R·26

THEORY OF GROUNDED GRID AMPLIFIERSby A. v. d. ZIEL 621.396.644

Summary '.In part I a p~ey of the existing triode theory at ultra h,l; is .given.Neglecting -léad effects, the four characteristic impedances of agrounded grid triode at u.h.f. can be described by the "cold" ~a,lvecapacities, the amplification factor p. and the transconductances $1and 82> in the cathode-grid lead and grid-anode lead, respectively(the moduli and the phase angles of these transconductances cal).hemeasured). Shot effect in triode valves can be completely, describedby assuming two mutually dependent fluctuating currents il and i2to be flowing in the cathode-grid lead and in the grid-anode lead,respectively; at u.h.f. i2 is delayed in phase with respect to il (theintroduetion of these mutually dependent fluctuating currents is adirect consequence of the Fourier analysis of the shot effect). It isshown that the introduetion of the "equivalent noise resistance"of the valve may cause serious errors in the' calculation of the signalto noise ratio:In part II this theory is applied to grounded grid amplifiers. The inputresistance RI of the valve when the output is short-circuited, and theoutput resistance R2 of the valve when the input is short-circuited,are of special importance in this case. Dènoting the transformed an-tenna rç,sistaÎl.ceby Rt' and the trarisformed input resistance of thenext stage by R2', the 'power gain g is calculated as-a function of:RI'/Rl and R2'/R2• It is shown that the internal feed-buck of the valvé .makes it impossible to match at the same time the antenna to tlîeinput of thé amplifier and its output: to the input of .the next stage.The best results 'are obtained by using a high value of Rl'fRI (Ioose 'antenna coupling) and matching the output of the amplifier t6 'the next-stage; thè theoretical gain limit is (p. + I), values between 0.5 (p. +,1)and 0.8 (u + 1) may easily be obtained. For wide band amplifiefsRI'/Rl = 1 for-maximum gain, whereas it is shown that a wide,anode-grid spacing will give a higher gain. It is shown that electronictransit times cannot account for the drop in power gain at u.h.f.;this drop must be due to the impedance of the electrode leads. Atu.h.f. instability may occur; a stability condition is' given, from t,

which it can be seen that careful shielding and narrow electrode ,spacings result in a better stability of the amplifier. Finally thesignal to noise ratio of the grounded grid amplifier is calculated andit is shown that the grounded grid amplifier contributes only slightlyto the noise, especially. for large. values of RI'/Rl' This 'result isverified experimentally.

1. TRIODE THEORYWe shall con'~ider t1\"o'kinds of triode amplifiers: a) with grounded ca-

thode, b) with grounded grid. 'I'he corresponding equivalent networks areshown ~nfigs la and lb. If Vg and Va denote the input and output signaloltages, Ig and la the corresponding signal currents, YIlc', Yag', S' and

Yac' on the one hand and Ygc", Yas", SIt and Y~c~'on .the other hand theour characteristic admittances of the grounded' cathode triode and groun-ded grid triode, respectively, we have the equations: .

(1,1)

• I

382 A. VAN DER ZIEL

for a. grounded cathode triode, and for a grounded grid triode:

~t,= Yge" Vg + Yae" (Vg + Va)( Ia = S" Vg + Y~/' Va + Yae" (Vg + Va)(1,2)

Fig. la) Equivalent network of a triode with grounded cathode. Ygc', Yog', S' and Yo/denote the four characteristic admittances of the circuit, Vg and Va denote the input andoutput signal voltage and Is and la die input and output signal current. c cathode, g grid,a anode. . _l_ - . .,. .

b) Equivalent network of a triode with grounded grid. Same notation as in a), but thefour characteristic admittances are now denoted by Yge", Yog", S" and Yoe".

When no direct current is flowing through the valve ,~e have:"UI' "Uil 'C "{TI y'" 'C y" "{Til 'e~ ge = ~ge =]00 gc, s: ag - og =]00 ag' ae = s: ce =}oo oe,

, .where Cge, Cog and Coc denote the "cold" valve capacities, 00 the angularfrequency and j = 1"-1; but when current is flowing through the valve,these characteristic admittances are changed. It is the object of this partto investigate the values of the four characteristic admittances in that caseand to give a description of the. noise currents in the various electrodeleads ..

1~Ultra h.f. vacuum tube electronics

. In order to discuss the influence of current flow upon the characteristicadmittances of triodes it is important to have a clear picture of the current,flow in such a valve. Such a picture is given in figs 2a-2d. In thesefigures Tcg and Tga denote the electronic transit times from cathode togrid and from grid to anode. When a signal voltage Vg is applied betweengrid and cathode; the ,problem is somewhat more complicated than when,

GROUNDED GRID AMPLIFIERS \

Fig.2a) Electron moving between parallel plates. The electron of charge -e is moving witha velocity v perpendicular to two parallel plates at distance d, thus giving rise to a currentevld in the outer leads. 'b, c) Currents in the cathode-grid and grid-anode lead due to the emission of an electron

by the cathode. '. I

The electron is emitted at t = to, thus giving rise to a current I1(t) from cathode to gridduring the time .interval to < t < (to + Teg) and to a current I2(t) from grid to anode'during the time interval (to + Te~) < t < (to + Teg + Tpa). J I1(t) dt = J 12(t) dt = e,Due to the almost perfect screening of the grid 11(t) and I2tt) Howat different time intervals. _

d) Alternating currents in the outer leads due to the emission of rui alternating current'by the cathode. ' .' '. .~. • . . •. . .. :The alternating convection current I is emitted by the cathode and gives rise to alternatingcurrents IJ and 12 from 'cathode to grid and from grid to- anode. Due to the -electronictransit times 11 is delayed inphase with respect to I and 12 is delayed ~ phase with respect,to IJ (the value 0.£ 11 at the time t can be calculated by adding the currents due to the,individual electrons moving between cathode and grid at that time, similar for 12),

383

-e.:--y

!!vjd

~~

!L ~;I;v47679

'1

according. to fig. 2d, an alternating convection current is emitted by thecathode, because the electrons after leaving the cathode still move in an _alternating electric field, which gives rise to velocity modulation. However,the following general statements still hold (comparefig. 3a):a') If an alternàting voltage Vg is applied between grid and cathode, an

alternating current SlVil will flow from cathode to grid, an alternatingconvection current SeVg through .the grid, and an alternating currentS2Vg from grid to anode. These' transconductances will generally becomplex.

b) An alternating voltage' Va applied between anode and cathode will cause-the same' currents in the cathode-grid and grid-anode leads as an alter-

. ' nating voltage Va/p between grid and cathode, where J1 denotes. the.amplificátion factor of the triode *). Hence. that alternating voltage

*) This is not exactly true. because the voltage Va between grid and anode will give riseto an additional velocity modulation in the grid-anode space. The currents due to thiseffect are so small, however, that they can be neglected here. I .

