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ON THE NOISE OF TRANSISTORS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE DELFf OP GEZAG VAN DE RECTOR MAGNIFICUS, IR. H. J. DE WIJS, HOOGLERAAR IN DE AFDE-LING DER MIJNBOUWKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP WOENSDAG 28 OKTOBER 1964, DES NAMIDDAGS OM 4 UUR

DOOR

ARIE BAELDE

NATUURKUNDIG INGENIEUR

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7 -:·

I " ;

DJT PROEFSCHRIFf IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DRS D. POLDER

-

.

..

' ... " .

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, '.

Aan mijn Ouders Aan Nita

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Aan de Directie van het Natuurkundig Laboratorium der N.V. Philips' Gloeilampenfabrieken te Eindhoven betuig ik mijn grote dank voor de wel-willende medewerking bij het verrichten van het onderzoek en het verschijnen van dit proefschrift.

Veel dank ben ik ook verschuldigd aan mijn collega's en aan allen die aan de totstandkoming van dit proefschrift hebben meegewerkt, in het bijzonder de heren D. Brand en J. Prins voor het conscientieus uitvoeren der vele metingen. Tenslotte dank ik de N.V. Eindhovensche Drukkerij voor de voortreffelijke wijze waarop dit proefschrift typografisch is verzorgd.

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INTRODUCTION . . . .

1. SMALL-SIGNAL PROPERTIES

1.1. Introduction. . . .

1.2. Power gain and stability of linear networks 1.3. The intrinsic transistor . . . : . . . . . 1.3.1. Formal description . . . . 1.3.2. Current-amplification factors aj and a'j

1.3.3. Equivalent-circuit representation of common-base config-Page 1 3 5 9 9 14 uration . . . 17 1.3.4. The TPTt matrix . . . 19 1.3.5. Equivalent circuit of common-emitter configuration 20 1.4. The practical transistor . . . 22 1.4.1. Equivalent circuit. . . 22 1.4.2. Admittance matrix in common-emitter configuration 24

1.4.3. Admittance matrix in common-base configuration 26 2. NorSE THEORY OF INTRINSIC TRANSISTOR

2.1. Introduction . . . 28 2.2. Noise of some fundamental processes. 33

2.2.1. Thermal noise . 33

2.2.2. Shot noise 34

2.2.3. I/J noise. . . . 35

2.3. Noise properties of the intrinsic transistor 36 2.3.1. Fundamental sources of transistor noise. 36 2.3.2. Short-circuit noise currents . . . 38 2.4. Noise in semiconductors in a non-equilibrium steady-state 41

2.4.1. Charge-carrier-density fluctuations in a three-Ievel semi-conductor. . . . . 41 2.4.2. Fluctuations in the generation-recombination ra te of the

Shockley-Read model. 43

2.5. Discussion . . . 47

3. TRANSISTOR-NOISE THEORY 3.1. Introduction . . . . 3.2. Linear noisy fourpoles . . . .

3.2.1. Equivalent noise generators 3.2.2. Noise matrix and noise quantities

50 51 51 52

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guration . . . . 3.3. Noise matrix of the intrinsic transistor. . . .

3.3.1. Linear recombination of injected carriers . . 3.3.2. A simple model of non-linear recombination. 3.4. Noise matrix of the practical transistor

3.4.1. Noise quantities . . . . 3.4.2. Asymptotic approximations . . 3.4.3. Influence of some additional elements 3.5. Discussion .

4. EXPERIMENTS 4.1. Introduction.

4.2. Methods of determining the noise quantities from noise-factor 55 57 57 59 62 62 67 72 73 75 measurements . . . 75 4.3. Experimental equipment . . . 77 4.3.1. Measurements of small-signal properties 77

4.3.2. Noise measurements . 79

4.4. Experimental results . . . 82 4.4.1. Small-signal properties . . . 82 4.4.2. Noise measurements and comparison with theory 87 4.4.3. Experimental investigation of non-linear recombination

model . . . . . 98 5. NON-LINEAR SURFACE RECOMBINATION

5.1. Introduction. . . . 5.2. Recombination on the surface of the base region

5.2.1. Homogeneous injection on the surface . . 5.2.2. Exponential decay of injection along the surface 5.3. Comparison with experiments and discussion

6. CONCLUSJONS Appendix A Appendix B Appendix C Appendix D REFERENCES. LIST OF SYMBOLS SAMENVATTING . 102 103 103 111 113 116 118 119 121 122 124 126 130

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Abstract

The noise performance of transistors is studied, both theoretically and experimentally. In the theoretical study the noise associated with generation and recombination of injected minority carriers in tbe base region is investigated, assuming that the process occurs via one kind of recombination centres (Shockley - Read process). It is shown that the noise can be represented by the independent shot noise of a recombina-tion current and a generarecombina-tion current, except for some specific situarecombina-tions. The representation is also allowed when the process is non-linear, i.e. when the lifetime of the injected carriers depends on the injection level. However, in the latter case the auto- and cross-spectral densities of the short-circuit noise currents at emitter and collector of the so-called intrinsic transistor deviate from their values for a linear recombination process. The experimental work, carried out on germanium transistors at room temperature, generally bears out the theory based up on alinear recombination process, except for the noise conductance at low frequencies. In order to explain the observed discrepancies it is necessary to assume that the recombination process is non-linear, even at a low injection level. A model of non-linear surface recombination is presented, which fairly well explains the experimental observations.

INTRODUCTION

Af ter transistors had become valuable elements in amplifying electrical circuits their noise properties became more and more important. In the begin-ning transistors exhibited a large amount of noise, which increased with decreas-ing frequencies. However, according as preparation techniques improved and the upper frequency of operation increased it grew clear that the noise of tran-sistors would be comparable with that ofvacuum tubes, at least within a certain frequency region. Attempts were th en made to understand the origin of transis-tor noise.

