Introduction

Plasma start-up is an important part in tokamak operation and has been studied in many tokamaks [1–6]. Successful Ohmic start-up at low electric field has been achieved with E ~ 0.25 V/m and electron cyclotron heating (ECH) assisted start-up with E ~ 0.15 V/m has been demonstrated in D-IIID [7]. Plasma current start-up to 100 kA was achieved successfully in the JT-60U tokamak without the use of the center solenoid [8]. Non-inductive current start-up based on pressure-driven (bootstrap) currents has been demonstrated on CDX-U [9]. Successful plasma start-up is achieved by the application of radio frequency (RF) power in START and MAST spherical tokamaks [10]. As an important aspect in start-up, the vacuum field needs to be accurately estimated to optimize plasma breakdown successfully, especially in EAST tokamak experiments. EAST is Chinese National Project designed to develop scientific and technological basis on the steady state operation of advanced tokamak. It is the first fully superconducting tokamak in the world and the integrated design was applied for the poloidal field (PF) system [11]. EAST has a D-shaped vacuum vessel structure and 14 individually charged superconductive poloidal field coils which allow the plasma to be flexibly shaped with an elongation factor of 1.5–2.0, triangularity of 0.3–0.6, major radius of 1.8 m and minor radius of 0.4 m. It should be noted that PF’s 7 and PF’s 9 are connected in series as are PF’s 8 and PF’s 10. Twelve independent superconducting PF coils are driven by 12 independent PF power supplies [12]. 76 poloidally aligned magnetic probes and 35 flux loops were fixed on the vacuum vessel to measure the poloidal flux and the poloidal fields. A good vacuum field null can only be obtained by adjusting the poloidal field (PF) coils current properly. For the PF system of EAST, there are no special coils for compensation to produce an appropriate null field due to the integrated design [13]. Therefore, it is crucial to numerically calculate the vacuum field due to the PF system.

Various methods to calculate the vacuum field have been applied in different tokamaks. The methods which have been successfully developed: (a) The traditional method of Green’s function which can precisely reproduce a complete vacuum field. (b) The new method based on Cauchy condition surface method which needs magnetic sensors to reproduce vacuum field, and is firstly proposed to use an analytical exact solution for reproducing vacuum field. (c) Eigen-modes method, which reduces the fitting parameters and pre-calculated eddy current based on a lump parameter circuit equation, and has been applied to reconstruct the vacuum field. (a) and (b) have been used for EAST. Recently, the eddy current effects are considered as a forward problem simultaneously with vacuum field reproduction.

In this paper, a traditional method which is used to reproduce vacuum field is briefly introduced in Sect. 2, and the method for calculating eddy current is also given in this section. In Sect. 3, the algorithm based on Cauchy condition surface method is discussed. The results of the vacuum field reconstruction in EAST are presented in Sect. 4. Finally, a summary is given in Sect. 5.

Traditional Algorithm for Vacuum Field Reconstruction

Vacuum field reconstruction of the traditional algorithm, simulates vacuum field based on Green’ function, is widely used to reconstruct the vacuum field. It uses both magnetic flux loop and magnetic probe to calculate the vacuum field. For EAST, the available magnetic diagnostics include 38 two-dimension magnetic probes, 35flux loops and 2 rogowski coils. The algorithm is discussed in Sects. 2.1 and 2.2 in detail.

Eddy Current Calculation

Eddy current effects is very important for vacuum field reconstruction, and it can’t be ignored and should be considered as an important problem during reproducing vacuum field. During plasma discharge, the magnetic field is produced by PF coils currents IC, eddy currents IV and plasma current Ip. So the magnetic sensor signals are combined by PF coils currents, plasma current and eddy currents. For the flux loop, it satisfies the following relationship:

$$ \left[ \varPhi \right] = \left[ {G_{fc} } \right]\left[ {I_{C} } \right] + \left[ {G_{fp} } \right]\left[ {I_{P} } \right] + \left[ {G_{fv} } \right]\left[ {I_{V} } \right]. $$
(1)

