OPT S7 
 
 The Unilateral Dynamic Characteristics 
 of Three-Electrode Vacuum Tubes. 
 
 BY 
 
 JOHN G. FRAYNE 
 
 
 A THESIS 
 
 SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE 
 
 UNIVERSITY OF MINNESOTA IN PARTIAL FULFILLMENT OF 
 
 THE REQUIREMENTS FOR THE DEGREE OF DOCTOR 
 
 OF PHILOSOPHY. 
 
 Reprinted from the PHYSICAL REVIEW, N. S., Vol. XIX., No. 6, June, 1922. 
 
The Unilateral Dynamic Characteristics 
 of Three-Electrode Vacuum Tubes. 
 
 BY 
 
 JOHN G. FRAYNE 
 
 A THESIS 
 
 SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE 
 
 UNIVERSITY OF MINNESOTA IN PARTIAL FULFILLMENT OF 
 
 THE REQUIREMENTS FOR THE DEGREE OF DOCTOR 
 
 OF PHILOSOPHY. 
 
 
 Reprinted from the PHYSICAL REVIEW, N. S., Vol. XIX., No. 6, June, 1922. 
 
Q 
 
[Reprinted from THE PHYSICAL REVIEW, N.S., Vol. XIX., No. 6, June, 1922.] 
 
 THE UNILATERAL DYNAMIC CHARACTERISTICS OF 
 THREE-ELECTRODE VACUUM TUBES. 1 
 
 BY JOHN G. FRAYNE. 
 
 SYNOPSIS. 
 
 Unilateral dynamic characteristics of vacuum tube when plate circuit includes 
 resistance, inductance or capacity. (i) Theoretical equations. For the case of pure 
 resistance (R), the Van der Bijl parabolic relation between plate current and effec- 
 tive grid voltage is expressed as a power series in e sin pt, the impressed simple har- 
 monic grid voltage. The coefficients of the various harmonics involve R, the nth 
 harmonic reaching a maximum value when R equals (n-2) /3 times Ro the tube re- 
 sistance. For the fundamental the maximum energy output for a given plate bat- 
 tery is secured when R 0.8 1 l?o. The dynamic characteristic was obtained by 
 compounding the harmonics into a single curve; it approaches a straight line as the 
 resistance is increased. For the case of pure inductance, the plate current is ex- 
 pressed as a Fourier series. The dynamic characteristic is a closed loop whose area is 
 proportional to the energy in the inductance. This loop reduces to an ellipse for 
 small values of e, in which case the tube functions as an alternator \vhose internal 
 impedance is a function of the external load. The insertion of a condenser instead of 
 an equivalent inductance gives identical results except that the phase angle of the 
 various harmonics is shifted. (2) Experimental verification. The effects on the 
 plate current of varying the alternating grid voltage e, the static grid voltage E c 
 and the plate voltage Et,, for a given value of resistance R or inductance I, and the 
 effect of varying R or I with constant Eb, E c and e (15 or 20 volts), were determined 
 and are shown in curves together with the corresponding theoretical values. A. W. E. 
 205B tube was used. The results show that the equations predict the harmonic 
 constituents of the plate current as high as the fourth, for values of e up to 15 or 20 
 volts (depending on j), the range for which the fundamental equation holds. For 
 this range the coefficients of the various harmonics in the equation are proportional 
 simply to e n . The fundamental becomes greater while the other harmonics di- 
 minish as we approach the straight portion of the static characteristic and as we in- 
 crease the plate potential. 
 
 Circuit for producing pure sine wave electromotive force with frequency of 200,000 
 cycles. The oscillating circuit and filters used are shown diagrammatically in Fig. i. 
 
 Pure resistance for high frequencies. A platinized[quartz fiber (diameter o.oi mm.) 
 with a resistance of 100 ohms per inch will carry 0.06 ampere when immersed in acid- 
 free paraffin oil and has a negligible skin effect. 
 
 f T is a well-known fact that the current flowing from a hot filament to 
 * the plate of a three-electrode vacuum tube does not vary as the first 
 power of the plate potential. With a view to determining what this 
 relation really was, theoretical and experimental investigations were 
 undertaken by Langmuir, 2 Bethenod, 3 Vallauri, 4 Van der Bijl, 5 Latour 6 
 
 1 Presented at the Chicago meeting of the American Physical Society, December, 1920. 
 
 * P. I. R. E., 3, 261-93, Sept., 1915, and PHYS. REV., 2, p. 457, 1913. 
 
 1 La. Lum. EL, 35, 25-31, Oct. 14, 1916. 
 
 4 L'Elettrotecnica, Vol. 4, Nos. 3, 4, 18 and 19, 1917. 
 
 6 PHYS. REV., n, p. 172-198, 1918. 
 
 6 La Lum. El., Dec. 30, 1916. 
 
' '-'JOHN G. FRAYNE. 
 
 and others. The second degree equation obtained by Van der Bijl lends 
 itself more easily to mathematical treatment than any of the others, 
 and agrees very closely with experimental evidence over a certain range 
 of plate and grid potentials. 
 
 The curves obtained by plotting the plate current against the grid 
 voltage for given plate potentials are usually referred to as the static 
 characteristics of the tube. The term "dynamic characteristic" is used 
 when the grid potential is of an oscillating nature. The latter charac- 
 teristic is usually referred to as being "unilateral" when there is no 
 external coupling between the grid and plate circuits, as distinguished 
 from "regenerative" when such coupling exists. Van der Bijl has shown 
 that the insertion of a resistance between the plate and the plate battery 
 changes the form of the dynamic characteristic from a parabola to a 
 curve which approaches a straight line with increasing resistance. A 
 solution similar to that of Van der Bijl is obtained here, and in addition 
 the case where the resistance is replaced by an inductance is worked out. 
 We shall consider three cases here. First, with no resistance in the plate 
 circuit, secondly, with a resistance inserted, and finally with the latter 
 replaced by an inductance. 
 