In the older, phenomenological, theories as presented by Van der Ziel58), Guggenbühl and Strutt43) four sources of noise were assumed, viz. diode noise of the emitter-base diode, full shot noise of the collector saturation current, partition noise due to the partition ofthe emitter current into a collector current and a base current and, finally, therm al noise associated with the ohmic resist-ance of the material of the base region. The theory gave good results at low frequencies (except for the so-called l/fnoise), but it failed at high frequencies. Experiments showed a large increase of the noise at high frequencies, whereas the theory only predicted a slight increase.

Then Becking40) and Van der Zie142) studied the fiuctuations in the principal processes taking pi ace in a transistor, viz. diffusion, recombination and genera-tion of charge carriers. They calculated the auto- and cross-spectral densities of the short-circuit noise currents fiowing out of the so-called intrinsic transistor in common-base connection. The theory, which did not require a detailed knowledge of the generation-recombination process, was generally borne out by experiments.

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2

-transistors of the type OC44 and OC45, generally were in agreement with the results of other investigators. However, some of our observations in the lower part of the considered frequency range, showed differences with theoretical results which were beyond the region of possible measuring errors. These results gave ri se to a thorough investigation, both theoretical and experimental, of the noise properties of transistors, which is presented in this thesis.

The first problem to which attention has been paid is the relation between noise properties and smaJl-signal performance. This may be elucidated by noting th at any noise voltage or noise current, produced somewhere in the interior of the transistor, has to pass a number of elements before it reaches the extern al terminals. Furthermore, the transport of noise within a narrow fre-quency band iJf around a frefre-quency f proceeds similarly to that of a pure signal of the same frequency. It is known that a number of transistor elements have negligible influence on the noise properties, provided that they are small. We want to relate the conditions for the neglect of these elements to some criterion of usefulness of the transistor. For that purpose chapter I starts with a brief discussion of power gain and stability of transistors used in practical circuits. The remainder ofthe chapter is devoted to a study ofthe small-signal properties. Much attention has been paid to the question wh ether a detailed knowledge about the process of generation and recombination of charge carriers is required or not. It is shown in chapter 2 that, although the basic assumptions about the noise of this process, made by the afore-mentioned authors, is usually valid for a commonly occurring type of recombination process, the resulting spec-tral-density expressions of the short-circuit noise currents are only valid when the process is linear. The result in itself may not be surprising, since both au-thors have restricted their theory to cases in which the level of injection is much below the concentration of majority carriers (mostly occurring in practice), thus tacitly assuming linearity of the recombination process. However, measure-ments of the d.c. base current as a function of the d.c. emitter-base voltage displayed some, and sometimes even astrong, deviation from the behaviour expected from Shockley's well-known theory.

The deviations may be attributed, at least formally, to a non-linear-recombi-nation process. Furthermore, it is shown in the chapters 3 and 4 that such a non-linear-recombination process can account for at least a large part of the observed discrepancies between our measurements and the results ofthe theory, based upon linear recombination. However, in this way the problem is shifted towards the existence of a generation- recombination process that depends on the injection level when it is much below the majority-carrier concentration. A possible °answer to this problem is given in chapter 5, where a model of non-linear surface recombination is presented.

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1. SMALL-SIGNAL PROPERTIES 1.1. Introduction

In order to describe its small-~igna(behaviour, the transistor is considered as a two-port device with small a.c. voltages appearing across input and output ports and small a.c. currents entering through the ports. If the interrelations between the four currents and voltages are linear, th en the two-port, or fourpole, is said to be a linear one. In the subsequent study it is assumed that a transistor can be considered as a linear fourpole. Generally the coefficients of the linear relations depend on the applied d.c. voltages.

The fourpole receives a signal from a signal source connected to its input terminals and delivers a signal to a load connected to its output terminals. The treatment of the small-signal properties of a transistor will be restricted to the situation where the a.c. signals at input and output terminals have the same frequency. In accordance with Thévenin's theorem the signal source may be represented by either an ideal voltage generator es in series with the source impedance Zs or an ideal current generator is in parallel with the source admit-tance YB = l/zs (cf. fig. l.l).

Linear

fourpole i1 =>11 ~+Y1212 !2 =.J2t ~+J!12 ~

Fig. 1.1. Representation of a linear fourpole, which receives a signal is from a signal souree Ys and delivers a signal to a load YL.

Usually the quality of an amplifier is determined by its ability to amplify the entering signa\. The rate of amplification may be defined in several ways. We will follow the most usual way and define it in terms of available power, extend-ed to exchangeable power later on.

The available power of a twopole, which contains at least one current or voltage source, is de'lned as the maximum power th at can be delivered to a load by arbitrary variations of the load admittance (or impedance). The availa-bIe power gain of a fourpole is defined as the ratio of the available power at the output terminals to the available power of the signal source. It depends on the admittance of the signal source, and optimization may be possible by varying this admittance. In this way the maximum available power gain is obtained at particular values of source and load admittance, respectively. The maximum available power gain only depends on the small-signal properties of the fourpole and therefore it is often used as a figure of merit of the small-signal performance of an amplifier. It must be emphasized that the definition of the available power of an arbitrary signal source is only meaningful when the adrnittance (or imped-ance) of the signal source has a positive real part. Hence, the maximum

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availa-bIe power gain of a fourpole is only defined when the optimum values of both source and load admittance have a positive real partI). Usually this requirement is met by transistors in the low-frequency region, but at high frequencies, where the internal feedback is increased, the condition may be violated. In order to include adntittances having a negative rea I part, Haus and Adler defined the exchangeable power of a source, having a root-mean-square short-circuit cur-rent is and an intern al admittance Ys

=

gs

+

jbs as folIows:1)

lisl2 Pe=- .

