PFC/JA-89-28
Filament-Circuit Model Analysis of
Alcator C-MOD Vertical Stability
D. A. Humphreys and I. H. Hutchinson
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
June, 1989
This paper submitted to Nuclear Fusion.
This work was supported by the U. S. Department of Energy Contract No. DE-AC0278ET51013. Reproduction, translation, publication, use and disposal, in whole or in part
by or for the United States government is permitted.
Filament-Circuit Model Analysis of
Alcator C-Mod Vertical Stability
Abstract
The rigid displacement, current conserving, multifilament plasma-circuit approach to
vertical stability analysis has been applied to the control system design for Alcator CMOD, a high performance tokamak under construction at MIT. For the purpose of rough
benchmarking of the multifilament model, comparison has been made with analytical calculations.
For walls conformal to D-shaped C-MOD equilibria, marginal wall positions
agree well with analytical results for the onset of ideal instability in corresponding elliptical equilibria with conformal walls. For Alcator C-MOD equilibria, velocity feedback is
observed to be necessary for active stabilization. A simple model including one passive and
one active coil is shown to exhibit the same phenomenon. A single filament representation
of the plasma is inadequate for modeling Alcator C-MOD equilibria, but 3-5 filaments are
found to be sufficient to come within 10% of the converged many-filament growth rate.
Application of the model to Alcator C-MOD indicates that n=2 equilibria can be stabilized
satisfactorily provided active power supplies with response time less than .5 ms are used.
2
Introduction
Many modern tokamaks possess strongly shaped, highly elongated plasmas in order
to exploit the higher betas and current densities achievable with high elongation. Such
a vertically elongated plasma is intrinsically unstable to vertical axisymmetric displacements, however, and must be actively controlled in order to maintain equilibrium [1,2,3,9].
Some degree of analysis of vertical stability is necessary for every tokamak possessing an
elongated plasma, but it is especially important in the next generation of machines for
which elongations near 2 are commonplace. For such devices, wall proximity and conformality are critical factors. In particular, if the vacuum vessel wall cannot be designed close
enough and conformal enough to the plasma, there may not be a great deal of stability
margin for highly elongated equilibria, and the issue of accurate modeling of the problem
becomes extremely important.
Because these are resistive wall modes, and machine geometries are in general quite
complex, some form of numerical analysis must be used to evaluate the stability of actual
tokamak plasmas and design feedback systems. Furthermore, the difficulty of the full MHD
and resistive wall problem has resulted in the common use of simpler models to analyze the
vertical control problem for real machines. In fact, since the process of design inherently
involves iteration, the algorithm must be rapid, reinforcing the need for simple models.
Traditionally the most popular simple approach has been to represent the plasma as an
array of current-carrying filaments and surrounding conductors as axisymmetric resistive
loops [4,5,6,7]. The linearized coupled set of circuit equations for the conductors can then
be solved together with the plasma equation of motion. The result is a set of eigenvalues
and eigenmodes describing both plasma motion and induced currents.
As machines are operated closer to their vertical stability limits at ever-increasing
elongation, it becomes very important to have a model which is sufficiently accurate to
design a satisfactory control system. At the very least, the model used must yield conservative results, if not accurate ones, so that the lack of passive stability margin can be
3
somewhat compensated by sufficient active control margin.
Fortunately, there is some
reason to believe [8] that the circuit approach provides such conservative results, and for
that and other reasons is well-suited to analyzing the vertical stability problem. In the
present work we propose to examine this approach, the further simplifying assumptions
often made in using the circuit method, and its application to the design of the Alcator
C-MOD tokamak.
I. The Vertical Instability
Vertical instability in tokamaks arises from the quadrupole magnetic field which must
be applied in order to create a yertically elongated plasma. When added to the vertical
vacuum field required to maintain radial equilibrium, the quadrupole field results in a
vacuum field which is concave toward the outboard side over much of the plasma. This
inherently destabilizing configuration of field lines is characterized by having a predominantly negative decay index [9], n, defined by :
R OBZ
R OBR
B. 8R
B,,
Oz
Regions of negative decay index must necessarily be formed in order to create elongated
equilibria. Typically this curvature will vary widely across the plasma, and the net destabilizing effect is a result of the field curvature profile "sampled" by the current profile over
the entire plasma. It should be noted that decay index is only a measure of curvature,
not of destabilizing force. To compare net destabilizing forces in different equilibria, one
must sum over the appropriate local field gradient force quantities, not over the local decay
indices. The local decay index is effectively the gradient force term normalized by the local
B., so the "average" (weighted by current density) destabilizing gradient force cannot be
simply related to the "average" decay index.
The nature of the vertical mode is such that unless a conducting wall is sufficiently
close to the plasma in some average sense, implying some degree of conformality, the mode
will be unstable on the ideal MHD timescale. In the presence of a resistive wall which is
sufficiently close, the mode will grow much more slowly, roughly on the wall L/R timescale.
