The Electrostatic Field in Vacuum Tubes With ArbitrarilySpaced Elements

By W. R. BENNETT and L. C. PETERSON

VACUUM tubes with close spacing between electrodes have become ofincreasing importance in recent years. The higher transconductances

and lower electron transit times thus obtained combine with other features toraise both the frequency and band width at which the tube may operatesatisfactorily as an amplifier. Specific designs have been discussed in papersby E. D. McArthur and E. F. Peterson', and by Fremlin, Hall and Shatfordi.The important contributions to structural technique made by E. V. Neherhave been described in the Radiation Laboratory Series", Further im­portant advances in the art have been recently announced by J. A. Mortonand R. M. Ryder of the Bell Laboratories at the recent I.R.E. ElectronicsConference held at Cornell University in June, 1948. The material of thepresent paper represents work done by the authors over a decade ago, andnaturally there has been considerable publication on related topics in theintervening years. It has been suggested by our colleagues, however, thatsome of the results are not available in the technical literature and are ofsufficient utility to warrant a belated publication. These results have to dowith the variation of the electric intensity, amplification factor, and currentdensity which takes place along the cathode surface because of the nearbygrid wires.

We shall deal mainly with the approximate solution which neglects theeffect of space charge. The correction required to take account of spacecharge is in general relatively small as shown by both qualitative argumentand experimental data in an early paper by R. W. King". More recenttheoretical work" extending into the high frequency realm has confirmedthe minor nature of the modification needed. The problem is thereby re­duced to one of finding solutions of Laplace's equation which reduce to con­stant values on the cathode, grid, and anode surfaces. The original workon this problem was done by Maxwell! who calculated the electrostaticscreening effect of a wire grating between conducting planes long before thevacuum tube was invented. All subsequent work has followed the methodsoutlined by Maxwell, In particular he suggested the replacement of theconducting planes by an infinite series of images of the grid wires anddescribed an appropriate solution in series for the case of finite size wires.The useful approximation obtained when the diameter of the grid wires is

303

304 BELL SYSTE},[ TECHNICAL JOURNAL

assumed small compared to their spacing was discussed in detail only for thecase of large distances between the grating and each of the conducting planes.

Figure 1 shows the assumed geometry of the grid, anode, and cathode.End effects are neglected. The origin is taken at the center of one of thegrid wires which have radius c, and the X-axis is along the grid plane. Thespacing of the wires between centers is a, the distance from grid to anode isd2, and that from grid to cathode is d!. No restrictions are placed on thesizes of a, d2 , and d-, Above the anode and below the cathode is shown adoubly infinite set of images which may be inserted to replace the conductingplanes of the anode and cathode. By symmetry the potential from the

+qo 0 0 0

-q 0 0 0 0

~---

-q 0 0 0 0

+q 0 0 0 0

-q 0 0 0 0

Fig. I-Array of images for production of equipotential surfaces in planar triode.

array of charges there shown must be constant for all x when y = d'1. andalso for all x when y = -d1• The double periodicity of the array suggestsimmediately an application of elliptic functions. The solution of the sym­metrical case was actually stated in terms of the elliptic function sn z byF. Noethers. The extension to the non-symmetrical case shown in Fig. 2is fairly obvious. One of the authors worked out such a solution in termsof Jacobi's Theta functions in 1935, but abandoned any plans for publishinghis analysis in view of the excellent treatment appearing shortly after thattime in the Proceedings of the Royal Society by Rosenhead and Daymond",who applied Theta functions to both tetrodes and triodes, and both cylindri­cal and planar tube structures for the case of fine grid wires. Some of theirformulas were later included in a book by Strutt", Methods of calculating

ELECTROSTATIC FIELD IN VAC(!UM TURES 305

the case of thick grid wires in terms of expansions in series of elliptic functionswere discussed by Knight, Howland ani McMullen8- lo. The problem of afinite number of grid wires was treated by Barkas". More recently tubeswith close spacing between grid and cathode, but with anode and grid as­sumed far apart, have been analyzed in terms of elementary functions by

\ x=o GIVESt/=f04 t

\ Jt-a--~2

\2C ~ X d.

1'-, - t"

~-!>O, \ a-.\ !!J =04a.

.£.=J..., \ a 14

\ J.I.o=21Td2

\0.5 \ a LOG 2;'C

\--~ \\~~ ,

..0.64 '....... \l\...\.....

-- t.O

~- -.~~-j1o !!J=oo ~.,,- . --- -.-f---

a , " ...

