\subsection{The GOLEM tokamak diagnostics}
%<*DiagnosticsOverview>
\begin{figure}[ht]
\centering
\includegraphics[width=0.6\linewidth]{BasicScheme/drawing.pdf}
\caption{Tokamak diagnostics used in this assignment.}
\label{fig:diagnostics}
\end{figure}
In tokamak research, the plasma is closely watched and controlled by a set of diagnostics. Figure \ref{fig:diagnostics} shows four basic tokamak diagnostics:
\begin{itemize}
\item A wire loop laid toroidally along the plasma ring: measures the \textbf{loop voltage} $U_{l}$.
\item A small coil attached to the vessel, its axis in the toroidal direction: measures the time derivative of the \textbf{toroidal magnetic field} $dB_t/dt$.
\item A Rogovski coil tied around the vessel: measures the time derivative of the total poloidal magnetic field $dB_p/dt$. The poloidal field consists of two contributions: (i) the field generated by the \textbf{plasma current} $I_p$, and (ii) the field generated by the current $I_{ch}$ running through the tokamak chamber, induced by the loop voltage along with the plasma current.
\item A photodiode with an $H_\alpha$ filter: measures the radiation intensity of the $H_\alpha$ spectral line (a dominant line in the hydrogen spectrum).
\end{itemize}
An example of the time evolution of these quantities is shown in figure \ref{fig:discharge-evol}, with the addition of the line-averaged electron density measured by an interferometer. The quantities marked in the bold face are the ones you will process for each discharge to calculate the plasma parameters. Each of the three diagnostics, the wire loop, the measuring coil, and the Rogowski coil, has its own particular signal processing. This is explained in the following sections.
\begin{figure}[ht]
\centering
\includegraphics[width=0.6\linewidth]{DischargeCharacteristics/drawingQS.pdf}
\caption{Time evolution of a well executed GOLEM discharge. From top to bottom - loop voltage $U_{l}$, toroidal magnetic field $B_t$, plasma current $I_p$, $H_{\alpha}$ spectral line intensity and line-averaged electron density $n_e$.}
\label{fig:discharge-evol}
\end{figure}
%</DiagnosticsOverview>
%<*LoopVoltage>
The wire loop signal requires no post-processing (beside offset removal if needed; see section \ref{part:data_processing}). The loop voltage $U_{l}$ is the direct output of channel 1 measurement.
%</LoopVoltage>
%<*BtCoil>
According to Faraday's law of induction, if the magnetic flux passing through a conductive loop changes, a voltage $U$ is induced on it. Assuming that the loop is small so the magnetic field inside it is uniform, the voltage magnitude is
\begin{equation}
\label{eq:Faraday}
U = NS\frac{d B_\perp}{dt}
\end{equation}
where $N$ is the number of the coil threads ($N=1$ for a single loop), $S$ is the loop area and $B_\perp$ is the magnetic field component perpendicular to the loop area.
Electromagnetic induction is also the principle of the $B_t$ loop measurement: the coil is simply placed into the magnetic field, its axis pointing along the toroidal direction, and its signal (the voltage $U_{B_t}$) is integrated in time and calibrated. The calibration constant is theoretically equal to $NS$; however, in this assignment you will calibrate the signal by comparing the toroidal magnetic field $B_t$ measured by the standard GOLEM diagnostic set to your own measurements of $\int_{0}^{t}U_{B_t}(\tau) d\tau$.
\begin{equation}
\label{eq:CBt}
B_t (t) = C_{B_t} \int_{0}^{t}U_{B_t}(\tau) d\tau
\end{equation}
This calibration may be done individually for every individual discharge, but it is more convenient to calculate the calibration constant $C_{B_t}$ once and then reuse it. (Note, however, that every $B_t$ coil is placed differently and so calibration constants of distinct coils will be different.)
%</BtCoil>
%<*RogowskiCoil>
\begin{figure}[h]
\centering
\includegraphics[width=0.2\linewidth]{RogowskiCoil.pdf}
\caption{Rogowski coil scheme.}
\label{fig:rogowskicoil}
\end{figure}
The Rogowski coil is the most complicated of the three self-implemented diagnostics whose signal you will post-process. It is a "coil loop" --- a one-metre long thin coil which is wrapped around the tokamak chamber poloidally. As seen in figure \ref{fig:rogowskicoil}, one of the Rogowski coil ends is directly accessible, while the other leads through the coil to negate the toroidal magnetic field contribution in its signal. As a result, the coil only picks up the poloidal magnetic field via electromagnetic induction,
\begin{equation*}
U_{RC} \propto \frac{d B_p}{dt}.
\end{equation*}
The poloidal magnetic field has two components: the field $B_{p,p}$ generated by the plasma current $I_p$ and the field $B_{p,ch}$ generated by the toroidal current $I_{ch}$ induced by the loop voltage in the tokamak chamber. The respective currents are then proportional to their magnetic field according to the Biot-Savart law. The chamber current contribution is unwanted and has to be removed in order to find the plasma current $I_p$. Luckily, $I_{ch}$ can be easily calculated using the loop voltage and the chamber resistivity, $I_{ch}(t) = U_{l}(t)/R_{ch}$ where $R_{ch}=0.0097 \, \Omega$.
