Contains research objectives.U. S. ArmyLincoln Laboratory, Purchase Order DDL B-00306U. S. Air Force under Air Force Contract AF19(604)-5200U. S. Nav

Haus, Hermann A.

Penfield, Paul L., Jr.

Rafuse, R. P.

DSpace@MIT

XXVI. NOISE IN ELECTRON DEVICESProf. H. A. HausProf. P. L. Penfield, Jr.Prof. R. P. RafuseRESEARCH OBJECTIVESThe study on noise performance of linear amplifiers and frequency-convertingsystems will be continued. The immediate objective is to describe the effect onnoise performance of the varactor series resistance in frequency-converting systems.H. A. Haus, P. L. Penfield, Jr., R. P. RafuseThis work was supported in part by Purchase Order DDL B-00306 with LincolnLaboratory, a center for research operated by Massachusetts Institute of Technologywith the joint support of the U. S. Army, Navy, and Air Force under Air Force Con-tract AFl9(604)-5200.237

Noise in Electron Devices

Contains research objectives and reports on one research project.Lincoln Laboratory, Purchase Order DDL B-00306U.S. ArmyU.S. NavyU.S. Air Force under Air Force Contract AF 19(604)-740

XIII. NOISE IN ELECTRON DEVICESProf. H. A. HausProf. P. L. Penfield, Jr.Prof. R. P. RafuseW. D. RummlerRESEARCH OBJECTIVESA detailed study of the noise performance of linear amplifying systems having mul-tiple input and output signal bands will be carried out. The object is to find the limitson the noise performance of such devices when the excitations in the signal bands arepartly or fully correlated.H. A. Haus, P. L. Penfield, Jr., R. P. RafuseA. WAVE REPRESENTATION OF AMPLIFIER NOISEA wave representation of noise in a linear two-port, which has been given by Bauerand Rothe,1 apparently has not been described in English. The representation has theadvantages that both equivalent noise generators are at the input to the amplifier (orlinear two-port), and that they are uncorrelated.1. Previous RepresentationsSeveral schemes have been used to represent noise at a given frequency f in a lineartwo-port.2 By Thivenin's theorem, the noise generated internally by the two-port canbe represented by two noise voltage sources in series with the input and the output.These two generators are, in general, correlated, and four real numbers are necessaryto specify the noise properties of the two-port; these may be the mean-square valuesof the two voltages and their complex correlation coefficient.,,,, " e NOISELESS OUTPUTFig. XIII-1. The Rothe-Dahlke noise model for a noisy two-port hastwo correlated generators at the input. The advantageis that the noiseless part can be disregarded in calcu-lating the noise figure of the amplifier.Unfortunately, in this representation there is one equivalent generator at each port.This work was supported in part by Purchase Order DDL B-00306 with LincolnLaboratory, a center for research operated by Massachusetts Institute of Technologywith the joint support of the U.S. Army, Navy, and Air Force under Air Force ContractAF 19(604)-7400.(XIII. NOISE IN ELECTRON DEVICES)For many purposes (particularly calculating noise figures of amplifiers) it is advanta-geous to have both generators at one port, the input. Rothe and Dahlke 3 have shown howto do this, with the equivalent circuit of Fig. XIII-1. The two generators are correlated;one can define the four real numbers associated with this representation as the mean-square values of en and in, and their complex correlation coefficient p or, alternatively,the quantitiesenl 2R = n (1)n 4kToAf. 2G = 4kTOA (2)andi e_n n (3)lin 2 len 2The correlation coefficient p is less than or equal to 1 in magnitude.The advantage of this representation is the ease of calculating the amplifier noisefigure. The noiseless amplifier shown in Fig. XIII-1 does not affect the noise figurebecause it treats all noises and signals identically. The noise figure expression is,therefore, independent of the properties of the noiseless amplifier except insofar asthey help to determine the quantities R n, G , and p. If the source impedance is Z =2, 3 . sR + jXs , the excess noise figure ise +Zs in 2F-l= n sn4kT AfRRn+ I Zs2 Gn + 2R-Gn nRe (pZs)R (4)sAs Z s is varied over all values in such a way that R s is positive, the noise figure goesthrough the minimum value(F-1)min = 2,G ( 1 - (Imp) + Re p). (5)min n nThis minimum occurs for the particular value of source impedance Zs, op tRs, opt + jXs, opt(XIII. NOISE IN ELECTRON DEVICES)IZ I oJ 2 = n (6)Zs, opt2 GnXs,opt =  Im p. (7)nIt has been suggested2 that the noise characteristics of a linear