Contains research objectives and reports on one research project.Lincoln Laboratory, Purchase Order DDL-B222U.S. Air Force under Air Force Contract AF19(604)-520

Adler, Richard B.

Rafuse, R. P.

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XXIV. NOISE IN ELECTRON DEVICESProf. R. B. AdlerProf. H. A. HausProf. J. B. WiesnerR. P. RafuseRESEARCH OBJECTIVESWhile our work on the general problem of noise in linear networks came to a con-clusion with the publication of Technology Press Research Monograph No. 2, "CircuitTheory of Linear Noisy Networks," by H. A. Haus and R. B. Adler (published jointlyby the Technology Press of Massachusetts Institute of Technology and John Wiley andSons, Inc., New York, 1959), related special problems continue to emerge. These areconveniently handled by the general methods developed in the monograph, and we shallcontinue to present applications of the general theory to cases of current interest asthey arise.R. B. AdlerA. MEASUREMENT OF ABSOLUTE NOISE PERFORMANCE OF PARAMETRICAMPLIFIERSAny measurement procedure that is used to determine the absolute noise perform-ance of a negative-resistance amplifier must necessarily include the behavior of theamplifier in cascade. For practical considerations (bandwidth, nonlinearity, and soon), one of the members of the cascade usually is chosen to be a unilateral, and inher-ently noisy, amplifier of a class whose normal performance is not as good as thecapabilities of the parametric amplifier. This situation is, in fact, encountered if anegative-resistance amplifier is added to an existing system in order to improve itsnoise performance.It should be noted that devices with negative-real input and/or output impedances,as well as positive-real devices driven from sources with negative-real impedances,are capable of being characterized and analyzed within the framework of exchangeablegain, noise figure, and noise measure (1). A suitable model for a generalized unilat-eral noisy amplifier (2, 3) is shown in Fig. XXIV-1. The excess exchangeable noisefigure for this representation is(F-1) = (R /R ) + G R + 2p(R G n)/2 (1)whereThis research was supported in part by Purchase Order DDL-B222 withLincoln Laboratory, a center for research operated by M. I. T., which is supported bythe U.S. Air Force under Air Force Contract AF19(604)-5200.205(XXIV. NOISE IN ELECTRON DEVICES)e in nP=_ _2 .2 2Figure XXIV-2 illustrates the behavior of (F-1) versus source resistance. The twominimum magnitude excess noise figures are(F+ -1) = 2(1+p)(RnGn)/2 (2)(F -1) = -2(1-p)(R G )/2 (3)o g, opt (R/Gnll/2for Rg opt = ±(R/G)/, respectively. Penfield has derived a simple model for thesignal-frequency behavior of a lossy varactor (4, 5). He also has shown that the bestperformance is obtained with the diode series resistance as the total idler circuit load.Figure XXIV-3 gives two models for the signal-frequency behavior of a lossy varactorfor whichm = Penfield modulation ratio (m < 1/Tr)maxOc = diode cutoff frequency = 1/RsCminw. = idler frequencyWs = signal frequencyR = series losssC = average capacity2 2m wR _ c R (4)d w. C s1 s2 2m Cc 22R 2C2W. O s s oG s 2m c22 c1 + -2 C2 m c Rs o s .W s1 s(F-l) Co T. O T RM s 1 P i se - (l1/G) wi T0  wi T R +Rd dThe best value of Me is obtained when Rd = Rd, max (strong pumping).206e n = 4kTo RnAfR ' =4kTGn Af [T]=NOISE NETWORK NOISELESS NETWORKNOISY AMPLIFIERFig. XXIV-1. Characterization of a noisy amplifier. Any linear noisy two-port canbe separated into a noiseless two-port preceded by a pair of correlatednoise generators.(F- I)+ Rg(F_- I)Rg,opt+ /G nFig. XXIV-2. Exchangeable excess noise figure. The two branches correspondingto ±R are particularly convenient when cascading. When the outputgimpedance of the preceding stage is negative, its exchangeable gainis also negative, and a definite contribution to over-all noise-figureresults.RdRsCo-00OIRFig. XXIV-3. Signal-frequency varactor models.A parallel equivalent circuit for thePenfield model is derived.207+C(XXIV. NOISE IN ELECTRON DEVICES)NEGATIVE - RESISTANCEAMPLIFIER enL -G0  s R yNI iIDEAL IDEALSECOND STAGEFig. XXIV-4. Model of a negative-resistance amplifier in cascadewith a noisy second stage. A signal-frequency modelof a parametric amplifier is cascaded with a general-ized noisy second stage. Two separate over-all noisefigure expressions are necessary for the two regions.Figure XXIV-4 presents a typical signal-frequency equivalent circuit for a "two-tank" parametric amplifier. The behavior of the cascade noise figure versus Rg is-1characterized by two regions for R (N2G, max) For R > 1/N 2 Gg \o, max ~ g o,max(F12-1)G = OR + (F/R ) + A (7)o, opt gand for R < /N 2 Go, max(F12-1)G = 2R + (/R g) + X (8)o, max gwhere(FZ-1)(F12 -1) = (F1 -1) +e, 1It is apparent from the expression for noise measure of any cascade (1),G -1M =M + AM (9)el2 el e\G - 1e12that if Me, 1 is optimized by making Rd = Rd, max' then Me, 1 2 is optimized by makingAMe a minimum and Ge, 1a maximum. This may be achieved in practice by pumping thevaractor as hard as possible (achieving m max), adjusting the turns ratio of the firstmaxtransformer, N, in such a way that N2R (Go, )- (G oo), and simultaneouslyadjusting the second transformer turns ratio 'yN - oo, with the result that the impedanceseen by the second stage is Rg, opt (from Fig. XXIV-2). The over-all exchangeablegain will be infinite in the limit, and Me, 12 reduces to (F 1 2 -1).The calculated values of 0, F, A, 4, 0£, and X necessary to predict the cascade208(XXIV. NOISE IN ELECTRON DEVICES)noise figure for non-optimum conditions are:2 G n T. RG T. i a(R )2S= (1-p 2 n P 1 N2 aR 2R3 n si o s+S s s/ 42 n T 2y 1 o y 1 o 2y RI n n aR 4N2Rw2 2 4\aR - 2a Rs sY 2 Ry n /2(coT.)1 s i2 R n W. T 04y R n1+P si To [yN] RnoTP il R. T 2 soNP i s1+ 1i o [y N]2R nT. Co FaRZ 2pR ( )/Js T P 1 2 ns 1/W. + -- aR + (G /R1P i sWiT o (yN) 2 RwhereW2 2coCsol + W2C 2 R2SOS(10)(11)(12)These values are calculated under the constraint that G is always adjusted (by varyingthe pump amplitude) so that the over-all noise figure is a minimum for any Rg (providedthat G is not reached). The other parameterso, max2n(13)Co T.2 = s - N2Go. T1 oC T (2+ P 2 Gmax 1 oT max+ aR N 2 - aR2N2Gs s Oma x+ 2 N4R G2n omax+G - 2p(R G n ) 1/2 N 2 Gn nnX = 2p(R G )1/2 - 2(yN) R Gmax209ToW. Ti\ 1 o/max(14)(15)a T.+ (G n/R n)1 I2 n2 . Ty 1 o12_ Ay=co210-p0 i/N 2 Go,maxFig. XXIV-5. Theoretical behavior of cascade noise figure versus R and y.g2.01.51.0 L0 100 200 300Fig. XXIV-6. Excess noise figure for an experimental second stage; (F 2 -1) versussource resistance was measured for a typical 108-mc receiver. Thevalues of R n , G n, and p are calculated from the measured values, andthe curve is drawn from these values.(F 12 -I )2.0 o MEASURED0 \ R:I/N2 GR / 2 o,m ax1.0 - I'100 200 300Fig. XXIV-7. Theoretical and experimental behavior of over-all excess noisefigure. The theoretical noise-figure performance for an experi-mental cascade is compared with the measured performance.210(XXIV. NOISE IN ELECTRON DEVICES)-1hold for values of Rg < (N2G max)1Illustrated in Fig. XXIV-5 is the behavior of (F 1 2 -1) versus y and Rg.tical reasons, it is necessary to know the extent to which the optimizationshould be continued for a given varactor and a given second stage. These(Eqs. 10-15) are all that are necessary.For prac-procedurerelationsTable XXIV-1.= 65.7= 1.4 X 10 - 2= -1. 28Numerical values of A,elements characterizingfier, and those derivedvalues for the six basicfor two values of y.r = 4.5 X 10-20 = 5. 3 X 10-3A = 0. 140, r, X, £2, and (D. From the necessaryan experimental negative-resistance ampli-from the second stage, a set of numericalparameters are derived. These are showny=2i = 252.8 r = 4.5 X 102£ = 3.0 X 10 - 2  0 = 1.32 X 10 - 3X = -5.33 A = 0. 14Figures XXIV-6 and XXIV-7 and Table XXIV-1 contain data on a specific experi-mental cascade. It is apparent that (F12-1)best is nearly achieved by a combination ofR = 100 ohms (with N2 = 50) and y = 2. It is interesting to note that, if the signal cir-gcuit is loaded too heavily (N small), a much poorer performance results. The agree-ment between theory and experiment indicates that the models chosen are adequate; theoptimization procedure is evident from the curves of Fig. XXIV-7.In conclusion, it can be shown that, as y becomes very large, r, 0, and A reduce too T.Rp 1 s.T N210co T.N 2  i (aR aR)W. T w. T s1 0 1 0These values result in(16)(17)(18)oT OT 41/2(Fs P (aR2-a2 R4F12 op t o. T 0. T saR -a1 0 1 0 (19)o-1/2for R - N2(a-aZ 2)] R 2g, opt I s/R. P. Rafuse211y=l(XXIV. NOISE IN ELECTRON DEVICES)References1. H. A. Haus and R. B. Adler, Circuit Theory uf Linear Noisy Networks (TheTechnology Press, Cambridge, Mass., and John Wiley and Sons, Inc., New York, 1950).2. A. G. Th. Becking, H. Groendijk, and K. S. Knol, The noise factor of 4-terminalnetworks, Philips Research Reports 10, 349-357 (1955).3. H. Rothe and W. Dahlke, Theory of noisy four-poles, Proc. IRE 44, 811-817(1956).4. H. A. Haus and P. Penfield, Jr., On the noise performance of parametric ampli-fiers, Internal Memorandum No. 19, Energy Conversion Group, Department of Elec-trical Engineering, M.I.T., Aug. 11, 1959 (unpublished).5. P. Penfield, Jr., Interpretation of some varactor amplifier noise formulas,Internal Memorandum, Microwave Associates, Inc., Burlington, Massachusetts, Sept. 1,1959 (unpublished).212

Noise in Electron Devices

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