BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS WITH ROTATIONAL SYMMETRY: SOME IMPORTANT PROPERTIES AND METHOD OF CALCULATION

Transcription

1 R812 Philips Res. Repts 27, , 1972 BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS WITH ROTATIONAL SYMMETRY: SOME IMPORTANT PROPERTIES AND METHOD OF CALCULATION Abstract 1. Introduction by J. HASKER For electron-gun design in television-display tubes, camera tubes and oscilloscope tubes, it is essential to know the properties of the beamcurrent characteristic and to be able to calculate beam current and cathode loading as a function of gun geometry and electrode potentials. Three different definitions of cut-off are considered and the relationship between the corresponding cut-off voltages is discussed. This gives rise to a recommendation for the determination of the gamma of the beamcurrent characteristic. The latter may be of interest for television-display tubes in particular. It is made clear that a simple approximation is needed to obtain quick insight in the dependence of beam current and cathode loading on gun geometry and electrode potentials. To arrive at it, the shortcomings involved in some known simple methods are discussed. The result is that scaling properties and the experimental behaviour of a reference gun must be used. Then, for a great variety of guns the beam current, cathode loading and current-density distribution at the cathode can easily be calculated with sufficient accuracy if only the cathode properties (T andjs) and the Laplace field-strength distribution at the cathode are known. After the last-named distributions have been : calculated, the calculation of the above-mentioned quantities for twelve different bias situations of a gun takes only about one minute on an EL-X8 computer. The results are compared with experiments and the relationship between maximum cathode loading and mean cathode loading is discussed. The appendix is a guide to the computer calculation and contains all necessary information. For electron-gun design it is essential to know the properties of the beamcurrent characteristic and to be-able to calculate this characteristic and the cathode loading for space-charge-limited operation as a function of gun geometry and electrode potentials. In the present paper Wehnelt-type electron guns, as employed in television-display tubes, oscilloscope tubes and camera tubes will be considered. For these guns the Laplace field-strength distribution at the cathode is "nearly" parabolic. In principle, to calculate the beam-current characteristic, Poisson's equation must be solved for a system with rotational symmetry. This computer calc~lation has been carried out taking into account the transverse velocities at emission 1). However, it has been shown that the axial initial velocities are essential to the beam-current calculation 2). Introduetion of the axial velocities, which gives rise to a space-charge minimum in front of the cathode; is very corn-

2 514 J.HASKER plicated and would require much computer time. Therefore, to obtain any quick insight into the dependence of beam-current characteristic and cathode loading on gun geometry and electrode potentials, a simple approximation of the problem is needed. The idea, already introduced by Ploke 3), is to use the Laplace field-strength distribution at the cathode - which can easily be determined from analogue measurements 4) or with the aid of a computer 1) - as a starting point for the calculation. However, this datum is not sufficient 2.5). In addition, scaling properties and measured beam currents of a reference gun must be used. Then, as will be shown, the beam-current characteristic and the cathode loading of any practical gun can be simply calculated. For the reader's convenience and as an introduetion to the method for calculating the beam current and cathode loading described in sec. 3, some important properties of the beam-current characteristic will be summarized and discussed in sec. 2. The results ofthe calculations, which are compared with experiments, will be discussed in sec. 4. The appendix is a guide to the computer calculation. It contains a quick calculation method and the necessary data. 2. Important properties of the beam-current characteristic Figures la and lb show a schematic representation of a triode gun and a tetrode gun of the type examined, respectively. The gun can be driven by a positive voltage Vc on the cathode (cathode drive, Vg = 0) or by a negative voltage - Vg on the grid (grid drive, Vc = 0).. Figure 2 shows a schematic representation ofthe beam-current characteristic. Three different cut-off voltages - i.e. Vo, Ve and Vs - which are defined in sec. 2.1, are shown in this figure. The maximum beam current, fm, is obtained Grid,-Vg 1~ om'1 Accelerating electrode, '0J i1l}r,)5mm_._._ Cathode, Vc:, : c~:. a} ~ Accelerating Grid,-lIg Anode,I{, electrode, '0J ~I ~ ç ~~ rrs. _l_::bl+l~._._._._.- Cathode, Vc i :: I' ta :.~~: cg" : I, b}.1,,1_ tg ga =4mm Fig. 1. Schematic representations of a triode gun and a tetrode gun.

3 BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS 515 If(~)- ~~----~~~~--~ Retarding- Space-charge field region region Fig. 2. Schematic representation of the beam-current characteristic of an electron gun. at zero bias. Obviously, Im has the same value for grid drive and cathode drive. The drive factor, k, is defined as the ratio of Im to the three-halves power of the cut-off voltage. The drive voltage Vd is the potential difference between cut-off and Vc or Vg, while the reduced drive Vd is the ratio of Vd to the cuto off voltage. It is noted that - in contrast to a planar diode - the~e is no sharp transition between the retarding-field region and the space-charge region of the beam-current characteristic (see also sec. 2.3). Sections 2.1 to 2.3 deal with cut-off and with some problems involved in the known approximate methods for calculating the beam.current *). In sec. 2.4 a simple approximate method for calculating the maximum beam current is briefly described **). As an example, the drive factor k, = Im/Vs3/2 will be calculated for the anti-moiré gun of the shadow-mask color display tube in sec Cut-off.. In the field of television display particularly, the situation with respect to cut-off is somewhat confusing because of different definitions. These definitions and the relationship between the different cut-offvoltages are considered below: The geometrical cut-off voltage, Vo, is defined as that value of Vc or Vg at which the Laplace field strength at the centre of the cathode is zero. Because of the initial velocities of the electrons emitted by the cathode, the beam current is not zero at geometrical cut-off. Nevertheless Vo can be determined accurately *) For more-detailed information, see ref. 2. **) For more-detailed information, see ref. 5.

