Computer aided design of magnetic deflection systems
Citation for published version (APA):Osseyran, A. (1986). Computer aided design of magnetic deflection systems. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR250546
DOI:
10.6100/IR250546
Document status and date: Published: 01/01/1986
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Computer Aided Design of
Magnetic Deflection Systems
Computer Aided Design of
Magnetic Deftection Systems
Computer Aided Design of
Magnetic Deftection Systems
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven,
op gezag van de rector magnificus, prof. dr. F. N. Hooge, voor een commissie
aangewezen door het college van dekanen in het openbaar te verdedigen op dinsdag 2 september 1986 te 16.00 uur
door
Anwar Osseyran
Contents
CONTENTS
1. INTRODUCTION . . . 1 1.1. Electron beam magnetic deflection . . . 1 1.2. Recent investigations of magnetic deflection systems . 1 1.3. Purpose and contents of this thesis . . . 5 References . . . 7
2. BASIC ASPECTS OF MAGNETIC DEFLECTION 9
2.1. Introduetion . . . 9
2.2. Deflection in a uniform magneti field 9
2.3. Deflection aberrations . . . 11
2.3.1. Geometrical aberrations . . 11
2.3.1.1. Non-linearity or S-distortion 11
2.3.1.2. Raster distortion. 12
2.3.2. Distortions of the spot . . . . 13
2.3.2.1. Astigmatism . . . . 13
2.3.2.2. Curvature of the field . 14
2.3.2.3. Coma . . . 14
2.3.3. Deflection errors in colour picture tubes 15 2.3.3.1. In-line systems . . . 16 2.3.3.2. Landing errors . . . 16 2.4. Other deflection performance characteristics . 17
2.5. Small-angle deflection theory 18
2.5.1. Gaussian deflection 18
2.5.2. Third-order errors 19
2.5.3. Fifth-order errors. 20
2.6. Deflection yoke symmetry 21
2. 7. Deflection field multipoles 23
2.8. Deflection yoke types . . 24
2.9. The design of deflection units 25
2.10. Magnetic deflection field shapers 27
2.11. Magnetostatic field equations . 27
2.11.1. The magnetic vector potential 28
2.11.2. The magnetization scalar potential 28
2.11.3. The magnetization vector . . . . 29
2.11.4. The total magnetic scalar potential 30
2.11.5. The magnetic charge density 30
References . . . 33 3. TWO DIMENSIONAL STUDY OF MAGNETIC DEFLECTION 34 3 .1. Introduetion . . . 34 3.2. Field calculation using the magnetic scalar potèntial . 34 3.2.1. Field of a circular current sheet . . . 36
3.2.2. The reileetion factor of the magnetic core . . . . 36 3.2.2.1. Influence of the coil position on the reileetion
factor. . . 37 3.2.2.2. Influence of the thickness and permeability
of the core . . . 37
3.2.3. The transmission factor of the core . 38 3.2.4. Calculation of the inductance . . . 39 3.2.4.1. Inductance of a saddle coil . 40 3.2.4.2. Inductance of a toroidal coil 41 3.2.5. Stored magnetic energy . . . 42 3.2.6. Hysteresis losses . . . 43 3.3. Field calculation using the surface magnetic charge 43 3.3.1. Magnetization field of the core. . . 43 3.3.2. Magnetization field of a permeable object of arbitrary
shape . . . 44 3.3.2.1. Fourier expansion of the scalar poteptial. 44 3.3.3. Calculation of the magnetic charge distribution 45 3.3.3.1. Application: field of a deflection yokie. 46 3.3.4. Numerical considerations . . . 47 3.4. Field calculation using the vector potendal . . . 48 3.4.1. Application: influence of the coil thickness 50 3.4.1.1. Estimation of the error . . . . 51 3.5. Two dimensional electron optica! analysis 52 3.6. Appendix A. Two dimensional solution using the scalar
poten-dal . . . , 54 3.6.1. Solution . . . . , 56
3.7. Appendix B. Solution for infinite permeability . 56
3.8. Appendix C. Field of a straight line segment 57 3.9. Appendix D. Field calculation using the integral formulation 60 3.10. Appendix E. Some examples of magnetization field
calcula-tions . . . 62 References . . . 66
4. COMPUTER SIMULATION OF MAGNETIC DEFLECTION 67
4.1. Introduetion . . . 67 4.2. Computation of the magnetic field of deflection yokes . 67 4.3. Calculation of the incident field . . . 67 4.3.1. Field of a three dimensional current distribution 67 4.3.1.1. Field of a differential element of cumint . 69 4.3.1.2. Field of a straight line current segment 70
4.3.1.3. Field of an are . . . 71
4.3.1.4. Field of a current loop . . . 72 4.3.1.5. Field of a solenoid . . . 72 4.3.2. Fourier expansion of the magnetic field strength 73
Contents
4.3.2.1. The discrete Fourier transform 73 4.3.2.2. Analytica! calculation . . . . 74
4.3.2.3. Numerical considerations 75
4.3.3. Field of a rotationally symmetrie current sheet . 76 4.3 .3 .1. Derivation of the matrix relation 77 4.3.4. Fourier expansion of the scalar potential 80 4.3.5. Description of the deflection coil . . 81 4.3.5.1. Continuons wire distribution 81
4.3.5.2. Discrete wire distribution 82
4.3.6. Synthesis of the winding distribution 83 4.3.6.1. Continuons wire distribution 84
4.3.6.2. Discrete wire distribution 84
4.3.6.3. Winding connections . . . 84
4.4. Calculation of the magnetization field . . . 86 4.4.1. Fourier expansion of the core magnetization potendal 87 4.4.2. Multipole separation of the integral equation 88 4.4.3. Simulation of magnetic field shapers . . . 89 4.5. Determination of the electron beam trajectory . . . 90
4.5 .1. Ray-tracing with given start conditions and deflection currents . . . 91 4.5.2. Ray-tracing with given start conditions and landing
point . . . 92 4.5.3. Determination of the magnetic field in the picture
tube . . . . . . . 93 4.6. Determination of the deflection system performance . 94 4.6.1. The quality factor . . . 94 4.6.2. Stored magnetic energy and self-inductance . 94
4.6.3. Total pull-back 96
4.6.4. Resistance . . . 96 4.6.5. Colour purity . . . 97 4.7. Appendix A. Evaluation of the elliptic integrals lee, I"' and Ie 97 4. 7 .1. The power series expansion . . . . 98
4. 7 .2. Upward-going recurrence relation 99
4.7.3. Downward-going recurrence relation 100
4.7.4. Accuracy check . . . 101 4.8. Appendix B. Geodeticalline on arevolving surface 101 4.9. Appendix C. Evaluation of the elliptic integrals lee, J .. and Je 103 References . . . 104
5. COMPUTER OPTIMIZATION OF THE DEFLECTION SYSTEM 106 5.1. · Introduetion . . . 106
5.2. Pormulation of the optimization problem. 107
5.2.1. The deflection system model 107
5.3.
