The Dependence of InterelectrodeCapacitance on Shielding*LEONARD T. POCKMANt, NONMEMBER I.R.E.

Summary-Although it has been known for many years that theinterelectrode capacitances of a vacuum tube with a glass envelopedepend upon the shielding of the tube, no quantitative informationon the extent and character of this dependence has been published.In the present work the general theory is developed with the help ofMaxwell's coefficients of capacity and induction. A simple mathe-matical transformation shows why these cofficients may be used forpractical measurements in which all potentials are measured withrespect to the earth rather than infinity. The relation between theMaxwell coefficients and the interelectrode and electrode-to-earthcapacitances is also developed. The effect of changes in shielding isnot necessarily small. The experiments reported show that Cf forthe particular triode studied can be made,to vary from 0.16 micro-microfarad to 0.41 micromicrofarad by changes in the geometry ofthe external environment. Furthermore, ungrounding the shieldwith which Cf was 0.16 micromicrofarad made the "effective" C,f1.40 micromicrofarad.

A LTHOUGH in 1929 Loughren and Parker1 pointedout that reproducible results could be obtainedin the measurement of interelectrode capaci-

tances only by the use of a standardizable fixture orshield, they did not report any measurements to showhow changes in the environment of the tube were cor-related with the changes in interelectrode capacitances,nor did they point out the theoretical reasons for thedependence of the interelectrode capacitances on the ex-ternal environment.Since that time, no further work has been published on

this aspect of the subject of interelectrode capacitance.The IRE Standards on Electronics, 1938, point out

on page 39 that "For most precise results, it is necessaryto mount the vacuum tube in a specified way as regardsshielding." This statement, though quite correct, never-theless gives the impression that the effect of changes inshielding on interelectrode capacitance is a small one.Nor, does this statement and the accompanying discus-sion emphasize the fact that these "most precise results"are not precise in an absolute sense, but only precise forprecisely the kind of environment seen by the tube dur-ing the measurements. More recently,2 Barco hastouched upon the subject of standardization in themechanical construction of tube adapters, but gives noquantitative data. The question of the standardizationof the total environment of the tube is not discussed.The lack of quantitative information on the depend-

ence of interelectrode capacitance on the total environ-ment of the tube electrodes, has resulted in considerable

* Decimal classification: R139XR262.6. Original manuscript re-*ceived by the Institute, January 13, 1943; revised manuscript re-ceived, August 9, 1943. Presented, San Francisco Section, December4, 1942; Portland Section, February 2, 1943.

t Heintz and Kaufman Ltd., South San Francisco, California.1 A. V. Loughren and H. W. Parker, "The measurement of direct

interelectrode capacitance of vacuum tubes," PROC. I.R.E., vol.:17,pp. 957-965; June, 1929.

2 Allerl A. Barco, "An improved interelectrode capacitance me-ter," RCA Rev., vol. 6, pp. 434-442; April, 1942.

confusion on the subject of interelectrode capacitancemeasurement among practising radio engineers, and isalso reflected in the Government specifications on thesubject.The measurements to be reported here have all been

carried out on one particular triode. It is not the inten-tion of the writer to give the impression that the largevariations.found with this triode are necessarily typicalof all triodes with glass envelopes, but rather to presentthe measurements as a striking example of how stronglythe interelectrode capacitances of a particular tube maydepend upon the external tube environment. The ap-proximate physical dimensions of the tube and the cir-cuit layout will be given later, and from a comparison ofthese with those of any other tube and with the help ofthe general theory to be presented, an estimate can bemade of the relative magnitudes of the variations to beexpected.On the theoretical side, it is illuminating to analyze

the problem with the help of the Maxwell3 4 coefficientsof capacity and induction. If this is done, the analysisis simple. and straightforward from a mathematicalpoint of view, and brings out sharply the physical as-pects of the problem. And this approach, through itsemphasis on the actual geometrical system involved,avoids the blurring of important physical details. Onthe other hand, an electrically equivalent network,unless applied with full knowledge of its limitations isapt to give the false impression that the interelectrodecapacitances are independent of the capacitances toearth.The results of this analysis may then be easily trans-

formed into a mathematically equivalent network of six2-plate condensers5 (for a triode). which are needed forthe application of the results in circuit analysis. Thecorrelation of :the equivalent network of condensers andthe Maxwell coefficients also will make it possible totake advantage of a theorem about the coefficients whichhas been proved by Maxwell, and which turns out to beof singular importance for the present problem.