4 516 J. HASKER from measured beam-current characteristics 2). This experimental value is denoted by Vo ex' On the other hand, Vo can be calculated from simple potential measurements on a resistance-network analogue or by solution of Laplace's equation with the aid of a computer. A possible deviation of Vo ex with respeet to Vo provides information on deviations in the dimensions of the actual gun with respect to the nominal gun. Vo can be written as where Va and VII are the potentials and Dl and D 2 are the "durchgriffs" of anode and final anode, respectively. Just like Vo, both Dl and D 2 can be determined from experimental beam-current characteristics and calculated. The agreement between experimental and calculated values has been found to be very good 2). It is well known that different values for Vo are found for cathode drive and grid drive. These values are denoted by Voc and VOg, respectively. Writing it can be shown that (1) (2) (3) The spot cut-off voltage, Vs, can be easily determined from experiments as follows: at a beam current of a few [LAthe beam is focussed in ~ static spot on the screen in a "dark" room; next Vc (or Vg) is increased up to the value at which there is just no visible light output. As this corresponds to a very low beam current of the order of A 2), the difference Vs - Vo is relatively large. The practical values 60 V and 50 V for Vs and Vo respectively, show its order of magnitude. Due to this large difference the depth of the Laplace minimum in front of the centre of the cathode at Vc (or Vg) = Vs is large with respect to ktfe (T is the cathode temperature). This means that at spot cut-off the influence of space charge on the potential field in front of the cathode is negligible. Hence, the beam current at spot cut-off is determined by the Laplace field in front of the cathode. With the aid of the experimental values of Vo and " Vs and analogue measurements, it has been shown that at constant VII the depth of the Laplace minimum,in front of the centre of the cathode at spot cut-off is nearly independent of Va. Using this property, it can be shown that - except for the unusual case that Vo :- the relationship between Vs and Vo is to a good approximation given by 2) where c is a co~siant dependent on gun geometry and cathode properties. Dif- (4)

5 BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS 517 ferent values for Vs and c are found for cathode drive and grid drive; It can be shown that 2) (5).. Moreover - as has been shown for a tetrode gun provided with an L-cathode at 1350 K - Dl can be determined simply by measuring both V Sg and V sc as a function of Va at constant VII 2). It is noted that for this gun with V h = 18 kv the depth of the Laplace minimum in front of the centre of the cathode at spot cut-offis about 1 55 V. As this quantity will be nearly the same for other geometries it can be used to calculate Vs. The extrapolated cut-off voltage, Ve, is determined from experiments as follows: 1 '4 is plotted versus Vc (or Vg) for beam currents 1 which are small with respect to 1m (see also sec. 2.2); Ve is defined as the value of Vc or Vg found from extrapolation of the obtained approximately linear relationship towards the Vc or Vg axis, respectively (see fig. 4). This procedure originates from the Ploke model for calculating the beam current which will be discussed in sec The relationship between Vc and Vo is one of the items of secs 2.2, and 3. The raster cut-off voltage, V r (not shown in fig. 2), refers to just no visible light output of the scanned raster. If - as is convenient for practical use in television display - the beam current is written as (6) \ where Cl is a constant and y increases with increasing Vd' the raster cut-off voltage must be used in the determination of the drive voltage Vd 6). However, because Vc nearly coincides with Vr for "normal" ambient light and because Ve can be determined objectively from some simple beam-current measurements, the use of Vc instead of Vr is recommended. It is noted that this quantity makes sense also for oscilloscope tubes and camera tubes. Because of the use of diaphragms, 1is the cathode current in these tubes Ploke's model for calculating the beam current In Ploke's model the following assumptions are made. Assumption 1. The initial velocities of the electrons and the fact that the saturation current density of the cathode has any finite valueare neglected. Under these conditions and for space-charge-limited operation the current density j in a planar diode is given by Child's law: (_E)3/2 j=. d l / 2 (7)

6 518 I.HASKER where d is the distance between cathode and anode and E the Laplace field strength at the cathode (jin A/cm 2, d in cm and E in V/cm). Thus, the current density for space-charge-limited operation is determined by the Laplace field strength at the cathode. Assumption 2. Just as in a planar diode, the space-charge-limited beam current in an electron gun is determined by the Laplace field at the cathode. The validity of this assumption will be discussed in sec Because of the rotational symmetry, the Laplace field-strength distribution at the cathode can be represented by E(r) = E(0)f(r 2 ), (8) where f(o) = 1. Assumption 3. To calculate the beam current the actual gun can be replaced by a system of concentric planar diodes which are mutually independent. All diodes have the same diode distance de. Assumption 4. The Laplace field strength at the cathode of a ring with radius r must be equal to E(r) for an anode potential which is given by where, because of the absence of initial velocities, the geometrical cut-off voltage Vo is the reference point for the drive voltage Vd' Hence, in virtue of eqs (8) and (9), where Eo is the Laplace field strength at the centre of the cathode at zero bias. Obviously, according to these assumptions, the beam current is given by (9) (10) RO [_ E(0'f(r2]3/2!'= 2' J 6 J 2 in: r dr,.. d// 2 o (11) where, if E(O) > Eo, Ro is the geometrical radius of the emitting area. The latter quantity is given by the condition that for r = Ro the Laplace field strength at the cathode is zero. At zero bias the integration is carried out in practice up to that value of r for which f(r 2 ) is sufficiently small. Now the following remarks must be made. (a) The definition of de is not unambiguous because the field strength at the cathode for the ring with radius r remains equal to E(r) when the right-hand parts of eqs (9) and (10) are both multiplied by the same constant. This results, / however, in a change of the beam current given by eq. (11).