For the magnetic probe, it satisfies the following relationship:

$$ \left\{ {\begin{array}{*{20}l} {[B_{N} ] = [G_{nc} ][I_{C} ] + [G_{np} ][I_{P} ] + [G_{nv} ][I_{V} ]} \hfill \\ {[B_{T} ] = [G_{tc} ][I_{C} ] + [G_{tp} ][I_{P} ] + [G_{tv} ][I_{V} ]} \hfill \\ \end{array} } \right., $$
(2)

where \(\varPhi\), BT and BN are flux loop, tangential magnetic probe and normal magnetic probe signals respectively. G is Green’s function among them.

The expression for eddy currents can be obtained according to Eqs. (1) and (2). Before plasma breakdown, there is no plasma current produced, the magnetic sensor signals are produced by both PF coils currents and eddy currents. So the eddy currents can be expressed as follows:

$$ \left[ {I_{V} } \right] = {{\left( {\left[ {\begin{array}{*{20}c} \varPhi \\ {B_{T} } \\ {B_{N} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {G_{fc} } \\ {G_{tc} } \\ {G_{nc} } \\ \end{array} } \right]\left[ {I_{C} } \right]} \right)} \mathord{\left/ {\vphantom {{\left( {\left[ {\begin{array}{*{20}c} \varPhi \\ {B_{T} } \\ {B_{N} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {G_{fc} } \\ {G_{tc} } \\ {G_{nc} } \\ \end{array} } \right]\left[ {I_{C} } \right]} \right)} {\left[ {\begin{array}{*{20}c} {G_{fv} } \\ {G_{tv} } \\ {G_{nv} } \\ \end{array} } \right]}}} \right. \kern-0pt} {\left[ {\begin{array}{*{20}c} {G_{fv} } \\ {G_{tv} } \\ {G_{nv} } \\ \end{array} } \right]}}. $$
(3)

To get the distribution of the eddy currents induced in the vacuum vessel, the vacuum vessel is divided into 40 conducting rings, the eddy currents can be modeled by these rings. Green’s function for EAST has been calculated. From Eq. (3), the eddy currents can be calculated according to the magnetic diagnostics (flux loop and magnetic probe). The total eddy current is measured by rogowski coils. To verify the calculation result of the eddy current, a comparison between the measured value and the calculated value about the eddy current is carried out. A test shot is used to test the calculation result. The result is shown in Fig. 1.

Fig. 1
figure 1

Eddy current comparison between the measured result and the calculated result in a test shot. The blue line indicates the calculation result of the eddy current using the traditional algorithm and the black line indicates the measurement result of the eddy current (Color figure online)

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For a test shot, no gas is injected into vacuum chamber, so there is no plasma current in the test shot. Seen from Fig. 1, the measured value and the calculated value of the eddy currents agree well with each other.

Vacuum Field Reconstruction

Vacuum field reconstruction consists of calculating magnetic field distribution from the measurements, including magnetic field and flux distribution in the vacuum region. In order to calculate the magnetic field and the flux distribution in the vacuum region, the vacuum chamber is divided into 33 × 33 rings. The method is summarized below.

In vacuum region, the relationships between magnetic field and currents are given as follows:

$$ \left\{ {\begin{array}{*{20}c} {\vec{B}_{r} = [G_{rvC} ] \times [I_{C} ] + [G_{rvV} ] \times [I_{V} ]} \\ {\vec{B}_{z} = [G_{zvC} ] \times [I_{C} ] + [G_{zvV} ] \times [I_{V} ]} \\ \end{array} } \right.. $$
(4)

For the flux in vacuum region, there yields

$$ \varPhi = \left[ {G_{gc} } \right] \times \left[ {I_{C} } \right] + \left[ {G_{gv} } \right] \times \left[ {I_{V} } \right]. $$
(5)

According to Eqs. (3), (4) and (5), the magnetic field and the flux distribution in the vacuum region before breakdown can be calculated, as is shown in Fig. 2. The magnetic field and the flux distribution in the vacuum vessel at t = −16 ms of shot 20,299 in EAST are calculated. Results are shown in Fig. 2a, b respectively. In Fig. 2, contour lines labeled by the values with the unit Gs for the magnetic field and with the unit Wb for the flux represent the distribution of the magnetic field and flux respectively. The null field region is the region in which the magnetic flux density B is less 5Gs for EAST tokamak. It is shown that there is a sufficiently large null field region (B < 5Gs) in the center for plasma breakdown.

