Negative L and C in Solid-State Masers

PROCEEDINGS OF THE IRE

and (9) cani be rewritten in the form

(G -1) (27)

4fla

to give the uncertainty in phase produced by the ampli-fier. If we refer these quantities to the input we have

1Anfai2 = Ana2 = (1 - 1/G)nj (28)

and

A'ka2 = (I- 1G) (29)4n1

From (26) and (27) we see that if the amplifier gain ishigh, the output uncertainties introduced by the ampli-fier are considerably larger than those of even a poormatched detector. In this case, the total uncertainty

An, in the inferred measurement of the input photonnumber aind AO1 in the inferred measuremenit of inlputphase are closely given by (28) and (29), which relateto the amnplifier alonie. In the limit of high gaini, theproduct of these uncertainties is

AnIA4u 'faiAX4a _ 2' (30)

This is, of course, the minimum value allowed by theuncertainty principle. Thus the minimum noise ampli-fier allows us to use a poor detector, one which intro-duces uncertainties considerably larger than the mini-mum necessary, and still mneasure an incoming signalwith an accuracy approaching the best allowed by theuncertainty principle. There still remains a question ofwhat limitation is put on the rate of informlation trans-mission by this maximum allowable accuracy of detec-tion. The answer to this question, however, imust awaitthe development of a quantum theory of communica-tion.

Negative L and C in Solid-State Masers*R. L. KYHLt, MEMBER, IRE, R. A. MCFARLANEt, MEMBER, IRE, AND M. W. P. STRANDBERG§,

FELLOW, IRE

Summary-The analysis of solid-state cavity masers is extendedto include the reactive component of the paramagnetic resonance.This reactance is inverted (in opposition to Foster's reactancetheorem). A two-cavity network makes use of this negative fre-quency dependence of reactance to obtain a broad-band flat-toppedamplifier response. In verification of this theory a ruby maser hasbeen built which has a 95-Mc bandwidth at 14-db gain and operatesat 9000 Mc and 1.50K. This performance is comparable to that of pub-lished, tapered magnetic field traveling-wave masers. General net-work limitations on cavity maser amplifiers are derived. Broad-banding techniques that have been published for parametric ampli-fiers are essentially equivalent. The tuning of the broad-band ampli-fier is critical. The same performance can be achieved in a unilateraltransmission maser by using circularly polarized cavities, but theproblem of circuit design and tuning with the increased number ofparameters has thus far prevented successful operation.

* Received February 19, 1962; revised manuscript received, May1, 1962. This work was supported in part by the U. S. Army SignalCorps under Contract DA36-039-sc-87376, the Air Force Office ofScientific Research, and the Office of Naval Research.

t Department of Electrical Engineering and Research Laboratoryof Electronics, Massachusetts Institute of Technology, Cambridge,Mass.

t Bell Telephone Laboratories, Inc., Murray Hill, N. J. For-merly with Research Lab. of Electronics, M.I.T.

§ Fulbright Lecturer, University of Grenoble, France, 1961-1962;on leave from the Dept. of Physics and Research Lab. of Electron-ics, M.I.T.

I. INTRODUCTION

r HE STIMULATED emission behavior of the ac-tive material in a solid-state maser can be char-acterized satisfactorily by its contribution to

the complex electric or magnetic susceptibility of thematerial. (Beam masers are somewhat more complicatedin this respect.) Typically this susceptibility showsa sharp resonance at a frequency corresponding tothe quantum transition involved. The imaginary com-ponent of the susceptibility is, of course, responsiblefor the maser gain, but the real, or reactive, componentmust also be present. In narrow-band systems this re-actance may be masked by the larger reactive effects ofthe microwave cavity or circuits. As larger gains andbandwidths are obtained with better substances and de-sign configurations, the reactive component Imlust betaken into consideration to obtain a correct analysis ofthe circuit behavior. When the population distribuL-tion between the quantum levels is inverted to achievemaser amplification, both components of the suscepti-bility reverse sign. The resulting frequency dependenceof reactance corresponds to the situation that wouldobtain in conventional circuit analysis if the symbols

1608 Jutly

Kyh1, et al.: Negative L and C in Solid-State Masers

L alnd C could be assigned negative values.1This report describes the construction and successful

operation of one scheme for utilizing the negative L andC properties of inverted populations to achieve im-proved gain-bandwidth performance from a cavitymaser. We call this "reactance compensation." Otherefforts in this direction have been reported.2 The samecombination of negative R, L, and C appears in para-metric amplifiers, although for entirely different rea-sons.3

This analysis will be entirely classical, althoughquantum phenomena are important in determining thesusceptibilities that we take as our point of departure.

Large-signal nonlinearities may be safely ignored atthe signal levels for which masers are useful. Saturationeffects at the signal frequency are adequately treated bypermitting slow time variation of the susceptibility.The pumping mechanism by which the inverted

population is achieved does not enter the discussion,although it should be recognized that the circuit per-formance will be affected by failure to achieve inversionover the entire volume of the active material.

II. BASIC THEORY

We introduce a simplified model for detailed calcula-tions. A resonant cavity containing a substance showingquantum-mechanical resonance absorption can be wellrepresented near resonance by the equivalent circuit ofFig. 1. The series circuit represents the cavity resonance(including any nonresonant susceptibilities of thematerial), and the parallel circuit represents the reso-nance susceptibility of the material. There are severalapproximations and pitfalls in this model. The cavitymust, of course, have a single isolated high-Q resonance.

Fig. 1-Equivalent circuit for cavity containingresonant susceptibility.

1 R. L. Kyhl, "Negative L and C in solid-state masers," PROC.IRE (Correspondence), vol. 48, p. 1157, June, 1960; Errata, vol. 49,p. 519, February, 1961.

2 J. J. Cook, L. G. Cross, M. E. Bair, and R. W. Terhune, "Low-noise X-band radiometer using maser," PROC. IRE, vol. 49, pp.768-778; April 1961. See also Goodwin and Moss.'7

3 H. Seidel and G. F. Herrmann, "Circuit aspects of parametricamplifiers," 1959 IRE WESCON CONVENTION RECORD, pt. 2,pp. 83-91.

G. L. Matthaei, "Study of optimum wide-band parametric ampli-fiers and up-converters," IRE TRANS. ON MICROWAVE THEORY ANDTECHNIQUES, VOI. MTT-9, pp. 23-38; January, 1961.

A. G. Little, "Wide-band single-diode parametric amplifier usingfilter techniques," PROC. IRE, vol. 49, pp. 821-822; April, 1961.

G. Schaffner and F. Voorhaar, "A nondegenerate S-band para-metric amplifier with wide bandwidth," PROC. IRE, vol. 49, pp.824-825; April, 1961.