7 BEAM-CURRENT CHARACfERISTIC AND CATHODE LOADING OF ELECTRON GUNS 519 (b) In spite of the difficulty involved in the choice of de, eq. (11) apparently _ accounts for the linear relationship between 1 4 and Vc or Vg which has been used to determine Ve This can be shown as follows. In any case near cut-off and generally for a great part of the range between cut-off and zero bias, the function f(r 2 ) can be approximated by 7) Then, in virtue of eq. (11), where the emitting area A is a function of Vd and C z is a constant. Again writing I cc VdY, it can be understood that y increases with increasing Vd since d 2 AldV/ > 0 2 ). For small values of Vd' i.e. for small beam currents, so that, in virtue of eqs (13) and (14) Hence, at small beam currents, 1 4 is proportional to Vc (or Vg). It is noted that this behaviour also applies to guns provided with an elliptical grid hole 6). (c) The experimental determination of V e shown in fig. 4 applies to the reference gun - a tetrode gun which will be specified in sec with practical values of the electrode potentials, i.e. Va = 450 V and V,. = 18 kv. For these values of Va and Vh the maximum beam current is about 2 ma. The values of I used in the plot correspond to values of VdlV o ;<; 0 05 to rule out the influence of retarding-field current, while on the other hand II/m ; If. for this part of the beam-current characteristic initial velocities were not essential Ve would nearly coincide with Voo However, in spite of the linear relationship found, this is not the case: the difference V,,- V o is about 3 V. If, at constant VhlVa, Vh is scaled down by a factor of 6 this difference remains approximately constant. (d) In virtue of eqs (10) and (11) the beam current is multiplied by n3/2 if all electrode potentials are multiplied by n. This agrees with the behaviour prescribed by Poisson's equation when initial velocities are negligible. However, comparing the experimental maximum beam current of the reference gun with Va = 450 V and V,. = 18 kv with /m at Va = 75 V and V h = 3 kv, the latter is 1 4 times as large as would correspond to this three-halves-power law. (e) According to eqs (10) and (11) the beam current does not change when all gun dimensions are multiplied by the same factor. This agrees with Poisson's equation. Though experimental deviations occur, they are much less serious than the deviations mentioned at (c) and (d). I (12) (13) (14) (15)

8 520 J.HASKER The deviations mentioned at Cc)and (d) indicate that initial velocities must be taken into account in the beam-current calculation. This is done in sec Approximate beam-current calculation including the influence of initial velocities The assumptions 2 and 3 of the Ploke model are maintained. Assumption 4 will be replaced by a different choice of de. Instead of using eq. (7), the current density in the concentric diodes is calculated with the aid of Langmuir's theory for the planar diode B). Due to the axial initial velocities there is a space-charge minimum in front of the cathode. Only electrons emitted with axial velocities larger than what corresponds to the depth of the minimum face to face with the point of emission contribute to the beam. The current density at the cathode of these electrons (j) can be calculated with the aid of the well-known dimensionless quantities ç and 1] introduced by Langmuir when the cathode temperature (T), the equivalent diode distance (de), the saturation current density (js) and the Laplace field strength at the cathode (E(r» are known. Both the depth of the minimum (V m ) and its distance to the cathode (zm) increase with increasing r because j decreases with increasing r. The integration for obtaining the beam current is extended up to that value ro of r for which the space-charge minimum is situated on the fictive anode; thus '0 > Ro. For further details on the calculation reference is made to the appendix (sec. A.2). Now the following remarks must be made. (a) It is noted that both V m and Zm are fully determined when j., Tand j are known. (b) For the reference gun with Va = 450 V and V h = 18 kv the diode distance de is chosen such that at I = 100!LAthe calculated beam current is about equal to the experimental value. Then, for the major part of the emitting area, the calculated current-density distribution at the cathode is similar to the distribution found experimentally for I = 100!LA2). (c) Due to the assumption of independent concentric diodes, anomalous velocity selection by the curved potential field in front of the cathode is left out of consideration. However, for the case considered at (b), it has been shown by calculations that only near the edge of the emitting area gives this effect rise to deviations in the current density 9). There is no special reason that a different result would have been obtained at higher beam currents. The deviation is caused by the fact that near the edge of the emitting area the model of concentric diodes does not apply..the reason for this is that de is of the order of magnitude of the grid-hole radius. Hence de ~ z",(o) which results in a fast increase of Zm near the edge of the emitting area. For decreasing beam current zm(o) increases (zm(o) CZ [-1/4) whereas ro decreases. Obviously, the model goes wrong when zm(o) R:i ro. In all calculations which are to follow in this paper zm(o) ;5 ro/3.

9 BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS 521 (d) Using the value of de found at (b), the plot considered in sec. 2.2(c) has been calculated. It is, to a good approximation, linear. The value of V e found from extrapolation is indeed greater than Vo. Moreover, the calculated value of Vc - Vo remains approximately constant when the electrode potentials are scaled down as described in sec. 2.2(c). However, Vc - V o ~ 1 5 V instead of 3 V. This has been ascribed to beam electrons emitted near the edge of the emitting area with great transverse velocities directed towards the gun axis which pass the potential barrier in the vicinity of the axis 2). As has been remarked at (c), this effect is not taken into account in the calculations. Another, much more important, reason for the discrepancy between the experimental and calculated values of Ve - Vo will be discussed in sec. 3. In spite of this discrepancy it may be concluded that V e is greater than V o due to the influence of initial velocities. (e) The experimental behaviour of Im on scaling of electrode potentials _ as described in sec. 2.2(d) - has been verifiedby calculations with a constant value of de which is chosen so that at Va = 450 V, VI. = 18 kv, the calculated value of Im agrees with experiments. The result is that the calculated maximum beam current at Va = 75 V, Vir = 3 kv is about 10 % higher than the experimental value. Though - obviously due to the greater distance between space-charge minimum and cathode - de must be greater at Va = 75 V, Vir = 3 kv than at Va = 450 V, Vh = 18 kv to obtain agreement with experiments, it may be concluded that the measured deviation with respect to the three-halves-power law is a specific initial-velocity effect. It is caused by the fact that Tand i, must be kept constant when varying the electrode potentials An accurate approximate calculation of the maximum beam current As in sec. 2.3, it is assumed that - for fixed values of T andjs- Im is determined by the Laplace field in the vicinity of the cathode. The validity of this assumption will be discussed below. The difficulty involved ill the choice of the equivalent-diode distance can be avoided if the known properties of a reference gun are used in the calculations. The quantities of this gun, the principal dimensions of which can be found from fig. lb and table I, will be denoted by an asterisk in the following. TABLE I gun cg tg Rg ga Ra ta (urn) (urn) (um) (um) (urn) (um) reference low-drive triode anti-moiré

