 22}) and the x-axis rotates CW, it shows that the minimum voltage position is shifted toward the antenna side. Conversely, if the angle rotates CCW, it means that the minimum voltage position is shifted away from antenna side. A ±180° rotation in S_{a_minV_ 11} and S_{a_minV_ 22} corresponds to the shift of the voltage node. The distance between the minimum voltage position and the antenna tip is an integral multiple of half the wavelength. It should be noted that the minimum voltage position simulated in the paper is the voltage node closest to the EVIT. Figures 4 and 5 provide clear evidence that when the density increases to a certain value with B_ax = 2.75 T for the HAS antennas (figures 4(a) and (b)) and B_ax = 1.0 T for the HAS and FAIT antennas (figures 4(c), (d) and 5(c), (d)), the minimum voltage position stops moving away from the antenna side and begins to move toward it. Furthermore, as the density increases, the minimum voltage position gradually moves away from the FAIT antenna ends for B_ax = 2.75 T (figures 5(a) and (b)). At lower plasma densities, the position change of the minimum voltage is more sensitive than that at high densities (figures 4 and 5). Figures 6 and 7 show the influence of coupling distance on the measured S-matrices at the minimum voltage position with vacuum injection for the HAS and FAIT antennas, respectively. For both the HAS and FAIT antennas, with the increase of coupling distance, the position of the minimum voltage gradually moves to the antenna side. When magnetic field changes from 1 T to 2.75 T, |S_{a_minV_ 11}| and |S_{a_minV_ 22}| decrease.

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Complementary to the experiments on the LHD, a theoretical simulation for ICRH antenna S-matrices in the presence of plasma not only provides a unique opportunity for verification of the simulation tools, but it also makes the comparison and assessments of measured and simulated S-matrices more straightforward. Therefore, we developed a hybrid circuit/3DLHDAP code based on COMSOL to study the S-matrix when the 3D ICRH antenna interacts with the LHD plasma.

3.1. Models and equations for the interaction between ICRH antenna and plasma

The model of interaction between the 3D ICRH antenna and the LHD plasma in the hybrid circuit/3DLHDAP code is shown in figure 8(a), where R, Φ, and z are the radial, toroidal, and axial directions of the LHD, respectively. The 3D magnetic configuration is reconstructed by VMEC [16], and the magnetic field is calculated from Biot–Savart's law [17]. The 3D antenna is located between the LCFS and the wall. The amplitude and flux surfaces of magnetic field, and the projection of the antenna for the Φ = 0 cross section are shown in figure 8(b), where the magnetic radius of the magnetic axis and the strength of the magnetic field on the axis are fixed at 3.6 m and 1.0 T, respectively. The material of the LHD wall is stainless steel, and the antenna head models are from the PA, HAS, and FAIT antenna heads in the LHD [2, 12], which are mainly composed of a Faraday shield (FS), two straps, protectors, and a portion of the input transmission lines. To provide a more precise description of the behavior of the antenna, the antenna geometry is curved; the materials of the components marked in blue (side protectors), red (top protectors), gray (FS, current straps, transmission lines), and green (FS, current straps) are respectively carbon fiber composite, graphite IG-430U, stainless steel, and molybdenum (figure 9).

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The radial profile of the electron density is obtained by fitting the measured results of Thomson scattering systems installed at the 3-O port (Φ = 18°) (figure 10). In the plasma region, the dielectric tensor only includes the collision effect (collision frequency $\upsilon$ is approximately equal to $0.1\omega$) and does not include the temperature effect. The cold plasma dielectric tensor $\varepsilon$ can be obtained by rotating ${\varepsilon _{stix}}$ [18] from local coordinates (Stix frame) to the laboratory frame with the method introduced in [19]. Using impedance boundary conditions, the finite element method, and the S-matrices of the antenna head S_{a_head} , the fields in the whole plasma-vacuum domain can be solved. Then, the total power absorbed by plasma is given by [20]

$$\begin{align}{P_{{\text{abs}}}} = \int_V {\frac{{\omega {\varepsilon _0}}}{2}} \operatorname{Re} \left[ {E_p^*\left( { - j} \right) \cdot \varepsilon \cdot {E_p}} \right]{\text{d}}V\end{align} \tag{ 8 }$$

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where ${\varepsilon _0}$, $\varepsilon$, and E_p are vacuum permittivity, the dielectric tensor, and the wave electric field in the plasma, respectively. The antenna radiation energy is [21]

$$\begin{align}{P_{{\text{rad}}}} = \frac{1}{2}\operatorname{Re} \int_S {J_{{\text{an}}}^*} \cdot E{\text{d}}S\end{align} \tag{ 9 }$$

where S denotes the surface of the current straps, and J_an is the antenna surface current density. Heating efficiency is defined as [22]

$$\begin{align}\eta = \frac{{{P_{{\text{abs}}}}}}{{{P_{{\text{rad}}}}}}.\end{align} \tag{ 10 }$$

