Projection engineering (Jan-Dec 1931)

Page 24 PROJECTION ENGINEERING The calculation of supply voltage, and oad impedance for a vacuum tube d Ip IP By E. K. Brown* WHILE investigating the characteristics of vacuum tube loads from the Ep-Ip curves it occurred to the writer that it should be possible to derive by mathematical analysis an expression which would give the value of load impedance and necessary supply voltage to be used under any given values of grid swing and requisite plate voltage. The final result was a simple formula which the writer has not seen previously published, and which, on account of this simplicity and its adaptability to other conditions, it is considered might be of interest. It will be noted that expressions are developed for two cases, i. e., a load impedance equal to the tube impedance and a load impedance equal to double the tube impedance, but the expressions can be easily extended to include load impedances having any given relation to the tube impedance. The result of the analysis and its application suggest some limits to grid swing apart from considerations of a positive grid, and of this it is hoped to treat in a future paper. The plate current (Ip) in a tube is a function of the grid voltage (Eg) and the plate voltage (Ep). -/ Ip = T (Ep,Eg) (1) The impedance (Zp) of a tube working as an amplifier may be taken, to a sufficient approximation,1 as the reciprocal of the slope of the Ip-Ep characteristic or, 1 Zp = ■ (2) / Ep.Eg) To secure maximum power output from a tube it can be shown that the load impedance (ZL) must be equal to the tube impedance (Zp). Now the slope of a load line on the Ip-Ep graph is the reciprocal of the impedance, or : 1 d Ip (3) ZL d Ep , For maximum power output then from a tube the slope of the load line through the working point on the IpEp curve must equal the slope of the Ip-Ep curve at that point, or, 1 d Ip 1 £• (Ep,Eg) d Ep Zp ~f (4) * Western Electric Company (NZ) Ltd., Auckland, N Z. The load line intersects the x axis at the point of supply voltage (Es) and therefore, Starting with the functional expression for the plate current a relation is developed between the expressions for tube impedance and load impedance when these are equal, and this relation is developed into an expression giving the value of supply voltage necessary under these conditions. The value of the impedance is found and an example of the application of the expression given. . . . The expression is further developed to include the case of a load impedance double the value of the tube impedance, and an example of its application given. (5) d Ep Ep-Es It will be noted that this slope is negative, indicating its direction, but since this point is of no concern in what follows we shall use the absolute value only, or : d Ip d Ep substituting in (4) '(Ep.Eg) Ip Ep-Es Ip Ip / Ip Es-Ep (Ep,Eg).(Es-Ep) Es-Ep (5A) (6> (6A) Van der Bijl1 gives the equation for Ip, Ip = a(Ep + ye Eg + e)2 (7) As pointed out by Van der Bijl, this equation, while not strictly true,2 is sufficiently accurate for practical purposes. (6A) then becomes, d Ip = a(Ep 4 ji Eg 4 £)2 = [a(Ep 4 (xEg4-£)2] (Es-Ep). d Ep (8) or, neglecting the small quantity e, a(Ep 4 \i Eg) = 2a(Ep 4 \x Eg). (Es-Ep) (8A) which can be written, 3 Ep 4 (i Eg Es = ? (9) where /j = amplification factor. It can be easily shown that the actual amplification constant (p) of a tube is related to the maximum amplification constant (fim) by the relation ZL jx = [im (10) zP 4 zL Thus the higher the load impedance the more nearly the actual amplification factor ( ju.) approaches the theoretical or maximum amplification («m). The expression (9) gives the value of supply voltage necessary to secure any given plate voltage on a tube at a given value of Eg when connected into a load equal in impedance to that of the tube. As an example of the application of this equation take the curves given in Fig. (1) for a Western Electric 205-D tube. Assume it is desired to work the tube at a plate voltage of 300 v. and as a peak grid swing will not exceed, say 25 volts, allow — 25 v. grid bias. This '■"Thermionic Vacuum Tube," Van Der Bijl, p. 151, p. 165. 2Equation (9) is strictly true only for that part of the characteristic curve over which (Ip)% = (Eg -f ..Eg) is a straight line.