# Presentation: 014179

## RFA Experiments on the T2R RFP

***Open loop control experiments***
- J.R. Drake 1), D. Gregoratto 2), T. Bolzonella 2), P.R. Brunsell 1),
-  D. Yadikin 1), R. Paccagnella 2), Y.Q. Liu 3), M. Cecconello 1),
- G. Manduchi 2), L. Marrelli 2), S. Ortolani 2), A. Bondeson† 3)
*1) Alfvén Laboratory, KTH, EURATOM /VR Association VR, Stockholm, Sweden*
*2) Consorzio RFX, EURATOM/ENEA Association, Padova, Italy*
*3) Dept. of Electromagnetics, CTH, EURATOM /VR Association, Gothenburg, Sweden*
- †* **Anders Bondeson deceased*

## Open loop control experiments  in EXTRAP T2R RFP

**Outline**
**The theme of the workshop is back to the basics.**
**Topics aimed at in this talk are:**
**•  ****Assumption of ****mode rigidity**** **
**•  ****Applicability of the ****linear model**
**Data shown:**
**•  “****Open loop” constant resonant control harmonic**
**	***(qualitative information) *
**•  “****Open loop” ****pulsed**** resonant control harmonic**
**	***(**quantitative estimates** of transfer functions, mode**	**growth rates and mode damping rates)*

## Basics - Resistive- wall mode stability in the RFP

- Cylindrical plasma bounded by a thin wall: radius ***r******w*** and (long) penetration time ******w**.
- Natural growth rate, *********m,n***, of a non-rotating m=1 RWM is then defined by the stability index given by the discontinuity in the logarithmic derivative of the perturbed field at *r**w*.[ref C G Gimblett, Nuc Fus 26 (1986) p 617].

-      **2********m,n********w**** = r****w********m,n********.**

- An important assumption is that the mode structure in the plasma is **rigid**.
- Therefore the plasma can be specified by the single parameter, the growth rate,* ****m,n*, of the RWM.

## Resistive- wall mode stability in the RFP

- An externally produced **control field **_**or**_** a field error**, at *r**f* > *r**w*, modifies ******m,n********and thereby the growth rate** for that (m,n)-harmonic.

- To compare experiment and theory, examine the time dependence of the **RWM radial field** perturbation evaluated on the inside surface of the thin wall, **which is the location of the sensor coils in the experiment**.

## Linear model for the (m,n)-harmonic of the RWM including external error and control field harmonics

**where,**
**	**•	**b****m,n** is the perturbed field measured at the sensor coil.
- 	•*	*******m,n** is the growth rate of the mode.
- 	 •*	*******m,n,w**=(******m.n,w**)-1 describes the diffusion rate of the harmonic at the  		thin wall and is determined only by the mode number and 		the wall parameters.
- 	•	**b****m.n****error** is that part due to an external inherent field error.
- 	•	**M****m,n****I****m,n** is the saddle-coil-produced control field 		(I**m,n**=current & M**m,n**=ratio field to current, [*Tesla/Amp*])

- 		    *d**t**b**m,n **= ****m,n**b**m,n** - ****m,n,w**(M**m,n **I**m,n** + b**m,n**error**)*

## T2R Coil system(s)

- Active coils:
- twice the width of the sensor coils
-  4 (poloidal) x 16 (toroidal) positions
- 50% surface coverage

**top**

**inboard**

**outboard**

**bottom**

- Br sensor coils
-  4 (poloidal) x 64 (toroidal) positions
- full surface coverage (limited acquisition)

-   343.125 ̊      0 ̊         16.875 ̊              39.375 ̊          61.875 ̊         84.375 ̊

## Control system

**Sensors**
**32x2 one-turn, m=1 flux loops measure radial magnetic flux through shell**
**16x2 saddle-coil current measurements**** **
**Controller**
**digital controller, 64 sensor inputs, 32 coil current inputs**
**real-time spatial FFT for toroidal harmonics**
**Input feedback laws in the form of a matrix of complex gains **
**16x2= 32 ****preprogrammed ****control voltage outputs drive the saddle coil currents**** **
**Actuators**
**m=1 connected saddle coils, L/R time constant 1 ms**
**coil current 20 A, magnetic field 1 mT ( 1% of equil. poloidal field)**
**high-bandwidth audio amplifiers, output power 700 Watt**

