--- format: markdown author: Ing. Kateřina Jiráková categories: Diagnostics title: Plasma current measurements using the Rogowski coil ... The plasma current $I_p$ is measured with a Rogowski coil (see picture below). The Rogowski coil is wound around the tokamak chamber in the poloidal direction.
 Rogowski coil
Note that the Rogowski coil is back-wound - one of its ends is folded and threaded back along the loop. This is necessary to negate the toroidal field contribution to the coil signal. As a result, both of the Rogowski coil contacts are at one end of the loop. #Theory of Rogowski coil measurement The Rogowski coil measures the plasma current via Amper's law (magnetic field around a conductor is directly proportional to the current running through it) and Faraday's law (changes of magnetic field cause voltage induction on surrounding loops/coils). In the case of tokamak plasma, Amper's law is $$\oint_l B_p \mathrm{d}l = \mu_0 I_p$$ where the integration path $l$ poloidally encircles the tokamak chamber as in Fig. 1, $B_p$ is the poloidal magnetic field, $\mu_0$ is the vacuum permeability and $I_p$ is the plasma current. The total magnetic flux $\Phi$ in the Rogowski coil is then: $$\Phi = n \int_S \mathrm{d}S \oint_l B_p \mathrm{d}l = nS\mu_0 I$$ where $n$ is the number of coil turns per unit of length and $S$ is the area of a single coil loop (given that all the turns are of the same area). During a GOLEM discharge, the plasma current ramps up from zero to some maximum value and then begins falling again, which changes the magnetic flux in time and cause a voltage $U$ to be induced between the Rogowski coil ends: $$U = -\frac{\mathrm{d}\Phi}{\mathrm{d}t} = -\mu_0 n S \frac{\mathrm{d}I_p}{\mathrm{d}t}$$ To calculate the plasma current $I_p$ from the Rogowski coil voltage $U$, the signal must be integrated and calibrated (see the next two sections). #Rogowski coil signal processing General aspects of magnetic coil signal processing, such as numeric integration and offset removal, are described [here](http://golem.fjfi.cvut.cz/wiki/Diagnostics/Magnetic/Theory/Magnetic_coil/description). In the case of Rogowski coils another specific complication appears - the coil doesn't pick up only the plasma current, but also the current running through the [tokamak chamber](/Tokamak/Chamber/chamber). Since the chamber is toroidally conductive, it responds to the transformer action by developping a toroidal current of its own. The total current $I_{total}$ encircled by the Rogowski coil is then $$I_{total} = I_p + I_{chamber}$$ and the chamber current $I_{chamber}$ must be subtracted from the current obtained by integration. Luckily the chamber current can be easily calculated via Ohm's law, $$I_{chamber}(t) = \frac{U_{loop}(t)}{R_{chamber}}$$ where $U_{loop}$ is the loop voltage and $R_{chamber}= 9.7 \times 10^{-3} \Omega$ [[source]](http://golem.fjfi.cvut.cz/shots/30001/basicdiagn/config.py) is the chamber resistivity. # Rogowski coil parameters The coil parameters are summarized in the following table: | --- | ---| | coil length | $l = 230 \, \mathrm{cm}$ | | wire diameter | $D_{wire}=0.3 \, \mathrm{mm}$ | | coil diameter | $D_{coil} = 7.9 \, \mathrm{mm}$ | | number of turns per metre | $n = 3000 \, \mathrm{m}^{-1}$ | Finally, the calibration factor $K$ in the equation $$I_{total}(t) = K \int_0^{t} U(t') \mathrm{d}t'$$ is $K = 5.3 \times 10^6 \, \mathrm{AV}^{-1}\mathrm{s}^{-1}$ when the signal is sampled with $\Delta t = 10 \, \mu s$. Please note that this value was determined by empirical means and thus it differs slightly from the theoretical value which can be calculated from the coil parameters.