3.2. Models and equations for S-matrix simulation

The FAIT and HAS antennas are mainly composed of an antenna head, an IVIT, a feedthrough, and an EVIT. Figure 11 shows a schematic view of the upper and lower FAIT antennas. The distances between the EVIT and the minimum voltage position with vacuum injection for the upper antenna (point A) and the lower antenna (point B) are respectively L_{1_V} and L_{2_V} , respectively. The upper and lower antennas can be modeled as a two-port network consisting of a cascade of an EVIT, a feedthrough, an IVIT, an antenna head, and ports 1 and 2 at points A and B, respectively. The ICRH antenna head's S_{a_head} is obtained by hybrid circuit/3DLHDAP (section 3.1), and the S parameters of the EVIT, feedthrough, and IVIT are calculated by the high frequency structural simulator (HFSS) [23]. Using equation (11), the S parameters of each part are combined one after another in sequence to get the antennas S parameters S_{a_minV} at the minimum voltage position with vacuum injection, starting from point A and ending with the point B

$$\begin{align}{S_{{\text{combine}}}} = \left( {\begin{array}{*{20}{c}} {S_{11}^{\left( 1 \right)} + S_{1\_c}^{\left( 1 \right)}}&{\frac{{S_{12}^{\left( 1 \right)} \cdot S_{12}^{\left( 2 \right)}}}{{1 - S_{22}^{\left( 1 \right)} \cdot S_{11}^{\left( 2 \right)}}}} \\ {\frac{{S_{21}^{\left( 1 \right)} \cdot S_{21}^{\left( 2 \right)}}}{{1 - S_{22}^{\left( 1 \right)} \cdot S_{11}^{\left( 2 \right)}}}}&{S_{22}^{\left( 2 \right)} + S_{2\_c}^{\left( 2 \right)}} \end{array}} \right)\end{align} \tag{ 11 }$$

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with

$$\begin{align*}S_{1\_c}^{\left( 1 \right)} = \frac{{S_{12}^{\left( 1 \right)} \cdot S_{21}^{\left( 1 \right)} \cdot S_{11}^{\left( 2 \right)}}}{{1 - S_{22}^{\left( 1 \right)} \cdot S_{11}^{\left( 2 \right)}}}\end{align*}$$

$$\begin{align*}S_{2\_c}^{\left( 2 \right)} = \frac{{S_{22}^{\left( 1 \right)} \cdot S_{12}^{\left( 2 \right)} \cdot S_{21}^{\left( 2 \right)}}}{{1 - S_{22}^{\left( 1 \right)} \cdot S_{11}^{\left( 2 \right)}}}.\end{align*}$$

3.3. Benchmarking of the code hybrid circuit/3DLHDAP

We benchmark our code with the experiment results of the LHD. In the ICRH, resonance ions are accelerated in a direction perpendicular to a magnetic field; then, ions with a larger perpendicular velocity slow down through the Coulomb collision with low-temperature background electrons and non-resonant ions [24]. As PA antennas have a poloidal array, the peak ${k_{||}}$ mainly locates near zero, direct electron heating by Landau damping and transit time damping is very small, and absorption occurs mainly through ion cyclotron resonances. Therefore, our collision model can approximately simulate the minority ion heating efficiency of PA antennas. Figure 12 shows the experimental [22] and simulation results for the heating efficiency of PA antennas without an FS at different current phases. The heating efficiencies of the experiments results were estimated from the power modulation experiment. Although the modulation amplitude of the stored energy has been used to derive the heating efficiency with a simple model [25], absolute errors may occur. However, trends with respect to phase can be clearly observed. The tendency of the simulation is consistent with the experimental results, which verifies the reasonableness and correctness of the code. The difference of absolute values between the experiment and the simulation may be due to the artificial collisional damping implemented in the simulation instead of real minority ion heating. Next, we use this code to calculate the S parameters of the FAIT and HAS antennas and compared them with experimental ones.