~"

", ~~, --- -~' ' ........

'~ - --

--r---20

o O.O!> 0.10 o.rs 0.20 0.2!> 0.30 0.3!> 0.40 0.4!> O.!>ODISTANCE ALONG CATHODE, X/a

Fig. 2-Variation of amplification factor along the cathode surface of a triode.

24

28

32

::I.. 56

II:ot 52~

~ 46~u!!; 44oJQ.::Ii« 40

60

64

66

72

76

36

Fremlin". A solution based on the Schwartz-Christoffel transformationhas been given by Herne" for the case of grid wires of finite size and approxi­mately circular in shape.

Since the derivations have been adequately covered in the references cited,we merely state the final formula here and indicate how it may be verifiedas correct. Let V(x, y) represent the potential function corresponding to

306 BELL SYSTEM BECHNICAL JOURNAL

(1)

Fig. 1, the planar triode with fine grid wires. The potential of the cathodeis set equal to zero. Then in the space between anode and cathode,

AV(x, y) = [21r d2(y - d2)! a + (dl + d2)!(x, y)]V"

+ [B(y + dl ) - 21r d2y/a - dd(x, y)]Vp ,

where

(x ) = tn It1dll-(x + iy - 2i d2)! a] ),y t?l [1r(x + iy)!a]

A = (dl + d!)B - 21r dVa

B = tn IatM21r~ dda) I1r£t?I(O)

(2)

(3)

(4)

Here we have used Jacobi's notation for the t?l-function, as explained byWhittaker and WatsonU

, rather than the Tannery-Molk notation used byRosenhead and Daymond. We write t?l(n) for their t?l(Z), In our notation..

t?l(Z) = 2 L (- )"ei (1l+l /2)2,. sin (2n + l)z,,-0 (5)

where the parameter T in the above formulas is given by:

T = 2i(d l + d2)! a (6)

By t?~(z) is meant the derivative with respect to s:..t?;(z) = 2 L (- )"(2n + 1)ei

(" +l/ 2)2,. cos (2n + 1) z (7)..-0

Verification of the solution is straightforward. The resulting Vex, y) isseen to be the real part of a function which is analytic in the complex variablex + iy except for logarithmic singularities at the points where the Thetafunctions vanish. Hence Vex, y) satisfies Laplace's equation in two dimen­sions in the region excluding the singular points. Since the zeros of t?J(z)occur at Z = m1r + 1t1rT, where m and n take on all positive and negativevalues as well as zero, the singular points of the solution are at

and

x + iy = ma + 2in(d l + dl ) - 2irh

x + iy = ma + 2in(dl + d2)) (8)

which coincide with the centers of the image circles of Fig. 1. The logarith­mic singularities represent line charges with the first set arising from a t?l­function in the numerator, yielding a positive charge, and the second setfrom the t?l-function in the denominator giving a negative sign. Theequipotential curves are approximately circular in the neighborhood of the

ELECTROSTATIC FIELD IN VACUUM TUBES 307

charges and hence V(x, y) gives a constant potential on the surface of eachgrid wire if the radius of the grid wire is small compared with the spacing.

We may show by direct substitution that V(x, y) becomes equal to VI' atall points of the anode and equal to zero at all points of the cathode. Onthe anode we have y = d2 which, when substituted in the expression forf(x, y), gives the logarithm of the absolute value of the ratio of conjugatecomplex quantities, and hence

f(x, d2) = 0

Substituting in (1), we then readily verify that V(x, dt ) = VI'" On thecathode we make use of the quasi-periodicity of the tYt-function, as ex­pressed by

to prove

21f'~j (x -dl) = -, a '

(9)

(10)

from which it follows that V(x, - d.) = O. To show that all grid wiresare at the same potential, we make use of the other periodicity of thetY.-function,

which shows that

f(x ± ma, y) = f(x, y), m = 0, 1, 2, ..•

(11)

(12)

(13)

(14)

It remains to prove that V actually approaches the value V, in the neigh­borhood of the typical wire, which may be taken at the origin since thesolution repeats periodically with the wire spacing. We let

+ . ,6X t.y = ce

and assume cia «1. Expanding in power series in cia, we find that thefirst order terms are included in:

( . ItM-2rid2Ia)If c cos 8, c sin 8) = (n tY:(O)1f'ce"la = B

The sign of the argument of the t7.-function in the numerator is of no conse­quence since it does not affect the absolute value. Substituting back in (1),we then find

Lim V(c cos 8, c sin 8) = Vg./a....O

The solution is thus completely established.

