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A method for the design of axially symmetric magnetic fields for image tube systems
NUCLEAR
INSTRUMENTS
AND
METHODS
17 (1962)
132-.136;
NORTH-HOLLAND
PUBLISHING
CO.
A METHOD FOR THE DESIGN OF AXIALLY SYMMETRIC MAGNETIC FIELDS FOR IMAGE TUBE SYSTEMS B. ZACIIAROV
C E R N , Geneva, Switzerland
R e c e i v e d 25 J u n e 1962
A n u m e r i c a l m e t h o d is described for c o m p u t i n g t h e c u r r e n t d e n s i t y d i s t r i b u t i o n in a m u l t i s e c t i o n coil, to r e p r o d u c e a given m a g n e t i c field possessing a x i a l s y m m e t r y . E x a m p l e s are given
of calculations for h o m o g e n e o u s a n d i n h o m o g e n e o u s fields of t h e order 10-3-10 -1 W b / m 2.
1. Introduction
2. Formation of the Magnetic Field It is only necessary, for systems of axial symmetry, to reproduce the required field distribution B(O, z) on the axis since the field components off the axis depend upon and can be expressed simply in series expansions of B(0, z) and its derivatives. Thus, if z is the axis of symmetry,
In nuclear physics and, in particular, in image intensifier systems employing magnetic focusing, it is frequently I - 3) necessary to produce axially symmetric fields with magnitudes of the order 10-2 __. 10-1 Wb/m 2. The magnetic fields required in these applications may be produced by permanent magnets, iron-clad coils or air coils. However, in the following, we shall consider only the last method. There have been m a n y approaches to the problem of designing air coils to reproduce a given field [see, for example Zworykin*)]. In particular, a method exists 5) for determining the distribution of windings on a cylindrical surface which will give rise to a given axial field distribution. However these methods are often of little use in practice since they yield ideal solutions and one then has to make approximations in the construction of coils leading, frequently, to impermissible errors in the fields produced. There is need therefore for a method which can yield a design for air coils which is practically applicable, and it is the purpose of this paper to indicate one method which has been found useful in treating certain classes of problem. t) B. Z a c h a r o v a n d S. D o w d e n , A d v a n c e s in E l e c t r o n i c s 12 (1960) 31. 2) W. L. Wilcock, D. L. E m b e r s o n a n d B. W e e k l e y , I . R . E . T r a n s , Nuc. Sc. NS-7 (1960) 126. 3) E. K. Zavoiskii et al., Dokl. A k a d . N a u k , U S S R 100 (1955) 241. 4) E l e c t r o n Optics a n d t h e E l e c t r o n M i c r o s c o p e - - Z w o r y k i n et al. (Wiley, N e w York, 1946). '~) W. Glaser, Z. P h y s i k 118 (1941) 264.
B~(r,z) = ~ (- 1)"B~(2")(O'z) (2) o=o
(n!):
2" '
1
17'
(1)
where B(")(O, z) is the nth-derivative of the field on the axis with respect to z. We must, initially, make some assumptions as to the form of the air coils we require to produce the desired field. Clearly they should be axially symmetric and, for practical reasons, we may take the coil to consist of a set of m coaxial annular sections. The size of any individual section m a y be different from the others, in general, and for a given section we can choose the inner and outer radii r. and R. and the first and second boundaries z, and Z. to conform with any desired geometry. The problem then is to determine what current density distribution is required in the coil to reproduce B(O, z) to a given accuracy. Now the magnetic induction on the axis of symmetry of an annular coil section (fig. 1) at some 132
AXIALLY SYMMETRIC MAGNETIC
point P(O, z) can be derived in terms of the current density I(n) in the coil. Thus, we m a y write simply P2 d~ dP z. [p2 + ~ -- ~)2]~" ".
(.)B(0, z) = T
,.
r.
(2)
FIELDS
FOR IMAGE TUBE SYSTEMS
133
the axial region over which we require the field fit. We then obtain a set of m simultaneous linear equations which can be solved to yield the values of I(n). The above method has been applied in several specific cases, two of which are given here as examples.
