MAY 1946 149

FLAT CAVITIES AS ELECTRICAL RESONATORS'

by C. G., A. von LINDERN t and G~ de VRIES.

Mter some introductory remarks about the characteristic vibrations which may o~curin Lecher systems which are short-circuited at one end, this article deals with conicalflat cavity resonators short-circuited around their outer edge ru;tdwhose behaviour,as regards the .rotation-symmetrical vibrations, corresponds entirely to that of the

,Lecher systems first mentioned. Subsequently the rotntion-symmetrieal characteristicvibrations of flat cavity resonators of more general forms are discussed and the variationof current and. voltage with the radius is drawn for a flat cavity resonator. Resonanceresistance and quality factor are calculated .. Further it is indicated how the qualityfactor and the resonance resistance can be improved, by making the cavity resonatorsthicker than those for which the theory given here applies unconditionally. In conclusion'somè examples are given of flat cavity resonators and their employment in high-frequency technique, for example for frequency stabilization and as output and inputelectrodes for short-wave transmitting valves. .

. In the excitation of high-frequency oscillations inradio transmitters, especially for very short wavesuse is made of specially formed resonators; becauseat very high frequencies the ordinary. oscillationcU:cuits Iconeisting of conce~trated capacities andself-inductions possess too low a resonance resis-tance and too small a' quality factor 1). In-additiontó the so-called resonance cávities, Lecher systemsconsisting of two parallel conductors of uniformdiameter as well as empty spaces with metal walls,which we call cavity resonators, are often usedfu. high-frequency technology. In a Lecher systemof a given length 1 stationary waves call' he

- I

generated having a wave length of 41, providedthe Lecher system is exci ..ed at one end .by an'electromagnetic oscillation, while the other end isshort-circuited 2). We shall inow examine the freeoscillations which-may: occur in cavity rés~natorsof a general form,' but in this article' shall confineourselves to f'l a t . cavity.·resonators, whichhave one dimension much smaller thanthe other two.When such flat cavity resonators are constructedin the form of solids of revolution, -electromagnetic-rihrations may occur .in them which do )not dependupon the angle 'of revolution,' but only upon theradius. Such rotation-symmetrical oscillations thusactually' depend upon only one coordinate. They cantherefore still be treated in practicallythesame wayas the oscillations of a Lecher system, which areof course also considered as depending exclusivelyon one coordinate, namely the length, since thetransverse dimensions may he neglected comparedwith the length. .1) See for example our article "Resonance circuits for very

high frequencies", Philips techno Rev. 6, 217, 1941, in:which, inter alia, different equivalent definitions of thequality factor are explained in more detail._

2) The behaviour of a Lecher system .with respect. totravelling and stationary waves was discussed extensively .by us in Philips techno Rev. ,6, 241. 1941..' .

. ,.

538.565 :538.566.5

"

We shall first discussthe characteristic vibrationswhich may occur in a homogeneous Lechersystem short-circuited at one end. Then we' shallextend our discussion to 'include flat cavityresona tors with, double conic cross-section,after which we shall have something to say about'the characteristic vibrations of flat cavity resonatorsof more general forms. After that we shall examinethe influence of the resistance, giving values forresonance resistance and quality factor. Thesetheoretical eonsiderations will then be concludedwith a discussion of the possibility of building up ,cavity resonators not having one dimeneion muchsmaller than the others, by piI.i.ngup flat cavity reson-.ators one upon another. Finally some examples aregiven of the practical use made of the cavity reson-ators -here .discussed in high-frequency' technology,

Charaeteristlc vibrations ~f L~cher systems

Let us consider a section of a Lecher systemconsisting of two parallel' conductors of uniform 'cross-section and a given length 1, which are con-nected at one end. We shall ignore the energy lossesfor the present,' while for the cápacity and' self-induction per unit oflength we introduce the symbolsCl and LI. For -the current i "and the voltage Vasfunctions of the time t and the coordinate oflength x, the following e~ations hold:

oi I 0V àV. . ~ài-='-C -and-=-L -. (1)Ox . Ot Ox . Ot

By differentiating the-se equations' with respect tox and t four equations are obtained:

o2i .. 02V' o2i 02V.Ox' ~ -C' OXO'r OxO' ~ .-C' át2 ; l (2f

. 02V . I o2i . '02V o2i .--=-L.--; __-_L1_. OX~ . . oxot oxOt - .Ot2 ' .

150 PHILIPS TECHNICAL REVIEW VOL. 8, No. 5

from which by combination two correspondingdiffere~tial equations of the second order (the so-called vibration equations) for i and V can hederived:

o2i _ LICI02i d 02V _, LIClo2Vox2 - + Ot2 an ox2 - + o'i2'

If now for the sake .of shnplicity we assign thecoordinate x = 0 to the short-circuited end of theLecher system (fig. 1) and to the open end x = Z,

, ~o

we require only' those solutions of the vibrationequations (3)' for which the voltage V is zero forx = O.For the characteristic vibrations of theshort-circuited section of Lecher system there is, the addi.tional condition that at the open end x = Zthe current i must be zero. The simplest stationarywave,' which satisfies, the vibration equations (3)and also these two conditions, is, except for a con-stant factor, the following 3): .

i = sin 2Z;;_ICI cos ~; , l. (4), lW nt st»

V= - V. cr cos ~ZYLICI~in2ï'

The variation of current and voltage along theLecher system according to (4) is representedin fig. 1. .

