THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS* By G. LIEBMANN, D.Phil., F.Inst.P.f {The paper was first received 3rd February, and in final form 10th October, 1945.) SUMMARY It will be necessary to review briefly the existing design theory The anode-current/grid-voltage characteristics of valves are deter- and to bring it into a form most suited for our purpose before mined with the help of diagrams and design charts based on Langmuir's proceeding to consider a closely spaced plane triode and more data on the current flow in plane diodes with consideration of initial complicated electrode structures. electron velocities, and on Oertel's and Herne's equations for the ampli- fication factor. The theory is extended to multi-grid valves and to (2) CURRENT FLOW IN A DIODE valves possessing a more complicated shape. Special attention is given The electrostatic field is determined, in the presence of space to the "variable-mu effect," which represents one of the limiting factors 2 in practical valve construction. A simple expression describing this charge, by Poisson's equation S/ V = — Anp. For a plane effect is derived, and a chart of the "variable-mu constant," a, is diode this equation can be simplified to presented, in which space charge is taken into account. Measure- 2 2 ments on specially made experimental valves and on several types of d V V modern mass-produced valves confirm the treatment of the variable-mu = — 477/ _ 2 a effect and show that the methods outlined in the paper, forming a dx \m complete design system, allow the prediction of the static valve If the potential at the cathode is taken as zero and the initial characteristic with good accuracy even in closely spaced modern electron velocity is also assumed to be zero, the solution of this valves. Finally, the influences of a change in control-grid wire differential equation is Child's7 law: diameter, of a statistical variation of control-grid pitch, and of cathode misalignment, are discussed. 2-34 x 10-6 3/2 2 W Va amp/cm (1) (1) INTRODUCTION where /. is the anode current density, V. the anode voltage and The problem of calculating the anode-current/grid-voltage bx the electrode spacing. characteristic of an amplifier from the given valve dimensions This formula has most frequently formed the basis for calcu- has received wide attention owing to its theoretical interest and lating the characteristics of radio valves with plane electrodes. its great practical importance. The position of the art has been It is, however, only a first approximation: it gives a fairly correct reviewed by Benjamin, Cosgrove and Warren,1 and more current prediction for valves with large electrode clearances and recently by B. J. Thompson.2 These authors reach the con- high accelerating voltages, but underestimates the cathode clusion that the existing standard valve design formulae are very current very considerably for such small clearances as are used useful in explaining the principles of operation of an amplifier valve, but cannot be more than an approximate guide to actual I0QO valve design, and they indicate that development of new valve types has to rely very largely on accumulated experimental material. There are three reasons for the difficulty encountered in pre- dicting quantitatively the characteristic of a modern closely spaced valve. The first is the necessity of taking the initial electron velocities into account. The solution of this problem was given by I. Langmuir,3 but in its customary form this solu- tion involves lengthy numerical calculations. Secondly, the close spacing between valve electrodes leads to a variation of the amplification factor along the cathode surface. Some implications of this problem were recognized fairly early, but only more recently was further attention given to it (Oertel,4 Glosios,5 Fremlin.6). The available solutions lead to involved formulae, which are moreover of only limited applicability. Thirdly, modern radio valves are rather complicated structures, and the simpler theory derived from valves with plane-parallel electrodes requires some adaptation. The object of this paper is to present a valve design method which allows the prediction of the static amplifier-valve charac- teristic with greater accuracy and with considerably less com- putation than was possible hitherto, particularly if the electrode 13 14 spacing is close and the valve structure is complicated. Special "Anode voltage V3. volts attention will be given to the unwanted variable-mu effect, as Fig. 1 this appears to be one of the factors limiting further improve- ment in certain amplifier valves. ; • Radio Section paper. t Cathodeon, Ltd. L without initial velocities. [138 ] LIEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS 139 in most modern amplifying valves. This is shown by Table 1, muir3 are compared with those determined from Child's law for where the correct current values calculated according to Lang- a few typical clearances and current densities. A further comparison is given in Fig. 1, in which diode Table 1 currents according to Child and to Langmuir are plotted against anode voltage for two electrode distances, the cathode tem- Electrode spacing, mm 0-5 0-3 0-2 01 01 perature being taken as 1 050° K and the saturation current as 1-0 amp/cm2, values representative of modern oxide cathodes.8 Va, volts 5-53 2-26. 0-94 -0-30 0-36 The dashed curve, Fig. 1, indicates that an appreciable change ia according to Lang- in cathode temperature or saturation current has only a moderate muir,3* mA/cm2 .. 200 200 200 200 400 influence on the diode current. ia according to Child,7 mA/cm2 120 8-8 5-4 0 50 Langmuir's solution was given in the form of series and tables and requires the numerical evaluation of several equations for Cathode temperature •- 1 050a K. Saturation current density = l-0amp/cm>. each pair of Va and ia values, making the prediction of the valve
Anode voltage)Va, volts 1 0 I- 5 \ ii 6 1' X } 10 11 12 13 14 K oft 10 ' • i \ f>n \ ooo 1 . i »v \ --—\ y^ \_y —-— V \ 1 i i A y A ^y^ ^^ ^—^ i /1 / y ^--- 500 1 ''/ ^y^ \ ^ \-*~~"~~' i / i /\ y V/ / i—^^"^ i/ / \ ^ -—' /I/ n A /\ 200 17 / ' A s J
