the arc path and prevents mercury, sputtered from thecathode, reaching the anode. The cross-sectional areaof the restriction is large, hence, provides a high cur-rent-carrying capacity.

In starting the rectifier it is necessary to preheat thecathode with a flame. After the tube has been placedin operation, the heat liberated at the cathode may besufficient to maintain the correct cathode temperature.The circulating water in the cathode water jacket isprovided either to heat or cool the cathode chamber,the requirements in this respect being fixed by theeffective current through the rectifier.

For supply voltages not exceeding 10,000 the waterin the cathode water jacket may be connected to theanode. For higher supply voltages, a separate controltransformer should be provided.

In all of the models described, contact with thewater in the water jacket has been made by insertinga short metal nipple in the rubber tubing near thewater jacket. A 6-foot length of tubing has offered asufficiently higher resistance to insulate the waterjacket from a grounded water supply or another recti-fier tube, when using ordinary tap water. When sev-eral adjacent rectifier tubes are used, as in a multi-phase rectifier, the cooling water is circulated throughall tubes in a series connection, 6-foot lengths of rub-ber tubing coiled on insulated spindles being used forthe interconnection of the rectifier tubes.

HIGH-VOLTAGE SWITCH APPLICATIONThe new rectifier has been used as a switch on the

highest alternating voltage available, namely 75,000volts effective. To close the mercury switch, the con-trol electrode is connected to the anode. The switch is

opened by transferring the control electrode to thecathode. Two rectifier tubes are connected in a reverseparallel arrangement so as to conduct current in bothdirections. After the circuit has been closed by themercury switch, the switch may be short-circuited bya knife switch until the circuit is to be opened.One of the advantages of this form of switch over

other types is that this switch always opens the circuitwhen the current is zero and thus produces no switch-ing transients.The rectifiers described may likewise be used as

high-voltage switches on direct-current circuits, theswitch being closed by connecting the control electrodeto the anode and opened by transferring the controlelectrode to the cathode and connecting in parallelwith the rectifier a high-voltage condenser, charged tothe applied voltage and poled in opposition theretoacross the rectifier, so as to reduce the current throughthe rectifier to zero.

ACKNOWLEDGMENTSThe majority of the tests and the experimental

rectifier designs described herein were made in thePhysics Department, University of Washington, withthe collaboration of Professors Joseph E. Hendersonand Donald H. Loughridge to whom the writer isdeeply indebted. The writer also wishes to express hisappreciation of the assistance rendered by his son,T. Marx Libby, in obtaining much of the test data.

Bibliography(1) F. Penning, Proc. Royal Akad., (Amsterdam), vol. 34, (1931).(2) T. L. R. Ayers, Phil. Mag., vol. 45, p. 363, (1923).(3) L. Smede, Elec Jour., vol. 25, p. 403, (1928).(4) Hull and Brown, Trans. A.I.E.E., vol. 50, pp. 747-753, (1931).

A Distortion-Free Amplifier*P. 0. PEDERSENt, FELLOW, I.R.E.

Summary-Distortion in tube amplifiers may be eliminated byusing an auxiliary amplifier in which a fraction of the main-amplifierinput voltage combined with a fraction of the distorted main-amplifieroutput voltage produces a correcting component which, combined withthe total output, restores this to the same shape as the input.

I. INTRODUCTION AND GENERAL THEORY

fl[IHE generally known amplifiers are afflictedj with various deficiencies as (1) the nonlinear

distortion that is mainly due to the curvatures ofthe tube characteristics; (2) the dependence of theamplification on the voltage of the power sourcesused; (3) amplifier noise; (4) and finally, inductionfrom outside sources into the amplifier itself whichmay cause disturbances.

* Decimal classification: R363.1. Original manuscript receivedby the Institute, July 10, 1939.

t Principal, Technical University of Denmark, Copenhagen,Denmark.

These deficiencies may most seriously affect ampli-fiers which are to handle wide frequency bands, orwhen several frequency bands belonging to differentsignal circuits (talking channels) are to be handled byone amplifier.A considerable improvement is attained in later

years by the widely used amplifiers with negativefeedback, about which already exists a very compre-hensive literature. This solution is not perfectly idealthough. The following deficiencies may be mentioned:(1) the nonlinear distortion is but partly eliminated;(2) the feedback principle itself involves an inevitabledifference in time (phase displacement) between theinput voltage and the feedback voltage which is in-convenient for a correct reproduction of rapid currentand voltage variations; (3) the feedback may causeinstability problems; (4) the amplification is reducedby the feedback.