5.2.3. The design parameters . . . . 5.2.4. The design constraints . . . . 5 .2.4.1. Functional or equality constraints 5.2.4.2. Regional or inequality constraints Optimization procedure . . . .
5.3.1. The sectioning method. . . . . 5.3.2. The 'damped' sectioning method . 5.3.3. Constrained optimization
5.3.4. The augmented Lagrangian method . Referenc~s . . . .
,.
108 109 109 110 112 116 119 120 124 127 6. COMPUTER-AIDEO DESIGN AND PRACTICALREALIZA-TION OF SOME PROTOTYPES 128
6.1. Introduetion . . . 128 6.2. System configuration . . . 128 6.3. Evaluation of the deflection unit performance 131 6.4. Optimization process . . . 131 6.4.1. Start point . . . 131 6.4.2. Optimization parameters and constraints 133
6.4.3. Optimization procedure and results 134
6.5. Design time and computer costs . . . 142
6.6. Practical realization and measurements 142
6.6.1. Magnetic field measurements . 143
6.6.2. Deftection aberration measurements . 147 6. 7. Appendix. Automatic raster and convergence errors
measure-ments . . 147 References 148 7. CONCLUSIONS 149 ACKNOWLEDGEMENT . 151 Summary . . . 152 Samenvatting. . 155 Curriculum vitae 158 Dankwoord . . 159
Introduetion
1. INTRODUCTION 1.1. Electron beam magnetic deflection
DeOeetion of electron beams using magnetic fields is applied in a variety of display instruments such as television receivers, data ~isplay tubes, electron probe instruments, etc. In all of these devices, the electron beam is focused on the screen before deftection, and its intensity is modulated during the picture scanning to display optically perceptible information. The resolution of the image depends among other factors on the moving spot. Inherent in magnetic deOeetion systems are 'defocusing' and 'distortion' effects resulting in a lossof some displayed information and distortion of the image. In colour display tubes, deOeetion defects result forthermore in colour purity defects and mis-convergence of the three beams.
From the designer's point of view, it is important to relate the deOeetion aberrations to the parameters of the deOeetion system. This can be done by calculating the magnetic field generated by a given deOeetion current con-tiguration and analysing the electron-optical effects of the electron beams traversing this field. An optimization technique can then be used to minimize systematically the deOeetion aberrations within practical realization constrains and according to predefined specifications.
1.2. Recent investigations of magnetic deflection systems
In this section, we shall review brieOy recent investigation methods of mag-netic deOection. General descriptions of the progress in electron beam deOee-tion were publisbed by Hutter1), Kasper2•3) and Ritz4•5).
As stated above, the investigation of electron beam magnetic deOeetion pro-ceeds mostly in three steps: the first step is the calculation of the deOeetion field, the second is the evaluation of the deflection unit performance and the third is the systematic improvement of this performance.
Magnetic dejiection field calculation
To calculate the magnetic field produced by deOeetion yokes, a three dimen-sional boundary value problem must be solved. A straightfoward approach would be to carry out a full three dimensional solution of the field equations, but this is cumhersome and expensive. For small-angle deflection, Haantjes and Lubben 6), and Kaashoek 7), expanded the magnetic field in a power series
around the picture tube axis. Different functions descrihing the magnetic field distribution along this axis were needed. These functions were mostly deter-mined by measuring the field along the tube axis. Kaashoek 7
) and Vonk 8)
rotationally symmetrie boundary shapes, a better approach is to reduce the general three dimensional problem, by Fourier transformation, into a finite set of two dimensional ones which are computationally much more convenient to handle. Glaser 9
) showed that the magnetic deflection field is fully determined by a set of functions expressing the behaviour of the field multipoles near the symmetry axis. These 'axial multipoles' are independent solutions of the equations of the deflection field and they offer a one to one relation between the field and its sourees also expressed in a Fourier series. They can 'pe determined by direct calculations or experimentally by measuring the field 1near the axis.
I
For large deflection angles, both approaches fail to supply accurate field values because of bigher order derivatives needed in the field expressions and which are numerically extremely difficult to determine with suftkient precision using calculated or measured data. Magnetic field calculations c~n thus better be performed by using differential or integral methods in terms of either the vector or the scalar potentials 10). The advantage of a scalar potential is of course that there is only one unknown quantity at each field pohlt. Neverthe-less, it should be recognized that the magnetic scalar potential can only be defined for regions where there is no current. Noticing that the deflection coils occupy a space sufficiently small to be considered as thin rotationally sym-metrie sheets, Nomura u) introduced the concept of potential gi'J,p to express the discontinuity of the magnetic scalar potendal across the current sheets. He developed the current density in a Fourier series and calculated the potential distribution using a finite difference method. Ximen and Chen 12
) also applied the finite difference method to calculate the field harmonies of a cylindrical toroidal deflection yoke. Use of the finite element method was reported by Munro and Chu 13
). They described the calculation of deflection field har-monies for both toroidal and saddle yokes, also in the presence of rotationally symmetrie ferromagnetic materials. Magnetic saturation effects and com-plicated geometries could be easily handled.
Compared to differential methods, integral methods permit to avoid the complexity of the mesh needed for the solution of the boundary .value proh-l em. In this case, the discretization is required at the fieproh-ld sourees onproh-ly and no far-field boundary conditions have to be imposed 14
). Deflection coil field in
free-space regions is mostly calculated using the Biot-Savart law. Heijoe-mans 16
) and Fye 16•17) used equivalent magnetic charge densities on the sur-face of the iron to evaluate the magnetization of present ferromagnetic mate-rials. They obtained Fredholm integral equations of the first kind. A large decrease in computer time and memory size was also reported. Tugulea et al. 18
) and Kuroda 19) used the continuity of the normal component of the flux density to develop Fredholm integral equations of the second kind, Kasper
Introduetion
and Scherle 20) employed coaxial rings carrying a harmonie souree distri bution in order to evaluate the deflection field, and Kasper 21
) presented an interesting algorithm for the solution of the one dimensional Fredholm equations gener-ated by integral methods.
Evaluation of the dejlection unit performance
In order to evaluate the electron optica! performance of a deflection unit design, it is necessary to solve the Lorentz equations of motion of an electron in the generated magnetic field. In general, these equations can not be solved analytically. To approximate the solution, Glaser22), Haantjes and Lubben 6) and Kaashoek 7
), expanded the Lagrangian function in a power series expan-sion. The resulting equations of motion were solved by successive approxima-tions. The first approximation, defined the so-called ideal or Gaussian deftec-tion. This first correction to the landing point of the undeftected electron beam is proportional to the deflection current. The next two corrections define the so-called third- and fifth-order aberrations. Corrections of higher order are not used because of number and complexity.