A ^^ Wf ' / ^—' "/ y^ -— ^--^ / <^ 100 IA 'A ' ^y^ \_^^~ ^^,— —' Fig. 2.—Chart of anode current-density and slope as functions of electrode distance and anode voltage. T = 1 050° K; /, «= 1 0 A/cm*. /„. gjta. 140 LIEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS characteristic a very tedious process. One can, however, reduce "equivalent diode." It may be noted that there is only one these lengthy calculations to a mere reading of graphs, once a position of the equivalent grid plane and only one value of VG certain number of standard values have been computed. which can be common to both diodes; this differs from the way A set of curves showing the anode-current/anode-voltage in which the equivalent diode is introduced by other authors relationship for several typical anode-cathode clearances, which (Thompson,2 Benham,10 Dow11)- the author prepared some years ago for his own use from Eqn. (5), which is not restricted to the cathode/control- Langmuir's data, is reproduced in Fig. 2. The almost vertical grid/anode structure, proves also useful in carrying out the broken lines in this Figure connect points on the different reduction of a multi-grid-valve and in evaluating inter-electrode iJVa curves of equal gjia ratio, gm being the mutual con- capacitances. For a triode (K, = 0, V2 = Va), eqn. (5) can be ductance; these were obtained by graphical differentiation. rewritten in the familiar form V +-V (3) THE ELECTROSTATIC FIELD OF A TRIODE * ft a Maxwell9 derived an expression for the electrostatic field of a (6) charged grid G (potential V.) between two plane-parallel elec- trodes P, (potential Vt) and P2 (potential V2) for the case in which the grid pitch a is large compared with the wire diameter 2c, In most modern amplifier valves, the condition 2c/a <^ 1 is but small compared with the distance b between the grid-wire not fulfilled, and the simple expression, eqn. (3), for /x has to be x modified. Expressions for /u, which cover the whole range of centres and the cathode, and the distance b2 between the grid- wire centres and the anode (Fig. 3). As shown in the Appendix, relative grid wire diameters 2c\a used so far in radio valves were derived by Oertel4 and by Herne12 (Herne's paper discusses also the ranges of validity of some of the earlier expressions). Using Oertel's and Herne's equations, a chart of a "standardized" amplification factor fi0 (for b2 = 1 mm) was computed for a Pitch, turns per inch T • 0 : r b, y • 0 •. 300 200 It j 1 X)80 tH 50 40 30 25 20 15 0 \ \ \ \ \ I I \ \ \ \ \ \ \ . \ \ \ \ \ -®- Grid G \ \ \\ \ \ \ \ \ \ \ \\ \ \ \ \ ^>-Cathode P, 200 y 1 \ \ \\ Fig. 3 Y \ \ \\ p-osyw 40O5005 •r ©-151 this leads to the following expression for the field near the 100 \ \ \ cathode P, \ \ \ [ \ \ \ t \Z \\ \'L \ \ \ \\ y\ \ V \ \\ v \ (2) 50 \ VV 1 \ v\ A V \ \N \ \ T: \ L \ Mi \ \ \ \ \v\VV\\ \ and to a similar expression for the field near the anode P . \ \ 2 \ \V r\ \ ^ The two amplification factors fj,{ and fi2 are connected by the 20 \ V \ V i \ relation /x, = b \x. lb . /n is the amplification factor /A as 0 x 2 2 2 1 \ \ v usually defined. Its value is 2 \ \ A \ a = n, — . . (3) 10 * a log 2 sin (TTC/O) sV \\ k\ s\ A \ In the absence of space charge the factor F, given by V v\ s. \ v \ N \\ 1 50 \ (4) s\ VW \ ^^ indicates the reduction of control action of the grid voltage Vr [ One can also see from eqn. (2) that the same field near plane Pj 20 is obtained if the grid is now replaced by a full plane ("equivalent grid plane") passing through the centres of the grid wires, having \^ k an "effective grid voltage" VG applied to it, where 10 \ v\ 1 v^ V2 M2 (5) n-5 1 1 0-50 01 0-2 0-5 10 20 50 Mi M2 Pitch, mm The triode can now be considered as consisting of twp inde- Fig 4>_chart of amplification factor as function of grid pitch and pendent diodes, the diode P,G being frequently called the grid-wire diameter. LIEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS ftl range of grid wires that are most likely to be met in practice Fremlin:6 (Fig. 4). Any values of fix or /x2 can be easily obtained by lirb. sinh multiplying fa taken from the chart by bx or b2 (mm) respec- tively. , / , 2TTZ>, 2TTX\ fri + W ( cosh i — cos ) (4) CURRENT FLOW IN A TRIODE a a 13 V / It was proved by Tellegen that the concept of an equivalent 2 sinh bx smh diode can also be applied to a triode if current flows through the valve. Owing to the comparatively high electron speeds in 1 + . 2irbx 27TX sin2 — cosh *. — cos the grid-anode region, in general the space charge in the grid- a a cathode region only need be considered, and Tellegen showed W that its effect is equivalent to an apparent decrease of the grid- Oertel's and Fremlin's formulae are both very unwieldy and their cathode distance by a factor 3/4 in the expression for VG use does not appear very practicable in actual valve design work. [eqn. (6)]. Moreover, both formulae have a limited range of application, Thus we obtain the current in a triode by calculating the and they also disregard the influence of space charge; this leads current for an equivalent diode with an electrode distance bx to an underestimate of the influence of the variable-mu effect and an effective grid voltage VG: on the valve characteristics. Starting, as Fremlin does, from Maxwell's description of the V 4- - V field of a charged grid, the author derived the following rather (7) simpler expression for the variation of the amplification factor 1 + - (1 + 4b2l3bx) (see Appendix): ... (10) 2TTX By introducing this value of VG into eqn. (1) a formula for the 1 — a cos prediction of the current passed by a triode can be obtained which has been* widely used in the past.* In view of the discussion in Section 2, this formula would be where a is a constant depending on the relative grid-cathode only a first approximation, and the author prefers to determine distance bxla and relative grid-wire diameter 2c/a. A chart of the current values from the graphical representation of the this variable-mu constant has been plotted in Fig. 5 for the iJVa relationship, Fig. 2, except that the expression for V'G [eqn. (7)] takes the place of Va noted on the chart. 2c/a = If we are interested in the mutual conductance, gm = VJbVg 0-25 0-3 of a triode, we take the gjia values from Fig. 2, but, remem- 5-0 bering that the values read off refer to the effective grid voltage VG and not to the impressed grid voltage Vv we have to multiply by the factor 0-OZVy x\\\\ F'1 . .- ._. (8) •kl 4b2l3bx) . l-o Hence gm = F'iaA(gJia) mA/V . (9) I \\ A being the cathode area in square centimetres. l\\\\N\ (5) THE VARIABLE-MU EFFECT In most radio valves made to-day, the grid-cathode distance is so small that the variation of the electrostatic field along the cathode surface, described by a periodic term cos (27rxla), must 01 be taken into account. This variation along the cathode surface leads in turn to a variation of the "aggregate" amplification K\^V factor for the lower part of the valve characteristic, and the \\\\\\ appearance of a marked "tail," for which the mutual con- \\\\\\s ductance is greatly reduced. If we call the local amplification factor /z2(x), as it varies with the position x along the cathode, and /*2, as before, the average amplification factor (taken from Fig. 4 and multiplied by b2), 0-01 these two expressions were given by the previous authors (in the 0-5 M) 1-5 bja (full space charge) notation of this paper) as follows: 4 0 0-5K) F5 Oertel: b^ja (no space charge) Fig# 5.