Proceedings of the I.R.E.Fe-bruary, 1940 59

Proceedings of the I.R.E.

A couple of years ago the author' became interestedin designing an amplifier along different lines in whichthese deficiencies should be eliminated or at leastreduced.The amplifier system to be described aims at avoid-

ing or at least reducing the deficiencies quoted: (1) byreducing feedback to a value which is of little or nosignificance; (2) by combining the output from themain amplifier with the output from an auxiliaryamplifier whose input voltage is determined by the

A4

aIZ r o=

Z'Z/5 eb7=Fig. 1-Principle schematically. A= main amplifier,

B = auxiliary amplifier.

where F(I) and r(t) are dependent on the curve formof ei(t) but independent of its amplitude in that sensethat F(t) and r(t) are the same functions of t for theinput voltages ei(t) and aei(t), where a is any numberdifferent from 1.The principle of the amplifier system is shown in

Fig. 1 while Figs. 2 and 3 are simplified diagrams to beused for calculating the input voltage of the auxiliaryamplifier and of the magnitude of the feedback, re-spectively. In these figures, Z3 is equal to (Z3+Z') ofFig. 1.Applying the symbols2 used in Fig. 2 we have

ea . ei-ea ea+ ec1 i2 + i3, 3 = 1 = 112 = (3)

ZB Z4 Z3

where ZB is the input impedance of the auxiliaryamplifier measured from the point a. Hence we get

(4)Z3 ZB ( Z4. )

Z3Z4 +Z3ZB +Z4ZBV Z3J

difference between (a) a certain, constant, or fre-quency-dependent fraction of the main-amplifier out-put voltage and (b) the main-amplifier input voltage;(3) by arranging the transmission time through themain amplifier to the input terminals of the auxiliaryamplifier to be equal, or at least very nearly equal, tothe time through the direct path from the main-ampli-fier input terminals to the auxiliary-amplifier inputterminals.

In this manner we obtain an amplifier system, Fig. 1,that is almost completely distortion-free, so that theratio F between the total output voltage eo(t-r) andthe input voltage es(t), that is, the amplification, willvery nearly be constant

F = eo(t -)et(t)

= constant,

If in this we put Z4/Z3=n, Z3/ZB=P then (4) maybe written

where

1eu = (ei - nec) = K(ei - ne,)

1 + n + np

ZB 1K==

ZB(1+ n) + nZ3 1 + n + np

(5)

(6)

In most cases it will be advantageous that the ration between the impedances Z4 and Z3 be constant,independent of frequency, and the same should beaimed at for the factor K. There are, however, caseswhere it is desirable that these factors be dependent onfrequency in a predetermined manner. Such a de-pendency may be obtained by a suitable choice of theimpedances Z3, Z4, and ZB.

indepeindent of the amplitude and the shape of theinput voltage; r being the transmission-delay constantof the amplifier system.

If (1) is satisfied then both the nonlinear and thelinear distortion have been eliminated.

There may, however, be cases where it is desirableto eliminate only the nonlinear distortion withoutdisturbing the linear distortion which does not giverise to other frequencies, cause cross modulation orthe like. In this case the condition (1) is changed to

eo(t - T(t))- e (t) (2)

1 Application for Danish patent filed March 16, 1938. After a

preliminary experimental investigation of this new amplifier prin-ciple, which verified its merits, it was found that a U.S.A. patentNo. 2,043,587 issued in 1936 to W. W. Macalpine, was based on

similar principles. In consequence hereof I dropped the patent case;but since the U.S.A. patent does not give either a theoreticalfoundation nor an experimental verification of the correctness of itsprinciple and, since I have found no such material published else-where, a brief account of my investigations, as a whole in accord-ance with the said Danish patent application, may be of sufficientinterest to justify its publication.

zlI

el,

01Z3 i2

Fig. 2-Simplified diagram forcalculating the input volt-age of the auxiliary ampli-fier. The impedance Z3in this figure is equal to(Z3+Z') in Fig. 1.

Fig. 3-Simplified diagram forcalculating the magnitudeof the feedback. Z3 in thisfigure is equal to (Z3+Z')in Fig. 1.