Fig. 2
figure 2

The null field and flux distribution in vacuum region obtained using Green’s function before breakdown. a Magnetic field distribution in vacuum vessel. b Flux distribution in vacuum vessel

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Algorithm Based on CCS Method

Cauchy Condition Surface method, which is based on the boundary integral equation, is an analytical exact solution for magnetic field in the multiply connected vacuum region [14]. As an important mathematical method, it is widely used in tokamak. On the basis of this consideration, we have developed the method of Cauchy condition surface (CCS) to reconstruct the vacuum field. The concept of this method is summarized below.

According to the following Maxwell’s equations,

$$ \left\{ {\begin{array}{*{20}c} {rot\vec{B} = \mu_{0} \vec{j}} \\ {div\vec{B} = 0} \\ \end{array} } \right., $$
(6)

and by introducing the vector potential \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {A} (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} = \nabla \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {A} ) \), we have

$$ rotrot\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {A} = \mu_{0} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {j}. $$
(7)

Using the flux function \(\phi\) which is defined as \(\phi = r{A_{\phi}}\), the Eq. (7) can be converted to the following scalar equation:

$$ div\left[ {\frac{grad\phi }{{r^{2} }}} \right] = - \frac{{\mu_{0} j_{\phi } }}{r}. $$
(8)

Note that the flux function \(\phi\) can be also written as \(\phi\) = ψ/2π, where ψ is total magnetic flux. For the vacuum region without the toroidal component of plasma current density [15], Eq. (8) can be reduced into

$$ div\left[ {grad\phi /r^{2} } \right] = 0. $$
(9)

The Eq. (9) is an elliptic non-linear homogeneous second-order partial differential equations (PDE) [16]. The Cauchy-Condition is defined as a closed surface ∂Ω p where both Dirichlet condition \((\phi)\) and the Neumann condition (B t ) are unknown. It is located inside the surface of magnetic sensors and is assumed mathematically that there is no plasma outside CCS, i.e. there is a vacuum everywhere outside the CCS [17]. The outer region Ω boundary by ∂Ω p is defined as the analyzed region. There are two kinds of magnetic measurement outside the CCS, flux loops and magnetic probes. The topological concept is shown in Fig. 3.

Fig. 3
figure 3

Main concept of Cauchy-Condition Surface method

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The poloidal flux function \(\phi\) and Green’s function G are scalar functions, they satisfy:

$$ \begin{aligned} & div\left[ {{{G(r_{x} ,r_{y} ,z_{x} ,z_{y} ) \cdot \left( {grad\phi (r_{y} ,z_{y} )} \right)} \mathord{\left/ {\vphantom {{G(r_{x} ,r_{y} ,z_{x} ,z_{y} ) \cdot \left( {grad\phi (r_{y} ,z_{y} )} \right)} {r_{y}^{2} }}} \right. \kern-0pt} {r_{y}^{2} }}} \right] - div\left[ {{{\phi (r_{y} ,z_{y} ) \cdot \left( {gradG(r_{x} ,r_{y} ,z_{x} ,z_{y} )} \right)} \mathord{\left/ {\vphantom {{\phi (r_{y} ,z_{y} ) \cdot \left( {gradG(r_{x} ,r_{y} ,z_{x} ,z_{y} )} \right)} {r_{y}^{2} }}} \right. \kern-0pt} {r_{y}^{2} }}} \right] \\ & \quad = G(r_{x} ,r_{y} ,z_{x} ,z_{y} ) \cdot div\left[ {{{\left( {grad\phi (r_{y} ,z_{y} )} \right)} \mathord{\left/ {\vphantom {{\left( {grad\phi (r_{y} ,z_{y} )} \right)} {r_{y}^{2} }}} \right. \kern-0pt} {r_{y}^{2} }}} \right] - \phi (r_{y} ,z_{y} ) \cdot div\left[ {{{\left( {gradG(r_{x} ,r_{y} ,z_{x} ,z_{y} )} \right)} \mathord{\left/ {\vphantom {{\left( {gradG(r_{x} ,r_{y} ,z_{x} ,z_{y} )} \right)} {r_{y}^{2} }}} \right. \kern-0pt} {r_{y}^{2} }}} \right]. \\ \end{aligned} $$
(10)