 CASE OF No EXTERNAL RESISTANCE. 
 Let E b plate potential, 
 /& = plate current, 
 EC = grid potential, 
 
 e sin pt = superimposed e.m.f. on grid. 
 According to the current-squared law 
 
 I b = A(E b + nE c + ne sin pt + e) 2 , (i) 
 
 where /* is the amplification constant, defined by 
 
 and A and e are constants depending on the structure of the tube. In this 
 case the grid potential has the value E c + e sin pt. The equivalent plate 
 potential is therefore /x (E e + e sin pt). 
 
 Before proceeding further it might be well to remark here that in order 
 that (i) may actually represent the true conditions the value of e must 
 lie within certain limits, namely 
 
 77, -L * 
 
 E c 
 
 e =i \E C 
 
 e s 
 
 A* 
 
 where g is the maximum positive voltage the grid can have before it 
 
VoL.^XIX.J THREE-ELECTRODE VACUUM TUBES. 63! 
 
 begins to attract many electrons. Also if e sin pt attains such a large 
 negative value in the cycle that the expression above is negative, the re- 
 sulting current wave will be flattened out at that part of the character- 
 istic curve. Equation (i) might be written generally as: 
 
 I b = f (fj.e sin pt) . 
 Expanding by Maclaurin's Theorem 
 
 /- - / (o) + W sin ptf (o) + Pt f" (o) 
 
 and since Ib is the same function of Eb as it is of sin pt 
 
 =2A ^ + MC + C 
 
 = 2 A. 
 
 \ d E b 2 J t =o 
 Therefore 
 
 . u.e sin pt A/J? e z . Ap? e z , , 
 
 I b = A (E b + fM E c + c) 2 + j " cos 2pt + -& . (20) 
 
 Thus in the simple case illustrated above where the plate potential is 
 kept constant throughout the operation, a pure sine wave on the grid gives 
 rise to a current of the same frequency (called the fundamental) in the 
 plate circuit, a first harmonic and a rectified current component. In 
 this case the actual dynamic and static characteristic curves will coincide. 
 
 CASE OF A PURE RESISTANCE. 
 
 Next we shall consider the case where the potentials on the grid and 
 plate vary simultaneously. Let a resistance R be connected between 
 the plate and the plate battery. Then 
 
 I b = A {E - R I b + M (E c + e sin pi) + e} 2 . (3) 
 
 This can be expanded as an infinite series. 
 
 J= /(O) | /'(o) Mg Gin^+ / " ( ) 
 
 2! 
 (tie sin pi) 2 + (pe sin pi) n (4) 
 
 Van der Bijl has shown that since the parabolic relation connecting 
 plate current and grid and plate potentials is only an empirical approxi- 
 mation, it is not to be expected that the higher derivatives in the series will 
 accurately represent the actual experimental values. However, the 
 
632 
 
 JOHN G. FRAYNE 
 
 [SECOND 
 
 LSEklES. 
 
 derivatives up to probably the third or fourth ought to be a close approxi- 
 mation and the higher derivatives ought to indicate, at least in a quali- 
 tative way, how the higher harmonics depend on the various tube con- 
 stants and on the properties of the external circuits. 
 
 Referring back to equation (4), the coefficients of the series are as 
 follows : 
 
 + 
 
 Ro ' 
 
 where 
 
 /(o) = 
 
 being defined as 
 
 2 A 
 
 /(o)=^ i-B- 
 
 = _ 2 
 
 -5/2 
 
 The general functional term is given by 
 
 fn ( ) = ( LL_ ^L. 
 
 The coefficient of sin () is therefore the value of this expression mul- 
 tiplied by (ve) n . If this coefficient is denoted by , then 
 
 Limit 
 
 n = . 
 
 n + I 
 
 2 ( + 2) 
 
 (* + I) S 
 
 In order that the series (4) may be absolutely convergent 
 
 2RA 
 ~B~ 
 
 < i. 
 
 Therefore 
 
 e < 
 
 B 
 
 2RA /*' 
 
 Thus for a given A, n and J?, the smaller the value of R , the greater e 
 may be. 
 
 Using the values of A, AC, R and J?o given later, e may have values 
 reaching up to 150 volts. However, it will be seen later that in practice 
 e cannot have a value larger than about 15 volts if equation (3) is to 
 represent conditions accurately. The physical limitations which the 
 tube imposes on the characteristic equation make it impossible to use 
 
VoL.^XIX.J THREE-ELECTRODE VACUUM TUBES. 633 
 
 grid voltages more than one tenth of the limiting value as given by (5). 
 It is very evident that for small values of e, the series (4) converges 
 rapidly and in consequence only a few terms need be evaluated in order 
 to find a close approximation to the actual current flowing under a cer- 
 tain condition of the amplifier. 
 
 Now /' (0) stands for the reciprocal of the total output resistance R' 
 when there is an external resistance in the plate circuit. 
 
 Therefore 
 
 i i 
 
 The total resistance of the complete plate circuit is thus a rather compli- 
 cated function of the external resistance and the internal output resist- 
 ance of the tube when there was no resistance in the plate circuit. 
 
 Since the series (4) is a power series in sin (pi) it is necessary to con- 
 vert the various powers of sin (pi) into first-power expressions of func- 
 tions of multiples of pt, and expressions corresponding to the rectified 
 currents. Since the series converges rapidly for values of input voltage 
 within the limits (2), all powers of sin (pt} beyond the fourth will be 
 omitted. 
 
 + 1 I ^ (i - B~ 1 ' 2 ) 1 - ^ A Z R M 3 e 3 ~ 5 / 2 l sin (#) 
 
 [I s 1 
 
 -A v?e 2 B~W + -A*R?n*e* B~ 7 / 2 cos (2 pt + TT) 
 2 2 J 
 
 + - \A* R M 3 e 3 5~ 5 / 2 1 sin 3 pt 
 I ^ 3 7? 2 /i 4 e 4 - 7 / 2 1 cos 4 pt + 
 
 Actual computation of the coefficients in this series show that for values 
 of e within the limits specified above, the series converges very rapidly. 
 