(15)

308 BELL SYSTEM TECHNICAL JOURNAL

(16)

The quantities in which we are specifically interested are electric field,amplification factor, and current density. The electric field is equal to thenegative gradient of the potential function. The amplification factor isfound by taking the ratio of partial derivatives of the electric field at thecathode with respect to grid and anode voltages. The current density maythen be studied for any assumed operating values of grid and plate voltages.

To calculate the gradient we note that since Vex, y) is the real part of ananalytic function W(z) = V + ill; it follows from the Cauchy-Riemannequations,

W '(=. ) = oV _ -i 0 V = <E; + iEyax ay

where E; and Ey are the x- and y- components of the electric intensity.From (1),

AW(z) = [(dl + d2) F(z) - 21rd2(iz + d2)/ alVu

+ [B(d1 - iz) + 21rid2z/ a - d1 F(z)lV p,where

F(z) = (n !?d1rSz - 2id2)/ a)171(1rZ/ a)

Calculating the derivative and making use of the relation,

!?~(z - 1l"T) = 11:(0 + 2it1b - 1l"T) I1 I (z) ,

we find at the cathode surface

F'(x - i d1 ) = 21ri [l + C(x)]a

where

It follows that when y = -dl , we must have E; = 0 and

aAE.,,/21r = [d1 + (dl + ~)C(x)]Vu

+ [d2 - dl -,aB/21r - d1C(x)]Vp

(17)

(18)

(19)

(20)

(21)

(22)

The amplification factor is then given by

iJEu/iJVo dl + (dl + d2)C(x)Po = iJEv/av;, = d;-~-d-;-- aB/21r _ dlC(x) (23)

Numerical calculation from these formulas can be made by means of (5)and (6). When d, and ~ are both large compared with unity, Eq. (23)

ELECTROSTATIC FIELD IN VACUUJf TUBES 309

reduces to the familiar approximate formula derived, for example, ill anearly paper by R. W. King l5

,

JI. -21f d2

a 'aln­21fc

(23a)

I e.. Va~

! /v -1---1----- ~--- ... --;/

/ /' '0.5.. --- -- V 1/

,,/ 0.64 ----"... '

1---- ._-->

j ....,'". - "...'

t·", ...A

...

,'" /1 ---.J:..o_ --- -- -- --, ,//1/......

J.........

J----a---~}_...... ,

,/ /, 2c-l \..~ i'/ //'

.J ~=5.0

/a

V' C I-a=i4

6 Vg= -2 VOLTS

/ Vp =100 VOLTS

\,/ VOLTSE =UNIT OF DISTANCE

0

1.4

o

1.2

1.0

0-8

1.6

06

-t.

-0.

-0.

-0.4

-0.2

on~ 0.4o>1; 0.2lJ.J

"0

o 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.4S 0.50DISTANCE ALONG CATHODE,'X./a

Fig. 3-Variation of cathode field strength in a triode.

Some calculated curves for JI. and EQ are shown in Figs. 2 and 3. Figure 2shows the amplification factor as a function of the distance along the cathodewith the ratio of grid-cathode separation to grid wire spacing as a parameter.The ratio of grid-anode separation to grid wire spacing is held constant atlive. Only half the grid spacing interval is included since the curves aresymmetrical. The increase in JJ.-variation as the grid-cathode separationbecomes small is clearly demonstrated. For negligible JI.-variationwe mustselect dt / a of the order of 2 or greater.

310 BELL SYSTEM TECHNICAL JOURNAL

Figure 3 shows the variation in field strength along the cathode for thetypical operating point, Vg = - 2 and Vp = 100 volts. It is to be notedthat for dlla less than 0.6, the electric field actually changes sign as we movefrom a point immediately below a grid wire to the midpoint between twogrid wires. In other words a part of the cathode will not emit at all in thesecases while the remainder emits in a non-uniform manner. In the ratherextreme case of dl/a = 0.4 only about a quarter of the cathode is emitting.It is worth noting how relatively rapid the "shadow" or "island" formationincreases between dlla = 0.64 and 0.5 as compared to the increase in theinterval from 0.5 to 0.4.

