PFC/RR-82-7
Contours of Constant Effectiveness For PF Coil Pairs
R.J. Thome and W.R. Mann
Massachusetts Institute of Technology,
Cambridge, MA 02139
March 1982
DOE/ET-51013-35
UC20-B
Contours of Constant Effectiveness for PP Coil Pairs
*
R.J. Thome and W.R. Mann
M.I.T. Plasma Fusion Center
Cambridge, Massachusetts 02139
M.I.T. Plasma Fusion Center
PFC/RR-82-7
Supported by U.S. D.O.E. Contract DE- AC02-78-ET51013
Contours of Constant Effectiveness for PF Coil Pairs
by R.J. Thome and W.R. Mann
M.I.T. Plasma Fusion Center
ABSTRACT
The process of designing a set of polidal field (PF) coils is iterative,
and involves detailed consideration of plasma requirements.
Invariably, ques-
tions arise concerning the impact of repositioning a current center on the
amp-turns, ampere-meters or energy stored.
Figures are given which illu-
strate contours of constant effectiveness for these quantities for a PF coil
pair in terms of the axial field, B , or derivative, (B z/3r), at a point in
the z = o plane or the flux coupled by a contour thru the point.
The figures
allow an insight to be gained and estimates to be made for how the amp-turn,
amp-meter and energy changes as the location of a coil pair is altered.
CONTOURS OF CONSTANT EFFECTIVENESS FOR PF COIL PAIRS
R.J. Thome and W.R. Mann
The process of designing a set of poloidal field (PF) coils for a
tokamak involves the determination of a current distribution among current
centers such that they provide a field distribution in the plasma with location
constraints imposed by interfacing equipment.
iterative.
The procedure is necessarily
Invariably, questions arise involving the impact on ampere turns,
amp meters and/or stored energy in the event that a current center is shifted from one position to another.
The enclosed figures illustrate contours
of constant effectiveness which allow estimates to be made for these changes.
The left side of Fig. 1 shows an arbitrary pair of PF coils providing
a specified z-directed field, Bz,
in the z = 0 plane at a radius r .
coil has a radius, a, and carries a current, I.
coils is 2d.
Each
The distance between the
The right side of the figure shows the coils in a normalized
space where dimensions have been normalized to ro, the distance from the axis
to the point of interest.
The field produced at (z = 0, r = r0 ) may be found
by superposing the fields from two circular current filaments (eg
Smythe [1])
see
and written in normalized form as
2
B
at
where
-
=.
E I
TT2
a(1 + p)
p
=
(ro/a)
T
=
(d/a)
K, E
P)
+
2
+2
nJ
T1 1/2
F (p,T)
complete elliptic integrals of 1st and 2nd
kind with modulus, k
(1)
-2-
k2
(1
B
at
=
4p
+ p)
(2)
2
+ j
r(3)
y1 I
I = amp-turns in one coil
Bat is a normalized field which, for a given r0 , may be viewed as the field
per amp-turn in the coils.
It is a function only of p and n or may be con-
sidered to be a function of (p 1, ni/P) = (a/r , d/r ).
The latter form is
particularly useful in plotting contours of constant Bat using Equation 1.
These contours are shown in Figs. 2a and 2b with different contour intervals.
In these figures, a PF coil on any given contour will provide the same Bz per
unit amp-turn at a specified point r = r0 , z = 0 in real space or at (d/r0 ) = 0,
(a/r ) = 1 in this figure.
If, far example, a specified Bz is required at a
specified r0 , the amp turns for a coil on a contour through its location at
(a/r , d/r ) can be estimated from the contour value for Bat.
is changed to another contour while holding B
If the location
and r0 the same, then the
change in amp-turns required is proportional to the change in contour values.
Note that a B
at
=
duce no Bz at r .
0 contour exists, hence, coils located on this contour proCoils on negative contours produce Bz < o at r
on positive contours produce Bz > 0 at r0 .
The Bat
=
and coils
0 contour defines the
upper bound for location of a PF coil which can produce negative
Bz at r0
without reversing its current direction.
In some cases, coil costs may be related to the amp-meters utilized
-3-
rather than the amp-turns, hence,
it is useful to consider contours of rela-
tice effectiveness in terms of the ampere-meter requirement.
The ampere-
meters necessary for the coil pair in Fig. 1 is
A-m = 4faI
Equations (1),
(3) and (4) may be combined to yield:
B
where
B
(4)
B
amn
am
=-
47r
F(PT)
(5)
B r 2
= z 0
(6)
p (A-m)
is a normalized field at r per ampere-meter and contours of constant B
are shown in Figs. 3a and 3b in a manner similar to that used earlier.
The
contours may be used to determine an amp-meter requirement or change in ampmeter requirement in the same way as the previous figures were used for ampturns.
