The magnetic measurements of FTU are based on the standard technique of axisymmetric equilibrium magnetic measurements, in which full voltage loops and saddle coils measure the poloidal flux function and poloidal pick-up coils measure the normal derivative over a contour enclosing the plasma cross section. These two data are the Cauchy conditions for the ill-posed magnetostatic problem of finding the plasma boundary, starting from the measurement contour.
One of the most suitable methods of solving the equilibrium problem is the use of multipolar expansion in particular geometries. In this paper is expanded in series of toroidal multipoles, by using the fully toroidal co-ordinate system
The plasma boundary is derived by following the truncated multipolar expansion
that fits the measurements and by assuming that the last plasma magnetic surface is the first closed magnetic surface that intersects any material object.
All the magnetic probes are located outside the circular cross-section vacuum vessel, on a toroidal countour with major radius Rpr= 0.935 m and minor radius apr= 0.356 m. The L/R high frequency cut-off of the vessel for the equilibrium fields and fluxes is of the order of 1 kHz. The standard magnetic measurements are composed of 16 full voltage loops and of four replicas of 16 poloidal pick-up coils and 16 saddle coils at four toroidal positions around the vacuum vessel.
Usually the measurements of the 16 saddles and of the 16 pick-up coils laying at a fixed toroidal position are used. An average between these measurements and the corresponding ones obtained from the toroidally opposite sector is performed only in the discharges affected by quasi stationary modes, in order to analyse the axisymmetric part of the equilibrium.
The poloidal pick-up coils are all equal and are equally spaced in the poloidal angle of the fully toroidal co-ordinates, so their distribution is disuniform in the arclength of the measurement contour. The calibrated product of the area times the number of turns of each poloidal pick-up coil is within +/-1% of the design value. The main systematic error that affects their measurements is the spurious pick-up of toroidal field caused by misalignments: to remove it, one has to subtract the signal of a purely toroidal field reference shot. The agreement between the plasma current, as measured by three Rogowsky coils, and the plasma current, as derived from the line integral of the poloidal field measured by the pick-up coils, is within +/-1%, at least when a negligible current flows in the vacuum vessel.
The poloidal extension of the saddle coils is similarly disuniform in the arclength of the measurement contour; as a matter of fact all the saddles are poloidally evenly spaced in the fully toroidal co-ordinates. The toroidal contours of the saddles follow rigorously a section of the torus at =constant, but the poloidal contours provide an empty space in between all adjacent saddles to allow for the passage of the 16 voltage loops that run around all the machine. These empty spaces introduce geometrical correction factors between 1.04 and 1.09 upon the measured by each saddle. The main systematic error that affects the measurements of the saddle coils is the spurious pick-up of toroidal field, that results from the interplay between the misalignments of the saddles and the time evolution of the n=12 and n=24 components of the ripple field produced by the toroidal field magnet. The time evolution of the toroidal field ripple is maily due to skin currents in the magnet itself. To remove the spurious pick-up, one has to subtract the signal of a purely toroidal field reference shot. A less important systematic error is a very small non zero sum of the 16 saddle signals, which remains after the toroidal pick-up has been cured; this negligible error of closure is anyway redistributed evenly on all the 16 saddles.
The 16 loop voltages do not exhibit spurious pick-up of toroidal field. However, because of the obstructions offered by the various ports, not all of the loop voltages can run smoothly along parallel circles around the torus. So they are affected by unknown errors due to the detours they have to make around the ports, although a systematic compensation has been attempted by alternating the direction of the detour between adjacent ports. As a consequence, only the signal of one voltage loop that runs perfectly along a parallel circle of the torus and lays immediately below the inner equatorial plane, has been used as the reference point for the measurement. The variation of around the poloidal angle is obtained by the sequential addition of the 's measured by the 16 saddle coils.
The method by which the toroidal multipolar expansion of the magnetic configuration is derived is a Fourier analysis that determines separately, order by order, the multipolar moments. A fixed toroidal co-ordinates centre Ö(Rpr2-apr2)=R0=0.864 m has been used.
In the open vacuum magnetic configurations, that are obtained without plasma by feeding currents into the various poloidal windings of FTU, both the flux measurements alone as well as the field measurements alone provide two indipendent boundary conditions to the well posed Dirichlet or Neumann magnetostatic problems for the domain inside the measurement contour. In both these cases, one can solve indipendently the two problems by using the appropriate measured boundary condition and then one can calculate the other boundary condition; the result is that the correspondence between fields and fluxes is within +/-1-2% .
In presence of the plasma, the multipolar expansion that starts from the multipolar order m=0, must be truncated at a certain order m=mmax. The determination of the m=3 internal multipolar moment should be accurate within +/-12.5% over the whole range of , assuming an uncertainty of +/-1% on the measured values of and Bpol. On the other hand, the accuracy on the m=4 moment is less than +/-50%, when is greater than 0.5, and less than +/-25% only when is greater than 0.8. Remembering that all these resulting uncertainties scale up linearly with the uncertainty on the measurements of and Bpol, the correct determination of the m=4 multipolar moment remains quite dubious in FTU.
On the other hand, the m=4 multipolar moments are not used in the equilibrium reconstruction codes and, as a Fourier method is used, the uncertainties on the high order multipolar moments do not influence the determination of the lower order ones. Therefore only the uncertainties in the low order multipolar moments (m=0 to m=3) propagate into uncertainties in the parameters inferred from the equilibrium reconstruction codes. As a consequence it is the level of the actual experimental errors on and Bpol that has to be assessed; the best way to infer it is to compare the measured data with the best fit obtained by truncating the multipolar expansion at the order mmax=3 or mmax=4.
The quality of the fit for the flux measurements in shot #4023, obtained by the toroidal multipolar expansion in presence of the plasma, is shown.
The correspondence between the flux measurement as inferred from the saddles and as directly measured from the voltage loops is quite good. The effect of moving the truncation of the multipolar expansion from the order mmax=3 to mmax=4 produces an uncertainty of +/-3% upon the fit of .
The quality of the corresponding fit for the poloidal field measurements is shown.
The effect of moving the truncation of the multipolar expansion from the order mmax=3 to mmax=4 produces an uncertainty of +/-3% upon the fit of Bpol.
However, from the point of view of the equilibrium recontruction, the relevant uncertainty is not the one of the fit at the position of the measurement contour but the resulting uncertainty at the position of the plasma boundary. The uncertainty on the position itself of the plasma boundary turns out to be at most +/-2.5 mm, so +/-1% of the minor radius of the discharge.
The uncertainty on the value of Bpol on the plasma boundary is also +/-1%.