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3.4. S-matrix simulation for FAIT and HAS antennas with hybrid circuit/3DLHDAP

Figures 13–15 respectively show the simulated S parameters at the minimum voltage position with vacuum injection for the HAS and FAIT antennas heating the plasma under different densities, magnetic fields, and coupling distances. The simulated trends are agree with the experimental measurement results (figures 4–7). And the simulation results confirm these facts. When B_ax is 2.75 T for the HAS antennas (figures 13(a) and (b)) and 1 T for the HAS and the FAIT antennas (figures 13(c), (d) and 14(c), (d)), there is indeed a turning point with a threshold density, where the S_{a_minV_ 11} and S_{a_minV_ 22} turn CCW to CW, and |S_{a_minV_ 11}| and |S_{a_minV_ 22}| increase. Reversing the magnetic field direction makes the mutual coupling S_{a_minV_ 12} and S_{a_minV_ 21} exchange (e.g. figures 13(a) and (b)). The mutual coupling is sensitive to low density, changing significant fluctuation with it. The measurement results of the experiments also confirmed that the impact of density on mutual coupling was greater at low densities (green dots in figures 4 and 5) than at high densities (red dots in figures 4 and 5).

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Figure 16 shows the simulated mutual coupling |S_{a_minV_ 12}|² for the FAIT and HAS antennas under the same conditions, with ${\overline n _e}$ = 0.4–10 × 10¹⁹m^–3, B_ax = 1.0 T, and CW. The mutual coupling between the upper antenna and the lower antenna of the HAS is greater than that of the FAIT antennas over a wide density range. This difference is due to the different placements of the FAIT and HAS antennas: the FAIT antennas are poloidally aligned [12], whereas the HAS antennas are placed in the toroidal direction [26]; the former is preferable for the weak mutual coupling between two antennas [27]. Therefore, the impedance matching can be easily done over wide parameters changes for the FAIT antennas.

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Figure 17 shows a schematic diagram of the ICRH antenna system. A and B are the minimum voltage points of the upper and lower antennas, respectively (shown in figure 11), and T is the connection point between the transmission line |TC| and the conjugate-T circuit. The lengths of the two legs of the conjugate-T are |AT| = L₁ and |BT| = L₂, respectively, which can be determined by equation (12) if two S_{a_minV} values are obtained experimentally. We choose two S parameters, S_{a_minV 1} and S_{a_minV 2} from all measured S parameters. Impedance matching at point D is assumed to be performed at the S-matrix S_{a_minV 1} and S_{a_minV 2}. To keep the reflected power zero even if the plasma condition changes, the following conjugate-T condition must be satisfied [28]

$$\begin{align}Y\left( {{S_{a\_\min V1}},{L_1},{L_2}} \right) = Y\left( {{S_{a\_\min V2}},{L_1},{L_2}} \right)\end{align} \tag{ 12 }$$

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where Y is the admittance at position T.

The impedance at the minimum voltage position on line |TC| in vacuum injection is R_m when the reflection coefficient is zero, and Z is the impedance at the same position when S_{a_minV} is changed. Then, assuming that the condition of matching device is the same in figure 17, the reflection coefficient |V_r /V_f | at position D is [28]

$$\begin{align}\left| \Gamma \right| = \left| {\frac{{Z - {R_m}}}{{Z + {R_m}}}} \right|.\end{align} \tag{ 13 }$$

According to the rule of S-matrices obtained in experiments and simulations, we know that the S_{a_minV_ 11} parameters at low density and high density are located near the lines imaginary part of S_{a_minV_ 11} equals to zero and real part S_{a_minV_ 11} is zero in the third quadrant of figure 18, respectively. To keep the reflection low with the designed conjugate-T device, we choose one S parameter, S_{a_minV 1}, at low density, and another, S_{a_minV 2}, at high density, and attempt to ensure that most measured S parameters exist between them. By solving equation (12), several sets of solutions can be obtained. To obtain high heating efficiency in minority ion heating, we choose a solution that ensures the current phase difference between the two points A and B of the conjugate-T circuit (figure 17) close to 180° to excite a large parallel wave number for the HAS antennas. As any current phase can achieve a high heating efficiency for FAIT antennas, a solution with the lowest reflection coefficient among the several solutions is chosen. For example, if we choose S_{a_minV 1} and S_{a_minV 2} respectively at positions 1 and 2 for the HAS antennas (figure 18(a)) and FAIT antennas (figure 18(b)), conjugate-T device's optimal lengths—L_{1_HAS} = 4.047 m, L_{2_HAS} = 4.660 m for the HAS antennas and L_{1_FAIT} = 1.236 m, L_{2_FAIT} = 4.074 m for the FAIT antennas—can be respectively obtained with the equation (12) to get a low reflection coefficient over a wide parameters range. If we select S_{a_minV 1} and S_{a_minV 2} at positions 1' and 2', respectively, for the FAIT antennas (figure 18(b)), only a small number of S-parameters are located between these two S-parameters, and the lengths of the conjugate-T circuit are L'_{1_FAIT} = 3.880 m, L'_{2_FAIT} = 1.780 m, respectively, as determined by equation (12). This type of conjugate-T device makes the reflection coefficient even higher than the previous one over a wide range of parameters (figure 19). Therefore, we should choose S_{a_minV 1} and S_{a_minV 2} carefully to obtain the optimized conjugate-T system.