4gs (1.1)

The definition is identical with the available-power definition when gs > 0, but Pe remains finite for negative values of gs. Since any two-terminal source of power (in general, any port of an N-port device) can be represellted by its root-mean-square short-circuit current is in parallel with its internal admittance Ys, the exchangeable-power definition also holds for the output power of a four-terminal network. Then the exchangeable gain Ge is defined as the ratio of the exchangeable output power of the network to the exchangeable power of the source. Though in this way an elegant and general treatment of the small-signal performance of linear networks can be given, the maximum exchangeable power gain (defined as the optimum value of the exchangeable power gain by varying the source admittance) cannot replace the maximum available power gain as a figure ofmerit, because negative values of Ge are prohibitive when sta-bIe operation in practical circuits is desired. Therefore, another quantity has to be found as a figure of merit of the small-signal performance of transistors. The use ofthe quantity IY2

1

/Y

1

21

has been proposed for th is purpose2,3). This quanti-ty, which is son'letimes called the gain-stability factor, will be discussed later.

All information about small-signal properties of a linear fourpole is contain-ed in the admittance matrix (or any other matrix such as impcontain-edance matrix, transmission matrix, etc.). For instanee the maximum available power gain can be expressed in terms of fourpole admittances, though this involves rather complicated calculations. Haus and Adler have extended the matrix represen-tation of linear networks to the powers entering and leaving the network. The exchangeable power gain may then be expressed in a simple way in terms of the elements of a new matrix, of which the determinant equals the quantity

-IY

12

/Y

21

12.

Therefore, besides the admittanee matrix, this new matrix will also be considered in this chapter. Another reason for studying this matrix is that it plays an important role in chapter 3 where transistor noise is considered.

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5 -1.2. Power gain and stability of linear networks

Let a linear four-terminal network be represented by its transmission matrix T (fig. 1.2). The transmission matrix is related to the admittance matrix in the following way:

1 (-Y22 - 1 ) 1 (t22

T = -

andY=-Y21 -detY -Yu t12 -1

- detT) , tu

(1.2)

where det Tand det Y are the determinants of the transmission matrix and the admittance matrix, respectively. Note that det T

=

Y12/Y21.

~ =tl1 v2 -ft2 i2

i , = t2/ 12 -t22 i2 Fig. 1.2. Transmission-matrix representation of a linear fourpole.

The network is driven from a signal source represented by its r.m.s.

short-circuit current is and its internal admittance Ys. Henceforth it is assumed that the internal admittance of the source always has a positive real part. The following "source equation" holds1):

Ï!

+

Ys Vl

=

is.

Now detine the following column vectors:

the "source vector" Y = (Ys*) 1 '

the "input vector"

the "output vector" m

=

(V~ )

,

-/2

and the "permutation" matrix P =

G

~)

.

The asterisk denotes the complex conjugated quantity. The source equation now reads:

ytl = is, (1.3)

where the daggert denotes the Hermitian conjugate.

The available power from the source now reads in matrix notation, according to (1.1):

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6

-The input and output vectors are mutually re\ated by the transmission matrix T, so the "output equation" giving the relation between i2, V2 and the current source is reads:

yt T m

=

is. (1.5)

By analogy with eqs (1.3) and (1.4) the exchangeable power at the output terminals of the fourpole is

lisl2

P -

----'---'--e,o - 2 yt TPTt Y (1.6)

The exchangeable gain now follows from its definition and eqs (l.4) and (1.6):

Pe,o yt Py

Ge= -

=

.

PB yt TPTt Y (1.7)

It may be noted that Ge may achieve negative values for certain values of y, notwithstanding that the real part of Ys is a1ways positive.

Nowwewrite

TPTt

=

(2RT ST) .

ST* 2GT

(1.8) The elements RT, ST and GT may be expressed in terms of the elements of the transmission matrix T, or of the admittance matrix Y or impedance matrix Z, for instance: (1.9) (1.10) and yu

*

det Y + Yu det y* '22 GT

=

t(t22 t 21*

+

t22*t21)

=

1-

1

1

=

-I

-I '

(1.11) Y21 2 Z21 2

where g22 and '22 are the real parts of Y22 and Z22, respectively.

The determinant of the TPTt matrix may he expected to be an interesting quantity. lndeed it follows from eqs (1.9) to (1.11) that

(1.12)

This quantity is always negative, and equals zero for a unilateral network. Furthermore, its negative inverse value equals the square of the gain-stahility factor.

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Next we are interested in the maximum of Ge and under what conditions it occurs. This problem may be solved by determining the stationary values of ytTPTt y, subject to the constraint th at ytpy is constant1). By introducing the Lagrangian multiplier the following set of equations results:

~(yt

TPTt

y

-

-~

yt py) = 0

(i

=

1,2)

OYi ÀT

and

~(yt

TPTt y-

~yt

py) = 0

(i

= 1,2).

OYi* ÀT

(1.13) These equations can only be solved when ÀT is a real quantity. Assuming for the moment that ÀT is real, its values are deterrnined by the requirement

det (P-À T TPTt) = det{ (TPTt)-l P- ÀTI}

=

0, (1.14) where I is the identity matrix.

So the solutions of ÀT are the eigenvalues of the matrix (TPTt)-l P. It is easily calculated (cf. eqs (1.10) and (1.12)) th at

ÀT = kT

(1

±

1

/

1

-~

)

.

