1. Introduction

Usually tokamaks plasma equilibria are investigated as two-dimensional (axisymmetric) systems. Although this symmetry offers many advantages for its analysis, but realistic tokamaks consists of finite number of Toroidal Field (TF) coils. Then, this discreteness yields the toroidal field ripples (a periodic variation of the toroidal magnetic field). In other words, realistic tokamaks could not be axisymmetric configurations. Most of the TF ripple studies have been done on effects of the TF ripple on confinement of the high energy alpha particles, formation of internal transport barriers, plasma rotation, and H-mode performance. In IR-T1 Tokamak, which is a small, low Beta and large aspect ratio tokamak with a circular cross section (see Table 1), the number N of TF coils is 16, and then the period of the TF ripple was 22.5°. In this paper we present the effects of the TF ripple on the plasma internal inductance in IR-T1 tokamak. Determination of the internal inductance is essential for tokamak experiments and optimized operation. Also some of the plasma information can be deduced from this parameter, such as plasma toroidal current profile.

Magnetic diagnostics, in particular diamagnetic loop (toroidal flux loop) are commonly used in tokamaks to measure the variation of toroidal flux induced by the plasma and then the poloidal Beta. On the other hand, the magnetic fields distribution outside the plasma provides the measurement of the combination of poloidal Beta and internal inductance, via the Shafranov parameter (). Then measurement of from the magnetic probes and poloidal

Beta from diamagnetic loop gives a value of internal inductance[1-65]. In this paper we present experimental investigation of the TF ripple on this parameter. Because of dependence of the toroidal field on the TF ripple amplitude, therefore we expect that this parameter is also depending on TF ripple amplitude. Brief approach for determinations of the TF ripple and Shafranov parameter using the discrete magnetic probes will be present in section 2. Diamagnetic loop method for measurement of the poloidal Beta and internal inductance will present in section 3. Experimental results of effects of TF ripple on the plasma internal inductance will discuss in section 4. Summary and discussion will present in section 5.

Table 1. Main Parameters of the IR-T1 Tokamak

2. Determinations of the TF Ripple and Shafranov Parameter Using the Discrete Magnetic Probes

A simple analytic model of the toroidal magnetic field strength widely used in the analysis is[1]:

(1)

where is the toroidal magnetic field at center of the tokamak chamber, and are poloidal and toroidal angles respectively, is the inverse aspect ratio, is the number of the toroidal field coils, and is the amplitude of the TF ripple where defined as:

(2)

In the IR-T1 the number of TF coils is 16, then the period of the TF ripple was 22.5°, and the inverse aspect ratio is 0.278. From the Eq. (1) we can write:

(3)

where these values of the toroidal magnetic fields can be determined using the magnetic probes at above poloidal and toroidal angles. Our measurements show that the amplitude of the TF ripple in IR-T1 is 0.01, as shown in Fig. (1).

Figure (1). Dependence of the Toroidal Magnetic Field on the Poloidal and Toroidal Angles, TF Ripple is also observable

Also the Shafranov parameter 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)). Distributions of the tangential and normal magnetic fields are also can be written in the first order of the inverse aspect ratio as follows, respectively[2,3,5]:

(4)

(5)

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. These equations accurate for low plasma, large aspect ratio, and circular cross section tokamaks as IR-T1, and where:

(6)

where

(7)

and where is the poloidal Beta and is the plasma internal inductance. We can obtain and after compensating and integrating of output signals of the magnetic probes. The compensation done by fields discharge without plasma and receives output signals of the magnetic probes and subtract those from total output signals. Experimental results will present in the section 4.

3. Determination of the Plasma Internal Inductance Using the Diamagnetic Loop

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. In cases of the ohmically heated tokamaks (low beta) where the plasma energy density is small compared to the energy density of the magnetic field, the change in the total toroidal magnetic flux is small. Therefore a reference signal equal to the vacuum toroidal magnetic flux is usually subtracted from it, giving a net toroidal flux equal to the diamagnetic flux produced by the circular plasma. Relation between the diamagnetic flux and the poloidal beta derived from simplified equilibrium relation[2-7]:

(8)

by substituting the Eq. (1) in the Eq. (8) we have:

(9)

where and where is the toroidal magnetic field in the absence of the plasma which can be obtained by the magnetic probe, 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 fields discharge without plasma. It must be noted that compensating coil for diamagnetic loop is wrapped out of the plasma current, and only the toroidal flux (which is induced by the change of toroidal field coil current when plasma discharges) can be received. So the diamagnetic flux caused by plasma current can be measured from the diamagnetic and compensating coil using subtraction. Therefore, according to above two sections we can find the internal inductance. From Eq. (6) we have:

(10)

By substituting the Eq. (6) and (9) in Eq. (10), we can write:

(11)

where the effect of the TF ripple introduced in the Shafranov parameter.

Experimental results of effects of the TF ripple on the internal inductance will present in next section.

4. Experimental Results

According to above discussion, we determined the plasma internal inductance and the effects of TF ripple on it. Results present in Figs. (2) and (3). As shown, the difference between the internal inductance in presence of the TF ripple and in absence of the TF ripple is in order of the , and in the high field side region of tokamak chamber, the TF ripple effect is increasing of the plasma internal inductance, whereas the low field side has inverse situation.

5. Summary and Discussion

In this research we investigated the effects of TF ripple on the plasma internal inductance in IR-T1 Tokamak. For this purpose, array of magnetic probes and also a diamagnetic loop with its compensation coil were designed, constructed, and installed on the outer surface of the IR-T1. Amplitude of the TF ripple is obtained 0.01, and also the effects of the TF ripple on the plasma internal inductance presented. One of the results is that the difference between the internal inductance in presence of the TF ripple and in absence of the TF ripple is in order of the , and also in the high field side region the difference is positive, whereas in low field side the difference is negative. In other words, in the high field side region of tokamak chamber, the TF ripple effect is increasing of the plasma internal inductance, whereas the low field side has inverse situation.

Figure (2). Parameters in absence of the TF ripple, (a) plasma current, (b) poloidal Beta, (c) internal inductance, and (d) Horizontal Displacement (H.D.)

Figure (3). Effects of the TF ripple amplitude on the difference of internal inductance with and without TF ripple (Dli) at different poloidal angles. As shown, difference between the internal inductance in present of the TF ripple and in absent of the TF ripple is in order of the . Also in the high field side region () the difference is positive, but in low field side () the difference is negative