Capability Assessment of the Equilibrium Field System in KTX

Plasma Science and Technology, Vol.18, No.1, Jan. 216 Capability Assessment of the Equilibrium Field System in KTX LUO Bing ( ) 1, LUO Zhengping ( ) 2, XIAO Bingjia ( ) 2,3, YOU Wei ( ) 1, TAN Mingsheng ( ) 1, GUO Yong ( ) 2, BAI Wei ( ) 1, MAO Wenzhe ( ) 1, LI Hong ( ) 1, LIU Adi ( ) 1, LAN Tao ( ) 1, XIE Jinlin ( ) 1, LIU Wandong ( ) 1 1 CAS Key Laboratory of Geospace Environment, Department of Modern Physics, University of Science and Technology of China, Hefei 2326, China 2 Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 2331, China 3 Collaborative Innovation Centre for Advanced Fusion Energy & Plasma Science, University of Science and Technology of China, Hefei 2326, China Abstract Radial equilibrium of the KTX plasma column is maintained by the vertical field which is produced by the equilibrium field coils. The equilibrium is also affected by the eddy current, which is generated by the coupling of copper shell, plasma and poloidal field coils. An equivalent circuit model is developed to analyze the dynamic performance of equilibrium field coils, without auxiliary power input to equilibrium field coils and passive conductors. Considering the coupling of poloidal field coils, copper shell and plasma, the evolution of spatial distribution of the eddy current density on the copper shell is estimated by finite element to analyze the effect of shell to balance. The simulation results show that the copper shell and equilibrium field coils can provide enough vertical field to balance 1 MA plasma current in phase 1 of a KTX discharge. Auxiliary power supply on the EQ coils is necessary to control the horizontal displacement of KTX due to the finite resistance effect of the shell. Keywords: KTX, plasma equilibrium, circuit model, eddy current PACS: 52.55. s DOI: 1.188/19-63/18/1/16 (Some figures may appear in colour only in the online journal) 1 Introduction The Keda Torus experiments (KTX) [1] is a middlesized reversed field pinch (RFP) [2] with major radius R = 1.4 m, minor radius a =.4 m, and plasma current I p 1 MA. The plasma equilibrium which mainly depends on active control coils (equilibrium field winding) and passive conductor (copper shell and vacuum vessel) must be taken into consideration for KTX. KTX needs a toroidal passive conductor shell, which must be as close to the plasma as possible to reduce MHD instabilities and to contribute plasma equilibrium. The conductor shell must have poloidal cuts to allow the toroidal electric field to penetrate the plasma quickly. Radial equilibrium of the KTX plasma column is affected by the vertical field, which is produced by the external equilibrium field (EQ) coils and toroidal eddy currents induced on the passive conductor shell [3]. An ideal equilibrium matching condition is that the EQ coils can balance plasma shift themselves, which cause the eddy currents to vanish on passive conductor to reduce the error field. However, it is difficult to satisfy the conditions. Thus the eddy currents, which provide part of the vertical field and connect with the error field control in the cuts, should be taken into consideration for KTX devices. In order to calculate the toroidal eddy currents density on the passive conductor, both theoretical modeling [3,4] and experiment results [3,5] have been demonstrated. Generally, the finite element electromagnetic analysis [6] is the most practical method to analyze the eddy current distribution on the passive conductor. In this paper, it is mainly studied the dynamic performance of EQ coils and stability copper shell during the KTX discharge phase, which is based on the KTX final design configuration and the power model. In section 2, the Poloidal Field (PF) system [1,7,8] of KTX is described briefly. The dynamic capability of EQ coils to equilibrate plasma has been studied in section 3, which is based on an equivalent couple circuit model between the PF coil system and plasma considering the topology structure of the poloidal field circuit of KTX [1,9]. In section 4, the finite element method is used to calculate the dynamic performance of the toroidal eddy current density on the copper shell, due to the topology structure of the KTX circuit model. supported by the Ministry of Science and Technology of China (No. 211GB16) 9

