Contains reports on three research projects.Lincoln Laboratory (Purchase Order DDL-B187)Department of the ArmyDepartment of the NavyDepartment of the Air Force under Contract AF19(122)-45

Adler, Richard B.

Haus, Hermann A.

Hilibrand, J.

Verma, J. K. D.

DSpace@MIT

XII. NOISE IN ELECTRON DEVICESProf. R. B. Adler Prof. J. B. Wiesner Dr. J. K. D. VermaProf. H. A. Haus J. G. IngersollA. THE CHARACTERISTIC NOISE MATRIX FOR EQUILIBRIUM NETWORKSAND MASER AMPLIFIERSIn the Quarterly Progress Report of April 15, 1956, pages 87-88, and in publishedwork (1) it was pointed out that a 2 X 2 matrix can be ascribed to a linear two terminal-pair network containing noise sources. This "characteristic noise matrix" expressedin terms of the rms open-circuit noise voltages E l and E 2 and the impedance matrix Zhas the formN =- (Z + z) EE (1)where E is the column matrixE=E 2and the bar over EE indicates an ensemble average.Alternatively, a characteristic noise matrix can be defined in terms of the scatteringmatrix S of the same network and the noise waves Pl and p2 that would be fed by thenetwork into matched loadsN' = (SS - 1) pp (2)whereand N' has the same eigenvalues as N defined in Eq. 1. The characteristic noisematrix N pertaining to a passive two terminal-pair network at an equilibrium tempera-ture T has the simple formN = - lkTAf = N' (3)Therefore, we might be led to suggest that the coincidence of the eigenvalues and thediagonal form of the N matrix are characteristic of equilibrium networks. But,*This research was supported in part by Purchase Order DDL-B187 with LincolnLaboratory, which is supported by the Department of the Army, the Department of theNavy, and the Department of the Air Force under Contract AF19(122)-458 with M. I. T.112(XII. NOISE IN ELECTRON DEVICES)conventionally, equilibrium is defined only for passive networks for which, as we seefrom Eq. 3, the eigenvalues are negative. Thus, coincidence of the eigenvalues andthe diagonal form of N can be interpreted as representing an equilibrium in the con-ventional sense only if the eigenvalues are negative.Recent suggestions that the maser may be considered as consisting of a mediumwith a negative conductance at a negative temperature provide an interesting interpreta-tion of the diagonal characteristic noise matrix with positive, coincident eigenvalues.Indeed, it is easy to show from Wittke's (2) work that for the maser consisting of alossless waveguide filled with a medium with inverted energy-level populations, we haveN = (S - 1) pp = IkTAf (4)The temperature T is so defined that it provides the proper occupation N2 of the higher-energy states and the occupation N 1 of the lower-energy states, according to an invertedBoltzmann distributionNN -exp(-tw/kT) (5)1with T < 0. In Eq. 4 the assumption is made that l << kT I .A corresponding relatively simple expression is found for the maser that containslossy material. We shall derive this expression in detail in a manner that brings outclearly the significance of the characteristic noise matrix as a measure of "equilibrium"(in a broader sense). Consider, first, a microwave transmission cavity or waveguidewith lossy walls at temperature T. The characteristic noise matrix of the cavity whenconsidered as a two terminal-pair network is given by Eq. 3. It is clear that the sameresult would be obtained with a cavity with lossless walls and filled with a medium ofconductivity 0- so as to give the same loss as the cavity in the preceding example. Thisstatement can also be proved by direct computation if we assign to every differentialrectangular box of volume dv, at the position T of the material, a mean-squareshort-circuit current-density fluctuation J in the x-direction, according to the NyquistxformulaJ 40-kTAf6() (6)x2 2and corresponding expressions for J and J . Here 5(F) is the three-dimensional Diracy z5-function. The correlation between current densities in two different directions iszero, e.g., J J = 0.xyEquation 6 is particularly convenient for generalization to maser analysis.According to Wittke's paper, a negative conductivity (o-m < 0) can be assigned to a vol-ume element occupied by maser material.113(XII. NOISE IN ELECTRON DEVICES)The short-circuit noise current i2 produced by the resulting equivalent negative con-ductance Gm(< 0) of the lossless maser fulfills the Nyquist formula2i = 4G kT Af (7)m mwhere T m < 0, and is defined by Eq. 5, with T = Tm. Equation 7 suggests that aNyquist generator can be assigned to every volume element dv of maser material at apoint F according