 For values of e below 10 volts, actual computations show that the co- 
 efficient of sin (pt} is practically a linear function of e; beyond ten volts 
 the term involving e becomes appreciable and the relation becomes more 
 complex. Similarly the coefficient of cos (2 pt + TT) varies as the square 
 of e up to about i o volts. Since we have taken no powers higher than 
 sin pt the coefficients of the third and fourth harmonics vary directly as 
 the cube and fourth powers respectively of e. 
 
 The relation between the coefficients and R, when the latter is variable, 
 can best be shown by examining the condition for maxima. If we take 
 
634 JOHN G. 
 
 the wth derivative as representing the coefficient of the general term, 
 then the latter will be a maximum when 
 
 d 
 
 dR 
 
 i.e., when 
 
 " R or (i + 2 R/R )- 1 = o. 
 
 The latter equation has a solution, R = <x> . This is, obviously, the con- 
 dition for a minimum. The first relation shows that R must be a nega- 
 tive quantity for n = I. Hence the fundamental has no real maximum. 
 For n = 2, the maximum occurs at R = o. For n 3, the maximum occurs 
 when the external resistance is one third of the tube resistance. For 
 the higher derivatives, the position of the maxima occur at continuously 
 increasing values of R. 
 
 If the amplitude of the impressed e.m.f. is less than 15 volts, equation 
 (3) holds good for all values of R, when the plate voltage is maintained 
 at 200 volts, and the grid voltage is 7.5. For values of e over 15 volts, 
 using the same grid and the plate potentials, equation (3) no longer holds. 
 Hence the amplitudes of the harmonics as experimentally found for 
 values of e over 15 volts depend on other features of the amplifier. Since 
 E c is 7.5 the grid will be raised to a positive potential of 7.5 volts 
 during this cycle. In Fig. 5 it will be noticed that at this value of E c 
 on the 2oo-volt parameter, the static characteristic begins to lose its 
 parabolic nature and tends to flatten out. From the nature of the static 
 characteristic it may be seen that the higher the plate voltage is raised 
 the greater the values e may have and remain within the proper limits. 
 This amounts to saying that the smaller R is, the greater the input volt- 
 age on the grid may be. 
 
 The dynamic characteristics for this case may be obtained as follows: 
 The instantaneous values of the various harmonics for values of pt be- 
 tween o and 27T are plotted, and then these constituent sine waves com- 
 pounded to give the actual wave shape. If now the resulting periodic 
 current is plotted along the /& axis and the input voltage plotted on the 
 EC axis, the resulting curve will be the so-called dynamic characteristic 
 of the tube under the specific conditions. It will be seen that if all terms 
 but the fundamental had been neglected, the characteristic would have 
 been a straight line. Addition, however, of the first harmonic causes 
 the characteristic to have a definite curvature. The smaller the value 
 of the external resistance, the more nearly does the curve approach the 
 parabolic relation holding in the static case, and, of course, in the limiting 
 case when R is zero, the two characteristics coincide. 
 
Na6 XIX '] THREE-ELECTRODE VACUUM TUBES. 635 
 
 CONDITION FOR MAXIMUM OUTPUT. 
 
 In connection with the expression for the internal resistance, it may 
 be pointed out that the usual statement that the maximum power is 
 obtained from a tube when the external resistance in the plate circuit is 
 equal to the internal output resistance of the tube needs clarification. 
 If by maximum power is meant the greatest power obtained from the 
 fundamental frequency, the following is valid. 
 
 Power = RP = ^ \ i B~ (6) 
 
 Therefore 
 
 dP _ _ /A 
 dR " jR 
 
 <fP ^r 
 
 = G> tor maximum r. 
 
 dR 
 
 Therefore 
 
 . = i 
 
 3/2 = 2B _ ! or 
 
 > 
 RO 4 
 
 The condition for a maximum dissipation of fundamental current 
 energy is that the ratio of R to the internal resistance when R was zero 
 is .81, This condition holds in the case where the maximum power is 
 desired with a certain fixed-plate battery, and a variable resistance is 
 available. The usual condition for maximum power, that the internal 
 and external resistances be equal, is only true in this case if the actual 
 plate potential is kept constant while the plate resistance is varied. The 
 condition under which the above relation was obtained is the one most 
 commonly met with in practice. 
 
 CASE OF AN INDUCTANCE. 
 
 When an inductance, /, is placed between the plate and plate battery, 
 the equation for the plate current may be written as follows: 
 
 7 = A \E b - l~ + p (E c + e cos p) + e f or (9) 
 
 B 2 - 2 B L~ - 2LF cos pt ~ + 2 BFcos pt + L 2 (~\+ F 2 cos 2 pt 
 at dt \dt ) 
 
 where 
 
 B = A 1 ' 2 (E b + vZc + ) L = A 1 ' 2 , F = A l '*ne. (10) 
 
 A rigorous solution of this differential equation for 7 is very difficult. 
 but an approximate method of solving it may be legitimately utilized. 
 Experimental evidence shows that / is a rapidly converging Fourier 
 
636 JOHN G. FRAYNE 
 
 series, and that the frequency of the fundamental is the same as the fre- 
 quency of the input e.m.f. on the grid. 
 We can therefore write : 
 
 oc oc 
 
 J = a /2 -f- ^ a n sin npt + ^ /3 n cos npt (n) 
 
 n=l 7i=l 
 
 In terms of the exponential values for the sine and cosine, 
 
 2a n sin npt + 2/3 n cos npt = (p n - ia n } e inpt + (ft, + ia n ) e - inpt . 
 