(1.l5)

<5

r

kT2

Obviously the eigenvalues are re al when <5 - ,,; I.

kT2 (1.16)

This means that maximization of the exchangeable power gain is only possible under the constraint (1.l6). The re sult can also be obtained when I/Ge is expressed in terms of the elements of the TPTt matrix:

(l.17) For finite values of Ge the stationary points of both Ge and I/Ge occur at the same values of the source admittance Ys. It follows from

(l.17)

that the station-ary points of I/Ge occur at

(G) (G) (G) 1 ( )

Ys, opt

=

gs opt

+

jbs, opt

=

2RT

±

V

4 RTGT -1'1,2

+

jlT . (1.l8) They only exist wh en

(1.l6a) which is identical with (1.16) . .An analysis of (1.17) with the aid of (1.16a) shows that the stationary value of Ge at gs > 0 means a maximum and at gs < 0 a minimum, provided that RT > O. Hence the optimum values of Ge are

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8

-Ge opt

=

1

=

kT

(1

T

1/

t-

~)

.

, kT

±

V

4 RTGT-/T2 c5

V

kT2 (1.19)

Both RT and GT are positive for transistors of practical interest (cf. section 1.4). Furthermore we are interested in the extremum of Ge at gs >

o..

Thus

Any value of Ge may occur in cases where no extrema of Ge exist, which indi-cates th at condition (1.16) must be related to some stability criterion. Llewellyn has studied under what conditions the output conductance of an active linear fourpole r~mains positive under arbitrary variations of the source admittance, provided th at gs > 0.4,5). His stability criteria may be written as

Vb

gll > 0., g22 > 0. and 0. .,,; - .,,; 1.

kT (1.21)

Of course kT has to be positive for stabie operation, and then our condition (I. I 6) is identical with Llewellyn 's third stability criterion.

Expression (1.20.) may be written in a somewhat different form: 1

Ge, opt

=

(l/v;5)

k V ·

T kT2

Vö+

-y-1

(1.22)

The second factor in the right-hand side of (1.22) is completely determined by the quantity Yb/kT, which represents the stability of the fourpole under arbitrary terminations. Venkateswaran and Boothroyd introduced the "inherent

stability factor"6,7)

(1.23)

Equation (1.22) may now be rewritten as

Ge, opt

=

I

Y211

~

. (1.24)

Y12 Si

From this notation the name "gain-stability factor" of the quantity IY21/Y121 may become cleat. The sign of Si depends on that of kT, which may be either positive or negative. Negative values of Si, however, imply negative values of

Ge,oPt, which cannot be realized with a positive load conductance. Therefore, to ensure stabie operation, Si has to obey the inequality

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A final remark concerns linear fourpoles that violate (1.16) and/or for which

kT < O. Then a new linear fourpole may be created by connecting positive con-ductances to the input and/or the output terminals ofthe original fourpole. The input and output conductances of the new fourpole are the sum of the original values and the added conductances, whereas the values of Y12 and Y21 remain upchanged8 ). Now the added conductances can be chosen such that the new fourpole obevs condition (1.16) and that its kT value is positive. Then the new stability factor is found from (1.23) after substituting the new values kT and <5, and stabie operation ofthe new fourpole is obtained when St ~ 1.

The problem of gain and stability returns in chapter 3. In the next section the physical properties of transistors will be considered.

1.3. The intrinsic transistor

1.3.1. Formal description

Consider a hypothetical p-n-p transistor of which the capacitances of the depletion layers existing ori either side of both p-n junctions are negligibly sm all (cf. fig. 1.3). Furthermore it is assumed that generation and recombination of charge carriers within these depletion layers can be ignored.

n p ~ p

i~

~ c

!~

? I ~T~ EmUfer Col/ector dep/etion dep/et/on /ayer h' /ayer

Fig. 1.3. Schematic representation of an intrinsic transistor.

Finally the voltage drop due to the majority-carrier flow fr om the base contact to the inner region between emitter and collector is assumed to be negligible. This transistor is called the intrinsic transistor.

For convenience a transistor with axial symmetry will be considered, having equal areas of emitter and collector cross-sections. The x-axis is taken as the symmetry axis, the emitter junction being situated at x = 0 and the collector

junction at x

=

wo.

Let the emitter be positively biased with respect to the base contact and the collector negatively. Then holes are injected by the emitter and are removed by the collector and, in general, electrons flow in the opposite direction.

The transport of free carriers through the base region is governed by the transport equations for holes and electrons, respectively9),

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and dp(x) J p = -qDp - -

+

qf-tp EbP(X) dx dn(x) J n = qDn - -

+

qf-tn Ebn(X). dx (1.26) (1.27) Let the emitter be ideal, i.e. the current is completely carried by holes, th en

Jn

=

0 and the internal electric field follows from eq. (1.27).

The electric field is negligibly sm all when the donor concentration in the base region is homogeneous. Then the injected holes move towards the collector by diffusion only.

A homogeneous electric field directed towards the collector exists when the donor concentration decreases logarithmically with the di stance from the emitter junction. The electric field may then be indicated by the drift field param-eter

q

Ab

=

fJWOEb, where fJ

=

-

.

kT (1.28)

In practice Ab does not exceed the value 8; then the field strength is 200 V/cm when the base width is 10 microns. The electric field accelerates the movement of the injected holes towards the collector, thus decreasing their transit time. Simultaneously with the hole injection an approximately equal amount of electrons is supported through the base contact in order to maintain a high degree of space-charge neutrality. In general, the electron density and the electric field strength depend on the donor concentration and on the level of injection. As long as the latter is low, i.e. the injected concentrations are small compared to the equilibrium concentration of electrons, both the electron density and the electric field strength are independent of the injection level.