Unfortunately, as usually presented, these coefficientare treated much more abstractly than need be, bytalking about "n" conducting bodies isolated in freespace with their potentials all measured with respectto infinity. Thus the impression that these coefficientsare of little earthly use is easily obtained. Or, if an

3 James Clerk Maxwell, "Electricity and Magnetism," vol. 1,Clarendon Press, Oxford, England, 1892, chapter 3.

4 L. Page and N. Adams, "Principles of Electricity," D. VanNostrand Company, New York, N. Y., 1931, p. 63.

' E. T. Hoch, "A bridge method for the measurement of inter-electrode admittance in vacuum tubes," PROC. I.R.E., vol. 16, pp.487-493; April, 1928.

Proceedings of the I.R.E.February, 1944 91

9Froceedtngs of the I.R..e

application of the coefficients is made, the potential ofthe earth is quite arbitrarily taken to be zero. In orderto get rid of both the abstract and arbitrary aspects ofthese coefficients, it is only necessary to measure thedifference between the potentials of the "'n" bodiesunder consideration and the potential of the earth (letthe earth be the (n+ 1) st body) and to forget: about theabsolute charge on the n+ 1 bodies and think only aboutthe change in the charge on each body corresponding tothe changes of potential of the "n" bodies with respectto the potential of the earth. In actual experiments, ofcourse, these changes in charge and differernces of po-tential are what are actually measured.

In order to understand this use of the coefficients withpotentials measured with respect to the earth andcharges measured as the difference between the chargeson the bodies when earthed and the charges on the samebodies at potentials different from earth potential, it isfirst necessary to understand how they are used whenpotentials are measured with respect to infinity and thecharges are the absolute or net charges on eaich body.The main ideas can be introduced without loss of

generality by considering the system of four iconductingbodies formed by the elements of a triode and the earth.The charge on each of the four conducting bodies canbe written as a linear function of the potentials of eachbody with respect to infinity. If the cathode, grid, plate,and earth are denoted respectively by the subscripts 1,2, 3, and e, the sytem is described by the following equa-tions:

Qi = CllVl + C12V2 + C13V3 + C6lV.o

Q2 = C21V1 + C22V2 + C23V3 + C2V,1 (

Q3 = C31V1 + C32V2 + C33V3 + C3eVe

Q6 = ColV1 + C62V2 + Ce3V3 + CeeVewhere Q =net charge on body 1; Q2= net charge on body2; etc.

V1 potential of body 1 with respect to infinity;V2 potential of body 2 with respect to infinity; etc.The coefficients having identical subscripts are called

"coefficients of capacity" and those having two differ-ing subscripts, "coefficients of induction." Both kinds ofcoefficients are constants independent of charges andpotentials and depend only on the geometry of thesystem and the dielectric constant of the medium.The physical meaning of the coefficients mnay be seen

by supposing, for example, that the system is chargedin su'ch as way that all potentials are zero except V1which is greater than zero. Lines of force will then startfrom body 1, with some going to infinity and the re-maining going to the other three bodies. Since lines offorce start on positive charge and terminate on negativecharge, this means that Q1 is positive and Q2, Q3, and Qeare negative.From these considerations it follows that

C11= Ql/V1 with VT2=V3=T I0=0.Or in words, Cul is equal to the charge on body 1 di-

videcd by the potential of body 1 when the other bodiesof the system are at zero potential. Since Q,> 0 and V '.> 0,C11 >0. Similarly, C21 = Q2/ V1 with V2 = V3 = Ve == 0. OIr inwords C21 is equal to the charge on body 2 divided by thepotential on body 1 when the potentials of all bodies inthe ;system are zero except that of body 1. Since Q2 <0 ifV1>0, then C2, < 0.Thus, it can be shown quite generally that all coeffi-

cients of capacity (Cii's) are greater than zero and thatall coefficients of induction (Cij's) are equal to6 or lessthan zero. It can also be proved quite generally thatCij= C,,.Maxwell has shown that for a general "n" body sys-

tem the coefficient of capacity of any given body islarger than or equal to the sum of the absolute values ofthe coefficients of induction of this body with respect tothe other n-1 bodies of the system. Thus, in the case ofC11 in the above four-body system, this means thatCl _ C12J + I C131 + C181 -

By inspection of (1), it can be seen that, physically,this inequality means that if the system is charged insuch a way that Vi>O and V2V3=Ve=O then,Q_- (Q2+Q3+Qe), i.e., in all but special cases some ofthe lines which start from Q, fail to terminate on(Q2+Q3+Qe) anrd go off to a negative charge at infinity.

In all practical problems bodies 1, 2, and 3 are verysmall compared to the earth and are located at a distancefrom the earth's surface which is also small comparedto the radius of the earth. Because of this fact, C11 isvery nearly equal to I C121 + | C131 + Cie|. In fact by asimple application of the theory of electrostatic imagesit can be shown that C11, C22, or C33 is larger thanI C12j|+| C13| +1 Cie I C211|+ | C231 +1 C20j , or Gil+ C321 + C3e , respectively, by a factor of the order ofmagnitude (R+L)/R where R= radius of earth andL-means distance of body from surface of earth.