—Variable-mu constant a as function of relative grid-cathode 1 - distance bi/a and relative grid-wire diameter 2c]a. Points measured in electrolytic tank: ' x Icja «= 0-1; o 2c/a = 0-2; + 2c/a = 0-3 cases where space charge is present and where space charge is both expressions also become formally identical under certain conditions (62 Tellegen's approach is preferred here as it can be interpreted more easily. negligible. 14? LIEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS For a given set of valve dimensions and electrode voltages, larger grid-wire diameters and small grid-cathode clearances the value of the effective grid voltage V'G, and hence the anodeinadequately covered by theoretically derived formulae, and current, becomes a function of the position of the current- further information needed for designing Fig. 5 was obtained emitting cathode element as expressed by eqn. (10). The valve experimentally by measurements on models in an electrolytic must therefore be considered as composed of a large number tank. of elemental valves working in parallel, each with a different Values of a near or above 1 • 0 taken from Fig. 5 should be value of V'G and different current yield. used with discretion because, for such inhomogeneous field If the constant a is not very small, the unevenness of electron distributions as are indicated by very large values of a, the emission from the various parts of the cathode may become periodic variation of the field with x is given by a Fourier series appreciable, and the current may even become cut off entirely of which a is only the coefficient of the first cosine term. How- for certain cathode areas. Although the current deficit due to ever, fjLmirl is given fairly well by /x.2/(l + a) even if a > 1. this local reduction in electron emission is fully compensated The value of a where there is no space charge is of some by a greater amount of current supplied by other areas [for each interest, for instance in the case of a suppressor grid in the cathode element with a local amplification factor /^/(l + K) proximity of the anode, but the more important case is that in there exists one of equal size with fi = /x^O — K)], the aggregate which space charge is present between grid and cathode. Taking mutual conductance may be reduced considerably. space charge into account exactly would require a full point-to- It can be shown that this loss of current slope is not serious point solution of Poisson's equation for the field between cathode so long as the current cut-off point is not reached for any cathode and grid, and the mathematical difficulties seem insuperable. area, but becomes progressively apparent from this point To some extent a reasoning similar to that used by Tellegen13 onwards. The variable-mu constant a has a fairly small value, can be applied, and it appears reasonable that, in this case also, of the order of 0-1 to 0*2, only in rather widely-spaced valves, the effect of space charge can be described by an apparent e.g. most kinds of output valves. In many modern valves the reduction of the grid-cathode distance by a factor 3/4. The 14 relative grid-cathode spacing bx\a and relative grid-wire diameter analysis by Maloff and Epstein of the influence of space charge 2c/a are such that a is of the order of 0-3 to 0-5 or more, and on the very inhomogeneous field of a cathode-ray-tube gun seems the loss of slope may become quite marked, even near or at the to support this view (taken in conjunction with electrolytic tank specified operating point. The present-day tendency of de- measurements of the electrostatic field configuration in valves creasing the grid-cathode spacing of valves in order to obtain with small grid-cathode distances). Hence, in the presence of better ultra-high-frequency behaviour or higher gjia or gjc space charge, the value of a corresponding to eqn. (11) is given by values as desired for wide-band amplification is, amongst other factors, severely checked by the increase of the variable-mu (12) constant which appears to set a definite limit to further valve improvement. The basic facts underlying this limitation have, of course, been The effect of space charge is therefore a greater value of a and known for a long time, but no relatively easily-applied method a considerable increase of the deleterious variable-mu effect for a quantitative analysis seemed available before. By using which makes the valve characteristics even worse than was suggested by previous authors. Figs. 2,4 and 5, characteristics even of valves in which clearances are close and the undesirable variable-mu effect is serious can To check this point experimentally, the author analysed be evaluated fairly quickly. Numerical computations can be 16 types of valves (experimental and mass-produced) and found reduced to a minimum if it is borne in mind that the variable-mu that in 14 cases the cut-off grid-bias was at least as great or slightly greater than would be expected from values of a deter- effect need, in most cases, be taken into account only if the mined from Fig. 5, for the space charge case. For two valves current becomes cut off for some part of the cathode, i.e. if only (one of them the 6V6G), the cut-off point was not quite as Vg is more negative than approximately — (1 — ot)(Valfi3). remote as predicted from Fig. 5 with space charge, but still Further, the author found that it is quite sufficient for most considerably more remote than would have beea expected if no practical purposes to approximate the variation of the amplifica- account of space charge had been taken. tion factor fi2(x) according to eqn. (11) by a small number of steps, assigning a fixed value of cos (lirxla) to each step. The As a further check of the correctness of the quantitative treat- simplest approximation is the 3-step method, used for the ment of the variable-mu effect and the reliability of the other numerical examples given later in the paper. In this method, computing methods outlined above, five series of triodes were we divide the cathode area between two grid wires into four made which all had the same dimensions of cathode, anode, equal sections. For two sections we use the amplification factor grid mandrel and grid wire, but differed in their grid pitch. The cathode was of the oval, nearly flat type. The essential dimen- fj, = fo directly as derived from Fig. 4; for the other two sections sions were: we choose fx — /x2/(l'— 0-90a) and fj. = /i2/(l + 0-90a) respec- tively, where a is taken from Fig. 5 (space-charge case). Only Cathode area, A 2-90 cm2 rarely will it be necessary to sub-divide the cathode area further. Cathode-grid distance, by 0-50 mm Grid-anode distance, />2 1-45 mm Fig. 5 requires a few further comments. As shown in the Grid wire diameter, 2c 0-100 mm Appendix, the constant a is given, in the absence of space charge, by the equation The grid