Transistor Noise in Circuit Applications

PROCEEDINGS OF THE I.R.E.

Transistor Noise in Circuit Applications*H. C. MONTGOMERYt

Summary-Linear circuit problems involving multiple noisesources can be handled by familiar methods with the aid of certainnoise spectrum functions, which are described. Several theorems ofgeneral interest dealing with noise spectra and noise correlation arederived. The noise behavior of transistors can be described bygiving the spectrum functions for simple but arbitrary configurationsof equivalent noise generators. From these, the noise figure can becalculated for any desired external circuit. Illustrative information isgiven for a number of n-p-n transistors.

INTRODUCTIONA SPECIFICATION of the properties of any trans-

mission device to be complete must includeinformation regarding the way in which it con-

tributes background noise to a signal it transmits. Ingeneral, the noise behavior of such a device depends oncertain properties of the device itself, and also on thecircuit impedances with which it is terminated. In ap-plications where noise is an important factor in per-formance, it is usually of interest to be able to determinethe circuit conditions leading to the optimum signal-to-noise ratio.Simple circuit problems involving sources of noise

can often be handled on an intuitive basis by regardingthe noise as a group of sinusoids closely spaced in fre-quency. When the circuit contains several independentsources of noise the contribution of each can be calcu-lated independently and then combined on a mean-square basis, to give the total noise. However, in morecomplicated problems, especially those involving partlycoherent noise sources, the intuitive method becomesunwieldy and a systematic approach is desirable.The noise behavior of a transistor can be represented

by a pair of fictitious noise generators associated withthe input and output terminals, according to a theoremdue to Peterson and described later in this paper. Sincethese noise generators need not have the actual internalconfiguration of the noise sources, they are in generalpartly coherent. The noise produced by the transistorin connected circuits may be calculated from the proper-ties of the fictitious generators. This is the sort ofproblem which, in the general case, is not easy to workout by intuitive methods, but which is very straight-forward with the systematic approach to be described.The method of handling linear circuit problems in

noise will be developed first. It consists essentially ofdefining certain functions for describing noise currentsand voltages which can be manipulated in circuit equa-tions in the same manner as the complex steady-statecurrents and voltages of familiar circuit theory. This isfollowed by a number of useful theorems and relations

* Decimal classification: R282.12. Original manuscript receivedby the Institute, August 11, 1952.

t Bell Telephone Laboratories, Inc., Murray Hill, N. J.

based on this type of analysis, which are of a quitegeneral nature. The final part deals with specific appli-cations to transistor problems, including methods ofrepresenting noise behavior, conversion formulas, andnoise data on a limited number of junction-type tran-sistors.Throughout the paper it will be assumed that we are

dealing with stationary noise processes, by which wemean that the statistical properties which are used todescribe them do not change with time, except for sta-tistical fluctuations which tend to decrease as the timeinterval of averaging increases. The discussion will belimited to linear circuit problems. Except as occa-sionally noted, there is no implied restriction to Gaus-sian noise processes.

1. DEFINITION OF IMPORTANT NOISE FUNCTIONSThe functions required to describe the behavior of a

system of noise currents and voltages are a powerspectrum for each noise and a cross spectrum for eachpair. The power-spectrum concept is quite familiar. Thecross spectrum is less well known, although it is gen-erally recognized that the result of adding two noisesdepends in an important way on the degree of coher-ence between them. The coherence properties are con-veniently described by the cross spectrum. As a basisfor establishing the important properties of these spec-trum functions, we shall give an analytical derivationstarting with the time function representing the noisedisturbance.

Suppose y(t) is the time function which represents anoise current or voltage. We may define a Fourierspectrum over any finite time interval by

S t+TSl(f) = (2T)--1/2 y, (t) e7i2,rf 'di. (1)

By a suitable smoothing along the frequency axis (dis-cussed in the Appendix) we obtain a complex functionof frequency the square of whose magnitude is thepower spectrum

P1(f) = aver Sl(f) 12 = aver Si*Si, (2)where the star denotes the complex conjugate. The angleof the S function is distributed at random, and does notconstitute a significant description of the properties ofa single noise.From this definition it is seen that the term power

spectrum is to some extent a misnomer, since it is adescription of the statistical properties of a noise cur-rent (or voltage). However, it is the power which wouldresult if the noise current (or voltage) were applied to a

1952 1.461


resistance of one ohm. The appropriate units are am-peres squared (or volts squared) per unit bandwidth.The cross spectrum between two noise currents is de-

fined asP12(f) = aver S*S2, (3)

where suitable smoothing of both magnitude and anglealong the frequency axis is assumed. This is a complexfunction of frequency, whose existence depends onthere being a systematic phase relation between SI andS2, which in turn implies an element of coherence be-tween yi and y2.The cross spectrum can be measured by sending the

two noise currents through identical narrow-band filtersand applying the outputs to a device, such as a dyna-mometer, which indicates the product of the instantane-ous values. The average reading of this device is the realpart of the cross spectrum at the frequency passed bythe filters. The imaginary part is found by shifting thephase of one of the noise currents by 90 degrees. Al-ternatively, the magnitude and phase can be deter-mined directly by adjusting the phase of one currentuntil a maximum reading is obtained from the productdevice. The equivalence of this procedure to the ana-lytical definition may be seen by resolving correspond-ing long portions of each noise current into Fourierseries, and considering the products of the terms in thetwo series (as discussed in the Appendix)

For many purposes it is convenient to use a normal-ized form of the cross spectrum, given by

P12'(f) = P12/(PlP2)"12. (4)This function is closely related to the correlation be-tween the noise currents in a narrow frequency band,which may be seen as follows:The correlation between two noise currents xl(t) and

X2(t) is defined asr12 = XIX21(Tl2-X2 2)1/2.