This approach bas led to a better understanding of the magnetic deflection effects and it gave very valuable design information (see for instanee Heijne-mans et al. 23) and Hutter 24)). Ho wever, the numerical application of the theory was seriously hampered by the problem of speed of convergence of the power series, which is only high if we are close to the picture tube axis (small deftection angle), and which coefficients involve cumhersome integrals of Gaussian deftections and field functions along the tube axis. On the other hand, the theory is completely inadequate for the description of deftection aberrations at deftection angles larger than 45° because of the divergence of the power series expansion used. A new metbod had therefore to be used in the design of 90 or 110° picture tubes.
A common approach nowadays to solve the path equations is the use of numerical integration methods. Yokota et al. 25
) and Carpenter et al. 26) used a fourth-order Runge-Kutta method, Hauke27) prefered Hamming's modified predictor-method and Kasper 2
) reported a more accurate version of this method. For the calculation of the deftection defocusing effects, Hutter28
) described an approach in which the central ray, defining a curved optical axis, is integrated numerically and any other trajectory is considered as to describe a perturbation of the central ray. Lucchesi and Carpenter 29
) computed a large
number of electron trajectodes and represented the deflected spot by a scatter of their landing points at the screen. Later, Ritz 30) reported a technique to reduce the number of trajectodes to be calculated. In all three approaches, the influence of space charge on the deftected beam was ignored.
Finally energy storage and inductance have to be evaluated in order to esti-mate the electrical performance of a deftection yoke. Ritz31
), described the calculation of the stored magnetic energy and compared energy storage in saddle and toroidal yokes. Dasgupta 32
), calculated the inductance directly by means of flux linkages for saddle yokes only. The finite difference method was used in both cases to evaluate the magnetic field.
Improvement of the dejiection unit performance
' In early deftection systems, defects resulting in misconvergence were reduced by a separate array of auxiliary magnetic fields (by means of dynamic convergence coils for instance). Recent approaches to reduce the deftection defects have been directed towards special requirements of the picture tube and appropriate shaping of the deftection field itself. Different yoke design techniques have been developed to provide criteria for minimizipg the deftec-tion aberradeftec-tions on the entire screen.
A technique widely used in actual design is the systematic elimination of the third order aberrations (see for instanee Heijnemans et al. 23
)). Starting point is formed by a 'third order inspection' of the display system. Tijis inspeetion provides picture tube specifications concerning matters relevant for deftection (as starting conditions of the electron beams), information about the field dis-tribution needed to minimize the third order aberrations and .information. about the functions of the electric circuitry which drives the unit.
Elimina ti on of the deftection aberrations is mostly done by a proper shaping of the magnetic field. An optimal trimming of all multipoles which determine the extreme higher-order aberrations can only be achieved by th~ use of sys-tematic optimization techniques. The defect function to be minimized is mostly defined as being the sum of squares of the weighted deftection aberra-tions. Hosokawa 33
) developed firstly a procedure for systematic elimination of third-order aberrations and later he reported (Hosokawa and Morita 34
)) the use of optimization techniques. Heijnemans et al. 23
) used the method of least-squares for the field optimization. Munro 35
) used firstly a Powell algo-rithm and later he reported (Munro and Chu 36
•37)) the use of the damped least-squares method. Ohiwa 38
), presented an algorithm for under~determined non-linear simultaneous equations, which was developed for computer-aided design of electron-optica! systems.
Field optimization can be used to realize target specifications but tech-nological constraints may cause difficulties. The problem of correct field reali-zation is one of the main problems of deftection units development. From a design point of view, it is thus important to relate the deftection aberrations to the winding distribution (Ohiwa 39
Introduetion
and the generated field must be known. These relations are indispensable for the application of the field optimization results in the practical design of the deflection yoke. Carpenter et al. 26
), developed a yoke winding algorithm to
assign the wires to correct positions to synthesize a winding pattem having the desired performance characteristics. Vassell40
-42) gave a theoretica! basisfora numerical procedure to calculate the winding distribution of a defined target yoke based on the linear integral relation between field and current, and he used curve fitting techniques to study the criteria of feasibility of the winding density. Dasgupta 43
) studied the influence of coil thickness on the magnetic
field and reported 44-47) some analytica! methods to calculate the deflection field and to estimate the produced performance. He presented 48
) also
analyt-ica! calculations to demonstrate how the mechananalyt-ica! constraints of a winding process limit the harmonie content of a toroidal deflection yoke and how one can change the various harmonies beyond this limitation by actding auxiliary turns to the coil.
1.3. Purpose and content of tbis thesis
The purpose of this thesis is to present a new computer-aided design metbod of magnetic deflection systems in catbode ray tubes. The metbod is based on an accurate simuitation of the deflection unit and systematic minimization of the aberrations within practical realization constraints and according to pre-defined specifications. The deflection yoke model used in this method, allows new designs to be synthesized by the computer to produce more nearly ideal performance. These generated designs are entirely practical and all realized prototypes showed a gratifying correlation between the expected performance and measurement results.
In this chapter we have made a. brief review of some of the methods that have been successfully applied to analyse and develop deflection system signs. The emphasis was made on three aspects: calculation of magnetic de-flection fields, evaluation of the dede-flection aberrations and systematic optimi-zation of the performance.
In eh. 2 the basic aspects of magnetic deflection will be presented. Starting point is the deflection in a uniform magnetic field. After that a qualitative des-cription of deflection aberrations and deflection unit performance is given, an introduetion to the small-angle deflection theory shall be made. The symmetry properties and different types of deflection yokes shall presented and the use of field shapers shall be introduced. Finally, the basic magnetostatic equations governing the deflection field distribution will be briefly reviewed.
A two dimensional model of deflection yokes is developed in eh. 3. This model is a very useful tooi to understand and analyse differentaspectsof
mag-netic deflection problems. The deflection field will be calculated analytically by different methods. The influence on the generated magnetic field, of the deflec-tion coil thickness, and of the core shape and permeability will be analysed. The inductance of a saddle or a toroidal coil and the stored magnetic energy will be given in terms of the wire density harmonies, and a metbod to estimate the core hysteresis lossesis proposed. Finally, the simulation of field shapers is treated and some examples of two dimensional electron optical analysis are presented.
In order to minimize the deflection aberrations we have tó start by an accurate computer modelling of the deflection system. This is treated in eh. 4. An integral equation metbod basedon Biot-Savart law will be de~cribed and is used to calculate semi-analytically the Fourier analysis of the 'h1cident field' which is the field generated by the wire distribution in absence of permeable materiaL The winding distribution will be described on a rotationally sym-metrie sheet and expanded afterwards into a Fourier series. A matrix relation is then developed between the Fourier components of the currents and those of the field. Applying the integral form of Coulomb's law to the harmonie sur-face magnetic charge distribution on permeable materials, we will show how to calculate their 'magnetization field'. The use of the F ourier transformation of the wire distri bution allows us to calculate easily the inductance, in terms of the flux linkage through the turns of the coil. Three dimensional simulation of field shapers will be also treated. Using a third-order Runge-Kutta, quadrature method, we will integrate the path equations between the gun anq the screen, calculate the aberrations in different points on the screen and give, the perfor-mance of the deflection system in terms of convergence and raster errors, total pull-back, magnetic energy and colour purity.