%A single measurement of $R_{ch}$ is sufficient to calculate $I_{ch}$ for any subsequent discharge. This is done in task \ref{task:4b}, where a vacuum discharge is created ($I_p=0$), allowing the calibration of the Rogowski coil according to
%\begin{equation}
%\label{eq:CRC}
%I_{RC} = \frac{U_{l}(t)}{R_{ch}} = C_{RC} \int_{0}^{t}U_{RC}(\tau) d\tau.
%\end{equation}
The calibration constant $C_{RC}$ can then be defined with the relation
\begin{equation}
I_p(t) + \frac{U_{l}(t)}{R_{ch}} = C_{RC} \int_{0}^{t}U_{RC}(\tau) d\tau
\end{equation}
and calculated from the standard diagnostic output on the left-hand side and the oscilloscope data on the right-hand side. With $C_{RC}$ and $R_{ch}$ known, the plasma current can finally be calculated from the oscilloscope data as
\begin{equation}
\label{eq:Rogowski}
I_p(t) = C_{RC} \int_{0}^{t}U_{RC}(\tau) d\tau - \frac{U_{l}(t)}{R_{ch}}.
\end{equation}
%</RogowskiCoil>
\subsubsection{Plasma current}
%<*diagnostics:currentDrive>
The current is driven in the plasma by the toroidal electric field $E_t$
induced by the transformer. This field integrates along the plasma loop into the
easily measurable loop voltage $U_l=E_t2\pi R$.%
%</diagnostics:currentDrive>
A simple electrical model for the inductive current drive is a time-varying voltage source ($U_l(t)$) connected to the plasma and the vacuum chamber in parallel can be seen on Figure \ref{fig:circuit}. Both the vacuum chamber and the plasma are modeled by LR circuits. The main difference is, that while the internal inductance and resistance of the chamber are constant, and thus they can be measured separately, the parameters of the plasma differ in each discharge.
\begin{figure}[ht]
\centerline{\resizebox{120mm}{!}{\rotatebox{0}{\includegraphics{TokamakSubstituteCircuit/circuit.pdf}}}}
\caption{Model of the inductive current drive circuit}
\label{fig:circuit}
\end{figure}
The basic circuit equations are:
\begin{align}
\label{eq:chamberLR}
U_l(t) &= R_{ch} \cdot I_{ch}(t) + L_{ch} \dfrac{d I_{ch}(t)}{d t}\\
\label{eq:plasmaLR}
U_l(t) &= R_{pl}(t) \cdot I_{pl}(t) + L_{pl} \dfrac{d I_{pl}(t)}{d t}\\
I_{tot}(t)&= I_{pl}(t) + I_{ch}(t)
\end{align}
The chamber parameters have already been determined according to Section \ref{sec:vacuum}. Integration of the \eqref{eq:chamberLR} circuit equation using the initial condition $I_{tot}(t=0)=I_{ch}(t=0)$ is implemented in the routine GOLEM\_chamber\_current.m to arrive to $I_{ch}(t)$. This can then be used to determine the plasma current, as $I_{pl}(t)=I_{tot}(t)- I_{ch}(t)$ as shown in Figure \ref{fig:blockdiagramscurrent}. Plasma resistivity can be determined in turn from equation \eqref{eq:plasmaLR}.
\begin{figure}[ht]
\centerline{\resizebox{160mm}{!}{\rotatebox{0}{\includegraphics[trim=1cm 5cm 0cm 4cm, clip=true]{BlockDiagrams/blockdiagramscurrent.pdf}}}}
\caption{Block diagram showing the steps of data processing for the plasma current measurement.}
\label{fig:blockdiagramscurrent}
\end{figure}
Having calculated the plasma current, a threshold can be defined significantly exceeding the calculation accuracy to safely determine the beginning and end of the plasma discharge. Using this threshold in the "find" function, one can cut the time signals to the extent of the discharge for further processing. Time duration of the dsischarge is also an imprtant parameter.
\emph{This task needs some programming that should be done parallel to the task described in Section \ref{sec:breakdown}!}
It can be attempted to investigate the effect of the $L_{pl}\approx 0 \;\rm H$ approximation by a more careful integration of choosing $L_{pl}\approx L_{ch}$ in the time region with plasma. If significant differences are found, this latter approximation has to be implemented for all further data processing.
A suitable threshold in plasma current can be used to determine the discharge duration and cut out the interval of the measured signals relevant for plasma diagnostics.
Plasma current has to be calculated for all discharges with plasma and the maximum value and the discharge duration have to be included in the shot summary table.
Ohmic heating power has to be calculated for all discharges with plasma and the maximum value has to be included in the shot summary table.
\textbf{The diagnostics used during the session to be accessed online:}
\begin{itemize}
\item Time resolved measurement of loop voltage ($U_l$).
\item Time resolved measurement of total toroidal current by Rogowski coil ($I_t$).
\item Time resolved toroidal magnetic field by coil measurement ($B_t$).
\item Time resolved measurement of plasma radiation by photodiode.
\item Vacuum chamber pressure ($p_{ch}$).
\item The temperature of the vacuum chamber ($T_{ch}$).
\end{itemize}
Besides
these basic diagnostics the tokamak Golem is equipped with a spectrometer, 2
fast cameras, a rake probe with a set of Langmuir probes and poloidal ring of
Mirnov coils.