This means that the following equalities obtain witha far higher accuracy than that of the best possiblecapacitance measurements:

Cii = C12 |+1 C13 | + C1eC22 = C211 + C231 + C2e1 (2)C33 = C311 + C321 + I C38 1.

For practical problems the exact magnitude of Ce,ethe coefficient of capacity of the earth, is of no impor-tance and unlike the three other coefficients of capacityis tremendously larger than the sum of the absolutevalues of its three coefficients of intduction.

If the filament, grid, and plate are all earthed so thatVl V2= V3 = Ve, no lines of force join the four bodies,but all of them send lines to infinity. These lines cor-respond to the fact that the net charge of the earth is notzero.Although the net charge on the earth is certainly of

no significance as far as ordinary capacitance measure-ments are concerned, this does not mean that these

6 The coefficient of induction between any two bodies is obviouslyzero if a third conducting body:completely surrounds one of them.

92 Fe-brXalry

Pockman: Interelectrode Capacitance and Shielding

Maxwell equations must be discarded as of merely aca-demic interest. They can be brought down to earth verysimply. The equation for Q, can be rewritten as follows:Q-= C11(V1 - Ve) + C12(V2 - Ve) + C13(V3 - Ve)

+ C1,(V, - Vs)+ [C1lVe + C12Ve + C13V, + C16Vl.I

This equation is identically the same as the originalequation for Ql. Because it may be safely assumed thatVe is not sensibly altered7 by the slight redistribution ofcharge corresponding to the changing of filament, grid,and plate from earth potential to potentials differingfrom that of the earth by (V11- Ve), (172- Vs), and(V3 - Ve), the terms in the square brackets are evidentlyequal to the charge on the filament when filament, grid,and plate are earthed. If this charge be denoted by Qle,the equation can then be written

Ql - Qle = Cll(Vl - V) + C12(V2 - V.)+ C13(V3 - Ve).

Now if qi is taken to denote the change in the charge onbody 1 caused by changing bodies 1, 2, and 3 from earthpotential by amounts V17- V2, V2 -Ve, and V3- 1, re-spectively and v1, v2, and v3, are taken to denote thesedifferences in potential, the equation may be written

ql = Cllvl + C12v2 + C13v3and similarly:

q2 = C21V1 + C22v2 + C23v3q3 = C31v1 + C32V2 + C33v3 (3)q. = C61v1 + Ce2V2 + C63V3.

It should be noted that this transformation does notrequire that the potential of the earth be known or thatit be arbitrarily taken as zero. Furthermore, the coeffi-cients of capacity and induction retain all of the proper-ties which they enjoy when the potentials are measuredwith respect to infinity and the charges are taken to bethe absolute or net charges. Likewise the charges qi andthe voltages vi, though not absolute, are precisely whatare measured in all practical circuit problems.A case of particular importance for practical measure-

ments of interelectrode capacitance can be nicelytreated with the aid of the Maxwell equations as re-formulated above. It is the case in which the earthcompletely surrounds the other three bodies; i.e. thecase in which an earthed conducting shield completelysurrounds the tube and that portion of the tube leadwires lying within the main body of the shield. In thisvery important special case it can be seen with the helpof Faraday's ice-pail experiment4 that qi, q2, and q3 arenot only the change of charge on bodies 1, 2, and 3 re-spectively caused by the changes in potential of v1, v2,and v3, but are also the absolute or net charges on each

I With the help of the theory of electrostatic images, -it can beshown that if a charge is transferred from the earth (of radius R) toanother conducting sphere (of radius r) with its center at a distanceL from the earth's surface, then

change in V,-rL2f i.e., for all practical purposes, Ve does not change.I Loc. cit., pp. 7 and 8.

of(the three bodies. Likewise q,, while not the absoluteor net charge on the earth is the absolute or net chargeon the "inside surface of the earth." Also, for this specialsystem, the relations between the coefficients of capacityand the coefficients of induction, given by (2), holdexactly, because no lines of force can travel from any ofthe enclosed bodies to infinity. Furthermore, the earthseen by the tube in this case is ideal in the sense that itis conducting and therefore equipotential. Also, thegeometry of the system as a whole is entirely insensitiveto disturbances on the outside of the shield and this isof particular moment for purposes of standardizationas will be illustrated by the experiments to be reported.With the aid of the foregoing, the relation between

the Maxwell coefficients and the six two-plate capaci-tances of the equivalent capacitance network shown inFig. 1 can now readily be obtained. Suppose that bodies

Fig. 1-Equivalent six-condenser network of a triode.