The potentiometer E varies the fraction a of thevoltage ea which should be supplied to the auxiliaryamplifier. Generally the conditions will be so selectedthat a will be a constant value, independent of fre-quency, but, if desirable, it may be made dependenton frequency.

2 In all formulas, in accordance with Figs. 2 and 3, Z3 signifiesthe sum of (Z3+Z') in Fig. 1. In most cases Z3>>Z', so that thedifference between Z3 and (Z3+Z') is rather snmall.

60 February

Pedersen: Distortion-Free Amplifier

F is used for adjusting the phase of the input voltageto the auxiliary amplifier in order to secure a desiredphase relation between its output voltage e02 and theoutput voltage eol of the main amplifier. Only thefiner phase adjustment is carried out in this mannerthough, larger phase shifts of about 180 degrees aredone by other means as by using an odd or an evennumber of valves or by suitable transformers.The impedances, Z3 and Z4, are chosen so that the

transmission time from the input terminal x throughthe main amplifier, the potentiometer D, and the im-pedance Z3 to the point a is very nearly equal to thetransmission time from the terminal x through theimpedance Z4 to the point a. Further, Z4 iS SO chosenthat

the input voltage, and also on the impedances con-nected to the output terminals. F(ej) will generallyshow both linear and nonlinear distortion. N denotesthe output voltage resulting from amplifier noise andV the voltage resulting from variation in the batteryvoltages etc., as well as from induction by other cir-cuits into the main amplifier.At the potentiometer D a certain fraction 3 of eol

is branched off, that is,

(13)

where 3 may be constant or dependent on the fre-quency.The voltage eb put into the auxiliary amplifier is

according to (5) determined byZ4

=Am >> 1,ZA(7)

where ZA is the input impedance of the main ampli-fier3 and Z3/ZA =M. When (7) is satisfied the feed-back will be insignificant as is clearly shown in Fig. 3.Using the symbols from that figure we have

il = i2 + i3, i2Z3 + i1(ZA + Z4) = e,i2Z3 - i3ZB = ec (8)e' = i1ZA,

whence the main-amplifier input voltage correspondingto the voltage e, is determined by

(mnn + 1)(1 + P) + m(9)

In most cases the condition will be so chosen that|nej is only very little different from IeiI wherebythe working interval of the auxiliary amplifier willbe very small, and consequently the amplificationfactor of the auxiliary amplifier will be approximatelyconstant. By inserting ne = es in (9) we get a feed-back determined by

e 1. = . , ~~~~~~~(10)

ei - n[(in + 1)(1+ p) +m] (

and by choosing suitably large values of m and nit is always possible to obtain a condition where

el<<1, (11)

ei

so that the feedback will be negligible.The output voltage eCo of the main amplifier may be

expressed byeo= F(ej) + 1V + Vz, (12)

where F(ej) is the output voltage of the main ampli-fier, which is generally dependent on the design andsetting of the amplifier, on the amplitude and shape of

I No attention has here been paid to the impedance Zi of thevoltage source connected to the input terminals of the main ampli-fier, but since this impedance, as far as the question of feedback isconcerned, is in parallel to ZA, its presence will tend further to re-duce the feedback.

eb -aeaKae - neoiI= aK [e - f3n(F(ei) + N + V)].

(14)

The output voltage eo2 of the auxiliary amplifier isdetermined by

e02 = f(eb) = f{aK[ei - 3n(F(e1) + N + V)] , (15)

here assuming the auxiliary amplifier to be so wellshielded that we may waive attention to inductionfrom outside sources, and further to be so designedthat the inside noise at a relatively small load is insig-nificantly small.The total output eo is then

eo = eol + e02 = [F(ei) + N + V]+ f{XaK[ei - n(F(ei) + N + V)]}.

(16)

By suitable choice of the properties of the auxiliaryamplifier the values of the function f, as well as ofthe parameters a, (, n, K can be made constant orin a suitable manner dependent on frequency. Thenwe are within wide limits able to give the total outputvoltage eo any desired dependency on the input volt-age ei.The interval within which the auxiliary amplifier

is supposed to introduce corrective action is generallyvery small and we can arrange the auxiliary amplifierso that its amplification is constant with very goodapproximation within the active interval. We thenhave

eo = abKei + (1 - abK3n)(F(ei) + N + V). (17)

Also in this case we can to a great extent secure adesired dependency of the total output voltage eo onthe input voltage ei.