Besides the transport equation (1.26) for holes in the base region, the conti-nuity equation holds9):

op(x) p(x)-po(x) 1 oJp

(1.29)

ot Tt q ox

The quantity Tt denotes the effective lifetime of the holes, incorporating the effect of surface recombination. It is assumed that Tt is independent of the injec-tion level as long as it is low compared to the majority-carrier concentrainjec-tion. Equations (1.26) and (1.29) may be combined, yielding

op P- PO 02p op

- - = - - -D p- + f-tpEb- .

ot Tt ox2

OX

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1 1

-This equation must be solved under the appropriate boundary conditions. The boundary condition at the emitter junction follows from Shockley's hypothesis that the free-electron concentration in the conduction band obeys Boltzmann statistics. So wh en a d .. c. voltage Ve ~ 1/{3 and simultaneously a small a.c.

voltage Ve

=

\Ie expjwt (Ve ~ 1/{3) are applied between the emitter and the base contact, the boundary condition at x = 0 is

pe

=

ps,e

+

pa,e

=

pO,e (exp {3Ve)(1

+

(3Î1e expjwt). (1.31) At the collector boundary, x

=

w, the hole concentration p

=

O. However, the base width w depends on the collector voltage10) because the width of a deple-tion layer depends on the voltage across the juncdeple-tion. Let w = Wo wh en the collector is negatively biased: -

V

c ~

I

/

p

.

Wh en a small a.c. signal Vc

=

\lcexpjwt

is also present across the collector junction, the actual base width changes in

~ accordance to

dw

w

=

Wo

+

-

Vc expjwt.

dVc

(1.32) The quantity dw/d Vc depends on the impurity distribution in both the base

region and the collector region. The solution of (1.30) has been given by Krömerl l). It is usually split up into two parts, viz. the steady-state solution

(op

/

at

=

0) and the a.c. solution. The latter describes the response of the system to a small sinusoidal emitter voltage (the second term in (1.31)). Two possible steady-state solutions for the hole concentration p in the base region are shown in fig. 1.4 for different values of Ab. At Ab

=

0 the hole concentration decreases about linearly with the distance from the emitter junction. When an aiding drift field is present the hole-concentration pattern is more complicated.

Pe,5 ~

o

\ \ \ \ \ \ \ \ \ \ \

,

\ \

,

\

,

\

,

"

\

,

,

,

-..: _ x

Fig. 1.4. Hole-concentration pattern in the base region of a p-n-p transistor at two different values of the internal drift field. The broken line represents a possible situation at increasing

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The d.c. emitter and collector currents are obtained from the steady-state solution of eq. (1.30) and from eq. (1.26). When the positive sign is used to indicate a current flowing into the transistor, the following expressions are obtained:

Ie= qDppo,eAoAe [CeXP,BVe-1) (cothAOWO

+-i-~)

+

e~p ~bj2]

11.0 2wo smh oWo

(1.33) and

[

exp(-Abj2) 1 Ab]

Ic=-qDppo,cAoAc Cexp,BVe-l). +cothAoWo-- - ,(1.34)

slllh AOWO AO 2wo

where

AO

=

V_l

-

+

Ab2 Dp7:i 4 w02

Of ten the collector current is written in terms of the emitter current as follows:

where -Ic

=

ao Ie

+

Ico, Ab eXP

2

ao

=

- - - - --

-1 Ab

cosh AOWO + - -sinh AOWO AO 2wo

(1.35)

(1.36)

The current Ico is the collector saturation current which flows when the emitter is open-circuited. It is of little use to derive an analytical expression for it as in most practical transistors Ico is caused by surface effects, which are not consider-ed here.

Thea.c. solution of eq. (1.30) makes it possible to calculate the e1ements ofthe admittance matrix. A distinction must then be made between the possible con-figurations in which the transistor can be used in practical circuits, viz. the common-emitter, the common-base and the common-collector configuration. As the latter is hardly ever used in practice it will be disregarded here. Of the other two the common-emitter configuration is most used in practical circuits, but the common-base configuration is related more c10sely to the physical picture (cf. fig. 1.3).

.

(~)

t-_;lc Ic Ytb.i=l~veb' d

- a' Vcb'=O

i, = ~'b,i

veb

+ Yrb,i Vcb'

i

Y. (Ic ) vc6' f~I=\a~b'd

Ïc=.Ytb,i "eb i;Yob,iVcb'a~ 'éb' =0

r;--

t---9·L Yrb,i

=(~b')

' - - - -_ _ _ _ _ _ .J 0 ai (f~b'=O

Yob,i

=(fv;,)

dVeb'=O

Fig. 1.5. Admittance-matrix representation of the intrinsic transistor in common-base connection.

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The fourpole representation of the common-base connection is shown in fig. 1.5. The complete expressions of the matrix elements follow [rom Krömer's a.c. solution of (1.30) and may be written as follows:

1 2A Wo coth A Wo

+

Ab

Ytb,i

= -

,

re 2AOWO coth AOWO

+

Ab (1.37)

1 2AWO exp (Ab/2)

Y!b i

=

-

-

. _ - -

,

, re (2Aowo co th AOWO

+

Ab) sinh AWo (1.38) Aexp (-Ab/2) dw

Yrb,t

=

Ie . hA dV:

sm Wo e

(1.39) and

YOb,t

=

~

- Ie (A co th AWO

_~

)

_ qPo,eAe

~

dw,

( 2wo Ti ~dVc (1.40) where 1 _ 1/ Ab 2 1 + jWTi A -

V

-

+-

-

-4wo2 DpTl,

and re is the emitter differential re si stance

aVe 1

re

=

-

-

=

.

ale f3(le

+

le,s) (1.41)

The current le,s, used in this equation, follows from (1.33) when - Ve ~ 1/f3.

In almost all practical cases le,s ~

Ie.

With these equations the small-signal behaviour of a transistor is completely described, but in practice it is not easy to handle them. Therefore suitable approximations are used in practice, which will be considered in the next two sections. However the current-amplification factor and the feedback factor will first be derived in this section.