LUO Bing et al.: Capability Assessment of the Equilibrium Field System in KTX 2 Poloidal field system of KTX The PF system of KTX contains two mutually independent coils system: the Ohmic field (OH) coils and EQ coils. Toroidal electric field is induced by Ohmic field coils to breakdown the gas and then drive plasma current. During this process, the Ohmic coils must provide a field null on plasma regions. The EQ coils providing the vertical field can restrain plasma expansion and maintain plasma equilibrium. The aim of utilizing the two independent PF coils system is to drive plasma current and control plasma equilibrium independently. The cross section [1,8] of KTX is shown in Fig. 1, including the Ohmic field coils (OH1-OH13), the EQ coils (EQ1-EQ6), the Toroidal Field (TF) coils and passive conductor (the stabilizing copper shell and vessel) around the plasma. The detailed KTX PF coil parameters [1] are listed in t Table 1 and Table 2. are constituted of 26 cake coils with up-down symmetry, and divided into three groups (OHA, OHB, and OHC). The 13 cake coils involve 3 turns, divided to OHA, OHB, and OHC respectively. The EQ coils are formed of 12 cake coils with up-down symmetry. Parameter turn stands for the turns of coils for each cake. The current ratio case of EQ coils is shown in Table 2 for 1 MA plasma current discharge. Table 2. Geometric parameter of KTX EQ coils Z low Z high R in R out Turn Current [ka] EQ1.69.282.711.729 36 1. EQ2.461.642.856.876 32 1.16 EQ3.7.769 1.75 1.129 26 4.25 EQ4.85.91 1.287 1.349 24 3.75 EQ5.667.737 1.925 1.976 19 6.84 EQ6.412.54 2.256 2.287 18 7., OH5U OH4U OH3U OH2U OH1U OH1D OH2D OH3D Table 1. OH4D OH5D OH6U OH6D Fig.1 OH7U EQ1U EQ1D OH7D OH8U EQ2U EQ2D OH8D OH9U EQ3U OH9D EQ3D EQ4U EQ4D OH1U OH11U EQ5U TF coil Shell EQ5D OH1D OH11D EQ6U EQ6D OH12U OH12D Schematic cross-section of KTX OH13U OH13D Geometric parameter of KTX Ohmic coils Z low Z high R in R out Turn Sector OH1.16.144.433.661 14 OHA OH2.176.34.433.661 14 OHB OH3.336.464.433.661 14 OHC OH4.5.6.499.741 11 OHB OH5.64.74.551.793 11 OHA OH6.844.887.659.91 11 OHC OH7.957 1.23.883.937 6 OHB OH8 1.141 1.31 1.73 1.127 6 OHA OH9 1.432 1.538 1.275 1.329 4 OHC OH1 1.48 1.52 1.89 1.891 3 OHC OH11 1.48 1.52 2.116 2.17 2 OHB OH12 1.48 1.52 2.571 2.599 1 OHA OH13 1.469 1.531 3.421 3.519 3 OHA,OHB, OHC Table 1 and Table 2 respectively show the geometric parameters of Ohmic coils and EQ coils. It can be seen that the cross-section of PF (Ohmic and EQ) coils are rectangular, which is determined by the position parameters (Z low, Z high, R in and R out ). The Ohmic coils 3 The dynamic performance of EQ coils 3.1 The couple between PF coils and plasma During the operation, the coupling always exists among PF coils, passive conductor and plasma, especially in the ramp up phase. A large pulse voltage may be induced in EQ coils in the ramp up phase due to current changes in Ohmic coils. As a result, it is a large technical difficulty to design the EQ coils power. Considering the design ideas [1] of RFX [11], the packet connection of PF coils is meant to avoid the surge voltage in EQ coils. The PF coils circuit connections diagram [1,9] is shown in Fig. 2. Three branches of Ohmic coils can also be seen here (OHA, OHB and OHC). The EQ coils are divided into six branches, with two of them connected in parallel to the Ohmic coils. For example, the EQ1U series is connected to EQ1D, and so is EQ6U to EQ6D; both of them are parallel connected to OHA. The series connection of EQ1U and EQ1D can be equally substituted by an EQ1 coil for convenience. In order to reduce the voltage to earth, the power supply of the PF coil system are also divided into three branches. The coupling relationship of PF coil, passive conductor and plasma is shown in Fig. 2. Ignoring the passive conductor, an equivalent circuit model is used to describe the coupling between the PF coils system and the plasma in Fig. 2. The corresponding circuit equation can be written as: M di + RI = U(t), (1) dt where, M is the matrix describing the mutual inductance between PF coils and plasma; R is the resistance matrix; and U(t) is the corresponding power voltage vector. M, R, I and U(t) are given in appendix A respectively. 91

