Cite this Article
Zhang Jia-Hui, Zhang Xin-Jun, Zhao Yan-Ping, Qin Cheng-Ming, Chen Zhao, Yang Lei, Wang Jian-Hua. Theoretical analysis of the EAST 4-strap ion cyclotron range of frequency antenna with variational theory. Chinese Physics B, 2016, 25(8): 085201  
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Theoretical analysis of the EAST 4-strap ion cyclotron range of frequency antenna with variational theory
Zhang Jia-Hui1, 2, Zhang Xin-Jun1, †, , Zhao Yan-Ping1, Qin Cheng-Ming1, ‡, , Chen Zhao1, 2, Yang Lei1, 2, Wang Jian-Hua1
Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, China
University of Science and Technology of China, Hefei 230026, China

 

† Corresponding author. E-mail: xjzhang@ipp.ac.cn

‡ Corresponding author. E-mail: chmq@ipp.ac.cn

Project supported by the National Magnetic Confinement Fusion Science Program, China (Grant No. 2015GB101001) and the National Natural Science Foundation of China (Grant Nos. 11375236 and 11375235).

Abstract
Abstract

A variational principle code which can calculate self-consistently currents on the conductors is used to assess the coupling characteristic of the EAST 4-strap ion cyclotron range of frequency (ICRF) antenna. Taking into account two layers of antenna conductors without lateral frame but with slab geometry, the antenna impedances as a function of frequency and the structure of RF field excited inside the plasma in various phasing cases are discussed in this paper.

PACS: 52.50.Qt;52.35.Hr;52.25.Os;52.55.Fa
Keyword:variational principle;ICRF antenna;impedance
1. Introduction

Heating in the ion cyclotron range of frequencies (ICRF) is one of the important auxiliary heating methods in the EAST tokamaks. A key issue is the improvement of the coupling between the ICRF antenna and the plasma. Over the past years, a lot of efforts have been made to analyze and design the ICRF antenna.[1–4] In this paper, we will apply a variational theory to evaluate the coupling characteristic of the EAST 4-strap ICRF antenna.

Theihaber and Jacquinot came up with the variational theory to calculate the self-consistent current distribution in the straps of the simple back-to-back ICRF antenna[5] and then improved the coupling code to take into account a more complicated trombone antenna geometry surrounded by an opaque frame[6] in JET. The coupling code was extended by Saigusa et al. so that it can be applied to a toroidal trombone antenna array with solid septa[7] despite its complication. Zhang et al. made some progress on Theihaber and Jacquinot’s coupling code to analyze the ICRF antenna array in HT-7U.[8] In the present paper, the EAST 4-Strap ICRF antenna impedances as a function of frequency and the structure of RF field excited inside the plasma in various phasing cases are presented by using a modification coupling code with the variational theory. For simplicity, the naked ICRF antenna model without lateral structure is employed in spite of the effect produced by the limiter frame while multiple current conductors are taken into account. The rest of this paper is organized as follows. In Section 2 the 4-strap ICRF antenna model is described, and the derivation of the wave equations in vacuum is also re-presented. In Section 3, we investigate the variation of the input impedance at the feeder point relating to antenna parameters and compare the RF field structures in different antenna phasing cases. Conclusions about the 4-strap ICRF antenna model are drawn in Section 4.

2. Model and variational theory
2.1. Description of the model

The model of a single antenna is shown in Fig. 1. The z direction of the coordinate is parallel to toroidal magnetic field BT, the y direction is in the poloidal field direction, and the x direction is in the radial direction pointing to the plasma. The antenna conductors are represented by infinitely thin metallic sheets. They radiate through an ideal electrostatic screen, which is perfectly conducting in the z direction and perfectly insulating in the y direction. This screen is completely opaque to the TM modes (Ez ≠ 0) and completely transparent to the TE modes (Hz ≠ 0), which carry the power into the plasma.

Fig. 1. Geometry of the coupling model: (a) schematic view, (b) poloidal cross-sectional view for the single ICRF antenna, and (c) plasma density profile.
Table 1.

Antenna parameters.

.

The back wall of the antenna is located at x = 0, the electrostatic screen at x = c and the plasma edge at x = d. The antenna consists of two main radiating conductors at x = b2 and two return conductors at x = b1. The feeder point connects to the return conductors at y = yF. The upper main radiating conductor short point is placed at y = y1 and the lower main radiating conductor short point is placed at y = −y2. The widths of the conductors in the z direction are both w.