10 522 J.HASKER When the Laplace field-strength distribution at the cathode is parabolic, the field in the vicinity of the cathode is given by [ '!Z2 - r2 ] V(r,z) - V(O,O) = - E(O) z 1 +. R02 (16) As,regard this field, the one gun can be obtained from the other by scaling the dimensions and the electrode potentials. This property will be used to calculate the maximum beam current. At zero bias the Laplace field-strength distribution at the cathode deviates from a parabola. This can be seen from fig. 3 where this distribution has been plotted for the reference gun and two other guns. The latter will be denoted as the triode gun and the low-drive gun (the principal dimensions of these guns can be found from fig. land table 1). Thus, at zero bias the one gun cannot be obtained from the other by means of scaling. The actual Laplace field-strength 2Iif)().. ~(O).".,,, }uv 'Ep 1\ " 1'>1V' \ ~ D" I - 1\ \ R* 0 \K o soo 500 _r(flm) Reference gun, 1&=450V, Vh=/8kV -/200 ",Ep r:.::::: -200 ~ ~'\~ ï\,\ \\ o o : SOD _r(pm) Rg Rp Low-drive rum, Va=50V, Vh=/6kV E-Z (V/cm r~ too ~ ) rvv:," Ep ~, x /6 \ M r 1?I'V, 1\'.I"V _M" '\ 0 '\'-- t-. Fig. 3. Laplace field-strength distributions at the cathode for zero bias.

11 , BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS 523 distribution is therefore approximated by a parabola which is characterized by Ep and Rp taking the places of E(O) and Ro respectively. This is done such that the current, as calculated with Ploke's method, and its first moment at the cathode 00 (= [ J r dim(r)jf1m) r=o I are made equal for the parabolic field and the actual field. This leads to the formulae and 0 ce o J ce [-E(r)]3/2rdr=Rp2(-Ep)3/215 J [-E(r)]3/2 r 2 dr = 11: Rp3 (-Ep)3/2132. (17) (18) Some results of calculations are shown by the dashed curves in fig. 3. In order to calculate Im of any gun with prescribed electrode potentials, cathode temperature (T) and saturation current density Us), the reference gun is considered. First, the electrode potentials of the latter gun are chosen such that (19) The maximum beam current of the reference gun for these electrode potentials, cathode temperature T and saturation current density js is L;*. Next, all dimensions of the reference gun are multiplied by the factor Rpi R; *. By virtue of Poisson's equation, the maximum beam current of this scaled reference gun is Im* when its cathode temperature is T and its saturation current density is js (Rp*IRp)2. The Laplace field in front of the cathode is the same as for the gun under investigation: tz2- r 2 ] V(r,z) - V(O,O)=- Ep z [ 1 + Rp2 (20) Thus the scaled reference gun differs from the gun under investigation only in that its saturation current density is larger by a factor (Rp *IR p )2. Now, Im for the gun under investigation can be calculated if it is known how the maximum beam current of the scaled reference gun depends on its saturation current density, i.e. how Im* changes when, at constant values of T andj.., the dimen-

12 524 J.HASKER sions of the reference gun are multiplied by RpJ Rp *. The rather slight correction is denoted by L1Im*. Thus Im = Im* + :LIlrn *, where Im * is known from experiments and L1Irn* iscalculated with the aid of the approximate beam-current calculation described in sec For a review of the successive steps in the calculation of Irn reference is made to the appendix (sec. A.l). Moreover, the calculation of Jrn where the cathode properties of the gun under examination differ from those of the reference gun will be dealt with in the appendix (sec. A.4). It has been shown that, for maximum beam currents greater than about 200 [J.A, the above method for calculating the maximum beam current applies to guns of widely varying geometry - compare the most critical normalized dimensions of the triode gun and the low-drive gun listed in table II - for values of RgJRg * between about 0 5 and 2 0 5). Thus, for most practical applications this method will give correct results. Section 2.5 deals with an example. TABLE II gun cg Ro*JRg tg Rg*JRg (urn) (urn) reference low-drive triode Finally, with regard to the assumption that the beam current is determined by the Laplace field in front of the cathode, two remarks must be made: (1) The above method by which the maximum beam current of any gun is expressed in the maximum beam current of the reference gun,.cannot be used to calculate the beam current for partly cut-off conditions. Due to the fact that the distance between the grid - at a fixed potential which cannot be decreased by space charge - and the beam is greater for partly cut-off than for zero bias, the calculated beam current for partly cut-off is higher than the actual beam current 5). In other words, the beam current for partly cut-off, obtained from the maximum beam current of the reference gun by means of scaling, is too high because the beam current is not only determined by the Laplace field in front of the cathode. (2) The described method for calculating Im will not apply to a rotationsymmetrical electrode system consisting of a flat cathode at zero volt, an electrode at zero volt corresponding to the zero-volt equipotential plane calculated from eq. (20), and a curved solid anode calculated from eq. (20) with an anode