pitch a, the calculated average amplification factor /n, the calculated variable-mu constant a and the average grid (11) control factor F' are given for these five series in Table 2. The complete static valve characteristics were calculated for This formula is only a first approximation, but its range of two anode voltages for each series of valves, which were made validity can be extended to greater values of a by replacing with great care. Special attention was given to alignment 2e-2nb1fa 5y i/sinh (277-6,/a). In Fig. 5, the term which, of the electrodes and the perfection of the grids. The anode- according to Oertel, corrects the simple expression, eqn. (3), current/grid-voltage characteristics and the contact potentials for the amplification factor for larger grid-wire diameters, has were carefully measured for each valve. The measured charac- also been taken into account. This still leaves the region of teristics were plotted together with the calculated curves, the LIEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS 143 Table 2 ' I 1200 Scries No. a V-av a F'av mm 1 0-40 580 006 0-922 2 0-50 31-9 014 0-867 3 0-60 20-8 0-22 0-811 4 0-75 131 0-35 0-729 5 1-00 7-6 0-58 0-610 points for each valve being shifted horizontally by the amount of the measured contact potential (see Section 8). The results are shown in Figs. 6-10. The agreement appears to be very -20' -15 -10 Grid voltage l£, volts Fig. 7.—Experimental triodes, series No. 2. 2c = 0-100 mm; a = 0-50mm. Calculated. O No. 1674, Ve= -O-35V. + No. 1675, J> = -0-65V. x No. 1676, Ve= -0-44V. Fig. 6.—Experimental triodes, series No. 1. 2c = 0-100 mm; a - 0-40 mm. Calculated. X No. 1671, Ve = - 106 V. + No. 1673, Ve = -0-40V. O No. 1685, Ve •= -0-97 V. A No. 1686, Ve = - 0-70 V (bright anode). • No. 1688, Vc = -0-85 V. good for small as well as for large anode currents, and the variations between valves of the same series are greater than the deviations of the individual measured valve characteristics from the calculated characteristics. The variations of the measured characteristics would have been still greater if the measured characteristics had not been corrected for contact potential. The calculated curves for the anode potential Va = 250 V are reproduced together once more in Fig. 11 to show the wide range of characteristics covered by the five series. One can, of course, abstract from the calculated characteristics, curves demonstrating the decrease of anode-current slope (for fixed anode current) with increasing grid pitch (Fig. 12), i.e. with increasing a. These curves are very similar to that measured by Benjamin, Cosgrove and Warren.15 (6) EXTENSION OF DESIGN THEORY TO VALVES WITH -30 -20 -10 NON-PLANE-PARALLEL ELECTRODES Grid voltage 1^, volts Fig. 8.—Experimental triodes, series No. 3. Most modern amplifier valves employ electrode structures 2c = 0-100 mm; a — 0-60 mm. which are neither plane-parallel nor cylindrical, but are of some Calculated. intermediate shape. It can be shown that in this case the x No. 1669; Ke- - 1-80V. + No. 1670; Ve= - 1-06V. curvature of the cathode (radius rk) increases the emitted electron O No. 168.5; Vt ---- - 1-40V. 144 LJEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS .-50 -40 -M -20 -10 -30 -20 -10 Grid voltage ^,volts Grid voltage l£ , volts Fig. 11.—Variation in characteristics of triodes with grid pitch. Fig. 9.—Experimental triodes, series No. 4. 2c = 0100 mm; b, = 0-50 mm; b2 = 1-45 mm; A = 2-90 cm*. 2< 0-100 mm; « - 0-75 mm. Va -=• 250 V. Calculated. No. 1678. Vt - - 0-40 V. • No. 1679, Vt =-- - I 50 V. ; Nq. 1677, V,. - 0-62 V. Fig. 12.—Variation in slope of triode current with grid pitch. Va = 250 V; 2c = 0-11 mm; bl = 0-50 mm; b2 = 1-45 mm. A = 2-90 cm*. rg being the radius of the surface passing through the grid-wire -.r>0 40 -30 10 centres. These corrections give a quantitative measure of the Grid voltage 1^", volts departure of a calculated characteristic from the observed one. Fig. 10.—Experimental triodes, series No. 5. If the radii of curvature of the valve electrodes are large com- 2c == 0-100 mm; a = 100 mm. pared with the electrode distance, we can consider the valve Calculated. structures as composed of a number of plane valves all working • No. 1680, Vc - 0-90 V. in parallel and all having the same voltages impressed on their O No. 1682, Vc - 152 V. > No. 1681, »••„ - -0-92V. electrodes, but differing more or less in their physical dimen- sions. Fortunately, three valves in parallel are quite sufficient current calculated on the basis of the plane electrode structure, to bring out the salient properties of most electrode structures. the upper limit of increase being A method of abstracting the parameters of a small number of valves in parallel representative of a valve structure of a 'i (13) complicated shape is best explained with the help of Fig. 13, 'a 2rk • showing a cross-section through a typical triode structure. Further, the amplification factor decreases by the amount Owing to the symmetrical construction we can usually confine our attention to one quadrant. We divide the cathode circum- (14) ference into n equal parts. These sections may be identified by ft the angle 6 between the horizontal axis and a line connecting L1EBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS 145 the centre of the section under consideration with the origin. The area of the section is dA(8). Next we draw two curves between cathode and grid and between grid and anode which approximately bisect the cathode- grid and the grid-anode distances. These curves are shown as dashed lines in Fig. 13. For each cathode element dA{6) we 8 • 00* Civs* st'ihmi oi flrctroili- structure •20 i -7 -6 -5 -4 -a -I Grid voltage lg , volts Fig. 14.—C302 experimental-type valves. Calculated. -i- No. 1692; Yc — 1 0 V. Fig. 13. -Cross-section through typical triode structure. O No. 1693; \ e - 1-9V. \ No. 1694; Ve - - 1-2V. measure its cathode-grid distance 6,(0) along a line originating considerable, we have to weight the values of b{ and b2 in in the centre of dA and perpendicular to the bisecting curve in forming the average values. As the current is very nearly pro- the grid-cathode region. From the point of intersection of this portional to lib], the average value of b will be near enough line with the centre of the grid wire we draw a line perpendicular x to the curve bisecting the grid-anode space. Along this line we measure the grid-anode distance b2{6) of the cathode element Ha, b\{ey dA(0). We now have n valves with the known design para- meters dA{0), 6,(0), b2(6), 2c and a. If this number is too great Similarly we obtain: V for easy manipulation, we group several of these valves together, ZJ b\(d) assigning to them the average values b and b for each group x 2 In this way we find the following three valves (Table 4) repre- of 6,(0) and b2(6) values. The work then proceeds as already described for plane-parallel electrode structures. senting the triode structure of Fig. 14. To illustrate this procedure by a numerical example, the dimensions used in the computation of the characteristics of a Table 4 series of experimental triodes, type C302, are given in Table 3, Relative a cross-section of the electrode structure and the calculated and Valve No. G cathode "ni-. av. a measured static characteristics being shown in Fig. 14. area 1 0°~30c' 0-333 0 140 0- 765 58 0•32 Table 3 2 30°-50° 0-222 0 147 0- 94 0•26 3 50°-90° 0-444 0 250 1- 10 83 0 •025 e mm mm In multi-grid valves it is sufficient to apply this somewhat on 0140 0-645 10° 0140 0-74 involved method only to the control-grid region, working with 20° 0 140 0-80 the average values of the electrode distances for the other parts 30° 0140 0-88 of the valve. 