Introduction

In the tokamak, the magnetic pressure of toroidal magnetic field acts together with the internal pressure (kinetic pressure) which is created by the hot gas to expand the plasma along the major radius [1]. In addition, the magnetic pressure of poloidal magnetic field further strengthens the outward power [1]. There is only the magnetic tension of toroidal magnetic field used as a reverse force to balance the outward force along the major radius, which is often not enough. In order to maintain equilibrium along the major radius, an external vertical magnetic field is necessary and available to present an inward force [2, 3]. The external vertical field is operated by the feedback control system, which bases on the estimation of the position of plasma current center [4]. Therefore, the determination of the plasma current center is very important for tokamak discharge experiments.

The plasma current center cannot be measured directly. We could only evaluate the position by external magnetic measurements. Firstly, we introduce and analyze the previous method on HT-7, which measure the plasma displacement by the single-turn flux loops [5]. This displacement was used in the vertical field feedback control replacing the plasma current center. Secondly, we used a new method, which has been successfully applied on EAST [4], to evaluate the plasma current center from magnetic diagnostic signals. At last, we analyzed the calculation results and estimated the measured plasma displacement value in HT-7 experiments.

Measurement of the Plasma Displacement by Single-Turn Flux Loops

In the tokamak, we can use a single-turn flux loop that wrap toroidally around the torus to measure the poloidal flux Ψ. In the large aspect-ratio approximation, the flux is written in the form [6]

$$\varPsi (\rho_{M} ,\omega_{M} ) = \frac{{\mu_{0} I_{p} R_{p} }}{2\pi }\left( {\ln \frac{{8R_{p} }}{{\rho_{M} }} - 2} \right) - \frac{{\mu_{0} I_{p} }}{4\pi }\left( {\ln \frac{{\rho_{M} }}{{a_{p} }} + \left( {\varLambda + \frac{1}{2}} \right)\left( {1 - \frac{{a_{p}^{2} }}{{\rho_{M}^{2} }}} \right)} \right)\rho_{M} \cos \omega_{M}$$
(1)

Here, (ρ M , ω M ) is the coordinates of measurement point, \(\varLambda = \beta_{\theta } + \frac{{\ell_{i} }}{2} - 1\) and (R p , a p ) are the plasma major radius and minor radius. Note that R p  = R 0 − Δx and \(a_{p} = a_{0} - \sqrt {\varDelta x^{2} + \varDelta y^{2} }\), here (R 0, a 0) are the tokamak major radius and the minor radius. To be convenient, based on the assumption that Δr < < a p  < a 0 < < R 0, the formula (1) can be expanded to the harmonic form [5] around (R 0, a 0):

$$\varPsi (\rho_{0} ,\omega ) = \varPsi_{0} + \sum\limits_{m = 1}^{\infty } {\varPsi_{X}^{m} \cos m\omega } + \sum\limits_{n = 1}^{\infty } {\varPsi_{Y}^{n} \sin n\omega }$$
(2)

Here, (ρ 0, ω) is the coordinates of measurement point in the quasicyclindrical coordinates.Then, we can derive that:

$$\left\{ \begin{array}{l} \varPsi (\rho_{0} ,\omega ) + \varPsi (\rho_{0} , - \omega ) = 2\varPsi_{0} + 2\sum\limits_{m = 1}^{\infty } {\varPsi_{X}^{m} \cos m\omega }\\ \varPsi (\rho_{0} ,\omega ) - \varPsi (\rho_{0} , - \omega ) = 2\sum\limits_{n = 1}^{\infty } {\varPsi_{Y}^{n} \sin n\omega } \\ \end{array} \right.$$
(3)

Neglecting second and higher order of Δx, Δy, Δr, the plasma displacement can be calculated by the equations below [4, 5]:

