Progress on Levitation in the

Levitated Dipole Experiment

D.T. Garnier, M.E. Mauel, T. Sunn Pedersen – Columbia University

A. Boxer, J. Ellsworth, I. Karim, J. Kesner, P. Michael, E. Ortiz, A. Radovinsky, – MIT PSFC

2006 Innovative Confinement Concepts WorkshopAustin, Texas

February 13-16, 2006

Columbia University

Synopsis

Levitation will likely greatly change operation of LDX‣Dominant loss channel removed -> better confinement

‣Higher background density with high beta -> more stable to HEI

‣Radial transport dominated (broader) profile -> more stable

Levitation system nearly complete‣Coil and control systems installation complete

‣Calibration and control algorithm development underway

‣ Laser detection system prototype complete

‣Catcher system under construction

Levitation system testing in progress‣ 3 major tests planned to give confidence in successful levitation

Bulk plasma must satisfy MHD adiabaticity condition

Fast electron stability enhanced due to coupling of fast electrons to background ions

Hot Electron Interchange Stability

Krall (1966), Berk (1976)...

Rosenbluth and Longmire, (1957)

!!

pb V ""

= 0

!

d ln pb

d ln V< !!1

V =!

d !

Bwhere

or

!d ln n̄ h

d ln V< 1+

m 2

"

24

!d h

!c i

n̄ i

n̄ h

Increased Neutral Fueling Stabilizes HEI

Stabilizes small HEI‣More background density

‣Smaller hot electron fraction

But loss of confinement‣Pitch angle scattering to

supports.

Levitation changes‣Pitch angle scattering gives

more isotropic distribution

‣Collisions lead to broader radial profile

‣Higher overall confinement

-10123456

kW

Heating Power (total)6.4 GHz 2.45 GHz

-1012345

kA

mp

s

Plasma Current

0.6

0.8

1.0

1.2

m

Current Centroid

-202468

a.u

.

Visible Intensity

-202468

10

a.u

.

X-Ray Intensity

01

2

34

10

10 c

m-3 Edge Density

0 2 4 6 8Time(sec)

0102030405060

10

-7 T

orr Neutral Pressure

LDX Levitation Basics

2 m

Levitation by upper lift magnet‣ Unstable only to vertical motion

‣ Mostly undamped stable secondary modes

HTS lift magnet‣ First in US Fusion program

‣ Much reduced power and cooling requirements

‣ AC heating introduces unique requirements for control system

Large 5 m diameter vacuum vessel‣ Eddy current times << levitation times

Laser position detection‣ Many secondary diagnostics

Digital feedback system

L-coil

F-coil

Generation II Catcher

New catcher under construction‣ Lightweight cone that will accelerate to

match F-coil fall without large impulse

‣Partial F-coil deceleration while launcher mass accelerates

‣ Limit all accelerations to less than 5 g

Upper space frame‣ Limit upward motion

‣Align radial motion for fall to catcher

L-Coil Design

High Temperature Superconductor.

‣ Negligibly power consumption compared to resistive equivalent.

‣ Nominal 105 A current, with ± 1 A, 1 Hz position control ripple.

‣ Easier to manage position control ac loss than for LTS.

‣ Funded by SBIR, first HTS coil in US fusion energy program.

Optimized, disk-shape geometry for F-coil levitation.

‣ Double pancake winding.

‣ Center support and cooling plate.

Conduction cooled coil.

‣ Low maintenance, moderate cost,

high conductor performance.

‣ Estimated 12 W hysteresis loss.

‣ One-stage cryocooler for coil. 20W @ 20K

‣ Liquid nitrogen reservoir

for radiation shield.

Levitation Control System

LDX Control System Description

150A, +/- 100V Power Supply‣ Integrated dump resistor for rapid discharge

Realtime digital control computer‣ Matlab/Simulink Opal-RT development environment

‣ 5 kHz feedback loop

Failsafe backup for upper fault

Programmable Logic Controller‣ Slow fault conditions

‣ Vacuum & Cryogenic monitoring

‣ PS user interface

Optical link to control room

User interface

LDX data system

Levitation Control System Schematic

Optical Position Detection System

Position/Attitude Sensing‣ Occulting system of 8 beams Provides measurement of 5 degrees of freedom

of coil with redundancy in each measurement

‣ Specification ± 1 cm detection range

5 µm resolution

5 kHz frequency response

‣ Keyence LH-300 COTS units Exceed all specifications

Procured with 2 channels installed on prototype mounting hardware• Require plasma test for final mount production OK

Rotation Sensing‣ Reflecting system to sense final degree

of freedom

‣ Remove Nonaxisymmetry systematic noise correction

Levitation Physics

x

y

z

L-coil

F-coil

Fgrav

Fmag

We can choose a Lagrangian formulation of the equation of motion so the constraints above can be easily incorporated:

€

L = 12 mi ˙ x i

2

i=1

6

∑ −MLFIFIL −12 LFIF

2− 1

2 LLIL2−mgz

F-coil is a superconducting loop, so its flux is conserved, whereas we can vary the flux in the L-coil by applying our control voltage:

€

ΦF = MLFIFIL + LFIF = constant

€

ΦL = MLFIFIL + LLIL = VL( t)dt∫

€

MLF = MLF (r x 1→5)Where:

And:

Feedback stabilization

IL (t )= I0 − a0 ε (t)dt − a1ε (t)− a2ε'∫ (t)

Automatic correction to I0 Damping term, acts like friction

The upward force on the F-coil is proportional to the radial magnetic field at

its position, generated by the L-coil.