1, 2, and 3 are charged to potentials v1, V2, and v3 withrespect to earth. Then, in terms of the equivalent net-work,

ql = VlCf e + (V1 - V2)Cfg + (Vl - V3)Cfpq2 = V2C,e + (V2 - V1)Cfq + (V2 - V3)C,p (4)q3 = V3Cv,, + (V3 - V1)Cf, + (V3 -2)Cpgq a = VlCf e + V2C e + V3Cp e

Taking (3) for the q's and substituting for the Cii'sin terms of the Cij's with the aid of (2), gives the follow-ing set of equations for the q's irn terms of the Maxwellcoefficients:

ql= v( -C1 e)+ (vl- V2) (C12)+ (V1 V3) (-C13)q2=v2(-C2e) + (V2 -v1)(C21)+ (V2-V) (-C23) (5

q3 V3(-C3e)+ (V3-V1) (-C13)+ (V3- V2) (-C32)- q =V1( CI,)+ 2(V C20) +V3( C30)By comparing this set of equations for the q's with

the preceding set of equations, it can be seen that thecoefficient of induction between two conducting bodies isequal to the negative of the equivalent capacitance be-tween the bodies.8 Thus, for the case of a triode andearth,

8 This relation was first pointed out by G. A. Campbell in hisclassic paper on "Direct capacity measurement," Bell Tech. Jour.,vol. 1, p. 18; July, 1922.

1944

Proceedings of the I.R.E.

C21= C12 = - C10C1 C1e - CfC31 C13-CfpCf =C2Ce c-C e (6)C32 C23 - Cgp Ce3 C3s - Cpe

and, although C10, C2e, and C3e cannot be measureddirectly by any feasible experiment, they can be ob-tained in terms of measured coefficients with the helpof (2). Consequently,

Cf= Cll + C12 + C13Cge = C22 + C21 + C23 (7)CPO C33 + C31 + C32.

Combining these relations with those above gives thefollowing relations between the coefficients of capacityand the equivalent capacitances of the six-condensernetwork:

Cll = Cf. + Cfg+ CfpC22 = Cg6 + C0f + CapC33 = Cpe + Cpf + Cpgo

(8)

The foregoing relationships between the inter-electrode and electrode to earth capacitances and thecoefficients of capacity and the coefficients of inductionmake it possible to take advantage of a theorem provedby Maxwell3 which is of singuilar importance for theunderstanding of the variations of interelectrode capaci-tance found in the measurements to be reported. Thetheorem states that the addition of a new- conductingbody to a system of conducting bodies increases all ofthe coefficients of capacity of the original system and de-creases all of the coefficients of induction. The proof isnot only general and rigorous, but is beautifully sim,ple.In order to follow the argument, however, it is necessaryto recall that the coefficient of capacity of a body isequal to the charge on the body divided by its potentialwhen the potentials of the other bodies are all zero. It istherefore numerically equal to its charge when its poten-tial is unity and the potentials of all the other bodies arezero. Similarly, the coefficients of induction between thebodies at zero potential and the body at unit potentialare numerically equal to and have the same sign as theirrespective charges. Maxwell's theorem follows:

"Let us suppose that A (any body in original system)is at potential unity and all the rest at potential zero.Since the charge of the new conductor is negative, itwill induce a positive charge on every other conductor,and will therefore increase the positive charge of A, anddiminish the negative charge of each of the other con-ductors."

Since placing a shield around a tube is equivalent toadding a new body to the original system, it follows,therefore, that the interelectrode capacitances of ashielded tube are smaller than those of the same tubeunshielded. Likew'ise, decreasing the size of the shieldwhich surrounds the tube is equivalent to keeping theold shield and adding a new body (the smaller shield)between the original shield and the tube. Consequently,if one shield, which completely surrounds the tube, is

Loc. cit., p. 118.

replaced by another shield, which completely surroundsthe tube and whose inner surface lies everywhere withinthe inner surface of the original shield (or in partcoincides with it), the interelectrode capacitances willbe smaller with the smaller shield. And since C1l,for example, will be increased and as shown aboveCjj=Cfe+Cfg+Cfp it follows that the increase in thecapacitance of the filament to earth more than offsetsthe sum of the decreases in the interelectrode capaci-tance between the filament and grid and plate, re-spectively. The corresponding statement may also bemade about the increase in Cqe and C,.The foregoing considerations are quite general in the

sense that they apply to the measurement of inter-electrode capacitance irrespective of the particular cir-cuit that happens to be used.

. LO,^CATMlJDED AT BEAT

OCILLATOR C ULA'TOR

0 Ha. .. SCALE

Fig. 2-Scale drawing of tube and shielding used in first series ofmeasurements.