The optimization procedure is developed in eh. 5. After the formulation of the deflection optimization problem, we will discuss briefly the use of some optimization methods, and we will describe the applied sçctioning algorithm. This iterative algorithm takes into account the design requirements and leads to new synthesized yoke designs which are entirely practical. Advantages and limitations of this metbod will be further exposed and the use of another algo-rithm based on the augmented Lagrangian metbod will be briefly presented. In order to illustrate the use of the computer-aided design metbod exposed in this thesis, we will present in eh. 6 results which have been obtained during the development of a deflection unit designed for a 110° 26" flat square picture tube. At different stages of the optimization process, practical units have been made, and the computer results were checked by conducting accu-rate measurements of the magnetic field and of the electron optical perfor-mance.
Introduetion
REPERENCES 1
) R. G. E. Hutter, Actvances in image pickup and display, Vol. 1, Kazan B, Academie Press, New York, 163 (1974).
2) E. Kasper, Magnetic electron lenses, Hawkes PW, Springer-Verlag, Berlin, 57 (1982).
3) E. Ka sper, Proc. of the third Pfefferkorn Conference, 63 (1984).
4
) E. F. Ritz, Actvances in electranies and electron physics 49, 299 (1979). 6) E. F. Ritz, Proc. of the third Pfefferkorn Conference, 97 (1984). 6) J. Haantjes and G. J. Lubben, Philips Res. Rep. 12,46-68 (1957). 7) J. Kaashoek, Philips Res. Rep. Suppl. No. 11 (1968).
8) R. Vonk, Philips Techn. Rev. 32, 61 (1971).
9) W. Glaser, Grondlagen der Elektronenoptik, Springer, Wien, 100 (1952). 10) C. F. Iselin, IEEE Trans. on Magn. MAG-17, 2168 (1981).
11) T. Nomura, Electrical Engineering in Japan 91, 147 (1971).
12
) J. Y. Ximen and W. X. Chen, Optik 57, 259 (1980).
13) E. Munro and H. C. Chu, Optik 60, 371 (1982).
14
) C. W. Trowbridge, IEEE Trans. on Magn. MAG-18, 293 (1982). 15
) W. A. L. Heijnemans, private communication, Philips Eindhoven, Nat. lab. Rep. 5119 (1975).
16) D. Fye, J. App!. Phys. 50, 17 (1979). 17
) D. Fye, IEEE Trans. on Magn. MAG-16, 1265 (1980).
18) A. Tugulea, C. Nemoianu, V. A. Maries and D. Panaite, Rev. Roum. Sci. Techn.
Electrotechn. et Energ. 24, 403 (1979). 19
) K. Kuroda, Optik 64, 125 (1983).
20) E. Kasper and W. Scherle, Optik 60, 339 (1982). 21
) E. Kasper, Optik 64, 157 (1983). 22) As ref. 9, pp. 467-486.
23) W. A. L. Heijnemans, J. A.M. Nieuwendijk and N. G. Vink, Philips Tech. Rev. 39, 154
(1980). 24
) R. G. E. Hu tter, Society forinformation and display and university of Illinois 1979 Seminar lecture Notes (SID Int. Sym.), Vol. 1, Society for information and display, Los Angeles, 42 (1979).
25) Y. Yokota, T. Li, T. Toyofuku and K. Tabatake, IEEE Trans. on Cons. Elec. CE-25, 468 (1979).
26
) M. E. Carpenter, R. A. Mombergerand T.W. Schultz, IEEE Trans. on Cons. Elec.
CE-25, 22 (1977).
27) R. Ha uk e, Theoretische Untersuchungen rotadons symmetrischer Elektronenstrahlerzeu-gungssysteme unter Berucksichtigung von Raumladung. Dissertation, Univ. of Tubingen, 1-54 (1977).
28) R. G. E. Hutter, IEEE Trans. on E!ec. Dev. ED-17 1022 (1970).
29) B. F. Lucchesi and M. E. Carpenter, IEEE Trans. on Cons. Elec. CE-25, 468 (1979).
30) E. F. Ritz, 1981 SID Int. Sym. Digest Tech. Papers, Lewis Winner, Coral Gables, Florida,
126 (1981).
31) E. F. Ritz, 1980 SID Int. Sym. Digest Tech. Papers, Lewis Winner, Coral Gables, Florida, 52 (1980).
32) B. B. Dasgupta, IEEE Trans. on Cons. Elec. CE-28, 455 (1982). 33) T. Hosokawa, Optik 56, 21 (1980).
34) T. Hosokawa, J. Vac. Sci. Techno!. B 1, 1293 (1983). 35) E. Munro, J. Vac. Sci. Techno!. 12, 1146 (1975).
36) E. Munro and H. C. Chu, J. Vac. Sci. Techno!. 19, 1053 (1981). 37) E. Munro and H. C. Chu, Optik 61, 213 (1982).
38) H. Oh i wa, J. Inst. Math. Appl. 21, 189 (1978).
8") H. Oh i wa, J. Phys. D. Appl. Phys. 10, 1437 (1977). ~ M. 0. Vassell, J. Appl. Phys. 52, 6357 (1981). 41
) M. 0. Vassell, J. Phys. D. Appl. Phys. 14, 523-573 (1981). 42) M. 0. Vassell, IEEE Trans. on Cons. Elec. CE-30, 650-656 (1984).
48
) B. B. Dasgupta, J. Appl. Phys. 54, 1626 (1983). 44
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) B. B. Dasgupta, RCA Review 45, 461 (1984).
Basic aspects of magnetic dejlection
2. BASIC ASPECTS OF MAGNETIC DEFLECTION 2.1. Introduetion
The characteristics of the magnetic deflection system determine to a con-siderable degree not only the sharpness and geometrical accuracy of the tele-vision image, but also a whole series of parameters of the teletele-vision system. The present chapter contains a brief survey of the basic aspects of magnetic deflection in catbode ray tubes (CRT). It provides also information and guide-lines necessary during the design phase of a deflection unit.
2.2. Deflection in a uniform magnetic field
A highly idealized deflection field configuration is a uniform magnetic field transverse to the axis of the undeflected electron beam and existing only over a finite volume of space.
The path of the electrous in the uniform field region is a circle and the elec-tron beam leaves this region following the tangent to the circle at its end (see fig. 2.1). The angle of deflection(/) is formed between this tangentand the
unde-Fig. 2.1. The dellection of an electron beam in a uniform magnetic field.
flected electron beam. Their point of intersecdon is called the deflection point. For increasing deflection angle, the deflection point moves toward the screen.