$$\left\{ \begin{array}{l} {{\left[ {\varPsi \left( {\rho_{0} ,\frac{\pi }{4}} \right) + \varPsi \left( {\rho_{0} , - \frac{\pi }{4}} \right) - \varPsi \left( {\rho_{0} ,\frac{3\pi }{4}} \right) - \varPsi \left( {\rho_{0} , - \frac{3\pi }{4}} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {\varPsi \left( {\rho_{0} ,\frac{\pi }{4}} \right) + \varPsi \left( {\rho_{0} , - \frac{\pi }{4}} \right) - \varPsi \left( {\rho_{0} ,\frac{3\pi }{4}} \right) - \varPsi \left( {\rho_{0} , - \frac{3\pi }{4}} \right)} \right]} {\frac{{\sqrt 2 \mu_{0} I_{p} R_{0} }}{{\pi \rho_{0} }}}}} \right. \kern-0pt} {\frac{{\sqrt 2 \mu_{0} I_{p} R_{0} }}{{\pi \rho_{0} }}}} = \varDelta x - \frac{{\rho_{0}^{2} }}{{2R_{0} }}\ln \frac{{\rho_{0} }}{{a_{0} }} - \frac{{\rho_{0}^{2} - a_{0}^{2} }}{{2R_{0} }}\left( {\varLambda + \frac{1}{2}} \right)\\ {{\left[ {\varPsi \left( {\rho_{0} ,\frac{\pi }{4}} \right) - \varPsi \left( {\rho_{0} , - \frac{\pi }{4}} \right) + \varPsi \left( {\rho_{0} ,\frac{3\pi }{4}} \right) - \varPsi \left( {\rho_{0} , - \frac{3\pi }{4}} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {\varPsi \left( {\rho_{0} ,\frac{\pi }{4}} \right) - \varPsi \left( {\rho_{0} , - \frac{\pi }{4}} \right) + \varPsi \left( {\rho_{0} ,\frac{3\pi }{4}} \right) - \varPsi \left( {\rho_{0} , - \frac{3\pi }{4}} \right)} \right]} {\frac{{\sqrt 2 \mu_{0} I_{p} R_{0} }}{{\pi \rho_{0} }} = - \varDelta y}}} \right. \kern-0pt} {\frac{{\sqrt 2 \mu_{0} I_{p} R_{0} }}{{\pi \rho_{0} }} = - \varDelta y}}\\ \end{array} \right.$$
(4)

Here, ρ 0 is the radius of the single-turn loop. Based on the formulas (4), we can use four single-turn loops, which are distributed equally in the toroidal direction, to measure (Δx, Δy). Here (Δx, Δy) presents the plasma displacement relative to the tokamak center (R 0, a 0). What calls for special attention is that the formula (2–4) can be applicable only if Δr < < a p  < a 0 < < R 0.

As shown in formula (4), the horizontal displacement Δx cannot be determined directly by using the four single-turn loops. There is an offset value \(\frac{{\rho_{0}^{2} }}{{2R_{0} }}\ln \frac{{\rho_{0} }}{{a_{0} }} + \frac{{\rho_{0}^{2} - a_{0}^{2} }}{{2R_{0} }}\left( {\varLambda + \frac{1}{2}} \right)\), which is known as Shafranov shift [1], between the measured value and Δx, The Shafranov shift is not a constant and hard to be measured directly during discharge, especially in the plasma breakdown phase. On the other hand, based on the calculation by formula (4), the vertical displacement Δy can be directly measured by using the four single-turn loops. In the results section below, we will give the simulation results about this method and evaluate its application.

Estimation of the Position of Plasma Current Center by Magnetic Measurement

As mentioned above, we could evaluate the position of plasma current center from magnetic diagnostic signals in EAST [4]. In the following, we will introduce the proven method of estimating the plasma current center by using magnetic measurement.

The Plasma Current Center Prediction

As we known, the total poloidal magnetic field measured by the poloidal magnetic probe array on HT-7 is induced by the toroidal current that flows inside the plasma and external poloidal field coils.

$$\left( \begin{aligned} {{B}}_{t} \hfill \\ {{B}}_{n} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} {{G}}_{tc} \cdot {{I}}_{c} + {{G}}_{tp} \cdot {{I}}_{p} \hfill \\ {{G}}_{nc} \cdot {{I}}_{c} + {{G}}_{np} \cdot {{I}}_{p} \hfill \\ \end{aligned} \right)$$
(5)

Here, B t and B n are the magnetic field in the tangential and the radial direction, (G tc , G tp , G nc , G np ) are the response matrices which are used to calculate the magnetic field by multiplying current, the subscript t, n refer to the tangential and the radial direction respectively and c, p refer to external poloidal field coils and plasma respectively, I c is the external field coils current vector, and I p is the plasma current vector on the grids inside the vacuum vessel.

