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THE CALCULATION of AMPLIFIER VALVE CHARACTERISTICS* by G

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THE CALCULATION of AMPLIFIER VALVE CHARACTERISTICS* by G 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^ \_^^~ ^^,— —' <yI/I yy /r ^> 'A ' w-fi- ' /] /• ^0^^ 6 / y \,y*^" y^ ^y^ 1 ^ 1 wU /y ^^ . 1 //J v ' ^ III if/ / <A _^y WlfiA k y^ 1 ^ 1 /\ y' 20 A ^y* ill77AA A A 1 1 "8 7/ I 1 10 01 As AI 3 ft j A I • 1 Wi i / t III A 1 WIN// / \ / I M /1 / ' 1 1 1 1 • 5 III/,/ / 1 1 I X 1 1 1 1 i 1 1 1 | J 1 1 1 1 tW 1 1 1 It 1 2 1 1 1 1 1 1 1 | 1 Mi 1 1 1 1 1 1-0 j 1 • 1 1 001 1ti.i. 1 , 3-0 15. 10 06 04 0-3 025 02 015 01 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 •.
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