Theoretical and Experimental Investigation of Effects of Toroidal Field Ripple on Poloidal Beta, Internal Inductance and Shafranov Shift in IR-T1 Tokamak

In this paper we p resented an investigation of effects of the toroidal field ripple (TF ripp le) (due to fin ite number of the toroidal field coils) on the poloidal beta, internal inductance, and Shafranov shift in IR-T1 Tokamak. For these purposes, array of magnetic probes and also a diamagnetic loop with its compensation coil were designed, constructed and installed on outer surface of the IR-T1. Amplitude of the TF ripple is obtained 0.01, and also the effects of TF ripple on the poloidal beta, internal inductance, and Shafranov shift were discussed. In the high field side reg ion of tokamak chamber, the TF ripple effects is the decreasing and increasing of the poloidal beta and internal inductance, respectively, whereas the low field side have inverse situations. Also no sensible variation observed in the Shafranov shift due to TF ripple on IR-T1 plasma.


Introduction
In most of toro idal plasma equilib riu m studies, tokamak eq u ilib ria are in ves tigated as axis y mmetric s ystems (two-dimensional systems). Although this symmetry offers many advantages for its analys is, but realistic to kamaks consists of fin ite number of toroidal field coils (TF). Then, this discreteness yields the toroidal field ripples (a periodic variation of the toroidal magnetic field) (TF ripples). In other wo rds, realistic to kamaks co u ld n ot b e axis y mmet ric configurations. Most of the TF ripple studies have been done on effects of the TF ripple on confinement of the high energy alpha part icles, format ion o f internal t ransport barriers, p las ma rot at ion , and H-mo d e p erfo rmance. In IR-T1 Tokamak, which it is a small, low beta and large aspect ratio tokamak with a circu lar cross section (see Tab le 1), the number N o f TF co ils is 16, and then the period of the TF ripple was 22.5°. In this paper we presented the effects of the TF ripple on determinations of the polo idal beta, p las ma internal inductance, and therefore the Shafranov parameter and Shafranov shift in IR-T1. Determinations of the poloidal beta, internal inductance, and Shafranov shift are essential for to kamak experiments and opt imized operat ion. Also some of the p lasma info rmations can be deduced fro m these parameters, such as plasma toroidal current profile, plas ma energy, plasma energy confinement time, and magnetohydrodynamics (MHD) instabilities.
Magnetic diagnostics, in particular d iamagnetic loop are commonly used in tokamaks to measure the variation of toroidal flu x 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 the Shafranov parameter fro m the magnetic probes and poloidal beta from the diamagnetic loop gives a value of the internal inductance. Also Shafranov shift can be determined fro m the Shafranov parameter. Although the questionable parameters in this work usually analy zed as a global plasma parameters, but we present theoretical and experimental investigation of the TF ripple on the estimate of these parameters fro m localized measurements. Because of dependence of the toroidal field on the TF ripple amplitude, therefore we expect that these parameters are also depending on TF ripple amplitude . Brief approach for determinations of the TF ripple and Shafranov parameter by the discrete magnetic probes will be presented in section 2. Diamagnetic loop method for measurement of the poloidal beta, internal inductance, and then the Shafranov shift will be presented in section 3. Details of design and construction of magnetic probe and diamagnetic loop will be presented in section 4. Experimental results will be discussed in section 5. Summary and conclusion will be discussed in section 6.

Determinations of the TF Ripple and Shafranov parameter by the Discrete Magnetic Probes
A simple analytic model of the toroidal magnetic field strength widely used in the analysis is the following: (1) where is the toroidal magnetic field at center of the tokamak chamber, and are po loidal 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 co ils is 16, then the period of the TF ripple was 22.5°, and the inverse aspect ratio is 0.278. Although the amplitude of TF ripple is no constant at the different poloidal angles, we can define the average value of TF ripp le amplitude fro m the Eq. (1): where these values of the toroidal magnetic fields can be determined using the magnetic probes at above poloidal and toroidal angles. Our measurements using the magnetic pick-up coils on outer surface of the IR-T1 tokamak show that the amplitude of TF ripple on the sensor position is 0.01, which is close to the result of modeling as shown in Fig. (1) (Fig. (1) is a plot of the Eq. (1) at the edge of IR-T1 tokamak plasma).
Also the Shafranov parameter and therefore Shafranov shift relate to the distribution of magnetic fields around the plasma current. Therefore, those can be written in terms of the tangential and normal co mponents of the magnetic field on the contour (see Fig. (2)). Distributions of the tangential and normal magnetic fields can be written in the first order of the inverse aspect ratio as follows, respectively [1][2][3][4][5]: (4) (5) Figure 1. Dependence of the toroidal field to the poloidal and toroidal angles (T F ripple) where (7) and where is the poloidal beta and is the plasma internal inductance. We measured and which are the contribution of the plas ma current to the equilib riu m field, after co mpensating the vacuum field and integrating of the output signals of magnetic probes. The major appro ximations in our wo rk are that we suppose the plasma minor rad ius defined by limiter radius, namely is constant, and also the TF ripple amp litude at different poloidal angles is constant, whereas the toroidal flu x is no constant as a function of phi, and then we have to have field lines which loop back on themselves, in which case we have to have localized poloidal currents. Experimental results will be presented in the section 5.

