--- format: markdown title: Magnetic field measurements using coils categories: Diagnostics ... Diagnostics based on magnetic coils are presently the standard method for measuring magnetic fields in tokamak experiments. Coils are (in the first approximation) easy to manufacture, simple to implement and straightforward to interpret. However, as usual, things get more complicated in a detailed view. The most important requirements which every useful probe-type diagnostic must satisfy are: * reasonable sensitivity (signal-to-noise ratio); the probe must provide signal high enough to overcome the electronic noise associated with impulse devices * a good frequency response, so as to follow the most rapid fluctuations present in the system * minimum perturbing effect on plasma, which means the smallest possible size and appropriate vacuum-friendly and plasma-resistant construction materials It is unfortunate that these requirements conflict directly with each other. To improve the frequency response (collect high frequencies without attenuation), the coil should be as small as possible. However, small effective area means small signal and bad signal-to-noise ratio. #Theory of magnetic field measurement using coils Technically measuring coils do not measure magnetic field itself; they measure the time changes of the magnetic field. (As a result, coils do not measure stationary magnetic field.) More precisely, they react to the time derivative of the magnetic flux $\Phi$ passing through their turns by inducing a voltage $U_\mathrm{p}$ upon itself. $$ U_\mathrm{p} = - \frac{\mathrm{d}\Phi}{\mathrm{d}t} $$ The magnetic flux $\Phi$ [Wb] passing through a single loop is defined as $$ \Phi = \int_{A_{\mathrm{loop}}} \textbf{B} \cdot \mathrm{d}\textbf{A} $$ where $A_{loop}$ is the loop area. If the loop is small enough to consider the magnetic field $\textbf{B}$ inside it uniform, we may simplify the formula to $$ \Phi = B_\perp A_{\mathrm{loop}} $$ where $B_\perp$ is the magnetic field component perpendicular to the loop. Now since a coil typically has $N$ turns, the voltages induced on top of them stack. Denoting the coil effective area $A_{\mathrm{eff}} = N A_{\mathrm{loop}}$, we may write $$ U_\mathrm{p} = - A_{\mathrm{eff}} \frac{\mathrm{d}B_\perp}{\mathrm{d}t}. $$ Finally the coil is typically oriented so that the measured magnetic field is perpendicular to its turns; that is, the coil axis lies parallel to the magnetic field. For instance, a coil measuring the toroidal magnetic field $B_T$ will be oriented in the toroidal direction. Under this assumption we may finally write $$ U_\mathrm{p} = - A_{\mathrm{eff}} \frac{\mathrm{d}B}{\mathrm{d}t} $$ where $B$ is the measured magnetic field. # Circuit scheme The following figure shows a simplified scheme of a measuring coil circuit.
// input and output arrays
values[LENGTH] //array of raw data from magnetic coil (input)
B_tor[LENGTH] //the magnitude of the magnetic field will be written in this array (output)
//magnetic coil's constants
DELTA_T //time between two samples (1Mhz = 1e-6s, 100kHz = 1e-5s)
CALIBRATION //calibration constant to convert the voltage to magnetic field
integr = 0
for (i = 0; i <<<> LENGTH; i++)
{
integr += values[i]
B_tor[i] = integr * DELTA_T * CALIBRATION
}
## Offset removal