Plasma Science and Technology, Vol.18, No.1, Jan. 216 3.2 Vertical field inducted by the EQ coils Magnetic confinement fusion devices, such as RFP, have axial symmetry or approximate axial symmetry. The equilibrium of RFP has two basic aspects: firstly there is the internal equilibrium between the pressure of plasma and the forces due to the magnetic field, secondly there is the position and shape of the plasma, which are determined and controlled by currents in EQ coils. Therefore, the dynamic performance of EQ coils should be verified. The required vertical magnetic field [12] to maintain the plasma equilibrium is given by, Fig.2 Equivalent circuit diagram of KTX. The coupling of PF coils, passive conductor and plasma is described In appendix A, where I p is the plasma current, U p (t) is the total power voltage of Ohmic coils, and UEQ is the auxiliary power on EQ coils. In the case that U p (t) and R p (t) are given, without other auxiliary power input to EQ coils (UEQ=), the time evolution of U p (t) and R p (t) are shown in Fig. 3. The waveform evolution between PF coils and plasma current can be calculated by solving Eq. (1). Respectively, current waveforms of Ohmic coils, EQ coils and plasma are shown in Fig. 4, and so is the toroidal voltage. It is indicated that the direction of current between plasma and PF coils is opposite, and the plasma breakdown voltage can reach to 45 V with 1 MA plasma discharge current inducted in PF coils. R p (V/A) U p (V) 6 x 14 4 2.1.2.3.4.5.6.4.2.1.2.3.4.5 tims(s) Fig.3 The time evolution of PF coils power voltage (top) and plasma resistance (down) B V = µ I p (β p + ln 8R 4πR a + l i 2 3 ), (2) 2 where I p is the plasma current; R and a are major radius and minor radius of the device respectively; β p is the poloidal beta, which is approximately 6% for RFP [13,14] ; and l i is plasma internal inductance that is associated with the distribution of plasma current. Generally speaking, l i is equal to 1 [12], since the plasma current distribution is appropriately described by the zero-order Bessel function. The vertical field calculated by Eq. (2) is provided by EQ coils. The geometric parameters of EQ coils are shown in Table 2. The EQ coils can be equivalently treated as a series of circular current loops with different heights and radii, and the center is the Z-axis. The mutual inductance of two coaxial loops is given by [15], and, M = µ (rr).5 [( 2 k k)k(k) 2 E(k)], (3) k k 2 = K(k) = E(k) = 4rR Z z 2 +(r+r) 2 π/2 dφ (1 k 2 sin 2 φ).5 π/2 (1 k 2 sin 2 φ).5 dφ, (4) where (r, z) and (R, Z) are respectively the radius and height of two circular filaments loops. The poloidal magnetic field of the grid point generated by EQ coils in the plasma region can be written as, B R = 1 Ψ 2πR Z = I M k 2πR k Z B Z = 1 Ψ 2πR R = I M k 2πR k R, (5) Fig.4 The time evolution of PF coils current (left) and plasma current and voltage (right) where Ψ is poloidal magnetic flux. With Eqs. (2) and (5), the dynamic capability of the required magnetic field (black) to maintain 1 MA plasma equilibrium and the vertical field by EQ coils (blue) are shown in Fig. 5. Without passive conductor and input auxiliary power on EQ coils, these results show that the vertical field generated by the PF coils mutual inductance is not sufficient, the eddy current should also be considered. 92