Measurements of the Poloidal Beta and Internal Inductance with Diamagnetic Loop
Diamagnetic loop measures the toroidal diamagnetic flu x for the purpose of measurement of the poloidal beta and thermal energy of the plas ma. The toroidal flu x produced by the plasma is related to the total perpendicular thermal energy of the plasma. This diamagnetic flu x is usually measured with the d iamagnetic loop. It is consists of a simp le loop that links the plasma colu mn, ideally located in a poloidal direct ion in order to minimize poloidal field pick-up. Relation between the diamagnetic flu x and the poloidal beta is [2][3][4][5][6][7]: (8) where is the diamagnetic flu x.
By substituting the Eq. (1) in the Eq. (8) we have: where and where is the toroidal magnetic field in the absence of the plasma and center of chamber wh ich can be measured using the Ampere law, is the plasma current, is the toroidal flu x because of toroidal field coils, and are the passing flu x through loop due to possible misalignment between oh mic field and vertical field and the diamagnetic loop, and is the toroidal field due to eddy current on the vacuum chamber. These flu xes can be compensated with compensation coil (see Fig. (3)) and also using dry runs technique. It must be noted that compensating coil for diamagnetic loop is wrapped out of the plas ma current, and only the toroidal flu x (wh ich is induced by the change of toroidal field coil current when p lasma discharges) can be received (see Figs. (2),(3)). Therefore, according to above two sections we can find the internal inductance. Fro m Eq. (6) we have: By substituting the Eq. (6) and (9) in Eq . (10), we can write: (11) Also, the Shafranov shift is determined fro m rearranging the Eq. (4): (12) where the effects of the TF ripple introduced in the last expression.
Experimental results of effects of the TF ripple on measurements of the poloidal beta, internal inductance, and Shafranov shift, will be presented in the section 5.

Design and Construction of the Magnetic Probes and Diamagnetic Loop
In general, a 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.
In the IR-T1 tokamak an array of four magnetic probes were designed, two magnetic probes were installed on the circular contour of the radius in angles  Diamagnetic loop and its compensating coil also were constructed and installed on outer surface of the IR-T1 tokamak, as shown in Fig. (3).  Table 2.
After measurements and fro m magnetic probes, fro m d iamagnetic loop, fro m Rogowski coil and substituting them in to Eq. (9), Eq. (11), and Eq . (12), the poloidal beta, internal inductance, and Shafranov shift and also the effects of TF ripple on them measured. Experimental results presented in the next section.

Experimental Results of the Effects of TF Ripple on the Plasma Parameters
We used the electric circu it as shown in Figures (4) and (5), for measurements of the magnetic fields and diamagnetic flu x, respectively:  According to above discussion, firstly, we measured the poloidal beta fro m the Eq. (9) and then the internal inductance and Shafranov shift fro m the Eq. (11) and Eq. (12). Results presented in the Figs. (6), (7), (8), and (9).
In the Figure (6), we plotted the plasma parameters in the absence of TF ripple. Plas ma current is observable in the Figure (6a). As shown in the Figure (6b), the measured poloidal beta is closes to one which acceptable for the  In the Figure (7), the effects of TF ripple amplitude on the difference of polo idal beta with and without TF ripple (DBetap) at d ifferent poloidal angles presented. As shown, difference between the poloidal beta in present of the TF ripple and in absent of the TF ripple is in order of the .
Also in the high field side reg ion ( ) the d ifference is negative, whereas in lo w field side ( ) the difference is positive.  In the Figure (9), the effects of TF ripple amplitude on the Horizontal Displacement (H.D.) at different poloidal angles presented. No difference observed.

Summary and Conclusions
In this paper we presented theoretical and experimental investigation of effects of the TF ripple on the poloidal beta, internal inductance, and Shafranov shift in IR-T1 tokamak. For these purposes, array of magnetic probes and also a diamagnetic loop have been designed, constructed, and installed on outer surface of the IR-T1. Then, poloidal and radial co mponents of the magnetic fields and also diamagnetic flu x measured. A mplitude of the TF ripp le is obtained 0.01, and also the effect of TF ripple on the poloidal beta, internal inductance, and Shafranov shift were investigated. One of the results is that difference between the poloidal beta 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 negative, whereas in the low field side the difference is positive. Another results is that 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 h igh field side region unlike the poloidal beta case, the difference is positive, whereas in low field side the difference is negative. In the Shafranov shift, no difference observed. Also no sensible variation observed in the Shafranov shift due to the TF ripple on IR-T1 tokamak plasma. The major appro ximations in our work are that we suppose the plasma minor rad ius is constant and also the TF ripple amp litude at different poloidal angles is constant, whereas the toroidal flu x is no constant.