4
II. The Circuit Model
The circuit model approach combines physical simplicity and rapid execution with
the potential for a high degree of accuracy in the conductor model. Unfortunately, the
weakness of the circuit model is that the plasma itself is not easily treated correctly.
Typically the simplifying assumption of rigid uniform vertical displacements is used, along
with conservation of plasma current rather than the flux conservation appropriate for an
MHD description. Further simplifying the plasma model, many formulations represent the
plasma as a single filament rather than an array. Current profile effects are therefore lost,
except as they are reflected in the equilibrium field gradient at the chosen location for the
single filament. Even if many filaments are used, the rigid shift is not necessarily the correct
energy minimizing displacement. At best, without allowing filament currents to vary, a
multifilament plasma model will simply give a better estimate of the net destabilizing field
gradient force across the entire plasma. This is reason enough to use a multifilament model,
however, since a priori it is far from clear where a single filament should be located to
produce the most accurate results.
In the following we shall describe the circuit model form appropriate for a uniform
rigid vertical shift and retained finite plasma mass [4,5], modified to include a multifilament
plasma specification. The coils representing stabilizing conductors can of course be driven
according to control laws modeling a feedback response due to plasma motion. Plasma
filament currents are kept fixed, so that all plasma-coil induction is due to the motion of
the plasma. The filament self inductances are thus effectively infinite. This circuit model
yields a set of M voltage equations for the M resistive loops representing the conductors
external to the plasma. These are of the form:
N
L
dlj +rI,+E
t
MJJdIJI,+2
1 dt'
Where:
Ij are the perturbed coil currents
(j
= 1, M),
Mj, is the inter-coil mutual inductance matrix,
5
19Z = Vappled
Mij is the coil-filament mutual inductance matrix,
rj is the resistance of coil
j,
Lj is the inductance of coil
j,
N is the number of plasma filaments,
Vpplied
is the voltage applied to coil
3
j.
If the coil in question is a driven control coil, the applied voltage is given by a control
law depending on the plasma displacement. For passive (undriven) coils this voltage is
simply zero.
The system of equations is closed by including the equation of motion of the plasma
for vertical displacements. This is simply the force balance including the force due to
non-zero decay index (which is destabilizing in the absence of feedback), and the damping
force due to eddy currents induced in surrounding conductors. These are denoted FD'
and FFddy respectively, and the resulting force equation is:
M
= FEddy
dt2
M
N
ii
M
ip
j
3 i3
OBR
N
Z
+ FDI
Z
Z27rRIpi OzZ
SB Z
Where:
Ip are the plasma filament currents (i = 1, N),
BRi is the radial vacuum field at filament i,
We now look for normal mode solutions of the form:
z --+ zet*,
I1 e-'
1j-
(etc...)
The final form of the problem is that of a generalized eigenvalue problem:
6
where the vector F contains the perturbed quantities z, C = i, and the fi. The solution
of the system thus gives us the normal modes and associated growth or damping rates.
This collection of normal modes includes many modes which do not contribute to plasma
motions of interest. Many of these represent decay modes of the conductor array with
essentially no plasma motion. In addition, if the plasma mass is non-zero, two high frequency oscillatory roots appear which are highly damped on the slow time scale of the
instability. These therefore represent transients which can be ignored given sufficient wall
stabilization. Depending on the order of the feedback control law, complex conjugate root
pairs appear. One of these pairs includes at least one unstable root at zero gain. This
pair is strongly affected by feedback (hence referred to here as the "feedback root"), beginning unstable at low gain, possibly becoming stable as gain increases, and often growing
unstable again as gain continues to increase.
III. Discussion of Various Analytic Models
Once one has chosen the circuit model approach to analyzing the vertical stability
problem, there are still many possible simplifying (or complicating) variations. A single
filament or many filaments can be used to represent the plasma. The plasma mass can be
included or neglected. The plasma can be taken to be current conserving or flux conserving.
Many considerations contribute to the validity and desirability of such assumptions, and
we shall discuss some of these in detail. To illustrate some of these features, we shall use
simple cases in which one or two coils are used to represent the conductors in the system.
To begin with, consider the case of a single resistive coil in the vicinity of a plasma
represented by an array of N filaments which are taken to conserve current rather than
flux, so that the circuit equations given in the previous section are applicable:
Mi = ScIe + SBZ
Leic
+
Sci
+
rc I
= 0
After Laplace transforming, we obtain a third order characteristic equation
s +Ve8
+ (Vw
VB)S - VbV, = 0
7
where v
the coil, vz
rc/Lc, vi ,
Se/Lem, vz
=
SB/r.
c is thus the characteristic decay time of
is proportional to the destabilizing field gradient force term, and V.2 reflects
the eddy current stabilization effect.
The zero-mass limit of this equation yields one root, which is unstable and consists of
the characteristic decay frequency with a modification due to eddy current stabilization.