On HT-7, the poloidal magnetic field is measured by magnetic probe array. The array consists of 12 magnetic probes distributed equally in the circle poloidal direction [7]. Each probe can measure the magnetic field in the tangential direction (B t ) and the radial direction (B n ). The currents of poloidal field coils, including ohmic field coils, vertical field coils and horizontal field coils, are measured by separate Rogowski coils outside the vacuum vessel [7]. The formula (5) can be turned to formula (6).

$${{I}}_{p} = \left( \begin{aligned} {{G}}_{tp} \hfill \\ {{G}}_{np} \hfill \\ \end{aligned} \right)^{ - 1} \, \cdot \, {{M}} \, \cdot \, \left( \begin{aligned} - {{I}}_{c} \hfill \\ {{B}}_{t} \hfill \\ {{B}}_{n} \hfill \\ \end{aligned} \right)$$
(6)

The diagram in Fig. 1 shows the schematic flow of the matrix M. Blocks (a) and (b) in the matrix M contain the response matrices G tc , G nc . Block (c) is a unit matrix, and its order is the amount of the magnetic measurement elements used for the prediction of plasma current center. On HT-7, we divide the vacuum vessel into 33 × 33 segments. According to the typical plasma current distribution, we calculate the plasma filament current in the segments. Each element of the plasma current vector I p corresponds to the plasma filament current on one sub grid inside the vacuum vessel.

Fig. 1
figure 1

A schematic flow of the matrix M

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By using the plasma current vector I p calculated by formula (6), the major radius R center and vertical position Z center of plasma current center can be predicted using the following formula.

$$R_{center} = \frac{{\sum\limits_{i = 1}^{n} {r_{i} I_{Pi} } }}{{\sum\limits_{i = 1}^{n} {I_{Pi} } }};Z_{center} = \frac{{\sum\limits_{i = 1}^{n} {z_{i} I_{Pi} } }}{{\sum\limits_{i = 1}^{n} {I_{Pi} } }}$$
(7)

Here, n is the amount of the plasma sub grids, (r i , z i ) is the position of each plasma sub grid and I Pi is the plasma current on each sub grid.

Determination of the Response Matrices

According to formula (6), the response matrices need to be pre-calculated for the prediction of the plasma current center. The response matrices are independent of the experimental diagnostic data. We can calculate the response matrices based on the physical parameters of the device.

HT-7 is an iron-core tokamak. We can calculate the poloidal magnetic flux of HT-7 with a spool model for the iron core, which was developed for the TEXT-Upgrade [8]. The shape of the iron core and the geometry of the poloidal field coils in HT-7 is shown in Fig. 2. The iron core spool model is shown in Fig. 3. In the spool model, the radius R Fe of the cylindrical central post is the actual radius of the iron core in HT-7. The radius R d of the two disks on the top and bottom is chosen beyond any poloidal field coils. The distance between the top and the bottom disks, Z Fe , is a free parameter. We chose the actual height of the iron core in HT-7 for Z Fe . The choice is made by comparing experimental HT-7 results with our calculation based on this model.

Fig. 2
figure 2

Shape of the iron core and the geometry of the poloidal field coils in HT-7

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Fig. 3
figure 3

Spool model of the iron core

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Based on the spool model, we suppose a current I s flowing in a coil at (R s , Z s ), where R s is the radius of the coil and Z s is the vertical distance from the mid-plane of the tokamak. The poloidal magnetic flux Ψ at (R, Z) is given by [8]

$$\begin{aligned} \varPsi (R,Z) = L_{m} I_{s} - \frac{{\mu_{0} \pi I_{s} }}{{Z_{Fe} }}(R_{s}^{2} + R^{2} ) + \frac{{\mu_{0} \pi I_{s} }}{{Z_{Fe} }}\left\{ \begin{array}{l} R^{2} ;R_{Fe} < R < R_{s} \\ R_{s}^{2} ;R > R_{s} \hfill \\ \end{array} \right\} \\ + \frac{{4\mu_{0} \pi I_{s} RR_{s} }}{{Z_{Fe} }} \times \sum\limits_{n = 1}^{\infty } {\cos \left[ {n\left( {\frac{\pi Z}{{Z_{Fe} }} + \frac{\pi }{2}} \right)} \right]} \cos \left[ {n\left( {\frac{{\pi Z_{s} }}{{Z_{Fe} }} + \frac{\pi }{2}} \right)} \right] \\ \times \left[ {\frac{{I_{0} \left( {\frac{{n\pi R_{Fe} }}{{Z_{Fe} }}} \right)}}{{K_{0} \left( {\frac{{n\pi R_{Fe} }}{{Z_{Fe} }}} \right)}}K_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)K_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right) +\,\left\{ \begin{array} {l}I_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)K_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right);R_{Fe} < R < R_{s}\\ K_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)I_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right);R > R_{s}\\ \end{array}\right\}} \right] \hfill \\ \end{aligned}$$
(8)