The plasma is assumed to have a linear density profile over a length ln from a low edge density of n01 at x = d to a maximum density of n0 at x = d + ln. The radiation condition is imposed at x = d + ln so that all outgoing waves inside the plasma are absorbed without reflection.

2.2. Derivation of the wave equations

We normalize Maxwell’s equations in the calculations as done in Ref. [5]. The subscript ‘p’ denotes the physical quantity, then: x = (ω/c)xP, E = EP, H = (μ0/ε0)1/2HP, and J = (1/ωε0)JP. With these normalizations, expressions for the impedance should be multiplied by (μ0/ε0)1/2 = 377 Ω to recover the physical result. We then have Maxwell’s equations:

From Maxwell’s equations, the equations for Ez and Hz are

where

The current distributions on conductors of the antenna are modeled by

where γ(x) is the step-function and δ(x) the Dirac delta. The solution for Ez (0 ≤ x ≤ c) consistent with the boundary conditions (Ez(x = 0) = Ez(x = c) = 0) is:

where

The boundary conditions for Hz are imposed as follows:

where YP is the plasma admittance defined as

at x = d. The solution for Hz satisfying the boundary conditions can be written as

where

and

Having obtained the solutions for Ez and Hz, we can proceed to find the other field components through the procedures in Appendix A of Ref. [5] as follows:

Furthermore, the feeder point impedance can be written in the variational form as

We can calculate the integration by using Parseval’s theorem

In Eqs. (19) and (20), D denotes the surface of the central conductors, J and K are trial functions for the currents, and IAJ and IAK are the total currents flowing at the feeder point, which are consistent with J and K, respectively.

The condition δZA = 0 will lead to the physical current J flowing in the conductors and the corresponding antenna impedance ZA at the feeder point.

2.3. Choice of trial functions for a single strap antenna

For simplicity, we assume that the current on the conductor in the z direction is uniform because the conductors are much longer than width, and the width is much shorter than a wavelength at the operating frequency. Then

where M(z) = 1 on the conductor (|z| ≤ w/2) and is 0 otherwise.

In order to satisfy continuity conditions at the connections and the short-circuit condition dJy2(y)/dy = 0 at y = y1 and y = −y2, we take advantage of the current distributions corresponding to the transmission line model in the upper and lower conductors, respectively. The upper and the lower antenna loop are balanced (yF = (−y1 + y2)/2), so we can write down the trial functions as follows:

where γ(x) is the step-function γ(x) = 0, 1/2, 1 for x < 0, x = 0, x > 0, respectively. We could express each current in terms of a linear combination of these trial functions:

Here, an and bn are the coefficients for J and K, where n = 0, 1, …, N. N = 4–5 is large enough to result in a rather small calculation error for ZA. With these trial functions, equation (19) can be rewritten as

where Pn = 2wcos [nk(yF + y2)] and the matrix elements are as follows:

Requiring an extremum δZA = 0, we will obtain an input impedance at the feeder point and the physical current coefficients on conductors:

where L = ‖Lmn‖ and P = (P0 ⋯ PN)T.

2.4. Extension for the antenna array

The situation with four identical current straps in the z direction can be treated by rewriting the current distribution and resolving Maxwell’s equations. The current distribution for the N-strap antenna array can be written as

where J(x,y,z) is the current distribution for a single strap antenna, Δz is the period of the antenna array in the z direction, and φn is the current phase of the n-th current strap. If the mutual coupling between current straps is ignored, we can obtain the impedance for each current strap by modifying Eq. (19) into the following expression:

3. Simulation results

In this section, first we give the frequency scans in the presence of the plasma for a single ICRF antenna. Then, we investigate the antenna behavior dependent on the antenna parameters at 30 MHz around which the EAST ICRF antenna operates frequently. Finally, the field structures corresponding to the single ICRF antenna and the ICRF antenna array are provided.

3.1. Frequency characteristics

Frequency scans with and without the plasma for a single strap ICRF antenna are shown in Fig. 2. The antenna parameters are listed in Table 1. The plasma edge density is assumed to be n01 = 3 × 1012 cm−3, the central density n0 = 2 × 1013 cm−3, and the linear profile length ln = 0.1 m. In the following figures, the line with black squares represents the resistance of the impedance R = Re(ZA) and the line with white squares refers to the reactance X = Im(ZA). According to the definition of the code, the negative and the positive signs of X represent the inductive and capacitive reactances, respectively. Figure 2(a) shows that the single strap ICRF antenna in vacuum becomes resonant at f = 31 MHz, where R reaches a maximum value and X starts to approach to zero rapidly. The minor peak of R occurs at f = 28 MHz because of the larger numerical calculation error near the resonant point. As shown in Fig. 2(b), the presence of the plasma smooths the frequency responses of R and X in vacuum. R is much larger and inductive reactance is smaller than their counterparts of vacuum condition in Fig. 2(a).