The present measurements of capacitance have beenmade on one and the same triode surrounded withvarious kinds of shielding. The method of measurementwas first used in this laboratory by P. A. Ekstrand9 forthe determination of interelectrode- capacitance and forthe simultaneous determination of the capacitance ofeach electrode to earth. This method of measurement isa substitution method in which a radio-frequency oscil-lator, adjusted to a particular frequency with the tubeout, is brought back to the initial frequency with thetube in, by reducing the capacitance of a variable stand-ard condenser. The return to the initial frequency is de-termined by listening to the beat frequency between thisoscillator and another oscillator loosely coupled to it. InFig. 2 the circuit diagram is shown. Six measurementsare required for the calculation of the six constants ofthe system. From the same set of measurements, theMaxwell coefficients- can also be determined. As amethod for the determination of the three interelectrodecapacitances only, it is considerably more tedious than

9 Now with the Navy Department.

February94

Pockman: Interelectrode Capacitance and Shielding

any of the methods described in the IRE Standards,10but it makes possible the determination of the six Max-well coefficients or of the equivalent network by a mini-mum number of measurements.A seventh measurement serves as an internal check on

the readings and the computations.The shield and the triode are both drawn to the scale

indicated in Fig. 2. The shield shown is one of theshields used in the measurements to be reported-astandard five-gallon gasoline can, cut so that the top isremovable.When the oscillator was adjusted to zero beat with

the tube out, the portion of the lead wires within thecan and, of course, their coaxial shields, were placed soas to be in the same position with respect to each otherand the main body of the shield, i.e., the inner surface ofthe can, as they would be when connected to the tube.The three single-pole double-throw switches facilitatethe seven distinct connections used for the seven meas-urements. As pointed out in the theoretical discussion,the shielding or lack of shielding of these switches' andlead wires lying outside of the grounded equipotential,'surface which surrounds the tube, have no influence onthe coefficients of inductioon and of capacity of thisclosed system. These coefficients and hence the inter-electrode capacitances and the earth-electrode capaci-tances are completely determined by the geometry andthe dielectric constant of the closed system. Thegrounded equipotential surface which surrounds thetube in Fig. 2 is evidently that surface formed by theinside of the five-gallon can and the outside of that por-tion of the lead-wire shields lying within the can.

It should be noted, however, that if the circuit wireslying outside of the tube shield are not shielded bygrounded conductors, the operator must be careful thatthe frequency is not altered by the motion of his body.Care was always taken to see that this was the case.For routine work, of course, such shielding of externalleads is a necessary convenience.

If the seven directly measured capacitances arede-noted by Cl, C2, .. . C7 the connections correspondingto each can' be obtained from Table I.

TABLE I

Elements above ElementsGround Potential Grounded

C, p, g,f (or 3, 2, 1)C2 p g,fc. g p,fC4 f g,pCs P, g fCs g,f pC7 p: f g

Seven simultaneous equations, of which any six are

independent, may be set up by equating the C's to theappropriate combination of Maxwell coefficients ';or ofthe six equivalent capacitances.

10 IRE Standards on Electronics, 1938, pp. 38-44.

When set up in terms of the Maxwell coefficients, theequations assume an extremely simple form and may besolved by inspection of the last six equations: the firstequation then provides an over-all check.

Thus, for example, the first equation is obtained bynoting that C1 = (ql+q2+q3) /v.

Substituting for the q's with the help of (3), and not-ing that in the measurement of Cl, v1 = V2 = V3 = V, leads to

Cl = C1l + C22 + C33 + 2C12 + 2C13 + 2C23.Similarly,

C2 =

C3 =

C4 =

C33C22C1l

(9)C5 = C22 + C33 + 2C23C6 = C1l + C22 + 2C12C7 C33 + C11 + 2C13.

Solving'for the six coefficients in terms of the meas-ured capacitances and making use of the relations be-tween the coefficients of induction and the interelectrodecapacitances given in (6) leadIs to the following equationsfor-the coefficients of induction and the interelectrodecapacitances -in terms of the measured capacitances:

C5 - (C2 + C3)C23 = -C2g =

2

(10)C6- (C3 + C4)

C21 = - Cgf =- 2

C31 = -Cf =C7 - (C2 + C4)2

Similarly, the following expressions for the threecapacitances to earth are obtained by combining (7),(9), and (10):

C6 + C7 - (C2 + C3)Cf = 2

Cb + C6 - (C2 + C4)C0e = 2 11)