M. Gilden and G. L. Matthaei, "A nearly optimum wide-banddegenerate parametric amplifier," PROC. IRE, vol. 49, pp. 833-834;April, 1961.

The representation of the quantum-mechanical reso-nance by a single degree of freedom needs justification,since there are, in fact, at least 10's active ions allresonating near the frequency range that is of interest.Some of the subtleties of this view are discussed inAppendix I and a more complete discussion is presentedelsewhere.4 The resulting simplification in Fig. 1 can beused to represent a Lorentz-shaped line. In many casesof interest the observed line shape is more nearlyGaussian, with the resulting reduction of absorption oremission (but not reactance) in the wings of the line.Notwithstanding these criticisms, it is felt to be best toanalyze the simplest possible circuit that exhibits thephenomena in which we are interested.Other simple choices of circuit could have been made

with a suitable shift in reference plane, including thedual of Fig. 1.4 The present choice fits well intuitivelywith the use of a paramagnetic crystal such as rubyfor which the permeability near resonance (Lorentzianline shape) is given by

(1)1 jXrnmax']

1A = AO IL 1 + jT2i\@

where Xmax" is the peak value of the absorptive com-ponent of the magnetic resonant susceptibility. Thefull linewidth is 2/T2, and Aco represents the deviationfrom center frequency.

This equation "describes" the parallel circuit andpart of the series inductance of Fig. 1. The remainder ofthe series inductance is to be associated with RF mag-netic fields in parts of the cavity that do not containactive material. Alternatively, one could define an effec-tive susceptibility Xeff" =fxmaxt, where f is a fillingfactor. The circuit model presents an open circuit farfrom cavity resonance. We might have preferred a shortcircuit, but our choice will be shown to be convenientlater. As is customary for microwave circuits, the im-pedance level is left arbitrary.We now introduce an inverted-population, stimu-

lated-emission situation by suitable "pumping" of theactive material. We then have

(2)jXmax1c/I = 4o + _+jT2A

and the equivalent circuit of Fig. 2. The -L and - C inFig. 2 express the fact that the dependence of reactanceon frequency is the reverse of the usual case. These re-sults derive from the solution of the equation of motionifor the quantum-mechanical system in question. In thecase of paramagnetic resonance, the Bloch equations'

4R. L. Kyhl (paper being prepared for publication).6 F. Block, "Nuclear induction," Phys. Rev., vol. 70, pp. 460-

474; October i and 15, 1946. See Section III.

1962 1609


show that, for every solution for the macroscopic mag-netization vector, there is another solution in the sameapplied fields with the direction of magnetization re-versed. This indicates an inverted population distribu-tion among the magnetic sublevels, and also a reversalof both in-phase and out-of-phase magnetic currents.Bloch's analysis provides additional insight into the re-sult that could have been obtained by substituting anegative population difference in the susceptibilityformula.6 Feynman7 has shown that the Bloch analysiscan be extended to other quantum-mechanical transi-tions by replacing the magnetization vector with astate vector in an abstract space. If the discussion isconfined to linear systems, the same result can be ob-tained from the general relations between real andimaginary parts of a complex response function.8

Fig. 2-Equivalent circuit for cavity containinginverted resonant susceptibility.

III. ELEMENTARY PROPERTIES OF -L AND -C

The use of the negative L and C notation does not re-quire any modification of the fundamental circuit equa-tions. Interpretation requires a little care. For example,a resonant circuit composed of +L, +-C, and -R isunstable; but the combination of - L, - C, and - R isstable. It becomes unstable with heavy loading, that is,with tight coupling to a resonant circuit.

Perhaps the most peculiar formal property of negativeL and C circuits is negative stored energy. There is noreal paradox because the devices that exhibit thisproperty have additional stored energy that is availablefor maser amplification, and the total energy in thesystem remains positive. Total storage of energy in anetwork is associated with frequency dependence ofreactance. By combining elements with negative storedenergy and elements with positive stored energy onemight hope to obtain improved frequency response.

It is not really necessary to introduce these newsymbols. The circuit of Fig. 3(a) is equivalent to thecircuit of Fig. 3(b) which has only negative R. However,since the negative R of the maser is found only in con-junction with the negative L and C, it is felt that theintroduction of additional concepts is justified.

6 A. M. Clogston, "Susceptibility of the three-level maser," J.Phys. Chem. Solids, vol. 4, pp. 271-277; 1958.

7 R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwarth,"Geometric representationi of the Schr6dinger equation for solvingmaser problems," J. Appl. Phys., vol. 28, pp. 49-52; January, 1957.

8 J. R. Macdonald and M. K. Brachman, "Linear-system in-tegral transform relations," Rev. Mod. Phys., vol. 28, pp. 393-422;October, 1956.

+

+

LB,.

(a) (b)Fig. 3-Negative R synthesis of negative L and C.

IV. REACTANCE COMPENSATION

If the series impedance in Fig. 2 is very low, the cir-cuit of Fig. 4 suggests itself. Here, except for the seriescircuit, cancellation of the reactance as well as greatlyincreased bandwidth can be achieved. The equivalentcircuit of Fig. 4 corresponds physically to two cascadedcavities with resonant permeability in the second cav-ity, as shown in Fig. 5. For use as a maser the circuit ofFig. 5 would be used with a circulator as in coniventionalcavity masers. Section VIII of this report presents ex-perimental results from a circuit of this type. Ignioringcavity losses we calculated the theoretical voltage gainicurves shown in Figs. 6-8. In Section V general nietworktheory is extended to this stituation.From the circuit of Fig. 4 (but with cavity losses

omitted) synchronously tuned at wo, we have nearresonance

Y, 2j(co -cO)CIZ2- 2j(co - coo)L2Y3 _ - G3 -2j(co- Wo)C3. (3)

Here the symbols G3 and C3 are taken to be positivenumbers. It is preferable to express the equivalent cir-cuit in terms of measurable quantities. By elementarycircuit analysis we derive the midband voltage gain

ql/2 = G3ZO + 1

G3Zo- 1(4)

in terms of the characteristic impedance of the trans-mission line. The various coupling coefficients are in-volved. For G3Zo < 1, the circuit is unstable. The para-magnetic (full) linewidth between half-power poinlts isgiven by

Awpara = G3/C3. (5)The magnetic Q, defined, as is customary, in terms ofstored energy in the cavity fields and power dissipatedby the paramagnetic material, is given by

QM = -cooL2G3. (6)

1610 Jutly

Kyhl, et al.: Negative L and C in Solid-State Masers

VOLTAGE GAIN

yI Z. Y3

R2,L 2 'C2 -G3,-L3!-C3

aQo0M(AWPARA"'O)o0.5

Fig. 4-Two-cavity reactaiice-coimipeiisation circuiit.