384' A,. VAN DER ZIEL'

S,Vg s,Va/fl SaVg saVa/fl(s,-SiM, E-

VglS,Vg

O,-ia)

Fig. 3a) Alternaring voltages Vg and Va applied between grid and cathode and betweenanode and cathode. Due to the alternating voltage Vg alternating curreûts SI Vs and S2Vg.flowin the cathode-grid and in the grid-anode lead, whereas a conveenion current SeVgflows through the grid. Due to the alternating voltage Va altemating currents SlVa/p,ana .82Va/p, flow in the cathode-grid and grid-anode lead. i1 and i2 denote the currentfluctuations in those leads due to shot effect. .

b) Mutual phase relations .between, Vs' SI VG and S2 VE• -rpI and -rp2 denote the phaseangles of SI and S2. S2 is more delayed in phase than SI' hence a current {Sl-S2) Vg flowsto the grid.

'e) Mutual phase relation between the fluctuating currents i1 and i2• i2 is delayed in phase.. (phase angle -rp) withrespect to 4; hence a noise current (i1-i2) flows to the grid.

will give rise to alternating currents SI Va/p and S2Va/p in the cathode-grid and grid-anode leads, respectively. , .:The mutual phase relations 24) between Vil' SI Viland S2Vg are shown in

fig. 3b. According to' .theory 16) the following relation exists between S2and Se:

(1,3)

1«PaUW-lla) I is shown as, a function of frequency infig. 4. In.order to avoidtoo large a decrease of IS21, W_ga should be chosen such that:

W_ga < n or V_ga <.t (w = 2nv) ,or, because the transit time 7:ga is given by the equation:

GROUNDED GRID AMPLIFIERS 385'

Oga = 2 dga (2eV/m)J/. = 0.33 X 10-7 dga V-l/. '(dga in cm, Vin volts)

(i~ which lIga denotes the anode-grid spacing, V the direct anode voltageand in the electronic mass), IS~Iwill not be much decreased, if .(I, 3a) 'jJ < 15 Vl/./dga Mc/s. '

Hence if V = 400 V and dga = 0.1 èm, (I, 3a) 'yields l' < 3000 Mc/s.The phase angle of (/Ja (jwoga) is in good approximation equal to-t OJoga, as long as wOga < si. '

f,OO

l<f>i;wTga)

Î0.50

..I I1 .-r---I 1

1 i' .......-

I'. r-

f- 1"::1.;-- -

r- r---~-00 7[/2 n 3rrj2 27[ 5rrj2 '31T 71Tf2 41T

0.0

_Wrga47~1JI

Fig. 4. 1<p3(jw't'ga)1 as a function of w't'''a' Note that 1<p3(jw't'g,,)1 does not decrease verymuch as long as w't'ga < n, 0 • ,

. The chief difficulty, however, is the calculation of SI and Se; up to nowit lias only been carried out under the assumption, that the influence of theinitial velocity distribution of the electrons can be neglected 1, s, ,4,12,14,16).Formulae are given from which the values of1S11, ISel and the correspondingphase' angles can be calculated; according to that theory ISel should hardlydepend upon frequency. Without questioning the accuracy of the calcu-lations, it can he said that SI and S2 really exist, that they have negativephase angles -'PI and ~'P2 ('P2 > 'PL clue to the fact that the transit timeto the anode is longer than the transi~ time to the grid), that ISll = IS21= Sfor moderately high frequencies (but ISll and IS21.will decrease in the dmwave range), whereas 'PI and 'P2 are both proportional to the angular.frequency w. 'PI depends on the electronic transit time ieg only (the sameholds for the phase angle of Se), whereas CfJ2 also depends upon 19~ (compare(I, 3)). Moreovèr SI and S2 can he measured 21);though measurements in,the dm wave range are stilllacking *)'. '

In order to calculate Yge", Yag", S" and Yaó" for a grounded grid triodee have to' combine figs lb and 3a and to -apply the above two prin-

ciples. Substituting Va = 0 in (1,2) we find: .

19 = (Ygc',' + Yae") Vil' Ia = (S" + Ya.") Vg,

*) Preliminary measurements were carried out at ~OMcj!Vç,na triode having de/!, = 0.1 cmand dga = 0.5 cm. At la =,8 mA we found S = 4; mA/V and 'PI = 9°. ,

386 A. VAN DER ZIEL

whereas 19 and la can also be found from the currents through the valvecapacities and the currents shown in fig. 3a, so that (Y&c" +Yae") and(S" + Yae") can be calculated. By doing so it has .to be borne in mind thatthe voltage Vg is not only applied between grid and cathode, but also he-s.tween anode and cathode (this accounts for the occurrence of the factor

, (1 + I/ft)). Setting Vg= 0 we can calculateYa," and (Yag" + Ya.") in asimilar way. Hence we have:

~19= [~w(Cg. + Ca.) + S~ (1+ I/ft)] :g + (jwCa• + Sl/ft) Va(Ia = [JW Ca.+ 82 (1 + I/ft)] Vg+ [Jw(Cag + Ca.) + S2/ft] Va(~, 2a)

and:

(I, 4)

In a similar way we may calculate the four characteristic admittances ofa grounded cathode triode:

(I, 5)

and:

~I, la)

~Ygc' = jwCg• + (Sl-S2) (1 + I/ft) .