The coefficient of the time t in the expression (4)is the angular frequency (J) = 2n'V. For the fre-quency 'VI of the simplest characteristic vibrationthe following thus holds:

1'V1= 4ZYLICI'

In-the same way it may be seen from the expressions(4) that the length Ä.1of the largest stationary wavealong the Lecher system short-circuited at oneend is four times as great as the length Z. Thedependence of x on 'c:urrent and voltage can be

3) The magnitude and sign of the coefficient in the expressionfor the voltage Vis of course 50 chosen that the·expressions(4) also satisfy the differentlal equations (1) of the firstorder for current and voltage from which we started inthis discussion. VLIjCI is oftèn called the "waveresistance"(cf. Philips techno Rev. 6, 24.1, 194.1).

..represented by a cos or sine of 2nx/ Ä.1,so that from(4) it does indeed follow that:

(6)

(3)If (5) and (6) are multiplied byeach other thewell-known result follows that the product of thecharacteristic frequency 'VI and the corresponding

wave length Ä.l is equal to 1JYL1C1, and this is, thevelocity of propagation v of the travelling waveswhich may occur along the Lecher system:

3 . 1010'VIAl = V =.1Il'LIC1= Y cm/sec.. (7)

e(.L .

The expressions represented by (4) for current'and voltage of the short-circuited Lecher systemrepresent the so-called fundamental vibrationof the latter. In addition to this, overtones also occuras characteristic vibrations of the same Lechersystem (fig.2), which also satisfy the vibrationequations (3) and the conditions for voltage andcurrent mentioned:

Kal

where k' may assume the values of the wholenumbers. The charaèteristic frequencies 'V2k+l ofthese overtones lie harmonically and are the oddmultiples of the fundamental frequency 'VI' namely:

2k + 1'V2k+l = ,1 . = (2k + 1) 'VI • (9)

4Z yLICI

The corresponding wave lengths are:

4Z Al ( )Ä.2k+l = 2k+1 = 2k+1 ' . '10

(5)

so that the length Z of the Lecher system isequal to an odd number of quarter wave .lengths.In the case of homogeneous Lecher systems weare therefore only concerned' with the familiarharmonic overtones which also occur with ia

vibrating string. This, however, is no longer the

I i'jO ' /.0 ;.0

it,i--~~-=~-~,~I.+~_~'_~I+-~~_~._~I;_-.~-.~i._'=-~~:~.!j.....-._ - - - -11- _ .. 1 - -4- :--- iI . - : I x~lx.o -XI : : I 1

V I: : :"\: : : i: It . If i 11 +t : i: : ; '\:'v~o I v~o I v~o -:140~ .i.. I

·1.1Fig. 2. Behaviour of current i and voltage V for the third'characteristic vibration J. of a Lecher system short-oir-cuited at one end. (k = 2 in equations (8), (9) and (10»,in the case of fundamental vihration.

MAY 1946 FLAT CAYITIES AS ELECTRICAL'RESONATORS 151

case when the capacity and the self~induction dependupon the position, i,e. when the Lecher system isnot homogeneous.

Then the position of the overtones is anhar-monic, as we shall see later in this article. '

Conical Hat cavity resonators

If we were to imagine the homogeneous Lecher, system short-circuited at one end, which we haveconsidered until now, to be rotated so as to form aHat box, with or without a hole in it (fig.3), we

Zf:P==== I ;:::;;:::::==:::=1I. ft ~

10.3:4180Jl

Fig. 3. Flat cavity resonator with thickness z and radius Tl

of the short-circuited outer edge. The radius of the hole is TO'

The Hat cavety resonator oIiginatcs by the revolution of ahomogenious Lecher System, short-circuited at one end.

might expect that the rotatory-symmetrical charac-teristic vibrations of such a flat cavity reso-na tor would exhibit a far-reaching similarity withthose of a Lecher system short-circuited at oneend. However, this does not give us the two-dimen-sional cavity resonator with the simplest charae-.teristic vibrations, since in the vibration equationsfor current and voltage as functions of the radius r,the capacity. and self-induction per ring of 1 cmwidth play a part in this rotatory-symmetricalcase and these quantities, which we, shall againcall Cl and LI, are not constant as in the homo-geneous Lecher system, but depend upon the radius,r of the ring being considered. It will he- obvious, that" Cl is directly proportional to T, while LI isinversely proportional to it; the expressions are as .follows:

I r I 2z (11)C --:-- and L = -. .'. .2z c2r '

where z represents the thi~kness of the flat cavityresonator, the .dielectric is a vacuum and c is thevelocity of light.A case, which is indeed mathematically entirely

analogous to the homogeneous Le ch e r system, isa conical flat cavity resonator, for which thedistance z between the two conductors is propor-tional to the radius r of the ring being ~onsidered(fig. 4). Accordi.ri.g to the ~atios (11) Cl and LIthen do indeed again beco~e independent of r.For the conical flat cavity resonator, therefore, weobtain exactly the same harmonic characteristic

vibrations as for the homogeneous Lecher system,only in this case we ,are not dealing with a coordinatex which passes from x = 0 at the short-circuited

418/0

Fig.4. ConicalHat cavity resonator with thickness zatradius T. '

Outer radius Tl' inner radius TO'

end to x = l at the open end, but with a coordinater which passes from r = ro at the open inner edgeto T = r1 at the short-circuited outer edge. Becauseof this the solutions of the vibration equations be-come somewhat more complicated than (4) and (8),but one is still dealing with an odd number ofquarter sirie waves of current and voltage (fig. 5),which now lie along the radius of the conical ,flatcavity resonator' instead of along the length of thehomogeneous Lecher system, so that in generalthe following is valide

, 2k+lTl - TO =-- À2k+l., 4 (12)

'which is quite analogous 'to relation (10) for thehomogeneous Lecher system. ",

Characteristic vibrations of othec flat cavity, reso-nators

. Flat cavity resonators whose' thickness z dependsin any arbitrary manner on r do not behave, as faras the electromagnetic characteristic vibrations' areconcerned, entirely like a Lecher system of uni-form cross-section, as is the case for ;the conicalflat cavity resonator just discussed.