40° 0-145 0-96 50° 0160 1-00 The analysis of a complicated electrode structure by assuming 60 0-220 105 several valves working in parallel is necessary if the details of 70' 0-320 112 the valve characteristic are required, particularly for lower 80 0-320 1-30 current values. However, it is frequently desired to investigate 90° 0-350 1-40 approximately the relative influence of a change of control-grid wire diameter or grid pitch only, leaving all other dimensions These figures were obtained from a scale drawing enlarged unchanged. In this case the process of compounding the 20 : 1. As the variations of 6,(0) and b2(Q) in this case are elementary valves into a small number of representative valves 146 LIEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS can be carried to a final step—the derivation of a single equiva- the contact potential usually observed is of the order of — 1 volt, lent valve. This process is identical with determining a "shape although varying considerably from valve to valve. Considering factor" to correct for the deviation of the valve structure from that the low contact potential is due to a "contamination effect," the ideal. this variation is not surprising. The author is not aware of much published material on this subject, the only recent publica- 17 (7) EXTENSION TO MULTI-GRID VALVES tion being an article by J. A. Darbyshire. The reduction of a multi-grid valve to an equivalent diode In order to obtain more information, the contact potentials can be carried out in successive steps in a way similar to that of 32 experimental valves (including cathode-ray tubes) and of in which eqn. (5) was derived. The value of the effective control- 26 mass-produced valves of various makes (British and American) were measured by Schottky's method.18 The distribution of grid voltage VQX, applied to a substitute anode in the position of the control-grid wire centres is given for a pentode, within contact potentials for the two groups of valves is shown in 1 to 2%, by the following formula: Fig. 15. It can be seen that the mass-produced valves show a lOr -5,_> 2(\ mass-produced valves (15) if! In the case of closely wound grids, as used in many high-frequency pentodes, eqn. (15) can be further simplified (with Vg3 = 0) to 11 3? experimental valves (16) In these two equations, the amplification factor JU<^ is that of the mth grid, using the distance bmn between the mth and nth electrodes for multiplication by the value of JHQ taken from Fig. 4. The grid control factors Fn are defined by 1 F.. ~ 1 + 0 -02 :04"06'-tog-KT-l-2 ' -|-4 "-1-6'-18'- and Vgl, Vg2, and VgZ are the control-grid, screen-grid and Contact potential. volts DVer suppressor-grid voltages respectively. The current density obtained with the value of V'GX calculated Fig. 15.—Distribution of contact potentials. from eqn. (15) and used in conjunction with Fig. 2 is, of course, the cathode current density (ig2 + ia). To find the anode contact potential varying from — 0-40 V to — 1 • 30 V, if a series current density ia and screen-grid current density ig2, the of six 6V6G beam tetrodes with unusually high contact potentials cathode current has to be split up as shown by Spangenberg16 is disregarded. The average value for this group (excluding the (with ^2"3/(l + ^)~1 and Vg2 = Va): 6V6G series) is - 0-83 V; the average for the series of 6V6G valves is — 2-33 V. The distribution curve for the group of experimental valves ig2 + ia a2 extends to a slightly lower value of contact potential, — 0'35 V, and includes some valves with rather greater contact potentials, 2c2 and a2 being the wire diameter and pitch of the screen grid. up to — 1 • 8 V. This is probably due to the fact that these experimental valves were made under less standardized con- (8) CONTACT POTENTIALS ditions; but the possibility cannot be excluded that the distribu- To complete our system of amplifier valve design we have to tion for the mass-produced group (apart from the 6V6G series) know the contact potential Vc between control grid and cathode. is more uniform, because valves with very high or very low The effect of the contact potential is to increase the negative contact potentials have a greater chance of being rejected by grid bias by an amount Kc, and for accurate forecasts of valve the makers' final test. The average value of the experimental characteristics the term Vgl in all our earlier equations has to group is — 0*90 V, not very different from that of the mass- be replaced by (Vgl + Vc). Particularly in high-slope valves produced series. with short grid-base the variation of contact potential is one During the production of the experimental valves no special of the major causes of variations in valve characteristics. attention had been given to the question how different pro- The contact potential is given, apart from a small contribution cessing may influence the contact potential, although it was due to the Peltier effect, by the difference of the work functions noticed that there was a certain tendency for a series of valves (in volts) of the cathode material and the material of the control to have either a high or a low average contact potential. A grid. The grid wire consists in most cases of molybdenum, and similar tendency was found in the various series of mass-produced the difference of work functions of clean molybdenum and a valves. barium-oxide cathode is — 3 • 27 volts.17 Owing to the evapora- The contact potential also appears to depend to some extent tion of barium from the cathode on to the control-grid wire on the ageing process, as shown in Table 5, which gives the during the manufacturing process and during later operation, contact potentials of a series of four experimental diodes after the work function of the grid wire is lowered considerably, and normal ageing and after prolonged further ageing. LIEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS 147 Table 5 As an example, the following is a list of the average