,,_ I ,,_ I ,,_ r ; ,,_ ,. IYag - Yag ,8 - S ,Ya• - Ya• " Ygc - Ygc + S ,

Upon comparing a grounded cathode triode with a grounded grid triode,the only difference consists in the fact that in the latter case the currentS'Vs flows through the input circuit to the anode (figs la and lb) as is

". mathematically expressed by (I, 5).In many cases the input and output circuits are tuned. This means

that in the expressions: . i

, I

[jw(Cgc + Cac).+ Sl(l + lift)], [jw(Cag + Cac) + Sl/ft][jw(C8C + Cag) + (Sl-S2)] and [jw(Cag + Cac) + S2/ftL

only the real ,parts have to be taken into account. Denoting these rè~lparts by I/Rl' 1/R2' I/Ra and 1/R4' respectively, we have:

(I, 6)

l' .R = IS11(1+ I/ft) cos CP1= S (1+ I/ft) + ...1 ...

1 _IS11 _ S/ 'R - - cos CP1- ft + ...2 ft1 '.'

R == f S11cos CP1-IS21cos CP2= yw2 'rei 8 +...a '1 _IS21 . _R +: - cos CP2- SIft + ...4 ft , .

in which y is a positive constant. For moderately high frequencies the firstterm in the four series expansions is sufficient, in the dm 'wave range the

GROUNDED GRID AMPLIFIERS 387

full equations (I, 6) have to he taken into account Ilo): In (I, 6) Rl' denotesthe input resistance of a grounded grid triode when the output is short-circuited, .R4 the output resistance of the same valve when the input isshort-circuited, whereas Ra denotes the input resistance of a. groundedcathode triode when the output is short-circuited. RI > Ra, if n/2 < f{J2 <3 n/2; even negative values of the input resistance .should be possible la).2. Other causes of losses in valves at ultra h.f..Dielectric losses in the glass and in other insulators can often be neglected,

esp.ecially when they are suitably chosen. We wish to pay attention to thefact, however, that the oxide coating of the indirectly heated cathode maybe the cause of a considerable input conductance 20) at ultra h.f. The oxidecoating is not too poor a dielectric at room temperature, but at the emissiontemperature ofthe cathode it shows an appreciable conductivity. Hence thegrid-cathode space can be representéd by the equivalent network shown injig. 5, in which Cl denotes the capacity between the grid and the outer. '.

C,

:: \f¥• 4766Z

Fig. 5. Equivalent network of the cathode-grid space. Cl denotes the capacity between the :.grid and the outer surface of the coating, whereas C2 and R2 describe the dielectric pro-perties and the conductivity. of the coating itself. '

surface of the coating whereas, C2 describes the dielectric properties ofthe coating itself and R2 its conductivity. The thinner the oxide coating,the smaller R2 and the larger C2, hence the input conductance will beconsiderably decreased by decreasing the thickness d of the coating (theinput conductance is proportional to d). For normal valves d= 50-75 fJ.,but for ultra h.f. valves the cathode department of this laboratory has suc-ceeded in preparing cathodes with an oxide coating only 5 fJ. thick, havingnormal emission and a sufficiently long life. .

*) According to our notation the electronic part of Yso" is given by 81and that of Ysc'by (81-82), if IIp is neglected with respect to unity. Obviously: .

81= 82+ (8c82)·

Jon e s 9) and Dis h a I G) neglect the phase angle -rp2 of the transconductance 82andhence use for moderatelyhigh frequencies the equation:

. 1Rl = 8+ yw2

7:0828.(I, 6a)

Of course this is incorrect, for according to (I,6) RI will increase with increasingfrequency (increasing rpl)' whereas according to the latter equation Rl will decrease.Furthermore Dis h a I extrapolates the first term yw2 7:0828 of the series expansion ofl/R3 to very high frequencies, so that it grows without limit; this cannot be correct,because 1811and 1821decrease with frequency according to theory, so that it is obvious.that the maximum value of the real part of .(81-82) is less than 28. . ' .

38B A. VAN DER ZIEL

The infh~ence of the self-induction 22) and of the resistàn~e (chieflydue to skin effect) of the electrode leads may be serious. If the input andoutput circuits have separate leads to the electrodes, the self-induction

- and the resistance of the leads can be considered as forming part of thecircuit. In recently constructed tubes in which cathode, grid and anodeform parts of cavity resonators or concentric lines 5, 15) this difficulty hasbeen greatly overcome, though it is still important there to use metalswith high conductivity and low magnetic permeability.

3. Ultra h.f. noise currents due to shot effect in a triode

It is well known that the fluctuations produced by a resistance R at atemperature aT, where T denotes normal room temperature, can- be des-cribed by assuming a fluctuating e.m.f. VR to be connected in series with R,such that its mean square value in a small frequency interval Llv is(Ny qu is t's theorem): . _ .' .'.

VR2 -:- a X 4kTRLI~1 (k is Bol tzmann's constant) .

-In order to discuss u.h.f. noise currents due to shot effect, we have torefer to our picture of current flow in triodes. According to fig. ·2a eachelectron starting from the cathode and. arriving at the anode will first giverise to a current from cathode to grid and then to a éurrent from grid toanode;' the shape of the current versus time graph is the same for each elec-tron. (fig. 2b). Applying a Fourier analysis to the currents in fig. 2b, it isobvious that a phase relation exists between the corresponding Fouriercomponents of the two currents. This phase relation is the same for eachelectron, and it is independent of the time of emission; as the totalfluctuating currerrts'vin the cathode-grid lead and in the grid-anode leadare cónstituted from the currents' due to the individual electrons, the same-holds for the corresponding Fourier components of these fluctuatingcurrents *). .

We shall denote the Fourier component of the fluctuating current in thecathode-gridIead for a small frequency interval Llv by il and the c~rres-ponding Fourier component of the fluctuating current in the grid-anodelead for that interval by i2 (compare fig. 3a ).-The calculation of il and i2 is amatter of mathernaties and. has only succeeded under several res-trictions 2, 17), except for lower frequencies for 'which rigorous calculationsexist 19). But without questioning the accuracy of these calculations, it issufficient to know that il and i2 and their mutual phase relation. reallyexist. This phase relation means that due to the finite transit time of theelectrons t« will be delayed in phase with respect to i1; this phase angle willbe denoted by-e-ç (fig.3c). For moderately high frequencies lill= li21 = i (butlill and li21 may change in the dm wave range), whereas tp is proportionalto the angular frequency co and dependent upon the transit times 'ieg and'iga' In ~ triode with grounded cathode, due to this phase difference,a fluctuating cUrrent. (il-i2) which is proportional to w flows from the

*) This is true as long as the influence of the electrons returning to the cathode in frontof the potential minimum and the fluctuations in emission velocity can be neglected.These effects become important in the dm wave range and .will be dealt with in aforthcoming paper on this subject; they have hardly any importance for moderatelyhigh fréquencies.