In most cases complete proportionality betweenthe total output voltage and the input voltage, orbetween the input voltage and the output current,or between the input voltage and a total number ofampere turns, is desirable.

Considering a distortion-free amplifier we must,according to (17), have

.rbK(3n = 1, (18)

1940 61

ec = 6Oeo,

Proceedings of the I.R.E.

thenes

eO = abKei = - (19)on

The resulting amplification is consequently

1F-abK =-* (20)

n,B

The parameters a, 1, b, n, and K can be constant,independent of frequency, positive or negative, or two

Fig. 4-Arrangement for distortion-free voltage amplification,A =main amplifier, B =auxiliary amplifier.

or more may be dependent upon frequency, but theirproduct must, using the positive directions of voltagesshown in Fig. 1, always be positive and equal to 1 ifthe system shall act as a distortion-free voltage ampli-fier.

If here we choose n = 1 then

eo 1 Z' +Z"F=-= f _

ei ,B ZI

where Z' and Z" are the two sections of the potentiom-eter D (see Fig. 1). If the ratio Z"/Z' is constantand independent of frequency, then the same willapply to the amplification. Z' and Z" may be pureohmic resistances or capacitances. If the latter arefree of losses the amplification will in both cases beindependent of frequency. Z' and Z" may also be morecomplex impedances that may give F any desired de-pendence on frequency.

If n is different from 1 and is a real constant, inde-pendent of frequency, then according to (19),

1 Z'+Z/"

n,B nZf(22)

II. EXAMPLE AS TO THE DESIGN OF ADISTORTION-FREE AMPLIFIER

An example of an arrangement for distortion-freevoltage amplification is shown in Fig. 4. The mainamplifier A consists of a single valve having theamplification factor ,t and, see Fig. 5, a static char-acteristic acb whose point c is representing the gridvoltage eoO and the anode current iao. The input volt-age es is reckoned from ego and the variable anode

(21)

current ia from iaO. Assume that the input voltagevaries between ± eio. The maximum voltages, thatwill act upon the auxiliary amplifier, will depend onthe amplitude ejo of the input voltage, on the shapeof the characteristic, and on the resulting amplifica-tion F. For the sake of simplicity put n-1 which inFig. 4 corresponds to the condition that C3 = C4 andr =fR1. According to (21) we then get F=1/,B.

If the main amplifier alone should produce thisamplification at the operating point c then the plateresistance poo at this point would be determined by

1 R

3 poO +J?or

Poo = (OA - 1)Rl.

(23)

(24)If the slope of the characteristic at this point cor-

responds to the straight line cc' then at an arbitrarypoint x on the characteristic we get an amplificationF, determined by

deol pRiF d= - = I

dei p,x + RI(25)

where px is the plate resistance of the valve in thepoint x; that is,

1 1 / dia\-=_- x.Px 4 \ de/

(26)

The share Aeo1l of that part of the voltage eol whichis to be compensated for, which is the voltage interval

d a

Ia I a6ioWI II!

_ -ezO - -.-- ejO-J +Fig. 5-Characteristic of the main amplifier.

from ei, to (e.+dej,), is accordingly

Aeo1 == (Fx - Fo)deixPoo - Pzx

(PR + R1)(poo + R1)(27)

If the amplification factor of the auxiliary amplifieris b then the corresponding contribution to the input

62 February

Pedersen: Distortion-Free Amplifier

voltage AebX of the auxiliary amplifier at the point xwill be

Aeo x iAR, Poo Px,Abx-=~ = - - dei (28)

b b (px + Ri)(poo + R1)x

(

Since Px does not vary much within the part of thecharacteristic used we have, with approximation

Aeb- constant (poo- px)deix.

tional to the input voltage of the auxiliary amplifier.It follows that

dUx= 0 for ei = ± /A1 -A,deix v 3A3

2 (AI- AO)3 2and Uxmax = +2 (A - A+3AoA 1(3A3)1

(29)Consequently : should be so chosen that PQO, de-

termined by (24), on the average deviates as little aspossible from the plate resistance p, over the usefulpart acb of the characteristic4 since otherwise unneces-sarily large compensation voltages will be obtained.