In order to describe the axially symmetrical

1[::' j .---!: }

~ ~CJ'-: .~;l .. -;:::;:I -.: !.:;::c.-<' , -..;:: ,::;:;.- -...;::: .

" . " j. . r, .1

~!~,'vl---rr I fT--Jj41769

Figv S. Variation of current i and voltage V with the radiusfor a conical Hat cavity resonator i: TO inner radius, Tl outerradius. '

. 152 PHILlPS TECHNICAL REVIEWI

VOL. 8; No. 5

characteristic vibrations which may occur in a flat,cavity resonator with a given cross-section, we referto the treatment of the characteristic vibrations of aso-called non-homogeneous Lecher system 4), forwhich the cross-section and/or the distance betweenthe two conductors 'depends upon the' position(fig.6). With Heaviside we shall now assume

a), 4'811

Fig. 6. Different forms of generalized Lecher systems con-sisting of: .

a) two parallel strips of varying Width,.· .b) two strips of uniform width at varying distance apart

andc) two strips' wliich not only become widerbut also diverge.

r ,

that the quantities Cl and LI vary only slowlywith the coordinate, which for the sake of simpliclty

, 'we shallimmediately call T, in order to obtain equa-tions, which are also valid for the flat cavity reso-nators. We then obtain the following for the varia-tions of current i ánd voltage V:

. ai ."_Cl oV and av = _Llai ... (13)aT ot aT at

These equati~ns (13) are now again differentiatedwith respect to Tand t:

, .02i . aClov I a2v 02i I a2v-=--~-C --and-=-C _'or2 aT ot aTat orot . Ot2 '

02V aLL 0 i a2i 02V 02i_. = _:_ -LI--and __ .= _L1 - •or2 ar ' ot aTat orot Ot2

By comb~ation of .these equations, in which (13)has to be substituted for the first derivatives, aboutthe same differentlal equations ofthe second ordercanagain b~ found as vibration equations for i and V:

a2i lOCI ai· I I 02iOT2 - Cl or. a, - L C il,' ~ 9 ~ . (14)

02V _ _!:_ oLI av _ LICI 02V = 0a~ L1

OT aT o~It-may be seen that compared with the vibrationequations (3) there is a middle term added due to theslow change of Cl and LI with the coordinate r.In order' to obtain closed solutions of (14) we

;, ;~. shall now assume as a special case that the thickness.,» ,

4) For a recent review of the theory of non-homogeneousLecher systems see K. W. Wagner, Die Theorie un-

o gleichförmiger Leitungen, Arch. Elektrotechn, 36, 69, 1942.

z of our flat cavity resonator is proportional to anarbitrary power n of the radius T (fig.7):

z 'h c)n ... ; .According to (ll) capacity and, self-induction perring 1cm ...vide in a vacuum are: .

','

(15)

1 T (Tl')n I .2h(T)n -C = 2h -; and L = C2T Tl "

thus, respectively, inversely and directly prop or- 'rional to rn-I. For this special case therefore thevibration equations (14) for current and voltagebecome:

(16) .

02i n-1'ai' 1 02ior2+-T- aT - c2 Ot2 - ,0, l02V 1-n av 102V'-or-2-t -T- -a-T- c2-ot-2 = O.

..• (17) ,

Just as for the, simple vibration equations (3),we now look for solutions of (17) which are harmonicÏI;L the time t. For current i and voltage V we again .take as variation with time sin and cos 2'JT.ct/ Ä,respectively. For the amplitudes of current (im)and voltage (V~), which depend only upon thecoordinate r, when we introduce x = 2'JT.r/Ä as newindependent variable we obtain the followingequations:

02im n-1, aim '. . ~-+----+Lm=Oox: x ax .,(18)

02Vm 1-n avm. v. .--+---+ m=O.ox2 x ax .For the case where n = 1 the middle term becomesequal to zero, so that the vibration equation (3) is'again obtained for current and voltage. For n - 1we have in fact the simple case, already discussed,of the conical flat cavity resonator with i and Vvarying harmonically in tand T.

Just as it is well known that the solution of (18)without the middle term are sines and cosines of

'41812

Fig. '7. Flat cavity resonator with thickness h at the outer'edge, o,uter radius Tl and inner radius TO' . •• ~', •

MAY 1946 FLAT CAVIT~1j:S AS ELECT,&ICAL RESONATORS 153

x, it is also known that Bes s eL functi.ons _of the'first and second sorts (Jp and Np, where p denotesthe "order" of the Bessel function) multiplied bypowers of the coordinate x are solutions of (18).It is clear that nor all solutions of (18) also satisfythe differential equations (13) with which we be-gan. Those, for -which this is indeed the case, aregiven by the following expressions:

'im = x1-n/2 [A Jn/2-1 (x) + B Nn/2-1 (x)], (, (19)

Vm:= - HC x1-n/2 [AJn/2 (x) +BNnJ2 (x)]; ,

A und B are arbitrary constants 'here and C is ,thewave resistance (cf footnote 4)):

llLI 60 h, ,C = '-= -- Tn-10hm.,Cl T1n

We must now choose A and B in (19) such thattheir' expressions satisfy the boundary conditions.