dimensions of the EL50 valve:— Contact potential Ve (volts) = 0-48 a,(av.) = 0-625 2c, = 0 060 Valve No. 1699 1700 1701 1702 bit = 1 13 «2= 1-20 2f2 - 0080 After normal ageing -0-58 - 105 - 1-52 -0-70 l2 After prolonged ageing -0-50 -0-43 -0-60 -0-62 = 3-3 a, = 4-5 2r3 = 0-125 = 1-5 From these data we obtain (9) THE CALCULATION OF CHARACTERISTICS OF MASS F, =0-730 PRODUCED AMPLIFIER VALVES >v = 0-687 The method of predicting the characteristics of radio valves #/ " — 1*7 n" = 1O*Q F — 0*714. described in this paper is sufficiently accurate for the calculation LI",' =1-85 LI'" = 0-84 F3 = 0 • 366 of the characteristics even of complicated mass-produced valves in considerable detail. For the control grid, a was found to be 0-20, and the measured As typical examples of modern valves the following types average control-grid pitch variation was expressed by the were chosen: standard deviation a = 5% (see Section 12). The effective average control-grid voltage is then (for Vg3 = 0): EL50 (output pentode) EF50 (h.f. pentode) V'GX = 0-698 [Vgl + VC + 0-0826(^2 + 0-040KJ] EF9 (variable-mu pentode) The numerical factors 0-698 and 0-0826 are, of course, 6V6G (beam tetrode). modified for those calculations which involve considerations of Of each type, 3 to 6 good, new specimens were carefully a or a. measured for static characteristics and contact potentials. The The measured and calculated characteristics of the four types valves were then opened and their internal dimensions measured of valve are reproduced in the self-explanatory diagrams, with the help of a projecting microscope. The deviations of Figs. 16-19, which show quite satisfactory agreement. the dimensions of the individual component parts and of the alignment from the average values were noted, but the compu- tations of the valve characteristics were based on the average values. For this reason the calculated characteristics show in some cases a sharper current cut-off than was measured (see also Sections 12 and 13). -30 -20 -10 -6 -5 -4 -3 "-2 -I Control grid voltage l^. volts Control grid voltage fy, volts Fig. 16.—EL50 valve. Fig. 17.—EF50 valve. Calculated, a - 0, Ve = - 0-8 V. 250 V; Vt3 = 0 V; 25OV;Ke= -0-8V. — • —• Calculated, with correction for grid-pitch variation, o = 5 ° Calculated (o 0). Makers' published curve. Limits of measured characteristics. — Limits of measured values. Makers' published characteristic (old type EF50). Va = V,% = 275 V. -•—•- Calculated, with correction for grid-pitch variation, a 5%. 148 LIEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS always involves the aggregate values, and does not permit this 10000 insight. Another advantage of the theoretical method is that certain defects, such as grid pitch errors, misalignment of electrodes or deformation of electrodes (if these are not too great) can be introduced deliberately, and their effect on the valve charac- teristic can be studied better and more speedily than can be done experimentally. Further, the theoretical method permits the design, on paper, of valves which would not be practicable, because for instance the grid wire becomes too thin or the electrode clearances too small. Although not immediately practicable, it is frequently of interest to know what the characteristic of such a valve would be. Similarly, the valve characteristic can be calculated for conditions in which the practical limits of anode dissipation or other limiting factors are disregarded. Of the many possible applications, only three problems will be considered, namely the influence of variation of grid wire diameter, the effect of statistical grid-pitch variation, and the effect of cathode misalignment. (11) THE EFFECT OF GRID-WIRE VARIATIONS This effect depends to a certain extent on the other design parameters. For this reason the following plane, compromise -70 -60 ^50 -40 -30 -20 -10 0 Control £rid voltage I£i .volts triode was chosen as a basis: b1 = 0-200 mm, b2 = 0-600 mm, JU2 = 35-5, cathode area Fig. 18.—Mass-produced variable-mu valve (type EF9); V - 250 V. a = 1 cm2, Va = 250 volts. Manufacturers' published curves. O Calculated; maximum pitch 0-50 mm. The characteristics were calculated with the following grid- ~. Calculated; maximum pitch, 0-49 mm. wire diameters: 2c - 0 (ideal grid), 0-030, 0-040, 0-060 and 0-100 mm. In each case a grid wound with perfect evenness (a = 0) was assumed. The result is plotted in Fig. 20. Fig. 19.—6V6G valve. Limits of measured characteristic. Calculated (Ve = — 2-3 V), to a first approximation. G Typical operating conditions, according to makers. (10) SOME SPECIAL APPLICATIONS OF THE DESIGN THEORY The method described of accurately predicting the static anode- current characteristic of an amplifier valve is essentially synthetic. Fig. 20.—Effect of variation of grid-wire diameter. The characteristic is built up from the current contributions of b-i = 0-200 mm; *2 = 0-600 mm; A — 1-00 cm*; no= const. = 35-5; Va 250 V. many elements of cathode area, and it is instructive to see how Curve No. 1 Ideal grid. certain changes in a design parameter affect the various parts 2 2c = 0-030 mm, ./ = 0164 mm, a •= 0053. of the cathode to different degrees, particularly if the electrode 3 2c = 0040 mm, a = 0195 mm, a = 0128. 4 2c = 0060 mm, a - 0-250 mm, a = 0-330. structure is at all complicated. The measured characteristic 5 2c = 0-100 mm, a = 0-350 mm, a = 0-930. LIEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS 149 It can be seen that with increasing grid wire diameter the Judging by the grids taken from opened valves, mass-produced anode current curve becomes more rounded near cut-off. The and experimental, the author ventures the following opinion: point where the curve starts to depart from the ideal moves gradually up to higher current values. For the grid-cathode excellent grid distance of 0-200 mm, the characteristic becomes noticeably good grid worse if the grid wire diameter is increased beyond 0-050 mm. a = 10% usable grid (limit) For a grid-cathode distance of 0-120 mm, as employed in some o -20% bad grid modern valves, the critical wire diameter is as small as 0-030 mm; For most grids taken from opened valves, a was found to lie the characteristics of some modern valves would be improved if near 5 %, but a larger value was sometimes found in small-pitch the grid wire diameter were smaller than it is. grids made from thin wire. The effect of changes in the grid wire diameter is still better It would not be practicable to calculate valve characteristics demonstrated by the anode-current slope curves obtained by on the basis of the continuous distribution of the Gaussian error differentiating the anode current curves. A plot of the change curve; this curve can be replaced with sufficient accuracy for the of slope is given in Fig. 21. This plot refers to the specific purpose by a five-step approximation, as given in Table 8. Table 8 Step Nominal value No. Range of deviations of deviations Relative weight 1 Under a(l - 1-5CT) a{\ - 2a) 0-067(1 - 2a) 2 o(l - l-5a)too(l -0-5cr) a{\ - a) 0-242(1 - a) 3 a{\ -0-5 Grid wire diameter 2c, mm The relative weights were obtained from Peirce's Tables,20 the Fig. 21.