 Write 
 
 2a n = p n ia n , and 2b n = /3 n + ict n . (12) 
 
 Then 
 
 / = a /2 + f; a n e inpt +^b n e *i". (13) 
 
 -=! 71 = 1 
 
 If we substitute this value of I in equation (10), we can arrange the 
 resulting terms in ascending orders of e ipt and in descending orders of 
 e ipt . The expression will not be given here as it is very lengthy and 
 cumbersome. Since we have terms involving e lpt and e lpt and corres- 
 ponding higher powers of e on both sides of the equation, the coefficients 
 of the like powers on either side may be equated. Hence we obtain a 
 series of 2n equations from which the a's and b's can theoretically be 
 determined. The general solution of these equations, while ideally pos- 
 sible, is impracticable without further assumptions as to the nature of 
 the coefficients. We saw in the case of the resistance of the plate cir- 
 cuit, that only the first few terms of the series were of importance for 
 values of e within the limits of equation (2), and that for values of e up 
 to 10 or 15 volts, the amplitude of the fundamental varied approximately 
 as the first power of e. If all the coefficients other than a , ai, and bi, 
 are negligible, we have: 
 
 BAWpe 
 
 bi = - , (14) 
 
 i 2BLip 
 
 showing that, for this case, a\ and &i are linear functions of e. Substi- 
 tuting the above values of a\ and 61 in the expressions for a z and b 2 , we 
 have : 
 
 4(1 + ( } 
 
 ~ 4 (i - 2 BLipY (i - 4 BLip) ' 
 
 The values of a z and b% were found on the assumption that all the higher 
 coefficients were negligible. Similarly o 3 and 63 may be found and so on. 
 
THREE-ELECTRODE VACUUM TUBES. 637 
 
 From relation (22) the values of ai, /3i, az, /3 2 , etc., may be found, and 
 
 fj, e cos (pt a) . _. Ip 
 
 ai sm pt + ft, cos pt = . . , where a = tan l:3 (16) 
 
 (jKo + 
 
 and RQ = 
 
 e) ' 
 
 .4ju 2 e 2 cos (2 pt - j8) 
 sm 2 * + ft cos 2 * = , (17) 
 
 where 
 
 /3 - 
 
 Similarly a 3 sin 3 pt + /3 3 cos 3 / may be found and so on for the 
 higher terms. 
 
 Since I = a /2 + 2a n sin (/) + 2/3w cos w^>^, the addition of the 
 various quantities found above will give the resulting current 7. 
 
 It is obvious that as soon as the values of a% and b% become appreciable 
 compared with a\ and bi, the values of the latter obtained above can no 
 longer be correct, since they were determined on the basis that all the 
 other coefficients were negligible. 
 
 If the values obtained for o 2 and b 2 are substituted in the equations 
 for ai and bi, the following is the value of 
 
 fj.e cos (pt (pt a) 
 
 where 
 
 2 R 5 A*y?e* Ip cos (pt - e) . . 
 
 (R<? + p p*y (R Q Z +4 pp) ilz ' 
 
 . _. .. ? -2R 2\ 
 
 a = tan : - e = tan 1 ~ - and - ) . 
 
 <?-5PP 2 J 
 
 2 lp 
 
 In order to obtain a numerical value for a\ sin (pt) + jSx cos pt, it is best 
 to evaluate each term separately and then compound the results by the 
 parallelogram law. Similarly if the values of o 3 and b 3 become com- 
 parable with a z and & 2 , we find for the corrected value of 
 
 R 3 Atf& cos (2 pt - |8) 
 2 sin 2 pt + focos 2 / = 
 
 2 
 
 12 RJ A 3 fji 4 e 4 1 2 p 2 cos (2 ^ - X) 
 
 
 where /3 is the same as defined in (27) and 
 
 3 ^ ~ I7 
 
 X- an-* 
 
638 JOHN G. FRAYNE. 
 
 By making successive approximations as many terms as desired may 
 be included in the expressions for any particular harmonic. It will be 
 noticed that the resulting angle of lag of each harmonic depends on the 
 number of terms we include in the coefficient, and thus there arises a 
 peculiarity in a vacuum-tube generator, namely that the angles of lag of 
 the various output harmonics are dependent on the amplitude of the 
 input wave on the grid. The larger the amplitude for the coefficients of 
 the various harmonics, and the consequent shifting of the angles of lag 
 results. 
 
 AREA OF THE CHARACTERISTIC LOOP. 
 
 Since the fundamental plate current lags behind the grid voltage by 
 an angle a = tan" 1 (Ip/R) , it is evident that if this current be plotted against 
 the alternating grid voltage an elliptical characteristic will be produced. 
 If, however, the first and higher harmonics are plotted in addition to the 
 fundamental, and the curves thus formed compounded into a single 
 curve, it is evident that the characteristic will no longer be a true ellipse. 
 Since the amplitudes of the harmonics are small compared to that of the 
 fundamental, the resulting curve will not seriously depart from an ellipse. 
 This curve is what is usually referred to as the dynamic characteristic. 
 The a /2 term of the series gives the point of operation on the static 
 characteristic, and it is obvious from the expression for the latter, that 
 the larger the harmonics become, the greater is the shift of this point 
 of operation. In practice this shift is noticed by the increased reading 
 of a direct-current milliameter. 
 
 If all but the fundamental current is omitted, the area of the loop may 
 be easily found. 
 