In the first place the voltage must be zero at theshort-circuited outer edge of the :flat cavity reso-'nator, i.e. Vm = 0 for T = Tl' or x = 2:n;Tl/A = b,This, boundary .condition thus gives:

AJn/2 (b) + BN~/2 (b) , -0 . (20)

,',By filling .in the value of B from (20) in (19) weobtain:

,im=Ax1-n/2 [Nn/2 (b) Jn/2-1 (X)-:Jn/2 (b) N~/2-i(x)],(2Ia)

Vm ' --V-lA Cx1-n/2 [NtÏ/~(b) Jn/2(x),-Jn/2 (b) Nn/2 (x)]. . (2Ib)

In the second place in the case of a :flat cavityresónator with a hole in it (i.e. To> 0) the cur-rent must be zero at the inner edge., This gives usthe relation ,

1Vn/2 (b) Jn/2-1 (a) -Jn/2 (b) Nn/2-1 (a) = 0, (22~

where a = 2:n;TO/A.In-the case of a :flat cavity reso-nator w it.ho ut a hole (i.e. TO= 0), the current in.the centre (i = 0) for n ;» 0 need not be zero be-cause the top an~ bottom planes do not. then toucheach other. It certainly cannot, however, be infiniteat that point, as-would follow for x = 0 and n ~ 2from expression (2Ia).

It is found by closer inspection of this expressionthat the current remains finite for, x = 0 andn ~ 2 provided-

(23)

For n < 2, except for n = 1, -1, -3, ... , the samecondition (23) results in the fact that the current(2Ia) at the centre not only re:m~ins finite bul; is

'..

moreover equal to· zero'. In' order to attain this forn = 1, -1, -3, ... , instead of (23), the condition,must .he fulfilled that

Nn/2 (b) = 0.' ..•..• (24)I

No~ since a = 27&Tov/e and b = 2:n;T1v/e and TO'

rl and n are fixed for a given :flat cavity resonator,equation (22) (resonator with hole) or (23) or (24)(without h~le) are only satisfied for certain values ofthe frequency v. These special values of v, which arethus the roots of'equations (22) (23) or (24), res-pectively, represent the series of characteristicfrequencies of the :flat cavity resonator in question.The smallêst positive root represents the funda-mental tone. '

It remains to he mentioned, that the valuesof the characteriatic frequencies - calculated from(22) pass over continuously into those 'calculated-from (23) or (24) when a in (22), i.e. the radiusof the:: hole, is allowed to approach zero.

const C =:J ,',

a05 0.1 0.15 0.2__. "o/I'!'-"

• '41770

Fig. 8. The quantity b = 2=1/)" which is proportional to thecharacteristic frequency, as a function of the ratio rO/r1 ofthe radii with different values of n for fiat cavity resonators. ,For n';:; 1 and ro = 0 the quantity b becomes n/2, so thatthe wave length J. is then equal to four times the outer radius rr-.

Infig: 8 it is now shown how at different values ofn according to (22) the value .of 2:n;'l/A for thefun da men t al tone varies with the quotient ofinner and outerzadih TO/T1 = a/b. In the limitingcase of à conical :flat cavity resonator the wavelength Al of the fundamental tone is four times thedifference' hetween outer radius Tl and inner radiusTO' It is evident fró:.p.' fig. 8 th~t this differenceT1-TO is, however, in general by no means .equal to a

154-: PHILIPS TECHNICAL' REVIEW 'VOL. 8, No. 5

quarter wave length. For instance in the case ofa flat cavity resonator (fig. 3) with no hole (n = 0and TO = 0), for which at the fundamental tone thefollowing holds: b = 2nT1/À1 =.2.405, the wevelength becomes

, 2nT1À -:- --5 = 2.61 Tl'

2.40

If we now wish to know the wave 'lengths ofseveral overtones for such a flat cavity resonatorwe can easily find them froin the higher roots ofequation (23) where n must he set equal to zero,so that we obtain:

The first three zero points of (Jo) lie respectivelyat 2nT1/À = 2.405; 5.520; 8.653 (cf. fig. 9), fromwhich then follow for the three longest wave lengthsat which a flat cavity resonator with no hole can,execute characteristic vibrations:

Ài = 2.61 Tl; À2 = 1.14 Tl; À3=0.73 Ti. (25)

The quotient of À2 and À3 is;1.56, which is not muchsmaller than 5/3 = 1.67; which would be the ratiowith a harmonic position of the characteristicfrequencies. The frequencies of the higher overtonesfor the flat cavity resonator with no hole ,are indeedfound to be progressively more nearly in the ratiosof 'the successive odd numbers. Thus for example"À3/À4 = 1.36 and 'À4/).5 = 1.27 while 7ï5 = 1.40and 9/7 = 1.29. It is only the ratio ofthe fundamen-,tal tone to' the overtones which is far from har~'monic, since À1/À2= 2'-29, which differs very much'from 3!

