Introduction

The University of Washington in collaboration with the Princeton Plasma Physics Laboratory has proposed a three-year program to implement, develop, and study coaxial Helicity injection (CHI) on NSTX as a method of formation and sustainment of a spherical Tokamak. This research will be conducted at the Princeton Plasma Physics Laboratory (PPPL) by Professor Thomas R. Jarboe, Dr. Roger Raman, and a graduate student from the University of Washington; Dr. Mike Schaffer from General Atomics; and Dr. Stanley Kaye and Dr. Dennis Mueller from PPPL.

NSTX is a low aspect ratio Tokamak that should have higher b and better confinement than a normal Tokamak. However, the low aspect ratio restricts the area available for the current drive transformer, thus some other plasma formation and sustainment method is needed. CHI is a steady state method of current drive that does not require a transformer and is compatible with low aspect ratio. The Helicity Injected Torus (HIT) experiment at the University of Washington has achieved plasma currents of up to 250kA with an injector current of about 20kA in a low aspect ratio (A=1.5) geometry. Assuming Helicity balance, the HIT experiment has achieved current drive energy efficiencies of 40%. CHI is predicted to sustain Tokamak reactors with efficiencies of a few tens of a percent as well. The other methods of steady state current drive now being pursued, RF and neutral beams, have efficiencies of less than 1%. These inefficient methods greatly increase the recirculating power fraction and capital cost of the reactor and adds the additional constraint that the equilibrium must have a high bootstrap fraction. The spherical Tokamak and normal Tokamaks have a high b-poloidal and, therefore, should have a high nuclear power to current drive power ratio and very little recirculating power should be needed for current drive in a reactor. Using CHI, the capital cost and recirculating power fraction needed for current drive should become insignificant and the high b-poloidal advantage of the Tokamak would be realized. CHI is also a very cost-effective method of starting up NSTX.

CHI is an enabling technology for the low and very low aspect ratio Tokamaks. On NSTX it is planned to use CHI to start up the spherical Tokamak (ST) and to sustain the edge plasma during sustained operation. NSTX needs a warm 500kA plasma as a target for RF and neutral beams heating and current drive. This is a very modest extrapolation of the 250kA achieved on HIT. To achieve very high beta and very high bootstrap fraction, edge current drive is needed. CHI is ideally suited for this type of current drive and tends to produce a hollow current profile similar to that needed for this type of equilibrium.

Helicity injection current drive is based on the observation that Helicity is the best constant of the motion for the magnetic field of magnetized plasma. Since the magnetic field energy decays faster than the Helicity, the configuration tends to relax toward a state of minimum energy while conserving Helicity. Helicity is the linkage of magnetic flux with magnetic flux ,and in toroidal geometry is given by

where phi is the toroidal flux inside a flux surface and psi is the poloidal flux. In general the rate of change of Helicity is given by

where the first integral is calculated over the plasma volume but with no plasma current and with the same boundary conditions as that of the second integral, the actual plasma case. The first integral is thought of as the injection term and is equal to 2*Injector Voltage*Injector Flux for CHI where the Injector-Voltage is the voltage between the coaxial electrodes and the Injector Flux is the flux that penetrates both electrodes. The second integral is thought of as the resistive dissipation of Helicity.

A very important implication of the minimum energy principle is that:

However, this does not imply that locally

For transformer current drive the Helicity injection rate is 2*Loop-Voltage*Phi-wall. The requirements for successful CHI current drive are the following:

1. the boundary conditions are such that relaxation toward the minimum energy state is also toward the desired equilibrium,
2. the boundary provides a Helicity barrier so the injected Helicity is confined,
3. sufficient Helicity is injected to form and sustain the equilibrium,
4. the energy per unit Helicity of the injected Helicity must be higher than that dissipated by the equilibrium ,and
5. the injected linked flux must flow into the equilibrium volume.

Helicity injection on NSTX essentially would work the same way that it is done on HIT. Since the minimum energy state has jp/B constant throughout the volume where jp is the parallel current density and B is the magnetic field. With CHI the current on the open field lines at the edge is driven to a high value producing a very hollow current profile. Relaxation toward the minimum Energy State will flatten this profile, giving a Tokamak equilibrium with current drive in the closed flux region. As on HIT the conducting shell on NSTX will provide a Helicity boundary so the injected Helicity will be confined within the Tokamak volume.

Poloidal Flux Feedback Control Mechanism

HIT-II and NSTX both utilize a feedback mechanism to control the poloidal flux boundary conditions through the use of their poloidal field coils / flux-loops. However, the actual method by which this is achieved is quite different between the two experiments. In order to simulate the time-development of the magnetic geometry in the two experiments, a mathematical model representative of their respective feedback-mechanisms was developed and coded in IDL.

HIT-II: Pulse Width Modulation

• Voltage is constant
• Coils are driven by voltage pulses of varying duration
• Pulse length is determined by the error between flux demand and measured flux at any given time-step.
• Error is 'sampled' in time-intervals determined by a carrier frequency.