By using the delta function into Green’s function G, it has

$$ \nabla \cdot [{{\nabla G(r_{x} ,r_{y} ,z_{x} ,z_{y} )} \mathord{\left/ {\vphantom {{\nabla G(r_{x} ,r_{y} ,z_{x} ,z_{y} )} {r_{y}^{2} }}} \right. \kern-0pt} {r_{y}^{2} }}] = \gamma \cdot \delta (r_{x} ,r_{y} ,z_{x} ,z_{y} ). $$
(11)

where \( \gamma \cong \mathop {vol\varOmega \to 0}\limits_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} \subset \varOmega }}^{\lim } \oint\limits_{\varOmega } {\nabla \cdot [{{\nabla G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} \mathord{\left/ {\vphantom {{\nabla G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} {r_{y}^{2} }}} \right. \kern-0pt} {r_{y}^{2} }}]} \cdot dV(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) = - 8\pi^{2} ,\quad \phi (r_{y} , z_{y} ) \) is the flux function at the point \( \left( {r_{y} , z_{y} } \right),\quad G\left( {(r_{x} , r_{y} , z_{x} ,z_{y} } \right) \) is the Green’s function between the two points \( \left( {r_{x} , z_{x} } \right) \) and \( \left( {r_{y} , z_{y} } \right) \), which is defined as:

$$ G\left( {x,y} \right) = \frac{{\mu_{0} \sqrt {r_{x} r_{y} } }}{\pi k}\left\{ {\left( {1 - \frac{{k^{2} }}{2}} \right)K\left( k \right) - E\left( k \right)} \right\}, $$
(12)

where k 2 = 4r x r y /(r x  + r y )2 + (z x  − z y )2, K and E being the complete elliptic integrals of the first and second kind, and \( dV(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) \) is the infinitesimal volume element at point \( \left( {r_{y} , z_{y} } \right) \),\( \delta (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) \) is the delta function, which is given by:

$$ \left\{ {\begin{array}{*{20}l} {\delta (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) = 0, \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} \ne \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} } \hfill \\ {\delta (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) = \infty , \, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} \, } \hfill \\ \end{array} } \right.. $$
(13)

Multiply both sides of the Eq. (8) with \( G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) \), and both sides of the Eq. (11) with \( \phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) \), we can obtained

$$ \left\{ {\begin{array}{*{20}l} {\phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) \cdot \nabla \cdot [{{\nabla G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} \mathord{\left/ {\vphantom {{\nabla G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} {r_{y}^{2} }}} \right. \kern-0pt} {r_{y}^{2} }}] = \gamma \cdot \phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) \cdot \delta (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} \hfill \\ {G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) \cdot \nabla \cdot [{{\nabla \phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} \mathord{\left/ {\vphantom {{\nabla \phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} {r^{2} }}} \right. \kern-0pt} {r^{2} }}] = - {{\mu_{0} j_{\phi } (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} \mathord{\left/ {\vphantom {{\mu_{0} j_{\phi } (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} {r_{y} }}} \right. \kern-0pt} {r_{y} }}} \hfill \\ \end{array} } \right.. $$
(14)