GROUNDED GRID AMPLIFIERS 389

cathode to the grid; this was shown theoretically and experimentally by.Bakker 2) *). ~According to Spenke and North 19,26) the following relation holds

for the mean square value of i up to moderately high frequencies: .

(I, 8)in good approximation, in which the constant e has a value of about 3 **).

The fluctuating current i2 in the grid-anodelead can also be representedby an "equivalent fluctuating voltage" in the grid-cathode lead, or by an"equivalent noise resistance" Re, such that ***):(I, 9) i2 = vS2; v2 = 4 k'I'R; Llv .

Representation of the spontaneous fluctuations of a valve by means of its"equivalent noise resistance" may lead to serious errors, because it doesnot takë into account the phase relations between i2 and il•

We first consider a triode with grounded cathode. The fluctuating cur-rent (i1-i2) which is flowing to the grid, will give rise to a fluctuatingvoltage Vg across the input circuit. As the fluctuating current i2 in the anodelead can be represented by the equivalent fluctuating voltage v. of (I, 9),the total "equivalent" voltage v' between grid and cathode, representingall fluctuations due to shot effect, is:

,(I, 10) . v' = (Vg +v).But itis generally false to set:

(I, 10\11) V'2 = vi +v2,becau~e'this relation only holds when Vg is 90 degrees out of phase with res-pect to v. Because (i1-i2) has a phase advance of about 90 degrees withrespect to i2 (fig. 3c), this condition is satisfied for a tuned input circuit.A much better signal to noise ratio would be possible if the input circuitwere so detuned-that Vg were 180 degrees out of phasewith respect to v.This was theoretically predicted by Strutt and van der ZieI2~) andexperimentally verified by Kl een P] at 1 m wavelength, This experimentmust be considered as a decisive proof of the validity of the above theoryeven a~ 1 m wavelength. For higher frequencies a complete suppressionof all shot effect noise should be possible under favourable phase con-ditions; 25) •.*) According to Bakker this fluctuating current can be taken into account byassuming

the input resistance Rato be fluctuating at a temperature about five times absolute roomtemporature. Hence in a small frequency interval Lfv, we have for the mean squarevalue of (i1-i2):

(. ')2 5 4kT Lfvt1-t2 = X --.

R3**)More exactly, e= 3 (I-n/4). Tenth/aT, in which Tenth denotes absolute cathode tempe-

rature and T absolute room temperature, whereas the value of a is generally between1/2 and 1. .

***) For a triode with grounded grid it is better to use the equation:(I, 9a) i2 = vS2 (1 + I/tL)because the voltage v is not only applied between grid and cathode, but also.hetweenanode and cathode (compare (I, 2a». Due to the fact that grounded grid trio des havea large amplification factor tL, this difference is only small.

I'

.: 390 A. VAN DER ZIEL

In the second place we consider a triode with grounded grid. In that casethe fluctuating current il is flowing through the input circuit, thus causing a, fluctuating voltage VIbetween grid and cathode. Due to the phase relationbetween il and i2 an analogous phase relation will exist between VIand the"equivalent 'noise voltage" v. Hence the total equivalent noise voltage v"between grid and cathode is: '

(I, 11) v" = (VI+ v) .il and i2 have only a small phase difference for moderately high fre-

quencies, and if the input circuitis tuned, VIwill be nearly 180 degrees outof phase with respect to v, so that VI and V may partially compensate oneanother.

Hence in this case it is false, even for lower frequencies, to neglect phaserelations between il and i2• In three recent papers 6,9,10), the authors neglect. the fluctuating current il f!.owingthrough the input circuit, owing to theintroduetion of· an equivalent noise resistance. Of course this gives a fartoo l~w signal to noise ratio *), as .will be shown in part II of .this paper.

II.' APPLICATION· TO GROUNDED GRID AMPLIFIERS• • j .'

. In section I, 1the basic equations for the discussion of the power gain ofgrounded grid amplifiers are already given in (I, 2a). As the input andoutput circuits are assumed to be tuned (that is, we shall assume that theinput circuit is tuned when the output is short-circuited and the outputcircuit is tuned when the input is short-circuited), the capacitive parts ofsome of the four coefficients can be neglected. For this reason the fourresistances Rl" ~2' Ra and R4 of (I, 6) were introduced. Furthermore,even ifjwC •.c cannot be neglected compared with Sl/"" it is often permissibleto neglect it compared with S2 (1+ 1/",). Introducing RI and R2 and thephase.angles CPland CP2 in (I, 2a) we have: .

, ) I, ~ ~,lVg, :- ~~2 (l~jtan CPl)+ jwC.c~ Va '

(11 1) I _ IS21cos CP2 ~ 1 ( .' ) V. 1 T)' ~, ,a-ISI R l-Jtancp2 g+R' Ya;1 COS CPI 1 2

, R2/RI = (",+ 1) .

1. The grounded grid triode as a radio frequency amplifierIn this section we neglect the influence of the anode-cathode capacity

Coc and the influence of electronic transit times (CPI= CP2 = 0). Equation(II, 1) now reads:

( 1 1 'II,2) 'Ig = Ia = R Vg +-R Va; R2/RI = (",+ 1)

I 2. .

in which RI denotes the input impedance of the amplifier when the outputcircuit is short-circuited, and R2 the output impedance of the amplifierwhen the input is short-circuited. ,', .

*) 'FUrthermore Dishal8) assumes the erroneously introduced additional input resis-, tance Ra (compare (I, 6a», which was considered to be connected parallel to the

input resistance Rl' to be fluctuat~ng at, about five times absolute. room temperature.