Suppose that xi and x2 are the currents which resultwhen the noise currents Yi and Y2 are passed throughthe pair of identical filters used to determine the crossspectrum. From the method of measuring the crossspectrum it is evident that the real part of the normal-ized cross spectrum is the correlation between the noisecurrents Yi and Y2 in a narrow band at frequencyf.Similarly, the magnitude of the cross spectrum is themaximum correlation which can be achieved. This willbe referred to as the intrinsic correlation P12, and is real-ized by shifting the phase of one noise current by theangle of the cross spectrum at the frequency in question.Thus we may write

r12 = actual correlation = real part of P12', (5)P12 = intrinsic correlation = magnitude of P12'. (6)

An extensive discussion of the power spectrum andrelated matters, with many references, is given by Rice.1

1 S. 0. Rice, 'Mathematical analysis of random noise," Bell Sys.Tech. Jour., vol. 23, pp. 282-332; July, 1944 and vol. 24, pp. 46-156;January, 1945.

The cross spectrum is discussed by Phillips,2 and themathematical background is treated in great detail byWiener.32. USE OF NOISE FUNCTIONS IN CIRCUIT EQUATIONSConventional steady-state circuit theory is based on

linear equations giving the relations between complexquantities which describe the magnitude and phase ofsinusoidal voltages and currents. These relations arestated in terms of complex impedances, admittances,or transfer functions. We will now show that the samecircuit equations can be used to describe the noise be-havior of a system merely by substituting the S func-tions of the preceding section for the steady-state com-plex currents or voltages.We note first that for the system of noise currents

y3(t) = yd(t) + y2(t),it follows directly from the definition (1) of the S fun-tion that

S3(f) = Sl(f) + S2(f). (7)This is parallel to the relation for steady-state sinusoidalcurrents, where in complex notation

13 = I1 + I2.Thus, the additive property of the complex S functionsis established.A second basic relation deals with the effect of com-

plex impedance, admittance, or transfer operators onthe S functions. If YB(t) is the noise current which re-sults when yA(t) is passed through a linear networkhaving a complex transfer constant A (f), it follows that

SB(f) = A (f)SA(f). (8)This relation was pointed out by Fry4, and is discussedby Phillips2 and Guillemin.5 This is parallel to thesteady-state complex relation

IB = A(f)IA.These two properties are sufficient for all the operationsof linear network theory. Hence it is possible to use theS functions in circuit equations in place of steady-statevoltages or currents.

In order to distinguish easily between noise currentsand voltages, we shall introduce the notation

I =S(f) for a noise currentV S(f) for a noise voltage. (9)

The bold-face notation will distinguish noise functionsfrom sinusoidal or dc functions.

2 R. S. Phillips, 'Theory of Servomechanisms," Rad. Lab. Series,McGraw-Hill Book Co., Inc., New York, N. Y., chap. 6; 1947.

3 N. Wiener, "Generalized harmonic analysis," Acta Math., vol.55, pp. 117-258; 1930; also N. Wiener, "Interpolation, Extrapolationand Smoothing of Time Series," John Wiley and Sons, Inc., NewYork, N. Y.; 1949.

4 T. C. Fry, 'The solution of circuit problems," Phys. Rev., vol.14, pp. 115-136; August, 1919.

6 E. A. Guillemin, "Communication Networks," vol. II, chap.XI, John Wiley and Sons, Inc., New York, N. Y.; 1935.

1462 Novoember

Montgomery: Transistor Noise in Circuit Applications

The general procedure for linear circuit problems isto write the circuit equations in the same form as for asteady-state situation, using noise spectra I or V forthe noise currents or voltages. The equations are manip-ulated in the customary manner, and usually as a laststep desired power and cross spectra are obtained by ap-plying-definitions (2) and (3) of these functions.

3. THEOREMS OF GENERAL INTERESTSeveral relations will now be derived which are gen-

erally useful in circuit work involving noise. These willalso serve to illustrate the procedure described in thepreceding section.Addition of Noise Voltages or Currents

Suppose that two sources of noise voltage havingpower spectra P1 and P2 and a cross spectrum P12 areplaced in series. We wish to find the power spectrumP3 of the resulting voltage. The equation of instantane-ous voltages is

V3(t) = Vl(t) + V2(t).From relation (7), and using the notation (9), we seethat the relation of voltage spectrum functions is

V3 = Vl + V2.Taking a product with the complex conjugate, and using(2) and (3),

V3* V3 = V1*VI+ Vi*V2 + V2*Vl + V2*V2P3 = Pl + P12 + P21 + P2.

From (3), (4) and (5) it follows thatP21 =P12*

P12 + P21 = 2 X real part of P12= 2(PlP2)'12ri1. (10)

Using this relation, it is seen thatP3 = P1 + P2 + 2(PlP2)'12r12, (11)

which is a simple and fundamental relation. A relationidentical in form would result if we had started withthree noise currents.Theorem on Terminal Noise in a Three-Terminal Net-work

Relation (10) may be used to establish a useful theo-rem relating the terminal voltages or currents in a three-terminal network containing noise sources. Referring tothe two networks of Fig. 1 we see that

V1(t) + V2(t) + V3(t) = 0,ii(t) + i2(t) + i3(t) = 0.

Except for a change in one sign, which is of no conse-quence in this case, each of these equations is like theequation of the preceding section, and it is easily veri-fied that the power-spectrum relation in either case isjust (11). This closely resembles the formula for one sideof a triangle in terms of the other two sides, and theincluded angle

X32= X12 + X22 - 2X1X2 COS 0.

The power spectrum at frequency f is proportional tothe mean-square voltage or current in a narrow bandabout f. Hence the root-mean-square noise voltages ornoise currents may be represented by the sides of atriangle, where the correlation between any two is thenegative of the cosine of the included angle, as shownin Fig. 1. The sign of the correlation will be as statedif one uses the sign conventions shown in Fig. 1 for thecurrents or voltages.