In conclusion, for the simple, case of the flat

, t2im=~J,(~)I

Fig. 9. Variation of the current amplitude Ïm and the voltageamplitude Vm with the radius T of a fiat, cavity resonator.The behaviour from x = 0 to 2.405 gives current and voltageof the fundamenta1 tone, from x = 0 to 5.520 the same for thesecond c1iàiácteristic vibration, from' x ,;,. 0 to 8.653,for thethird characteristic vibration, and so on. In the same way atthe .,zero points of the current curve xJI lie the maxima forthe .voltage Jo' ' , ''r·· •

cavity rèsonator with no hole (n = 0; Tl = 0) weshall examine the way in which the current andvoltage are distributed along the radius on thebasis of equations (21) and (23'). When we make useof the relation J_1 (x) = -J1 (x) and omit the con-stant factors: we obtain the following:

im = X J1 (x), ~ . . . . . (26). Vm = Jo (x). ~

The variation of current and voltage with th~ radiusaccording to (26) is' drawn in fig. 9. In the variationof the voltage we find the zero points of Jo alreadymentioned in the derivation of the 'characteristicfrequencies 5), which are at the same time thepoints of the maxima of the current curve xJI'

The formulae given here lose their physical signi-ficance as soon as the absolute value of n becomestoo large. The quantities Cl and LI can then 'nolonger be consideréd as varying slo w1y with T (cf.(16», so that the assumption for which the initialequations (13) we_re derived is no longer satisfied.

Resonance resistance and quality factor

In the foregolng considerations we have assumed.,that we were concerned with flat cavity resonators'having' no ohmic resistance. Upon resonance nocurrent then flows at the open inner edge, indepen-dent of the voltage acting on it, so that we would bedealing with an infinite resonance resistance. Now,however, we wish to calculate the resonance re sis:tance and the quality factor which can be obtainedin practice' with flat cavity, resonators, in whichcase, therefore, the energy losses should actuallybe taken into account.

The quality fact~r' Q of a resonance circuit, exceptfor a factor st, is the reciprocal of the logarithmic'decrement and is thus a measure of the time inwhich a 'free vibration dies .out in that circuit.The energy losses cannow be taken into account byfirst determining the distribution 'of current andvoltage as if :there were no losses, and afterwards 'calculating the corresponding losses with, thiscurrent and voltage distribution, without allowingthem to influence that distribution. We shall followthe same method for the flat cavity resonators.

Like the self-induction LI and the capacity Clper cm width of ring (cf. fig. 7), the resistance RIper cm width of ring now depends upon the ramus T

of the ring in question. It becomes1 ' ,

R1= __ ,2nTO'(j

(27)

6) The short-circuited outer edge of the fiat cavity resonatormay of course he situated at each of these zero points.

MAY 1946 FLA.T CAVITIES A.S ELECTRICAL RESONATORS 155

-where (1 is the specific conductivity and {J the depthof penetration due to the skin effect for the fre-quency in question, which is given by the following~xpression:

. ,

It is evident from this that for a conical Hat cavityresonator also the resistance is by no means con-stant, but varies along the radius. The analogy withthe homogeneous Lecher system thus indeed holds

:1 3

[11 fi2

t f '

, i.2 '2

J •

II

for the character of thevibrations and waves, butnot for the energy consumption by the conical Hatcavity resonator. Upon. calculating the qualityfactor Q complicated formulae are obtained. Forthe fundamental vibration the quality factorcan be ~itten in thé following form: .

Q=~ (L1 ;{J h

(L2+-Tl

.(L1 and (L2 here fepresent numerical factors which wehave represented in fig.IOa and b tor differentvalues ofn, as functions of TO/T1•

In the derivation of equation (28) we beganwith the fact that the quality factor Q is equal to2:n;times the quotient of fieid energy and energydissipated per period (see footnote 1)).

The resonance resistance of the Hat cavityresonator may be considered as' the quotient of thesquare of the effective voltage V for T = TO and theheat developed ill it per second. Since the latter,

except for' a factor. 2:n;, 'is itself the quotient ofthe field energy and the quality factor Q, theresonance resistance (in ohms) becomes:

1 " QV2 h.Z = - = 120 Q - (La, .: (29)

2n field energy Tl

where (La again represents a numerical factorwhich is shown in fig. lOc for different values ofn as a function of TO/T1' .

In fig. 11 for the case of a Hat cavity resonator(n = 0) the quality factor Q is shown as a function

°0~--::~~05-=---::~.L.'--O::':'J5=---=0,2 00 0,05 0,1 0,15. q2 00 0,05 DJ• ~15 0,,2----,.. rO/r, - ro/r, - ro/r,a) bl ' . ct· 41771

Fig. 10: The auxiliary quantities !Ll' !L2.and !L; for different values of n as functitns of TO/Tl'

3

!

--.....r-,.,"- <,

""r--r-- -....... :--...t

v-

•

2

n.-o,S

n.t,

n.2

of the ratio Ta/Tl of the radii. The continuous (almost)straight line which is nearly horizontal is calculatedaccording to equation (28). The small circles indi-cate the values determined experimentally. They

(28)100001-----..,... _

Qt 8000

6000

4000

2000

oo'

o

Fig. H. The quality factor Q as a function of the ratio Tofrlof the two radii of a Hat cavity resonator (n = 0) with athickness of 4 cm and an outer 'radius' of 40 cm. For largeholes it is found that smaller values of Q'are measured thanthose calculated. .

