1. Introduction

Tokamak plasma equilibrium is one of the important problems of the fusion programs. Plasma equilibrium is a condition which plasma pressure is balanced by electromagnetic force (Lorentz force). Tokamak plasma equilibrium is a significant fraction of the fusion program studies in order to achieve tokamaks optimized operation and become close to Lawson criterion. Determination of precise plasma position during confinement time is essential to transport it to the control system based on feedback. Magnetic diagnostics, in particular toroidal flux loop (diamagnetic loop) are commonly used in tokamaks to measure the variation of toroidal flux induced by the plasma. From this measurement, the total diamagnetic energy content and the confinement time of the plasma can be obtained as well as the poloidal beta. If the internal inductance is known from the anyway, then measurement of the Shafranov parameter and poloidal magnetic field gives a value of the plasma horizontal displacement. The value of is determined by the radial distribution of toroidal current profile of the plasma[1-13].

In this paper we presented combination of the diamagnetic loop and magnetic probes for determination of the poloidal Beta and poloidal magnetic field, and moreover an approximate calculations for determination of the plasma internal inductance, and therefore the plasma horizontal displacement in IR-T1 Tokamak, which is a small, low and large aspect ratio tokamak with a circular cross section (see Table 1). Details of the theoretical approach for determination of the plasma horizontal displacement based on a diamagnetic loop and magnetic probes measurements will be presented in section 2. Details of approximate calculations for determination of the internal inductance will be presented in section 3. Experimental results will be discussed in section 4. Summary and discussion will be present in section 5.

Table 1. Main parameters of the IR-T1 tokamak

2. Theoretical Approach for Determination of the Plasma Horizontal Displacement Based on a Diamagnetic Loop and Magnetic Probes

Shafranov parameter and therefore the plasma horizontal displacement relate to the distribution of magnetic fields around the plasma current. Therefore, those can be written in terms of the tangential and normal components of the magnetic field on the contour (see Fig. (2)). Distribution of the poloidal and radial magnetic fields are can be written in the first order of the inverse aspect ratio as follows, respectively[1]:

(1)

(2)

where is the major radius of the vacuum vessel, is the Shafranov shift, is the plasma current, and are the minor plasma radius and minor chamber radius respectively, and is the Shafranov parameter. These equations accurate for low plasma and circular cross section tokamaks as IR-T1, and where:

(3)

where is the poloidal beta, and is the plasma internal inductance.

Rearranging of the Eq. (1) give us:

(4)

where

(5)

Therefore, with combination of the poloidal magnetic field and poloidal beta measurements, and also calculation of the internal inductance, the plasma horizontal displacement can be determined from Eq. (4).

Magnetic probes consist of a coil in solenoidal form, which whose dimensions are small compared to the gradient scale length of the magnetic field. A total magnetic flux passed through such a coil is , where n is the number of turns of coil, A is the average area of cross section of coil, and B is the local magnetic field parallel to the coil axis.

On the other hand, diamagnetic loop measures the toroidal diamagnetic flux for the purpose of measurement of the poloidal beta and thermal energy of the plasma. The toroidal flux that produced by the plasma is related to the total perpendicular thermal energy of the plasma. This diamagnetic flux is usually measured with the diamagnetic loop. The relation between diamagnetic flux and the poloidal beta derived from simplified equilibrium relation is[2-4]:

(6)

where

and where

Where is the toroidal magnetic field in the absence of the plasma which can be obtained by the magnetic probe or diamagnetic loop, is the plasma current which can be obtained by the rogowski coil, is the toroidal flux because of toroidal field coils, and are the passing flux through loop due to possible misalignment between ohmic field and vertical field and the diamagnetic loop and is the toroidal field due to eddy current on the vacuum chamber. These fluxes can be compensated either with compensation coil or dry runs technique. Extremity, calculation of the plasma internal inductance is discussed in the next section.

3. Approximate Methods for Calculation of the Plasma Internal Inductance

The internal inductance of the plasma per unit length, normalized to can be determined from the conservation of zeroth order magnetic energy:

(7)

For typical profile of the poloidal field which correspond to flat current density profile (usually accurate for low beta tokamak), as:

(8)

where

Then first approximate value for the internal inductance can be easily obtained by substituting Eq. (9) in Eq. (8):

(9)

where this relation for IR-T1 tokamak parameters equal to value of 0.994.

Second approximate value for the internal inductance can be determined from the well-known Bennett current density profile, as:

(10)

therefore, the poloidal magnetic field profile can be obtained:

(11)

and then second approximate value for internal inductance can be obtained:

(12)

where this relation for IR-T1 tokamak parameters equal to value of 0.332.

In general case, for the large aspect ratio and circular plasma, the current density distribution is[2]:

(13)

The poloidal magnetic field profile can be obtained:

(14)

where

(15)

If we assume a more peaked current profile with central safety factor, then the values of the internal inductances can be determined from substituting the Eq. (14) in Eq. (7), as a function of the . Results presented in Table 2 and Figure (1).

Table 2. Dependence of the Internal Inductance to the values of       for IR-T1 tokamak parameters

Figure (1). Dependence of the Internal Inductance to the values of for IR-T1 tokamak parameters

Figure (2). Combination of the Diamagnetic Loop and Magnetic Probe Results: (a) Plasma Current, (b) Effective Edge Safety Factor, (c) Toroidal Magnetic Field, and (d) Shafranov Parameter

Our experiments show that the value of which proportional to the edge safety factor reduced from 8 to 1 along time interval of plasma current (see Figure (2)). Therefore, according to recent method, IR-T1 tokamak plasma internal inductance reduced from 2.5 to 1.2 along the time interval of plasma current.

4. Experimental Results

According to above discussion, two magnetic probes were designed and installed on outer surface of the IR-T1 in radius of in angles of and to detect the tangential component of the magnetic field. Also a diamagnetic loop with its compensation coil were constructed and installed on outer surface of the IR-T1. Design parameters of the magnetic pickup coils present in Table 3. Diamagnetic loop and its compensating coil also were constructed and installed on the IR-T1 tokamak. Its characteristics are also shown in Table 3.

After measurement of from magnetic probes, from rogowski coil, poloidal beta from diamagnetic loop and substituting them and the approximate values of internal inductance in to Eq. (4), the plasma horizontal displacement were obtained.

Results presented in the Fig. (3). As shown, by increasing the internal inductance from one, plasma shifted inward.

Table 3. Design parameters of the magnetic probe and diamagnetic loop

Figure (3). (a) Plasma current, (b) Horizontal Displacement (H.D.) obtained with combination of the magnetic probe, diamagnetic loop, and approximate values of the internal inductance (Eq. (4)), for li=1, (c) H.D. for li=2, (d) H.D. for li=3, and (e) H.D. for li=4. As shown, by increasing the internal inductance from one, plasma shifted inward

5. Summary and Conclusions

Array of magnetic probes and also diamagnetic loop have been designed, constructed, and installed on outer surface of the IR-T1 tokamak chamber. The poloidal component of the magnetic fields and also diamagnetic flux signal obtained, and therefore the poloidal field and poloidal beta were measured from them. Then, a few approximate values of the internal inductance calculated. Therefore the plasma horizontal displacement determined.

Results show that by increasing the values of internal inductance from one, plasma column shifted inward.