/,

GROUNDED GRID AMPLIFIERS

If, however, the output of the amplifier is loaded by a resistance R2'

so that la = ,--Va/Ra' (fig. 1b), the input impedance of the stage is given bythe; equation.

(II, 3)

and the voltage amplification is (compare (II,2)):

[ R 1-1Val VB= (p, + 1) 1+ R:' ,(II, 4)

as can be shown by substituting the above value of la in (II, 2), elimin-, ating Va in these relations and expressing Vg/Ig and Va/Vg in terms of {J.Ra and R2'. Equations (II, 3) and (II,4) were already given by J ones 9).If, on the other hand, the input of the amplifier is loaded by a resistance

RI' so that Ig = -Vg/R1' (compare fig. 1b), the output impedance of thestage is given by the equation:

(II, 5) . Va/Ia = (p, + 1) (Rl + Rl')'as can be shown in a similar wayas (II,4)., ' ,

In practice Rl' will be the transformed internal resistance of an antenna(with e.m.f, Vant and internal resistance Rant), coupled to the input bymeans of a transformer with negligible loss, whereas R2' will be the trans-formed input resistance of the next amplifier stage, coupled to the output,of the first stage by means of a transformer with negligible loss.

, 1 Vant2 • ' •, Let E8J1t = -4 -R denote the maximum amount of power (available

. antsignal power7)) which can be delivered by the antenna, then an e.m.f, V'has to be assumed in series with Rl' such that:

V'2(II, 6) R' = 4 Eant.

I

In order to calculate the power gain of the amplifier we first consider theoutput of the amplifier to be short-circuited. In that case the grid voltageVg' is given by the equation (compare fi8' 6a): '

(II 7) V. ' Rl V' ,, g = (Rl + Rl') ,

so that a current I' = S Vg' flows through the lead, short-circuiting the output,

, ffl ' TrOj~~+rr=.È.. ' .1l. ' 41611:E

Fig. 6a) Equivalent network for the input of a 'grounded grid amplifier having the outputshort-circuited. .

b) Equivalent network for the output of a grounded grid amplifier ..

391

, \

392 A. VAN DER ZIEL

When that short-circuiting lead is removed, the anode voltage Va can befound by assuming the current

(I~, 8) I' = S (1+ I/~) Vg' = ~: ,= (RI :' RI')

to be flowing into the output of the amplifier. As the output impedanceof the amplifier is given by (11, 5), we have for the current [2' flowingthrough R2': ' • .

(11, 8a) I ,_ (~+. 1) (Rl + RJ.') I'2 - (p + 1) (Rl + Rl') + R2' •

(~+ 1) V'= (~+ 1) (Rl + Rl') + R2"

so ,that the po~er I2'2R2' delivered to R2' is:

(I I '2 R ' _ . (I-' + 1)2 V'2 R2'I, 9) 2 , 2 - [(I-' + 1) (Rl +,Rl') + R2'J2

[(~+ 1) in, + Rl') + R2'J2'As the amount of power Ènnt can 'also be delivered to R2' by directly

coupling and matching the: antenna to R2', we have for the power gain gof the. stage: .

4 Rl' R2' (p + 1)2 4{J6(IT, 10) B = [(",+ 1) (Rl' + Rl') + R2']2 - (ft + 1) (1 + {J + 6)2

in !whichRl' IRI is denoted by {Jand R2'IR2 by 6. For a constant value of {J,g has a maximum value:

,> (11, lOa) gl = (~+ Ir f3.1(I + (J) for 6 = 1 + {J ;for a constant value of 6, g has a maximum value:

(11, lOb) g' = (p + 1),67(I.o+ 6) fo~ {3 = 1 + 6.

, From these equations it is obvious that {J aÎld 6 cannot be chosen suchthatg has an absolute maximum. For very large values' of {Jand 6, however,(1 + (J) is nearly equal to {J, and (1+ 6) equal to 6. Hence, due to the feed-back from the output to the input 'of the amplifier, it ~simpossible to match

, the antenna to the inp ut of the amplifier and at the same time the next stageto the output of the amplifier; for very high values of RI' and R2' inputand output will be practically matched, if Ri' IRI = R2' IR2• .

InfiB' 7, g/(p + 1) is shown as a function of {3 with 6 as a iparameter(because st (p + 1) is symmetrical in {Jand 6, (J and 6maybe interchanged inthis figure). Suitably chóosingRI' andR2' it must be possible to obtain valuesfor g somewhere between 0;5 (~ + 1) and 0.8 (~ + 1) (a further increasein RI' and R2' may sometimes be possible, but it will be shown that, espe-cially on shorter wavelength, the influence of Cae and electronic transittimes may cause instability of the amplifier). It will be shown that highvalues of RI' IRI are important for obtaining a better signal to noise ratio.Aacording' to, fig. 7 this is permissible and even results' in a higher gain,pr~vided the value of the output resistance R2' cap.be c~osen large enough.

393GROUNDED GRID AMPLIFIERS

O.

0 II IIIIII'" , - IIII

8 1\ ~ ~jl~~l/ =6 I ~ \. IIII.' # . , \0=4 IIII -,

I 1'" J IcJ"'2 I • R'

" If3=7/;" 1-;-.I--='rJ=..!!1. r.-2 IL r-, rJ=J R2

~ Ö=D:s 1 ""1 ~ ; ~=a251 Ill!fa . faO

--r-f3 .

r#tJt.Î o.

O.

O.

oos .

Fig. 7. g/{P + 1) as a function of P = (Rl'/RI) with D = (Rz'ÎR2) as a parameter. Asg/(p. + 1) is symmetrical in pand D, pand D ,may be interchanged.Dashed curve 4PJ(1 + P)2 as a function of p.

. I

From these calculations the importance of a high amplification. factor p ofthe triode is evident.