to Eq. 6. ThusS= 4m- kT Af5(F) (8)x m mand so forth, for the other current components.Turning now to the maser with loss, we note that the conductivities a- (attributed toloss) and a-m (attributed to the gain mechanism) appear in parallel and therefore givethe net conductivity0t = - m + a- (9)The noise currents are also uncorrelated and add up to2J = 4Af6 (F) [-T + -T ] (10)and so forth, for the other current components. But, Eqs. 9 and 10 show that the maserwith loss may be considered as consisting of a material with conductivity a-t and tem-perature T t defined bya-tTt = a-mT + -T (11)The characteristic noise matrix then follows directly asN = -kTtAfl (12)If the maser has gain, a-t < 0, and therefore T t < 0. Thus, N has two coincident posi-tive eigenvalues. The optimum noise figure is given directly by the positive eigenvalueskTtAf of Nt (1 tF- kTAf _( ) (13)e kTAf G T (13)where T is the temperature of the circuit and the lossy part of the maser (walls), andthe subscript e refers to the generalizations of F and G in terms of "exchangeablepower" (3).If we use Wittke's notation, we have114(XII. NOISE IN ELECTRON DEVICES)- am a -t  a - a -t  a - at g t gN-1 exp(+ f/kTm) 1 + 4f/kTm  F F; G = GN m m e eand thus we obtain-kT N (14)m N1-NIntroducing expression 14 into Eq. 13 and using Eqs. 9 and 10, we obtaina1+ g( 1 hvagT -I= -1 )G kT N a - a1-which is in agreement with Wittke's expression.Thus we conclude that a maser is a device with simple equilibrium properties. Thismakes the analysis of maser noise performance through the characteristic noise matrixparticularly attractive.R. B. Adler, H. A. HausReferences1. H. A. Haus and R. B. Adler, Invariants of linear noisy networks, IRE ConventionRecord, Part 2, 1956, pp. 53-67.2. J. P. Wittke, Molecular amplification and generation of microwaves, Proc. IRE45, 291-316 (1957).3. H. A. Haus and R. B. Adler, An extension of the noise figure definition, Proc. IRE45, 690-691 (1957).B. LOW-NOISE 30-MC AMPLIFIERConstruction of a low-noise 30-mec amplifier has been completed. This amplifieruses Western Electric 416B planar triodes in a cascode circuit. It is well known thatthe cascode combination (grounded cathode-grounded grid) gives the best results of anyof the nine possible combinations of two triodes with respect to noise figure, gain, andstability. The 416B tube was designed for operation at 4000 me, but recently it hasalso been used at lower frequencies; for example, at 60 me for low-signal operation.With an optimum value of 1620 ohms for source resistance, the theoretical noise figure115(XII. NOISE IN ELECTRON DEVICES)of the amplifier is 1. 06, or 0. 26 db. The corresponding gain and bandwidth are 32 dband 6. 2 me, respectively.This "preamplifier" is followed by a "postamplifier" that uses 417A tubes incascode and a 6AK5 tube in the final stage. The measured noise figure of the post-amplifier is of the order of 1.4 db. It can be easily seen that connecting the two ampli-fiers in cascade does not appreciably affect the noise figure of the preamplifier. Thenoise figure of the whole amplifier is given by the relationF -1F =- F +F2- (1)12 1 Gwhere F 12 is the noise figure of the whole amplifier, F 1 is the noise figure of thepreamplifier, F 2 is the noise figure of the postamplifier, and G is the availablegain of the first cascode. In this relation, G is of the order of 1000, and hencethe noise contribution of the second term on the right-hand side in Eq. 1 is negligiblysmall.The measured noise figure of the amplifier as a whole is of the order of 0. 6 db.The noise figure as measured by existing commercial equipment, for example, byRadaNode of Kay Electric Company, is approximate, since the accuracy of theseinstruments is poor in the lower ranges. Hence, the radiometer method (1) is beingused. The amplifier will be used to measure noise in semiconductors.J. K. D. VermaReferences1. G. E. Valley, Jr. and H. Wallman, Vacuum Tube Amplifiers, Radiation LaboratorySeries Vol. 18 (McGraw-Hill Book Company, Inc., New York, Ist ed., 1948), p. 718.C. CHARACTERIZATION OF PROBABILITY DISTRIBUTIONS FOREXCESS PHYSICAL NOISEThis work was completed and is being published as Technical Report 276, which isbased on a thesis that was submitted to the Department of Electrical Engineering,M. I. T., Sept. 1956, in partial fulfillment of the requirements for the degree of Doctorof Science.J. Hilibrand116

Noise in Electron Devices

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Noise in Electron Devices

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