two-port, such asan amplifier, be specified by the four numbers (F-1)mi n , R n , and Zs, opt (Zs, opt is acomplex number). Probably such a specification is more convenient than simply R n,Gn, and p, since the quantities of engineering interest are (F-1)min and the optimumsource impedance.2. Wave RepresentationAn alternate representation, suggested by Bauer and Rothe, uses wave, or scat-te,ring4 variables, and retains the advantages of the Rothe-Dahlke representation. Ithas the further advantage that the two noise sources are uncorrelated, so that theexpression for noise figure is particularly simple.5We define the noise wave generators an and b at the input of the amplifier, in termsof the parameters of the Rothe-Dahlke model,e +Zia n v n (8)n 2 e -Ze -Zib n vn (9)n 2NfReZwhere Z is a normalization impedance. Thus, if the scattering matrix of the two-portvis S, we obtainb 1 -b n S L S12 a 1 + a= . (10)b2  S21 S22 a 2An equivalent circuit of this representation is shown in Fig. XIII-2. The wave genera-tors are represented by directional couplers and ordinary sources; this schematic isaccurate in the limit of small coupling to the line and large strengths of the sources.Note that this "transmission line" has a complex "characteristic impedance," Z .The normalization impedance Z used in Eqs. 8 and 9 is chosen so as to makeV(XIII. NOISE IN ELECTRON DEVICES)a and b uncorrelated,n na b = 0.n n (11)It happens that this impedance is precisely Zs, opt given in Eqs. 6 and 7. This pointis of importance: If the optimum source impedance is used as the normalization imped-ance in defining noise wave generators from the Rothe-Dahlke noise generators, theINPUTProposed wave model of a noisy two-port consists of anoiseless two-port with uncorrelated wave generatorsat the input. The noiseless two-port can be disregardedin calculating the noise figure of the amplifier.resulting wave generators are uncorrelated.is entirely the result of this fact. The wavethat is, only when R n = 0, Gn = 0, or p = ±j.zero and is achieved with a reactive source.sary if it is properly placed.The strength of an defines a temperaturenThe simplicity of the formulas given belowrepresentation here fails when R = 0;s, optIn each of these cases, (F-l)min equalsOnly one equivalent noise source is neces-lan 2Taa k~fand similarly, the strength of bn defines a temperature Tb.lbn12b kAfIn terms of the Rothe-Dahlke model, these areT = 2T Ni ( 1 - (Im p + Re p)a 0 nn(12)(13)(14)Fig. XIII-2.(XIII. NOISE IN ELECTRON DEVICES)Tb = 2T 0 R-n( 1 - (Im p) 2 - Rep) (15)so that Ta is related to the minimum noise figure of Eq. 5:Ta = T 0 (F-1)min' (16)The expression for the noise figure F - 1 is very simple when written in terms ofthese wave variables. Consider the amplifier run from a source with impedance ZsR s + jX s . The reflection coefficient of the source, F s , isZ -Zss V (17)Z +Zs vwhen the wave variables are defined with Z as the normalization impedance. Then thevnoise figure is simplyT a + Tb slT 0 (I-_I' 2)T T +T b  ZZv 2- a +a T  s___S + aT (18)0 0 4R s Re (Z )The choice of source impedance to minimize the noise figure, from Eq. 18, is obvi-ously the value that sets F s to zero, or simply ZV. The minimum excess noise figureso achieved is Ta/T 0 . The temperatures Ta and T b can be easily measured, at leastin principle, even at microwave frequencies.It has been suggested 2 that Ta [or (F-1)min]' Zv and one other parameter (say, Rn)be used to characterize noisy two-ports, since these are the parameters of most engi-neering interest. Perhaps a more meaningful fourth parameter to specify would be Tb,or a quantity related to it, rather than R .The author is indebted to W. E. Dahlke, of Telefunken g.m.b.h., for calling hisattention to the paper by Bauer and Rothe.P. Penfield, Jr.References1. H. Bauer and H. Rothe, Der aquivalente Rauschvierpol als Wellenvierpol,Archiv der Elektrischen Ubertragung, Vol. 10, pp. 241-252, 1956.2. IRE Subcommittee 7.9 on Noise, H. A. Haus, Chairman, Representation ofNoise in Linear Twoports, Proc. IRE 48, 69-74 (1960).(XIII. NOISE IN ELECTRON DEVICES)3. H. Rothe and W. Dahlke, Theory of noisy fourpoles, Proc. IRE 44, 811-8 18(1956).4. H. J. Carlin, The scattering matrix in network theory, Trans. IRE, Vol. CT-3,No. 2, pp. 88-97, June 1956.5. D. C. Youla, On scattering matrices normalized to complex port numbers, Proc.IRE 49, 1221 (1961).

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Noise in Electron Devices

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