C5 + C7 - (C3 + C4)Cpe--= ~~~2

As mentioned earlier, the measurements reported inTable II have all been carried out on one and thesame triode by the method described above. In thetable, each row of data was obtained from measure-ments made on this triode surrounded by the particularphysical environment described in the first column. Thetube and its physical environment in the first series ofmeasurements are shown to scale in Fig. 2. The secondseries was made with the same orientation of lead wiresas on the interior of the can in Fig. 2, but with a one-gallon benzene can (4X61XO9 inches) instead of thefive-gallon gasoline can. The tube was also located at thesame distance from the bottom of the can and on thecentral axis of the can. The bakelite adaptor was usedfor the filament prongs. In one of the series oi measure-ments it was possible to eliminate completely all of the

1944 95

Proceedings of the I.R.E.e

TABLE IICAPACITANCE MEASUREMENTS ON ONE AND THE SAME TRIODE: TUBE COMPLETELY SURROUNDED BY A GROUNDED METAL SHIELD

"Earth" seen by triode Ct C,2 Cal Cf, Cue CpO Cgp =-C2S Cfg = -C1 Cpf =-C1, Cgj/Cpf1. Inside surface of five-gallon can plus those 5.34 6.84 4.52 1.29 0.70 1.61 2.50 3.64 0.41 8.9

parts of lead wire shields on interior of can(see Fig. 2)

2. Same as "1" but with a one-gallon can (see 5.54 6.91 4.80 1.61 0.87 2.01 2.45 3.59 0.34 10.6text)

3. Mainly the inside surface of a close fitting 7.15 7.78 5.93 3.67 2.29 3.60 2.17 3.32 0.16 20.7shield (see Fig. 3)

'EFFECTIVE" CAPACITANCE MEASUREMENTS ON SAME TRIODE: TUBE ENVIRONMENT HETEROGENEOUS

4. Exactly asin "3" but with shield electrically 5.98 7.31 4.91 0.48 0.35 0.65 2.86 4.10 1.40 -floating

5. Grounded metal plate (7 X9 inches) about 1 5.33 6.80 4.42 1.23 0.61 1.45 2.53 3.66 0.44inch below glass envelope plus groundedlead-wire shields, plus general environmentof laboratory

6. Same as "5," but with lead-wire shields con- 5.18 6.68 3.69 0.71 0.01 0.37 2.76 3.91 0.56nected to respective lead wires

bakelite without otherwise changing the system. Adirect comparison of measurements made with andwithout the bakelite revealed no difference greater than0.01 micromicrofarad, the approximate limit of error inall the measurements.The third series of measurements was made with a

close-fitting grounded shield essentially like that shownin Fig. 3. Since the tube had to be put into and removed

TO SW 3

| X T O~~~~~T SW2

it . ==-~~~BAKELIYE

TO SW I

.-*I SCALE

Fig. 3- Tube with close-fitting shield used in third and fourthseries of measurement.

from the shield, the actual shield was made up of severalparts which could be separated and reassembled quickly.The details of the tube construction are also shown toscale in this figure. The stem shield was of course elec-trically connected to one of the filament leads.These first three measurements are grouped together

under one main heading because only for these three

cases can it be said that the measurements determinethe coefficients of capacity and induction as definedabove. For these three cases the geometry and thedielectric constant of the system enclosed within theshields completely determine capacitative constants ofthe system.

In the fourth case, with the close-fitting shield elec-trically floating, the six measurements do determine aset of six possibly useful capacitative constants fordescribing this particular system with a floating shieldbut no longer are these constants determined solely bythe geometry and dielectric constant of the internalsystem but are determined also by the geometry anddielectric constant of the external environment as wellas by whether or not the conducting bodies in this ex-ternal environment are grounded or floating, and bywhether or not the lead wires are shielded or unshielded.

For purposes of standardization it would be possiblein principle to specify so completely both the preciseinternal and external environment of the shield that re-producible measurements of these "effective" capacita-tive constants determined with the shield floating couldbe obtained, but clearly it is far simpler for purposesof standardization to obtain reproducible results bygrounding the shield. This also retains the original simpledefinitions of the Maxwell coefficients and their relatedcapacitances.

Furthermore, there is a very practical reason forusing a tube with all of its shields (including any metal-base shell) grounded and not floating. Thus, with thehelp of the Maxwell coefficients it is easy to show quitegenerally that a floating conductor in the neighborhoodof the tube will always give "effective" interelectrodecapacitances larger than the interelectrode capacitancemeasured with these bodies or shields grounded.1'

Similarly, it can also be shown that the "effective"capacitances of the electrodes to earth will be decreased

11 Measurements made at Heintz and Kaufman on the feedbackcapacitance of the HK-257-B by E. L. Ung show that the feedbackcapacitance with base shell floating is nearly twice as large as withthe base shell grounded, the geometry being the same in the twocases.

