Sawtooth wave

Summary

The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ramp waveform. The convention is that a sawtooth wave ramps upward and then sharply drops. In a reverse (or inverse) sawtooth wave, the wave ramps downward and then sharply rises. It can also be considered the extreme case of an asymmetric triangle wave. The equivalent piecewise linear functions x(t) = t - \lfloor t \rfloor x(t) = t \bmod 1 based on the floor function of time t is an example of a sawtooth wave with period 1. A more general form, in the range −1 to 1, and with period p, is 2\left({\frac t p} - \left\lfloor {\frac 1 2} + {\frac t p} \right\rfloor\right) This sawtooth function has the same phase as the sine function. While a square wave is

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The sawtooth period control in tokamak is important issue in recent years because the sawtooth crash can trigger TM/NTM instabilities and drive plasmas unstable. The control of sawtooth period by the modification of local current profile near the q=1 surface using ECCD has been demonstrated in a number of tokamaks [1, 2] including KSTAR. As a result, developing techniques to control the sawtooth period as a way of controlling the onset of NTM has been an important area of research in recent years [3]. In 2012 KSTAR plasma campaign, the sawtooth period control is carried out by the different deposition position of EC waves across the q=1 surface. The sawtooth period is shortened by on-axis co-ECCD (destabilization), and the stabilization of the sawtooth is also observed by off-axis co-ECCD at outside q=1 surface. In 2013 KSTAR plasma campaign, the sawtooth locking experiment with periodic forcing of 170 GHz EC wave is carried out to control the sawtooth period. The optimal target position which lengthens the sawtooth period is investigated by performing a scan of EC beam deposition position nearby q=1 surface at the toroidal magnetic field of 2.9 T and plasma current of 0.7 MA. The sawtooth locking by the modulated EC beam is successfully demonstrated as in [3-5] with the scan of modulation-frequency and duty-ratio at the low beta (8N-0.5) plasma. In this paper, the sawteeth behavior by the location of EC beam and the preliminary result of the sawtooth locking experiments in KSTAR will be presented.

EDP Sciences2015

Demonstration of sawtooth period control with EC waves in KSTAR plasma

Timothy Goodman, Doohyun Kim, Olivier Sauter

E D P Sciences2015

From sawtooth to profile control: developing feedback control using ECRH/ECCD systems on the TCV tokamak

Stefano Coda, Federico Alberto Alfredo Felici, Timothy Goodman, James Paley

Real time control of heating systems is essential to maximize plasma performance and avoid or neutralize instabilities under changing plasma conditions. Several feedback control algorithms have been developed on the Tokamak a Configuration Variable (TCV) tokamak that use the electron cyclotron (ECRH/ECCD) system to control a wide range of plasma properties, including the plasma current, shape, profiles as well as the sawtooth instability. Controllers have been developed to obtain sawteeth of a pre-determined period, to maximize the sawtooth period using an extremum seeking control algorithm and finally to provide simultaneous control of the plasma emission profile peak and width using multiple independent EC actuators.

Plasma Phys. Control. Fusion2009