The contours have a slightly different shape since a coil must trace
a different line to produce the same field Bz at r0 with the same amp-meters
rather than the same amp-turns.
A zero contour also occurs in this case and
negative contour values correspond to negative B at the point r .
z
0
Another case of interest involves producing the same B at r0 with
z
coil pairs of equivalent energy. The self inductance of one of the loops
may be shown to have the following form (eg -
see Smythe [1]).
-4-
L = yi
where
r
(7)
a FL (rw/a)
is a characteristic radius of the PF
coil cross-section
F L = [Pn (8a/rw)}-
(7/4)
The mutual inductance between the loops may be shown to have the form (eg see Garrett [2]):
M = U
a FM 0
FM (r)
2 1/2
= (1 + ri)
(8)
[2 (K -E)
-k
2
K]
(9)
where, in this case, the complete elliptic integrals use the modulus k 0 such
that
k 2
21
Equation (7) is not very sensitive to variation in Cr /a) over the likely range
of interest, therefore, we will assume (r /a) = 0.05 for cases involving F .
The stored energy in the PF coil pair then takes the form
E = pj
where
a I2 FE (TI )
FE = FL + FM
(9 a)
-5-
Equations (1) and (9a) may be combined to produce, B
the dimensionless Bz
and energy relation.
[F (p, Ti)
BE
(10)
3
r
20E
where
= Bz
BE
(11)
Contours of constant BE are plotted in Figs. 4a and 4b.
For a given r
and
Bz requirement the contour value BE may be used with Equation (11) to find
the energy stored by a coil pair located by the contour.
If a coil set loca-
tion is altered by changing from one contour to another, then the change in
energy is proportional to the difference in the squares of the contour values.
The procedure described above may be used to create effectiveness contours for other field quantities of interest.
For example, the derivative of
axial field with respect to radius at the point r
found using (1).
in the z = o plane may be
This quantity may be cast in dimensionless form to yield a
dimensionless field derivative per amp-turn, Gat
2
at =r
(12)
G
whe re :
3Bz er:
Gt
at
(
r)
'
2
I
(13)
_0
-6-
F
[P
=
2
G
(p~1 + 1)]
-(k 2/2)
21/2
+ P)2
E (1
-k
p
1
2
2
2 + T2
- 2
-K
TI2
[+
1)2
2
+(1--P)
2 1
2
)
(1 -p)(1-p 2
2
2
fl
312]
n ] [(1 +p) +
{( -)+
(1 -P2
[K)
/
2
222
2I
+
[(1 +p)2
2
2 322
2
(1 +p)
(14)2-)(1-P2
+P
+ 2
SP)2
2E
(1.p)
~
-2E~~ ~2]~
+23/
Contours of constant Gat are plotted in Figs. 5a and 5b.
They may be
used with Eq. 13 to find the change in amp-turns for a change in location when
the point r0 and desired field derivative at the point is specified.
Equations (12),
(13) and (14) may be combined with Eq. (4) to develop
the relation between field derivative and amp-meters in the coil pair.
The re-
sult is
3
Ga am= 4
(15)
2 FGG
7T
r)3
(r
where
G
am
r
(A-m)p
0
(16)
-7-
Contours of constant G
are given in Figs. 6a and 6b.
The next case to be considered relates the derivative to the stored
energy by using Eqs.(12),
(13), (14) and (9a).
The equations which result
are:
5/2
E
T
F
FG1/2
E
where:
G = ('Bz j
E
3-r
(18)
E
00
Contours of constant GE are shown in Figs. 7a and 7b for the case of (rw/a) = 0.05
(see page 4).
A similar approach may be used to produce contours which locate coils
so as to generate the same flux through a circular loop in the z = o plane with
radius, r .
Coils on a given contour would then produce the same loop voltage
at ro if their current was ramped at the same rate.
The flux through a circular loop of radius ro in the z
o plane in Fig.
1 is given by
=
47roA
where:
A = vector potential at (ro,O)
due to two current loops
(19)
-8-
An expression for A may be found in Smythe [1] and combined with Eq.
4
Iat
-1/
P1/2 F
(19).
(p,n)
(20)
where:
=
F
at
(1 - k 2/2)K - E
(21)
(22)
P Ir
00o
Equation (22) defines a dimensionless flux per amp-turn and contours of constant Dat based on Eq. (20) are plotted in Figs. 8a and 8b.
All coils located
on a given contour with a given current will produce the same flux through a
loop through r
.
Using the same approach followed earlier, contours of constant flux per
amp-meter can be derived.
1/2
=k
=am
F4 (p, T)
23)
where:
am
p (A-m)
Contours based on Eq. (23) are given in Figs. 9a and 9b.