 Put x = E c + e cos (pt) 
 
 Limits for pt are o and 2ir 
 
 X2T /2T I 1 
 
 ydx = - { ao/2 + cos (pt - a) [ sin pt dt 
 
 Jo I (Ro + fW J 
 
 27r e z u sin a 2ir /j. e 2 Ip . _. Ip 
 
 . _ _ _ * _ _ ci t"| (*f* sy * T~O t" -*- ,- _ 
 
 ~ (J?o 2 + W 2 " CRo 2 + Pp 2 ) ' Ro' 
 
 If the curve be referred to the /&, E b axes, this expression must be mul- 
 tiplied by /i. Also since the maximum value 7 of the fundamental is 
 jue (J? 2 + I 2 p 2 )~ 1/2 , the area of the loop may be written as 
 
 A = 2 -xlf lp = 2 7 r^ /o 2 . 
 S-o 
 
No L 6 XIX '] THREE-ELECTRODE VACUUM TUBES. 639 
 
 Since // 2 / 2 = the maximum dynamical energy in the inductance, the 
 area of the loop is thus proportional to that quantity. If the inductance 
 were not present A = o, which means that the characteristic no longer 
 has the form of a loop, but reverts back to the type found when a resist- 
 ance was placed in the plate circuit. 
 
 CASE OF A CAPACITY IN THE PLATE CIRCUIT. 
 
 Since a condenser placed between the plate and the battery prevents 
 the direct current from flowing, it is necessary to place a choke coil across 
 the condenser. The choke coil will allow the direct current to pass, 
 but if made properly will offer almost an infinite resistance to the high- 
 frequency current. 
 
 The solution for this case is directly analogous to that for the induct- 
 ance problem, the only difference in the final result being that i/cp 
 always replaces Ip. 
 
 Thus the simple expression for the fundamental becomes iJ.ej(R^ 
 + i/c z p 2 ) ll2 cos(pt + a) whereon = tan" 1 i/R cp. Similar expressions for 
 the other harmonics may be found from comparison with the expressions 
 found for the inductance. 
 
 The dynamic characteristic for this case is similar to that found for the 
 inductance, the only difference being that it is traced out in the opposite 
 direction. 
 
 DESCRIPTION OF APPARATUS AND EXPERIMENTAL PROCEDURE. 
 In order to have an experimental set-up which could be used to verify 
 the preceding theory, the following conditions and requirements had to 
 be met. 
 
 (a) Production of a pure sine wave e.m.f. 
 
 (b) Use of a sufficiently low frequency that capacity effects might be of 
 
 small magnitude. 
 
 (c) Accurate measurement of the input e.m.f. on the grid of the har- 
 
 monic producing tube. 
 
 (d) Use of a pure resistance. 
 
 (e) Use of a pure inductance. 
 
 (/) Measurement of the amplitude of the harmonics produced, without 
 introducing extraneous resistances, etc., into the harmonic producer. 
 
 Fig. i is the complete circuit diagram of the entire collection of appar- 
 atus used in the experimental work. It may be divided into three main 
 sections, the oscillator, harmonic producer and the harmonic ana- 
 lyzer. The oscillator in the upper left corner is designed so as to pro- 
 duce as pure a sine wave as possible. The tuned circuit L l C l prevents 
 
640 
 
 JOHN G. FRAYNE. 
 
 [SECOND 
 [SERIES. 
 
 the fundamental frequency from passing into the battery circuit, thus 
 compelling it to travel to the filament through the inductance L s of the 
 oscillating circuit. The condenser Ci offers less and less impedance to 
 
 EJ05-B E 
 
 Fig. 1. 
 
 the higher harmonics, and the latter will pass down through Ci to the 
 filament terminal. As the first harmonic is always appreciable, it was 
 specially filtered out of the oscillating circuit L S C S , by means of the anti- 
 resonant circuit LzCz. The inductance L 2 = .52 M.H., of course offered 
 some impedance to the fundamental frequency, but that was negligble 
 compared with the impedance that L\C\ offered to the fundamental. 
 These filters thus helped to produce a pure sine-wave oscillation of the 
 same frequency as the fundamental in the circuit L S C S . The frequency 
 used throughout was 200,000 cycles per sec., or a wave-length of 1,500 
 meters. This frequency was high enough that it could be tuned very 
 sharply, and yet not so high that the internal capacities of the tubes 
 would be of any importance. Ballantine 1 has worked out expressions 
 for the input impedence of tubes under various conditions, and applying 
 his formulae to the W. E. 205 B tube at this frequency and under the 
 experimental conditions which will be described below, the input impe- 
 dance was found of the order of 100,000 ohms. 
 
 The inductance Z, 4 was loosely coupled to L 3 and connected by means 
 of a twisted pair with L 5 , which in turn was loo sely coupled to L 6 . These 
 latter coils were placed about seven meters away from the oscillating 
 circuit, in order that they might not pick up any of the harmonics. The 
 
 1 PHYS. REV., 15, p. 409-420, 1920. 
 
NoT6 XIX '] THREE-ELECTRODE VACUUM TUBES. 64! 
 
 loosening of the couplers already described resulted in maintenance of 
 the sine e.m.f. The combination of condensers C, C&, C& and Cy and the 
 inductance L 6 is tuned for the fundamental frequency. The arrange- 
 ment of these condensers is what is known as a potential divider and has 
 been described by Hulbert and Breit. 1 The object is to take a portion 
 of the alternating e.m.f. across L 6 and impress it on the grid of a tube. 
 When the thermocouple is in the dotted position the current passing 
 through C\ is measured, but the current passing through C 5 and Ce can 
 easily be calculated when the values of the different capacities are known. 
 The object of measuring the current in C 4 is that for small values of input 
 potentials, the currents passing through C 5 and C 6 would be too small to 
 be recorded by a low resistance thermo-couple. If I represents the am- 
 plitude of the alternating current passing through C$ and Ce then the 
 resulting input e.m.f. is 1/2 IT (Co + Ce) /, where / is the frequency of the 
 wave. For an e.m.f. of over one volt, the current / could be measured 
 directly by the low resistance thermo-couple. 
 
 The resistance R% is used to provide a leak for any charge that may 
 accumulate on the grid, and allow it to flow to earth. Its resistance 
 must be comparable to the input impedance of the tube. By means of 
 potentiometer R- A the p otential on the grid could be varied as desired. 
 