2. Grounded grid iriodes as wide band amplifiers

In the case of wide band amplifiers, the output capacity COg of the valvemakes it impossible to use high values for R2'. In that case feed-back fromthe. output to the input is negligible, and according to the preceding sec-tion the input rêsistance of the amplifier will be R2/(p + 1) = RI (becauseR2' ~ R2)· Hence the input of the ,amplifie! is so much damped that the'input impedance can be considered as a resistance for all frequencies' of theband. ,

In that case maximum gain is obtained for RI'/RL - 1 ({3= 1) and(H, 10) becomes approximately (R2/RI = (p + 1)): :

(Ir, il)

According to this equation a higher value of R2' results in 'a higher gain;ö~.l means a high value of p', But on the other hand the value ,of R2' ,

which can be allowed depends upon Cog: the smaller the value of Cog,the larger the permissible value of R2/• A small value of Cog can be obtainedby using a wide anode-grid spacing, indirectly this means a high p. Hencefor wide band grounded grid, amplifiers a wide anode-grid spacing and alarge amplification factor p are favourable for obtaining a high gain. , "Though maximum gain is obtained for RI'/RI = 1, an appreciable differ-

ence in the value ofR/ /RI from unity will only give rise to a slight differencein gain. This is shown in fig. 7, where 4{3/(1+ (3)2 is given as a function,of {3:for RI'/Ri = 2 the drop in gain is only 11%, and for RI'/Rl = 4 only36%. This is important. for obtaining a better signal to noise ratio, foraccording to section H, 4 the signal to noise ratio wi1l increase with aninçreasing value of RI'/RI .. ·" ,

394 A. VAN DER ZIEL

, '

3. The influence of the anode-cathode capacity Cs; and of the. electronictransit times

When the capacity Cae andelectronic transit times can be neglected, itfollows' from section 11, 1 that the amplifier remains stable' even for veryhigh values ofRl' jR1 and R2' jR2• But on the other hand it is well known thata grounded grid triode can be used as an oscillator (fig~ 8) provided thecapacity Cae i!?sufficiently large and the impedance between grid and ca-thode is capacitive. Hence it may be possible that the amplifier is stillstable for the frequency at which input and output circuit are tuned, butnot for a frequency for which the input circuit is slightly detuned. It willbe shown that this sets an upper limit for the values of Rl' jR1 and R2' jR2for which stability is secured. As it will be shown that the capacity Coc. and the electronic transit times have a similar influence upon stability,the two effects will he treated sim_ultaneously.

Fig. B.Grounded grid triode as an oscillator. L is a choke in the cathode lead, thus causingthe grid-cathode impedance to be capacitive.

In order to discuss stability of the amplifier we first assume the outputof the amplifier to be short-circuited and the input circuit to .,be tuned atthe desired frequency, and afterwards detune the input circuit by. intro-ducing. a capacity C (positive or negative). When thetransformed antennaresistance is again denoted by Rl" we have to introduce into (11, 1):

Ig = -Vg (;1'+ froC) •

We then remove the lead, short-circuiting the output of the amplifier,and calculate the internaloutput conductance of the amplifier as a functionof C. C is then chosen such that the internaloutput conductance of theamplifier is as. negative as possible. In order to secure stability of theamplifier under all circumstances the external output conductance 1jR2'

has to be chosen such that it over-compensates the negative internaloutput conductance.

Introducing the above value of Ig'into (11, 1) and eliminating Vg:Vg=_ Va (1' - j tan fP1+ jro Çae R2) ,

, , R2' (~1+ R~'+jroC)

and introducing this into the equation fo~ Ia, we have for the internal outputadmittance of the amplifier (R2~= ISll cos f{J1/P): . .

(Il, 1~) Ia/Va· IS21cos f{J~[1- (1-1tan <p, +iroC.; R,) (l-j ton'~')l=p ," I + Rl + 'roC R .

\ .R', J 1 "1 .

= IS21:os f{J2[(1 + ~~ r + ro2C2Rlfl X

X' [(~:' + tan f{J1tan f{J2-roCaeRztan f{J2)(1 + ~~,) + w2C2R12 ++ «c«, (tan ~l + tan f{J2-roCaeR2)] ,+ ·imaginary terms. ', I·. .

The real part is as negative as possible, if:

(Il, 13) , roCRl , - t (tan f{J1+ tan f{J2~Cae Rz), {:2 <i,roZCZR12~ I) ..', .. . (1 IS I cos f{J . 1 cos cP )in which case the real part of la/V" becomes -R = _2 __.__ 2= R __ 2

" , 4 p.' z cos f{J1(Il, 12a)

_!_1(1 t~) (~+tanf{JltanIP2-roCaeR2tanIP2)-~ (t~nf{Jl+tanlPz~rocaeRz)l

R4L. . (1 + ~:,r+t (tan f{J1+ tan IPz-roCae Nzr . . ,In order' to secure stability under 'all circumstances this has to be less

negative than -1/Rz'. If (RI/RI') ~ 1and (tan IPl + tan IPz-roCaeRz) ~ 1this leads to the stability condition:

(11 14)" Rl R4 1 ( ,... R ) a, , R--;+R' > 4' ro\'ae 2 + tan IPz-ta,n CPl .1 Z .If the conditions for RI/RI' and (tan CPl+ tan IP2-roC"e R2) are not ful-

filled, the full equation (Il, 12) has to be taken into account. .It might be of interest to knowat what frequencies roC"eR2 = 1. When

.stray capacities can he neglected, as is allowed in the case of perfect scree-ning (e.g. for disk seal tubes), we have: .

, ,

GROUNDED GRID AMPLIFIERS 395

' ......

and, because I/R2' Sip,roCaeR2 = roCge/S ' 1.

According to Langm uir:

S = 3.51 X 10-6 0 V6'" deg-2 A/V,whereas:

roCge= O.556x 10-120 'JI deg-1 A/V (ro= 2:7t1l),in which 0 is the' emitting cathode area, V. the mean voltage in the gridplane and deg the cathode-grid ~stance. Hence roCaeR2 - 1 means

11 = 6.4 V,'/. deg-1 Mc/s.