Similarly, contours can be related to the energy stored in the PF coil
pair.
The result is
(24)
-9-
NE
(25)
F1/2
FE
where:
were:E
(26)
yj E 0r0
Contous
onstanof $
Contours of constant
ba
E based on Eq. (25) are given in Figs. 10a and 10b.
Figure 10 was based on (r /a) = 0.05 as was Figures 4 and 7 since these three
functions depend on the self inductance of the
loops through FE and FL'
The contours of relative effectiveness in Figs. 2 to 7 allow estimation of changes in amp-turns, amp-meters or energy for a PF coil pair to be
made as it is moved from one position to another if the point of interest r
and desired Bz or (aBz/Dr) at that point are known.
On the other hand if a
point, coil location and Rz or ( Bz/@r) are selected then the required anpturns, amp-meters and energy can be found.
Figures 8, 9 and 10 can be used
in similar fashion in considering the flux generated at ro by the PF coil pair.
The procedure is straightforward and allows some insight to be gained into the
impact of location changes before detailed computations are performed.
-10-
REFERENCES
1.
W.R. Smythe, "Static and Dynamic Electricity", 3rd ed., McGraw-Hill, New
York, 1968.
2.
M.W. Garrett, "Calculation of Fields, Forces and Mutual Inductances of
Current Systems by Elliptic Integrals", Jour, of Applied Physics, Vol. 34,
No. 9, September 1963.
0
If
L
a
I -
0
0
Nb
0
'5-
h..
Nb
0
4w
0
Or
bm
g
N
ow
9
f
--
so
+
*1
Au
U
I
4
O
-40
of*
00
ii
-1
C
J0
Pb
II.
N
if
-,-----~
6.4U
U
F
U
-~
I
I
6
11-4u
B
at
=
Bz ro
I
0.
a
U
--.
0.5
.6
.19
0
1
0.7)
2
Radial CoIl Location (a/ro)
Figure 2a - Contours of constant Bat - see equation (3); A PF
coil on any given contour provides the same Bz per unit amp turn
at a specified point r-ro, z-0 in real space or at (d/ro)-B,
(a/ro)-I in this figure.
0.02
0.07
-..
0..1
0..1
5
L
0
at
Radial Colli Location (a/ro)
Figure 2b - contours of constant Bat - see equation (3); A PF
coil on any given contour provides the same Bz per unit amp turn
at a specified point rwro, z-0 in real space or at (d/ro)-O,
(a/ro)wl in this figure.
B zr02
B
am
L
p10(A-m)
C
0
J
U
0
x
02
0
±2
Radial ColI Location (a/ro)
Figure 3a - Contours of constant Bam - see equation (6); A PF
coil on any given contour provides the same Bz per unit amp-meter
at a specified point r-ro, z-0 in real space or at (d/ro)-B,
(a/ro)-I in this figure.
0.001
.m
SAW
8.
L
"o
U
.
0
0
x
4r
B r2
am
0
1
PO (A -m)
2
Radial Coll Location (a/ro)
Figure 3b - Contours of constant Bam - see equation (6); A PF
coil on any given contour provides the same Bz per unit amp-1eter
at a specified point r-ro, z-0 in real space or at (d/ro)-G,
(a/ro)-1 in this figure. i
~.
I
**~:~
i
I.
a
3
BE
zB.-77--2f
B
B.0
C
0
48J
0.06
0
w1.-t
16
0.10
.12
0 ,
0
I-
a
2
Radial Coll Location (a/ro)
Figure 4a - Contours of constant Be - see equation (11);
A PF
2
coil on any given contour provides the same (Bz)
per unit energy
at a specified point raro, z8S in real space or at (d/ro)=O,
(a/ro)-1 in this figure.
L
cy -+
C
U
E
Bz
0
W1-
x
(C
0
I
±
0
Radial CoIl
I
I
2
S
Location (a/ro)
Figure 4b - Contours of constant Be - see equation (11); A PF
2
coil on any given contour provides the same (Bz) per unit energy
at a specified point rwro, z-0 in real space or at (d/ro)-S,
(a/ro) 1 in this figure.
'V
______
G
at
roro 2
(
0
L
CY +
0
J
U
0.1
0
W4-f4
-0.3
0.0
.4
-- 0.5
x
(r
'-.1
-1.0
.3
0.1
1.0
00
1.0
0.2
\
I
I
i
2
Radial ColI
a
Location (a/ro)
Figure 5a - Contours of constant Gat - see equation (13); A PF
coil on any given contour provides the same derivative, (9Bz/Dr)rO,
per unit anp turn at a specified point r-ro, z-0 in real
space or at (d/ro)-O, (a/ro)-1 in this figure.
-0.02
-0.04
L
-.
06
-0.0
.
0-.0
0
.04
-0.10
U
.