 The upper tube to the right is the harmonic producer. By means of 
 the condenser potential divider a known value of input e.m.f. was im- 
 pressed between the grid and filament and then according to equations 
 (4), for a resistance EF in the plate circuit, and (9) for an inductance 
 EF, a plate current will result which is capable of being represented 
 as a series of harmonics. In order to get the results predicted in equa- 
 tion (4), EF must be a pure resistance. A straight wire immediately 
 suggests itself as a resistance which would possess a minimum inductance 
 and capacity. However, in order to obtain a resistance of the order of 
 3,000 ohms, so much wire would be needed, that inductive and capaci- 
 tive effects would become appreciable. Then again, it is a well-known 
 fact the conductivity of a wire diminishes with the frequency owing to 
 the skin effect, and consequently the exact value of the resistance at any 
 particular frequency is not easily determined. A resistance suitable for 
 high-frequency work should have a negligible skin effect, as well as hav- 
 ing negligible inductance and capacity. On the suggestion of Professor 
 W. F. G. Swann, the author tried out some platinized quartz fibers im- 
 mersed in acid-free paraffin oil, and found that they would carry currents 
 up to at least 60 milliamperes. From the formula for change in resist- 
 ance with frequency, 2 it can be shown that using fibers about .01 mm. 
 
 1 PHYS. REV., 4, p. 278, 1920. 
 
 2 J. A. Fleming, Wireless Telegraphy, p. 97. 
 
642 JOHN G. FRAYNE. [SECOND 
 
 LoKRIES . 
 
 in diameter, the skin effect can be neglected. As the fibers used had a 
 resistance of about 100 ohms per cm. only a short length of circuit was 
 needed, thus reducing the inductance and capacity. The inductance 
 EFwas wound with No. 16, D.C.C. copper wire, and the windings were 
 spaced about I mm. apart. The resistance of the coil was 10 ohms, 
 whereas the reactance was 2,700 ohms. 
 
 In order to detect the various harmonics, a fraction of the e.m.f. along 
 EF was impressed on the grid of a W. E. Co. V tube, this impressed e.m.f. 
 being always less than I volt. A loo-ohm slide wire of IAIA wire was 
 used for the variable portion of EF. It can be seen from equation (8) 
 that in order to obtain pure amplification without the introduction of 
 harmonics whose amplitudes are appreciable compared to that of the 
 fundamental, the input e.m.f. must be small (less than one volt), and 
 the value of the external resistance must be high. The resistance of an 
 anti-resonant circuit is R -f- (L 2 o> 2 )/.R, where Leo is the inductive reactance. 
 Since R, the ohmic resistance, is negligble, at radio frequencies, in com- 
 parison with (LV)/jR, the latter term may be taken as the value of the re- 
 sistance of an anti-resonant circuit. Therefore for any given co, L should 
 be made as large, and R as small as possible. Now in the plate circuit 
 of the tube which is used to separate out the harmonics, a series of anti- 
 resonant circuits are placed. The first one is tuned for the fundamental, 
 the second for the first harmonic, and so on. A vacuum thermocouple 
 is placed in each circuit on the capacity side. This is done so that the 
 D.C. plate current will not affect it. In order to keep the ohmic resist- 
 ance low, thermocouples with heater resistances of from 0.5 ohm to 5 
 ohms were used, the higher resistance thermo-couples being used to meas- 
 ure the weaker amplitudes of the higher harmonics. 
 
 The effective resistance of these circuits are as follows: fundamental, 
 120,000 ohms; first harmonic, 183,000 ohms; second harmonic, 95,000 
 ohms; third harmonic, 175,000 ohms. 
 
 Arrangements (not shown in Fig. i) were also made for measuring 
 higher harmonics than these, by changing the inductance Ln, and by 
 retuning C\\. Each thermo-couple could be connected successively 
 to a Leeds and Northrup galvanometer, and the deflection of the latter 
 indicated the root-mean square value of the alternating current passing 
 through the heater. Previous to placing the thermo-couples in the 
 circuits, they were calibrated using alternating current (60 cycles). It 
 will be seen that the harmonic analyzer is essentially a voltage amplifier, 
 picking out each frequency in the producer and magnifying its voltage. 
 For this reason a tube with a large voltage amplification constant was 
 chosen, in fact the value of n as given by equation (4) was 26. 
 
No"6 XIX '] THREE-ELECTRODE VACUUM TUBES. 643 
 
 Let 1 1 = the maximum current in the fundamental circuit. 
 
 Let RI = the effective resistance. 
 
 Let R = the internal output resistance of the V tube. 
 
 Therefore R\Ii = e.m.f. across L 8 . 
 
 Let e = e.m.f. across FG. 
 
 Let M 1 = actual voltage amplification factor. 
 
 , , ijRi 26 X 120,000 
 
 Therefore /x 1 = - = = 20.8. 
 
 R + RI 29,250 + 120,000 
 
 Rill 120,000 X 
 
 .p, r 
 
 Therefore e 
 
 Let r = resistance of FG. 
 
 Let i = amplitude of current of fundamental frequency 
 passing through FG. 
 
 , r , . 120,000. /i , N 
 
 Inereiore n = e and * = - . (15) 
 
 20.8 X r 
 
 Thus knowing /i from the galvanometer deflection, and r from the 
 Wheatstone Bridge, the value of i can be determined. For the case 
 worked out above i represents the amplitude of the fundamental fre- 
 quency produced by a pure sine wave impressed on the grid of a tube 
 having a pure resistance load in the plate circuit. Similarly, by measur- 
 ing the currents in the other tuned circuits we can work back to the 
 equivalent current in the harmonic producer. 
 