U

Fig. 5-Two-cavity reactance-compensation configLuration.

VOLTAGE GAIN

M( PARA/ 0)a=0.96

a I.49B A - bx 1.66

twPARA c' 1.79

(W- 0U)/ (OPARA

Fig. 7-Theoretical voltage gain-bandwidth curve, a =0.5.

VOLTAGE GAIN

WB -W b = 1.10

c -31.2AwPARA d - 1.73

e = 20

-a

-ba=-QM(AwPARA /wo)=0.2

(W - WO0 )/APARA

Fig. 6-Theoretical voltage gain-bandwidth curve, a = 1.0. Curve ccan be compared with dot-dash curve: single cavity with infinite-bandwidth ruby; and with dashed curve: single cavity withactual rutby.

a=2 19WBWA =-b= 2.36

AWPARA c=2.53

-a

0 0.5(w wo)/AW PARA

Fig. 8-Theoretical voltage gain-bandwidth curve, a= 0.25.

1.0

1962 1611


Historically this definition was introduced whenparamagnetic reactance was negligible compared withcavity reactance. For amplification we take QM to benegative. In terms of the resonanice susceptibility wefind that

Xmax (

for unity filling factor. In general the equality holdsif Xmax" is the effective susceptibility, which includes afilling factor.The coupling between the two cavities produces a

split in resonanit frequencies in the absence of para-magnetic resonance.

(B - =_A___VC,L2

(8)

Fig. 9 Modified two-cavity reactaince-compeinsation circiuit.

VOLTAGE GAIN

If we normalize this splitting by comparing it withthe paramagnetic linewidth, we obtain a circuit cou-pling parameter that is giveni by

XB A C3

A(Opara / G3,\L2Ci(9)

Finally we calculate the ratio of the electromiiagnieticenergy storage at resonance in the filled cavity to theenergy storage in the equivalent circuit that representsthe paramagnetic resonance. This ratio is called a. It isan indication of the possibility of significant reactancecompensation. We shall see that a should be as small aspossible.

L2G3' Aw/Apara \/Cwparaa = - QI = +

C3 co /XXmaxiWO(10)

In terms of 91/2, a, anid (WB-CA)/A(jpara, the equivalenitcircuit may be recharacterized, as shownl ill Fig. 9. Inthis figure

j1 /

a Wpara )2( (L - WO)agZo VCUB - &A 2V Apara

\ 2/Aparaf

Y3= 1 /Co-(12Ys= - 122/Wpara) (2

w B- AB 166t PARA

-o DETUNED 5% AwPARA

1.25%-

0.5 1.0(w -wo)/AtwPARA

Fig. 10-Change of voltage-gain curve with detuninigof resonanice susceptibility.

curve. Results are shown in Figs. 6-8. Fig. 10 shows theresult of detuning the magnetic field. In practice a 1-gauss shift of the nmagnetic field makes a significantchange in band shape.

V. GENERAL NETWORK THEOREMSConisider Fig. 11, the generalization of the circuit of

Fig. 4, in which an arbitrary network is used to adjustthe gain curve of the maser cavity.

TIhe gain will be represented by the reflectioin co-efficient, F, at the termiinals. Because G is niegative,

.> 1. The correspondinig low-pass problem with posi-tive G was solved by Fano.9 Our development willclosely parallel his analysis. The simpler problem shownin Fig. 12 was anialyzed by Bode'0 who gave the familiarrelationi for the low-pass nietwork

and the turns ratio n of the ideal transformer is suchthat Zaa=-n2Zbb, with

Sl2-28 =(13)91l/2 + 1

Numerical calculationis were made for values of a inthe range achievable with pink ruby completely fillingthe cavity and operating at 9 kMc at 4.2°K and 1.5°K.For each choice of gain the cavity splitting was chosento optimize the flatn-ess of the top of the band-pass

1 GIn dw <Irl - c

(14)

The equality holds wheni F has nio zeros inl the right half-plane.The low-pass equivalenit of Fig. 11 is shown in Fig.

13. There are two general relationships in this case. As

I R. M. Fano, "Theoretical limitations on the broadband match-ing of arbitrary impedances," J. Franklin Inst., vol. 249, pp. 57-84,January, 1950; vol. 249, pp. 139-154, February, 1950.

"I H. W. Bode, "Network Analysis and Feedback Amplifier De-sign," D. Van Nostrand Company, Inc., New York, N. Y., ch. 6;1945.

1612 July


above (16) are not the same as L2 and C3 in (3). The cor-respondence must be made on the basis of Y(w) orZ(w). This introduces another factor of 2. For the band-pass case,

Fig. 11-Generalized reacta nce compensation circuiit.

ARBITRARYr- (NO LOSS) C G

NETWORK

Fig. 12-Network for Bode Theoremii.

Fig. 13 Low-pass equivaleint of reactance-comtipenisation circuit.

L

z ~~~~SAME -C

zoa

Fig. 14-Circuit for definition of r'.

shown in Appendix LI, thev are

-+L lnI rdzo - +± XP-Xz (15)1r 0C

c2ln FrIdwG3 GG3 + G __ Ap-3 +- EAX . (16)3C3 LC2 3 3

The X's are defined in terms of the circuit of Fig. 14.Here, rF is defined relative to +G, and the X's are thevalues of complex s=.w at which r' has poles and zeros

in the right half-plane. Relations (15) and (16) differfrom the ones given by Fano in the following ways:

1) When G becomes negative the IrF is inverted.Compare (14) with (15) and (16).

2) There are appropriate sign changes caused by - C.3) The presence of - C permits the presence of both

poles and zeros of F' in the right half-plane. In-deed the network is not stable without at leastone pole to make the right side of (15) positive.

We now transform the expressions to the high-fre-quency maser case. The network is modified to shiftcenter frequency from zero to wo. Then 91/2 is substitutedfor I. We go from a single-sided integral from zero

frequency upward to an integral over the full bandwidthof the gain curve, although the limits are still 0 to oo.This introduces a factor of 2 on the right. The L and C

ln9112dW _- Awpara + 2(Xp- jwo)O (17)

( cwo)2 In 9112dco

2 AW\para\ A/para 2 2(3 cP2a)~+ (t ;2woXmaxff - (Xp-jwo)3 (18)

The expressions are now approximate because we

have ignored the pass band at negative fretquencies andalso the negative-frequency poles, as in fact we havealready done in (1) and (2). We have dropped the sum-

mations because (as is shown in Appendix III) in a

properly designed maser there are no zeros-just one

pole of I" in the right quadrant. The network has lostsome generality in the shift because the gain curve mustpresumably be symmetric about the center frequency

in order to correspond with the low-pass formulas.This is not a serious restriction and may not be com-

pletely necessary. The situation does not merit tooclose an inspection in any case, since in a practicalmicrowave maser there are other cavity resonances andother resonant susceptibilities that are ignored here.