In principle, the duration of the applied voltage pulse is controlled by two 'switches' (in reality these are pairs of IGBT transistors). Whenever the two 'switches' are in opposing states a voltage is applied. The state of 'switch1' changes every time the error-signal crosses the carrier-wave, the state of 'switch2' changes whenever the negative of the error-signal crosses the carrier-wave. Figure 1 illustrates these dependencies. The resulting code in IDL is as follows:

;carrier signal
xf = carrier_amp * $
carrier_fun( tsteps( i ) - tsteps( 0 ), $ carrier_per )   
yf = - xf

;demand and error signals
error = looptemp - $
gain * demand_val( tsteps( i ) - tstart, $
                        wave_grps, wave_freq ) 

;set new voltages on driven coils
vc(0:n_grps-1) = vapp * ( $
-1.0 * (( error GT xf ) AND ( error GT yf )) $
+1.0 * (( error GT xf ) AND ( error LT yf )) $
) / restc_grps(0:n_grps-1)

NSTX: Pulse Amplitude Modulation

On NSTX, the coils are spaced significantly further apart than on HIT-II. In addition, there is a larger amount of conducting material between the flux-loops used for feedback and the coils generating the field. As a result, it is not enough to feed back to the error signal alone, but the time-derivative of the error must also be considered to stabilize the feedback mechanism. The derivative term causes the system to 'anticipate' the evolution of the error, act accordingly and thus avoid growing (unstable) oscillations centered on the demand signal.

The voltage is applied to the coils continuously and the amplitude varied proportional to the error-signal and its derivative. Each relation has a factor referred to as the feedback gain.

A singularity arises when feeding back to a zero demand signal. To avoid this problem, the function is adjusted with an exponential function.

Lastly, the two contributions are expressed as a single gain factor and two percentages indicating how much gain is applied to the error and how much to the error derivative:

Coded in IDL the NSTX feedback mechanism looks like this:

;find percent error
demand_tmp = gain*demand_val(tsteps(i)- $
  				tstart,wave_grps, wave_freq )
error_old  = error
error      = looptemp - demand_tmp
error      = error / (ABS(demand_tmp)+ $
  				exp(-ABS(demand_tmp))*0.01)
error_del  = (error - error_old)

;determine voltage
temp 	 = INDGEN(n_grps)
vc(temp) = vapp(temp)*fbscale(temp)*(-1.0)*$
  	       ( (1.0-fbscale2(temp)) * error + $
  	             fbscale2(temp)  * error_del )

;clip voltage to power supply maximum
temp = WHERE(ABS(vc) GT ABS(vapp))
IF MIN(temp) GT -1 THEN $
  	vc(temp) = vapp(temp) * sign(vc(temp))

Development of Time-Varying Equations

The simulation code models the interaction between all conducting material elements on the experiments, assuming toroidal symmetry. There are 4 different cases which are considered:

• vessel elements (passive)
• driven coils (active)
• plasma current (modeled as a distributed driven coil - active)
• saddle coil elements (passive)

Once the currents in all these elements is known, the resulting flux can be calculated using previously determined green-functions for all flux-loops and a distributed spatial grid.

Consider the circuit elements shown in Figure 2 as a representation of all the sub cases involved. Each of the sub-cases can then be recovered by adjusting the properties of this circuit:

Vessel Element
Driven Coil
Saddle Coil

The equivalent circuit diagram for the general current element is shown in Figure 3. To find the current I generated by the two time-varying DC voltage sources, each voltage source is treated as the sole source in the circuit with the other replaced by a short, and the results are then super-imposed. The resulting expression for the current is:

Mutual Inductance Matrix

V is the voltage due to self and mutual inductances:

The expression for the total current in the element then becomes:

For notational convenience, the following definitions are made:

I squiggle has no physical meaning in that it cannot be measured with an ampere meter anywhere in the circuit. The final expression then becomes:

The solution to this ODE then yields the expression we're after:

In order to use the equation in a computation it needs to be discretized over each time-step, the external voltage is held constant over the time-step:

External Voltages

It was previously defined that I squiggle = V squiggle / R. V squiggle is the voltage across the resistor R squiggle. For a vessel element this resistor has a value of zero ohms and therefore I squiggle is equal to zero. For a coil connected to a power-supply, the resistance is infinite, and the voltage between these two points in the circuit simply becomes the voltage of the connected power-supply, and I squiggle is the applied Voltage divided by R. The most involved case arises, when the element is part of the saddle coil: In this V squiggle is the voltage drop of the equivalent resistance of all the gap-resistances of the saddle coil elements summed up in parallel (see Figure 5 ). In this case the resulting voltage V squiggle can be expressed by the sum of all the currents in the saddle coil elements:

In order to find a general expression, define a binary vector a that contains the value 1 for a saddle coil element and 0 for all other elements:

The final expression for I squiggle then becomes:

Where Vapp is zero for elements that are not connected to a power-supply. The resulting expression for the total current is then:

Solving this expression for the current I yields:

The final equation is:

If no saddle coil is present, the matrix S becomes the identity matrix, and the above equation changes back into the form originally derived for the HIT-II simulation by Dave Orvis.

Time Simulation of Poloidal Magnetic Field Geometry

The entire simulation code consist of several sub-components, grouped in the order in which they need to be executed to achieve the desired results. The required input for the code is the experiments geometry, the resistivity of all used materials, the flux demand values as a function of time (wave-forms) and the current distribution of the plasma current (generated by using EFIT). The output is the magnetic geometry in the experiment over time, as well as the current and voltages in the power-supplies. This serves as a valuable guideline in designing the power-supplies on NSTX.
The individual code modules are called from the main GUI and perform the following tasks:

1. Establish the geometry of the experiment.
2. Find the green-functions for all components and the spatial grid.
3. Prompt for information on coil / flux-loop grouping, and power-supply capabilities.
4. Prompt for simulation parameters (plasma current breakdown time / peak value / ramp-up time, length of simulation, etc.).
5. Present the results.

Results

This is a small sample of some results for typical experiment configurations.

HIT-II	NSTX
Vacuum Flux Plot	Vacuum Flux Plot
Plasma Flux Plot	Plasma Flux Plot

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Ralph Ewig
HIT Plasma Physics Lab
University of Washington