Hence, it is proportional to the current in the L-coil.

Without feedback, the vertical position is unstable because dBR/dz>0, so if the F-coil moves up, the upward electromagnetic force will increase, and the

coil will move even further up.

If we detect a small increase in vertical position, and decrease the L-coil current appropriately, we can bring the coil back to its original position.

Simple Approach: Use proportional-integral-derivative (PID) feedback:

Because of the L-coil inductance, we cannot change IL instantaneously. We can control the voltage=L*dIL/dt, instantaneously (or as fast as the power supply allows us to change its voltage):

Include an integral term to automatically adjust for DC losses:

The b parameters are optimized to get the best stabilization: Put feedback expression into equation of motion to find most stable, critically

damped solution

Technique used to estimate required currents / voltages for L-coil

Similar technique (using only derivative gain) used to determine required current for damping Rock & Roll motion using TSR coils ~ 200 Amp turns required…

Feedback: Optimized Voltage PID

VL (t )= −b0ε (t)− b1ε' (t)− b2ε' ' (t)

VL (t )= −b−1 ε (t)dt∫ −b0ε(t) −b1ε' (t) −b2ε ' ' (t)

Basic Simulink Levitation Model

This basic model simulates 6 degrees of freedom of F-coil with L-coil levitation using voltage feedback control.

1s

velocity

1s

position

MATLABFunction

gravity

VectorSelector

Selector3

VectorSelector

Selector1

VectorSelector

Selector

Scope

Product

PID

PID Controller

f(u)

Mlf(u)

MATLABFunction

Mlf'(u)

-C-

Mass vector

Mlf

Mlf'

Ll

Lf

FluxL

FluxF

Magnetic Forces

Magnetic Force

2

Ll

.6

Lf

1s

L-coil flux

.6*2070

F-coil Flux

du/dt

Derivative

em66

Accel

6

6

6

6

6

66

6

6

66

6

6

6

6

6

6

V

A

3

ZX

AX

Levitation Physics - Simulink Model

… solving for the magnetic force on the F-coil due to the L-coil in terms of the flux gives:

This equation translated to a Simulink model might look like:

€

Fmagnetic =r ∇ MLF

(MLFΦF − LFΦL )(MLFΦL − LLΦF )(LLLF − MLF

2)2

Basic Levitation Model Results

Control parameters as calculated from analytic optimization for voltage PID loop‣ Simulations stay within L-coil supply

specifications

Simulink works!‣ Results match previous numerical

simulations

‣ Analytic analysis eigenmodes are 1.0 and 0.4 Hz

On to implementation!

0

0.2

0.4

0.6

0.8

1x 10

-3 F-coil Z PositionF-coil Z Position

-1

-0.5

0

0.5

1x 10

-3 F-coil X PostionF-coil X Postion

-2

-1

0

1

2x 10

-3 F-coil Tilt about Y-axisF-coil Tilt about Y-axis

-20

-10

0

10

20

30

40F-coil Vertical AccelerationF-coil Vertical Acceleration

0 1 2 3 4 5 6 7 8 9 10-20

-15

-10

-5

0

5

10L-coil Control VoltageL-coil Control Voltage

LDX Control System State Machine

Design of simple levitation controllers requires linear approximation of dynamics near operational point

But magnetic forces are go like L3

Use a state machine to ensure safe operation of system while not near flight status

‣Simulink Stateflow handles state machine in LDX control design

c_pos

current_level

l_pos

reset

voltage_switch

l_posd

l_speed

eloce

enable

stateflow_block

l_pos

current

F_mag

F_launch

Z

scopes_block

l_speed

l_posdl_pos

launcher_block

Z_velocity

reset

P_gain

I_gain

D_gain

V_demand

feed_back_block

-5000

P_gain

l_pos

L_coil_volt

F_mag

F_launch

Z

Z_velocity

coil_pos

Motion_block

eloce

voltage_switch

Rtn

V-demand

enable

V+

Lcoil_supply_block

V+

Lcoil_volt

current_level

Rtn

Lcoil_block

-5000

I_gain

-200

D_gain

66

66

System Simulink/Stateflow Model

Feedback block

L-coil supply block(Simulated or idealized)