The current-amplification factor for the common-base connection of the intrinsic transistor is defined as

(ale)

ai - -

-ale dVc _ 0 It follows from fig. 1.5 th at

Y!b,t at = -

-Ytb,i

(1.42)

( 1.43)

and so this quantity can immediately be deduced from (1.37) and (1.38), yielding

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Ab À exp

->-2 2Àowo cosh Àowo

+

sinh Àowo

ai=

---

= ao--- ---2Àwo cosh Àwo

+

À sinh Àwo 2Àwo cosh Àwo

+

sinh Àwo The right-hand side of thi!l, equation is obtained with the aid of (1.36).

(1.44)

The feedback factor /-lee is defined as the ratio of the reverse and the forward transadmittance:

Yrb,i Ie 1 dw ( Ab)

/-lee

=

-

= -

-

-

ÀO coth ÀoWO

+ -

exp(-Ab) Y!b,i Ie

+

I e,8 ,8 d Ve 2wo

(l.45)

kT dw Ab

+

1

""" - - - exp (-Ab). q dVe Wo

It may be noted that /-lee is independent of frequency12) and changes into the

well-known Early factorIO) in the case of Ab

=

O. 1.3.2. Current-amplification factors ai and a' i

Af ter the rigorous treatment based upon the differential equation governing the behaviour of the holes in the base region, which has been summarized in the preceding section, a simpier and much less rigorous description will now be given of the main processes that occur in the intrinsic transistor. The treatment is based upon a simplified vector diagram13), shown in fig. 1.6. Let a sm all a.c.

fc,st=j~-

Rf,J

Vet,'

I c,/r

Fig. 1.6. Simplified vector diagram of the intrinsic transistor.

voltage Veb'

=

\leb' sin rot be applied between the emitter e and the base contactb'

of the intrinsic transistor of section 1.3.1. lt is assumed that the transistor is commonly biased (Veb' > 1/(3, - Veb' }> 1/(3). The hole concentration at the ernitter junction immediately follows the applied voltage, giving rise to a transit current ie,tr which varies in fase with Veb'. Some of these holes disappear by recombination between emitter and collector; hence the current ie, tr that reaches the collector is somewhat smaller than ie, tr. Furthermore, ie, tr and ie, tr are opposite in phase, owing to the sign convention.

However, tbe varying emitter-to-base voltage gives rise to a varying number of holes in the base region. For instance, at increasing ernitter voltage more holes have to be injected by the emitter than are removed by the collector. This is repr'esented by the broken line in fig. 1.4 in the case of zero drift field, Trus

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capacitive effect gives rise to "storage" currents ie,st and ie,st that are 90° out of phase with the transit currents. The ratio of both capacitive components to each other depends on the hole-concentration pattern in the base region and is indicated by the quantity Mt, which will be discussed below. The total capaci-tance associated with the varying number of holes is usually called the "diffusion capacitance" Ca. lts value may be estimated from fig. 1.4 when Ab

=

0, yielding

Fig. 1.7. Complex diagram of the current-amplification factor at of the intrinsic transistor. The angular frequency w = 2nf is indicated along the curves. The diagram holds for the

practical transistor when the subscript i is omitted.

The resulting a.c. emitter and collector currents ie and ie are now easily obtained. The phasedifference between thetwo currents is less than 180°, except at very low frequencies. The transit time of the holes has not been accounted for in the simplified treatment just given. Of course it is incorporated in the complete expression for ai, given by eq. (1.44), and represented in fig. 1.7. Usually, in-stead of the complicated expression (1.44) the following approximation is used, introduced by Thomas and Moll for frequencies up tofe,(14):

exp (- j1>i W/We,i) at "'" ao .

1

+

jw/we" (1.46)

In this expres sion We,i

=

2nfe,i denotes the angular frequency at which

la

i

l

=

!aov 2, and 1>, denotes the ph ase angle in excess of 90° at W

=

We,i. lt has been shown by Te Winkel15) that the excess phase angle is a linear function of the drift-field parameter, viz.

1>, = 0·221

+

0·098 Ab. (1.47) The frequency fe,i is mainly determined by the transit time of the injected car-riers through the base region. This may be illustrated in the following way. Let

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the transport of injected carriers (i.c. holes) completely be governed by diffusion. Then the time required by the injected holes to cover the distance Wo is wo2/Dp. In that case (Ab = 0) the frequency fe.i follows from (1.46) to be

1·21 Dp

le.i= - - .

n wo2

The numerical factor increases with increasing intern al drift field, which sweeps the holes more rapidly towards the collector.

Often it is not the cut-off frequency le.i that is used as a characteristic fre-quency of transistor performance, but the frefre-quency at which the real part of

a,

has decreased to the value 1/215.16). This frequency is called

11.i

and will be further discussed below. However, it may be noted here that the angular fre-quency W1.'

=

h/1.( very nearly equals the value (reCd)-l. Furthermore the frequencies/1.t and/e.t are mutually related by the re1ation

(1.48) which was analytically established15).

A comparison with eq. (1.47) shows th at

(1.49) Let us now consider the base current. It follows from the simplified treatment given above that it is built up from two components (cf. fig. 1.6):

the "recombination" component ib.r = ie.tr - ie.tr and the "storage" component ib.st

=

jwCdVeb'.

The recombination component may be written as ib.r

=

q iJp/Tt, where iJp

is the total number of excess carriers in the base region due to the a.c._ voltage Veb' and Tt their (average) lifetime. Then it follows th at the angular frequencyw-r at which both components are equal is identical with the reciprocal of the lifetime Tt, independent of the drift field.