In this limit the divergence corresponding to the ideal MHD stability boundary occurs
when v, = vB.
In general when mass is retained the roots of this equation consist of a stable complex
conjugate pair with imaginary parts on the order of Alfven frequencies, and one real
unstable root whose magnitude depends on the degree of stabilization. In this case the
ideal stability boundary is somewhat more difficult to define. If we nondimensionalize
the characteristic equation by dividing through by vB, defining x = s/v,
7 =-v/vB,
e
=
vcl/vB,
we obtain
X3+
Ex2 + (7 -
1)x -
C = 0
If we now plot the unstable real root as a function of 77, we find a family of curves as
e varies [Fig. 1]. The figure shows that although 7 > 1 is no longer such a hard ideal
stability boundary as in the massless case, it is still a good approximate boundary for
typical cases of interest (10-6 < e
from resistive frequencies (x
77
<
<
10-4). In such a case we see the growth rate change
1) for 71 Z 1 to Alfven frequencies (x
-
1) for 1 ~< 1.
1 is thus also the approximate boundary for validity of the massless approximation.
That is, if 17 is sufficiently greater than 1, the system is significantly well stabilized on
the Alfven timescale, and the massless approximation will be quite good. Of course, if
it should become important to follow the transition to ideal instability, for example in
analyzing highly elongated equilibria, mass must be retained. This is the reason mass has
been included in the VSC code and its applications described later.
Now consider the two coil massless current conserving plasma case. The equation of
motion is now a force balance equation:
0
= SlIl
+ S2
8
2
+ SBZ
and the circuit equations are
L 1i1l+ M12i2 +r1I
1
+ Sli = V
L 2 i 2 + M124 +r 2 12 + S 2 i
0
where coil 1 is the active control coil, to which control voltage V is applied.
Laplace
transforming we find a transfer function of the form
G(s) = z(s) _
V(s)
rs +1
2
As +Bs+C
That is, it contains one zero and two poles.
Expressing the passive transfer function in a standard form:
G
_)
Go(rs + 1)
(ris + 1)(r2s - 1)
where ri- is the unstable pole, we can now add a feedback law
V(s) = -Ko
0 (res + 1)z
representing proportional-derivative (PD) feedback. The system now appears as in Fig. 2,
in which H(s) = Ko(r.s + 1), so that the open loop transfer function is of the form
G(s)H(s) = a(rTs + 1)(,ras + 1)
(Tis + 1)(-r2 s - 1)
The corresponding root-locus equation whose solution gives the location of system poles
as a varies is
(aT7.7 - 7172)52 + (71 - 72 + a[r. - r.])s + (a + 1) = 0
which can be stabilized by an a which is sufficiently negative, even if r. = 0 (i.e. without
derivative feedback).
The above has assumed, however, that the system zero, -1/r., is a left-half plane zero
(i.e. r, > 0). This is not necessarily the case. Expansion of the full solution shows that
the intrinsic system zero is given by
1
r2
SIM
9
Now, since Si = I, am", we have
where "
' "
S2
Mi,
R2BR2
S1
M(,
R1BR1
has been used to indicate derivative with respect to z, and BR; is the radial
B-field at coil i due to the plasma. Clearly this can be arbitrarily large as the BR-coupling
between coil 1 (the control coil) and the plasma is reduced. An extreme example is placement of coil 1 near the same z-location as the plasma filament, where BR is equal to or
nearly zero. In particular, if the coupling between the passive stabilizing structure (represented here by coil 2) and the plasma sufficiently exceeds that between the control coil and
the plasma, a right-half plane system zero will result. Allowing r,
< 0, defining t
=
IrZ I,
we find for the new characteristic equation
(-at.r.
-
-ri1
2 )s
2
+ (-r1
-
r 2 + a[-r. - t.])s + (a + 1) = 0
For -i- > r 2 , true for a system which is not extremely well passively stabilized, this
cannot now be stabilized without derivative feedback (i.e. if -r = 0).
Since vacuum
vessels are in general better coupled to the plasma than to control coils, there is usually
a maximum elongation which a given machine can stabilize without derivative feedback.
This phenomenon is seen in code results to be discussed later.
IV. The VSC Code
In order to analyze C-MOD equilibria, and PF coil and vacuum vessel designs, a
code (known as VSC for Vertical Stability with Circuit model) was written which implements a current-conserving, non-zero plasma mass, multifilament plasma circuit model.