## ”Natural” mode amplitude and phase traces 

**for standard discharge **

** “****Error field” mode (e.g. n=-2):**
-  Linear growth
-  Wall locked
-  Reproducible phase
** ****RWM (e.g. n=-10):**
-  Exponential growth
-  Wall locked
-  Reproducible phase

- Time (ms)

## Open loop preprogrammed constant external 

**control field harmonic applied to n=6 unstable mode**

**What happens if a constant control harmonic is applied**
**• ****in phase with the natural mode (i.e. positive feedback orientation)?**
**• ****180 degrees phase difference relative to the natural mode (i.e. negative feedback orientation)?**
** **
**The data qualitatively demonstrates mode rigidity.**

## Open loop RWM compensation

- Amplitude [a.u.]

- Amplitude of the mode is ≈ 1.0 mT; a. of the external perturbation is ≈ 0.02mT

**n=6**

## Open loop preprogrammed constant external 

**control field harmonic applied to n=6 unstable mode**

**What happens if the control harmonic is turned on at****a later time when the mode amplitude is larger ?**
**• ****The data qualitatively demonstrates mode rigidity.**

## Open loop RWM studies

- Compensation can be obtained with a perturbation starting in the middle of the discharge but at the price of a higher amplitude.
- This case can be compared with feedback operations

- Time (s)

- Amplitude (a.u.)

## Open loop pre-programmed pulsed external control field harmonics applied to different modes:

**Open loop pre-programmed **_***pulsed***_** external ****control field harmonics applied to different modes:**
**	****•  ****unstable modes**
**	****•  ****marginally unstable modes**
**	****•  ****marginally stable modes**
**	****•  ****robustly stable**

**Measure growth and damping rates.**
**Give indication of applicability of linear model.**
**Analyze resonant field error amplification.**

- Resonant field error amplification?

## Open loop experiments with a pre-programmed pulsed external control harmonic

**      ****Comment on the spectrum of unstable modes in an RFP**

- n=+5

- n=+10

- n=-4

- n=-11

- n=+12

## Open loop preprogrammed pulsed external control field harmonics applied to different modes.

- n=-4 (marginally  unstable)
- n=+12 (stable)

**Black = current pulse to external saddle coil****(Short pulse with constant phase)**
**Green = b-external harmonic pulse measured by sensor coil.**
**Blue = b-plasma harmonic for a  reproducible reference discharge**
**Red = b-plasma harmonic for a discharge with external perturbation applied.**

**Open loop preprogrammed **_***pulsed***_** external control ****field harmonics applied to different modes.**

- Amplitude                                     Phase

- Time (ms) 	                                                        Time (ms)

## Open loop preprogrammed pulsed external control field harmonics applied to different modes.

**Blue = b-plasma harmonic for a **_***reference***_*** *****discharge**
**Red = b-plasma harmonic for a discharge **_***with***_** external perturbation applied.**

**Open loop preprogrammed **_***pulsed***_** external control ****field harmonics applied to different modes.**

**Harmonics for two modes are shown:**
- n=-4 (marginally  unstable)
- n=+12 (stable)

- Real part                                   Imaginary part

- Time (ms) 	                                                        Time (ms)

## Open loop preprogrammed pulsed external control field harmonics applied to different modes.

**Green = ****external perturbation****(Short pulse with constant phase)**
**Blue = ****(discharge with external perturbation applied) ****  **_***minus***__** **_**(reference discharge)**

**Open loop preprogrammed **_***pulsed***_** external control ****field harmonics applied to different modes.**

**Harmonics for two modes are shown:**
- n=-4 (marginally  unstable)
- n=+12 (stable)

- Amplitude                                   Phase

- Time (ms) 	                                                        Time (ms)

## Open loop experiments

- Vacuum shots

- Resistive wall diffusion rate, m,n,w =  1/m,n,w
- Ratio  M = field / current

- Growth & damping rates of RWM
- External Error field
- Resonant field amplification (RFA) studies

- Plasma shots(Reproducible mode *********n,o***)

## Open loop: Vacuum wall diffusion time, w     & Ratio Mm,n,    bm,nex      Im,nsc 

-            6
-  w    4
- (ms)
-            2
-            0

**m=-3**

**m=1**

-         0.06
-  Mn
- (mT/A)
-              0

**Theory: Calculations for a thin, smooth resistive cylinder**
**Experiment(+): Measurements in vacuum using pre-programmed current waveforms for saddle coil currents.**