96 Februm,

Pockman: Interelectrode Capacitance and Shielding

by ungrounding a near-by body. This, however, is anadvantage for which an unduly high price is paid interms of the corresponding increase in the interelectrodecapacitance. This is forcefully illustrated by a compari-son of the third and fourth series of measurements.

In the fifth series of measurements, the tube wasmounted above the center of a grounded metal plate ap-proximately 8 inches square. The same bakelite socketwas used for the filament prongs as that shown in Figs.2 and 3. The three lead wires were arranged in such away that their geometry with respect to each other andwith respect to the tube was approximately the same inthe near neighborhood of the tube as the correspondinggeometry of the lead wires on the inside of the five-gallon can. From the axis of the tube to the nearest partof the set of three single-pole double-throw switches (seeFig. 2) the shortest distance was 12 inches. The groundedshields of the lead wires exteided all the way to theseswitches. That the presence of these unshielded switchesat this distance caused no important change in themeasurements was established by running a series ofmeasurements with the set up exactly as in this fifthseries except that the switches a:nd high-voltage leadwires were surrounded by some grounded metal screens.No significant change was caused in the interelectrodecapacitances and the capacitances to earth were in-creased by only a few per cent.

Furthermore, the various and sundry floating bodiesin the general environment of the laboratory are at suchlarge distances from the tube compared to the dimen-sions of the electrodes that the "effective" constantsdetermined by this series of measurements differ by aprobably negligible amount from measurements whichcould be made with a shield two or three times as largeas the five-gallon shield. This view is supported by thesmall differences between the results of the first seriesof measurements and this fifth series. Also these differ-ences are in such a direction as to support this viewfurther; i.e., the interelectrode capacitances in the fifthseries are all slightly larger and the capacitances to earthsomewhat smaller than in the first series.A comparison of the fifth and sixth series shows that

the "effective" interelectrode capacitances measuredwith grounded lead-wire shields are all significantlysmaller and the "effective" capacitances to earth allsignificantly larger than the corresponding capacitancesmeasured with lead-wire shields connected to their re-spective lead wires. A detailed analysis made with thehelp of the Maxwell coefficients shows that the "effec-tive" interelectrode capacitance, such as that betweengrid and plate, determined from measurements madewith shields connected to lead wires is related in thefollowing way to the corresponding capacitance meas-ured with the lead-wire shields grounded:

Ca, (shields connected to lead wires)

where

p plate lead shieldg grid lead shield

ACp,, = Cp,gO (tube in) -Cp, (tube out).

Since Cp,g, is greater with the tube out, ACp,g, is lessthan zero. But as the measurements show this is morethan offset by the positive sum of Cp,,g and C0,,, so thatthe "effective" Cgp is greater with the shields connectedto the leads than with the shields grounded.

Similar analysis of the capacitance of grid to earthshows that:

C.. (shields connected to leads)= C,, (shields grounded) - (Cg0' + C1p, + Cgp,)+ AC e6. (13)

And neglecting the relatively small term AC,,, thismeans that with the shields grounded the shields formpart of the "earth" whereas when connected to the leadwires, they do not. And because the mean distance ofthe lead-wire shields from the tube is small compared tothe mean distance of the rest of "earth," this explainswhy the "effective" capacitances to earth are so verymuch smaller in the sixth series than in the fifth series ofmeasurements.

It might be supposed that the large influence of theunshielded leads can be mainly attributed to the largediameter of the leads in the sixth series of measurements.But that this is not so was proved by substituting No. 25wire, unshielded, and in the same relative position, forthe large lead wires. The results werre intermediate be-tween those of the fifth and sixth series, but consider-ably closer to those of the sixthThe results of the first three series of measurements

strikingly verify Maxwell's theoretical prediction'2 aboutthe variation o the coefficients of capacity and the co-efficients of induct on with the shielding of the system.They, of course, also show the corresponding dependenceof the interelectrode capacitances and the electrode-earth capacitances on shielding. Clearly the variationsare significantly large, the change from the five-gallonshield to the close-fitting shield, to take the extremecase, causing a two and a half fold change in C,f.Furthermore, it becomes quite clear that it is meaning-less to speak of one set of interelectrode capacitances as

being more nearly the "true" set than any of the others.And this means that the wide variations in the reportedinterelectrode capacitances of a given tube can only beavoided by strict standardization of the type of shieldingto be used in the measurements. If this is done concord-ant results will be obtained by different workers, re-gardless of which of the many adequate methods ofmeasurement happens to be used.The column showing the ratio of C,f to Cpf is of pecul-

iar interest if the ratios and the trend of these ratios is

(12) 12 See discussion following equation (8) for details.