G a.-
1.0
0..1
0.1
0
i2
a
Radial Coll Location (a/ro)
Figure 5b - Contours of constant Gat coil on any given contour provides the
per unit amp turn at a specified point
space or at (d/ro)-a, (a/ro)-1 in this
see equation (13); A PF
same derivative, (3Bz/9r)ro,
r-ro, z-0 in real
figure.
'I
0
r3
( Br
G
am
L
=-
(A-m)p
0
C
a0.
4J4
x
0
-0.0
0.
-0.0.03.
OD
0
0.01
.02
N
Radial CoIlI Location (a/ro)
Figure 6a - Contours of constant Gam - see equation (16); A PF
coil on any given contour provides the same derivative, ( Bz/Dr)ro,
per unit anp-meter at a specified point r-ro, z-0 in real
space or at (d/ro)-O, (a/ro)-1 in this figure.
-0.001
-0.002
L
-0.003
('I-f.
-0.004
-0.005
40
U
0
-0.010
-
rz
Gm
0 0
3
(A-m),p
0.002
.003
0.004
0.0100.010
0
0
Rd2
C
a
Radial Coll Location (a/,-o)
Figure 6b - contours of constant Gam - see equation (16); A PF
coil on any given contour provides the same derivative, (DBz/3r)ro,
per unit amp-1reter at a specified point r-ro,z-0 in real
space or at (d/ro)-u, (a/ro)-i in this figure.
G
L
00
CC
C
J
0
'U
-0.
-0.
x 00
0.1
0.
/.
0.1
-.
-0.2\
0.5
12
0
Radial CoIlI Location (a/ro)
Figure 7a
- Contours of constant Ge - see equation (18); A PF
coil on any given contour provides the same derivative, (DBz/Dr)rO,
per unit energy at a specified point r-ro, z-0 in real
space or at (d/ro)-8, (a/ro)-I in this figure.
-(.7
GE
=
'0
0
(U +
U
0
0
-0..
-i -
x
0.02
0.03
0.
0.04
1.01
t
0
00
0.10
0.10
i
I
±
2
Radial CoIl
Location (a/ro)
Figure 7b - Contours of constant Ge - see equation (18); A PF
coil on any given contour provides the same derivative, (3Bz/ r)r0 ,
per unit energy at a specified point r-ro, zm6 in real
space or at (d/ro)-6, (a/ro)-I in this figure.
L
0.s
-4'
111
('11+
at
ViIr
0 0
0
0
-
'- -
0
.1
0
£
2
Radial Coil Location (a/ro)
Figure Ba - Contours of constant 'at - see equation (22); A PF
coil on any given contour provides the same flux per unit ampturn through a circular loop with radius ro in the z-0 plane
in real space or a radius of (a/ro-i) in the (d/ro-O) plane in
this figure.
a
(9
0.1
0.2
0
0.
L
-W
.
0.7
0
(E3
'-'-I-
1.2
1.3
$
1.4
ai 0 t r 01.5
0
0
i
2
Radial Coil Location (a/ro)
Figure Eb - Contours of constant (at - see equation (22); A PF
coil on any given contour provides the same flux per unit ampturn through a circular loop with radius ro in the z-0 plane
in real space or a radius of (a/ro-1) in the (d/ro-O) plane in
this figure.
9
.
L
4
C
J
U
0
0.1
4 Raia
C
in ra s a Ir a rIu C ol
this figure.
o=to
(oa lo)n
(/
t(/ro-)
paei
0.005
.
5
L
(M
0..040
0..045
a:
TI1
0.05
0
0
.
2
Radial Coil Location (a/ro)
Figure 9b - Contours of constant IGam- see equation (24); A PF
coil on any given contour provides the same flux per unit ampmeter through a circular loop with radius ro in the z-0 plane
in real space or a radius of (a/ro-1) in the (d/ro-O) plane in
this figure.
a
10.1
-.
E
0E0
L
(V +
0.2
0
JJ
.3
4
0
'-i -
.5
'U
x0.6
a
0
£
Radial Col
2
S
Locat ion (a/ro)
Figure 10a - Contours of constant DE - see equation (26); A PF
coil on any given contour provides the same flux per unit energy
through a circular loop with radius ro in the z-0 plane in real
space or a radius of (a/ro-i) in the (d/ro-O) plane in this figure.
09
e.9(
/0.03
9.1
CE
0
-o
4j
0
0
-I
aE
jE r
02
Radial Coll Location (a/ro)
Figure 1b
-
Contours of constant
4'E
- see equation (26);
A PF
coil on any given contour provides the same flux per unit energy
through a circular loop with radius ro in the z-0 plane in real
space or a radius of (a/ro-) in the Cd/ro-B) plane in this figure.