1. November 3-5, 2003Feedback Workshop, Austin NORMAL MODE APPROACH TO MODELING OF FEEDBACK STABILIZATION OF THE RESISTIVE WALL MODE By M.S. Chu(GA), M.S. Chance(PPPL), A. Glasser(LANL), and M. Okabayashi(PPPL) Nucl. Fusion Vol. 43, 441 (2003) Acknowledgement to A. Bondeson, Y.Q.Liu

2. November 3-5, 2003Feedback Workshop, Austin MOTIVATION To develop a model for understanding results from experiments (DIII-D) on feedback stabilization and to evaluate performance of future devices (ITER) To develop a model beyond the usual model which includes only the geometrical effects from the slab or cylindrical geometry, i.e. Grad- Shafranov equilibrium To compare and benchmark with results from other codes

3. November 3-5, 2003Feedback Workshop, Austin NORMAL MODE APPROACH (NMA) BASED ON ENERGY CONSERVATION OF GENERAL PLASMA EQUILIBRIUM Perturbation energy of RWM for ideal plasma –General plasma equilibrium: axi- symmetric or helical –General plasma perturbation: axisymmetric or helical –Frequency dependent non-self-adjoint Plasma W P, K Vacuum W V DWDW E C Kinetic Energy Wall Dissipation Coil Excitation Energy

4. November 3-5, 2003Feedback Workshop, Austin NMA BASED ON THE NORMAL MODES OF THE OPEN LOOP OPERATION NMA applicable if open loop system can be represented as a set of normal modes –No plasma rotation –No plasma dissipation –A more conservative model than MARS-F The details of the system is completely described –Does not rely on Pade approximation

5. November 3-5, 2003Feedback Workshop, Austin THREE STEPS FOR FULL SOLUTION Open loop stability: Generalization of the ideal MHD stability problem (no feedback) Evaluate the excitation and sensor matrices of the feedback geometry Evaluate feasibility of feedback based on Nyquist diagram or characteristics equations Plasma W P Vacuum W V DWDW E C =0

6. November 3-5, 2003Feedback Workshop, Austin NMA IMPLEMENTED BY COUPLING DCON + VACUUM + TANK DCON expresses plasma free energy in terms of perturbed magnetic field at plasma boundary Extended VACUUM expresses vacuum energy in terms of perturbed magnetic field at plasma boundary and the vacuum tank Tank evaluates the energy dissipation in terms of the perturbed

7. November 3-5, 2003Feedback Workshop, Austin CURRENTS ON VACUUM VESSEL REPRESENTED AS A SET OF DISSIPATION EIGENFUCTIONS Flux leaking through the resistive wall excites dissipation eigenfunctions Odd even Poloidal position along the resistive wall Induced by toroidal efffect

8. November 3-5, 2003Feedback Workshop, Austin GRAD-SHAFRANOV SOLVER (TOQ) AND DCON ANALYSIS DETERMINES RWM STABILITY BOUNDARIES Equilibrium Flux Function Safety factor Pressure W from Dcon Plasma Vacuum Total W

9. November 3-5, 2003Feedback Workshop, Austin EDDY CUURENTS OF OPEN LOOP STABILITY EIGENFUNCTIONS Computed also by MARS Toroidal angle Unstable RWM 1st Stable Mode 2nd Stable Mode 3rd Stable Mode Poloidal angle

10. November 3-5, 2003Feedback Workshop, Austin CHARACTERISTICS EQUATION OF CLOSED LOOP SYSTEM DETERMINES RWM FEEDBACK Closed loop feedback stability described by a compact set of equations for open loop amplitudes i plus coil currents I C Diagonalization of the open loop response allow reduction of the dynamical variables to ( I, I c ) Response to Feedback Coils Open Loop Eigenfunction Excitation Matrix Sensor Matrix Gain Matrix Characteristics Equation Identity Matrix

11. November 3-5, 2003Feedback Workshop, Austin SINGLE INPUT AND SINGLE OUPUT CAN BE ANALYZED USING NYQUIST DIAGRAM Stablized if transfer function P( ) encircles (-1,0) Radial sensors are less effective and stabilize lower range of N Poloidal sensors stabilize the whole computed range of N C = 10% 22% 38% 67% 82% Poloidal Sensor Radial Sensor Less Effective Re[P(j )] Im[P(j )] 0 = No Wall 1 = Ideal wall C-Coils

12. November 3-5, 2003Feedback Workshop, Austin

13. November 3-5, 2003Feedback Workshop, Austin FEEDBACK MODELING SHOWS INTERNAL I-COILS ARE MORE EFFECTIVE THAN EXTERNAL C-COILS C-Coils I-Coils couple more effectively to the unstable RWM since closer to plasma E I and E C are elements of excitation matrix I-Coils Ratio of Effectiveness C-coil / I-coil 5.0 0.0 2.5 C 0.0 1.0 0.5 E I / E C

14. November 3-5, 2003Feedback Workshop, Austin COUPLING OF FEEDBACK COIL TO STABLE MODES IMPEDES STABILIZATION f=1 f=3/4 f=1/2 f=1/4 f=1/8 f=3/4 f=1/2 f=1/4 f=1/16 RiRi f R i for all stable modes C =42%C =83% (-1,0) Nyquist Diagram

15. November 3-5, 2003Feedback Workshop, Austin FOR REAL SYSTEM THE TIME CONSTANT OF THE EXTERNAL CIRCUIT IS IMPORTANT Solution of characteristic equation 0 30 -30 Voltage Amplification w RWM Circuit Stable Modes f=1 f=.15C =83% c =.03 w

16. November 3-5, 2003Feedback Workshop, Austin SCOPING STUDY FOR C-COIL EXTENSIONS Radial Sensor, Ideal Feedback Upper extension Lower extension C-Coil C 0 1 w 0 30 f 0 1 All Three Coils C-Coil Upper+ Lower

17. November 3-5, 2003Feedback Workshop, Austin SUMMARY / CONCLUSION Feedback with ideal plasma response formulated for general plasma equilibrium through energy conservation. Phase space of feedback system reduced to the normal modes of open loop eigenfunctions and currents in feedback coils (NMA) For tokamak geometry NMA has been implemented by coupling DCON with extended VACUUM to study RWM feedback stabilization –Poloidal sensors are more effective than radial sensors –I-Coils are more effective than C-coil MARS-F benchmarked against NMA for ideal plasma