All toroidally continuous conductors in the vicinity of the plasma are modeled as arrays
of axisymmetric loops with circular poloidal cross section and finite resistivity. Resistivity
is defined to be that of the appropriate material, and cross-sections are chosen to conserve
the total resistance of each wall section. Coils are located at evenly spaced points along
straight segments of the vacuum vessel. EF coils are generally represented by one or two
10
model coils of circular cross-section, but the OH stack model is typically composed of a
larger coil array. Figure 3 shows the code model of the plasma-conductor system. The
filament array representing the plasma is determined by reading the current density profile from ASEQ, a PEST-like equilibrium code [10], and further discretizing that (already
discrete) data into a number of filaments much smaller than the entire grid used by the
equilibrium code. This yields a plasma model which, though very simple, provides the
possibility for very accurate calculation of the net destabilizing force on the plasma. The
EF coil currents are taken from the same equilibrium code, and the EF coils themselves
are rediscretized much like the plasma. The accuracy of the rediscretized EF coil model
is checked by comparing flux values at the plasma resulting from the model with values
calculated by the equilibrium code.
The code allows the use of a variety of approaches to active feedback control. In
particular, an explicit double pole, single zero control law [5] of the form
dt
d2
_
dz
can be specified, describing a control voltage Vi(t) applied to coil i as a function of plasma
displacement z(t) and velocity i(t). This yields a transfer function of the form
V1(s)
Z(s)
_
-C(i + t.s)
(1 + tbS)(1 + teS)
This form is especially useful in analyzing the constraints of a realistic pulsed power supply.
Such supplies possess an inherent time delay in their response to a control signal. If this
response time is too long relative to the growth time of the phenomenon being controlled,
the power supply will not be able to stabilize that mode. The control law form above allows
us to model this intrinsic delay with one of the poles (for example tb, setting t = 0), and
then to determine the maximum possible delay time (tb) for which the system can be
stabilized.
To support the general equilibrium control analysis, including vertical stability control,
a data base is being created consisting of equilibria chosen to sufficiently span the space
of C-MOD operation. This data base can be used to interpolate required gain values for
controlling both vertical stability and the evolution of equilibria.
11
V. Benchmarking and Model Investigations
Several alternative methods of analysis are available with which to benchmark the
circuit model. Analytic solutions exist for simplified geometries, in particular for marginal
wall positions. Full MHD analysis codes (such as GATO) can provide marginal wall positions with more realistic geometries and complex displacement profiles. To benchmark
resistive growth rates, either actual machine data or a more detailed tokamak simulation
must be used. Here we shall consider marginal wall position comparisons with analytic
cases for rough qualitative benchmarking.
Haas [11] has demonstrated that there is a maximum distance at which a perfectly
conducting wall will stabilize the vertical mode for an elliptical plasma. This provides a
rough test of the general reliability of the code algorithm: if a resistive wall is used we
would expect to see a similar critical distance manifest itself. This would be the point at
which the mode becomes ideal unstable rather than unstable on the slow resistive time
scale. For low , cases to be stable on the ideal timescale this criterion requires that:
Here r, is the characteristic plasma minor radius, r, is the characteristic conducting wall
minor radius, both measured from the magnetic axis along the midplane in the outboard
direction, and . = b/a is the plasma elongation. For an elliptical equilibrium with r. of
1.8, this criterion for ideal MHD stability is:
( w
r.
<1.97
/
Assuming that this should be roughly true even for the unusual case of an Alcator CMOD equilibrium, the code was used to predict the critical distance for onset of ideal
MHD instability for two vacuum vessel wall shapes. Figures 4, 5, and 6 show the real
parts of all of the system eigenvalues, plotted on an Alfven-like frequency scale to reveal
the transition to ideal instability. An array of conductors modeling a given wall shape is
enlarged conformally from an initial position at the plasma surface. At a critical value of
12
rw/r,, the real parts of two of the modes go rapidly from the resistive frequency regime to
an Alfven growth/damping rate. One of these is the unstable root, and this is the critical
point we wish to compare with the Haas result.
Because the Haas criterion was calculated for a conformal (elliptical) wall, the first
vacuum vessel shape chosen was a D-shaped array of conductors approximately conformal
to the plasma surface for a typical Alcator C-MOD equilibrium. Using the multifilament
plasma model, this array was enlarged from an initial configuration lying on the plasma
surface [Fig. 4]. The ideal instability onset point was found to occur at:
(
W) = 2.00
Still using a multifilamentary plasma, the critical distance was also determined for the
actual C-Mod vessel, sweeping its size using the same algorithm as in the conformal case
[Fig. 5]. As expected, the resulting value,
( rr
=
1.80
indicates that such a highly non-conformal wall must be closer to the plasma than the
conformal wall must be. Nevertheless, even this critical distance is encouragingly close to
the Haas prediction.
The conformal shell case was repeated using a single filament located at the current
profile centroid [Fig. 6]. In this case the result is significantly more optimistic, yielding a
critical distance of
( rr
= 2.28
This greater optimism for the case of a single filament at the current centroid is consistently
observed and emphasizes the need for a multifilament model. Note that as the magnetic
axis is further out than the current profile centroid, placing the single filament on the axis
yields even more optimistic results.
The circuit model can itself be used to investigate the assumptions made in such a
model. In particular we shall address two issues raised above: convergence with increasing
number of filaments used in the plasma model, and location of a single filament. Figs.