156 PHILlPS TECHNICAL REVIEW. VOL...B, No. 5

are derived from the resonance widths Llw and theresonance frequencies wo: the quality factor mayalso he defined as wo/Llw. It may be seen' that forlarge .holes the theory yields too large values of Q.This need not be surprising because in the preceding.discussion the rad i a ti 0 n los s e s were comple-tely ignored.

The influence of. the radiation losses, like thatof the heat development, could now be taken intoaccount. beginning with the assumption that theradiation, like the ordinary ohmic resistance, 'doesnot change the established current distribution fora loss-free Hat cavity resonator. It would, however,. .be necessary to take mto account that the currentson the outside of the Hat cavity resonator can nolonger he disregarded as soon as the hole becomeslarger. In the foregoing we have only eonsidèredthe field inside the resonator. There the electricfield. is fairly, hOIIi.~geneou8land has a verticaldirection. 4r~~d th~ hole in t~e resonator, ~o~eyer,.the lines of force are somewhat bent (fig. 12a) sothat. a .small part' of 'them even passes "from thèl~wer' outside' s~face. to the upper out~ide' surface.There. are the~efore .also .charges. on the outsidë :ofthe resonator and iheir intensity. varies at the samefrequency-as that' at which the resonator vibratès.Consequently. .currents occur: which How: on theo'utside' hut also êOhtmue on :the inside (fig. 12b).. .e _.In, our. calculation of the .current distribution in theresonator, 'ho,vever,' we' assu~ed that i = 0 whenr ' ...r0' so' that a '~'~!recW)llmust actually be intro-duced here. .This results. not only in correctionsof the characteristic frequency but also in an: in-crease 'ill the radiation -josses;Wé shall .not-discuës. " . " . . .-

this 'rather,: complex problem 'any further' here.

linprovèmen~ of tlÎe quality factor,and the resonance• • • 6. . _....... ,..." ...... ~. .' •••••

, As may he seen from formula (28), for the qualityfactor (Q)" this quantity is in the first instanceproportional to the 'r~tio between the thiclfnessof the Hat cavity resonator and the depth of pene-

a)

h)

-4171'6

Fig. 12. a) Paths of die electricallines of force between th~'pl ates of a flat cavity resonator. b) "Electric currents inducedIDthe yic!IDtr,of the hole as a resJ!ltof the bending of the linesof force. . , . .' •

tratien (J due to the skin effect. Our previousconsiderations always referred to very thin cavityresonato~s for which h ~ r. If h ,is taken smallenough to justify this treatment the quality factor,will often be inadequate. This is not an insurmount-able difficulty, since the cavity resonator can. asrequired be made somewhat thicker and then beconstru~ted as being built up of. a number of thin 'resonators, for which the foregoing considerationsare immediately valid, By such a p ilm'g up ofHat cavity resonators, therefore, it is possible todeal also with thicker cavity resonators, without it ,being necessary to pass over to the general theoryof cavity resonators of three dimensions.

~11--==--------1a) b) ,

41776

Fig. 13. By the stacking of thin cavity resonators (a) moregeneral resonators can be built up (b) •

, A~ is shown diagrammatically in fig. 13, for thecase where n = 0 ~ cylindrical cavity resonator canvery easily be obtained by the piling up of thin.Fl a t cavity resonators, the height of the final reso.·nator no longer being small compared with' theradius. In this cylinder, however, there are still.par tit ions which are at the same time the topsurface, óf one of the thin cavities and the bottomsurface of the next, Since all these Hat cavityresonators vibrate with the same characteristic

, .frequency and all possess an equally intense verticalelecteic field, the situation is now simply that thecurrents belonging to the upper and lower cavitiesin such a partition are everywhere equal and oppo- 'site, so that the partitions could he removed from,the cylindrical cavity resonator. One point will:however, then he, altered; the 'heat development,which the electric currents in the partitions of thepiled up thin. cavities would produce, no longeroccurs! This rednetion of the losses, by the removal,of partitions is not merely an imaginary experi-ment but a true fact: as long as the partitions still'remain in our cylindrical cavity resonator the heatlosses "Wip. occur in it, because as a result of theskin effect the currents then actually How onlyin the top and bottom layers of such a partition,so that they will by no means cancel each other asfar as the development of heat is concerned.'Thus by theremoval of the partitions we reduce

the energy losses due to heat development comparedwith the total. vibration energy, and it is thereforeunderstandable that for such a more general cavity'

MAY 1946 FLAT CAVITIES AS ELECTRICAL RESONATORS 157

resonator obtained by "stacking" we can obtainbetter values for the quality factor Q and the impe-dance Z than were possessed by thc flat cavityresonator with which we began. If we calculate Qfor a cylinder of height h and radius r, we obtain

h hQ = _. . . . . . (30)Ö rl + h

a) b)

Fig. 14. Thin cavity resonators in the shape of conical shellscan also be stacked.

In a similar way, by the stacking of the thincavity resonators shown in fig. 14a in the form ofconical shells, we can obtain the cavity resonatorshown in fig. 14b, which has become very importantin short-wave technology. In the discussion ofseveral practical examples we shall also encountersuch a cavity resonator (see fig. 20).