The deflection angle (/) (for (/)
<
90°) can be expressed as 1 )Sin(/) = I·
I BI ·
V
where I is the length of the uniform field region, Bis the magnetic flux density vector, e and m the charge an mass of an electron and U is the CRT final anode voltage, with respect to the cathode.
As the deflection angle grows, the magnetic field must be made shorter to avoid 'neck-shadowing' due to the interception ofthe beam bythe glass ofthe CRT neck. The relation between the maximal angle (/)m, the field length I and
the neck diameterDis easy to derive2)
('Pm)
Dltan - =
-2 2 • (2.2)
The magnetic flux density Bis proportional to the deflection eliment I and to the number of turns N. The inductance L of the deftection coil is proportional toN squared. Using eqs (2.1) and (2.2) we can estimate easily the stored mag-netic energy
W=!L/2
• (2.3)
Wis proportional to the following expression 8 )
D sin2 (tp) tan (
'P;)
U. (2.4)Although deflection yokes do notprovide a perfectly uniform n'lagnetic field nor a field having discrete boundaries, the former expressions can be con-sidered as a first approach to the real solution. Some practical design problems can already be highlighted:
- When choosing the deflection angle, primary considerations are tube di-mensions and deflection power. Although wider angles permit the use of wider sereens without excessive depth, deftection losses which are propor-tional to the stored magnetic energy, increase rapidly with the deflection angle.
- The deflection is not linear with the current. Picture distordon and loss of resolution will also increase with the deftection angle.
- When selecting the neck diameter, a compromise has to bemadebetween electron opties and deflection power. Beam spherical aberrations decrease as the neck diameter increases 3
), but the stored magnetic energy also in-creases.
- A simHar compromise is encountered when prescribing the final anode voltage. Although higher acceleration voltage permits a higher brightness at lower beam current and consequently a better picture resolution, the stored magnetic energy is proportional to this voltage.
Basic aspects of magnetic dejfection
2.3. Deftection aberrations
The picture on the display screen must be a faithful reproduetion of the transmitted scene. Beside the requirement for a rectangular raster without noticeable variadons in sharpness, magnetic deftection must produce a picture possessing the same linear relationships as the original scene. Coloured pic-tures must satisfy the additional requirement of a correct reproduetion of the scene colours.
The real deflection deviates from this ideal deflection and the ditierences are called aberrations. As is done in light opties, deflection aberrations are defined in a manoer such that if each aberration was present alone it would describe a characteristic effect 4
-8). We shall now give a brief qualitative description of
the deflection aberrations. 2.3.1. Geometrical aberrations
As geometrical aberrations we distinguish non-linearity and raster dis-tortion.
2.3.1.1. Non-linearity or S-distortion
Non-linearity of the defiection is due to the flatness of the screen and can be separately observed when only one deflection field is present. A pattem of originally equidistant lines appears on the screen as compressed at the centre and expanded at the edges (see fig. 2.2). This error is eliminated by simply making the currents through the deflection coils slightly S-shaped (S-correc-tion) insteadof purely saw-toothed (see fig. 2.3).
y
~r----r---+--r-r-r--r---r----r---~x
I I \ I I I I I I I I / \ / / i / current
i
Fig. 2.3. S-shape of the defiection current.
2.3.1.2. Raster distortion
Raster distortion is observed when both deflection fields are operating simul-taneously. It is due to the flatnessof the screen or to the shape of the deflection fields. Due to raster distortion, a rectangle takes a barrel- or
a
pincushion-figure (see fig. 2.4). ·
a)
b) barrel c) pincushion
Fig. 2.4. a) Pincushion raster distortion on a fiat screen, due to uniform defiection fields. b) Barrel raster distortion due to cushion-shaped fields. c) Pincushion raster distortion due to barrel-shaped fields.
Basic aspects of magnetic dejiection 2.3.2. Distortions of the spot
Because of the finite cross-section of the electron beam, magnetic defiection causes spot distortions. These are astigmatism, curvatureof the field and coma. 2.3.2.1. Astigmatism
Astigmatism occurs when the defiection field exercises a focus effect in which electrons in different axial planes focus at different axial distances. The spot on the screen of a rotationally-symmetric beam becomes elliptical (seefig. 2.5). Astigmatism may be devided into isotropie astigmatism caused by line or frame field separately and anisotropic astigmatism which can only occur if
3
Fig. 2.5. Aberration due to astigmatism and curvature of the field.
3 3
4 2
both fields are operating simultaneously. Because of anisotropic astigmatism, the axes of the elliptical spot are no longer horizontal and vertical but may take up an oblique position (see fig. 2.6).
2.3.2.2. Curvature of the field
lf the deflection field focusses points off the axis on planes different from those on the axis, the focussed image of a flat object will be on a curved sur-face whose curvature will vary with the field shape. The im~ge will appear sharp all over only where the screen has the correct curvature. This aberration is called the curvature of the deflection field (see fig. 2. 7).
Fig. 2. 7. Curvature of field. 2.3.2.3. Coma
This distortion gives a comet-like appearance to the electrop beam spot. The rays from an object point close to the axis form an image of the point in the image plane. Rays which come from points further off the axis may be brought to another focus in the image plane but the focallength around the ray now varies with the height. Their intersecdon with the screen forms a circle. lts centreis shifted with respect to the axis. The image consists then of a sharp point with a series of overlapping circles increasing in radius with their distance from the axis (see fig. 2.8). As for astigmatism, coma is divided into
Basic aspects of magnetic dejlection
isotropie coma caused by line or frame fields separately and anisotropic coma observed only when both fields are operating simultaneously.
2.3.3. Dejlection errors in colour picture tubes
Most present-day colour CRT use three electron beams, one to display each of the three primary colours: red, green and blue. The three electron guns may be arranged in a 'delta' or 'in-line' formation (see fig. 2.9). Because of the shadowing effect of a perforated metal sheet, the shadow-mask, each beam strikes only the appropriate colour area at the screen (see fig. 2.10).
y y
delta in -line
Fig. 2.9. Schematics of a 'delta gun' and an 'in-line gun' arrangements.
mask screen R G B R G
Fig. 2.10. Two-dimensional diagram showing the geometry of convergence and landing of two beams in a shadow-mask tube.
The defects of magnetic deftection in colour tubes are basically of the same type as those of a single beam. The problem of deflection of three beams may therefore be treated as a problem of minimizing 'spot distortion' by searching for deflection fields of least amount of aberrations. The concept of
astig-matism and coma errors can be made by considering the individual beams to be a part of the circumference of a hypothetical thick beam focused at the screen in the undeflected state (see fig. 2.9). Spot distortions of the individual beams are an order of magnitude smaller than the distordons of the thick beam defined in fig. 2.9 and are therefore seldom considered.