Here, L m is the iron core’s magnetic resistance, I α (x) and K α (x) are the first and second kind modified Bessel functions. Then the magnetic field can be calculated by the formula below.

$$B_{R} = - \frac{1}{2\pi R}\frac{\partial \varPsi }{\partial Z} = G_{rs} I_{s} ;B_{Z} = \frac{1}{2\pi R}\frac{\partial \varPsi }{\partial R} = G_{zs} I_{s}$$
(9)

Here, G rs are G zs are the response matrices which are used to calculate the magnetic field in two directions respectively, the radial direction and the vertical direction. We can derive the formulas of response matrices by using (8–9).

$$\begin{aligned} G_{rs} (R,Z) &= \frac{{2n\mu_{0} \pi R_{s} }}{{Z_{Fe}^{2} }} \times \sum\limits_{n = 1}^{\infty } {\sin \left[ {n\left( {\frac{\pi Z}{{Z_{Fe} }} + \frac{\pi }{2}} \right)} \right]} \cos \left[ {n\left( {\frac{{\pi Z_{s} }}{{Z_{Fe} }} + \frac{\pi }{2}} \right)} \right] \hfill \\ &\quad\times \left[ {\frac{{I_{0} \left( {\frac{{n\pi R_{Fe} }}{{Z_{Fe} }}} \right)}}{{K_{0} \left( {\frac{{n\pi R_{Fe} }}{{Z_{Fe} }}} \right)}}K_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)K_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right) + \left\{ \begin{array}{l} I_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)K_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right);R_{Fe} < R < R_{s} \\ K_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)I_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right);R > R_{s} \\ \end{array} \right\}} \right] \hfill \\ \end{aligned}$$
(10)
$$\begin{aligned} G_{zs} (R,Z) &= - \left\{ \begin{aligned} 0;R_{Fe} < R < R_{s} \hfill \\ \frac{{\mu_{0} }}{{Z_{Fe} }};R > R_{s} \hfill \\ \end{aligned} \right\} + \frac{{2\mu_{0} R_{s} }}{{RZ_{Fe} }} \times \sum\limits_{n = 1}^{\infty } {\cos \left[ {n\left( {\frac{\pi Z}{{Z_{Fe} }} + \frac{\pi }{2}} \right)} \right]} \cos \left[ {n\left( {\frac{{\pi Z_{s} }}{{Z_{Fe} }} + \frac{\pi }{2}} \right)} \right] \hfill \\ &\quad\times \left[ {\frac{{I_{0} \left( {\frac{{n\pi R_{Fe} }}{{Z_{Fe} }}} \right)}}{{K_{0} \left( {\frac{{n\pi R_{Fe} }}{{Z_{Fe} }}} \right)}}K_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)K_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right) + \left\{ \begin{array}{l} I_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)K_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right);R_{Fe} < R < R_{s} \hfill \\ K_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)I_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right);R > R_{s} \hfill \\ \end{array} \right\}} \right] \hfill \\ &\quad+ \frac{{2\mu_{0} R_{s} }}{{Z_{Fe} }} \times \sum\limits_{n = 1}^{\infty } {\cos \left[ {n\left( {\frac{\pi Z}{{Z_{Fe} }} + \frac{\pi }{2}} \right)} \right]} \cos \left[ {n\left( {\frac{{\pi Z_{s} }}{{Z_{Fe} }} + \frac{\pi }{2}} \right)} \right] \hfill \\ &\quad\times \left[ {\frac{{n\pi I_{0} \left( {\frac{{n\pi R_{Fe} }}{{Z_{Fe} }}} \right)}}{{Z_{Fe} K_{0} \left( {\frac{{n\pi R_{Fe} }}{{Z_{Fe} }}} \right)}}K^{\prime}_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)K_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right) + \left\{ \begin{array}{l} \frac{n\pi }{{Z_{Fe} }}I^{\prime}_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)K_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right);R_{Fe} < R < R_{s} \hfill \\ \frac{n\pi }{{Z_{Fe} }}K^{\prime}_{1} \left( {\frac{n\pi R}{{Z_{Fe} }}} \right)I_{1} \left( {\frac{{n\pi R_{s} }}{{Z_{Fe} }}} \right);R > R_{s} \hfill \\ \end{array} \right\}} \right] \hfill \\ \end{aligned}$$
(11)

Considering the response matrices G ts , G ns shown in the formulas (5–6) and G rs , G zs calculated by formulas (10–11) are in different coordinate systems, we should transform the response matrices from Cartesian coordinate system to polar coordinate system using formula below.

$$G_{t} = G_{R} \cos \theta + G_{Z} \sin \theta ;G_{n} = G_{R} \cos (\theta - 90^{ \circ } ) + G_{Z} \sin (\theta - 90^{ \circ } )$$
(12)

Here, θ is the tangential angle of the probes, which are distributed counterclockwise along the poloidal circle on HT-7.