Eqs. (17) and (18) give the general limitations onlgain and bandwidth for a single crystal, Lorentzianline-shape maser-that is, single crystal in the sense

that the active material is describable by a single set ofelements in an equivalent circuit. A long, thin crystalfor use in a traveling-wave structure does not fit into thiscategory. (See also Appendix I and Section IX.) It isclear from (17) that the circuit design should maximize(X, -jcoo) for best performance; however, (X,-jwo) can-

not be made arbitrarily large because of the restrictionof (18). Since G is never less than 1 for the circuit inquestion, both integrals must remain positive to pre-

serve stability. Eq. (17) gives little insight into possiblegain-bandwidth figures until XXp has been evaluated.This can be done in some particular cases. If the maser

amplifier is to be a high-gain, narrow-band one, then we

may set the second integral (18) equal to zero. We may

then solve for X, and substitute in (17) so that, in thenarrow-band limit, we obtain

In9l2do = Aa[(1 +7 . AWxpara= ACpara[(l + 3/a)1/3 - 1]. (19)

It is seen that for poor filling factors or weak resonances

(a>3) the gain-bandwidth product is linear in 1/o. Forstrong resonance and filling factor near unity (a <3) thegain-bandwidth product may become arbitrarily large,but only as the cube root of (1/a).

+C' +L'

16131962


Another interesting situation is the square responsewith constant gain over a bandwidth of Aw and unitygain outside. The integrals of (17) and (18) can beevaluated directly to give

XmaxitWO 1 / Aw In §1/2 /Aw In §1/2 2AWpara ak 4O)para lA/Kwpara r )

observed at 4.2°K and 1.5°K. The 85-Mc anid 106-Ilclinewidth rubies were operated as single-cavity masers,and we determined the magnetic Q from gain-banidwidthmleasuremenits by using the relations 13

§1/2B 2

A1/para 1 + a(21)

I Aco In z2)rI- r 821law L l 1/l2AWpar 7r In

(20)

Although such an ideal response cannot be obtained witha finite network, (20) gives some idea of what can beaccomplished with broad-band operation. For example,if we wish to achieve a voltage gain of 23, (ln §1/2=7r)and a bandwidth equal to a typical ruby bandwidth of60 Mc, we need an a of 3/8 or less, which is entirelypossible at low temperature.

It should be recognized that 1/a is niot a completefigure of merit for maser materials because it only tellswhat can be achieved relative to the natural linewidth,and, other things being equal, a broader linewidth willgive broader band operation so that ACOpara/a, or(Xmax"co), is nearer to a true figure of merit.

VI. VERIFICATION EXPERIMENTAL DETERMINATIONOF OPTIMUM RUBY-CHROMIUM CONCENTRATION

A ruby maser at 9000 Mc was built to demonstratethe operation of reactance compensationi. The designwas, in other respects, a conventional three-level cavitymaser operated at liquid-helium temperature. One ofthe most important factors in achieving large gain-bandwidth operatioin is the choice of optimum rubyconcentration. Our selection was from several boules ofLinde Companiy synthetic ruby.The paramagnetic liniewidth of Cr3+ in A1203 increases

with Cr3+ concentration, and has a residual value ofapproximately 42 Mc at vanishinigly small concenitra-tions.'1 As the linlewidth lApara is more easily nmeasuredthan the density of paramagnetic centers N, it wasused as an indication of the relative Cr3+ concentrationsin the several ruby samples examined. The value ofXmax" depends on N/Avpara for a fixed inversion ratio,and it is apparent that there should be some optimumvalue for N which would make Xmax" as large as possiblewithout going to unduly high concentrations in whichdeleterious effects, such as cross relaxation, can de-crease the achievable inversion. 12

Five samples with linewidths from 67 Mc to 220 Mcwere measured. Of these only the three lowest concen-tration materials exhibited significant inversion whenpumped. Fig. 15 shows the value of 1/1 QmI which were

11 "Research on Paramagnetic Resonances," Res. Lab. of Elec-tronics, M.I.T., Cambridge, Mass., Eleventh Quarterly ProgressRept. on Signal Corps Contract DA36-039-sc-74895, pp. 1-3, May15, 1960; Tenth Quarterly Progress Rept., op. cit., pp. 17-27,February 15, 1960.

12 T. H. Maiman, "Maser behavior: temperature and concentra-tion effects," J. Appl. Phys., vol. 31, pp. 222-223; January, 1960.

anld

QM

WO/ ACOpara

The 67-Mic linewidth data are only approximate, asvalues of magnetic Q have beeni estimated from themeasuremenits of the compensated amplifier performn-ance.

0.03

1 0.02Q M

0.011

40 60 80 100RUBY PARAMAGNETIC LINEWIDTH - MO/S

Fig. 15 Experimental ruby magnetic Q vs paramagneticlinewidth for the samples used in this work.

Clearly, the available populationi difference for thesignal transition decreases very rapidly as the Cr3+ coIn-centration is increased beyond some optimum value.The change with temperature was only a factor of ap-proximately 2 for the lowest concentration materialstudied. In the absence of cross-relaxation processesthis should be nearer 3. We conclude that the 67-Mclinewidth material is still slightly high in concenitra-tion, the optimum apparently being near a linewidth of60 MVIc. Unfortunately, data for samples of lower con-centration were not available. In the low-concentrationlimit l/Qm must be proportional to concentration.A change in line shape from Gaussian for low concen-

tration to Lorentzian at high concentrationi was ob-served as the Cr3+ concentration was increased and thelinewidth varied from 67 Mc to 223 Mc. (The linewidthsquoted are half-power absorption full widths.)

VII. CAVITY DESIGN

The amplifier was designed to operate at a signalfrequency niear 9 kMc with a pump frequency of 23kMc; operation is in the push-pull pumping mode. This

13 M. W. P. Strandberg, "Unidirectional paramagnetic amplifierdesign," PROC. IRE, vol. 48, pp. 1307-1320; July, 1960.

1614 Jutly


required that the c axis of the crystal make an angle of54.7° with the dc magnetic field.To achieve the highest possible filling factor, we used

the ruby crystal to entirely fill the fixed tuned micro-wave cavity that resonated in the TEIo mode at 9kMc. It was, therefore, convenient to cut the rubycrystal as shown in Fig. 16. The Hrf was everywhereperpendicular to Ho, which gives strong coupling. Thecrystal was pumped at a higher mode cavity resonance.A number of resonances were present at 23 kMc, but itwas not necessary that one of these coincide exactlywith the push-pull pump frequency since saturation ofboth pump transitions is achievable through cross re-laxation for the 67-Mc linewidth material under ap-proximately push-pull pumping conditions. The cavitywas formed by enclosing the ruby with two pieces ofO.F.H.C. copper, as shown in Fig. 17, and brazing themtogether in a hydrogen atmosphere. The resulting blockwas cut down to provide a continuous 0.007-in wall of