13 BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS 525 potential Va '= V(r,z) - V(O,O) > 0, which is very near the cathode in the vicinity'.of the axis. In this case - though the Laplace field strength at the cathode is parabolic - the space-charge-limited current will be higher than in a gun with a "free" electron beam. Obviously, the calculation oîl; will apply if the distance between cathode and beam stop is larger than roughly the grid-hole radius Calculation of the spot cut-off drive factor for the anti-moiré gun in the shadow-mask tube The principal dimensions of the anti-moiré gun are shown in table I. The gun is provided with the usual oxide-coated cathode. Because focussing on the screen, which is at 25 kv, is performed by an accelerating lens, the potential V h in fig. lis about 4 5 kv. The maximum beam current will be calculated for Va = 300 V. However, due to the spread in cut-off in normal production, it is more convenient to characterize the gun by its drive factor 5). Generally, for commercial display tubes the spot cut-off drive factor is specified. Therefore - though it would be better to use the extrapolated cut-off voltage (see sec. 2.1) - k, will be calculated. Apart from an astigmatic element in the space between anode and accelerating electrode, the anti-moiré gun has rotational symmetry. This element can be considered as a cylindricallens acting in the vertical plane, i.e., in the direction perpendicular to the scanning lines on the screen. At low beam currents the real cross-over 7) is situated at the cathode side of this lens which shapes two virtual cross-overs: one in a vertical "line" and one in a horizontal "line". The focussing is adjusted such thàt the spot on the screen is the image of the vertical focal line. Hence the vertical dimension of the spot is increased when compared with the gun without astigmatic element. This makes the scanningline pattern invisible and thus eliminates the moiré at low beam currents without affecting the horizontal definition. This beneficial effect is maintained even at maximum deflection of the beam. The astigmatism of the beam diminishes automatically at increasing beam current since the real cross-over then 'gradually shifts towards the centre of the added astigmatic lens and so makes this lens ineffective. This is essential as it avoids a loss ofvertical definition at high beam currents. It is noted that the influence of the astigmatic element on the potential field in the vicinity of the cathode is negligible, i.e., this field remains rotationsymmetrical. For Va = 300 V and VI! = 4 5 kv the Laplace :field has been calculated. The geometrical cut-off voltage for grid drive at these values of Va and V h has been calculated and is 83 2 V. In spite of the facts that the cathode properties differ from those of the reference gun - which is provided with an L-cathode _ and that the screen potential is 25 kv instead of 18 kv, the spot cut-off voltage will be determined to a good approximation by the criterion that at spot cut-

14 526 J. HASKER I off the depth of the Laplace minimum in front of the cathode be 1 55 Y (see sec. 2.1). Calculation from the Laplace field gives Vs = 96 1 Y for grid drive. Using the calculated Laplace field-strength distribution at the cathode, Im has been calculated with the aid of the method described in sec The fact that the cathode properties differ from those of the reference gun has been taken into account in this calculation (see also the appendix, sec. AA). This calculation of Irn for a known Laplace field-strength distribution at the cathode _ including the time for compiling the program - takes less than 40 seconds when using the EL-X8 computer. The calculated value of ks (= I m jv s 3/2) is 3 05 (J.Ajy 3 / 2 which is in very good agreement with the nominal value (3'0 (J.Ajy 3 / 2 ). 3. Calculation of beam-current characteristic and cathode loading In principle, the method of sec. 2.3 will be used to calculate the beam-current characteristic. The equivalent diode distance de is chosen such that the maximum beam current equals the -value obtained in the way described in sec This value of de is denoted by deo. First, two remarks with respect to de must be made: (a) In secs 2.2 and 2.3, de was assumed to be independent of r. In fact, de will decrease with increasing r. However, because of the result described in sec. 2.3(b) and because the method for calculating 1m applies to differently shaped Laplace field-strength distributions at the cathode (see the distributions for the low-drive gun and the triode gun shown in fig. 3), this decrease is of minor importance and will be neglected. (b) Tt is much more important that, within Ploke's model, the assumption that de is independent of Vd is not consistent with assumption 2 in sec To show this, two different bias situations of the same gun are compared. The Laplace field-strength distributions are assumed to be parabolic. Hence the two, situations are characterized by (1 1, ROl' E(O)l) and (12, R02' E(0)2) respectively. The second can be obtained from the first by multiplying all dimensions by R 02 jr ol (then the beam current does not change), followed by a multiplication of the electrode potentials by R02 E(Oh/ Ro 1 E(O) i - Thus, in virtue of Poisson's equation, E(O) R )3/2 I _ I ( E(O)! ROL Hence, considering small beam currents, so that Ro cc Vl1 2, eq. (21) yields (21) I cc Vl'4, (22) instead of eq. (15). This means that in Ploke's model de must not be kept constant but taken proportional to Ro. Clearly, at low beam currents de,cc Ro