3. The Homogeneous Field For many photoelectronic devices and especially for most magnetically focused image tubes, it is ZR z n desirable to operate with a magnetic field which is as nearly homogeneous as possible. In the latter, 0 P iq z) the homogeneity is important if good geometrical rn --~ resolution 6) and freedom from distortions are to be obtained. Here the magnetic field is usually produced b y a solenoid of air coils, the over all Rn length and diameter of which is typically twice Fig. 1. n - t h a n n u l a r coil section. that of the cylindrical volume over which the iin M.K.S. units, where Po is the permeability of free uniform field is required; in this case, from the space. From eq. (2) it is elementary to show that point of view of the image tube, the solenoid is regarded as of infinite extent. (.)B(0, z) In a typical application, a solenoid of length to diameter ratio of about 4:1 was built for a magnetically focused image tube, and the length and 2 + + _ / the internal diameter of the solenoid were greater than that of the tube b y a factor more than 1.5. The ampere-turns distribution was uniform through(3) - (z - z.) log ~ - [(r,I 2 + (z - z.)2j * ] out the solenoid and the field on the axis of symmetry was measured with a temperature comso that, since we hay, made some definite assump- pensated Hall-plate. The field was found to be tions as to the form of the section-i.e, we have noticeably inhomogeneous, the magnitude of the axial field varying b y as much as + 7 per cent chosen r., R., z. and ,*.-we m a y write over the length of the image tube (fig. 2). The in(,)B(O, z) : ½t~oI(n) . (,)F(z) . (4) homogeneity of the field at the input end of the Thus the field at a ly point P(0, z) on the axis, tube, at the region of the first photocathode, was due to the complete coil of m-sections, is given particularly serious, since it is just there, where the ,fimply b y a superpo:, ition of terms of the type in photoelectrons have only low energies, that the maximum uniformity is demanded for good image eq. (4), so that quality. B(O, z) = -2-1~o ~ I(n) . (.)F(z) . (5) The solenoid (58 cm long) consisted of nineteen 1 annular sections, and it was decided to apply the vchich, for a given value of the parameter z, is a method outlined earlier to see if, b y using a nonlinear equation in I(n). Hence, in order to find the uniform current density distibution in these distribution I(n), to reproduce any given field sections, the field uniformity could be significantly B(0, z) we need to solve eq. (5). We can do this 6) B. Zacharov, Advances in Electronics (1962) to be readily b y choosing m different values for z within published.
Z
134
B. Z A C H A R O V
improved without having to change the design of the coils in any way. Nineteen equispaced points on the axis were chosen along a region of 43,2 cm at which to fit the uniform fieldt, and the results Solenoid of 19 ,'~ctions~.%
t~////////////////A J 'm~¢ tub¢
B(O, z) measured. The results, shown in fig. 3, were regarded as satisfactory since the departure from uniformity over the required region (about 1%) was comparable to the accuracy of the field measurement and the inaccuracy expected as a result of misalignment of the individual coil sections. Solenoid of 19 s t c t i o n s - - ~
I
k\\\\\\~\\\\\\\\\"
V/////////////A
[
~150
Ima9tt"
[
['..\\\\\\ \\\\\\\\\"
/>
200
q q
200
g ~so ~1oo
s~
.u 0
50
O
/
#.
\ 20
I 40
Fig.
50
I
I
60
80
distance
2. Field w i t h uniform c u r r e n t
IOO
/
~_
(cm) I 20
0
d e n s i t y distribution.
I 40
i 60
80 distance
are shown in fig. 4; a Ferranti Mercury computer was used, the time taken being approximately one minute for the complete calculation, The field on the axis of symmetry produced by the new current distribution was then calculated at 25 points on the axis other than those where the fit was made. The predicted field was then uniform to better than 0.1 per cent over a region of 44 cm. The new currents demanded in the coil sections were all of the same sign and the maximum current density was still within the limit set by the permissible temperature rise in the coil. However forced air or water cooling could have been used if a higher field had been required. The new current distribution was used and the resulting axial field
Fig. 3. Field with non-uniform c u r r e n t density distribution.
l
4o
N
V
~3o
J
~o 20 ~ ~ ~o u o
t This region was intentionally chosen to be slightly a s y m m e t r i c a l a b o u t t h e solenoid m e d i a n plane.
(cm}
2
4
6
8
10
12
14
16
le,
Fig. 4. C o m p u t e d c u r r e n t d e n s i t y distribution.