Usually the a.c. base current is considered with respect to the a.c. collector current ie:

(1.50)

Since ib. r ~ ie. tr the angular frequency W-r is usually rather low, so that the ph ase difference between ie and ie. tr is still negligibly smalI. Hence it may be expected that the phase difference between ib and ie will be about 450

at W

=

wr The analytic expression of

at'

can be deduced from (1.44) with the aid of (1.50). A good approximation may be obtained from (1.46), yie1ding

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l-jif>,w/we,t ao

at'

"'"

ao' ao' = -

-I

+

jao' W/Wl,t ' l-ao (1.51)

It is represented by a semi-circle in the complex plane (cf. fig. 1.8). The fre-quency fea',t, defined as the frequency at which

l

at'l

=

1-

ao'v2, is determined by the lifetime of the injected carriers. It follows from (1.51) that

fl,t fl,t

!ca',i

=

-Vr======if>=2= "'" -ao-' .

(ao')2 _

2 i

1

+

if>t2

(1.52)

Fig. 1.8. Complex diagram of the current-amplification factor ai' of the intrinsic transistor. The angular frequency w = 2nf is indicated along the curve. The diagram holds for the

practical transistor when the subscript i is omitted.

Finally it follows from (1.50) and the definition offl,i that

l

a/I

=

1 atf = fl,i. Thus it appears thatfl,i is a characteristic frequency of transistor performance in both common-base and common-emitter configuration. For this reason it is of ten regarded as characteristic of the high-frequency performance of the intrinsic transistor.

1.3.3. Equivalent-circuit representation of common-base configuration

The small-signal properties of a transistor may be represented by an equiva-lent circuit, provided that its admittance matrix is in accordance with the expressions (1.37) to (1.40). Figure 1.9 presents an equivalent circuit of the intrinsic transistor in common-base connection. The emitter-junction admit-tance Ye,1 is represented by the parallel connection of the differential resistance re

=

kT/q1e and the capacitance (l/Mt)Cà. Thelatterrepresentstheholestorage in the base region, supplied by the emitter current. It has been shown th at in this way a rather good approximati0n. l!~ ,to the frequency fl,t of the input ad-mittance Yib,t is obtained15) (cf. eq. 1.37): : "

1 ,~ -'\ " " -'

...,

---4 ~ ~:'J ... ,' ~ J {IU\\"

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Yib,i

=

Y

e

,j

=

~

+

jWCd=-.!..(1

+ j _ W_ ); Wl,i = -.

re Mi re MiWl,i reCd (1.53)

.----_~---9c

...ie

r

r

Fig. 1.9. Equivalent-circuit representation of the intrinsic transistor in common-base con-nectioD. 10 fr{.

j's

6 4 2 ~

--V

./' / /

--

V

o

0 1 2 3 4 5 6 7 S - A b 9

Fig. 1.10. The quantity Mi, i.e. the reciprocal of the fraction of the diffusion capacitance Cd which is measured at the emitter-base junction of the intrinsic transistor, versus the drift-field

parameter Ab.

The quantity Mi depends on the drift field only and is graphically represented in fig. 1.10. The part of the injected holes that reaches the collector is represented by the current generator aiie,1, where ai may be approximated by eq. (1.46). The

collector-junction admittance Ye,1 is represented by the parallel connection of the resistance re,1 and the capacitance Ce,1' It can be deduced from eqs (1.40) and (1.37) that

re,1

=

/-tee re (1.54)

at low frequencies. The relation between YOb,i and Yib,i and hence between Ye,1

and Ye,j is less simple at higher frequencies, bilt Ye,j always remains very small. The internal feedback, due to the base-width modulation, is represented by a current generator that may be expressed in terms of the feedback factor /-tee and the emitter-junction.admittance (cf. eq. (l.45»:

Yrb,i

=

/-tee ai Ye,1' (1.55)

The admittance matrix of the intrinsic transistor may now be written as follows:

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Yb,i

=

Ye,i ( 1 -ai

-aiflee) Ye,i/Ye,j

(1.56)

lts elements may be given by either the complete analytic expressions or by

the approximations discussed above.

One quantity still needs to be discussed, because it is needed in section 1.4.2

and in chapter 3, viz. the quantity

l

ai

Ye,il. Of course a suitable approximation immediately follows from (1.46) and (1.53). However, a close examination of (1.44) and (1.37) shows that it is less frequency-dependent than is suggested by

the product of both equations. lts value has numerically been calculated from (1.37) and (1.44) and is graphically represented in fig. 1.11. A very good approxi-mation up tof

=

/l,i

now is

ao2

l

ai

Ye,il2 "'" re2(1

+

b2 W 2/Wl,i 2) , (1.57) where (1.58) ~.~ /<XiYej/2 re2

t

~ 1·2 I Ab=B

~

, .~

~~

. /

~

~

~

V

~

~

~

V

V ~

---

0

-"""

~ ~ ~

-

I--I--~ 1-0 04 0·2 0·3 04 O·S 0-6 0·7 0-8 O·g 1·0 1-1 1·2

-tff,.;

Fig. 1.11. Frequency dependence of the modulus of the transadmittance of transistors in common-base and in common-emitter connection. The curve holds for both the intrinsic and

the practical transistor. 1.3.4. The TPTt matrix

In this section the TPTt matrix of the intrinsic transistor in common-base connection will be derived from the admittance matrix. Knowledge of this matrix is needed in chapter 3. Furthermore, as an illustration, the quantities kT and Cl will be considered briefty. It follows from (1.56) and (1.2) that

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Now the TPTt matrix immediately follows from its definition, yielding (TPTt)b,t

=

It is easily verified that det (TPTt)

= -

jdet" Tj2. Hence we find that

(1.60>,

(1.61) So the gain-stability factor of the intrinsic transistor in common-base connec-ti on is independent of frequency and completely determined by the feedback factor. This result is not surprising because it directly follows from the defini-tion. The quantity kT will only be discussed for low frequencies. Then it follows

from (1.60), (1.10) and (1.54) that

(1.62)

The maximum available gain is therefore strongly determined by the feedback factor. This result might have been expected, because the transistor in common-base connection is mainly an impedance-transforming device.

Finally the "inherent stability factor" follows from (1.23), yielding

St

=

2-ao2

+

V

(2-a02)2-1.

