Contains reports on one research project.Lincoln Laboratory, Purchase Order DDL-BZZZDepartment of the ArmyDepartment of the NavyDepartment of the Air Force under Contract AF19(122)-45

Haus, Hermann A.

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IX. NOISE IN ELECTRON DEVICESProf. R. B. Adler Prof. J. B. Wiesner Dr. J. K. D. VermaProf. H. A. Haus J. G. IngersollA. LANGEVIN EQUATION FOR MICROWAVE FERRITE WITH THERMAL NOISEThe Langevin equation for a ferrite at thermodynamic equilibrium is presented. Thenature of the statistical "driving term" in this equation is such that it seems reasonableto retain it unchanged, even in the presence of the pumping excitation which would beused in a parametric ferrite amplifier.The equation of a ferrite with loss (1), with the use of the Landau-Lifshitz represen-tation of the loss, isdM aydt I-y (Mx H) M X (MX H) (1)dt MINow suppose that the ferrite is exposed to a time-independent magnetic field for pro-ducing the saturation magnetization M o . The thermal agitation in the ferrite causes adeviation of the magnetization vector M from alignment with M . This fluctuation canbe taken into account by supplementing Eq. 1 with a random torque T (in units of coul/m)dM lay• -dt YI(IX H) - - M X (MX H)+T (2)t IMIDenote by T(r,w) the complex Fourier amplitude of frequency w at the point r. Thetorque at the point r' is denoted by T(r',0). Then, we can show that the correct equi-librium fluctuations of the magnetization vector will be produced if the mean-squarefluctuations of the three Cartesian torque components areTx(7,w) Tx( ',w) = T (F, w) T y(', w) = T (r, ) T* (Tý' w)x x y y z z= 41ayM kTAf 6(i - ') (3)and mutually perpendicular torque components are uncorrelated. Here, 5(r - T') is adelta function, indicating that the torques in adjacent volume elements are uncorrelated.It is noteworthy that the spectrum of the torque distribution indicated in Eq. 3 isuniform. Accordingly, the fluctuations of the torque are extremely rapid compared withany frequency for which Eq. 1 applies.* This research was supported in part by Purchase Order DDL-BZZZ 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.141(IX. NOISE IN ELECTRON DEVICES)The proof of Eq. 3 is accomplished by reduction of Eq. 2 to small-signal form. Forthe small-signal complex vector ml of the magnetization we obtain the relationjojom = jJ4oX • h + JM (4)where h is the small-signal complex magnetic field vector, JM is the magnetic noisecurrent density with a Fourier amplitude that is linearly related to T(r,w). Then, wemake use of the fact that, at equilibrium, the magnetic noise current density of a mediumcharacterized by an equation in the form of Eq. 4 must satisfy the relation (see QuarterlyProgress Report, July 15, 1957, p. 113, cf. Eq. 6, which was developed for an isotropicmedium)JM(r, ) JM(r', w) = Zjw4o(X - X ) kTAf 6(r - r') (5)in which the dagger indicates the complex-conjugate transpose of the tensor X. By com-paring the explicit expression for JM(r,w), as obtained from Eq. 2, with Eq. 4 we findthat the mean-square fluctuations of the torque components are perpendicular to thesaturation magnetization M . The component of T parallel to M does not affect theo osmall-signal equilibrium fluctuations of M, and therefore cannot be obtained from suchconsiderations. However, by symmetry, it is reasonable to postulate a component of thetorque in the direction of M that is equal to the component perpendicular to M . Theo oresulting self-fluctuations and cross fluctuations of T(r,w), in Eq. 3, are more easilygeneralized to parametrically excited ferrites.We can develop a theory of noise in parametric ferrite amplifiers with the assumptionthat the equilibrium distribution of the torque, in Eq. 3, is unaffected by the applicationof the pump. This is a reasonable assumption in view of the statistical independenceof the equilibrium torque at successive time intervals as indicated by its uniform fre-quency spectrum, and its statistical independence in two adjacent volume elements asindicated by the delta function in Eq. 3.H. A. HausReferences1. N. Bloembergen, Magnetic resonance in ferrites, Proc. IRE 44, 1259-1269 (1956).142

Noise in Electron Devices

Contains a report on a research project.Lincoln Laboratory (Purchase Order DDL-B222)United States Department of the ArmyUnited States Department of the NavyUnited States Department of the Air Force (Contract AF19(604)-5200