AXIALLY
SYMMETRIC
MAGNETIC
4. The Inhomogeneous Field The method described has also been applied to the design of inhomogeneous fields required for the focussing of a d.emagnifying image tubeT). In this
FIELDS
FOR IMAGE
TUBE
SYSTEMS
135
In one alternative, the coil system to produce this field was chosen to consist of 40 annular sections of successively decreasing radius (fig. 5) and 40 points were taken on the axis for the field fit in
r
~----Z 0
23
-y 2o
Fig. 5, C u r r e n t d e n s i t y d i s t r i b u t i o n ( a r b i t r a r y units) for the"conical" field B(O,z) = const./z 2.
application an axially symmetric converging magnetic field was needed over a region of about 20 cm, with magnetic induction varying along the axis between 0.01 and 0.25 W b / m 2. Here, in one particlflar case, it was necessary to produce a "conical" magnetic field over a limited domain. B y choosing tile co-ordinate origin suitably, such a field could be described 7) b y const. B(O, z) - - (6) g2
the region where the "conical" field was required. A current density distribution was obtained, shown in fig. 5, which contained terms of both signs. The field generated b y such a coil system with the derived current density distribution was calculated at 100 points over 25 cm in the required region along the axis, when an agreement with eq. (6) was obtained to better than 0.01 per cent. An unfortunate consequence of the requirement 7) B. Z a c h a r o v , R e v . Sci. I n s t r . 32 (1961) 1392.
136
B. Z A C H A R O V
of different signs of currents in different coil sections is that the absolute magnitude of current density was too high, for the magnitude of field demanded, for continuous operation of the coil system without forced cooling. In nuclear physics applications, however, where it is required to use the image tube to view events within a short period only (typically 100 msec, every 3 sec), it is possible to pulse the coil system without exceeding the practical dissipation limit. It should be noted that the conditions which must be satisfied in order to produce a "conical" field to the accuracy demanded are particularly stringent,
and it is this which results in a current density distribution with so many oscillating terms. Other convergent fields have also been considered, more like the fringe fields of short solenoids, where there is no longer spherical symmetry over the region of the demagnifying tube, and application of the method described has yielded practical solutions in which the coils can operate continuously without forced cooling.
Acknowledgement The author would like to thank M. Rousseau for the measurements described in the present paper.
INSTRUMENTS
AND
METHODS
17 (1962)
132-.136;
NORTH-HOLLAND
PUBLISHING
CO.
A METHOD FOR THE DESIGN OF AXIALLY SYMMETRIC MAGNETIC FIELDS FOR IMAGE TUBE SYSTEMS B. ZACIIAROV
C E R N , Geneva, Switzerland
R e c e i v e d 25 J u n e 1962
A n u m e r i c a l m e t h o d is described for c o m p u t i n g t h e c u r r e n t d e n s i t y d i s t r i b u t i o n in a m u l t i s e c t i o n coil, to r e p r o d u c e a given m a g n e t i c field possessing a x i a l s y m m e t r y . E x a m p l e s are given
of calculations for h o m o g e n e o u s a n d i n h o m o g e n e o u s fields of t h e order 10-3-10 -1 W b / m 2.
1. Introduction
2. Formation of the Magnetic Field It is only necessary, for systems of axial symmetry, to reproduce the required field distribution B(O, z) on the axis since the field components off the axis depend upon and can be expressed simply in series expansions of B(0, z) and its derivatives. Thus, if z is the axis of symmetry,
In nuclear physics and, in particular, in image intensifier systems employing magnetic focusing, it is frequently I - 3) necessary to produce axially symmetric fields with magnitudes of the order 10-2 __. 10-1 Wb/m 2. The magnetic fields required in these applications may be produced by permanent magnets, iron-clad coils or air coils. However, in the following, we shall consider only the last method. There have been m a n y approaches to the problem of designing air coils to reproduce a given field [see, for example Zworykin*)]. In particular, a method exists 5) for determining the distribution of windings on a cylindrical surface which will give rise to a given axial field distribution. However these methods are often of little use in practice since they yield ideal solutions and one then has to make approximations in the construction of coils leading, frequently, to impermissible errors in the fields produced. There is need therefore for a method which can yield a design for air coils which is practically applicable, and it is the purpose of this paper to indicate one method which has been found useful in treating certain classes of problem. t) B. Z a c h a r o v a n d S. D o w d e n , A d v a n c e s in E l e c t r o n i c s 12 (1960) 31. 2) W. L. Wilcock, D. L. E m b e r s o n a n d B. W e e k l e y , I . R . E . T r a n s , Nuc. Sc. NS-7 (1960) 126. 3) E. K. Zavoiskii et al., Dokl. A k a d . N a u k , U S S R 100 (1955) 241. 4) E l e c t r o n Optics a n d t h e E l e c t r o n M i c r o s c o p e - - Z w o r y k i n et al. (Wiley, N e w York, 1946). '~) W. Glaser, Z. P h y s i k 118 (1941) 264.