In the next section, we will evaluate and validate the response matrices, which are calculated by using formulas (10–12). Using the available response matrices and measured magnetic signals on HT-7, we will evaluate the plasma current center position by using formulas (6–7) for several typical discharges and analyze the comparison between the predicted values and the experimental measurement that mentioned in the Sect. 2.

Results and Discussions

Simulation of the Measurement of Plasma Displacement by Single-Turn Flux Loops

As described above, the measurement of plasma displacement by using single-turn flux loops has limitations. On the one hand, the formula (3–4) is applicable under the assumption that Δr < < a p  < a 0 < < R 0. It means the plasma center should inside of a small bounded neighborhood of the tokamak center. On the other hand, we cannot determine the Shafranov shift during discharge. To be convenient, we use the measured value shown in the left side of formulas (4) to represent the horizontal displacement of plasma approximately.

To evaluate the influence of the assumption in this technique, we simulate the condition that dissatisfied the assumption in EAST using EFIT code [9, 10]. The results are presented in Fig. 4. In EAST, the major radius is 1.80 m and the minor radius is about 0.4 m. The plasma horizontal displacement is 1.81 m in Fig. 4a and b, 1.71 m in Fig. 4c and 2.00 m in Fig. 4d. The Fig. 4a indicates the applicability of this technique under the assumption. The Fig. 4c, d shows that it failed to measure the plasma center by using this technique when the assumption is dissatisfied. By a logical extension of this point, if the plasma center is far away from tokamak center during discharge in HT-7, the measurement of plasma displacement is inaccurate. The Fig. 4b presents that the measurement results become more accurate while the flux loops approach the outmost magnetic surface.

Fig. 4
figure 4

The simulation of studying the influence of the assumption in the measurement of plasma displacement with the circles indicating the location of the flux loops. (a) and (b) The applicability of this technique under the assumption. (c) and (d) The prediction of this technique when the assumption is dissatisfied. The calculation results by using this technique (black x-mark) and the exact value (red point)

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Validation of the Response Matrices

Before we predict the position of plasma current center based on the magnetic measurement, benchmark tests are performed to evaluate and validate the response matrices calculated by using formulas (10–12). We selected a testing shot #96489 without plasma, during which only the vertical field coils has power supply. We compare the measured magnetic signals with our calculation based on the formula (13).

$${{B}}_{p} = {{G}}_{v} \cdot {{I}}_{v}$$
(13)

The comparison is shown in Fig. 5 below. Considering the measurement error of the probes, the calculated and the measured values consistent with each other perfectly and the calculated response matrices are evaluated to be valid and available.

Fig. 5
figure 5

Comparison of the measured magnetic signals (square points used) with the predicted values (circle points used) for a HT-7 testing shot 96489# in 2.3 s

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The Major Radius of the Plasma Current Center

Since the response matrices have been calculated and validated, we will evaluate the plasma current center based on the magnetic measurement, analyze the R center results and the plasma horizontal displacement values R p that measured by single-turn loops on HT-7.

In HT-7, the magnetic field induced by the ohmic field coils current is mainly confined in the iron core far away from the vacuum vessel. In this case, the vertical magnetic field in HT-7 is mostly induced by the current flowing in the vertical field coils. As shown in formula (9), the vertical magnetic field is directly proportional to the vertical field coils current. Moreover, in the large aspect-ratio approximation, the external vertical field can be derived in the form [1]:

$$B_{v} = - \frac{{\mu_{0} I_{p} }}{{4\pi R_{p} }}\left( {\ln \frac{{8R_{p} }}{{a_{p} }} + \varLambda - \frac{1}{2}} \right)$$
(14)

Thus the horizontal displacement of plasma can be expressed as the form:

$$R_{p} = - \frac{{\mu_{0} I_{p} }}{{4\pi B_{v} }}\left( {\ln \frac{{8R_{p} }}{{a_{p} }} + \varLambda - \frac{1}{2}} \right) \propto \frac{{I_{p} }}{{I_{v} }}$$
(15)

Here, I p is the plasma current and I v is the current that flows in the vertical field coils. When the plasma current maintains unchanged, with the absolute value of the vertical field coil current increasing, the strength of the vertical magnetic field enhances. As a result, the inward Lorentz force that against the plasma current strengthens. It means that the plasma current center will move toward inside of the vacuum vessel; conversely, it will move outside.