The following integral equation of (14) is given by

$$ \begin{aligned} & \sigma \phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ) - \int\limits_{\partial \varOmega } {\left[ {G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )\nabla \phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} ) - \phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )\nabla G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} \right]} \frac{{dS(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )}}{{r^{2} }} \\ & \quad = \int\limits_{\varOmega } {\mu_{0} j_{c} (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )G(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )} \frac{{dV(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y} )}}{{r_{y} }}. \\ \end{aligned} $$
(15)

Here the integral volume Ω is the entire region which cover CCS and all the known current sources, e.g. poloidal field coils, ∂Ω is the boundary surface of region Ω, j c is the current density and σ is a constant, defined as follows

$$ \sigma = \left\{ {\begin{array}{*{20}l} {8\pi^{2} \, (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} {\text{ in the }}\varOmega {\text{ interior)}}} \hfill \\ { 4\pi^{2}\, \,(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} {\text{ on }}\partial \varOmega )} \hfill \\ {0 \, (\overset{\lower0.5em\hbox{$\smash {\scriptscriptstyle\rightharpoonup}$}} {x} {\text{ in the }}\varOmega {\text{ exterior)}}} \hfill \\ \end{array} } \right.. $$
(16)

By applying Eq. (15) to flux loops, magnetic probes and CCS, three types of equations can be obtained.

$$ \begin{aligned} & \phi \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{f} } \right) = \mathop \varSigma \limits_{i = 1}^{M} W_{A} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{f} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right)\phi \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right) + \mathop \varSigma \limits_{i = 1}^{M} W_{B} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{f} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right)Bt\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right) + \\ & \quad \mathop \varSigma \limits_{j = 1}^{NC} W_{C} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{f} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y}_{j} } \right)I_{PF} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y}_{j} } \right) + \sum\limits_{k = 1}^{ND} {W_{D} } \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{f} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e}_{k} } \right)I_{E} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e}_{k} } \right). \\ \end{aligned} $$
(17)
$$ \begin{aligned} & B_{t} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{B} } \right) = \mathop \varSigma \limits_{i = 1}^{M} W_{A1} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{B} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right)\phi \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right) + \mathop \varSigma \limits_{i = 1}^{M} W_{B1} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{B} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right)Bt\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right) \\ & \quad + \mathop \varSigma \limits_{j = 1}^{NC} W_{C1} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{B} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y}_{j} } \right)I_{PF} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y}_{j} } \right) + \sum\limits_{k = 1}^{ND} {W_{D1} } \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{B} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e}_{k} } \right))I_{E} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e}_{k} } \right). \\ \end{aligned} $$
(18)
$$ \begin{aligned} & \frac{1}{2}\phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ) = \sum\limits_{i = 1}^{M} {W_{A0} } \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right)\phi \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right) + \sum\limits_{i = 1}^{M} {W_{B0} } \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z} } \right)B_{t} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right) \\ & \quad + \sum\limits_{j = 1}^{NC} {W_{C0} } \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y}_{j} } \right)I_{PF} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y}_{j} } \right) + \sum\limits_{k = 1}^{ND} {W_{D0} } \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e}_{k} } \right)I_{E} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e}_{k} } \right). \\ \end{aligned} $$
(19)

where \( \phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{f} ) \) is the flux loop signal at location \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{f} \) , \( B_{t} (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{B} ) \)  is the magnetic probe signal at location \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x}_{B} \), \( \phi (\vec{z}_{i} ) \) is the flux function of the i’th discretized point of CCS. \( B_{t} (\vec{z}_{i} ) \) is the magnetic field of the i’th discretized point of CCS. M is the number of discretized points on the CCS. \( I_{PF} (\vec{y}_{j} ) \) is the j’th poloidal field coils current in region Ω.\( I_{E} (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e}_{k} ) \) is the eddy current in region Ω. W A , W B , W C , W D , W A1, W B1, W C1, W D1, W A0, W B0, W C0 and W D0 are the coefficient matrices which can be calculated beforehand from the geometrical configuration of flux loops, magnetic probes, CCS, PF coils and eddy current model [18].