Fig. 2. (a) Frequency scans of the antenna impedance ZA = R + iX in vacuum. The line with black squares represents the resistance of the impedance R = Re(ZA) and the line with white squares refers to the reactance X = Im(ZA). (b) Frequency scans of the antenna impedance ZA = R + iX in the presence of a DH plasma: nD/ne = 0.9, nH/ne = 0.1, B = 2.3 T, n01 = 3 × 1012 cm−3, n0 = 2 × 1013 cm−3, ln = 0.1 m, the antenna parameters are indicated in Table 1.
3.2. Dependence on antenna parameters

Figure 3 shows the variations of real part (resistance) and imaginary part (reactance) of impedance with antenna parameters for a single antenna. The resistance of the impedance R = Re(ZA) and the reactance X = Im(ZA) first fall down rapidly as d − c increasing in Fig. 3(a). Then R and X remain almost unchanged as d–c increases beyond 60 cm. R in Fig. 3(b) is similar to that in Fig. 3(a) dependent on c − b2 while X first increases with c − b2 increasing. As shown in Fig. 3(c), R goes up and X falls down with the distance b2 − b1 increasing steadily. Figure 3(d) shows that R decreases with b1 increasing rapidly and X reaches a maximum value at b1 = 0.04 m approximately.

Fig. 3. (a) Antenna impedance versus distance d − c. y1 = 0.01 m, y2 = 0.4 m, w = 0.1 m, b1 = 0.02 m, b2 − b1 = 0.1 m, c − b2 = 0.01 m; (b) Antenna impedance versus distance c − b2. y1 = 0.01 m, y2 = 0.4 m, w = 0.1 m, b1 = 0.02 m, b2 − b1 = 0.1 m, d − c = 0.01 m; (c) Antenna impedance versus distance b2 − b1. y1 = 0.01 m, y2 = 0.4 m, w = 0.1 m, b1 = 0.02 m, c − b2 = 0.01 m, d − c = 0.01 m; (d) antenna impedance versus distance b1. y1 = 0.01 m, y2 = 0.4 m, w = 0.1 m, b2 − b1 = 0.1 m, c − b2 = 0.01 m, d − c = 0.01 m.
3.3. RF field structure

The three-dimensional (3D) distributions of the electric field |Ey| in the poloidal and toroidal direction at the plasma edge are shown in Fig. 4 for a single strap ICRF antenna and for 4-strap ICRF antenna array in different toroidal phasing cases. We can see that the electric field with toroidal phasing (0,π,0,π) is much smaller than that with the phasing (0,0,0,0,).

Fig. 4. (a) Distribution of |Ey| at the plasma edge for the single antenna; (b) distribution of |Ey| at the plasma edge for the 4-strap antenna array with toroidal phasing (0,0,0,0,) and period Δz = 0.2 m; (c) distribution of |Ey| at the plasma edge for the 4-strap antenna array with toroidal phasing (0,π,0,π) and period Δz = 0.2 m; all with parameters: y1 = 0.01 m, y2 = 0.4 m, w = 0.1 m, b1 = 0.02 m, b2 = 0.12 m, c = 0.13 m, and d = 0.14 m.
4. Discussion and conclusions

A general variational formalism is used to evaluate the coupling of the ICRF antenna. The antenna resonant frequency is around 30 MHz in vacuum, and the presence of the plasma smooths the peak of the frequency response of the antenna resistance. The antenna resistance in the case of the plasma is much larger than the resistance with the vacuum condition due to the fact that the ICRF waves are absorbed in the plasma. The relationship between the impedance and the antenna parameters can be used to optimize the design of the ICRF antenna by maximizing the resistance R and minimizing the reactance X. The RF electric field distribution of the EAST 4-strap ICRF antenna shows that the antenna array with (0,π,0,π) phasing has lower electric field than with the (0,0,0,0) phasing. The mutual coupling between adjacent straps will be considered in the variational method in the future.

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