-cos- (-r,2 )

OR I2

V3 OR I3.Fig. 1-Relations between terminal noise voltages or currents.

From this representation it can be seen that (a) iftwo voltages and one correlation coefficient are knownthe third voltage and the other two correlations can becalculated; (b) if one voltage and two correlations areknown the other items can be calculated; (c) if all threevoltages are known, all three correlations can be calcu-lated; and (d) if one voltage is much smaller than theothers, the others must be almost equal and almostcompletely (negatively) correlated. The last relation ap-plies to transistors, where the open-circuit emitter-basenoise voltage is usually less than one per cent of thecollector-base voltage.

Simple Circuit RelationsConsider the case of a network of several meshes,

containing independent noise sources I,, 12 forwhich we know the power spectra P1, P2--. Sincethe sources are independent, the cross spectra P12,P2i * * are all zero. The resulting noises in two meshes,A and B, are given by

IA= A1I + A2I2 + * .IB-B1Il + B2I2 +

where the A's and B's are complex transfer functions.We wish to find the power spectra of IA and IB and thecross spectrum between them.

For want of a general symbol, we have used I's inthis example to represent either noise voltages or cur-rents or some of each. Hence the A's may be imped-ances, admittances, or transfer ratios, as may be ap-propriate.

1952 1463


From (2) and (3) it is straightforward to show thatPA = A12P1 + A2 12P2 +PB = B112P1 + B212P2 +PAB = A1*B1P1 + A2*B2P2 + * -

= A1Bj eiolPl + A2B21 eiG2P2 +where 0E is the angle of B3, minus the angle of A,.A special case of this relation may be used to estab-

lish the following theorem: If two noise currents (orvoltages) are compounded partly of a common source,and partly from independent sources, and if a is thefraction of power in the first due to the common sourceand 3 the fraction of power in the second due to thecommon source, then the intrinsic correlation betweenthe two currents is the geometric mean of a and 3.This theorem gives some insight into the significanceof correlation between noise currents. Let

IA = AJ11 + AoIoIB = B2I2 + Bo1o.

ThenPAPB

PAB

= A1, P1+ |Ao 12Po= B2 12P2 +| BOj2PI= A o*BoPo/(PAPB)1/2 = (ai) 1/2ei-,

wherea -I AoI2Po/PA= I Bo 2P0P/PB

'y is the difference in angle of Bo and A o.

Since by (6) the magnitude of PAB' is the intrinsic cor-relation between IA and IB, the theorem is established.

Another relation which will be used in Section 4 forinterpreting the conversion formulas for equivalent-gen-erator representations follows. Let

IA = A1J1 + A2I2IB-= B11 + B2I2,

where, in contrast to the earlier example, 1, and I2 arenot necessarily independent. Then, by a similar pro-cedure,PA = A1A2P1 + A2 12P2 + 2 real part (A1*A2P12)PB =- B1 j2P1 + B2 12P2 + 2 real part (B1*B2P12) (12)PAB = Ai*B1P1 + A1*B2P12 + A2*B1P21 + A2*B2P2,which gives all the spectra of IA and IB in terms of thespectra of the component sources 11 and I2 and thecross spectrum.A second useful theorem may be established by con-

sidering a special case of the above relation. The in-trinsic correlation between two noise currents (orvoltages) is not changed by passing one or both currentsthrough linear networks; the actual correlation is notchanged by passing one or both currents through linearnetworks having real transfer functions. To show this,

letIA = Al1IB= BI2,

where I, and 12 are not independent. Then from (12)PAB' = A*BP12/(PAPB)112 = AB| P12ei'/(PAPB) 2

= P12'eiywhich shows that the magnitude of the two normalizedcross spectra is the same, and the angle is changed by y,the difference in angles of the two transfer functions.Hence the theorem is established.Peterson's Equivalent Noise Generator TheoremA theorem on the use of equivalent noise generators

which is of great utility in circuit problems was appar-ently first stated by Peterson in an unpublished memo-randum dated August, 1943, referred to briefly in apublished article6 and restated in an unpublishedmemorandum in 1949, from which we quote.

"It is well known that the signal performance of alinear active four-pole is completely determined by fourparameters, which depend only on the internal struc-ture of the four-pole. In order to describe the noiseperformance of the network two additional intrinsicparameters are needed, thus making six in all. These twoadditional parameters may take the form of two fic-titious noise generators, one of which is placed in serieswith the input mesh and the other in series with theoutput mesh of the four-pole, which in this representa-tion is itself entirely noiseless. This particular equiva-lent noise circuit is most suitable when used with theopen-circuit signal parameter set. On the other hand, ifthe signal parameters are taken as the short-circuitadmittance set, it is more convenient to select the noiseparameters in the form of two noise currents impressedacross the input and output terminals respectively."

It may be noted that the four transmission parame-ters are in general complex, so that eight real quantitiesare required to specify them. Likewise, the two noiseparameters require two (real) power spectra and a(complex) cross spectrum to specify them, so that fourmore real quantities are involved, making a total oftwelve.

Peterson's proof consisted of writing mesh equationsfor the noisy device and also for the quiet device andshowing that there is a unique relation which insuresequivalence. We shall give a somewhat different proofwhich has the advantage of making no assumptionsregarding the internal structure of the networks (otherthan linearity of transmission properties) and of show-ing quite clearly what sort of statistical properties ofthe noise sources must be equated to make the networksequivalent.