DIRECTION OF_

5i47 S ns

Fxig. 16-Ruby crystal dimensions.

copper around the ruby, and signal and pump couplingirises were machined out.The compensation cavity was made in a similar fash-

ion, but contained polycrystalline sapphire (WesgoType A-300), a high-density, low-loss material having adielectric constant of 10 at X band. The large couplingaperture to the signal waveguide lowers the resonantfrequencty of this cavity approximately 10 per cent,and this must be considered in the calculation of thecavity dimensions. The size of the aperture between thecavities was adjusted with each concentration to providefor the separation of the normal-mode resonant fre-quencies indicated for the desired operating performance.A movable knife-edge in front of the signal aperture

in the sapphire cavity provided a very fine control of thecavity center frequency, and an adjustable capacitivepost in the waveguide was used as a loading control tovary the amplifier gain. The individual cavities and thecoupled system are shown in Fig. 18.

Fig. 17-Assembly procedure for copper-clad ruby cavity.

Fig. 18-Two-cavity assembly with cover block containing pump guide removed.Enlarged views of the two cavities are shown above.

1962 1615


WAVEMETER

Fig. 19-Schematic drawing of test system.

TABLE ISINGLE-CAVITY MEASUREMENTS

magneti Temper- Centermagnewict ature Frequency 'm) a IQMILin(emwci)th (°K) (mc)

85 4.2 9300 53 2.2 2401.5 9300 70 1.43 155

106 4.2 9100 8 24.5 2100

Fig. 20-Experimental power-gain curve at 4.2°K; rubylinewidth, 85 Mc; 55-mc bandwidth.

VIII. AMPLIFIER OPERATIONThe test system is shown in Fig. 19. A broad-band

swept oscillator was used as a signal source, and gainmeasurements were determined by means of a precisioncalibrated attenuator. To avoid amplifier saturation, wemaintained output signals below -20 dbm. Sufficientvideo gain was available for oscilloscope presentation ofthe band-pass characteristics.The pump klystron supplied 20 mw of power, and it

was determined during the measurements that this wasat least 3 db in excess of that required to saturate thepump transition. Significantly lower pump power wasrequired for the low-concentration ruby.

ResultsThe single-cavity 91/2B measurements made on the

85-Mc and 106-Mc materials are summarized in TableI. The values of a given in Table I for the 85-Mc line-width material will be compared with the performancedata of the same material in the compensated amplifier.The 85-Mc linewidth ruby was operated in the

Fig. 21-Experimental power-gain curve at 1.5°K; rubylinewidth, 85 Mc; 48-mc bandwidth.

double-cavity configuration at 4.2°K and 1.5°K with acircuit-coupling parameter of 1.06 which represents aseparation of 90 Mc between the normal cavity modes.Fig. 20 shows the band-pass characteristic obtained at4.2°K. Midband gain was 7.4 db, and the half-powerbandwidth was 55 Mc. For a value of a=2.2 as de-termined above, the theoretical bandwidth is approxi-mately 70 Mc at this gain. Fig. 21 shows the amplifierperformance at 1.5°K. The gain was 10.6 db, and thebandwidth was 48 Mc. The theory predicts a bandwidthof 70 Mc for a value of a= 1.43.The results for the 67-Mc linewidth ruby are sub-

stantially better. They are summarized under a varietyof design conditions at 9 kMc, in Table II. The valueof a was not measured in a single-cavity maser for the67-Mc ruby. The assumed values of a which best fitobserved two-cavity operation were a= 0.5 at 4.2°K and0.25 at 1.5°K. The observed and theoretical band-passcurves are shown in Fig. 22 for the 9-db and 14-db gaincases. The quantity (fB-fA)/linewidth is under experi-mental control, but, as can be seen from Table II, the

July1616


TABLE 11RESULTS OF MEASUREMENTS FOR 67-Mc LINEWIDTH RUBY

Measured TheoreticalTemperature Gain Bandwidth fB -fA Bandwidth fB-fA Bandwidth fB -fA

(0K) (db) (Mc) (Mc)

Linewidth Linewidth Linewidth Linewidth

4.2 14 50 120 0.75 1.8 1.02 1.721.5 14 95 155 1.4 2.3 1.54 2.451.5 12 108 169 1.6 2.5 - 2.581.5 9 122 171 1.8 2.55 2.30 2.901.5 -6 145 200 2.2 3.0 - 3.46

I- IOOMC -- -->IOOMC|---E4 IOOMC |-

WU-WA .172APARA

C, - iS A 2 45

AWPARA

-) _... -..V.U W 11 -1.5 -10 -05 0 0. 1.0 1.5 -1.5 -10 -0.5 0 0.5 0 1.5(w%)/NwPARA (co wo)/AWPARA (w-wO)/'APARA

(a) (b) (c)Fig. 22-(a) Experimental and theoretical power-gain curve for 67-Mc linewidth ruby maser. T=4.2°K; gain, 14 db. (b) Experimental and

theoretical power-gain curve for 67-Mc linewidth ruby maser. T= 1.5°K; gain, 14 db. (c) Experimental and theoretical power-gain curvefor 67-Mc linewidth ruby maser. T = 1.5°K; gain, 9 db.

value used for the calculations was not precisely thevalue used in the experimental measurements of band-width and gain.

Fig. 23 (next page) summarizes all of the measure-ments on the compensated amplifier and shows thenecessary normal-mode splitting to achieve the flattestfrequency response at a specified voltage gain, with a asa parameter. The single-cavity measurements on the85-Mc linewidth material are substantiated here. Forthe 67-Mc line-width ruby at 1.5°K, the chosen valueof a = 0.25 for the theoretical calculations appears to beslightly less than that actually obtained during op-eration.

