Dependence of Tokamak Plasma Position to its Internal Inductance

In this contribution we will presented comparison of two techniques in order to investigate of the effects of internal inductance on tokamak plasma position, based on a toroidal flux loop (diamagnetic loop) and magnetic probes measurements. For this purpose, a diamagnetic loop with its compensation coil, and also array of magnetic probes were designed, constructed, and installed on outer surface of the IR-T1 tokamak chamber, and then the poloidal beta and poloidal and radial magnetic fields obtained. Moreover a few approximate values of the internal inductance for different possible profiles of the plasma current density are also calculated. Then, the Shafranov parameter and also the Shafranov shift were determined. Experimental results compared. Comparison of results on IR-T1, show that (1) by increasing the internal inductance from one, plasma column shifted inward, and also (2) IR-T1 plasma current density profile relate to the power of 3 ≈υ approximately.


Introduction
An equilibriu m p lasma position is one of the important problems of tokamak experiments. Equilibriu m is a condition for wh ich plasma pressure is balanced by electro magnetic force (Lo rentz force). To kamak plas ma equilibriu m 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 flu x loop (diamagnetic loop) are co mmonly used in tokamaks to measure the variation of toroidal flu x induced by the plasma. Fro m this measurement, the total d iamagnetic energy content and the confinement time of the plasma can be obtained as well as the poloidal beta. On the other hand, the magnetic fields distribution outside the plasma and then the Shafranov shift depend on the combination of Shafranov parameter (asymmetry factor) Λ . If i l is known fro m the anyway, then measurement of the Shafranov parameter and poloidal magnetic field g ives a value of the Shafranov shift. The value of i l is determined by the radial distribution of toroidal current profile of the plasma [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].
In this paper we present two magnetic techniques based on the diamagnetic loop and magnetic probe for determination of the poloidal Beta and poloidal magnetic field, and moreover an appro ximated method for determination of the plasma internal inductance, and therefore the Shafranov parameter and Shafranov shift in IR-T1 To kamak, wh ich it is a small, low p β and large aspect ratio tokamak with a circular cross section (see Table 1). Details of the theoretical approach for these magnetic techniques will present in section 2. Details of the diamagnetic loop and magnetic probe methods for determinations of the poloidal Beta and poloidal magnetic field will be d iscussed in section 3. Details of approximate method for calculation of the internal inductance will present in section 4. Experimental results for determination of the polo idal magnetic field, Shafranov parameter, and Shafranov shift will be discussed in section 5. Summary and discussion will present in section 6.

Theoretical Approach for Determination of the Shafranov Shift Based on the Diamagnetic Loop and Magnetic Probe
Shafranov parameter and therefore Shafranov shift relate to the distribution of magnetic fields around the plas ma current. Therefo re, those can be written in terms of the tangential and normal components of the magnetic field on the contour Γ (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 follo ws, respectively [1,5]: , cos 2 1 2 where 0 R is the major radius of the vacuum vessel, s ∆ is the Shafranov shift, p I is the plasma current, a and b are the minor p lasma radius and minor chamber rad ius respectively, and Λ is the Shafranov parameter. These equations accurate for low β plas ma and circular cross section tokamaks as IR-T1, and where: Also with co mbination of Eq. (1) and Eq. (2), we have: Therefore, with co mbination of the magnetic field and poloidal Beta measurements, and also calculation of the internal inductance, Shafranov shift can be determined fro m Eq. (4).
In order to investigate the effects of the internal inductane on determination of plas ma position, we used the Eq. (4) and then results compared with results of Eq. (5) in IR-T1.