Consider three linear networks N1, N2, N3 which haveidentical transmission properties in both directions for

6 L. C. Peterson, "Signal and noise in microwave tetrode," PROC.I.R.E., vol. 35, pp. 1264-1272; November, 1947.

1464 NOVemiber


external signals. N1 is the given network having internalsources of noise. N2 is a noise-free network with a voltagegenerator in series with the input terminals and anotherin series with the output terminals, each generatorproducing a voltage instantaneously identical with theopen-circuit voltage at the corresponding terminals ofN,. N3 is a noise-free network with generators whosevoltages may not be identical with, but have similarstatistical properties to those of N2. By a straightfor-ward extension of Thevenin's theorem, networks N1and N2 must supply identical currents and voltages tosimilar terminating impedances. Because of their iden-tical transmission properties, N2 and N3 must supplycurrents and voltages with similar statistical propertiesto similar terminating impedances. Hence the samemust be true for N1 and N3, which establishes the theo-rem. In particular, if the generators of N3 have the samepower and cross spectra as the open-circuit voltages ofN1, then the two networks will be equivalent in noisebehavior for all linear circuit applications.A similar theorem for short-circuit noise currents can

be proved in an analogous manner.The theorem is stated and proved for linear net-

works. It may often be usefully applied to small signalsin nonlinear networks, a procedure which is well knownand much used in transmission problems. In makingsuch an application one must certainly satisfy the usualcriteria for linear behavior with respect to externallyapplied small signals, but this is not sufficient. Whenthe device contains internal sources of voltage or cur-rent, one must also be sure that the small-signal require-ments are satisfied internally and that the external con-nections do not appreciably affect the generationmechanism.7 Although no cases have been found experi-mentally where noise behavior of transistors was notcorrectly predicted by small-signal theory, this possi-bility should not be overlooked, particularly with small-bias values and circuit connections providing largeamounts of feedback.

4. NoiSE BEHAVIOR OF TRANSISTORSTwo different methods of describing the noise proper-

ties of transistors have been found useful. The first isin terms of equivalent noise generators, as described inthe preceding section. This is the most compact way ofgiving a complete description of the noise properties.For linear circuit problems, a suitable description con-sists of the power spectra of two equivalent generatorsand a cross spectrum between them. These will befunctions of two dc bias parameters, but independentof the external circuit impedances. The second methodmakes use of the noise figure, which is a simple anddirect index of the noise behavior of the device as anamplifier. It depends on frequency and the dc biasparameters, and also on the generator impedance towhich the device is connected, but is independent of theload impedance. The noise figure can be easily calcu-lated from the equivalent-generator description. The

7 These considerations were pointed out by W. Shockley.

reverse process, while possible in principle, is not readilyaccomplished in practice.

Equivalent-Generator RepresentationThere are at least a dozen ways in which two voltage

or current generators may be associated with two of thethree pairs of terminals of a transistor to produce anequivalent noise network. These are equally general intheir ability to represent the noise behavior. Thus it isclear that the ability of an equivalent network to com-pletely represent the noise behavior of a transistor is noindication that it resembles the physical configurationresponsible for the noise. It is possible to secure moreelaborate representations by using generators at allthree terminals, or by using both a voltage and a cur-rent generator at each pair of terminals. A choice be-tween the various configurations depends on severalconsiderations, such as (a) the simplicity of the repre-sentation with respect to the sort of spectra obtained,the correlation between the noise generators, and thevariation of the noise parameters with dc bias values;

0 E C

B

(a)

t E Co-

t BB

(b)

(c)Fig. 2-Representations of noise behavior by

equivalent-noise generators.

(b) the ease and accuracy with which the noise parame-ters can be measured; (c) convenience for calculatingexternal-circuit behavior; and (d) degree of correspond-ence to the actual physical noise mechanism. For-tunately, it is not difficult to convert from one form ofrepresentation to another, and methods of doing thisare described in the following.Some representations used or seriously considered are

shown in Fig. 2. The one which has been used most ex-tensively to date is that of Fig. 2(a), which will be re-ferred to as the V-V representation, since it makes useof two low-impedance noise-voltage generators placedin series with the emitter and collector terminals. Thegenerator V. is equated to the noise voltage betweenemitter and base in the transistor, and V, to that be-tween collector and base, with all terminals open cir-cuited.The I-I representation of Fig. 2(b) makes use of two

high-impedance noise-current generators Ie and I,equated to the emitter-base and the collector-base noisecurrents of the transistor, with both pairs of terminalsshort circuited.A third representation, shown in Fig. 2(c), is a mixed

system, designated V-I. It uses a voltage generator V1

1952 1465


in series with the emitter terminal and a current gen-erator I2 connected from collector to base. These areequated to the corresponding noise voltage and currentin the transistor, with the emitter open circuited andthe collector shorted to the base. It should be noted thatin general Vi is not equal to Vi, nor is I, equal to I2,because the opposite-end terminating impedance is dif-ferent in both cases.

Before writing the conversion formulas between thevarious representations it will be useful to summarizesome of the methods of representing the transmissionproperties of a three-terminal network. In terms of open-circuit impedances

V1 ZllIi + Z1212,V2 Z21ITl + Z2212.

The Z's are simply related to the equivalent-T parame-ters often used to describe transistors.

Z,l = R. + Rb,Z21 = Rm + Rb,

Z12 = Rb.Z22= R, + Rb.

In terms of short-circuit admitances,11 Y11 Vl + Y12V2,12 = Y21V1 + Y22V2.

A mixed system which is especially suited to the V-Inoise generator representation is

V1 = H1-11 + 1112 V,I2 = H21I 1+ H22 V2.

With the exception perhaps of the last, these systems ofparameters are well known. Relations between them aregiven by Guillemin.IThe conversion formulas between the three equiva-

lent noise-generator representations can be writtendown by inspection in terms of the transmission parame-ters. The relation between the V- V and the I-I systemsis seen to be

VC = ZllIe + Zi2Ic,Vc= Z211. + Z22Ic. (13)

The converse relation isIe = YllVe + Y12Ve,

IC= Y21Ve + Y22Vc- (14)

The relation between the V-I and the V-V systems isgiven both in terms of the Z's and the H's.