According to Eqs. (17), (18) and (19), the \( \phi (\vec{z}_{i} ) \) and \( B_{t} (\vec{z}_{i} ) \) at discretized points on CCS, and eddy current \( I_{E} (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e}_{k} ) \) can be calculated.Then the flux value of any point \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} \) can be computed by using the following equation:

$$ \begin{aligned} & \phi \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} } \right) = \mathop \varSigma \limits_{i = 1}^{M} W_{A2} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right)\phi \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right) + \mathop \varSigma \limits_{i = 1}^{M} W_{B2} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right)Bt\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z}_{i} } \right) \\ & \quad + \mathop \varSigma \limits_{j = 1}^{NC} W_{C2} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y}_{j} } \right)I_{PF} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {y}_{j} } \right) + \sum\limits_{k = 1}^{ND} {W_{D2} } \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e}_{k} } \right)I_{E} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e}_{k} } \right), \\ \end{aligned} $$
(20)

where \( \phi (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ) \) is the flux value of any point \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} \), W A2,W B2,W C2 and W D2 are the coefficient matrices which can be calculated beforehand from the geometrical configuration of CCS, PF coils and eddy current model.

Results and Discussion

To verify these methods’ accuracy, a simulation for the two methods is performed in a virtual tokamak (not a real tokamak in reality), and the results are shown in Fig. 4.

Fig. 4
figure 4

The simulation results of the flux distribution about the two methods. a The virtual tokamak b The flux distribution obtained using Green’s function method. c The flux distribution obtained using CCS method. d The comparison of the simulation results about two methods

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Figure 4a is the virtual tokamak, which has two pf coils and 12 flux loops are mounted on the vacuum vessel. Figure 4b is the simulation result of flux distribution based on the traditional method in the virtual tokamak. Figure 4c is the simulation result of flux distribution based on CCS method in the virtual tokamak. Figure 4d is the comparison of the simulation results about the two methods. It is shown that difference exists in regions near pf coils and magnetic sensors, while the simulation results of the two methods agree well in other regions.

To testify the new method’s performance in experiment, it is applied to EAST tokamak to examine its validity. A very important part of the vacuum field reconstruction is to test the adequacy of the magnetic diagnostic data used for the vacuum field reconstruction since not all of magnetic diagnostic data can be used for vacuum field reconstruction. As is shown in Fig. 5, a set of measured flux loop and magnetic probe signals are compared with the calculated values in a test shot. There is no plasma current in the test shot. To ensure no eddy current is induced in the magnetic measurement, the currents in the PF coils are held at their flat-top values for a period of time. It should be note that not all the magnetic diagnostic data which satisfy the precision of measurement can be used to reconstruct vacuum field using CCS method. If the surface consisted of magnetic sensors is too rough, which will give poor fits in vacuum field reconstruction. In order to make sure the magnetic sensors surface is smooth, some of magnetic sensors are left out. Figure 6 shows the selected magnetic sensors which are used to calculate vacuum field with Cauchy condition surface (CCS) method.

Fig. 5
figure 5

Comparison of the measured magnetic signals with the calculated values in a test shot of EAST

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Fig. 6
figure 6

The selected magnetic sensors (magnetic probe and flux loop) used to calculate vacuum field with CCS method

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The results using the new method in EAST tokamak are shown in Fig. 7. Calculation results of the flux distribution based on the traditional method, and the CCS method are shown in Fig. 7a, b, respectively. Figure 7c shows an overlay of them. Comparing the results, we can see the results agree well in most of the region except some special areas in which the discrete nature of the magnetic sensors becomes obvious. Comparison of the calculation results in the mid-plane is shown in Fig. 8. The reasons of the difference mainly come from two aspects: one is the surface which is consisted of magnetic sensors to calculate the vacuum field is not smooth, the other is the derivation from error of magnetic measurement.