- Open loop: Vacuum wall diffusion time, w     & Ratio Mm,n,    bm,nex      Im,nsc

-    2
-    1
-    0

**m=-3**

-   0.01
-         0

**m=1**

-      -20          0       +20
- toroidal mode number n

-      -20          0       +20
- toroidal mode number n

-      -20          0       +20
- toroidal mode number n

-      -20          0       +20
- toroidal mode number n

## Open loop preprogrammed pulsed external control field harmonics applied to different modes.

- Shell diffusion time of (m=1, n) harmonics
- Ratio  M = field / current  (harmonics)

- Growth & damping rates of RWM
- External Error field
- Resonant field amplification (RFA) studies

- Plasma shots(Reproducible mode *********n,o***)

**Open loop preprogrammed **_***pulsed***_** external control ****field harmonics applied to different modes.**

## d[bn,with - bn,ref ]/dt =n [bn,with - bn,ref ] - n,wMn In

**• ****Use the linear model. **
**• ****Make a best fit to the data where the fitting parameters are the plasma growth (damping) rates and the ratio M = b****n****/I****n****.**

***d[b******n,with ******- b******n,ref ******]/dt =************n ******[b******n,with ******- b******n,ref ******]****** ******- ************n,w******M******n ******I******n***

**Open loop preprogrammed **_***pulsed***_** external control ****field harmonics applied to different modes.**

***black is I******n***
***blue is [b******n,with ******- b******n,ref ******]******(measured)***
***red [b******n,with ******- b******n,ref ******]******(simulation)***
*********n****** & M******n****** are fitting parameters***

## Red =Best fit simulation with n & Mn as fitting parameters

**Blue = ****(discharge with external perturbation applied) **
***minus***** **
**(reference discharge)**

**Harmonics for two *****unstable***** modes: n= -11  & n=+5 **

- Amplitude                                Phase (shifted)

- Time (ms) 	                                                      Time (ms)

**Preprogrammed **_***pulsed***_** external control harmonics applied to different modes.**

## Red =Best fit simulation with n & Mn as fitting parameters

**Blue = ****(discharge with external perturbation applied) **
***minus***** **
**(reference discharge)**

**Harmonics for a *****marginally stable***** mode (n= -2) & a stable mode (n=+14) **

- Amplitude                                Phase (shifted)

- Time (ms) 	                                                      Time (ms)

**Preprogrammed **_***pulsed***_** external control harmonics applied to different modes.**

## Red =Best fit simulation with n & Mn as fitting parameters

**Blue = ****(discharge with external perturbation applied) **
***minus***** **
**(reference discharge)**

**Harmonics for another***** unstable***** mode (n= -6) & a stable mode (n=+10) **

- Amplitude                                Phase (shifted)

**Preprogrammed **_***pulsed***_** external control harmonics applied to different modes.**

- Time (ms) 	                             Time (ms)

## Preprogrammed pulsed external control harmonics applied to different modes.

**Summary of growth (damping) rates and ratio M= b****n****/I****n****  determined from best fit to measured data.**

**Note!**
**Red ”o” are values for cases with plasma & control harmonic**
**Blue ”+” are values for control harmonic in vacuum**
**The mutuals, ****M****n****= b****n****/I****n****,****should be the same for best fit to vacuum data and to plasma data.**
***(Higher m numbers must be included to model current in saddle coils.)***

## 		    dtbm,n = m,nbm,n - m,n,w(Mm,n Im,n + bm,nerror) 

**Modeling of open loop control experiments  in EXTRAP T2R RFP using the well known form:**

- 		    *d**t**b**m,n **= ****m,n**b**m,n** - ****m,n,w**(M**m,n **I**m,n** + b**m,n**error**)*

**With fitting parameters:**
**  • ****Growth (damping rates)**
**  • ****Mutuals (field at sensor coil/current in saddle coil**
**Give the following values for growth (damping) and inherent error fields:**

**   ****n****	****   -11****	****      -6        -4****	****       -3        -2****	****            +5**
**  ********(ms****-1****)     ****	****0.22+i0.15   0.05    >0.02      <0.01     <0.02      0.08+i0.03**
**  ****b****err**** (mT)****	****	****	****   0.013    0.02       <0.01      0.1**
**  ****b****err**** (phase)****	****	****      ****	****      0 ̊       -145 ̊   ****	****      45 ̊**