—Reduction of anode-current slope with increase of grid-wire factors in brackets providing for the fact that the corresponding diameter. cathode area is changed if the pitch varies, the total number of bi - 0-200 mm; bt = 0-600 mm; Va = 250 V. grid turns remaining constant. dimensions of the assumed valve. If used with discretion, the To demonstrate the effect of statistical grid-pitch variation, diagram can yield more general information. The change of the characteristics of the same valve as in the last Section, with relative slope is chiefly a function of the variable-mu constant a, 2c = 0 • 060 mm and a = 0 • 250 mm, were computed for a = 0, which depends only on the ratio of dimensions and not on their absolute values. If, therefore, the linear dimensions are all changed by the factor K, the slope reduction curves are still valid (with 2cK as abscissa) except that, to a first approximation, the current densities in mA/cm2 used as parameters should be divided by K2. (12) STATISTICAL VARIATION OF GRID PITCH We consider now the effect of a variation of grid pitch around an average value. The pitch already varies in grids taken direct from the grid winding machine; in the subsequent manufac- turing operations they may get worse but they will rarely improve. It is reasonable to expect that in a continuously wound grid the pitch will vary statistically, and sometimes a systematic variation due to cam operation or similar causes may be superimposed on this. Of the many ways of describing pitch variations the author prefers to define them by the statistical standard devia- tion, which is defined by the equation: £82 (18) where n represents the number of turns counted and 8 is the deviation of the pitch of the individual turn from the average pitch of the grid. These definitions are taken from the theory of errors, based on the Gaussian probability curve.19 Fig. 22.—Effect of statistical variation of grid pitch. To check whether the representation of the pitch distribution bx = 0-200 mm; b2 = 0-600 mm; A = O-lOOcmZ; Va = 250 V; Yt - - OS V. Curve No. by a Gaussian curve is reasonably correct, a number of grids 1 Ideal grid (very thin wire, a -; 0° '). of 100 turns each were carefully made and the pitch distribution 2 a = 0%^ 3 n = 5% Uc -0060 mm was measured with a travelling microscope. The distribution 4 a -» 10°^ fa 0-250 mm 5 a =-- 20 %J curves approached a Gaussian error curve in every case as well a -- Standard deviation from average value of grid pitch. as could be expected for such small quantities. Similar tests were applied to the grids taken from opened mass-produced 5%, 10% and 20% (see Fig. 22). The effect of grid-pitch variation valves, and it was concluded that the standard deviation can be is quite marked, increasing rapidly with increasing a. The anode considered a fair measure of the perfection of grids. current appears increased for all parts of the curve; this increase 150 LIEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS is .most pronounced for low currents, where a long "tail" results 2c/a the reduction of relative slope for a given value of a is from any appreciable value of o\ greater or smaller. We can prepare from Fig. 22 a graph giving the reduction of the relative cathode-current slope for various current densities, (13) EFFECT OF CATHODE MISALIGNMENT taking a as abscissa [Fig. 24 (b)]. As before, this graph is of To investigate the effect of cathode misalignment, the same more general importance; and again the current density has to 2 valve as in the last Section was taken as an example, with a = 0. be multiplied by \/K if the linear dimensions are changed by a The valve was considered as composed of two equal halves with factor K. However, this set of curves refers to a specific value the only difference that one half was displaced by an amount of Icja = 0-24. For considerably smaller or larger values of A^t towards the cathode, the other by an amount J^bl away from it, leaving the average value of bl unchanged. The resulting set of current characteristics is shown in Fig. 23. It will be seen that small amounts of misalignment do not affect the characteristic appreciably; greater amounts of misalignment shift the characteristic to larger current values, but the slope appears only slightly reduced for moderately large misalign- 500? 1 i / / i / 4002- i *' • i' / ;/ / / // y 3 3002 1 / / / s / / / // / E L,. -lOinA/cni* I 2007. 1 100% Fig. 23.—Effect of misalignment of cathode. b2 = 0-600 mm. V, = 250 V. 2< = 0-060 mm. Kc= -0-8V. a = 0-250 mm. A -= 1-000 cm*. 0 10% ~~~Wl 30% Variation of distance between grid and cathode —rp1 . Curve No. Fig. 25.—Effect of cathode misalignment on cathode current density. 1 ± o 2 -t io% Ratio of maximum average current density to average current density. 3 ± 20% Ratio of maximum local current density to average current density. 4 ± 30% 6, = 0-200 mm; b% = 0-600 mm; 1c •= 0060 mm; a = 0-250 mm; V, = 250 V; Vc = — 0-8 V;/l - 1 OO cm.2 100% 50? (a) Cathode misalignment (b) Statistical variation of grid pitch 0 10% 207. 307. 0 5% 10% 157. 207. Variation of distance between grid and cathode -r-1 Standard deviation a (a) Cathode misalignment. (fi) Statistical variation of grid pitch. Fig. 24.—Reduction of anode-current slope by variations in manufacture. LffiBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS 151 ments. As expected, the greatest change takes place near (3) LANGMUIR, I.: Physical Review, 1923, 21, p. 419. current cut-off. In contrast to the effect of grid-pitch variation, (4) OERTEL, L.: Telefunken Roehre, April, 1938, No. 12, p. 7. the ensuing "tail" is not very marked. (5) GLOSIOS, T.: Hochfrequenztechnik und Elektroakustik, 1938, These facts are also shown by the cathode-current slope graph 52, p. 88. of Fig. 24(a). (6) FREMLIN, J. H.: Philosophical Magazine, 1939, 27, p. 709. A more serious consequence of bad misalignment of the (7) CHILD, C. D.: Physical Review, 1911, (I), 32, p. 492. cathode is shown in Fig. 25, giving the maximum average current LANGMUIR, I.: ibid., 1913, (II), 2, p. 450. density and the maximum local current density at the cathode (8) BLEWETT, J. P.: JournalofApplied Physics, 1939, 10, pp. 668 as functions of the relative misalignment A^i/^i • The maximum and 831. local current density is that found when the variable-mu effect (9) MAXWELL, J. C: "A Treatise on Electricity and Mag- is taken into account (for the particular grid 2c/a = 0-24, netism" (Oxford University Press), 1st ed., 1873, 1, bja = 0-80); the maximum average current