V1 = Ve - (Z12/Z22) V. - Vs - 112Vc,

I2 = (1/Z22) VC= H22Vc. (15)The converse relation is

V, = V1 + Z12I2 V1 + (H12/H22)I2hV, = Z2212 = (1/H22)I2. (16)

' E. A. Guillemin, "Communication Networks," vol. II, chap. IV,John Wiley and Sons,, Inc., New York, N. Y.; 1935. Relations givenby Guillemin must be modified somewhat to apply to active net-works.

It will be seen that all of the conversion formulas areof the form of the circuit equations leading to thespectrum relations (12). When the power spectra andcross spectrum for one equivalent-generator representa-tion is known, the corresponding spectra and crossspectrum for another representation can be calculatedfrom relations (12), where the A's and B's are takenfrom the appropriate conversion formulas. When thetransfer functions are real, the calculations are verysimple. With complex transfer functions, the process ismore involved, and some care is required to combine theangles in the proper sense.Noise Figure-Grounded-Base Connection

In this paper we shall make use of the narrow-bandnoise figure F as an index of the noise behavior of atransistor when used as an amplifier. This may be de-fined as the ratio of the total noise power appearing inthe load in a narrow band to that portion which is dueto the amplified Johnson noise of RG, the generator im-pedance to which the input is connected.9 Its valuedepends on the noise properties of the transistor, thedc bias conditions, the frequency, and the generatorimpedance, but is independent of the load impedance.The noise figure can be calculated from any of the

equivalent-generator representations described in thepreceding section by determining separately the con-tributions to the noise in the load from Johnson noisein RG and from the two equivalent noise generators.Since noise figure is independent of load impedance,any convenient value of the latter may be selected. Forexample, consider the V- V representation of Fig. 2(a),using Z parameters and terminating the input in ZGand the output in open circuit. If the open-circuit noisevoltages at the output are Vo due to Johnson noise inRG, VA due to generator V., and VB due to generatorV, we find that

Vo = (4kTR0G)1 2Z21/ (Z1l + ZG),VA = - V.Z21/(Z11 + ZG),VB = VC

where kT is the Boltzmann constant times absolute tem-perature and ZG is the generator impedance with realpart RG. The corresponding power and cross spectra are

Po = 4kTRG Z21/(Z11 + Za) 12,PA = P. Z21/(Z11 + ZG) 2,PB = Pe,POA POB = 0,PAB= - Pe [Z21/(Z11 + ZG) ]*

From this, the noise figure isF = (PO + PA + PB + 2 X real part PAB)/PO

= 1 + I:T EPe 4+ Pe zI Z lo4kTR0 , Z21+ 029 This is a narrow band form of the definition given by H. T.

Friis, 'Noise figures of radio receivers," PROC. I.R.E., vol. 32, pp.419-422; July, 1944.

1466 November


-2 X real part {P,c + (17)which is the expression given by Ryder and Kircher,'0except for inclusion of the correlation term, and ex-pression of noise in terms of unit bandwidth.

If the noise is expressed in volts-squared per cyclebandwidth, the constant 4kT is 1.64X10-20 at roomtemperature.

In terms of the I-I representation and the Y parame-ters, the noise figure is

F =1 + 4kGT P + PC L21

-2 X real part {Pe c }], (18)F2'where YG is the admittance of the generator with realpart GG. It should be noted that the power spectra inthis expression are not the same as those in (17), butare derived from I. and Ih in an analogous way.

In terms of the V-I representation and the H parame-ters

1 FHi, +ZG 2F = I + 1 PI + p21Bl+Z4kTRG[21

+ 2 X real part {P12 Hi + }] (19)

where Pi, P2, and P12 are the power spectra and crossspectrum of V, and I2.

In all three formulas the last term is simplified if thefrequency is low enough so that all the transmissionparameters are real. In this case the cross spectrum isreal and is equal to the geometric mean of the two pre-ceding terms, multiplied by the correlation between thetwo noise generators.The generator impedance RG required for minimum

noise figure can be determined from any of the expres-sions for noise figure. A particularly simple case occurswhen, in terms of the V-V representation, the noisegenerator Vv is so small that its contribution to the loadnoise is negligible, and all the impedances are pure re-sistances. In this case, minimum noise figure is obtainedwhen Ra equals Z11. Experience has shown that thisgives a fairly good approximation to the optimum RGactually required by nearly all transistors over the usualrequired by most transistors at low frequencies in theusual range of dc bias values. This is usually a good dealhigher than the generator impedance required for maxi-mum power gain into practical working loads, so acompromise has to be made. Fortunately, the noisefigure increases rather slowly when RG departs from itsoptimum value as shown in the following table:"RG/optimum RGIncrease in noise figure, db

a or 2 1 or 5 or 100.5 2.6 4.8

"1 R. M. Ryder and R. J. Kircher, "Some circuit aspects of thetransistor," Bell Sys. Tech. Jour., vol. 28, pp. 367-400; July, 1949.

11 The table contains asymptotic values which are good approxi-mations for nois figures greater than 10 db. For smaller noise figuresthe values are less than those shown in the table.

Noise Figure-Grounded Emitter or CollectorNoise-figure formulas were given by Ryder and

Kircher9 in terms of the V-V representation for all threegrounding connections. Inspection of these formulasshows that for the impedance values as they exist inpresent transistors, the noise figures do not differ sig-nificantly among the various connections except forbackward transmission in the grounded-collector case,where the noise figure may be quite large. The optimumvalue for RG is substantially the same for grounded-emitter and for grounded-base operation. In the case ofthe grounded-emitter connection the optimum RG fornoise is usually a good deal smaller than the optimumRG for maximum gain.