971944

.C, (shields grounded) + C",,V.+ c9Iv + ACV,Iv,

compared with the measured j. of the triode, which is28.5. Thus Chaffee,"' following van der Bijl and usingthe Maxwell coefficients, shows that

=C/C/ (14)or, in the notation used here,

CI=C13 (15)Hence, using the relations established earlier between

the Maxwell coefficients and the interelectrode capaci-tances, this leads to

= Cg/Cp. (16)Of course, this is true only for the idealized case in

which all of the lines of force corresponding to Cpf andCgf terminate on the active portion of the filament. Inthe case of the triode measured here this ideal conditionis never realized because some of the lines terminate onthe nonactive filament lead wires and the stem shield.Nevertheless this ideal condition is more closely realizedthe smaller the shield. This can be seen qualitatively by

13 E. Leon Chaffee, "Theory of Thermionic Vacuum Tubes,"McGraw-Hill Book Co., New :York, N. Y., 1933, p. 146.

inspection, of the geometry of the system and notingthat as the grounded shield approaches the tube it willhave a much more pronounced effect in reducing thenumber of stray lines of force from plate or grid (fora given potential) to the inactive parts of the filament,than in reducing the number of lines to the helix, whichis much better screened from the influence of the shieldthan are the inactive parts of the filament. Also, ofcourse, the percentage decrease in Cpf will be muchgreater than in Cgf because these strays form a muchlarger portion of 'Cf than of C0f. This accounts for therapid rise of the ratio toward the l of the tube.

ACKNOWLEDGMENTThe writer would like to acknowledge the valuable

assistance which he has received throughout the courseof this work from Mr. W. G. Wagener. It is also a pleas-ure to thank Heintz and Kaufman, Ltd., for its splendidco-operation on a project which the author hopes will beof some value for the problem of standardization in theinterest of the war effort.

Equivalent Circuits for Discontinuities inTransmission Llnes

J. R. WHINNERYt, ASSOCIATE, I.R.E., AND H. W. JAMIESONt, ASSOCIATE, I.R.E.

Summary-Exact equivalent circuits for several types of discon-tinuities in parallel-plane transmission lines are obtained by Hahn'smethod of matching electromagnetic-wave solutions. Values oflumped elements to be used in these are given in curve form, withrules for using results in the corresponding coaxial-line problems.Experimental checks are reported, which verify results of the calcu-lations and stress the importance of the discontinuity capacitancesappearing in the equivalent circuits.

I. INTRODUCTIONI N 1941, Hahn' presented a description of a new ap-

proach toward matching of electromagnetic-wavesolutions across discontinuities, with application to

the problem of cavity resonators as an example. Inmuch unpublished work, Hahn has shown that themethod when applied to transmission systems with dis-continuities leads to simple equivalent circuits withlumped-circuit constants representing the effect of thesediscontinuities. The equivalent circuits are simple, ex-act, and very useful for engineering thinking or calcula-tion. It is our purpose in this paper to present this pointof view with the following goals in mind:

1. To show the exact equivalent circuits for severalrepresentative transmission-line discontinuities.

* Decimal classification: Rl 16. Original manuscript received bythe Institute, April 26, 1943. Presented, Winter Technical Meeting,New York, N. Y., January 28, 1944.

t Electronics Laboratory, General Electric Company, Schenec-tady, New York.

W. C. Hahn, "A new method for the calculation of cavityresonators," Jour. Appl. Phys., vol. 12, pp. 62-68; January, 1941.

2. To present curves showing numerical values to beused in these equivalent circuits.

3. To discuss the physical pictures inherent in themethod and results, and the application to practi-cal problems.

4. To show, in enough detail to be readily followed,the mathematical steps for one simple example.

It should be poihted out at the outset that the pointof view in the equivalent circuits is identical with thatused by Schelkunoff2 in radiation problems, in which hehas shown that the "complementary" end-effect wavesof an antenna may be included in a principal waveequivalent circuit as a lumped admittance at the end ofthe antenna. Similarly, we find equivalernt circuits to beused with the principal wave on a transmission system,the effect of local waves at a discontinuity being in-cluded in lumped admittances.

In this paper, numerical values are given only forparallel-plane transmission lines. Although it is plannedto present corresponding results for coaxial lines in asucceeding paper, instructions are given in Part III forapplication of present parallel-plane results to coaxiallines. The experimental checks described were made oncoaxial lines. These not only check-the calculations, butemphasize the importance of knowing values for the dis-continuity capacitances at high frequencies.

2 S. A. Schelkunoff, "Theory of antennas of arbitrary size andshape," PROC. I. R. E., vol. 29, pp. 493-521; September, 1941.

Proceedings of the I.R.E.98 Februaiy, 1944