Several practical examples

For a copper conical flat cavity resonator (fig. 15)with n = 1, a thickness h = 10 cm at the outsideedge and radii ro = 0.6 cm and rl = 30 cm, at awave length of 122 cm, we measured a qualityfactor Q = 10 200, while the value 11 400 wascalculated. For the impedance Z a value of 273 600ohms was calculated. If in this flat cavity resonatora power of 25 watts is developed in the form of heat,the current at the outer edge is i = 130 A, the volt-age on the inner edge of the cavity amounts toVro = 2600 V, while the magnetic field strengthsat the inner and outer edges are Hl'O = 1.4 gaussand Hrl = 0.9 gauss. It was actually found that asmall transmitter which could deliver a power of25 W gave a current of about 100 A in the short-circuit ring, which could be concluded from thevoltage generated in a loop.

Fig. IS. Conical flat cavity resonator constructed with n = l.The dimensions are given in centimeters.

For the case of a flat cavity resonator (n = 0)with a thickness h = 4, cm and an outer radiusrl = 40 cm and different radii of the hole: ro = 0;3; 4 CJ;n,measurements gave the following values ofthe quality factor: Q = 10 350; 7 700; 6 100.

Frequency stabiZization with flat cavity resonators

Since a satisfactory quality factor Q can be ob-tained with flat cavity resonators, they can verywell be used for keeping the frequencies of oscillatorsconstant. The cavity resonator is then loosely cou-pled with the oscillator, so that in certain frequencyregions the wave length is determined much more bythe tuning of the resonator than by the rest of theconnections joined to it. Infig.16 a conical flatcavity resonator is shown coupled with aLecher system of variable length Zl. In.fig. 17 the

_,---------,~-;!lr-------~_,i---------:Jt"--; !v--'---t--

4181&

Fig. 16. A conical flat cavity resonator is coupled with aLecher system of variable length /1; in order to keep thefrequency of the oscillator constant. In the photograph theloop by means of which the current i1 is measured can justbe seen on the upper right-hand edge of the flat resonator.

158 PHILlPS TECHNICAL REVIEW

wave length;' at which this system vibrates and theamplitude i1 of the current flowing in the short-cir-cuit ring of the resonator are plotted as functions of

À

t

-1,Fig. 17. The wave length A. and the current i1 occurringwiththe connections shown in fig. 16 as functions of the length I1of the Lecher system. The figures represent a portion ofthe A.-11 plane and the ~-11 plane, respectively, the pointsA. = l1 = 0 'and it = l1 = 0 falling outside the figures.Over a wide range the influence of Il on the wave length isfound to he only slight, since the latter is mainly determinedby the wave length A.o of the conicalflat resonator. It is furtherevident from the figure that the state of oscillation is notdetermined unambiguously by the length ll' but that it alsodepends upon the way in which a given tuning is reached.

the length 11 of the Lecher system. Over a widerange the length 11 of the Lecher system is foundto have only very little effect on the wave length À.at which the whole system vibrates. The latterwave length is then determined almost exclusivelyby the wave length ;'0 of the free vibration of the

VOL: 8, No. 5

alone but also depends upon the way in which themomentary state of the circuits was reached, sothat the previous history of the state at a givenmoment also plays a part. This is a result of thenon-linearity of the transmitting valves used G).

Cavity resonators as output and input electrodes.with short-wave transmitting valves

When the high-frequency oscillation energy IS

taken from radio valves by means of the anodeupon which the electrons must themselves finallyimpinge with high v~locities, in the case of high-power valves a large amount of heat is developedin the anode. In the construction of such valves thisshould be taken into account, be it at some cost ofconsiderations connected with high frequency.For this reason transmitting valves for very shortwave lengths are at present often constructed sothat the electrons pass along a pair of rings in whichthey induce charges. They then give off high-frequency oscillation energy to the rings withoutthemselves striking the rings. The collision energy,however, is taken up by the anode, which is earthedfor high frequency and may thus have any desiredlarge dimensions.Infig.18 such a so-called induction tube oscil- .

la tor is shown. A beam of electrons whose intensityis controlled in a high-frequency rhythm (fig. 19)by a grid (hf) passes through a slit in a flat cavi ty

41817

conical flat cavity resonator. In this figure the dis-advantage is also evident that the oscillator fre-quency is not wholly determined by the circuits

,Fig. 18. Induction tube oscill-ator for waves of 65cm, equip-ped with a vertically placed,rectangular flat cavity reson-ator for taking off the high-frequency oscillation energy.On the left-hand side of the flatresonator may be seen one ofthe magnetic coils which serveto keep the electron beamconcentrated. The Lecher sys-tem extending to the left servesfor tuning the grid circuit, whilethe Lecher system extend-ing toward the rear conductspart of the oscillation energyexcited in the flat resonatorback to the grid, so that theelectrical oscillations are main-tained. The Lecher system tothe right takes from theresonator the energy dissipatedin the lamp serving as loadingimpedance.

6) Cf. for example Balth. van der Pol, Trillingshysteresishij de triodegenerator met twee graden van vrijheid(Vibration hysteresis in the triode generator with twodegrees of freedom). T. Ned. Radio Gen. 1, 125, 1921.