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To illustrate the optimized conjugate-T's performance over a wide range of plasma parameters, the reflection coefficient |Γ| of the antenna system with and without the conjugate-T is shown in figure 20. Without the conjugate-T, the upper or lower antenna system can only make the reflection coefficient very low near the matching point (point 2 for HAS antennas (figure 20(a)) and point 1 for FAIT antennas (figure 20(b)). With the optimally designed conjugate-T, the antennas system can maintain low reflection coefficients (less than 0.4) over a wide range of parameters without a controlling impedance matching device.

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ICRH heating experiments and simulations to design the load resilient conjugate-T system for the LHD have been carried out in this paper. ICRH power modulation and phase scan to measure S-matrices with real hydrogen plasma were firstly conducted on the LHD, and the change rules of S-matrices for HAS and FAIT antenna systems with density, magnetic field, and coupling distance during heating experiments were discussed in section 2. To further confirm the rules of S-matrices observed in experiments, a hybrid circuit/3DLHDAP code to predict the S-matrices in the presence of LHD plasma with PA, FAIT, and HAS antennas heating the plasma has been developed and benchmarked.

The geometries of the model in hybrid circuit/3DLHDAP are built according to the actual size of the LHD device and PA, FAIT, and HAS antennas, which takes into account their curvature and Ohmic losses or power losses of them. The 3D magnetic configuration is reconstructed by VMEC, magnetic field is calculated from Biot–Savart's law, and a cold plasma model including the collision effect is adopted. The FAIT and HAS antennas' IVIT, feedthrough, and EVIT are separately calculated by HFSS, and then held them together with 3DLHDAP in RF circuit modeling. This hybrid circuit/3DLHDAP combines ICRH antennas' impedance matching, coupling, and power deposition process, and can predict port S-matrices, port reflection coefficients, the electric currents on conductors, electric fields distribution in plasma, power coupled to plasma, and optimal matching settings, which are very important to evaluate and predict the ICRH system performance.

In this paper, we mainly use hybrid circuit/3DLHDAP to simulate S-matrices for the ICRH antenna system. The variation of S-matrices for HAS and FAIT antenna systems with density, magnetic field, and coupling distance during heating obtained by the calculations with the code hybrid circuit/3DLHDAP is in agreement with the experiment results (section 3). Both theoretical simulations and experimental measurements show that when the magnetic field on the magnetic axis equals to 2.75 T and 1.0 T, the rotation direction of the S_{a_minV_ 11} and S_{a_minV_ 22} for the FAIT and HAS antennas with increasing density (within a certain range) is opposite to that with increasing coupling distance, which implies that under certain conditions, controlling the coupling distance may compensate for the S-parameters changes caused by plasma density variation, and keep the minimum voltage position unchanged. In the LHD, the effective length is defined as the distance between the stub tuner closest to the antenna and the minimum voltage position [29], which was found to be increased during long-pulse ICRH operation [30]. By adjusting the coupling distance, it is possible to fix the effective length, and may make it easier for the ICRH antenna system to obtain impedance matching during long-pulse operation. The simulations also suggest that under the same conditions, the mutual coupling between the upper antenna and lower antenna of the HAS is greater than that of the FAIT antennas over a wide range of densities, which proves that the relative position between the straps has a great influence on the mutual coupling.

According to the rule of S-matrices obtained in experiments and simulations, we respectively optimize the HAS and FAIT antennas' conjugate-T system by the code hybrid circuit/3DLHDAP. The simulation shows that by using the optimized conjugate-T, reflection coefficients can be kept low without a controlling impedance matching device for both HAS and FAIT antennas over a wide range of plasma parameters. The hybrid circuit/3DLHDAP's ability to correctly predict the S-matrix in the presence of plasma and set the optimal impedance matching network is crucial in the antenna design phase, and the related results provide good confidence for future high-power long-pulse operation of the fusion devices' ICRH antenna system.

The authors gratefully acknowledge Dr Kamio S. providing original experiment datum of figure 12, and thank the LHD experiment group members and technical staff members for their contributions to this work. This work is supported by National magnetic confinement fusion energy development research project (2022YFE03070003), Key projects of Hunan Provincial Department of Education (20A432), Government Sponsored Study Abroad Program of Chinese Scholarship Council (202108430056), Natural Science Foundation of Hunan Province (2020JJ4515), National magnetic confinement fusion energy development research project (2019YFE03070000), the LHD budget ULRR703 from National Institute Fusion Science. The LHD data can be accessed from the LHD data repository at www-lhd.nifs.ac.jp/pub/Repository_en.html.