IX. EXTENSION TO CIRCULARLY POLARIZED CAVITIES

If the coupling of the quantum-mechanical transitionto the electromagnetic field is circularly polarized, andif a cavity that can be driven in a circularly polarizedmode is used, nonreciprocal gain can be achieved with-out the use of circulators.14 These conditions can bepartially met. With typical magnetic fields and crystalorientations as used in masers, paramagnetic resonanceis elliptically polarized with the ratio between right-and left-hand circularly polarized components being

14 Strandberg, op. cit., p. 1309.

1962 1617


z X 67MC/S 420K8 a=2.0 10 05 025 @67 MG/S 150K

wS

0> ~~~~~x *40

0 0.8 1.6 2.4 3.2 4.0

AWPARAFig. 23-SuLm-mniary of experimental results and theory: voltage gain

plotted against cavity coupling with ruby quality as a parame-ter. Optimtum flatness of the top of the gain curve is asstunied inthe design.

strotngl depenidenit onl operating coniditionis. 'fIn orderto obtaiin circularly polarized cavity imiocdes, it is nieces-sarv to have a cavity with two degenierate miiodes whichis coupled, as showIn in Fig. 24, by usii1g directional-coupler coupling.'6 A useful equivalenit cirCuit for amaly-sis is shownl ill Fig. 25.A signal in one port of the two-port circuit produces a

circularly polarized mode of onie sen-ise of rotationi; asignial in the other port produces the opposite senise ofrotation. The resultinig fields will be circularlY polarizedin some parts of the cavity anid will miiore generally beelliptically polarized. Such structures can be cascaded;inideed the comb-type, uniilateral, traveling-wave miiasercanl be thought of as a cascade of a large nlumber- of in-dividual resoniators.

Reactance comiipenlsatioin, in prinlciple, can be appliedto unilateral cavity systems. The transmissioni coefficienitwill obey the samiie coniditionis that have beeni (lerivedabove for reflectioni coefficients in simiiple cavities. 'theeffective susceptibility to be used is giveni by

H*X'"(r)Hd volume

(22)

H* Hd volume

TRANSMISSION LINE

Fig. 24-Schematic drawing for production of circUlarly polarizedresonance, with directional-coupler coupling to degenierate-mlodecavity used.

HORIZONTALLY POLARIZED CAVITYRESONANCE

PATH DIFFERENCE

IDEAL MAGIC TEE

VERTICALLY POLARIZED CAVITYRESONANCE

Fig. 25-Eqtivalent circtit of circuilarly polarized resonmmce.

RUBY.

Fig. 26-Possible conifiguration for unilateral reactance-compensated maser stage.

where x"(r) is a tenlsor. 'T'hetn Xeff" will be differenit forthe two directions of signal flow hopefully very differ-ent. Fig. 26 shows a possible geonmetry. Attempts tooperate such a system have not yet been successful be-cause the large number of simultaneous circuit adjust-metnts introduces great difficulty inl practice.

X. DiscUSSION

The experimenits reported here clearly demonistratethe effect of solid-state reactanices oni maser perfortmi-ance. Agreemenit with theory is generally satisfactorv.Numerical evaluationi of r in, %2dw fromi Fig. 21 gives460 Mc; this value is to be comiipared with a value de-duced fromii (17) and (18) anld single-cavitv imieasure-nments of 720 1\lc. The discrepanicy- is undoubtedly theresult of our nieglect of cavity losses, although depart-ture from the Lorentzian linie shape is a factor.A considerable increase in banidwidth has b)eei ob-

tainied over the mi-iaximl-uml value that is possible with it

sin1gle cavity.

No nioise-figure mieasuremiienits were made, but thesamiie effective nioise temperature shoul(d prevail as forotlher mnaser designis. Careful conistructioni should keepcavity losses low" eniough to avoidl serious Calverse nioiseconitributlionl

l\V S. Chanig anid A. E. Siegnia'i, "Characteristics of ruby formaser applicationis," Stanford Electronics Labs., Staniford, Calif.,Tech. Rept. 156-2; September 30, 1958. (Reprinted January 30,1959.)

16 M. Tinkham and M. W. P. Strandberg, "The excitationi of cir-cular polarization in microwave cavities," PROC. IRE, vol. 43, pp.

734-738; juine, 1955.

TWO-MODECAVITY

ff

1618 JItlly

192Kyhi, et al.: Negative L and C in Solid-State Masers

Some work carried out elsewhere has indicated thatimproved gain-bandwidth results from putting activeruby in the reactance-compensation cavity.'7 Somesimple numerical calculations made here indicate theopposite when the active material in both cavities hasidentical resonance behavior.

It is too early to say whether reactance compensationin the form described here will be useful in maser devel-opment. A small number of cascaded, reactance-com-pensated, unilateral cavity units, presumably includingsome unilateral, reverse-attenuation elements, andpossibly stagger-tuning of the paramagnetic resonance,can in principle match or exceed the performance ofother designs, including traveling-wave devices. Thedecision as to the actual circuit to be used will be madeon the basis of practical considerations. Here it must beadmitted that the adjustment of reactance-compen-sated circuits is critical. A slight change in cavity load-ing will produce either a single narrow-gain curve or twosharp gain peaks, as illustrated in Fig. 27. Additionalchange in loading in either direction leads to instability.The traveling-wave maser that is broad-banded bymeans of tapered magnetic field to provide tapered para-magnetic resonance"8 frequency may well be simpler tooperate. At present, solid-state reactance effects arenot ordinarily taken into account in traveling-wave maser

design.20 One of the advantages of traveling-wave masersis the possibility of tuning the operating frequency byadjusting the magnetic field without the necesssity ofretuning the structure. Use of reactance-compensationschemes, to achieve a flat gain-response, for example,would seriously interfere with this flexibility.

In any case, the inverted reactance behavior will be afactor in the analysis of any solid-state maser for whichlarge filling factors are used.

APPENDIX I

CIRCUIT REPRESENTATION OF PARAMAGNETICRESONANCE

The perturbation formula for initially lossless cavi-ties2'

AZ =jc,I d volume [H*AMH + g*AEg

1j2(23)

yields the desired formula immediately. Setting

Al=jXmaxf

1 +jT2Aw

and factoring it out of the integral, we obtain

d volume H*H'IAZ = 1,( Xmax I

1+ jT21AW iI12 (24)

SLIGHTLYOVERCOUPLED

A

CORRECTLYCOUPLED

B

SLIGHTLYUNDERCOUPLED

C

If we combine (24) with the impedance of the cavitynear resonance and assume that fd volume H*H/ I| 2 isconstant over the frequency range of interest (whichrequires use of the series resonance form for the cavity),we have an expression of the form

Z = R + jx(Aw) +( )OXmax - constant

UNSTABLE UNSTABLE

HT IMS (SMITH) CHART

Fig. 27-Impedance behavior of reactance-compensated maser,showing effect of coupling adjustment (the HTIMS Chart is used'9).

17 F. E. Goodwin and F. W. Moss, Hughes Aircraft Co., CulverCity, Calif. (private communication, November 12, 1959).

18 S. Okwit and J. G. Smith, "Traveling-wave maser with in-stantaneous bandwidths in excess of 100 Mc," PROC. IRE (Corre-spondence), vol. 49, p. 1210; July, 1961.