 When the resistance EF is replaced by an inductance, a portion FG 
 of the inductance is used to obtain the input on the grid of the analyzer. 
 In this case it will be noted that EF offers twice as much impedance to 
 the first harmonic, three times as much to the second harmonic, and so 
 on. This makes it possible to measure weaker harmonics than in the 
 case of the resistance. The inductance of GF in this experiment was 
 0.0587 milli-henry, whereas the whole inductance of EF was 2.14 milli- 
 henries. The input impedance of the analyzer to which EF was at- 
 tached was of the order of 50,000 ohms at 200,000 cycles per sec., whe reas 
 the impedance of .0587 henry is only 74 ohms. This showst hat the 
 impedance of the analyzer was practically short-circuited by the coil 
 FG, and consequently did not affect the nature of the external circu it of 
 the producer. For measurement of large output current values, the 
 value of FG was reduced to 0.04 M.H. To obtain, say the amplitude 
 of the fundament current with the inductance, we have an equati on 
 
644 
 
 analogous to (15), 
 
 JOHN G. FRAYNE. 
 
 [SECOND 
 [SERIES. 
 
 1 = 
 
 I2O.OOO / 
 
 20.8 (/w) ' 
 
 where I is the inductance of FG and co = 2 TT X the frequency. 
 
 EXPERIMENTAL RESULTS. 
 
 The following constants for the 2O5-J5 tube were determined from its 
 static characteristic. A .554 X io~ 6 , e = 7.5 volts, /* = 6.7. For E b 
 = 260 volts, E c = 7.5 volts, Ro = 1/2 A (E b + nE c + )*= 3,570 ohms. 
 
 200 volt* 
 <o. t CC--T* " R- 2700 Oh 
 
 Resistance in Plate Circuit. When a resistance of 2,700 ohms was 
 placed in series with the plate, and the value of E h reduced to 200 volts, 
 the plate current was 21.5 milliamperes. Using these values of poten- 
 tial and resistance the curves shown in Fig. 2 were obtained. The ampli- 
 tude of the input e.m.f. was varied from o up to 50 volts. The latter 
 maximum loaded the tube rather heavily, and the larger currents were 
 maintained just long enough to obtain the necessary readings. 
 
 Keeping the e.m.f. of the input at 15 volts, the actual plate potential 
 at 200 volts, and varying the static grid potential the curves in Fig. 3 
 were obtained, for ranges of E c between 30 and +12 volts. 
 
 With a static voltage of 7.5 on the grid and the alternating e.m.f. 
 kept at 15 volts, the variation of the harmonics with plate voltage was 
 determined, as in Fig. 4. 
 
VOL. XIX.1 
 No. 6. 
 
 THREE-ELECTRODE VACUUM TUBES. 
 
 645 
 
 - Fundamental 
 B I* Harmonic 
 C 
 
 Volts - Et 
 
 Ra 4 
 
646 
 
 JOHN G. FRAYNE. 
 
 [SECOND 
 
 [SERIES. 
 
 A -Fundamental. 
 3- ft Harmonic 
 
 E- zoo volts 
 
 K * 2700 Ohms. 
 
 Fig. 5 represents the wave-shape of the plate current obtained when a 
 pure sine wave e.m.f. of 15 volts is impressed on the grid, the plate and 
 grid potentials being the same as stated above. The fundamental and 
 first harmonic are the only components included in the wave-shape. The 
 
VOL. XIX.l 
 No. 6. 
 
 THREE-ELECTRODE VACUUM TUBES. 
 
 647 
 
 amplitudes of the other harmonics are too small to be shown on the same 
 scale. Fig. 5 also shows the dynamic characteristic for this case, where 
 the wave-shape thus obtained is plotted against the alternating input 
 voltage. 
 
 In Fig. 6 the variation of the harmonics with the value of the external 
 resistance is shown. 
 
 Inductance in Plate Circuit. Exactly similar experimental procedure 
 was undertaken for the inductances. A higher plate voltage and plate 
 current was used here, since there were no delicate platinized quartz fibers 
 to be dealt with. For this case 
 
 = 250 volts, EC = 10 volts, ^0 = 
 
 2A (E b 
 
 = 3,800 ohms 
 
 Theoretical 
 A Fundamental 
 Harmonic 
 
 The reactive load in the plate circuit was 2,700 ohms. Fig. 7 shows 
 the variation of the harmonics with the alternating e.m.f. on the grid, 
 under the conditions given above. 
 
 In Fig. 8 is shown the variation of the harmonics with the value of 
 the static grid potential, the alternating input e.m.f. being constant at 
 20 volts, and the plate voltage being 250 volts. Fig. 9 shows how the 
 variation of the static plate voltage affects the values of the harmonics. 
 
 The curves of Fig. 10 give the variation of the harmonics with the 
 magnitude of the inductance in the plate circuit. 
 
 Fig. ii represents the wave shape, using only the values of the fun- 
 damental and first harmonic. 
 
648 
 
 JOHN G. FRAYNE. 
 
 [SECOND 
 [SERIES . 
 
 A- fundamental 
 
 B- KHormonic 
 
 C-2* " 
 
 0-3* " 
 
 E-4* 
 
 .750 
 
 V-15 - .L-2-/4MH 
 
 I-nJuctanct -Z/4AW 
 
 s s s A -Fundamental 
 5 -v B - 1^ HaTTnoni'c 
 fs? C 
 
 t) 3" 
 E 4 
 
VOL. XIX.1 
 No. 6. 
 
 THREE-ELECTRODE VACUUM TUBES. 
 
 649 
 
 A- 
 
 B-K Harmonic. 
 
 C-pf F)( p- 
 
 Thcor. 
 
 C r . . ' " * 
 
 Fig. ia 
 
 Fig. II also represents the dynamic characteristics for this case, the 
 area of this loop being proportional to the amount of energy delivered. 
 
 A- Fundamental 
 C- |tf Hahmonic 
 B- Wave-Sha^e 
 
 D- Dfnam/c 
 J 
 ISO Volts 
 
 EC -10 ' 
 L ' X- lit 77) h 
 
 t" * 20 VbltS. 
 