15 BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS 527 will give erroneous results when initial velocities are taken into account because the current density at the centreof the cathode would then approach the saturation current density if de approaches zero. The effect of de OC Ro in the presence of initial velocities has been calculated for the reference gun. r--r r---,--r-,--,.-,---r---,r--r-,--,.-, (pa) Fig. 4. [0.4 versus Vc for the reference gun with Va = 450 V and V h = 18 kv; circles: experimental values; squares: calculated according to sec. 2.3 with de = deo; black triangles: calculated according to sec. 2.3 with de = deo [Rp(Vc)/Rp]; open triangles: calculated according to sec. 2.3 with de = deo [ Rp(Vc)/Rpl/2 68. Figure 4 shows 1 4 versus Vc for the reference gun with Va = 450 V and V» = 18 kv for 1;5 100!LA: (1) according to experiments; (2) according to the calculation described in sec. 2.3 with de = deo; (3) the same as (2) but with de = deo [Rp(Vc)jRp], where Rp(Vc) and Rp have been calculated with the aid of eqs (17) and (18) (for a parabolic :fieldstrength distribution RiVc) = Ro). It can be seen that, in spite of the above comments with respect to a constant value of de, the calculation with constant de gives the better approximation of the experimental results. This can be understood, because in the reasoning giving rise to eq. (21) the influence of the boundaries (see sec. 2.4) was left out of consideration. Its effect, however, is a relative increase of de with decreasing beam current because of the increasing distance between beam and boundary. Moreover, due to the influence of initial velocities (see sec. 2.3(e)), de relatively increases with decreasing cathode loading, i.e., with decreasing beam current. Because de OC Ro does not apply when the varying distance between beam and boundary and initial velocities are taken into account and because, on the other hand, there is no obvious reason to maintain Ploke's assumption of

16 528 J.HASKER constant de, the dependence of de on the bias situation has been determined from experiments pn the reference gun. Considering the results presented in fig. 4 it is obvious to take (23) giving de = deo/cl + oe) > 0 for RP(Vc) = O. For oe = 1 68 the calculated beam currents and the value of V e fit in well with experiments (see fig. 4). The beam currents of both the low-drive gun and the triode gun have been calculated with the method described in sec. 2.3 using the value of de from eq. (23) with IX = 1 68, whereas for deo, Rp(V) and R; the values obtained for these guns have been used. It is noted that deo depends on the electrode potentiais. For instance, for the low-drive gun deo is about 7 % smaller at Va = 50 V, rh = 16 kv than at Va = 12 5 V, VI. = 4 kv. The results of the calculations are compared with experiments in figs 5 and 6, respectively. In these figures V e - V V e (24) where V = Vc in the case of cathode drive (Iow-drive gun), V = Vg in the case of grid drive (triode gun) and V e is the experimental value of the extra- 5 I III Low-drive ~n T I I 11 LlVexp= (v., -l1j )exp =2 3V I (pa) Va=50V, 11!= 16kV:. l1j=32 8V 16=12'5\1,\1,=4kV (cathode drive) ) j 1 p t: 5 t: I ~ r4\fo/c =3 IV r Lllftrtr 'J)J ~>tof charabtjl1ic u~ed to drterrnine lie - I -Vd /}5 I IJ '5 Fig. 5. Experimental and calculated beam-current characteristics of the low-drive gun; drawn curves: experiments; circles: calculations. 10 5

17 BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS 529 I (pa) t : 5 I1 I Triode gun I I 1_11 L1v.,xp=(v., -li/)exp=3'ov r- r-l-h=25kv..li/=si 6V l-h=7 SkV_ (grid drive) j_ I 11 I(PA) : 1 2 rl if Ij If 2 : r--alfalc =3 4 V Lllfalc=3'W I.6-t:+ ~I+, '-- Part of characteristic_ used la determine IC D'5 1 I -vd '5 Fig. 6. Experimental and calculated beam-current characteristics of the triode gun; drawn curves: experiments; circles: calculations. j(o)(a/cm2) 6 j; -s o...l- r-- o V V 3 /fo' 16=5OV o 2 Vh=16kV - r- o,,/ Im=1300pA I1 0 o 0 2 j(o)(a/cm2) t:a '" 0 1j. ()O6 ()O8 -I/I m ooe ()oo ()oo / --V I- I-- ~ V I- liz=12 SV /' Vh=lIkV - r- Im=2I*1l_A 0 0 ()O2 ()ol!. 0 6.~I/Im (}o8 1 Low-drive gun,: j(o)(a/cm2) 5... I:::::: ê 0 V l- r-- I, 5./ 11,=2SkV 1 LL 0/ Iml'~7!,A l- ()o S" 0 0 0'2 ()olj. ()O I/I oi~ j(o)(a/cm 2 ) m 5 V- - o o 2 L!1,=7 SkV Im=29 lila t- '1 _L _L 1 o 011 o O l! ~I/lm Triode gun - Fig. 7. Maximum cathode loading versus the beam current as obtained from the calculations.

18 530 J. HASKER polated cut-off voltage in the representation of the experiments and the calculated value in the representation of the calculations. ' The calculated maximum cathode loading j(o) versus the beam current is shown in fig. 7. The results obtained and the relationship betweenj(o) and the mean cathode loading will be discussed in the next section. '4. Cónclusions and discussion 'For television-display tubes in particular the recommendation is made to use the extrapolated cut-off voltage Vc in the determination of y with the aid of eq. (6) (y is the slope of the double logarithmic plot of I versus Va) This implies that at small beam currents y = 2 5. When considering the behaviour of the equivalent-diode distance this is a reasonable approximation. The dependence of y on Va is determined by gun geometry and operating conditions. This dependence is found from a calculation of the beam-current characteristic; y > 4 at Va = 1 is not unusual. It can be seen from figs 5 and 6 that the calculated beam-current characteristic fits in fairly well with experiments. As has been stated in sec. 2.3(c) the calculation does not apply very near to cut-off where zm(o) F::i ro Moreover, a discrepancy between the calculated and experimental values of Vc remains (see figs 5 and 6). It has been ruled out in figs 5 and 6 by using the calculated value in the representation of the calculations and the experimental value in the representation ofthe experiments. It must be noted, however, that' this discrepancy gives rise to a deviation of the calculated value of j(o) with respect to the actual value. Since L1j(O)fj(O) F::i 2 5 L1valva, this is only serious for the low-drive gun at Va = 12 5 V and V h = 4 kv. For the other cases taken under examination the error in j(o) due to the discrepancy between calculated and experimental values of V e is certainly smaller than 5 % if I/Im ~ 0 1. Some calculated current-density distributions at the cathode are shown in fig. 8: curves 1, 2 and 3 correspond to the situations indicated by the same numbers in figs 5 and 6. Itis noted that 3practically coincides with (1- r 2 /r02)3/2. Table TIr shows various quantities calculated for the three situations. TABLE TIr situation I r02 r02/rl j(o) j=i/nr02 j(o)/j ([LA) (cmê) (A/cm 2 ) (A/cm 2 ) I _ _ _10-6 I-IS