2 -^za C - -- IsI

,Y

~~~~e, .-Fig. 6-The curve shows the values of U. under the assumption

that the maximum deviation in positive and negative directionsare of equal magnitude.

If the equation of the characteristic referred to thepoint c is

thenia = Aiei -A3ei,

1 1 dia 1- - - - = - (Al - 3A3eix2)PX , de, ,u

or

(30)

(31)

P x .=

A 1 - 3A 3e" x2

If we put poo= AI/Ao then

(Px- poo)deiAo -+3A3e2de=AUx. (32)

,, Ao(Al - 3A3e,,2)

If the applied part of the characteristic is so nearlystraight that 3A3ejO2 is much smaller than A1, we mayapply the following approximate expression for theincrease A U, in the input voltage of the auxiliaryamplifier, instead of (32),

Ao - A1 + 3A3ei.,AUX (Px - poo)deix AoA- de,. (33)

Here both A U, and Ux according to (29) and (33)should be multiplied by a constant dependent on thesystem. It follows that

fO=;u2 =Ju

1 reIjAUX = I (AO- A1 + 3Aa3ei,2)dei,

AoAl o1 (34

=AoA [(Ao- Aj)ej, + A3ei 3].

According to the foregoing U, is very nearly propor-4 If n=1 then poo' = (n,3j - 1)R1 should in maximum deviate as

little as possible from the values of p.

for

and

for

/A1-Ao3A3

2 (A1 - AO)32U11 m in =

1/2__________

3A oA 1 (3A3)12

ei= + ,,/A- AI' 3A3

while

Ux = 0 for ei = 0 and eix= +'VAl-AA3

(35)

(36)

(37)

When U for + eiQ has the same absolute value asUma. and UJ,., then the following equation is obtainedfor determining correlated values of (A1-A o) and e,o:

(Al-Ao)3I2-+ 332A31/2ejo(Aj-Ao)-_ 3!2A33'2eiO3=0,(38)

which is satisfied by

A1- Ao = 3A3e,02 or ejo = 2, -OAo

then

Poo = - =

AO A1-3A3ejol

(39)

(40)

The corresponding values of U,, are shown in Fig. 6.From the foregoing it appears that a value of poo

determined by (40) will be the most favorable, namely,the one that will give the auxiliary amplification thesmallest maximum amplitudes.From (40)

IKm= I + RI(A1- TA3eio2) (41)

The value of poo determined by (40) corresponds tothe straight line dcc" through c, that cuts the char-acteristic (30) at points corresponding to the voltages+X/3

III. CALCULATION OF THE RESULTING CHARACTERISTICAND OF THE REDUCTION OF THE "1KLIRR-FACTOR"

We shall now investigate how the system will workwhen the characteristi.cs of both the main and theauxiliary amplifier are curved and determined by theequations, respectively,

eo, = a1ei + a2e 2 - a3ei3 + a4e 4 + a5ei5 . . .

and(42)

eO2 = bleb + b2eb2 - b3eb3 + b4eb4 + b5eb5 * - - (43)

1940 63

A \

Proceedings of the I.R.E.

We shall assume the system to be adjusted so that theresulting amplification is equal to that of the mainamplifier at small input voltages, that is,

leo\= a,.

ei ei O(44)

This adjustment is not the most favorable for largervalues of the input voltage, but the following calcula-

eo eo0 + eO2

-= a1ei- 3 [- a2e62 + a3ei3- a4eI-bis

+ [ 34 + * * -

a23b3 a22a3b3- ale, +± e 6 -33 e7

b13 bis

Fig. 7-Curve I shows the main-amplifier characteristic: 10e -ei3,and curve II the resulting characteristic: 10ei- (1/103)ei9.

tions will be much simpler and clearer under thisassumption than for more favorable adjustments ofthe voltage amplitudes.

In accordance with (44) the input voltage eb of theauxiliary amplifier is so adjusted that

e - (aiei eoi)bi

1a2ei2 + a3ei3 a4e,4 - a5ei5 (45)

b,

The resultant output voltage eo is consequently

eO= eo + eO2

= a1ei + [- a2ei2 + a3e3- a4e4-a5ei-aeb,2

-b- [- a2ei2 + a3ei3 - a4e 4 - a5ei5-. . 3

bis

+ b4[]+b5|]5+** (46)

Generally the adjustment of the auxiliary amplifierwill be so chosen that b2 = 0; that is, the characteristichas its inflection point at the starting point. In thatcase (46) will become

so that the nonlinear part of the resulting character-istic in this case has for its first term ei6.

If the adjustment of the main amplifier is also chosenso that a2 =0 then (46) becomes

eO= eoi + e02

- a1e6 - -[+ a3e3 -a4ei4- a5e,5bi4

+ -[ 4 + .b,4a33b3 a32a4b3

=- ale - b e,9 + 3 eiio

a32a5b3+ 3 es

3

. . I

(48)

The nonlinear part of the resulting characteristichas under these assumptions for its first term ei9.

I tZa I

- 12