Magnetic Probe and Diamagnetic Loop for Determinations of the Magnetic Field and Poloidal Beta
In general the magnetic sensors (magnetic probe or diamagnetic loop) works by Faraday's law and measures component(s) of the local magnetic fields or magnetic flu xes for use in plasma control, equilibriu m reconstruction and detection of plas ma energy, poloidal beta and M HD instabilities.
Magnetic probe consists of a coil in solenoidal form, which whose dimensions are small co mpared to the gradient scale length of the magnetic field. A total magnetic flu x 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.
The induced voltage in the magnetic probe and then magnetic field is: where ω is the frequency of the fluctuations of the magnetic field. Therefore in order to measurement of the magnetic field distribution we must be integrating the output signals of the magnetic probe.
On the other hand, diamagnetic loop measures the toroidal diamagnetic flu x for the purpose of measurement of the poloidal beta and thermal energy of the plasma. The toroidal flu x that produced by the plasma is related to the total perpendicular thermal energy o f the plas ma. Th is diamagnet ic flu x is usually measured with the diamagnetic loop. It is usually a single wire which circling the plas ma co lu mn either inside or outside of the p lasma vacuum chamber. Intrinsical ly this loop will also pickup the toroidal magnetic flu x fro m the toroidal field co il and any current circu lating in the poloidal plane, in particular toro idal field coil current, eddy currents in the conducting vacuum chamber induced during transient changes in the plasma energy and plasma current. In other words, the diamagnetic loop consist of a simple loop that links the plasma colu mn, ideally located in a poloidal direction in order to minimize detecting the poloidal field. In cases of the ohmically heated tokamaks (lo w beta) where the plasma energy density is small co mpared to the energy density of the magnetic field, the change in the total toroidal magnetic flu x is small. Therefore a reference signal equal to the vacuum toroidal magnetic flu x is usually subtracted from it, giv ing a net toroidal flu x equal to the diamagnetic flu x D ∆Φ produced by the circular plasma. Relat ion between the diamagnetic flu x and the poloidal beta derived fro m simp lified equilibriu m relation[2,3,6] is: According to above discussion, we designed, constructed, and installed four magnetic p robes and also diamagnetic loop with its compensation coil, on outer surface of the IR-T1, in order to determinations of the magnetic fields distribution and poloidal Beta. Plasma current is also obtained with Rogowski coil. But, in determination of the Shafranov shift with Eq. (4), we needed to calculation of the internal inductance, where in next section approximate method presented.

Approximate Methods for Calculation of the Plasma Internal Inductance
The internal inductance of the plasma per unit length, normalized to  (11) where this relation for IR-T1 tokamak parameters equal to value of 0.994.
Second approximate value for the internal inductance can be determined fro m the well-known Bennett current density profile, as: therefore, the poloidal magnetic field profile can be obtained: and then second approximate value for internal inductance can be obtained: (14) 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 [4]: The poloidal magnetic field profile can be obtained:    Extremity, our final appro ximate method for determi -nation of the internal inductance is experimental method. In this method firstly Shafranov parameter is obtained fro m the magnetic probe measurements, and then the values of the poloidal Beta which obtained fro m diamagnetic loop, subtracted from it.
With combination of Eq. (1) and Eq. (2), we can find the value of the Shafranov parameter as: For this purpose, a diamagnetic loop with its compensation coil, and also an array of magnetic probes were designed, constructed, and installed on outer surface of the IR-T1 tokamak chamber (see Figures (4), (5)), and then the poloidal Beta and poloidal and radial magnetic fields obtained from them. As shown in Figure (

Experimental Results and Comparison between Them
In the IR-T1 tokamak an array of four magnetic probes were designed, two magnetic probes were installed on the circular contour Γ of the rad ius    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. We used the electric circu it as shown in Figures (6) and (7), for the measurements of induced voltage in the diamagnetic loop and magnetic probes, respectively:  Therefore for the steady state portion of the plas ma current and according to electric circuit, for output signals of the probes, we have: where, RC is the integrator time constant, and Eq. (19) is the integrating output of the probe signal. Moreover i V is the inductive voltage supplied by each one of the magnetic pickup coils or diamagnetic loop, where installed around the vacuum chamber of the IR-T1 tokamak.
Therefore, by substituting the Eq. (7) And in the case of diamagnetic loop, diamagnetic flu x can be obtained from:  According to above discussion, firstly we determined the Shafranov shift fro m the Eq. (5), which only depends to magnetic probes and Rogowski coil data (see Figure (8)).
Then, to compare the results we also determined the Shafranov shift fro m Eq. (4), wh ich depends to the magnetic probes and diamagnetic loop data, and also the appro ximate values of internal inductance (see Figure (9)). Results of comparing the two methods show that IR-T1 tokamak plas ma current density relate to li=2 (corresponding to ).

Summary and Conclusions
Array of magnetic p robes and also diamagnetic loop have been designed, constructed, and installed on the outer surface of the IR-T1 tokamak chamber. The poloidal and radial components of the magnetic fields and also diamagnetic flu x signal obtained, and therefore the Shafranov parameter and poloidal beta were measured fro m them. Then, a few approximate values of the internal inductance calculated, therefore the Shafranov shift determined.
To co mpare, the Shafranov shift obtained independently fro m the magnetic probes measurements. Results of comparing the two methods show that by increasing the values of internal inductance fro m one, p lasma co lu mn shifted inward, and also IR-T1 plas ma current profile relate to the power of 3 ≈ υ .