19 BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS 531 j/j(oj '~- t \~"~ ~'I~ ~" 0.- W@ ","A <, ~ <, G ~"'" r0, <, r-... ~ r-:t Fig. 8. Current-density distribution for three different situations (see the corresponding numbers in figs 5 and 6).. It is noted that 2 has been obtained from 1 by multiplying the electrode.potentials by a factor of 4. The facts that ro2jr/ deviates considerably from unity and that j(û), is much smaller than 4 3 / 2 j(o)1 clearly show the influence of initial velocities. Moreover, the ratio j(o)jï deviates from the value 2 5 which would be obtained in the Ploke approximation. The value 2 59 for this ratio in case 3 is due to a deviation of the Laplace field-strength distribution at the cathode from a parabola (the distribution shows some "tail"). It is remarked that in cases 1 and 2 the field-strength distribution is parabolic. If, for instance, the Laplace field-strength distributions at the cathode for twelve different bias situations of a gun and the cathode properties (T and js) are known, the calculation of the corresponding beam currents, cathode loadings and current-density distributions at the cathode take only about one minute with the EL-X8 computer *). This time includes the time for compiling the program. In conclusion, the method described will provide a quick and sufficiently accurate insight into the dependence ofbeam-current characteristic and cathode loading on gun geometry and electrode potentials. Appendix Eindhoven, January 1972 This section is a guide to the computer calculation of beam current and cathode loading and contains all necessary information. *) This is a small-size computer. Its addition time, for instance, is 5 0!lso See for further information and for comparison with other computers: Codex, Computer Oaten Extract 1970, P. Lohse und G. Sonnleitner, Informationsbüro für Datenverarbeitung, 403 Ratingen im Rot 12, W-Germanv.

20 532 J. HASKER The Laplace field-strength distributions at the cathode, as calculated for the various bias situations, are fed to the computer as arrays E[O:n], the elements of which represent the field strength in equidistant points on a radius along the cathode surface. Generally, dependent on the bias situation, 11 varies between about 4 and 15. In the calculations, the field strength for any value of r is obtained by means of quadratic interpolation using three consecutive elements of E[O:n] which are adjacent to r. The calculation of beam current, cathode loading and current-density distribution at the cathode proceeds as follows. A.I. Calculation of the maximum beam current (1) Rp and Ep are calculated with the aid of numerical integration and eqs (17) and (18), where E(r) is the Laplace field-strength distribution for zero bias. (2) Next, Ep* is calculated from Ep* = Ep (Rp/Rp *), where Rp* = 302 urn. (3) If 300 V/cm <-E p * < 2900 V/cm, the maximum beam current 1",* of TABLE A-I -Ep* 1",* -E/ In.* -E/ 1",* (V/cm) ((.LA) (V/cm) ((.LA) (V/cm) ((.LA) the scaled reference gun is found from table A-I with the aid of linear interpolation. If - Ep* > 2900 V/cm the extrapolation (A. I) (1",* in (.LA, Ep* in V/cm) can be used. It is noted that the condition j(o) < 10 A/cm 2 must always be satisfied. (4) Finally, using the known value of Rp/R/, LlI",* is calculated giving the maximum beam current I", = Irn* + LlIrn*. The correction L1Im* -- which is of importance only for -Ep * < 2900 V/cm -- has been calculated as

21 BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS 533 a function of E;* for various values of Rp/Rp * using the approximate beam-current calculation described in sec. 2.3 (see also sec. A.2) and a / properly chosen equivalent-diode distance 5). The result can be approximated by LlI".* =5'52I".*{I-exp[ -1,1 (~:*-1)J}(-E/)-O.765. (A.2) Obviously, this approximation facilitates the computer calculation of I".'l It must be remarked that the above calculation applies to a gun provided with a cathode for which j, = 10 A/cm 2 and T= 1350 K. Different cathode properties will be introduced in sec. A.4. A.2. Beam-current calculation according to sec. 2.3 First, the application of Langmuir's theory to a planar diode, with a diode distance of de cm, a cathode with a temperature of T K and a saturation current density of i, A/cm 2, will be considered. Starting with some current density j[i] < js which passes the space-charge barrier in front of the cathode, the calculation proceeds as follows. (1) Calculate 'YJ- from, 'YJ- = In (js/j[i]). (2) With the aid of 'YJ- the value of L is found (see sec. A.5). (3) Next, ~+ is calculated from (A.3) where al = T- 3/4 (j[ij)l/2. (AA) (A.5) It is noted that the distance Zm between space-charge minimum and cathode is given by (4) With the aid of ~+ the value of 'YJ+ is found (see sec. A.5). (5) Finally, the anode potential Va is found from (A.6) Va = (kt/e) ('YJ+ - 'YJ-). (A.7) The corresponding Laplace field strength at the cathode, El[i], is given by (A.8) Obviously, the current density j corresponding to a prescribed value E of the Laplace field strength at the cathode must be obtained by means of iteration.