13
7 and 8 show growth rates as a function of radial position of a single filament along the
plane at the z-position of the current centroid for equilibria eq6 and eq7 (see Table I).
Current centroids are denoted by RO in the figures, and in both cases the magnetic axis
is located at 66 cm. Multifilament model-derived growth rates are 388 sec'
for eq6 and
945 sec- 1 for eq7, whereas the corresponding single filament growth rates are 174 and 461
respectively, for filaments located at the current centroids. It is clear from these examples
that care must be taken when locating a single filament. While for some equilibria placing
a filament at the current centroid may yield growth rates relatively close to what one
finds with a multifilament model, for others (e.g. eq6 and eq7) this is not the case. In
fact, Figs. 7 and 8 illustrate that for eq6 and eq7 there is no location for which a single
filament will yield the same growth rate as the multifilament model.
The situation is
particularly bad for equilibria such as eq7, which are relatively close to the marginal ideal
stability point, or equivalently are very highly elongated. Such cases were also illustrated
in the benchmarking section, in which the marginal wall position was seen to be strongly
dependent on the choice of single or multifilament model.
Active analyses also tend to be very sensitive to choice of filament model, since using a single filament is essentially the same as drastically increasing the peakedness of an
equilibrium which may not originally have been highly peaked. In particular, this can significantly decrease the coupling between control coil and plasma, even though the vacuum
vessel coupling may not be as strongly affected. The result of this difference is seen in
Table II, showing tolerable delay times (described in Section IV of the text) for marginal
stability with fixed derivative feedback time constant for single and multifilament models
of equilibria eq6 and eq7.
Having argued that some degree of spatial distribution is important in a filamentary
model of a plasma used in circuit models, it is reasonable to ask how many filaments is
sufficient. Figs. 9 and 10 show convergence of growth rate values calculated with increasing
numbers of filaments modeling the plasma for equilibria eq6 and eq7. Equilibrium eq6 is
a double null, elongation 1.8 case, and eq7 is a single null equilibrium with an elongation
of 2.0. For these typical C-MOD cases, only 3-5 filaments need be used to get within 10%,
and roughly 20 filaments or more should be used to converge to within 1% of the growth
14
rate calculated with the maximum number of filaments used, on the order of 50-60. That
only 3-5 filaments are so effective is indicative of the low order of significant multipole
moments in the vacuum magnetic field.
VI. Alcator C-MOD Vertical Stability Analysis
Alcator C-MOD [12] is a high performance tokamak intended to be somewhat prototypical of the Compact Ignition Tokamak (CIT). As such it will provide a test bed for
many of the critical issues to be faced in the design and operation of CIT. The essential
machine parameters are as follows: major radius R=0.66m, minor radius a=0.21m, maximum toroidal B-field Bt=9T, maximum plasma current I,=3MA, and typical elongation
n=1.8. Machine geometry is shown in Figs. 3 and 11. C-MOD will possess a strongly
shaped diverted plasma, with maximum elongation in excess of 2. The complexity of its
geometry, including D-shaped single and double null plasmas and a roughly rectangular
vacuum vessel with a high degree of diagnostic access producing many interruptions in
toroidal current paths, has motivated this study of the vertical control problem.
The "standard" mode of the VSC code retaining plasma mass, conserving plasma
current, and using full multifilament models, has been used to analyze C-MOD equilibria.
Passive growth rates for a variety of equilibria spanning a wide range of parameters are
shown in Table I. As elongation increases from n = 1.65 to n = 2.0, the unstable growth
rate increases from y = 25 to -y ~ 950 sec~1 . It is interesting to compare these values with
some characteristic resistive decay times of the system. The diffusive penetration time of
the vacuum vessel,
TD
[13], is on the order of
TDE_
d
11 ms
where a is a rough equivalent radius for a circular model of the vacuum vessel wall, taken
to be 50 cm; d is the thickness of the vacuum vessel wall, taken to be 1.27 cm (.5 in);
and 1 is the resistivity of the vacuum vessel, taken to be that of stainless steel at room
temperature, 70 pO - cm. For the EFC control coil, characteristic of copper elements in
15
the system in general, the L/R time at room temperature is
= 80 ms
-L
(R cu
The vacuum vessel model itself has many possible decay modes due to the large number
of coils used to represent it. One among these of particular interest is the dominant stable
midplane-antisymmetric mode. This is the lowest order mode which couples to vertical
plasma motion. The current distribution in surrounding conductors for this mode is shown
in Fig. 12 for the "standard" C-MOD equilibrium (eq2). The figure shows the entire
vacuum vessel above the midplane, as well as a portion of it below. The area of each coil
marker in the figure is proportional to the current in that element, and positive or negative
currents are denoted by open or filled square markers respectively. As one would expect,
the dominant antisymmetric mode has a roughly sinG dependence of current distribution
on poloidal angle. The corresponding damping rate is
ms
() R Antisym = 10.8
Comparing this value to the actual growth rates for the range of equilibria in Table I,
we observe that C-MOD equilibria do not in a sense take full advantage of the stabilization
potential of the system. This is due largely to the plasma shape, irregularity of the vacuum
vessel, and the peakedness of actual current profiles. The lack of wall-plasma conformality
alone makes it extremely improbable to be able to reduce the growth rate to this optimistic
level for plasmas having the elongation of the vacuum vessel (K = 2).