5. NoiSE CHARACTERISTICS OF JUNCTION TRANSISTORSThe measurements now described are the result of

study of about a dozen n-p-n type 1752 transistors takenat random from recent product.'2 Some of the noiseproperties were measured in only a few units, others inthe whole group. The object was a general survey ofnoise properties, methods of measuring and represent-ing them, and their significance in helping toward aphysical understanding of the noise process. It is be-lieved that the data presented are reasonably represen-tative, but they are manifestly not extensive enoughnor sufficiently systematic to serve as a complete spec-ification of noise performance.

It is clear from the earlier part of this paper that agreat variety of methods of representing noise behaviorare available. Those used in this section were foundquite serviceable, but no claim is made for having ex-plored all the possibilities or foreseen all the require-ments in various applications. It is hoped that theinformation here presented may serve as a useful starton the problem, and that the general methods describedmay be a helpful basis for future development.

Methods of MeasurementIt does not seem worthwhile to give a detailed de-

scription of the measuring equipment, since high-gainlow-level amplifier construction is a pretty well under-stood art. A few comments will suffice.The measuring system consisted of six stages of wide-

band amplification, followed by a set of suitable filtersand a vacuum-tube voltmeter. The input stage was aWestern Electric 348A tube, pentode connected, for therange 20 to 15,000 cycles. A different preamplifier usinga Western Electric 403B tube in a cascode connection"was used for the high-frequency range 1 kc to 1 mc. Theamplifiers were operated from conventional power sup-plies using gas-tube regulated plate supplies and havinga rectified heater supply for the three stages of thelow-frequency preamplifier.

12 Some early measurements are reported by R. L. Wallace andW. J. Pietenpol, "Some circuit properties of n-p-n transistors,"Bell Sys. Tech. Jour., vol. 30, pp. 530-563; July, 1951.

13 R. Q. Twiss and Y. Beers, 'Vacuum Tube Amplifiers," Rad.Lab. Series, McGraw Hill Book Co., Inc. New York, N. Y., chap.13.10; 1948.

1952 1467


The filters were single-section constant-k structureshaving a ratio of upper to lower cutoff frequency ofabout 1.5. This has been found a convenient compro-mise between resolution and steadiness of output. Thefilter mid-frequencies are spaced at intervals of aboutan octave in most cases. For careful spectrum deter-minations at the lower frequencies, these filters were re-placed by a General Radio Wave Analyzer, type 736A,whose chief advantage was its ability to get in betweenharmonics of 60 cycles when power-line pickup wastroublesome.The system was ordinarily used as a bridging ampli-

fier, and was calibrated for gain and bandwidth with alow-level sinusoidal voltage. The effective bandwidthof the various filters was carefully determined by plot-ting the response in power units and integrating. Forcalibration it is found convenient to provide a sinusoidalsignal in an impedance of 2 ohms which will give thesame response on the measuring instrument as a noisevoltage of 10-12 volts-squared per cycle bandwidth.

Bias was applied to the transistor under test througha terminating resistance of 20,000 ohms at the emitterand 5,000 ohms at the collector. Since the input im-pedance is usually less than 2,000 ohms, and the outputimpedance greater than 0.1 megohm, these termina-tions provided substantially an open circuit at the inputand a short circuit at the output end. The noise-measuring amplifiers were bridged across the termi-nating resistances, thus measuring V1 and I2RL directly.

Ideally, noise voltage squared should be determinedby a square-law rectifier. In practice, it is much moreconvenient to use a vacuum-tube voltmeter such as theBallantine or the Hewlett-Packard model 400C. Theseinstruments are approximately linear full-wave recti-fiers. Theory shows that such a rectifier calibrated toread the rms of a sine wave will read low by a factor of0.886 in voltage (1.05 db) when used on Gaussiannoise. Transistor noise is approximately Gaussian andexperience shows that such a correction will give thetrue rms value to within a fraction of a db. Mostmeasurements were made with the Hewlett-Packard in-strument, with a 6,000 microfarad electrolytic con-denser shunted across the meter to give a larger timeconstant, a correction of 1 db being added to the read-ings. The peak-reading type of vacuum-tube voltmeteris very undesirable for noise-power measurements, andthe linear rectifier would have to be used with cautionunless the Gaussian properties of the noise had beenestablished.The measuring amplifiers and filters are provided in

duplicate as a means of determining correlation betweentwo noise voltages. Correlation measurements weremade by the following very convenient artifice, whichavoids the use of a product-measuring device. The defi-tion of the correlation between two noise voltages maybe put in the following form:

v1v2 (V1 + V2)2 - (V1 - V2)Asimpleswitc gat2bn]/2 he o

A simple switching arrangement between the outputs

of the two amplifier channels and the vacuum-tube volt-meter makes it possible to measure any of the followingvoltages, vi, v2, vl+v2, v1-v2. The squares of the readingsdiffer from the mean square of the instantaneous volt-ages by the constant factor 0.886 mentioned previously,which cancels out in the expression for r12. As shown bythe second theorem in Section 3, Simple-Circuit Rela-tions, the correlation is independent of the relative gainof the two amplifier channels, and the best accuracy isattained when the gains are adjusted to make the meterreadings equal for v1 and v2.

General PropertiesTransistor noise is relatively steady in character, and

aside from a characteristic difference in spectrum, is not

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Fig. 3-Typical noise spectra for n-p-n transistors.

unlike Johnson noise in a resistor. Short-time fluctua-tions of a rectifier measuring device are somewhatgreater than for Johnson noise, but of the same order.Drift is often observed for a few minutes after bias isapplied, more often downward than upward, andamounting usually do not more than 2 or 3 db. Meas-urements at intervals over a period of several weeksgenerally agree to within 2 or 3 db. Application oflarge reverse bias to either the emitter or collectorwith the other terminal floating tends to raise thenoise, sometimes substantially (10-15 db). This ap-pears to be a temporary effect which disappears afterminutes or hours. A small minority of units displaybursts of noise of a very irregular character. Units whichhave been damaged by excessive biases tend to behavein this way. The properties of noise (except the magni-tude) are so similar in junction and point-contacttransistors as to suggest strongly that the basic noisemechanism is the same in both devices.