396 A. VAN DER ZIEL

Setting V. = 4 volts and deg = 0:01 'cm we have v = 1200 Mcfs (25 cm)., Hence in many cases the influence of the anode-cathode capacity uponthe stahility of the amplifier can he neglected especially when the cathode-grid spacing is narrow. The influence of the phase angles CP1and CP2.of thetransconductances S1and S2 is small too, if the cathode-grid and grid-anodespacing are narrow. Hence narrow electrode spacings result in a betterstability of the amplifier. 'Up to now !lC) reason has heen found why the power gain of a grounded

grid amplifier should decrease with increasing frequency. In order to makethis point clearer, we shall show that for wide-hand amplifiers (R2' ~ R4)

electronic transit times do not cause a decrease in gain. We may then use(Il, 6), (Il, 7) and'(H, 8) with these restrictions, that: ,

RI = IS11 (1 + I/fA) cos CP1;Irl= ÎS21(1 + 1/;) Vg' = ,lsS2

11R 1 Vs' ,1 . 1 1cos CP1

. whereas 12' = I', hecause the whole current I' is how flowing through R2'

(R2' ~ R2). Hence we have for the power delivered to R2' instead' of(Il, 9):. . . .

12'2 R2' , 4 Eant[ S212(1+ 1/f1)2 R2' Rl' (Rl ~l~J2,so that the energy gain g is now:

(Il, 15) g _ IS211S21 (1 + 1/,11) R2' Rl' ( 2 Rl )2,- ISll cos CP1 u, s,+ Rl'

and in the ease of maximum gain (RI'/R1 = 1):

(Il, 15a) S (1+ 1/",) R2'g = cos CP1 '

hecause in goed approximation IS11=.IS21. S. Hence the power gainwill not decrease' with increasing frequency.

It is a well-known fact that grounded grid triodes used as oscillatorshave an optimum frequency at which they will operate. This means that atthat frequency the power gain is nearly equal to unity. According to theahove theory electronic transit times cannot account for- it. The chiefreason must he sought in circuit losses, especially in losses in. the electrodeleads. Eor it is well known that the tuned impedance Z of a circuit consis-ting of a capacity C, a selfinduction L and a resistance r (co2 LC = 1) is(compare fig. 9): . . " .

(Il, 16) Z = .!:_ = 1Cr co2 C2 r

..,1686,

Fig. 9. Tuned circuit. L, C and T denote the self-induction, capacity and resistance of the'. circuit, 002 LC =, 1.

GROUNDED GRID AMPLIFIERS 397

As the series resistance of thè leads d~e to skin effect increases as w'/.,~ is proportional to ,w-'I•. (Taking À = 100 cm, or w = 2,X108, C = 5 fLfLFand R = 2.Qwe have Z = 5000 .Q;this is generally smaller than the internal.resistance of a valve with high amplification factor fA.) The result is that themaximum values of RI' and R2' which can be obtained decrease with fre-quency and hence, according to fig. 6, the energy gain will decrease too.In this respect the newly constructed disk seal tubes 15), in which all elec-trode leads have heen replaced hy copper plates, will give a hig improvementin power gain. But even in this case an equal decrease of all valve dimen-sions hy a factor p will result in the same transconductance hut in 'a decreaseof the valve capacities hy a factor p and hence, according ):o.eq. (Il, 16), .the circuit impedance will increase hy a factor p2 *): Hence 'even in thiscase, though electronic transit times may 'not cause much harm, narrowcathode-grid, and ,g:r;id-:anodespacings are extremely importànt!

4. Signal to noise ratio in grounded grid amplifiers

Before calculating the signal to noise ratio of a grounded grid amplifierit will he shown that. the signal to noise ratio of an amplifier will not bealtered, tohenthe ozÛput-circ.u,it is short-circuited (or, more exactly, the signalto noise voltage ratio ,at the output of the amplifier is the same as. the signalto noise current ratio in a lead short-circuiting the output). The advantageof this theorem is that in order to calculate the signal to noise ratio of theamplifier the output may he considered short-circuited so that feed-hackcan he neglected. 'If the output ofthe ämplifier has an internal resistance R; asignal e.m.f,

Vo and a noise e.m.f, Vo (for a frequency interval Lt,,), then the ~quare ofthe signal to noise vfJltage ratio is V021v02. If the output is short-circuited, asignal current 10 . VoLR and a noise current io= VoiR will flo\Vthrough the ~lead short-circuitinJi. the output, so that the square of the sig:ï;tal"tonoise

.. 12/'2 V.2/2 ' . .current ratio IS 0 z'o = 0 vO' it .

According to section Il, 1 equations (Il, 6) and (Il, 7) the square oftheinput signal voltage Vg', if the input circ~it is tuned, is:

Vg'~ , 4 s.; R/ (Rl~lRJ2.With respect to noise we have to distinguish hetween antenna noise and

shot effect noise. The antenna noise can he described hy assuming the radia-tion resistance of the -antenna to produce fluctuations corresponding to a"noise temperature" of aant times normal room temperature T; Qant' 1,if the antenna noise is of terrestrial and not of stellar origin 8•.19). '

The available' noise power 7) in. a small frequency interval Lt·v isQant le'I'á»; this power will he delivered to every impedance to which theantenna is matched. Hence the mean square of the noise voltage hetweengrid and cathode due to antenna noise is (compare (Il, 17)):

(Il, 17)

(Il, 18) .

*) After what has been said in !, 2 it is obvious that the thickness of the oxide coatingmust be decreased by ihè same factor.

398 A. VAN DER ZIEL

According to section I, 3 the total equivalent shot effect noise voltage:

(Il, 19) v" .: v~+ v.In' this equation:

(II, 19a)

because the fluctuating current il is flowing through the tuned input circuit,whereas according to (I, 9a): "

(1I, 19b) . i2 • vS2 (1 + IIp),so that the mean square of the total equivalent shot effect noise voltage is:

(1I,- 22) , Rl *)a = aant ~ ,8 Rl' .