Fig. 8-Curve I shows the main-amplifier characteristic:10e +ei2-ei3 and curve II the resulting characteristic:1Oe, + (ei2 -ei) /103.

If the auxiliary amplifier is a 'distortion-free" ampli-fier of this latter type, while a new main amplifier is

adjusted to a2 = 0, then the nonlinear part of the result-

ing characteristic of such an amplifier system wouldhave for its first term e 27.

In order to illustrate this straightening out of the

resulting characteristic we shall calculate for a few

(47)

2,5 0 1 2 bz

70

20

February64

1940 Pedersen: Distort

simple cases the resulting characteristics. We assumethe auxiliary-amplifier characteristic in both cases tobe given by

eC2 = beb - b3eb3 = lOeb - eb31 (49)

while the main-amplifier characteristic in the firstcase is determined by curve I of Fig. 7.

eol = alei - a3ei3 = lOei-ei (50)

and in the second case by curve I of Fig. 8,

eo, = alei + a2e, - a3e 3 = 10ei + e,2 - ei3. (51)

The resulting characteristics consequently will be

1eo = 1ei-l- 3ei9-

103

eo = 1Oei +103

(52)

(53)

corresponding to the curves II in Figs. 7 and 8, respec-

tively.It may be of some interest to see whether other ad-

justments of the resulting amplification could be more

favorable. Suppose instead of having the auxiliary-amplifier voltage adjusted according to (45) we put

1

eb- [(a, - xa3)ei - eoj].bi

(54)

10

tion-Free Amplifier

a33b3

D = [x3Vo3 sin3 wt - 3X2V05 sin5 cot

+ 3xVc7 sin7 ct - Vo9 sin9 cot] (58)a33bN

= V3V E1 sin cot + E3 sin 3cot + E5 sin 5cot

+ E7 sin 7ct + Eg sin 9It} . (59)

Io te

Fig. 10-Reduction of the "klirr-factor" with thearrangement shown in Fig. 4.

The "klirr-factor" k is then determined by5

90_

?30

720 _1/0-

90

80

70-

60_

5,0

Jo~ -Wvl

02

Fig. 9-Value of the "klirr-factor" for the resulting amplification,given in decibels below the linearly amplified fundamentalnote (al-xa3) Vo sin wt.

Inserting this expression in the formula for eo we

get

eo = eol + eO2 = (a, -xa3)e + 3 (xei - ei3)3. (55)b1

Here the distortion D is represented by the last term

a33b3D = (xei e,3)3. (56)

If we putei = Vo sin cot, (57)

we get

a33b3k =

(a xa3)b3\/E2 + E32 + EL52 + E72 + E92. (60)

In Fig. 9 the ordinate gives the "klirr-factor" indecibels below the fundamental note, that is-20loglok, as a function of Vo and for a1=b,=2.3, fora3= b3 = 1/400 as well as for x = 0, 40, 60, and 80. CurveA represents the "klirr-factor" for the main amplifieroperating separately. For x=0, corresponding to the"klirr-factor" determined by (45), the curve declinessmoothly with increasing values of Vo; here curve Agives in a corresponding manner the "klirr-factor" forthe main amplifier. For Vo=3.5 volts the differencebetween the two curves, i.e., the reduction of the"klirr-factor," is about 90 decibels, and about 32 dec-ibels for Vo=10 volts. For the other values of x thecurves are somewhat irregular, but as a whole more

flat than for x=0; and it has been confirmed thatthe higher the maximum amplitude of Vo the higherthe value of x that should be chosen. It is further seen

that with the assumptions applied a reduction,of 40to 60 decibels, or even more, in the "klirr-factor" may

be obtained.There has been no opportunity so far for a thorough

experimental investigation of the system, but pre-

liminary tests with the arrangement shown in Fig. 4

5 We have here included the amplitude to sin wt, although thisis not measured by the common "klirr-factor" bridges. Since how-ever, this amplitude increases with the 3rd and higher powers ofVo it does rightly belong to the nonlinear distortion.

and

40 I

30

pni

65

dlb\

Olh

x.80xw60x-40

X=o

gave the reduction of the "klirr-factor" shown in Fig.10, using casually chosen valves type Philips A-415 forA and B, and for valve B1 type Philips A-425.

IV. CONCLUSION

The above is to be considered only as preliminarycommunication. I have not included anything aboutthe conditions for applying reactive output imped-ances or for using output transformers, which offer the

advantage that one of the valves in the auxiliary am-plifier may be omitted. My experimental investiga-tions of these points are not sufficiently advanced.