19 R. L. Kyhl, "Plotting impedances with negative resistancecomponents," IRE TRANs. ON MICROWAVE THEORY AND TECH-NIQUES, vol. MTT-8, p. 377; May, 1960.

which describes the equivalent circuit of Fig. 2. Thisderivation, although qualitatively correct, createsdifficulties. The use of a perturbation expression is validsince A/u is indeed small for all maser materials thatare now available. However, the constancy off| H| 2d volume/ I| 2 is open to question. The frequencyrange for the validity of the expression is not clear,and it is by no means obvious how to proceed in the case

of more complex cavity networks. By the method ofexpanding the cavity fields in the set of orthonormalmodes of the uncoupled cavity, several authors haveshown that the admittance or impedance of an air-

20 R. W. DeGrasse, E. 0. Schultz-DuBois, and D. Scovil, "Thethree-level solid state traveling-wave maser," Bell Sys. Tech. J.,vol. 38, pp. 305-334; March, 1959.

21 B. Lax, "Frequency and loss characteristics of microwaveferrite devices," PROC. IRE, vol. 44, pp. 1368-1386; October, 1956.Only trivial modifications of the equations are required.

(25)

1962 1619

PROCEEDINGS OF THIE [RE J

filled cavity can be expressed in the form2i'

constant,y = E (-w)

i (w w')(26)

in which thie comnplex wi represenits both the inifiniite set

of coimiplex niatural resonance frequencies of the cavitywith the inlput couplinig aperture short-circuited (open-circuited in the case of the corresponding impedatnceformula) anid also a dc ternm wi=O. The correspondinigequivalenit circuit is shown in Fig. 28.

For well-separated resonanices it is comiimon to truni-

cate the circuit, lumpinig the reactances of all termiisexcept one into Y0 (as showni in Fig. 29) to obtaini thesimple circuit that is valid in the nieighborhood of onie ofthe resonianices. If there are two closely spaced reso-

niainces in the cavity, one can similarly approximate theadmittanice behavior ini the neighborhood of the pairof resonianices by the equivalent circuit of Fig. 30(a)which, by algebraic nmaniipulationi, can be coniverted inltothe equivalent formns of Fig. 30(b)-(d). If it is possibleto choose a niew referenice plane in an input tranismiiissionlline, the circuits canl be reduced to the still simpler formiisof Flig. 30(e) anid (f).The extenisioin of the genieral expressioni (26) to include

other dynnamic variables, such as electric or imagnieticpolarizationis, is to be expected anid is demonstrated inanother paper.4 The presenit problem concerns the trun-cation of the expression for practical analysis.Out of the essentially infiniite number of degrees of

freedom introduced by the sample, how imlaniy must beretained to give an acceptably accurate equivalent cir-cuit? For ani empty cavity, all but onie or two of theresonanice frequenicies occur outside the frequenicy range

that is of interest. However, all of the resoniances ofthe material itself, associated with a particular para-mnaginetic transitioni, occur in the relevanit frequenicyinterval. Clearly the overwhelnming majority must rep-

resent spatial patternis of the maginetizatioin whiclh are

very complicated anid which, therefore, couple weakly,if at all, to the incident field. In order to make the prob-lem tractable we must introduce simplifying assuinp-

tions. We have assumed that

1) The material shall be homogeneous in the sense

that all portions show the same paramagneticresonanice frequency and linewidth. There is thena direct relation between the cavity-field patternand the polarization pattern. Clearly, a radicalviolation of this assumption leads to chaos. We are

not attempting to ainalyze tapered magnetic-fieldsituations.

22 J. C. Slater, "Microwave Electronics," D. Van Nostrand Cotim-pany, Inc., New York, N. Y., p. 74; 1950.

K. Kurokawa, "The expansion of electromagnetic fields in cavi-ties," IRE TRANS. ON MICROWAVE THEORY AND TECHNIQUES, vol.MTT-6, pp. 178-187; April, 1958.

2) The cavity-field patterni shall be assumned niot to beinifluen-iced bv the material. All maser mi-iaterialshave sm-all resonance sLusceptibilities. They arelarge eniough to ra(lically chanige network imll-pedanices, but thex represen-it only a small per-turbationi on ,u or e. Large niotnresonianit suscepti-bilities, such as the dielectric conistanit of the ruby,are conisidered to be part of the cavity systenLi

3) As the finial step we assert that the cavity-fieldpatterns remiiaini conistanit over the frequency ranigeof interest, beinig, in fact, just the molde patterni forthe particular isolatecl cavity resonianice inl ques-tioIn. This is a commuonlily imade assertion-i but is niotnecessarily a good oie.

Combiniing these three assumptionis, we conicludethat a single new variable is capable of describiing thestate of magnetic or electric polarizationi of the mnaterialin the frequency range of initerest. If the line shape isLorentzian, theni this new variable satisfies a harmonictype of equationi of miotioni and, when coupled with thecavity field equationis, produces just onie additionial nior-mnal miiode of the coupled systemii. The validity of theequivalen1t circuits in Fig. 30 follows directly. (Trhedifficulty with a Gaussiani linie shape is that the polari-zationi vector obeys nlo simuple equationi of ImlotioIn. TIhecase of a cavity filled comnpletely with a uniformii linlearimlediuIim anid havinig a Gaussianl resoinanlce cani be solvednumierically anid leads to anl infiniite set of niormual miiodesfor each cavity miiode.)

Thle niext higher approximationi is based on the factthat the fields in a cavityinear resonanice canl be rep-resenitedl with considerable accuracy by a linlear com-biinationi of two field patterns.4 For example, these ina)be the field patterns for a "short" at the window anid foranl "opet" at the window. For small- or moderate-sizedcoupliing winidows these patterns will differ primarily inthe regioni of the winidow, and thus another linear conm-biniationi suggests itself o-ne patterni being the differ-enice between the two mentioned above and essentiallythe pattern of the window friniging field, the other beingthe ordinary mode pattern, perhaps onie or the other ofthe modes mneintioned above.

Returninig to the argument for the filled cavity, wesee that to match this approximlationl we need two niewdynamic variables to describe the paramagnetic polar-izatioin. This is enough to describe the class of accept-able equivalent circuits. We give a physically intuitivesample in Fig. 31.The values of the parameters in the chosein equivalent

circuit can, in principle, be determinied from a solutionifor the normal modes of the system. It is mnuch betterfor the experimenter to relate the circuit parameters tothe physically measurable behavior of the structure,as was done in Section IVTFor this reason the choice,

Jtlll,)1620

KyhI, et al.: Negative L and C in Solid-State Masers

0 I 0 @.

0 ~ ~ ~ ~ @

Fig. 28-Conventional equivalent circuit for normal-mode expansion of cavity impedance.

Fig. 29-Approximate circuit near an isolated resonance.

(b)

(d)

(c)

(e)Fig. 30-Forms of approximate equivalent circuits near a pair of isolated resonances. (e) and (f) have the reference plane

chosen to simplify the representation.