2.3.3.1. In-line systems
Haantjes an Lubben showed 9) that an 'in-line' system offers simp Ier solu-tions to solve the convergence problem. Most colour picture tubes made by Philips nowaday have three guns arranged in in-line formation in the hori-zontal plane. The guns red (R), green (G) and blue (B) are ori~nted such that the three beams conver ge in the undeflected state at the centre, of the screen. Geometrical errors are defined for the central beam G. Conv~rgence errors that the beams R, G and B have are: astigmatism (half the vector difference of the deviations GRand GB) and coma (half the vector sum GRand GB) 10
) (see also fig. 2.11).
y y'
screen
Fig. 2.11. Convergence errors in an in-line colour picture tube.
2.3.3.2. Landing errors
The three electron beams must be so deflected that they not only converge properly but they have to reach the screen through the holes of the shadow-mask at such an angle that they land properly in every part of the picture. The beam landingangles depend among other factors on the deflection field. If any beam lands, even partly, beyond the edge of the correct phosphor, a colour error will result. These landing errors can be caused by the variation of the
Basic
spot size with its location, by the 'local doming' (local deformation of the shadow-mask due to heat production) and by manufacturing tolerances. Nominallanding errorscan he reduced e.g. by spacing the shadow-mask closer to the screen at the edges than in the centre or by an appropriate shaping of the deflection field itself.
2.4. Other deflection performance characteristics
The performance of a deflection yoke is not only defined by the electron opties of the picture. To scan the screen a sawtooth current is passed through each deflection coil and the stored magnetic energy (defined then by taking for I in eq. (2.3) the peak to peak value), is transferred back and forth between the deflection yoke and the circuit. This results in energy losses in the magnetic core (due to hysteresis), in the deflection coils (due to static and dynamic resis-tance) and in the circuit itself (dissipation in the electronic components). The electron beam's deflection sensitivity is specified in terms of deflection angle for a given stored energy and at a given final anode voltage.
For vertical deflection, the power losses due to the magnetic energy ex-changes between circuit and coil are small in relation to the resistance losses because of the low vertical scanning frequency (i.e. 50 or 60 cycles/s). The reverse is true, however, for horizontal deflection, where the scanning fre-quency is very high. Stored magnetic energy and resistance losses are therefore important performance characteristics of a deflection yoke.
Distributed capacitance in the yoke windings resonates with the yoke induc-tance (ringing). At frequenties near this resonance frequency, the yoke acts no longer as a pure transducer of current to magnetic field, but a portion of this current flows in capacitive elements. The line magnetic field due to this renance produces successive bright and dark verticallines. This results in the so-called 'curtain effect' on the screen (see fig. 2.12). Distributed capacitance is affected by the number of turns in the yoke, the mechanica} configuration and by encapsulation.
Over-all positioning accuracy on the screen for random scan applications is limited by hysteresis or residual magnetism of the magnetic core. Corona dis-charge and dielectric breakdown are affected not only by the voltage distri-bution inside the windings, but also by the mechanica! contiguration and by encapsulation. Finally, axial shift of the yoke is necessary to realize colour purity. The maximal shift that can be done before beam shadowing occurs, is called the total pull-back and is a performance characteristic of the yoke design.
2.5. SmaU-angle deflection theory
The small angle deflection theory (see for instanee Sturrock 11
), Tsukker-man 12
) or Kaashoek 13)) is basedon power series expansion around the axis of the CRT of the equations of motion of an electron beam in an electromag-netic field:
dv
m -
=
eE+
e(vxB),dt (2.5)
where v is the electron velocity vector at the timet, Eis the electric field and m, e and Bas defined in eq. (2.1).
No specific assumptions about the field distribution are mad.e other than that it obeys certain symmetry properties required to produce ~ rectangular raster by a simple polarity reversal of the energizing sources.
The first order approximation of eq. (2.5) gives rise to the 'Gaussian deflec-tion'. The next higher order approximations produce solutions ·of the third order, fifth order, etc. These solutions deviate from the Gaussian deflection and the differences are called aberrations of the same order.
2.5.1. Gaussian dejiection
The first order solution of eq. (2.5) possesses all desirabie characteristics: 1) The deflection magnitude on the target is proportional to the.current
ap-plied.
2) Spot and raster are free from distortion due to deflection.
Analogous to light opties this 'ideal deflection' is called the Gaussian deflec-tion. Physically two assumptions have been made to obtain the Gaussian de-flection:
1) The axial velocity of the electron beam remains unchanged during the de-flection.
2) The magnetic field is only function of the axial position and is transverse to the electron beam.
Basic aspects of magnetic dejfection
z
undeflected beam of slope x',
Fig. 2.13. Deflection of an electron beam in the X-direction.
Consiclering only the deftection in the horizontal direction (see fig. 2.13), the Gaussian path X g(Z) is given by 14
)
z
Xg(Z)
=x.+
x~
(Z- Z.)+
V
e/<z
u) By(U) du, (2.6) 2mUwhere
x.
andx;
are the landing point and slope of the undeftected beam on the screen situated atz.,
B1(Z) is the vertical component of the deftection field at the axial position Z, Zo is the starting axial position and e, U and m as de-fined at eq. (2.1).Similar expressions can be found for the vertical deftection.
2.5.2. Third-order errors
In the third-order theory, all terms higher than third degree in the power series expansion of the equations of motion are neglected. The result differs from the Gaussian deftection and the differences are called the third-order errors. The magnitude of these depend on:
1) The deftection amplitude.
2) The beam initial conditions (eccentricity and slope). 3) The deftection field distribution.
4) The screen farm and position.
From the symmetry properties of the field, it follows that the power series expansion contains no terms that are of even degree in the currents and the slopes tagether. As in light opties, third-order errors are classified according to the powers of the slope of the undeftected beam:
- S-distortion is independent of the slope but proportional to the third power of the deflection currents.
- Curvature of the field and isotropie astigmatism are proportional to the square of the deflection current.
- Coma is proportional to the deflection current and to the square of the slope. When the beam is deflected in both directions, two additiorial third-order errors occur and are proportional to the product of the two deflection cur-rents: raster distoriion and anisotropic astigmatism.
As an example we consider the third-order deflection aberrat,ions of an in-line system. We will denote by /hand lv the horizontal and vertical deflection currents and by a the slope of the undeflected red beam. The following table expresses the third-order errors 10
).
error horizontal vertical
S-distortion AX. =aal~ A 'Ys = aol~
east-west-distortion AXEw = adhl~
north-south-distortion AYNs = ad~lv
isotropie astigmatism AXa
=
(b2 I~+
bo /~)aanisotropic astigmatism AYa
= b1/h/va
coma AXc = c1/ha 2 AYc = c2lva 2
ao, a1. a2, as, bo, bh b2, c1 and c2 define the third-order aberratiot'J. coefficients. Most of them contain the sum of two or more integrals and depend on the de-ft eetion field distribution, the gun position and the screen position and form.