By contrast, Fig. 13 shows the current distribution for the feedback root (the dominant
mode which is affected by feedback and contains plasma motion) in an actively stabilized
case for equilibrium eq2. It is noteworthy that this mode clearly contains high poloidal
harmonics, and therefore cannot be accurately modeled by a low order mode. In fact, using
a lower order mode to estimate the growth rate will yield an unrealistically well stabilized
result, since the characteristic decay time increases with decreasing mode order.
It is also of interest to examine the dependence of growth rate on triangularity. Equilibria eq2, eq4, and eq5 in Table I were chosen requiring all other relevant parameters
16
to remain relatively fixed while triangularity varies. The data suggest that for the uniform displacement, current conserving, multifilament circuit model, triangularity does not
strongly affect the growth rate. Notice that the eq4 single filament calculation yields a
growth rate much higher than the single filament result for the other two equilibria having
the same plasma parameters. This is another manifestation of the unreliability of single
filament results. For eq4, the choice of the current centroid for the single filament location
proved to yield a growth rate fortuitously close to that calculated with the multifilament
model.
A fast circuit model code is ideal for studying the relative passive stabilization effects
of various control coil placements. These coils will typically be the closest copper coils
well-coupled to the plasma vertical motion, and should therefore have a strong influence
on passive stability properties. Many control coil positioning options were investigated
during the design of C-MOD, but we shall address the passive stability for two cases alone.
They correspond to control on the OH2 portions of the "notched" OH stack, or on coils
denoted "EFC" placed in the upper and lower outboard "shoulders" of the vacuum vessel
specifically for control purposes [Figs. 3 and 11 ]. Passive growth rates corresponding to
these cases are shown in Table I. The stabilizing effects of these two choices for control coil
are in general comparable. In light of this, and acknowledging the difficulty of engineering
a fast control power supply operating in parallel with the OH2 equilibrium supplies, the
outboard coils denoted EFC in Fig. 11 have been chosen for vertical control.
Passive analysis can provide a sense of the difficulty of stabilizing given equilibria, and
to some extent measure the relative quality of various stabilizing configurations. However,
on the basis of these data alone nothing reliable can be said about power supply response
time requirements. Active studies are required to determine actual limits on the control
system. In particular, while the passive growth rate may not be very sensitive to exact
control coil location, the calculated tolerable delay time may depend more strongly on it.
Active analysis of tokamak equilibria primarily addresses whether an equilibrium can
be stabilized at all given the available geometry and hardware constraints. This can also
find application in an iterative procedure attempting to optimize placement of control
sensors and coils. The VSC code is capable of implementing flux loops and B-field sensors,
17
as well as direct state sensors, which have immediate access to plasma displacement and
velocity. However, we shall only address the use of direct state feedback here.
Linear power supply dynamics in the form of frequency domain transfer functions
can be defined between demands to the power supplies and actual output to a control
coil, allowing power supplies to be modeled with a relatively high degree of complexity.
This transfer function definition can also be used to represent feedback compensation. For
the most part, these extra dynamics have been used to model the response delay time
intrinsic to a pulsed power supply, as described in Section II. A single pole has been used
to represent the delay, with a single compensating zero representing derivative feedback.
In general, elongated C-MOD equilibria have been found to require derivative feedback in
order to be stabilized at all, even with zero delay. This is the same phenomenon as that
illustrated in the discussion of analytical models above, and has been investigated by other
authors [14]. Fig. 14, for example, shows the result of sweeping gain in eq6 using direct
displacement feedback with no derivative term. The unstable root has a growth rate of
388 at zero gain, but becomes more and more stable as gain is increased. In the absence
of derivative feedback, however, this root never becomes stable. Fig. 15 shows the result
of adding derivative feedback. For this case the mode are stabilized. Addition of a single
pole modeling power supply delay makes it more difficult to stabilize, and in fact a given
equilibrium will exhibit a maximum tolerable delay which can be stabilized at all.
Stability boundary results showing maximum tolerable delays and necessary zero values for eq6 and eq7 are shown in Table II. Using this crude delay model, the maximum
tolerable delay time for the maximum elongation equilibrium (eq7; passive growth rate =
945) is found to be approximately
=
713
1 msec
for the actual EFC configuration (Fig. 3) with a lead value of 50 msec. The actual fast
power supply will have a response time well below .5 msec, ensuring a significant safety
margin. It is of interest to note that a previous EFC placement yielded a passive growth
rate for this equilibrium of 980 sec
1
and maximum tolerable delay of approximately
ra'
=
0.4 msec
18
with the same lead value. Thus the passive growth rate was affected relatively little by
this design change, but the acceptable delay was roughly doubled.