CAVITY

SPIN EXCITATION BYCAVITY FIELD

Fig. 31-Equivalent circuit for single ruby-filled cavity, including the action of the window-coupling field on the ruby resonance.

1962 1621

y0 yI Y2

(a)

(f)


when possible, of a circuit that bears a direct similaritvto the structure is extremely desirable.Our analysis does Inot use the more exact circuit of

Fig. 31. If we had iintenitionially nmachined out the por-tioIn of the ruby in the regioni of the winidow frinigingfield, the introduction of the additional circuit elemiienitwould have been unnecessarv. This was niot done in theexperiments, although it appears that it mnight have beena good idea.

APPENDIX I I

EXTENSION OF FANO'S ANALYSIS TO THE -LAND - C SITUATION

XVe do nIot nieed to repeat the entire inathematicalanalysis of Fano,9 but it is necessary to point out wherechanges or modificationis must be made. He begani hiisanialysis by investigatiing the reflectioni coefficient F' (seeFig. 14). Thein he argued that '] = IF as a corollaryof the reciprocal inature of the inetwork where F anid rFare defined by the positive G equivalents of Figs. 13 anid14. The impedanice levels at the two ports of the niet-work do not influenice the result directly, if it is under-stood that the r's are definied in terms of the respectiveconductancles, +G onl the right anid a choseni Zo oni theleft.

Ani essenitial step is a contour integratioin of ln F'(s)alonig the w axis and around the right-half s plane atinfinity. For ordinary nietworks ReY>0 in the right-half s planie. Therefore, F'| <1 in the right-half splanie, in which F' catn have zeros but niot poles anidIn F' canl have logarithmiiic sinigularities wherever F' haszeros. It is necessary to detour the conitour initegralarounid these singularities, as shownin Fig. 32.A ty,pical result of this iintegration- is giveni by (27).

Other results cani be obtainied bx a simiiilar contour in-tegrationi of w2 In F', w4 In F', anid so oni. WVe shall nieedonly the second expressioni givenl in (28).

J In dw= A-4A 0-2 (27)I l 2 .

COMPLEXs PLANE

Cf)x

3

x

cZ2 In (28)

These are Fanio's equations (21) anid (22).'9 TI'he A's arecoefficieInts in a Laurent expanision of the argumenit InF' about inifinity. A major poinit of the theory is thatA,' is a fuiictioni of only the first eleimienit of a ladderdevelopment of the entire inetwork, startinig at the right,and -3x is a funictioni of only the first two elemenits,anid so onl. This is nlot at all obvious; the proof is givenby Fano.9 IThe X's are the values of s at the logarithmicsinigularities in the right half-plane. There are n1o resi-dues at these poinlts, but they formii bratnch points of atspiral-staircase niature for the logarithmiiic funlctioll.Successive Riemiiannii surfaces have values of the funic-tioii differinig by 27wj at all poinits. The contributionl ofthe detour out anid back to the conitour integral is there-fore 2rjX.

WATe are niow able to mlake the necessarv modificationlsin the theory.

I'he argumiienit for Fr = F'I is still valid, for rec-iprocity is niot related to the sigin of C. Chanigiing thesigin of G results in the inversioni of Fr I. The tnew factorin the probleii is the presenice of poles of rF itn the right-half s=jw planie. These initroduce additional logarithmlicsinigularities in the argumenit. Now, however, at thesebranlclh points the Reimannii surfaces spiral the otherway, and the conitributionl to the integral clhaniges signi.Eqs. 27 anid 28 becomne

fIn Ir dw= -(Al- - 2 E,Z + 2 IX,

co2 In FrIdw(A,, E A' + 2_E AP" °

(29)

(30)

Inisertinig expressions for A1 and .1;' for the circuit ofFig. 13, we obtaini (14) and (15).

R \CD

Fig. 32 Contour integral tised in the derivation of the gain relations.

1622 Jltlly

2X,.,

3


APPENDIX III

OPTIMUJM POLE-ZERO CONFIGURATIONIt can be shown that the low-pass circuit that we

have been using must have just one pole of F' in theright-half s plane because the network starts with asingle negative element. A plausibility argument may bemade as follows: Looking in at aa' (Fig. 14) in the ab-sence of the negative capacitance we see that Y willhave a positive real part in the right half-plane and thusneither poles nor zeros. Now by adding -sC to theadmittance, we add a simple pole at oo but none in theright half-plane. However, far enough to the right in thehalf-plane, the real part must become arbitrarily largeand negative. There must be at least one point on thepositive real s-axis where Y= -1, and at this pointthere is a pole of I". To show the absence of any addi-tional poles of F', we consider the locus of Re Y= -1 inthe right half-plane. This contour cannot cross theimaginary axis, and it cannot form loops or be multiplyconnected because there are no singularities. All that ImY can do along this locus is behave monotonically, goingto infinity at s= + oo. Hence, there are no other pointswhere Y= -1, and no other poles of I".

Critics may object that this is a strange argument to

make about a circuit that is only claimed to be a validrepresentation over a small frequency range, but thebehavior of the structure should approximate the be-havior of the idealized network in this frequency range.A number of zeros of r' are possible, however. We

should maximize (Xp-2Xz) in (15) and keep (p3-2Xz)below a fixed limit in order to keep the expression in(16) positive. Now

p3- Z X] = [Xp- z

*[p2 + XpE XZ) + (ZE X 2) + (X) -Z X .(31)

This expression is to be minimized while (Xp-ZXz) ismaximized, hence the last expression in brackets is to beminimized. All terms are positive-it takes a littlealegbra to show that (IX2) 3-Xz3 iS positive. Therefore,for best gain-bandwidth performance, 2X = 0, and thereare no zeros in the right half of the s plane.

ACKNOWLEDGMENTThe authors would like to express their appreciation

to Dr. T. Ogawa of Doshida University, Kyoto,Japan, for his assistance with these experiments.

CORRECTIONW. Eckhardt and F. Sterzer, authors of "Microwave-Carrier Modulation-Demodulation

Amplifiers and Logic Circuits," which appeared on pp. 148-162 of the February, 1962, issueof PROCEEDINGS, have called the following to the attention of the Editor:

OnOnOnOnOnOn

p-

P.p-

P.P.P.

148, table, n sec replaces Nsec.153, (55), w82Co2R8, replaces W12C02R,P2.154, line 2, (Gp'-G8p,')/Gp' replaces (Gp'-G,,'IGp).158, Table I, the last column does not belong to the MA 450/460 series.161, in the line above (82), RI replaces Rt.161, (82) and (83), v0n, v02,, c, RI replace z z'2, w Rz.

1)2)3)4)5)6)

1962 1623

Documents

Negative L and C in Solid-State Masers