MAY 1946 FLAT CAVITIES AS ELECTRICAL1RESONATORS 159

resonator (5), upon which electric charges areinduced, while the electrons of the beam themselvesfinally impinge on the anode (a). Seen from thepoint of view of the cavity resonator, the inductiontube seems to possess a fairly high adaptationresistance. In order to ensure a good transmission

I TANT.==:=J-+-----~~-----l-a

h~ [JSt#/1I18

Fig. 19. Diagrammatic representation of an induction tubeamplifier with a flat cavity resonator S as output electrode.The electrons are controlled by the high-frequency grid h.fand move toward the anode a. The aerial (ANT) is coupledto, the resonator, which is excited by the pulsating electroncurrent passing through it.

of the high-frequency energy to the aerial, the impe-dance of the unloaded oscillation circuit must behigh compared with the adaptation resistance.Therefore a cavity resonator with its large impe-dance is especially suitable to play the part ofoscillator circuit here. The high-frequency energyis thereby taken from the cavity resonator by cou-pling the aerial to it with the help of a loop.The resonator then functions mainly as trans-

former between the high-frequency electro~ beamsupplying the energy and the aerial radiating it.If it is impossible to obtain an adequate quality

factor and resonance resistance with the thin,flat cavity resonator shown in figs. 18 and 19 in theinduction tube oscillator, these quantities cannot beeasily increased by using a somewhat thicker,flat resonator, since in order_to keep the electronbeam in the correct direction magnetic coils aresituated along it at fairly short intervals and aresonator has to be :fitted between the coils. It is,however, possible to obtain a larger quality factorand a higher resonance resistance by using a reso-nator like the one shown in fig. 14b whose thicknessat the centre remains sufficiently small while atdistances farther away from the electron beam theresonator becomes' much thicker. In fig.20 aninduction tube amplifier with such a thicker reso:nat or is shown.

For the generation of high powers at ve,ry highfrequencies much use is being made at present of acontrolmechanismwithwhichthe velocities oftheelectrons allowed to pass are varied and not the num-ber. In their further progress the faster electronswill then overtake the slower ones, so that accumu-lations occur whereby the same effect is attainedas with amplitude control. In the application ofsuch so-called velocity modulator valves differentcavity resonators must be introduced around the

Fig. 20. Induction tuhe ampiifier for 45 cm waves, equippedwith a thicker cavity resonator like that shown in fig. 14bwith which a larger quality factor and higher resonanceresistance can be obtained than is possible with thinnerresonators. In the cylinder on top of the cavity resonator isan artificial aerial in the form of an incandescent lamp. Thecoil below the resonator provides the magnetic field whichkeeps the electron beam concentrated.

pathfollowed by the electrons from cathode to anode.Therefore thin, flat cavity resonators areoften used here.In fig. 21 a velocity modulator amplifier

is shown diagrammatically. In this case instead ofbeing controlled by the griçl hJ of fig. 19 the elec-tron beam is controlled by the slit of the inputresonator 51' Some distance farther along the pathof the electrons is the output resonator 52' so that

o TANT..==:=J-------------+--a

[JSi [JS2",,"g

Fig. 21. Diagrammatic representation of a velocity modulatingvalve amplifier with flat input and output cavity resonators:Sl and S2' The electrons, on their path to the anode a, arecontrolled in a high-frequency rhythm by Sl' while then inpassing S2 they excite it in a high-frequency rhythm. The latterresonator passes on its oscillation energy to the aerial (ANT).

160 PHILlPS TECHNICAL REVIEW VOL. 8, No. 5

the electrons will arrive there with a variationin density sufficient to excite oscillations in S2with reasonable efficiency. The electrons themselvesfinally reach the anode a, while the flat resonatorS2 gives off its high-frequency oscillation energyto the aerial coupled with it.

Fig. 22. The flat input and output cavity resonators SI and S2are coupled to each other in order to make it possible for thesystem to-oscillate independently. The high-frequency oscilla-tion energy for the aerial (ANT) is obtained from the pulsatingelectron current on its way toward the anode a by means athird flat cavity resonator Sa.

If the flat input and output resonators (Sl and S2)are coupled with each other, this system can be madeto oscillate independently (jig.22). The high-frequency oscillation energy is then finally takenfrom this velocity modulator oscillator

with the help of a third flat resonator S3 coupledwith the aerial. A demonstration model of such agenerator for very short waves is shown in jig. 23.

Fig. 23. Velocity modulator for waves of 40 cm in which useis made of tbree flat cavity resonators (cf. diagram of fig.22).The cathode is on the left and the anode on the right. Thesize of the opening in the cavity resonator at the right Sacan be regulated by turning the spokes visible in the photo-graph in order to change the characteristic frequency. Themagnetic coils serve to keep the electron beam concentrated.

/~~

The~number of Philips Research Reports (No. 3 of volume 1, April1946)contains following articles:

R12: K. F. Niessen: On the error in the determination of the medianplane of a radiobeacon in a tilted airplane.

R13: B. D. H. Tellegen: Network synthesis, especially the synthesisof resistanceless four-terminal networks.

R14: H. B. G. Casimir: On Onsager's principle ofmicroscopic rever-sihility.

RIS: M. Gevers: The relation between the power factor and the tempe-rature coefficient of the dielectric constant of solid dielectrics.

R16: T. J urriaanse, F. M. Penning and J. H. A. Moubis: Thecathode fall for molybdenum and zirconium in the rare gases.

R17: G. W. Rathenau and J. L. Snoek: Apparatus for measuringmagnetic moments.

Anyone interested in these articles can apply to the Administration ofthe "Natuurkundig Laboratorium" Kastanjelaan, Eindhoven, Holland,where a limited numher of reprints are available for distribution gratis.For suhscription to Philips Research Reports application shou1d be madeto the publishers of "Philips Teohnical Review".