The extra dynamics facility is also useful for modeling the higher frequency dynamics
of the power supply. Since plasma mass is retained in our model, increasing gain will
eventually drive the MHD roots unstable. This destabilization requires the power supply to
respond on a microsecond timescale, which is clearly unreasonable. Addition of appropriate
filters to the power supply model can prevent this nonphysical destabilization of the MHD
roots.
The usefulness of the circuit model in determining power requirements is somewhat
limited. An initial state must be selected and expanded in eigenfunctions of the system,
and some assumption must be made about power supply limits. Using the control coil voltage and current components from the unstable eigenmode calculated by the VSC code,
the effective control coil impedance is found. Once the power limit is established, this
impedance allows demand currents and voltages to be determined scaled to the displacement. If a reasonable initial displacement can be determined, this procedure provides a
test of sufficiency of the power supply specifications. Results indicate that a 1 MVA power
supply will stabilize the n = 2 plasma of eq7 with approximately 30 V applied to the EFC
coil, yielding about 35 kA for a 5 mm displacement.
It should be noted that all of the above calculations have ignored the stabilizing effects
of various equilibrium field coil pairs which will routinely be operated so as to be passively
stabilizing. The most effective of these pairs is the OH2 upper and lower coils, which
are operated independently in order to produce single null plasmas.
As noted before,
the stabilizing effect of these coils is comparable to that of EFC, whose effect is in turn
quite large. Thus, the most pessimistic passive stabilization scenario has been consistently
chosen.
VII. Conclusions
The circuit model approach to vertical stability analysis has proven to be extremely
fast in program execution, easily understandable, and highly versatile. Much information
19
can be extracted from the eigenvalue analysis, and the method is useful for feedback control
system design as well as analysis. However, great care must be exercised in applying the
model. In particular, the use of a single filament to represent the plasma can produce
growth rates and active control properties which are significantly different from those
of a more realistic multifilament distributed plasma model. With a judicious algorithm
for filament placement, only 3-5 filaments are needed to substantially improve on the
performance of a single filament.
The vertical stability characteristics and control system requirements of Alcator CMOD have been analyzed using a multifilament circuit model code. Consistent with predictions of simple analytic models, C-MOD has been found to require velocity feedback in
order to stabilize highly elongated equilibria. Even for the most highly elongated equilibria
(n! ~ 2.0) planned for C-MOD, the final control coil and vacuum vessel designs have been
found to tolerate power supply delay times well above the actual power supply response
time, and to result in reasonable control system current and voltage demands. This is true
even for the most pessimistic of stabilizing structure models. Lacking the wall conformality
and closeness of other tokamaks in return for increased diagnostic access and other design
considerations, Alcator C-MOD will nevertheless be able to produce high performance
plasmas with elongations to rival the highly elongated conformal wall devices operating
today.
20
REFERENCES
[1] Mukhovatov, V.S., Shafranov, V.D., Nucl. Fusion 11 (1971) 605
[2] Laval, G., Pellat, R., Soule, J.S., Phys. Fluids 17 (1974) 835
[3] Rebhan, E., Salat, A., Nucl. Fusion 18 (1978) 1431
[4] Jardin, S.C., Larrabee, D.A., Nucl. Fusion 22 (1982) 1095
[5] Thome, R.J., Pillsbury, R.D., Montgomery, D.B., Politzer, P.A., Wolfe, S.M., Mann,
W.R., Langton, W.G., Pribyl, P., MIT Plasma Fusion Center Rept. PFC/RR-83-32
(1982)
[6] Mori, M., Suzuki, N., Shoji, T., Yanagisawa, I., Tani, T., Matsuzaki, Y., Nucl. Fusion
27 (1987) 725
[7] Nagayama, Y., Naito, M., Ueda, Y., Ohki, Y., Miyamoto, K., Nucl. Fusion 24 (1984)
1243
[8] Hutchinson, I.H., submitted to Nucl. Fusion
[9] Wesson, J.A., Nucl. Fusion 18 (1978) 87
[10] Strickler, D.J., Miller, J.B., et al ORNL Rept. ORNL/FEDC-83/10 (1983)
[11] Haas, F.A., Nucl. Fusion 15 (1975) 407
[12] Hutchinson, I.H., Becker, H., Bonoli, P., Diatchenko, N., Fairfax, S., Fiore, C.,
Granetz, R., Greenwald, M., Gwinn, D., Hakkarainen, P., Humphreys, D., Irby, J.,
Lipshultz, B., Marmar, E., Montgomery, D., Park, C., Parker, R., Pierce, N., Pillsbury,
R., Porkolab, M., Ramos, J., Rice, J., Schultz, J., Sigmar, D., Silva, F., Takase, Y.,
Terry, J., Thibeault, E., Wolfe, S., MIT Plasma Fusion Center Rept. PFC/RR-88-11
[13] Freidberg, J.P., Ideal Magnetohydrodynamics, Plenum Press (1987) 309
[14] Lazarus, E.L., Lister, J.B., Bull. Amer. Phys. Soc. 33 (1988) 1961
21
Figure Captions
Fig. 1. Growth rates normalized to the characteristic Alfven frequency (xa
s/yB, where s is
the actual growth rate), as a function of stabilization factor, 77 _ u.,/zv.