Stateflow model

L-coil Power Supply Simulation

Model of 12 pulse switch regulated power supply for L-coil‣ Uses Simulink Power System Blockset

Internal voltage control feedback loop

1

V+ +- v

Vc

VbVa

A

B

C

pulses

+

-

Universal Bridge1

A

B

C

pulses

+

-

Universal Bridge

+- v

Uc

+- v

Ub

+- v

Ua

acos

TrigonometricFunction1

A

B

C

a2

b2

c2

a3

b3

c3

Three-PhaseTransformer

(Three Windings)

T connector

Saturation4

Saturation

L connector

1s

Integrator

[0]

IC

Ground (input)

.8

Gain7

-K-Gain6

200

Gain4

-K-

Gain3

.0146

Gain2

-K-Gain

butter

AnalogFilter Design

1

alpha_deg

A

B

C

Block

PY

PD

Synchronized12-Pulse Generator

1

3 stop

2

Rtn

1

V demand

State System Simulation

Complete functional model Includes human check wait

statesAutomatic failure modes

tested

Z Position Graph Shows launcher in actionCurrent ramp of L-coil Pre-flight check (Premature) launch Free flight

More Full Simulation Results

Vibrating launcher spring L-coil supply voltage ripple

~ 40 volt ripple

Current ripple is < 50mA

No filtering required

Some state machine bugs…

LCX II: Digitally Controlled Levitation

Levitated Cheerio Experiment II Uses LDX digital control system

LCX I was analog demonstration

Modified PID feedback system Low pass filter added for high

frequency roll-off of derivative gain

Integral reset feature for launch transition

Dynamic model block replaced by I/O and estimators

Real-time graph shows position and control voltage Wiggles indicate non-linearly

stable rolling mode…

Noise reduction necessary

Noise reduction necessary for derivative gains

Multipole filter noise reduction limited due to added phase delay

2 4 6 8 10

-0.04

-0.02

0.02

0.04z!t" with Single!Pole Filters

2 4 6 8 10

103

104

105

I!t"

The DFT power spectrum from a simulation of the l-coil current fluctuations

including the effects from plate eddy currents, the l-coil short, and vacuum

vessel eddy currents. The current fluctuations increased significantly

(from Fig. 1) to ±91 mA.

Kalman Filter Simulation

Kalman filter can be used to reduce noise with minimal latency ‣ Uses a physics based predictor that tracks the real motion and is

updated with every time step

2 4 6 8 10

-0.1

0.1

0.2

0.3

0.4

Actual F!Coil z!t"

2 4 6 8 10

-0.1

0.1

0.2

0.3

0.4

Meas & State F!Coil z

2 4 6 8 10

-15

-10

-5

Volt!t"

2 4 6 8 10

103.2

103.4

103.6

103.8

L!Coil Current

Figure 3. Simulation of F-coil position control using a digital controller and

a Kalman filter. (“Wow!”)

Kalman Filter Reduces L-coil AC Losses

Kalman filter results‣ Improved filter greatly reduces

noise in system

‣Reduced noise leads to reduced AC heating of L-coil

Kalman filter with simple feedback sufficient‣Simulations show this method

should meet our requirements for stable levitation

2 4 6 8 10Hz

0.01

0.02

0.03

0.04

0.05!I"2L!Coil Current Power Spectrum

Figure 4.

The DFT of the l-coil current fluctuations from a simulation of the digital

control of f-coil using a Kalman filter with R = I! 1.

2 4 6 8 10Hz

0.25

0.5

0.75

1

1.25

1.5

1.75

2!I"2L!Coil Current Power Spectrum

Figure 2.

The DFT power spectrum from a simulation of the l-coil current fluctuations

including the effects from plate eddy currents, the l-coil short, and vacuum

vessel eddy currents. The current fluctuations increased significantly

(from Fig. 1) to ±91 mA.

Single Pole Filter

Kalman Filter

Control System Development

Integrated test results‣ System identification to ensure observed behavior

matches system model

‣ Identification of model parameters

Formal check of observability and controllability

Optimal Control Theory‣ MIMO System Simulation

‣ Optimal control with balance of minimization of noise and L-coil heating explicitly

‣ Ensure control system won’t add noise to stable modes

Further state machine testing

Output PointInput Point

In1 Out1

Floating Coil Dynamics

F-coil state measurementcoil voltage output

Feedback System

Levitation Test Program

✓ System integration test‣ Test inter-operation of cryogenic and two control systems

L-coil Integrated Performance Test‣ Test L-coil cryogenic performance under worst-case operation point

Gather data to determine HTS coil quench detection algorithm

‣ Calibrate “transfer function” of L-coil System Check state space model for unknown system variables

Integrated System Plasma Test‣ Characterize noise on levitation diagnostics in plasma environment

‣ Operate L-coil systems at 1/2 current with plasma present Calibrate system using measured lift forces

Levitation Test‣ First levitation with nearby supports and safety lines

Levitated Plasma Operations begin