If the equation for p. is solved for C(x) and the result substituted back inthe expression for Ell at the cathode we find:

(24)

where here of course p. varies with x. This is identical with the expressionderived by Benham's from Maxwell's approximate solution except that inthe latter case IJ. was a constant. Our colleague, Mr. L. R. Walker, haspointed out that the equation follows directly from the assumption of smallgrid wires without explicit solution for the potential function. Since thecharge density fT. on the cathode is proportional to the .field strength (thefactor of proportionality in MKS units is the dielectric constant E of vacuumor 9.854 X 10-12 farads/meter), Maxwell's capacity coefficients CgO andCp • may be calculated from

(25)

The minus sign is used here because we are taking the ratio of charge tovoltage at the negative plate of the condenser consisting of cathode, grid andanode surfaces. Hence

(26)

(27)

Since p. is variable, an integration is required to determine the total capaci­tance. From the periodicity of p. with grid spacing it is possible to expressthe result in terms of the average values of Cg. and Cpo over an interval oflength a along a direction parallel to the grid plane and multiply thesevalues by the total area of cathode surface.

ELECTROSTATiC FiELD iN VACUUM TUBES 311

Equation (24) may be interpreted in a number of different ways of whichwe shall mention the following two:

1. The "equivalent voltage" Vp + Vp/p. does not act at the grid but at adistance D from the cathode, where

(28)

Both the equivalent voltage and distance vary along the cathode surface.2. The "equivalent voltage"

V. = (V, + V,,!p.)/[l + (l + rh/d1)!IJo) (29)

acts in the grid plane and varies with distance along the cathode surface.As far as the cold tube is concerned the two formulas are equivalent at

the cathode, but not at the grid. When the tube is heated and completespace charge is present, the two formulas also differ at the cathode. Thecurrent density in the presence of space charge is, according to (28) andChild's law:

while, from (29),

I = K(Vp + V,,!p.)3f2j lJl (30)

(31)

In both, K2 = 32 E2e/ 81m, where elm is the ratio of electronic charge to mass.The value of current given by (31) is [1 + (1 + rh/d1)/ p. )1/2 times as largeas that given by (30). If u >> 1 + d2! d1 the two values are nearly thesame. In tubes with close grid-to-cathode spacing the inequality may notbe fulfilled. As to which viewpoint is more accurate, we note that Ferrisand North in their papers 17.18 on input loading adopted the latter, and thatat high frequencies where electron transit time must be considered the secondviewpoint is preferable because of the more accurate representation ofeffects at the grid. For a more complete discussion see Reference 19.Figure 4 shows curves of relative current density as a function of distancealong the cathode as computed from Eq. (31). The transconductance forunit area of cathode surface as computed from the same equation is given by:

d~gm. = d~ aI = ~ E' /2e( Vp + V,,) (cb)3ev, 3 'V m p. D

= 3.512(Vp+ V"jp.)1/2(d1!D)3/2 micromhos, (32)

The resulting variation with distance along the cathode is shown in Fig. 5.Defining the figure of merit M at a point x along the cathode as the ratio

312 BELL SYSTEM TECHNICAL JOURNAL

between the transconductance aI/aVg and the sum of C ge and Cpo at thispoint, we find from (30)

(33)

where J = elf m«, elm = 1.77 X lOll coulombs/kg. From (31), we findon the other hand

(34)

~=Q)------ L._1-- - v:,;;},- ,."

"..." ,;,V/'

.'~.> i!......~'

v' / V.--' ,~1

7i f'---J',,, ,, ,

,/J 2C-l ~ d,0.6~1 •,. /1 ~=so

" D.S

a .

~=.!., .' II a 14,// 0.4

Vg = -2 VOLTS,,, , ;; Vp = 100 VOLTS

" ~,oo O.OS 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

DISTANCE ALONG CATHODE,X./aFig. 4-Variation of current density in a triode.

t.1

a:~ i,ozwv

::! 0.9::>....a:~ 0.80(

offi 0.7

!(!( 0.6J:....g o.sw>~ 0.4~wa:~ 0.3111Z~ 0.2....Zwg; 0.':JV

(35)

Both formulas indicate that for a cathode capable of supplying a given cur­rent density the only means of improvement lies in decreasing the cathode­grid spacing. The improvement is extremely slow; doubling the figure ofmerit requires an eight-fold decrease in spacing.