ACKNOWLEDGMENTI thank Mr. J. P. Christensen, E. E., Mr. A.Morten-

sen, E. E., and Mr. J. Oskar Nielsen, E. E., for valu-able assistance in carrying out experiments and calcula-tions.

Frequency Modulator*C. F. SHEAFFERt, ASSOCIATE, I.R.E.

Summary-The frequency deviation obtainable in a reactance-tube frequency-modulator-oscillator combination is limited by two re-sistances which shunt the oscillator tank. The first is due to theplate resistance of the reactance tube and the second, to the fact thatthe voltage on the reactance-tube's grid is less than 90 degrees out ofphase with the plate voltage. A method is deduced from a simplemathematical analysis whereby the phase of the reactance-tube gridvoltage may be adjusted for cancellation of these two effects. Thiseliminates the tendency of the reactance tube to amplitude-modulatethe oscillator output, makes possible the use of somewhat higher re-actance values in the oscillator tank, and permits the satisfactory useof more powerful reactance tubes. The principle is of importance onlywhere wide frequency deviation may be required without resorting tofrequency multiplication, but will possibly find many uses as the artof frequency-modulated-wave transmission and reception progresses.

flF HE use of the so-called reactance tube as ameans of controlling the frequency of an oscil-lator has been discussed by several authors.1"2

Its use as a frequency modulator has also been re-ported.' This paper is concerned with the development

Rp

2 2

Fig. 1-Circuits which are assumed to have equivalentimpedances at terminals 1 and 2.

of a more-satisfactory means of producing wide-devia-tion frequency modulation, or frequency control.There are perhaps several possible uses for a wide-deviation frequency modulator, however the onewhich interested the author was the exploration ofseveral frequency-modulation transmitter-frequency-

* Decimal classification: R355.8 XR414. Original manuscriptreceived by the Institute, September 25, 1939.

t Tulsa Broadcasting Company, Tulsa, Okla.1 D. E. Foster and S. W. Seeley, "Automatic tuning, simplified

circuits, and design practice," PROC. I.R.E., vol. 25, pp. 289-313;March, (1937).

2 Charles Travis, "Automatic frequency control," PROC. I.R.E.,vol. 23, pp. 1125-1141; October, (1935).

3 I. R. Weir, "Field tests of frequency and amplitude modula-tion with U.H.F. waves," Gen. Elec. Rev., May and June.

stabilizing schemes which require the production ofwide-deviation frequency modulation without resort-ing to frequency multiplication.

Experiments conducted by the author have resultedin the deriving of a suitable means of using the higher-powered vacuum tubes of the pentode, or beam type,or if necessary, even using two or more such tubes inparallel for producing the controllable reactance.Reasonably wide frequency deviation may be obtainedby the system without making the oscillator undulyunstable, or subject to amplitude modulation.The procedure followed is perhaps best explained by

the following mathematical representation of the fac-tors considered.

Referring to Fig. 1, it is seen that the impedancelooking into the plate of a vacuum tube may beassumed under certain conditions to be dependentupon three values: the plate resistance, the platevoltage, and the value and nature of the grid excita-tion. The resulting impedance will be the ratio of theplate voltage to plate current.

(1)

If we now assume the grid voltage to be of the same

frequency as the plate voltage and to have some fixedphase relation with it we may write the equation forthe plate current

Ep + ukEp L 0Ip =

Rp(2)

where k is the ratio of grid voltage to plate voltage.The admittance looking into the plate is therefore

1 uk juky = -+-cos 0 +- sin 0.

Rp Rp Rp(3)

The presence of two real terms, one of which may bemade negative at will, at once suggests the possibilityof canceling these two terms so as to leave only thesusceptance term. The conditions required by (3) fordoing this are

I(COSU = 1/ut?. Vt)

Proceedings of the I.R.E.

Zin =EplIp,

February, 194066