LUO Bing et al.: Capability Assessment of the Equilibrium Field System in KTX B Z (T).2.15.1.5.1.2.3.4.5 times(s) B V B Z EQ Fig.5 The required vertical field (black) to maintain 1 MA plasma equilibrium and provided by EQ coils mutual inductance (blue) 4 Dynamic performance of copper shell The passive conductors of KTX are composed of a vacuum vessel (thickness th =6 mm) and a copper shell (th =1.5 mm) that can reduce MHD instabilities and control plasma equilibrium. In practice, the toroidal eddy currents on the shell also contribute to plasma equilibrium in the ramp-up phase. In order to evaluate the toroidal eddy currents density on the copper shell, the KTX 3D model (Fig. 6) has been established, which includes PF coil, copper shell, plasma and two vertical cuts lying in and 18 along the toroidal shell. The model is based upon the following simplifying assumptions: a. Only the toroidal eddy current on the copper shell is considered, since the conductivity of copper is greater than the stainless steel vacuum vessel. b. Plasma is modeled as a column whose radius is.31 m and its center 1.4 m to the Z axis. c. The thin copper shell is considered as one layer conductor with a uniform current density. section A are (, 1.4 m, ), and its radius r=48 mm, to make sure it is away from the two vertical cuts as far as possible. In order to calculate the temporal and spatial distribution of toroidal eddy currents density on section A, three steps are conducted. Firstly, the upper half part of section A is divided into 45 points uniformly, which means the upper part of section A is equivalent to 45 current filaments on the copper shell. The lower half part is processed similarly. Secondly, the loading excitation signals to PF coils and plasma are previously shown in Fig. 4: (a) corresponds to the loading currents in Ohmic coils, (b) corresponds to the loading in EQ coils, and (c) loading in the plasma current. Finally, the toroidal eddy currents density distribution for 45 tracking points can be estimated by finite elements analysis in the time domain solver. The toroidal eddy currents density distributions are differences on the tracking points and the corresponding normalized distributions are presented in Fig. 7(a) and (b) respectively. It is shown that the eddy currents direction of the inner copper shell is opposite to that of the outer shell, and the eddy currents density increases rapidly within 8.5 ms and then gradually decreases to zero. These results indicate that the major effect of the stability shell is to maintain the equilibrium of the plasma in phase 1 of the KTX discharge. It also demonstrates that the maximum of the error field may appear at 8 ms adjacent to the cuts, thus important guidance may be provided for the error field control of KTX. J T (A/m 2 ) J T (Normal) J T (A) 2 2 x 17 4 1.5.1.2.3.4.5 66 o 9 b o 1 9 o.5.1.2.3.4.5 5 x 17 time(s) The eddy current on cross section of KTX shell a c 66 o 9 o 17 o 17 o fitting 18 o o 27 o Fig.6 KTX 3D schematic diagram In this model, the toroidal eddy current density on the copper shell can be estimated by finite elements analysis methods, with active excitation signals in PF coils and plasma. Since the eddy currents can generate the error fields near the vertical cuts, we also consider the toroidal eddy currents on this small circuit crosssection A on the copper shell, with A lying in 9 along the toroidal. In other words, the center coordinates of 5 1 2 3 θ o Fig.7 The time evolution of toroidal eddy currents density (a), the normalized eddy current and cross-section of KTX shell (b) and eddy current distribution (c) for different track point on shell The normalized fitting curve of eddy currents density distribution (Fig. 7(b)) indicates that the toroidal eddy current has two time scales: τ 1 and τ 2. The time scales are supposed to be related to the power supply and the copper shell. A simple model which includes a cylindrical copper shell and external currents has been used 93