density shown in p. 248; 3rd ed., 1904, 1, sect. 203. Fig. 25 is that obtained when the variable-mu effect is neglected. (10) BENHAM, W. E.: Proceedings of the Institute of Radio The limit of the ratio of maximum average current density to Engineers, 1938, 26, p. 1093. average current density is of course 2-0; in this case one-half (11) Dow, W. G.: "Fundamentals of Engineering Electronics" of the valve supplies the whole current. (John Wiley and Sons Inc., New York, 1937). The ratio of maximum local current density to average current (12) HERNE, H.: Wireless Engineer, 1944, 21, p. 59. density rises steeply to a high value, of the order of 5 to 8, but (13) TELLEGEN, B. D. H.: Physica, 1925, 5, p. 301. does not continue this trend for the upper part of the curves (14) MALOFF, I. G., and EPSTEIN, D. W.: "Electron Optics in (not shown in the Figure). Television" (McGraw-Hill Book Co. Inc., New York, As far as the cathode is concerned, the maximum local current 1938), pp. 137 and 143. density is of importance; for the other electrodes the maximum (15) BENJAMIN, M., COSGROVE, C. W., and WARREN, G. W.: average current density plays the bigger part. The considerable loc. cit., Fig. 3. increase of current densities with an increase of £j>xlbx can (16) SPANGENBERG, K.: Proceedings of the Institute of Radio easily lead to a detrimental local overload of the cathode or of Engineers, 1940, 28, p. 226. other electrodes without exceeding the nominal rating of the (17) DARBYSHIRE, J. A.: Proceedings of the Physical Society, valve. 1941, 53, p. 219. (14) CONCLUSION (18) SCHOTTKY, W.: Annalen der Physik, 1914, 44, p. 1011. Many further considerations enter into the problem of radio SCHOTTKY, W.: Physikalische Zeitschrift, 1914, 15, p. 872. valve design. Some of these are concerned with technological (19) REDDICK, H. W., and MILLER, F. H.: "Advanced Mathe- questions, some with the co-ordination of circuit requirements matics for Engineers" (John Wiley and Sons Inc., New and valve characteristics, and many of the conditions which a York, 1938), chap. 9, sects. 88 and 89. valve has to satisfy simultaneously are somewhat contradictory. (20) PEIRCE, B. O.: "A Short Table of Integrals," 3rd ed. The method of valve design presented in this paper forms a (Ginn and Co., Boston), pp. 116-119. complete system which has proved a useful tool in arriving at the best compromise in each specific case. It has been used by (17) APPENDIX the author for several years in designing experimental valves, Maxwell9 gave the following expression for the potential near many of these ab initio. a charged grid (wire grating) at the position y = 0 between two The system appears to be capable of dealing with rather charged planes detailed questions concerning the shape of valve characteristics, and P2: provided enough time and trouble can be taken over the matter. In many cases, however, only approximate answers are required, V = — A log [ 1 —2e a cos f- e -**,+ c. and then the use of the various design charts should lead to a a speedy solution which will not be too far removed from the experimentally confirmed facts. assuming 2cfa <^ 1 and b2 > a; A and A' are the electric charges per unit length of grid and anode plane respectively; the other symbols have the same meaning as in Fig. 4. (15) ACKNOWLEDGMENTS We transfer the grid to the position y = b{, bringing the The author had an opportunity to check the design data used plane Pj to y — 0, and assume an electric image of the charged in the calculation of the Mullard valves with Dr. J. D. Stephenson grid at the distance — bx behind P,, thus making Pj a true of the Mullard Radio Valve Co., Ltd., and he was also able to equipotential plane even if near the grid. This leads to use some of Messrs. Mullard's valve design specifications. He wishes to record his indebtedness to Mr. T. E. Goldup and to . 2TT(> - b.) 2TTX Dr. Stephenson for the facilities given. cosh ~ — cos V = - A log The author also expresses his thanks to Messrs. L. Theobald . 2vy + b, 2TTX and F. Wade for their assistance with the experimental part of cosh — cos the paper, and to Mr. G. B. F. Goff for his help in the prepara- tion of the numerous diagrams. He further wishes to thank the Our boundary conditions, which determine the constants A, A' Directors of Cathodeon, Ltd., for permission to publish the and Cj are: paper. V = K, for y = 0, "| (16) REFERENCES bl + b2, I . . . . (19) (1) BENJAMIN, M., COSGROVE, C. W., and WARREN, G. W.: V =V for y = b x - c. J Journal I.E.E., 1937, 80, p. 401. g lt J (2) THOMPSON, B. J.: Proceedings of the Institute of Radio Eliminating A, A' and Cl with the help of eqns. (19) gives [with Engineers, 1943, 31, p. 485. log 2 cosh {Airbja) ~ log 2 sinh {Airbja) ~ 4TTZ>,/«]: 152 LEEBMANN: THE CALCULATION OF AMPLIFIER VALVE CHARACTERISTICS •cos h 2TT(>' - 6,) — cos — a a 2y, „ . trc V(x v) = - log r- log 2 sin — (20) TTC , 27r + bx) 2nx (b -f b )2 log 2 sin cosh — — cos l 2 a a To obtain the fields in the regions bounded by P, and G, For 27rxla = ± n/2 we have cos (27rx/a) = 0. Assuming and G and P2 respectively, eqn. (20) has to be differentiated tanh 27r6j/tf ~ 1, Q)Vft>y)y » 0 can then be written as with respect to .v. This leads to somewhat complicated expres- sions which will be omitted for sake of brevity. As we are 1 mainly interested in the fields near the plane electrodes P, (y = 0) and P2 (v =- 6, + b2), we write down here only those simplified equations resulting from putting y = 0 and y = 6j + b2 in the expression for 'dV(x,y)l'dy, and re-arranging, neglecting for the where [x2 has the former meaning. moment the term cos (2iTx)(a as compared with cosh For cos (2nxfa) # 0, we can write approximately, with an error of about 1 % for bja = 0-5), 7TC 1 .. 1 b.-ib, + 1 1 + and with a . TTC sinh -—! 477 y ft, -f- b a , _ . -c a 2TTX b2 - (6, -f cosh -—- cos —j- log 2 sin a a fl2 (X) = - 27T-6, 277-6, Putting sinh •2 log 2 sin — a log 2 sin — a , 27r6. 2TTX a cosh 1 — cos — and which can be simplified,' again with an approximate error of a log 2 sin — about 1% for bja = 0-5 [neglecting (bl/b2) cos (ZTTX/O) since a this term is much smaller than sinh (2.7Tbja)] to: these two equations become sinh 2TTX\ b. 2ttx cosh —- cos ) — -- cos a a J b-, a With tanh (lirbja) ~ 1, we can re-write this as [eqn. (10) of 1 -r the main text] and 2 1 — a cos a b y ~bi+ b-2 2 1 + b, with 27Tb, sinh The first of these is eqn. (2) of the main text. 2-b, We consider now the case bja <^ 1, where we can no longer As a rule we can put sinh Q-Trbja) ~ \e and as neglect cos {2irxla), but put for sake of simplicity V2 — Va and ?,/x2/62 = jUj we obtain our simple formula (11): Vx =- 0; i.e. we consider only the cathode-grid-anode system. We find in this case a = 2(1 + fJt^e—1 sinh - sin• hu 2jrbl' Ar.b, 477-6, 2 log 2 sin - 1 27T6, 27T.V ,cosh cos v 0 2(6. — b-,) log 2 sin — — a a