B~(r,z) = ~ (- 1)"B~(2")(O'z) (2) o=o
(n!):
2" '
1
17'
(1)
where B(")(O, z) is the nth-derivative of the field on the axis with respect to z. We must, initially, make some assumptions as to the form of the air coils we require to produce the desired field. Clearly they should be axially symmetric and, for practical reasons, we may take the coil to consist of a set of m coaxial annular sections. The size of any individual section m a y be different from the others, in general, and for a given section we can choose the inner and outer radii r. and R. and the first and second boundaries z, and Z. to conform with any desired geometry. The problem then is to determine what current density distribution is required in the coil to reproduce B(O, z) to a given accuracy. Now the magnetic induction on the axis of symmetry of an annular coil section (fig. 1) at some 132
AXIALLY SYMMETRIC MAGNETIC
point P(O, z) can be derived in terms of the current density I(n) in the coil. Thus, we m a y write simply P2 d~ dP z. [p2 + ~ -- ~)2]~" ".
(.)B(0, z) = T
,.
r.
(2)
FIELDS
FOR IMAGE TUBE SYSTEMS
133
the axial region over which we require the field fit. We then obtain a set of m simultaneous linear equations which can be solved to yield the values of I(n). The above method has been applied in several specific cases, two of which are given here as examples.
3. The Homogeneous Field For many photoelectronic devices and especially for most magnetically focused image tubes, it is ZR z n desirable to operate with a magnetic field which is as nearly homogeneous as possible. In the latter, 0 P iq z) the homogeneity is important if good geometrical rn --~ resolution 6) and freedom from distortions are to be obtained. Here the magnetic field is usually produced b y a solenoid of air coils, the over all Rn length and diameter of which is typically twice Fig. 1. n - t h a n n u l a r coil section. that of the cylindrical volume over which the iin M.K.S. units, where Po is the permeability of free uniform field is required; in this case, from the space. From eq. (2) it is elementary to show that point of view of the image tube, the solenoid is regarded as of infinite extent. (.)B(0, z) In a typical application, a solenoid of length to diameter ratio of about 4:1 was built for a magnetically focused image tube, and the length and 2 + + _ / the internal diameter of the solenoid were greater than that of the tube b y a factor more than 1.5. The ampere-turns distribution was uniform through(3) - (z - z.) log ~ - [(r,I 2 + (z - z.)2j * ] out the solenoid and the field on the axis of symmetry was measured with a temperature comso that, since we hay, made some definite assump- pensated Hall-plate. The field was found to be tions as to the form of the section-i.e, we have noticeably inhomogeneous, the magnitude of the axial field varying b y as much as + 7 per cent chosen r., R., z. and ,*.-we m a y write over the length of the image tube (fig. 2). The in(,)B(O, z) : ½t~oI(n) . (,)F(z) . (4) homogeneity of the field at the input end of the Thus the field at a ly point P(0, z) on the axis, tube, at the region of the first photocathode, was due to the complete coil of m-sections, is given particularly serious, since it is just there, where the ,fimply b y a superpo:, ition of terms of the type in photoelectrons have only low energies, that the maximum uniformity is demanded for good image eq. (4), so that quality. B(O, z) = -2-1~o ~ I(n) . (.)F(z) . (5) The solenoid (58 cm long) consisted of nineteen 1 annular sections, and it was decided to apply the vchich, for a given value of the parameter z, is a method outlined earlier to see if, b y using a nonlinear equation in I(n). Hence, in order to find the uniform current density distibution in these distribution I(n), to reproduce any given field sections, the field uniformity could be significantly B(0, z) we need to solve eq. (5). We can do this 6) B. Zacharov, Advances in Electronics (1962) to be readily b y choosing m different values for z within published.