2.5.3. Fijth-order errors
The next approximation of the equations of motion gives a result that differs from the third-order approximation by a series of contributions known as the fifth-order errors. In each term, the powers of the undeflected beam slope and the deflection current add up to five. Terms in which· the third or higher powers of the slope occur are discarded because of the fairly small slopes encountered in practice.
According to the ascending power of the slope, the fifth-order errors are:
1) Oeometrical distortion proportional to the fifth power of the horizontal or vertical deflection currents (S-distortion) or to their fifth-degtee products (raster distortion).
2) Astigmatism proportional to the slope and to the fourth power of the de-flection currents (isotropic) or totheir fourth-degree products (anisotropic).
Basic aspects of magnetic dejlection
3) Coma proportional to the square of the slope and to the third-degree pro-ducts of the two deftection currents.
Compared with the third-order theory, the formulation of the fifth-order error coefficients as functions of the field distribution is much more difficult to develop. Many more error coefficients are present and their expressions are far more complicated. Besides, not all fifth-order aberration coefficients are available in explicit analytical form. Especially, the fifth-order coma coeffi-cients are missing. At the time that the fifth-order theory was developed (Kaas-hoek 13)}, the delta-guns arrangement was used. The deftection fields were rather smooth and the coma error was negligible even in 110° picture tubes. For in-line arrangement, the coma error is already of importance for 90° pic-ture tubes!
Moreover, the validity of the small-angle deftection theory is limited to de-ftection augles not exceeding 45° 15) and the power series expansion of the equations of motion becomes slowly convergent for augles between 35° and 45°. The calculation of the fifth-order errors has the effect that the results be-come more accurate but only for small angles. Extrapolation of the results of third and fifth-order theories for augles above 45° is thus of dubious value.
2.6. Deflection yoke symmetry
As stated before (sec. 2.5), the deftection field must fulfil certain symmetry requirements. The nominal yoke has therefore two planes of symmetry and its axis must coincide with the picture tube axis 13
). The deftection current and the generated magnetic field can thus be sufficiently described in one quadrant (see fig. 2.14). Because of the rotational symmetry of the core and of the deftection coil form, the deftection unit is mostly described in the cylindrical coordinates system (R,O,Z), in which the Z-axis coincides with the yoke axis. By a suitable choice of the 0-origine 16
) (on the X-axis for line coils respectively Y-axis for frame coils), the Fourier expansion of the componentsof the current density vectorjon the circle (R,Z) (see fig. 2.15) will be
jR(R,O,Z) =
L
jR,n(R,Z) COS (n ()) n=l, 3, ... js(R,O,Z) =L
h,n(R,Z) sin (n 0) (2.7) n=1,3, ... jz(R,(),Z)=
L
h,n(R,Z) cos (n ()). n=l, 3,. ~. RemarkIt is evident that the generality of the results obtained in this thesis will not be affected by the choice of the 0-origin. Sinusoidal series can be always con-verted into cosinusoidal series by a rotation of n/2 and vice versa.
x
Fig. 2.14. Cross-sections of a saddle coil (left) and a toroidal coil (right). Stray fields are present oUtside the toroidal yoke.
y
y
Basic aspects of magnetic dejlection
2. 7. Deftection field multipoles
The classieal approach for the calculation of aberration coefficients consists of representing the deflection field by a power series expansion about the pic-ture tube axis 13
). For practical coil design, the direct applicability of the aber-ration coefficients obtained is rather limited. The relation between the coil parameters (winding distribution for instance) and the aberration coefficients is not obvious and difficult to establish.
Recent approaches have been directed toward the use of the Foorier trans-formation of the deflection field 17
). Each harmonie forms a multipole com-ponent of the field. The deflection aberrations become simply and explicitly related to the harmonie composition of the magnetie field and there is a direct correspondence between the field multipoles and the spatial distri bution of the field sources. Harmonie ratio's of the deflection field may be specified in terms of criteria whieh eliminate astigmatism and coma at different points on the screen.
Moreover, the representation of the deflection field in multipoles offers the following additional advantages:
1) Field multipoles represent magnetic fields which are physieally independent from each other. Every multipole can be independently altered without atfecting the other field harmonies.
2) A multipole has no effect on deflection errors of lower order than its multi-plicity. This permits to design a deflection unit systematically and in steps. The Gaussian deflection is then determined by the dipole only. Third-order errors are corrected with dipoles and hexapoles, etc. A qualitative analysis of the principal electron optieal action of every multipole is easy to de-velop 10). This is very helpful especially when the designer desires to get some insight in the physics of the deflection problem.
The magnetic scalar potential is mostly used to describe the deflection field inside the picture tube. lts multipole expansion of the circle (R,Z) is given by
(/>(R,O,Z) =
L
q>,.(R,Z)sin(nO), (2.8)niiil, 3,. ~.
q>,.(R,Z) is then the amplitude of the potential harmonienon the circle (R,Z). Since (/> satisfies Laplace's equation (see eq. 2.11), (/>,. satisfies the following equation 18
)
a2q>,.
a
2q>,. 1 aq>,. n2az2
+
aR2+
/i
aR - R2if>,.
0. (2.9)This Bessel's differential equation is solved using the metbod of separation of variables to give 19
)
where An, Bn and k are constauts and Jn(k R) is the Bessel function of order n of the first kind.
The deflection field of every multipole is fully determined by the behaviour of f/Jn(R,Z) near the Z-axis. f/Jn(R,Z) varies in terros of R as Jn(k R). There-fore higher order multipoles decrease rapidly with decreasing
R,
so that the total number of multipoles which are of importance inside the tleflection area is rather limited.2.8. DeOeetion yoke types
I
Deflection yokes are classified in three types: toroidal, saddle and hybrid.
I
Each type bas advantages and disadvantages regarding electron-optical per-formance, ease of manufacture, power consumption and compatibility with other components in the system.
A toroidal coil (see fig. 2.16) is wound around the core and generates stray fields at the outside of the deflection area. The winding packagps of a saddle
co re
Fig. 2.16. Toroidal coil wound around a ferrite core.
Basic aspects of magnetic dejlection
coil lay on a rotational symmetrie jig and are strained between different pin positions ( see fig. 2.17). They are surrounded by the co re. A hybrid yoke con-tains a mixture of mostly a toroidal frame coil and a saddle line coil.
Toroids are inherently less sensitive than saddle coils, because of the wasted energy stored in the extensive stray fields. Hysteresis losses being more than proportional to the square of the magnetic induction, arealso more important because of the higher load of the co re in the toroidal case. Further, in order to avoid local reduction of the permeability due to saturation, the co re of a toroid must be made thicker, because it is leading more flux than that of a saddle.