22 534 J. HASKER The transition b~tween space-charge-limited and retarding-field current is given by the condition that the space-charge minimum is situated at the anode. Then, the current density je is to a good' approximation given by (10-3 T)3/2 je = d/ (A9) I The corresponding value of El is denoted by El[m]. It is noted that El[m] > O. Now, an electron gun with equivalent-diode distance de is considered. The calculation of de will be described in sec. A3. Further, it is assumed that the array element E[n] which has been fed to the computer obeys the condition E[n] > El[m]. The inherent difficulty occurring at zero bias will be considered in sec. A.3. To obtain the current-density distribution at the cathode and the beam current, j(r) must be calculated for prescribed values of r, i.e., for prescribed values of the Laplace field strength E(r). To avoid the time-consuming iteration for each value of r, use is made of interpolation from an array EI[O:m] and the corresponding values j[o:m]. This is done as follows. (1) The arrayelements El[i] are calculated with Langmuir's theory as described above for j[i] =jmax - Cilm) (jmax - je), (AIO) where i = O(I)m and jmax is greater than }(O). The choice of jmax will be discussed in sec. A.3, while the value of the integer m is considered below. (2) The radius r» of the emitting area is calculated with the aid of interpolation from the array E[O:n] and the condition (3) E(r) is calculated from E[O:n] for E(ro) = El[m]. r 2 = (kis) r02, (All) (AI2) where k = O(1)s and s is an integer. This is so done becausej(r) versus r 2 is a smooth curve (see fig. 8), thus facilitating the numerical integration for obtaining the beam current (see (5». The value of s is considered below, (4) To obtain the current density j(r) corresponding to E(r), the difference j[i] - je is written as }[i]-je =A(-E/[i] +El[m])P. (A.I3) The constants A and (3 are determined by taking in succession i = pand i = p + 1, where p follows from Then, El[p + 1] < E(r) < El[P]. (AI4) j(r)- je = A {-E(r) + El[m]}P. (A.I5)

23 BEAM-CURRENT CHARACTERISTIC AND CATHODE LOADING OF ELECTRON GUNS 535 (5) Finally, the beam current 1= ro n Jj(r) dr 2 is easily obtained with the aid o of quadratic Simpsou integration. With regard to the values of m and s, it can be remarked that the results of the calculations are sufficiently accurate if m = s = 25, provided that jmnx is properly chosen. A.3. The calculation of the equivalent-diode distance According to sec. 3 the value of the equivalent-diode distance at zero bias, deo, must be so chosen that the beam current calculated by the method described in sec. A.2 equals the maximum beam current calculated according to sec. A.I. However, to avoid the difficulty involved in the definition of the emitting area for zero bias, which is caused by the fact that E[n] < 0, the procedure described above is carried out at Vd < 1. This gives correct results if the deviation of Vd with respect to 1 is small, because then the error in the beam current calculated according to sec. A.l is sufficiently small (the error is due to the increased distance between beam and boundary, see sec. 2.4(a)). The results presented in this paper have been obtained by calculating deo at Vd ~ Consequently, the value of Rp in eq. (23) - which corresponds to zero bias - has been replaced by the value corresponding to Vd = The constant ct in eq. (23) is taken equal to The calculation of deo proceeds as follows. (1) The beam current] at Vd = 0 95 is calculated by the method described in sec. A.I. Obviously, now E(r) is the Laplace field-strength distribution at the cáthode for Vd = (2) Next, deo is estimated with the aid of Ploke's theory described in sec Hence, in virtue of eq. (11) deo is found from 6! ]=2'33.1Oo R O [ _ E(r) ]3/2 d 1/2 co 2nrdr. (A.16) Because this estimate is worse as E(O) is larger, it is multiplied by {25OO/[-E(0)]}1/2 if E(O) > V/cm., The Ploke current density at the centre of the cathode, jio), is given by [ B(0)]3/2 h. (0) = d--- 1 /- 2 - Now, the current density jmax> introduced in sec. A.2 is chosen as ' jmax = WjiO), where w is a safety factor which is somewhat larger than 1. co (A.17) (A.18)

24 536 J. HASKER (3) Using the above values of deo and jmax> the beam current I is calculated by the method described in, sec. A.2. Obviously, l, =.10 Afcm 2 and T = 1350 K in this calculation. (4) If 1(I-I)fII > 0,5.10-2, the calculation' at (3) is repeated with (A.19) It is noted that the Algol-60 symbol : = is used for brevity and that generally three repetitions are sufficient to obtain the required value of deo The corresponding value of the current density at the centre of the cathode is denoted by i.; After deo has been calculated, the value of jmax in the beam-current calculation according to sec. A.2 is chosen as W j'm {E[O]}va Jmax = {E[O]}Va=l ' (A.20) where the value of i.; obtained above is used while the safety factor w is taken equal to 1 2. The factor {E[O]}v a f{e[o]}va=l has been introduced to decrease jmax with decreasing beam current, so that the interpolation according to eq. (A.15) remains sufficiently accurate at low values of Vd' A.4. Introduetion of different cathode properties The preceding calculations of the maximum beam current and deo are carried out for cathode properties corresponding to those of the reference gun, i.e., i, = 10 Afcm 2 and T = 1350 K. The beam-current characteristic of a gun provided with a cathode with l, =1= 10 Afcm 2 and T =1= 1350 K can be calculated when use is made of the value of deo obtained in sec. A.3, eq. (23) with if. = 1 68 and the method described in sec. A.2. In the latter calculation jmax is again given by eq. (A.20). Obviously, as described above, the beam-current characteristic for different cathode properties is obtained for values of v«~ The corresponding maximum beam current can be obtained by means of an adequate extrapolation. Finally, it is noted that for most practical applications - as in practice the permissible variations in T are relatively small - the influence of the change in cathode properties on the equivalent-diode distance will be of minor importance (see sec. 2.3(e». A.5. The calculation of ~_ and 'YJ+ The relationship between Langmuir's dimensionless quantities ~ and 'YJ - obtained with the aid of Poisson's equation - is given by as