We again emphasize that the calculated current densities and figures ofmerit are functions of .r, the distance along the cathode. The total currentbetween the two grid wires is found from (30) to be

rIT = 2K (Vg + V pl p.)312dxlD2

"'0

ELECTROSTATIC FIELD IN VACUUM TUBES 313

~b,,"~..... ,

r: ,

/ "w~

7,f!,,

/j,~=I.O

7 1-I- ~-- 1------~--f--- --~- .• 9mo= TRAN5CONDUCTANCE PERI

I UNIT AREA OF CATHODE 5URFACEI

/ , ANODE/

JI

0.64/ --J;--a---U GRIDI ,I iI

I 2C~ f.-~ d lI I ! CATHODE

I 0.5jI d2 = 5.00.4 aI . £ =1-/ ;I a 14

IVg = -2 VOLTS

I II ; Vp-=IOO VOLTS

I;0.2

0.4

0.6

0.6

2.6

2.6

1.0

1.6

2.0

2.2

2.4

IIIoi 1.6oIl:~~ 1.4?;o~ 1:2

"'-~

ao 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

DI5TANCE ALONG CATHODE, X/a

Fig. 5-Variation of transconductance along the cathode surface of a triode.

(36)

where Xo is given by

(37)

On the basis of several reasonable assumptions it may be shown that both(35) and (36) lead to an approximate 5/2 power law instead of 3/2 powerlaw. Such a law has actually been observed in cases where shadow forma­tion was suspected.

We wish to express our appreciation to Messrs. R. K. Potter, J.A. Morton,

314 BELL SYSTEM TECHNICAL JOURNAL

and R. M. Ryder for their encouragement, and to Miss M. C. Packer foraid in the numerical computations.

REPERENCES

1. E. D. McArthur and E. F. Peterson, "The Lighthouse Tube; A Pioneer Ultra-High­Frequency Development," Proc,NaJ. Electronic Conference, Chicago, Oct. 1944, Vol.I, pp. 38-47.

2. J. H. Fremlin, R. N. Hall, and P. A. Shatford, "Triode Amplification Factors,"Electr. Comm, Vol. 23 (1946), pp, 426-435.

3. Hamilton, Knipp, and Kuper, "Klystrons and Microwave Triodes, Radiation Labora-tory Series, New York, 1948, p. 153.

4. J. C. Maxwell, "A Treatise on Electricity and Magnetism," Vol. 1, pp. 310-316.5. Riemann-Weber, "Differentialgleichungen der Physik," Vol. 2, p. 311.6. L. Rosenhead and S. D. Daymond, "The Distribution of Potential in Some Thermionic

Tubes," Proc, Roy. Soc., Vol. 161 (1937), pp. 382-405.7. M. J. O. Strutt, "Moderne Mehrgitter-Elektronenrohren," Berlin, 1940, S. 154.8. R. C. Knight, Proc. LondonMatll.Soc. (2) Vol. 39 (1935), pp. 272-281.9. R. C. J. Howland and B. W. McMullen, "Potential Functions Related to Groups of

Circular Cylinders," Proc. Cambro Phil. Soc., Vol. 32 (1936), pp. 402-415.10.· R. C. Knight and B. W. McMullen, "The Potential of a Screen of Circular Wires

between two Conducting Planes." Phil. Mag. Ser. 7, Vol. 24, 1937, pp. 35-47.11. Barkas, "Conjugate Potential Functions and the Problem of the Finite Grid," Phys.

&11., Vol. 49 (1936), pp. 627-630.12. J. H. Fremlin, "Calculation of Triode Constants," Phil. Mag. SeT. 7, Vol. 27 (1939),

pp. 709-741; also Electr. Comm., Vol. 18 (1939), pp. 33-49.13. H. Herne, "Valve Amplification Factor," Wireless Engineer, Vol. 21 (1944), pp. 59­

64.14. Whittaker and Watson, "Modem Analysis," Third Edition, Cambridge (1940),

Chapter XXI, p. 462.15. R. W. King, "Thermionic Vacuum Tubes," Bell System Technicaf JOUI"nal, Vol. II

(1923), pp, 31-100.16. W. E. Benham, "A Contribution to Tube and Amplifier Theory," Proc, I. R. E., Vol.

26 (1938), pp, 1093-1170.17. W. R. Ferris, "Inf.ut Resistance of V~cuum Tubes at Ultra-High Frequencies,"

Proc, I. R. E., Vo .21 (1936), pp. 82-10:1.18. D. O. North, "Analysis of the Effects of Space Charge on Grid Impedance," Proe.

1. R. E., Vol. 201, (1936), pp. 108-136.19. F. B. Llewellyn and L. C. Peterson, "Vacuum Tube Networks," Proc. I. R. E., Vol. 32

(1944), pp. 144-166.