The analysis in practical cases, must however consider the contribution of the stray fields to the deftection. Because of the extension of their fields, espe-cially at the gun side, toroids produce more deftection than saddles at equal currents. Therefore, the toroidal current must be reduced to give the same de-ftection as the saddle. The stared energy wich is proportional to the square of the current is thus also reduced. This means that the energy starage level will depend strongly on core and winding geometry20
). Further, the average capper turn-length for toroids, is smaller than for saddle coils. Toroids have therefore lower resistance to inductance ratio, so that they are more suitable for vertical deftection.
2.9. The design of deftection units
In general deftection aberrations are affected by:
1) The picture tube parameters as gun eccentricity, screen form and position, deftection angle and picture tube profile.
2) The horizontal and vertical deftection coil positions with respect to the screen and with respect to each other.
3) The distribution of the deftection field.
Ani deal approach would be to optimize all these parameters simultaneously in order to get the best picture performance. In practice, before starting the design of a deftection unit, many of these parameters have already been de-fined. The most important way to keep deftection aberrations within allowable limits is the proper shaping of the deftection field itself. This depends not only on the wire distribution of the deftection coils, but also on the form and posi-tion of static magnets and permeable field shapers.
The relation between deftection-field sourees and deftection errors is mostly treated in two parts:
- The relation between field souree and field distribution and the relation between field distribution and deftection errors.
Third-order aberration theory can be used to provide accurate relations between deftection errors and deftection-field distribution only for modest
angles (up to 30°). Representing the magnetic field by a Fourier series permits to obtain a direct correspondance between the field and the spatial distribution of its sources.
From a design point of vue, finding an aberrationless system with the aber-ration formulas is an important subject. Starting from the analytica! expres-sions of the third-order coefficients, it is generally possible to 1 determine the
field distri bution which is needed for the desired third-order behaviour. Heijnemans for instance21
) derived conditions which have to he satisfied by
I
the deftection field in order to eliminate convergence errors. , Hosokawa 22) developed a system of equations based on a parametrie representation of the axial field distribution, in order to eliminate third-order deftection aberra-tions. Manual trimming based on the qualitative relations between the field sourees and the generated multipoles can be performed to reálize the found
I
field distribution, but technological constraints may cause diffirulties. A better approach would be to use an optimization technique in which the various design parameters for deftection yokes are refined to yield a design which minimizes an evaluation function such as the sum of the squares of the aberration coefficients. This approach is widely used in the design of electron beam optica! systems for submicron lithography23•24). ·
As stated before (sec. 2.5), the small angle deftection theory is limited to deftection at angles not exceeding 45°. Another approach that retains the elec-tron optical characteristics of the smalt angle deftection theory and that could be used for the description of spot distordon at large angles, is :based on the concept of 'central trajectory'. This is the trajectory of an electron at the centre of the beam which is assumed to be of circular cross-section before entering the deftection field. The local fields in the neighbourhood of the central trajectory will be expressed in terms of power series expansion in powers of coordinates transverse to the central trajectory. The beam distor-don effects can then be described by aberration coefficients of different orders also in the neighbourhood of the central trajectory thereby avoiding conver-genee problems. However, this approach can not be used for the determina-tion of deftecdetermina-tion aberradetermina-tions in colour picture tubes where the distance be-tween the three electron beams is not sufficiently small to use the paraxial con-cept (see fig. 2.9). An accurate calculation of the deftection field using the developed power series expansion around the central trajectory would suffer in this case from convergence problems.
The direct approach for problems involving beam deftection at large angles is the use of numerical methods. The calculation of the deftection field in-volves a three-dimensional magnetostatic problem. Differential or integral methods using scalar or vector potentials can be used. The equations of
Basic aspects of magnetic dejiection
motion of the individual electrous in these magnetostatic fields can be solved using accurate numerical integrations. The performance of the deftection system can then be exactly evaluated and with any number of coil harmonies. Optimization techniques can also be used to minimize an evaluation function defined as the sum of the squares of the calculated aberrations.
2.10. Magnetic deflection field shapers
As stated above, deflection yokes require often the use of magnetic field shapers, mostly plates of soft magnetic material of very high permeability, to correct aberrations at the screen due to coma, astigmatism and raster distor-tion. The field shapers are designed to alter the deflection field in order to minimize these aberrations.
The necessity of the use of field shapers in deflection yoke design is due to the fact that winding technology imposes often limitations on the winding pat-tem modulation needed for correcting coma, astigmatism and raster in the coil at the same time. The use of field shapers in the electron gun for instanee permits to correct coma errors by introducing a difference between the deflec-tion field that acts on the electron beam from the central gun and that acts on the beams from the outboard guns 26). Field shapers can also be placed in the coil to correct deflection aberrations 26
). Moreover, they often permit to in-fluence the electdeal performance of the deftection coil (higher indoetanee over resistance ratio), or to avoid high production spread by making strong winding pattem modolation unnecessary. Some disadvantages of field shapers are, the spread on their position, their placement costs and their power losses due to the dynamic field.
The exact geometry of field shapers needed for a given picture tube-yoke combination is mostly determined by using simplified field analysis techniques, often consictering two-dimensional models, coupled with a considerable amount of direct experimentation. The time, processing costs and potential needed for experimental trial and error methods associated with empirica! design techniques make it desirabie to develop a highly accurate computer simulation technique of field shapers.
2.11. Magnetostatic field equations
For future reference in this thesis, we shall briefly review the basic equations of magnetostatics. For a full derivation of the equation see refs 27 and 28, but also the references indicated in the text.
The basic equations of magnetostatics start with the Maxwell equations
VxH j, (2.12) where H is the magnetic field strength, j is the current density vector and B is the magnetic flux density vector.
B and H are connected by the relation
(2.13) where !Jo is the permeability of free space and /Jr is the relative permeability which is in general a function of
I
HI
in magnetizing materials.2.11.1. The magnetic vector potential
The magnetic vector potential is defined by
B=V xA. (2.14)
B satisfies automatically eq. (2.11). According to eqs (2.12) and1 (2.13), A must satisfy
1
V
x
(_!_V
x
A)
= j./JO /Jr (2.15)
To make the solution of eq. (2.15) unique, a further condition must be im-posed. Usually one uses the Coulomb gauge29-31)
V·A 0. (2.16)
In two dimensions A bas only one component A. and is normal to the plane
considered. A. is given by the differential equation
V·
(:r
VAz)
=
-tJojz,
(2.17)where j. is the appropriate component of the current.
2.11.2. The magnetization scalar potenfiat
It is convenient to express the magnetic field strength H as the sum of the field produced by the current He and the magnetization field Hm produced by the permeable materials 29•30
•32-34)
H=He+Hm. (2.18)
He obeys the Maxwell eqs (2.11) and (2.12); i.e.
V·He 0 (2.19)
and