 Fig. 11. 
 
 DISCUSSION OF RESULTS. 
 
 The theoretical curves as shown in the various figures, are plotted from 
 the values of the coefficients given by equations (3) and (8) . In the case 
 of the resistance the experimental fundamental values check up very 
 closely with the theoretical values up to an input voltage of fifteen volts. 
 
650 JOHN G. FRAYNE. 
 
 Beyond that voltage equation (3) no longer accurately represents con- 
 ditions. It will be noticed that the maximum positive potential to which 
 the grid is raised in this operation is 7.5 volts. Fig. 5 shows that at this 
 voltage the static characteristic begins to flatten out, due to the passage 
 of electrons to the grid instead of to the plate. For voltages higher than 
 fifteen the theoretical first and third harmonics fall below the experi- 
 mental values, whereas the second harmonic is greater than the experi- 
 mental values would indicate. This would seem to be due to the flat- 
 tening out of the wave at the upper end of the characteristic. In the 
 case of the inductance good agreement between theoretical and experi- 
 mental values were found for input voltages as high as 20 volts. This 
 is due to the fact in this case that the operation was carried out over the 
 25o-volt static characteristic, and it will be seen from Fig. n that this 
 curve does not begin to flatten out until a positive grid voltage of ten 
 volts is reached. This point of operation coincides with the maximum 
 positive voltage to which the grid was raised with an input e.m.f. of 
 20 volts, E c being 10 volts. 
 
 The curves of Figs. 3 and 8 show, in an emphatic manner, that the 
 further we move towards the straighter portion of the static character- 
 istic, the greater the fundamental becomes while the other harmonics 
 continue to diminish. At 40 volts the higher harmonics were still 
 very much in evidence although the fundamental was rapidly approach- 
 ing zero. The second harmonic in the resistance curves and the third 
 in the inductance curves show rather peculiar irregularities. The sharp 
 maxima and minima would at first seem to point to some sort of internal 
 resonance in the tube. It will be noticed, however, that these are pro- 
 duced by simply varying either the grid or plate potentials, and are 
 probably due to irregularities in the static characteristic which are 
 smoothed out in the ordinary process of plotting. It will be recalled 
 that in the case of the resistance the coefficient of the second harmonic 
 is principally determined by the value of the third derivative of the cur- 
 rent function. A small unnoticeable irregularity in the static charac- 
 teristic might produce a large variation in the third derivative, and this 
 effect is magnified by this method of analysis. In the case of the induc- 
 tance the coefficients cannot be expressed as simple derivatives, but some 
 such explanation as that given above will probably hold good here also. 
 
 The curves of Figs. 4 and 9 show that the greater the plate potential 
 the greater the fundamental becomes while the harmonics continue to 
 diminish. Beyond a certain plate potential, about 300 volts, the funda- 
 mental becomes nearly constant in value. The curves which exhibited 
 the irregularities in the grid-variation series, also exhibit similar irregu- 
 
No L 6 XIX '] THREE-ELECTRODE VACUUM TUBES. 65! 
 
 larities here. Further they occur at the same value of (Et, -f- /j.E c ) 
 showing that the effect is independent of whether the grid or plate poten- 
 tial is increased provided the sum as given is the same. 
 
 The curves of Figs. 6 and 10 show how the harmonics depend on the 
 external impedance. Using this method of analysis it was impossible 
 to use an impedance much less than 500 ohms. Even at this value, the 
 input impedance of the analyzing tube cannot be neglected. Hence, 
 beyond the first harmonic the experimental and theoretical curves do 
 not agree very closely in this region. Although the experimental curves 
 show signs of flattening out at this low impedance, they should have been 
 rapidly approaching zero. For zero resistance or inductance, the funda- 
 mental and first harmonic have the values given by equation (20), and 
 all the other harmonics are zero, which is also in accordance with the 
 same equation. 
 
 If all the harmonics were neglected the wave-shape for the resistance 
 would be a pure sine wave in phase with the impressed e.m.f. These 
 two, when compounded, would give a straight-line dynamic character- 
 istic. It may be seen that with a sufficiently high resistance in the plate 
 circuit this condition may be nearly reached. However, if the higher 
 harmonic were also taken into consideration, the dynamic characteristic 
 would be no longer linear but would have a curvature, which would be 
 much less than that of the static curve. With the inductance, if all but 
 the fundamental had been neglected the current would have been a pure 
 sine wave lagging behind the impressed e.m.f. by an angle 6 = tan~ l lp/R. 
 The dynamic characteristic under those conditions would have been an 
 ellipse. Addition of the other harmonic components tends to flatten 
 out the sine wave, and consequently distort the purely elliptical charac- 
 teristic. 
 
 In calculating the theoretical values of the coefficients of the various 
 harmonics, up to a range of 15 or 20 volts, the coefficients of the funda- 
 mental were practically proportional to the first power of the applied 
 e.m.f.; the first harmonic was proportional to the second power; and so 
 on. It was only for values of the input voltage beyond 20 volts that the 
 more complicated expressions for the coefficients had to be evaluated. 
 
 In conclusion, the writer wishes to express his very sincere thanks to 
 the Western Electric Company, Inc., of New York, for their kindness in 
 loaning necessary tubes and vacuum thermo-couples, and also to Profes- 
 sor W. F. G. Swann of this department for his many helpful suggestions 
 and criticisms and his invaluable encouragement at all times. 
 
 DEPARTMENT OF PHYSICS, 
 
 UNIVERSITY OF MINNESOTA, 
 November n, 1921. 
 
IHX8B ^ s ^TyffED^ ^ g 
 
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 OVERDUE 
 
Gay lord Bros. 
 
 Makers 
 Syracuse, N. Y. 
 
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 487359 
 
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