The Fig. 6 shows the result of a typical ohmic discharge #103965. The plasma horizontal displacement R p including Shafranov shift that measured by single-turn loops directly is presented in blue line, and the red line presents the estimation of the major radius of the plasma current center R center . During discharge, the plasma current and position are affected by the vertical field coils current in HT-7. According to the analysis above, shown in the formula (15), the plasma horizontal displacement should change following the ratio of plasma current and the vertical field coils current I p /I v when the plasma current ramps up to the flattop after 180 ms.

Fig. 6
figure 6

Prediction of the major radius R of plasma current center and comparison with the plasma horizontal displacement values that measured by single-turn loops for ohmic discharge #103965

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For the ohmic shot #103965, during 180–350 ms, the ratio of plasma current and the vertical field coils current increase obviously. Following the change, the plasma should move from outside to inside toward horizontal direction, which means the horizontal displacement R p should increase during this phase. However, the measurement by using single-turn flux loops failed to reflect the tendency. On the other hand, the major radius of the plasma current center R center we estimated reflected the trend of the vertical magnetic field. After 350 ms, the plasma current center turns into the steady state as the plasma current and the vertical field coils current retain in the equilibrium state. In HT-7, the plasma current center is nearby the magnetic axis and the magnetic axis is outside the plasma outmost magnetic surface about 1–2 cm (the major radius of HT-7 is 1.22 m and the minor radius is 0.27 m). The R center stays outside of the plasma horizontal displacement which is the measurement R p plus the Shafranov shift (about 2 cm in HT-7) about 1 cm, which is also reasonable.

In addition, we select three discharges with auxiliary heating and current drive on HT-7 to verify this method. As shown in Figs. 7 and 8, the R center results are consistent with the trend of the ratio I p /I v very well. In particular, during the current flattop phase at about 280 ms in these three discharges, the ratio I p /I v reaches its bottom and then increases. The horizontal displacement R p should reach bottom after that moment and then increases, which means the plasma reach the innermost position and then move towards outside. Unfortunately, the R p measured by using single-turn flux loops failed to reflect the change and kept unchanged. Furthermore, as the shot #104750 shown in Fig. 8, the ratio I p /I v shifts significantly from 550 to 650 ms. Relatively, the R center calculated based on the magnetic measurement reflect the tendency expressly. About the invalidation of measurement by single-turn loops, as simulated and analyzed in Sect. 4.1, this technique will become invalid when the assumption Δr < < a p  < a 0 < < R 0 is dissatisfied.

Fig. 7
figure 7

Prediction of the major radius R of plasma current center and comparison with the plasma horizontal displacement values that measured by single-turn loops for ohmic+LHCD discharge #104700

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Fig. 8
figure 8

Prediction of the major radius R of plasma current center and comparison with the plasma horizontal displacement values that measured by single-turn loops for ohmic+ICRF discharge #104750

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The Vertical Position Z of the Plasma Current Center

By now, we cannot get the effective experimental data of the horizontal field coils current. Its change mainly affects the trend of the vertical position of plasma current center, as the vertical field coils current to the horizontal position of the plasma current center. Here we could not discuss the calculation of vertical position of the plasma current center. In future, we could gather the experimental data of the horizontal field coils current for further analysis.

Summary

In this study, we found out the plasma horizontal displacement value, which was measured directly by single-turn loops, had failed to reflect the change of plasma current center on HT-7, especially during the plasma breakdown phase. We did the simulation of this measurement method and analyzed the limitation of measuring the plasma displacement by using single-turn loops. In addition, we attempted to evaluate the plasma current center by using external magnetic measurement on iron-core tokamak HT-7. The calculation showed that this simplified method was validated to evaluate the horizontal position of plasma current center for the discharges on HT-7. The prediction of vertical position of the plasma current center need to be further examined. Besides, we need to modify the response matrices to improve the absolute value accuracy of the prediction during practical discharges. Further, this simplified method could be used for the determination of plasma current center on the iron-core tokamak.