Z
134
B. Z A C H A R O V
improved without having to change the design of the coils in any way. Nineteen equispaced points on the axis were chosen along a region of 43,2 cm at which to fit the uniform fieldt, and the results Solenoid of 19 ,'~ctions~.%
t~////////////////A J 'm~¢ tub¢
B(O, z) measured. The results, shown in fig. 3, were regarded as satisfactory since the departure from uniformity over the required region (about 1%) was comparable to the accuracy of the field measurement and the inaccuracy expected as a result of misalignment of the individual coil sections. Solenoid of 19 s t c t i o n s - - ~
I
k\\\\\\~\\\\\\\\\"
V/////////////A
[
~150
Ima9tt"
[
['..\\\\\\ \\\\\\\\\"
/>
200
q q
200
g ~so ~1oo
s~
.u 0
50
O
/
#.
\ 20
I 40
Fig.
50
I
I
60
80
distance
2. Field w i t h uniform c u r r e n t
IOO
/
~_
(cm) I 20
0
d e n s i t y distribution.
I 40
i 60
80 distance
are shown in fig. 4; a Ferranti Mercury computer was used, the time taken being approximately one minute for the complete calculation, The field on the axis of symmetry produced by the new current distribution was then calculated at 25 points on the axis other than those where the fit was made. The predicted field was then uniform to better than 0.1 per cent over a region of 44 cm. The new currents demanded in the coil sections were all of the same sign and the maximum current density was still within the limit set by the permissible temperature rise in the coil. However forced air or water cooling could have been used if a higher field had been required. The new current distribution was used and the resulting axial field
Fig. 3. Field with non-uniform c u r r e n t density distribution.
l
4o
N
V
~3o
J
~o 20 ~ ~ ~o u o
t This region was intentionally chosen to be slightly a s y m m e t r i c a l a b o u t t h e solenoid m e d i a n plane.
(cm}
2
4
6
8
10
12
14
16
le,
Fig. 4. C o m p u t e d c u r r e n t d e n s i t y distribution.
AXIALLY
SYMMETRIC
MAGNETIC
4. The Inhomogeneous Field The method described has also been applied to the design of inhomogeneous fields required for the focussing of a d.emagnifying image tubeT). In this
FIELDS
FOR IMAGE
TUBE
SYSTEMS
135
In one alternative, the coil system to produce this field was chosen to consist of 40 annular sections of successively decreasing radius (fig. 5) and 40 points were taken on the axis for the field fit in
r
~----Z 0
23
-y 2o
Fig. 5, C u r r e n t d e n s i t y d i s t r i b u t i o n ( a r b i t r a r y units) for the"conical" field B(O,z) = const./z 2.
application an axially symmetric converging magnetic field was needed over a region of about 20 cm, with magnetic induction varying along the axis between 0.01 and 0.25 W b / m 2. Here, in one particlflar case, it was necessary to produce a "conical" magnetic field over a limited domain. B y choosing tile co-ordinate origin suitably, such a field could be described 7) b y const. B(O, z) - - (6) g2
the region where the "conical" field was required. A current density distribution was obtained, shown in fig. 5, which contained terms of both signs. The field generated b y such a coil system with the derived current density distribution was calculated at 100 points over 25 cm in the required region along the axis, when an agreement with eq. (6) was obtained to better than 0.01 per cent. An unfortunate consequence of the requirement 7) B. Z a c h a r o v , R e v . Sci. I n s t r . 32 (1961) 1392.
136
B. Z A C H A R O V
of different signs of currents in different coil sections is that the absolute magnitude of current density was too high, for the magnitude of field demanded, for continuous operation of the coil system without forced cooling. In nuclear physics applications, however, where it is required to use the image tube to view events within a short period only (typically 100 msec, every 3 sec), it is possible to pulse the coil system without exceeding the practical dissipation limit. It should be noted that the conditions which must be satisfied in order to produce a "conical" field to the accuracy demanded are particularly stringent,
and it is this which results in a current density distribution with so many oscillating terms. Other convergent fields have also been considered, more like the fringe fields of short solenoids, where there is no longer spherical symmetry over the region of the demagnifying tube, and application of the method described has yielded practical solutions in which the coils can operate continuously without forced cooling.
Acknowledgement The author would like to thank M. Rousseau for the measurements described in the present paper.