Plasma Science and Technology, Vol.18, No.1, Jan. 216 to verify the finite element analysis method. Two assumptions are made in this model: 1) the outside magnetic field of the cylindrical shell satisfy B 1 = B (1 exp( t/τ 1 )), which is induced by the external coil current, such as PF coil current; 2) the characteristic time of the cylindrical copper shell is τ 2. The field penetration methods are applied to calculate the eddy current α f = α τ 2 /(τ 2 τ 1 ) [exp( t/τ 1 ) exp( t/τ 2 )]) on the shell. The characteristic time scales τ 1 5 ms, τ 2 18 ms, the normal α f is well consistent with the fitting curve of KTX toroidal eddy current (black line in Fig. 7(b)). In this situation, the eddy current reaches to maximum at t = 8.8 ms. The vertical fields induced by EQ coils and the copper shell are shown in Fig. 8. The black line is the required vertical field distribution over time for 1 MA plasma; the blue line is the vertical field created by EQ coils; the green line is the vertical field by toroidal eddy currents induced on the copper shell; the red line is the total vertical field by EQ coils and copper shell. The EQ coils and copper shell are shown to have sufficient capability to achieve equilibrium for 1 MA plasma within 3 ms, which attains the design goal of the first phase. After the 1st phase, it needs the additional power provided in EQ coils to control the horizontal displacement of the plasma, due to the finite resistance of the shell. The spatial distributions of the vertical field are given in Fig. 9, (a) for EQ coils, (b) for copper shell, and (c) for the combination of EQ coils and copper shell. The vertical field decreases gradually for EQ coils with the increase of R, and this distribution trend is opposite to the copper shell. Fig.8 The time evolution of vertical field by EQ coils (blue), and shell (green) Finally, the vertical fields generated by EQ coils and shell are related to the plasma internal inductance, which is mainly determined by the plasma radius in this model. The relationship is presented in Fig. 1. Bz(T).2.15.1.5 EQ Shell.5 1 1.5 ι i Fig.1 The vertical field by EQ coils (green) and shell (blue) for different internal inductance l i 5 Conclusion In this paper, verification of the dynamic performance of EQ coils and copper shell is mainly achieved. The vertical fields generated by EQ coils are calculated using the equivalent electric circuit model, which is based on the coupling between PF coils and plasma, without considering the passive conductor and auxiliary power on EQ coils. The results show that the vertical field generated by PF coils mutual inductance is not sufficient. Therefore, the toroidal eddy currents on the copper stabilizing shell must also be considered in this case. The KTX 3D model is established with PF coils, stability copper shell, plasma and two vertical cuts included. The coupling of PF coils, shell and plasma is considered in this model. By loading excitation signals of PF coils and plasma, the temporal and spatial distribution of eddy current on the copper shell is calculated. Utilizing the track point toroidal eddy current density, further calculations can be made for the vertical field induced on the passive conductor. It indicates that the vertical fields induced on the shell can remedy the deficiencies of EQ coils. The KTX 1 MA plasma equilibrium can be maintained under the combined effects of EQ coils and passive conductor within 3 ms. Then, it needs additional new power in EQ coils to control the horizontal displacement to maintain the long-term equilibrium of KTX plasma, since the rapid decrease of the toroidal eddy currents on the shell. It is also emphasized that the maximum error field may appear at about 8 ms in cuts. Acknowledgment The authors are grateful for the assistance and support of the KTX groups. The authors would acknowledge Dr. ZHAO Zhenling gratefully for their helpful discussion. Fig.9 The induced vertical field spatial distribution in the plasma region by EQ coils (a), copper shell (b) and EQ coils and shell combined (c) Appendix A The corresponding circuit equation described cou- 94

LUO Bing et al.: Capability Assessment of the Equilibrium Field System in KTX pling between PF coils and plasma can be written as Eq. (1): Where, M is the matrix describing the mutual inductance between PF coils and plasma; R is the resistance matrix; and U(t) is the corresponding power voltage vector. M, R, I and U(t) are given in appendix A respectively. Ip is the plasma current, Up (t) is the total power voltage of Ohmic coils, and UEQ is the auxiliary power on EQ coils. In the case that Up (t) and Rp (t) are given. 95

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