XXII. NOISE IN ELECTRON DEVICESProf. R. B. AdlerProf. H. A. HausProf. J. B. WiesnerWhile our work on the general problem of noise in linear networks was declaredclosed in Quarterly Progress Report No. 52 and a Research Monograph is in the proc-ess of being published, related special problems continue to emerge. These are con-veniently handled by the general methods developed in the monograph, and we shallcontinue to present applications of the general theory to cases of current interest.A. GENERAL NOISE ANALYSIS OF PARAMETRIC AMPLIFIERSIn three-frequency parametric amplifiers that use one or more solid-state diodesfor the nonlinear energy-storage element, the major contribution to the noise seems tobe the thermal noise associated with the linear dissipative parts of the equivalent circuitANTENNAT. p " s LOSSLESS LINEAR p +sPARAMETRICI (3) SATISFYING : (4) ITo s M-R RELATION Wp- WS ToINPUT (1) (2) OUTPUTFig. XXII- 1. Small-signal schematic diagram of parametric amplifier.and the noise contributed by the idler termination. Under such conditions the amplifiernoise performance can be analyzed by using a scattering matrix representation similarto the one that was successful in the noise analysis of electron beam amplifiers (1, 2).From this approach, information for the necessary steps leading to noise optimizationcan be obtained with little effort. The representation is easily extended to the analysisof parametric amplifiers with more than one idler frequency, frequency-up-or-downconverters, and parametric amplifiers under double-sideband operation.A system under a large-signal pumping excitation at frequency w and a small-signal excitation at frequency ws leads to excitations at frequencies mwp + ws, whereThis research was supported in part by Purchase Order DDL-B222 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(604)-5200 with M.I.T.206(XXII. NOISE IN ELECTRON DEVICES)m may range over all positive and negative integers. We shall follow usual analysesby assuming that the loss of the system can be separated from the pumped nonlinearpart of the system and can be represented by linear circuit elements. The losslessnonlinear part of the system is assumed to satisfy the Manley-Rowe relations. Thelinear operation of a three- (signal, idler, pump) frequency, negative-resistance para-metric amplifier with the idler circuit coupled to the antenna and with a parasitic exci-tation at the upper sideband wp + os can then be represented by the circuit of Fig. XXII-1.Terminal pair (1) is the input, (2) the output, the 1-ohm resistors at terminal pairs (3),(4), and (6) represent the losses at signal, idler, and parasitic upper-sideband frequen-cies at room temperature To , terminal pair (5) represents the idler coupling to theantenna with a noise at the idler frequency represented by the temperature Ta. Thenormalized incident waves a (al,a ,... a 6 ) and reflected waves b(b l , b2 ,... b ), normal-ized to transmission lines of 1-ohm characteristic impedance, are related byb = Sa (1)Here, the lower sideband o - w is characterized by the complex amplitudes at the neg-p sative frequencies (- co + w ). The network has to satisfy the Manley-Rowe relation, andthereforebtPb = atPa (2)where (f) indicates a Hermitian conjugate andP = diag 1 1 1 1 p1 (3)P = da , ,' , (3' +( s s p s p s p sCombining Eqs. 1 and 2 and noting that a may be chosen arbitrarily, we obtainSPS = p (4)Equation 4 multiplied by SP - 1 from the left and S-1p - 1 from the right givesSP-1St = p- 1  (5)whereP- 1 =diag (,ws ,s , s - +  , - +  , + )The 22 element of the matrix equation, Eq. 5 isW(S Z I 2 S 2 + IS2 3 1) -(W p s) (IS24 22+ S251Z2+ (wp + s) IS2 6 2 =s (6)The noise figure of the parametric amplifier is207(XXII. NOISE IN ELECTRON DEVICES)lbz iF-l-=lb 2 2whereiS 2 3 a 3 + S 2 4 a 4+ S2 5 a 5 + S2 6 a 6 12Ib 2 t= 2112 kT 0 ofNoting that all crosscorrelations of the a's are zero andj3 2 = a4 2 =a 5 2 = kTaAfand using Eq. 6, we haveW To +TC -w Tp s ap ws Toa 6 2 = kTo0 AfS -o TS -p s o+2 1S26] 21S$21T -To aToS 24$21Equation 12 shows the harmful effects of the parasitic upper sideband (S26), the loss at thesignal frequency (S23), and the loss at the idler frequency (S24). The best noise figure isachieved if none of these noise contributions enter into Eq. 12. Furthermore, noting thatIs2112=G1 - IS2 2 2  ewhich is the exchangeable gain (3), we have for the optimum noise figureF - 1 = s a 1 (13)S-wT G (13)p so eThe optimum noise measure under any input termination (3) isw TM S a(14)e - T (14)p s o208b2 b i =and(10)(11)+ 1 S23$21(12)(XXII. NOISE IN ELECTRON DEVICES)and therefore (see ref. 3) the excess noise figure at high gain of any over-all twoterminal-pair amplifier employing the parametric amplifier in conjunction with other"noisier" amplifiers cannot be better than Eq. 14.H. A. HausReferences1. H. A. Haus and F. N. H. Robinson, The minimum noise figure of microwaveamplifiers, Proc. IRE 43, 981-991 (1955).2. L. D. Smullin and H. A. Haus, Editors, Noise in Electron Devices (The Tech-nology Press, Cambridge, Massachusetts, and John Wiley and Sons, Inc., New York,1959).3. H. A. Haus and R. B. Adler, Optimum noise performance of linear amplifiers,Proc. IRE 46, 1517-1533 (1958).209

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Noise in Electron Devices

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