Fig. 7
figure 7

The comparison of calculation results (flux distribution) between the traditional method and Cauchy condition surface method before breakdown. a the flux distribution obtained using Green’s function method, b the flux distribution obtained using CCS method, c overlay of the two methods with Green’s function in black and CCS in red (Color figure online)

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Fig. 8
figure 8

The comparison of calculation results (flux distribution) in mid-plane. The black line indicates the result of Green’s function method and the red line indicates the result of CCS method (Color figure online)

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Both methods have been applied to vacuum field reconstruction in EAST, In order to show the reconstructed vacuum field agrees with a series of phenomena in the experiments, a reconstruction example is presented to illustrate the result of the vacuum field reconstruction in EAST. Some measurement signals (the poloidal coil current, plasma current, H-αline and loop voltage) for shot 33,017 are shown in Fig. 9. As seen from Fig. 9, the currents in poloidal coils change at −17 ms, the plasma current and the loop voltage begin to increase after −17 ms. According to the theory of plasma breakdown, the plasma formation stage is usually divided into two phases, the avalanche (Townsend) and Coulomb phases. According to the signal of H-αline, the transition from the avalanche phase to the Coulomb phase happens at 5 ms. After the transition, the H-α emission goes through its maximum at 8 ms and the plasma current starts increasing rapidly. A high speed camera is used to record the whole process of plasma discharge. The sampling rate of the high speed camera is 1000 frame/s in experiment. As shown in Fig. 10a, high speed camera images show the process of breakdown during the start-up of EAST. It should be noted that any light can’t be seen from the high speed camera if the signal of plasma current is less than 20KA in EAST, which agrees with the theory of plasma breakdown (The transition occurs when plasma current is roughly 20 kA). Seen from Fig. 10a, the obvious phenomenon is that the moment of plasma formation (the transition from the avalanche phase to the Coulomb phase) is about 5 ms, but the exact location of the plasma is not clear. In order to obtain the exact location of the plasma, image processing technology is used to process the images of plasma discharge. The processed images of shot 33,017 at 4–6 ms are shown in Fig. 10b. The proportion of images to the original size is calibrated to obtain the real position of plasma. One pixel is equal 0.48 × 10−2 m. Comparing with the standard image, we can obtain the exact location of the plasma. As shown in Fig. 10b, the plasma is located in the middle of low field side. Seen from Fig. 11, the null field from −16 to 5 ms is located in the middle of low field side, indicating that is the most likely region of breakdown and this agrees with the first location of plasma light from the fast camera images.

Fig. 9
figure 9

The poloidal coil current, plasma current, H-αline and loop voltage change with time during start-up phase

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Fig. 10
figure 10

The process of breakdown is recorded by high speed camera. It is shown that the moment of plasma formation is about 5 ms. a The original discharge images of shot 33,017 at 4–6 ms. b The processed images of shot 33,017 at 4–6 ms by image processing. The pink light indicates the distribution of the plasma and the white point indicates the centre of the plasma (R = 2.04 m, Z = −0.09 m at 5 ms) (Color figure online)

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Fig. 11
figure 11

The calculation results about magnetic field distribution from −16 ms to 5 ms for shot 33,017 in EAST. The unit is Gs. Green’s function method was used for this

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Summary

In conclusion, it has been shown that the above two methods can be used to calculate the vacuum field before breakdown properly. On the basis of our studies, as discussed in the previous sections, we have reached the following conclusions.

The first method is used to reconstruct the vacuum field based on calculating the eddy currents accurately. The second method is based on an analytical exact solution for magnetic field in the multiply connected vacuum region and is used to numerically calculate the vacuum field. The differences in the eddy current calculation have an influence on the accuracy of the vacuum field calculation. By comparison with the results of different methods, the trustworthiness of both methods is enhanced. The accuracy of the first method relies upon calculating the eddy currents accurately. For the second method, the smooth surface defined by magnetic sensors having small separation is necessary to guarantee the accuracy of the calculation. Since the two methods have different parameters that determine their accuracy, the combination can give results that are less dependent upon either the eddy current measurements or the density of sensors.