1000 ImALoiI II 4-

1468 November

1 20 40 60 100


SpectraAt frequencies from 20 cycles up to around 50 kc the

spectrum of the noise at emitter or collector terminalsfor a variety of bias conditions seems universally to beof the form

P(f) = KIP,where the exponent is a little greater than unity, usuallyabout 1.2. At higher frequencies, the spectrum of col-lector noise sometimes shows a rise of 5 or 10 db fromthe extrapolation of the low-frequency law; in othercases it follows the above law pretty closely to 500 kc,which is as high as our measurements have been carried.Several examples are shown in Fig. 3. It has not beenpossible to carry measurements of the emitter noise tothe higher frequencies because the levels are so low.It is not clear whether the anomalies at high frequencyare a part of the noise mechanism or are caused bychanges in the transmission parameters, which begin tobe important in this frequency range.

0.2 0.4 0.6 0.8 1 2 4 6 8 10 21COLLECTOR BIAS IN VOLTS

Fig. 4-Equivalent-noise generator voltages at 1 kc in the V-Vsystem, for an n-p-n transistor. These values were calculated fromthe measurements in Fig. 5.

Since the spectra at frequencies below 50 kc are sosimilar in form, it is usually sufficient to give a value ata single frequency to represent data over this frequencyrange; 1 kc is often used as a reference frequency.

Equivalent-Generator RepresentationIt has been customary to represent the noise be-

havior of point-contact transistors by equivalent gen-erators in the V-V system of Fig. 2(a). A good deal ofinformation is available on the variation of these gen-erator voltages with dc bias. A few measurements havebeen made of the I2 generator in the V-I system of Fig.2(c). In most cases it has been found that the 12 gen-erator behaves in a less simple way as a function of dcbias than the generators of the V-V system Since theopen-circuit noise and transmission properties are easilymeasured in the case of point-contact transistors, the

V-V representation seems to be generally satisfactory.In the case of junction transistors, the situation seems

to be different. Because of the high-collector impedance,it is not easy to make measurements directly in the V-Vsystem. However, at low frequencies it is quite easy toeu

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Fig. 5-Equivalent-noise generator voltage, current and correlationat 1 kc in the V-I system, for an n-p-n transistor.

calculate values from data in the V-I system, and thishas been done in Fig. 4, which shows the equivalent-generator values in the V-V system for the same datawhich is shown in Fig. 5 for the V-I system. A few

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Fig. 6-Equivalent-noise generator current 12 at 1 kc in theV-I system for two other n-p-n transistors.

points on the V, curves of Fig. 4 have been checked bydirect measurement by means of a comparison voltageintroduced in series with the collector circuit. Fig. 6shows partial data for several other transistors in the

1952 1469

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V-I system, to give some idea of the variability amongdifferent units. For junction transistors, it appears thatthe V-I system represents the noise behavior in asomewhat simpler manner as a function of bias thandoes the V-V system. Although the theory of noiseproduction is in a rather rudimentary state, there issome reason to believe that the noise should be propor-tional to the bias current rather than the voltage, andthe V-I representation fits in with this picture betterthan the V-V representation. On the whole, it seemsthat the V-I system of equivalent-noise generators hasseveral advantages in the case of junction transistors.Noise Figure

It was pointed out above that, with certain simplify-ing assumptions, the noise figure should vary ratherslowly with the generator impedance RG to which atransistor is connected and should have its minimumwhen RG=ZlZ. Since the simplifying assumptions arenot always met in practice, Fig. 7 gives the extreme

RATIO OF RG TO R11

Fig. 7-Variation of noise figure with generator impedance RG.Solid curve shows simplified theory. Dashed curves are extremelimits of experimental data for several n-p-n transistors undervarious bias conditions and at 1 kc and 200 kc.

range of experimental data on several junction-typetransistors under various bias conditions and at fre-quencies of 1 and 200 kc. It is clear that the behavior isin reasonably good agreement with that calculated forthe simplified case.

COLLECTOR BIAS IN VOLTS

Fig. 8-Noise figure of an n-p-n transistor at 1 kc forvarious bias conditions.

In Fig. 8 the noise figure at 1 kc for a transistor withthe grounded-base connection is plotted as a function ofthe bias parameters I. and Vc, with RG adjusted forminimum-noise figure at each point. Fig. 9 shows simi-lar data for another transistor, and includes one curvefor the frequency 200 kc. While not representing ex-treme cases, these two sets of data are representative of

the variation among units. Similar information for 13transistors at three selected bias conditions and at 1and 200 kc is given in Table I. The average difference

0.2 0.4 0.6 0.8 1 2 4 6 8 10COLLECTOR BIAS IN VOLTS

Fig. 9-Noise figure of another n-p-n transistor at 1 and 200 kcfor various bias conditions.

in noise figure at the two frequencies is 14 db, whereaswe would expect a difference of 27.5 db from a simpleextrapolation of the low-frequency behavior accordingto the spectral law. The discrepancy is partly due to theanomalies in the spectrum shown in Fig. 3 and partlyto reductions in gain at the higher frequency. It will benoted that operation at low bias values usually makesan appreciable improvement in the noise figure. Theoptimum value for I. is probably between the twovalues used in the table, probably round 0.1 ma for anaverage unit.

TABLE INoise Figures of n-p-n Transistors at 1 and 200 kc

Noise figure in db for three selected bias conditions. Generatorimpedance picked approximately to give lowest noise figure;

for Is - 1.0 ma, Ra 500 ohms;for IB 0.030 ma, RG - 1,000 ohms.