a) e = 10-3,
b) e = 0.1, c) e = 0.55, where e = vl/vB. e is typically less than 10-4.
Fig. 2. Block diagram of the closed loop feedback system.
Fig. 3. Code Model of the Alcator C-MOD machine geometry. Solid squares represent stabilizing conductors, open squares marked with "X" are equilibrium field (EF) coils, and
filaments modeling the plasma current distribution are denoted by open circles. EF
coils which are considered to be stabilizing are represented as solid squares.
Fig. 4. Growth rates as a function of wall distance, illustrating the critical distance for onset
of ideal instability for a conformal d-shaped wall and multifilament plasma model.
Fig. 5. Growth rates as a function of wall distance, illustrating the critical distance for onset of
ideal instability for a C-MOD vacuum vessel model and multifilament plasma model.
Fig. 6. Growth rates as a function of wall distance, illustrating the critical distance for onset
of ideal instability for a conformal d-shaped wall and single filament plasma model.
Fig. 7. Growth rate for a single filament plasma model as a function of filament radial position
(equilibrium eq6). Ro denotes the current centroid, at RO=65.Ocm, while the magnetic
axis is located at R=66.5cm.
Fig. 8. Growth rate for a single filament plasma model as a function of filament radial position
(equilibrium eq7). RO denotes the current centroid, at Ro=65.4cm, while the magnetic
axis is located at R=66.5cm.
Fig. 9. Growth rate as a function of number of plasma filaments for equilibrium eq6, . = 1.8,
6 = .35. Roughly 3-5 filaments are necessary for the calculated growth rate to come
within 10% of the converged value.
Fig. 10. Growth rate as a function of number of plasma filaments for equilibrium eq7, . = 2.0,
6 = .32. Roughly 3-5 filaments are necessary for the calculated growth rate to come
within 10% of the converged value.
22
Fig. 11. Alcator C-MOD machine geometry. Arrows indicate equilibrium field (EF) coils and
the ohmic heating (OH) stack.
EFC is the vertical control coil pair.
Note the
"notched" OH coil design, consisting of the single large OH1 segment and two independently driven OH2 coils.
Fig. 12. Currents induced in the vacuum vessel for the dominant passive antisymmetric mode
with equilibrium eq2. The area of each coil marker in the figure is proportional to
the current in that element, and positive or negative currents are denoted by open or
filled square markers.
Fig. 13. Currents induced in the vacuum vessel for the stabilized feedback root (the dominant
mode which is affected by feedback and contains plasma motion) with active control
of equilibrium eq2. The area of each coil marker in the figure is proportional to the
current in that element, and positive or negative currents are denoted by open or filled
square markers.
Fig. 14. Root-locus plot for eq6 with no derivative feedback (t. = 0). The figure shows that for
this equilibrium in the absence of derivative feedback, the plasma is never stabilized.
Arrows indicate the direction of motion of the feedback root (the dominant mode
affected by feedback containing plasma motion) as gain increases. Note that the plot
also contains stationary roots which remain fixed as gain varies, representing modes
which do not couple to plasma motion.
Fig. 15. Root-locus plot for eq6 with derivative feedback (t.
$
0).
The figure shows that
the addition of derivative feedback stabilizes equilibrium eq6. Arrows indicate the
direction of motion of the feedback root (the dominant mode affected by feedback
containing plasma motion) as gain increases. Note that the plot also contains stationary roots which remain fixed as gain varies, representing modes which do not couple
to plasma motion.
23
Table Captions
Table II. Table of equilibrium data. m is the elongation, 6 is the triangularity, and
-y is
the
passive growth rate for the given equilibrium. "mf" refers to multifilament plasma
model, "sf" to single filament model, "vv+efc" to both vacuum vessel and EFC passively stabilizing, and "vv+oh2" to both vacuum vessel and OH2 passively stabilizing.
See Fig. 11 for machine geometry.
Table II. Table of active feedback law data. t. is the lead compensation (velocity feedback)
time constant, and
t
b
is the lag time constant modeling power supply delay time. As
in Table I, "mf" and "sf" refer to multifilament and single filament plasma model
respectively. For all Table II cases the vacuum vessel is used for passive stabilization
and EFC for active control.
24
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29
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30
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31
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32
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34
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37
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