Assuming the antenna noise to .be of terrestrial origin, wehave aant = l.raking 8= 3 (according to section I, 3), the shot effect contribution to thesignal to noise ratio of thë grounded grid amplifier is only three rimes thermalnoise for Rl' IRI = 1, equal to thermal noise for RI' IRl = 3; and negligiblefor large values of RI' IR!" It has already been shown in section 1I, 1, thatfor low values of R2' IR2 the power gain of the amplifier is only slightlyreduced for appreciable values of RI'/RI, whereas for high values of R2'IR2

maximum gain is obtained for RI' IRI = R2' IR2 + 1. Hence relatively largevalues of Rl' IRI can be allowed, so that a grounded grid amplifier is prac-tically an ideal amplifier with respect to the signal to 'noise ratio•

•) If the fluctuating current il flowing through the input circuit had been neglected(V1 = 0), we would have found: •

(1I, 20b) V"2 =12R12 = e 4kTLlv Rl' ,Introducing this into (1I, 21) we would have obtained:

\. (1I, 22a) a = a.nt + s ~; (Rl ~lRly,differing markedly from (H, 22), especially for large values of R{/Rl'

GROUNDED GRID AMPLIFIERS 3.99

.Measurementa of the signal to noise ratio. of a grounded grid amplifierstage were carried out at a wavelength of 7 m. Using a triode for which I

. S = 6.9 mAjV and fI = 173, so that Rl _:_14~n,we found:

(a-a.nt)28

As according to (Il, 22) (a-a.nt) is proportional to (R1')-1, these .resultsmay be considered as a direct proof of the validity of the above theory ..Moreover these measurements yield /3 = 4, which is also a reasonable value ..The signal to noisè ratio for higher frequencies, for which phase angles

have to be taken into account, will be dealt with in a forthcoming paperwritten in collaboration with Dr Strutt. It will be shown there thatunder favourable phase conditions a complete shot effect suppressionshould be possible. _ . .. But even if those phase conditions mightnot prove to be such that a corn- .plete shot effect noise suppression would be possible, it can be stated thatthe phase angles will remain small, even at higher frequencies, providedthat the electrode spacings (and hence the electronic transit times) arechosen as small as possible. Hence for grounded grid trio des with narrowelectrode spacings (Il, 21) should remain valid up ~overy highfrequencies.

Eindhoven, December 1945

REFERENCES

1) C. J. Bakker and G. de Vdes, Physica 2, 683, 1935.2) C. J. Bakker, Physica 8, 23, 1941.3) W. E. Benham, Phil. Mag. 11, 457, 1931.4) W. E. Benham, Proc. Inst. Rad. Eng. 26, 1093, 1938.5) N. D. Deviatkov, M. D. Gurevich and N. K. Khokhlov, Proc. Inst. Rad.

Eng. 32, 253, 1944, .6) M. D. Dishal, Proc. Inst. Rad. Eng. 32, 276, 1944.

. 7) H. T. Friis, Proc. Inst. Rad. Eng. 32, 419, 1944. .8) E. W. Herold and L. Malter, Proc. Inst. Rad. Eng. SI, 423,491,501,567,575,

1943..9) M. C. Jones, Proc. Inst. Rad. Eng. 32, 423, 1944. .10) L. Katz, Proc. Inst. Rad. Eng. 32, 641, 1944.11) W. Kleen, Telefunkenröhre, Heft 23, 273, 1941.12) F. B. Llewellyn, Proc. Inst. Rad. Eng. 21, 1532, 1933.13) F. B. Llewellyn and A. E. Bowen, Bell Syst. Techn. Journ. 18, 280, 1939.14) F.. B. Llewellyn and L. C. Pet er son, Proc. Inst. Rad. Eng. 32, 144, 1944.15) E. D. Ma cá.r thur, Electronics, Febr. '45 p. 98.16) D. O. North, Proc. Inst. Rad. Eng. 24, 108, 1936.17) A. J. Rack, Beil Syst. Techn. Joum, 17, 592; 1938.18) G. Reber, Proc. Inst. Rad. Eng. 30, 367, 1942.19)· F. Spenke, Wiss. ·Veröff. SiemensWerke, 16, 19, 1937.20) M. J. O. St rut t and A. v. d. Ziel, E.N.T. 12, 347, 1935.21) M. J. O. Strutt and A. v. d. Ziel, E.N.T. 15, 103, 1938.22) M. J. O. Strutt and A. v. d. Ziel, Proc. Inst. Rad. Eng. 26, 1011, 1938.23) M. J. O. Strutt and A. v. d. Ziel, Physica 8, 1, 1941.24) M. J. O. Strutt and A. v. d. Ziel, Physica 8, 81, 1941.25) M. J. O. Strutt and A. v. d. Ziel, Physica 9, 1003, 1942.26) B. J. Thompson, D. O. North and W. A. Harris, R.C.A. Review4,269,441,

1940; 5, 106, 240, 371; 595, 1940/1941; 6, 114, 1941.

400

A new series of small radio valvesThe use of. isotopes as tra~ers

G. Alma a~d F. PrakkeA. H. W. Aten'Jr. andF.'A. HeynP. C. van der Willigen

PHILlPS TECHNICAL REVIEW

Philips Technical Review deals with technical descriptions of new Philips products,and reports on investigations relating thereto. It gives information concerning .the appli-cations of these productsand in addition gives technical articles of a more general natureon various questions, which may be of interest to the reader in connection with the abovementioned applications. ' , '

The endeavour is made' to present the subject matter as simply as possible. Mathe-matical treatment is resorted to as far as necessary. Additional explication in simplelanguage, however, makes it possible for those who do not wish to follow the mathematics,to gather the gist of the ~ticle from the text alone or even from the subscripts of theillustrations. ,,'

Philips Technical Review appeared for the first time on January 1, 1936 and was.continuated until June 1~42. In 19,42 the publication was prohibited by the occupying: power. In January 1946 wemade a new start with the edition of this journal, We hope thatthe reappearance of Philips Technical Review will receive in a broad circle of readers thesame attention as in former years.

Here follow the contents' of the September-October issuesVo18, Nr. 6 (September 1946).

The multireflec~on tube, a new oscillator for very shortwaves F. CoeterierLiving room lighting with tubular fluorescent lamps L. C. Kalif and J. VoogdA voltage stabilizing tube for very constant voltage T, JurriaanseImpedance measurements with a non tuned Lecher.system J. M. van Hofweegen

Vol 8, Nr. 10 (October 1946)

Penetratien and welding speed in 'c~ntact are-weldingAutomatic change-over to' an emergency apparatus ina communication system'Non-ferrous copper wire for moving-coil meters

G. Hepp'A. Rademakers

Philips Technical Review appears al~o in French (Revue Technique Philips), in Spanish. (Revista técnica Philips), in. Oerman (Philips Technische Rundschàu) and in Dutch(Philips Technisch Tijdschrift).