FrequencyI kc 200 kc

I. 1.0 1.0 0.03 1.0 1.0 0.03V. 4.5 0.5 0.5 4.5 0.5 0.5

1 20.1 18.1 19.1 7.2 5.5 5.02 20.2 19.7 24.1 5.8 5.2 5.43 19.1 16.9 21.3 5.5 5.5 8.04 23.3 19.9 17.0 5.2 4.2 3.95 22.3 22.3 14.0 5.9 6.7 5.46 25.6 24.7 13.0 6.6 6.5 8.77 21.7 22.2 11.6 6.2 6.7 4.48 24.4 21.6 21.1 7.0 6.2 5.19 20.8 21.8 13.5

10 45.7 24.7 23.3 22.3 8.1 8.111 26.9 17.2 26.8 10.3 5.8 10.312 22.6 20.3 16.2 6.5 6.8 10.113 27.4 16.2 20.0 13.6 6.2 11.8

Average 24.6 20.4 18.5 8.5 6.1 7.2

CONCLUSIONSThe systematic method of dealing with linear circuit

problems in noise set forth early in this paper is quitegeneral, and should prove useful in many types of

1470 November


problems. The discussion of applications of the methodto the noise behavior of transistors makes it apparentthat many forms of description are possible, amongwhich a choice may be made based on convenience. Wehave indicated certain choices which have suited ourapplications, and have endeavored to present the meth-ods in sufficient generality to enable the reader to makesimilar choices. The specific information on noise be-havior of n-p-n transistors, while based on a small num-ber of units, is believed to be reasonably indicative ofthe behavior of the present product, and should serveas at least a rough guide in circuit-design work.Nothing has been said regarding the mechanism of

noise production in transistors. This is a subject which isnot at all completely understood at present, and dis-cussion of it was felt to be beyond the scope of thispaper. The interested reader is referred to a forth-coming paper."4

ACKNOWLEDGMENTTo many of my associates I am indebted for helpful

discussion of matters presented in this paper, and forprovision of the transistors whose noise behavior is re-ported.

APPENDIXThe function S(f), defined formally by (1), has de-

tailed properties which depend on the nature of thenoise function y(t). Generally speaking, both the mag-nitude and anglettof S(f) fluctuate along the frequencyaxis in such a way that there is little correlation in thevalues over frequency intervals of the order of 1/T. Toobtain a well-behaved function it is desirable to smooththis function over frequency intervals much larger thanI/T, limited of course by the desired frequency resolu-tion in the spectrum. We shall give a definition of thesmoothed functions required for (2) and (3) which de-pends on a Fourier series expansion of the noise currentover a finite time interval. With this approach it isnot hard to show that the definition is in accord withthe quantities measured by the procedures outlinedpreviously.

Let y, (t) and y2(t) be two noise currents for whichpower and cross spectra are to be defined. Let yl' andY2' be the result of cyclic repetition of that portion ofy, and Y2 lying in a time interval T. If T is much longerthan the reciprocal of the bandwidth of the measuringfilters, transients due to discontinuities at the boundariesof the time intervals will have a negligible effect on theaverage measurements If the noise currents are sta-tionary, the response of the measuring system to yl'and Y2' will not differ significantly from the response toyi and Y2. The primed currents are described in completedetail by their Fourier series expansions

Y = ak cos (2irkt/T Ok),Y2I = bk cos (27rkt/T - 'kk).

14 H. C. Montgomery, "Electrical noise in semiconductors," BellSys. Tech. Jour., vol. 31, pp. 950-975; September, 1952.

If the magnitude of Si(f) is obtained by rms smoothingof the ak's over the frequency interval of the measuringfilter (multiplied by T-112, which is the number ofcomponents per unit frequency interval), it is evidentthat the power spectrum defined by (2) is just the powercontained in those Fourier components passed by thefilter.

(1/2T) Z ak2 or (1/2T)Zk2.The cross spectrum defined by (3) is a smoothed ver-

sion of(1/2T) akbkeit(k-kAk)

The measuring process described for obtaining the crossspectrum involves filtering y, and Y2, taking the product,and dividing by the bandwidth. This is equivalent to(l/8f) E ak cos (2irkt/T - ok) X E bk cos (2irkt/T - k)where each sum is carried over those terms passed bythe filters. By a well-known property of trigonometricseries, products of terms of unlike frequency average tozero, so the above is equivalent to

(ll/f)E akbk cos (2irkt/T -ok) cos (2wrkt/T -Pk)= (1/28f), akbk[cos (4rkt/T - 0,^ - ck) + cos (k - Ok)]= (1/28f) E akbk cos (Qk - Ok),since the first term in the bracket averages to zero. Thesummation involves 3f/T terms, so the average value is

(1/2T) > akbk cos (Ok - Ok)@This is the real part of the cross spectrum, as deter-mined experimentally, and this relation may be takento define the smoothing process required in (3). Theimaginary part of the cross spectrum is determined by aquadrature measurement, and is evidently

(1/2T) E akbk sin (kk - Ok).For the special case of two identical noise currents

Yl'=Y2', it is seen that ak=bk and Ok=1'k; hence thecross spectrum is real and equal to the geometric meanof the two power spectra. If yl' and Y2' are derived froma common source through linear networks whose trans-fer functions vary slowly over the interval 3f, we haveapproximately ak= const. X bk and Ok - Pkk= const. Inthis instance the cross spectrum is complex, but itsmagnitude is again the geometric mean of the powerspectra. Either of these cases represents completely co-herent noise currents. In the case of completely incoher-ent or independent noise currents the angles of the indi-vidual terms are completely random, and the average isnegligibly small compared to the power spectra, and nofixed phase shifts in the system can increase it. It may benoted that apparent incoherence (with respect to themeasuring system as described) can be produced by sub-jecting one noise current to phase shifts which vary sub-stantially over the frequency interval 8f. However, thiseffect can be removed by a complementary phase shift.

1952 1471

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Transistor Noise in Circuit Applications