Determination of Plasma Position using Poloidal Flux Loops and Comparing with Analytical Solution in IR-T1 Tokamak

In this contribution we presented comparat ive measurements of tokamak plasma position. In the first technique, two poloidal flux loop were designed and installed on outer surface of the IR-T1 tokamak chamber, and then the plasma displacement obtained from them. To compare the result obtained using this method analytical solution of the Grad-Shafranov equation based on expansion of free functions as quadratic in flux function is also experimented on IR-T1. Results of the two methods are in good agreement with each other.


Introduction
In oh mically heated (lo w β ) tokamaks, rad ial pressure balance is achieved by the poloidal field, and toroidal force balance is achieved by equality between the external vertical field force (Lorent z force) and outward forces due to toroidal configuration. But, in toroidal force balance problem, the two opposite forces may be not equal and therefore plasma intend to shift inward or outward, which it is dangerous for tokamak p lasma. Therefore, plasma equilibriu m study is one of the fundamental problems of the magnetically confined plasmas. There are many availab le experimental methods and analytical solutions of the steady state magnetohydrody namics (MHD) equations, in particular, the Grad-Shafranov equation for the plasma equilibriu m problem. Control of plasma position has important role in plas ma confinement and to achieve optimized tokamak plas ma operation. Determination of accurate plasma position during confinement t ime is essential to transport it to a control system based on feedback. Over the years different methods have been developed to analysis the tokamak plas ma equilibriu m problems .
In this paper we present two experimental and analytical methods for determination of plasma co lu mn center in IR-T1 Tokamak, which it is a s mall, air core, lo w β and large aspect ratio tokamak with a circular cross section (see Table  1). Details o f the po lo id al flu x lo o p s techn iq ue fo r determination of Shafranov Shift will present in section 2.

Poloidal Flux Loops Method in Measurement of Plasma Position
Poloidal flu x loop is a simple toroidally loop wh ich measure the polo idal magnetic flu x and usually array of them use in control and reconstruction of plasma equilibriu m states. The magnetic flu x passing through such a loop is equal to πψ 2 , where ψ represent to magnetic polo idal flu x. In the ohmically heated tokamaks, ohmic coils field is the main fraction of po loidal flu x wh ich passing through the flu x loop. Therefore to obtain net poloidal flu x due to plasma, compensation will require for all excessive flu x. Because of large area of the flu x loop, the inductive voltage is also large and then it consists of usually one turn. According to relation for frequency response, it is obvious that because of small self inductance, frequency response of flu x loop usually is higher than which desired. Although magnetic probe suitable fo r measurement of plasma position only in circu lar cross section plasma and not for elongated one, but the flu x loop either in elongated and or circular cross section tokamaks can be used. Therefore we used these two techniques for the IR-T1 tokamak with circular cross section.
The plasma boundary is usually defined by Last Closed Flu x Surface (LCFS). In the LCFS polo idal magnetic flu x is constant, if we install so me flu x loops at some distance in vicinity of LCFS, then we can find plasma displacement fro m difference in polo idal flu xes that received with flu x According to Figure (1) the poloidal flu x is obtained:  Table (2)).   As mentioned above, the ohmic field is the significant fraction of the poloidal flu x which passing through the flux loop, therefore essentially co mpensation is needed. Co mpensation is done with all fields discharge without plasma and subtraction them fro m the total flu x that received with flu x loop. Experimental result for measurement of plasma position using this method will present in section 4.

Analytical Solution of the Grad-Shafranov Equation
For axially symmetric configurations, Maxwell's equations together with the force balance equation fro m MHD equations, for stationary and ideally conducting plasmas, reduce to the two-dimensional, nonlinear, ellipt ic partial d ifferential equation, or Grad-Shafranov equation (GSE) [1]. In some of the work, authors solved the Grad-Shafranov equation by expanding the free functions ( ) (ψ p , and ) (ψ F ) in different order in ψ (flu x function). In this section, we regarded quadratic order (which proposed by Guazzotto [3]), and examined on the IR-T1. The GSE is: If we choose the free functions to be quadratic in ψ as where 0 / R a = ε , is the inverse aspect ratio, and . aR , The solution of Eq. (7) in cylindrical coordinates ( ) Z , R can be written as: where ( )ρ  (13) and condition for right convexity on the inboard midplane is: also two conditions, one defining the location of the magnetic axis ( axis R ) and the other the normalization for ψ on magnetic axis:  (16) where subscripts ( A and T ) indicates the analytical and traditional plas ma shape parameters respectively, and sum in minimu m include 3 angles ( π π /2, , 0 ) for our purpose (circular plasma).
In general by min imizing the error function (e.g. by Mathematica) as possible to zero, and finding optimal values for α , 2 k , 3 k , and also solving seven equations for the boundary conditions (Eqs. (13), (14), (15)

Experimental Results and Comparison Between Them
For determination of the plas ma position using the first method, we needed for determination of the poloidal magnetic flu x around the plasma. Therefore we designed and installed two poloidal flu x loops on outer surface of the IR-T1 chamber. Positions of the flu x loops were shown in figure (1). Also, plasma current and average vertical field, were obtained fro m the Rogowski co il and magnetic probe, respectively.
According to the Faraday's law, output of all magnetic diagnostics proportional to derivative of the magnetic flu x which passing through them, therefore we needed to integrate the output of the flux loops and magnetic probes after co mpensating their output.
The integrator output o V is given by: (17) where RC is the integrator time constant, and where i V is the inductive voltage supplied by the flux loops and each one of the magnetic probes, which were p laced around the IR-T1 tokamak vacuum chamber.  We used the two methods to determine the horizontal displacement in IR-T1 as shown in Figure (4). These figures show that two methods give us a same horizontal displacement. Moreover we plot the magnetic flu x surfaces or Eq. (12) for plas ma parameters at t=15ms in target shot on IR-T1 tokamak, as we expect there is displacement of plas ma column center as shown in Figure (3).

Summary and Conclusions
In this paper we determined the Shafranov shift by two experimental and analytical methods in IR-T1 tokamak. In the first method we designed and installed two poloidal flu x loops on the outer surface of the IR-T1 tokamak chamber, and then plasma displacement determined fro m them. To compare the plas ma position obtained using this method, the analytical solution of the Grad-Shafranov equation based on expanding of the free functions, quadratic in ψ is also presented, and experimented on IR-T1. Results show that two methods are in good agreement with each other. The acceptable differences between them are because of (1) approximation in measurement of poloidal flu x on LCFS, (2) the approximate values chosen for γ , and (3) the errors do not become zero during minimizing the error function.