In most cases the expression only slightly exceeds unity, hence a stabie operation of the intrinsic transistor can hardly be expected.

1.3.5. Equivalent circuit of common-emitter configuration

The fourpole representation of tbe common-emitter configuration is sbown in fig. ·1.12, wbere the relation between the extern al currents and voltages are given in terms of the admittance matrix Ye,t. Obviously a close relationship

must exist between the elements of the matrices Y b,t and Ye,t. This relationsbip

can be determined by making use ofthe following relations between the extern al currents and voltages:

ie

+

Îb

+

ie

=

0,

Vb'e

+

Vee

+

Veb'

= 0,

Veb' = -Vb'e, etc.

(1.63) (1.64) (1.65)

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e

c ie

' - - - - _ - A l _ ib =.J1.,; ~'e+Yr.,i vee

ie =Jfe,i ~'e +Yo.,i Vee

e ~ l~,l-~bl!>'~ dv. .-flli) ("~ie) ~ -0 Jf~,i =\hvbI! ~. ah -0

Y,.e,i=~~t

biC)

Se-O X, . -f_ O~,l -~bvUd vSe -0

Fig. 1.12. Admittance-matrix representation of the intrinsic transistor in common-emitter connection.

After substituting these equations in the definitions of the elements of the Ye,i matrix (cf. fig. 1.12), they have to be identical with the corresponding elements ofthe Yb,i matrix (cf. fig. 1.5), yielding

Yie,i

=

Yib,t

+

Ylb,t

+

Yrb,i

+

YOb,t "" Yib,t

+

Ylb,i

,-Yle,i

=

-(Ylb,t

+

YOb,i) "" - Ylb,i ,

Yre,i

= -

(Yrb,i

+

YOb,i)

and

Yoe,i

=

YOb,i.

The admittanee matrix now immediately follows from (1.56), yielding

ai!-lee-Ye'i/Ye'i) "" Ye,i/Ye,i ( I- ai "" Ye,i ai ai!-lee-Ye'i/Ye'i) . Ye,i/Ye,i (1.66) (1.67) (1.68) (1.69) (1.70)

The approximations in (1.66), (1.67) and (1.70) are justified because !-lee and

Ye,i/Ye,i are very small quantities. The most severe approximation is made in

Yie,i at low frequencies wh ere the condition ao' !-lee ~ 1 must hold. However, this condition is fulfilled in most practical cases.

Before giving an equivalent circuit of the common-emitter connection we

shall discuss the input admittance Yie,i. Using the approximation (1.53) we obtain

(1.71)

A fairly good approximation of Yie,t is obtained when (1.46) is substituted,

yielding

l- ao w

Yie,i ""- - + j - - l- -- ao + jwCa. (1.72)

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The input capacitance is now equal to the tot al diffusion capacitance, illus-trated by the vector diagram of fig. 1.6. In fig. 1.13 an equivalent circuit is shown that represents the admittance matrix (l.70) in a good approximation.

The following identity has been used in deriving the circuit:

Y!e,i ,. ,

Y!e,t Vb'e

= -

Ye,1 Vb'e

=

ai Zb . Yte,i

. b'

L

11--__.---",...-...., .---_ ... _ _ c __ ~

e e

Fig. 1.13. Equivalent-circuit representation of the intrinsic transistor in common-emitter connection.

1.4. The practical transistor

1.4.1. Equivalent circuit

The intrinsic transistor studied in the preceding section in fact extends be-tween the base boundaries of the depletion layers of the emitter-to-base and the collector-to-base junctions (cf. fig. 1.3). Consequently the elements th at have to be added first in order to describe a practical transistor are the capacitances of these depletion layers, called Ceb and CCb, respectively. The emitter depletion-layer capacitance Ceb is situated in the equivalent circuit in parallel with the emitter-junction admittance Ye,i (fig. 1.14). The situation of the collector depletion-Iayer capacitance will be discussed below.

h c

e L _____________ IntrinsIc transistor ~ ______ ~ e

Fig. 1.14. A possible equivalent-circuit representation of the practical transistor (both alloy and diffused-base transistor) in common-emitter connection.

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Another effect that has to be considered is the voltage drop caused by the current carriers on their way from the contacts to the active part of the transistor. Commonly the ohmic resistances of the emitter and the collector regions are neglected because both regions consist of low-reslstivity material. This is not the case, ho wever, with the ohm ic re si stance of the base region, the resistivity of this material being considerably higher in order to ensure a good emitter efficiency and a high breakdown voltage. The base resistance Rb is situated in

the equivalent circuit between the base contact bi of the intrinsic transistor and the extern al practical base contact b.

In fact the base resistance. is not quite constant, as the effective base width varies with the a.c. collector potential. So the base current, flowing through a varying resistance, causes a varying voltage drop across it. This effect is repre-sented in the equivalent circuit by a voltage sOUrCe,ubeVeb' in series with the base resistance17). The collector depletion-Iayer capacitance Ceb must be situated in parallel with the collector leakage re si stance re,l. The situation of this com-bination in the equivalent circuit depends on the geometry of the transistor. In the case of an alloy transistor the emitter and the collector are situated opposite to each other, the emitter being smaller than the collector. A useful equivalent-circuit representation is th en obtained by dividing the base resistance into two parts Rbl and Rb2 and connecting the collector admittance between bil and e

(fig. 1.14). The geometry of a diffused-base transistor is generally such that the collector is situated opposite to both the base contact and the emitter. The in-fluence of the collector then extends to close to the base contact and therefore in the equivalent-circuit representation the collector admittance is divided into two parts, one part being connected between bil and e and the other part be-tween bi and CIS). Sometimes an additional leakage current flows along the

.surface from the collector to the base contact. This effect can be incorporated in the resistance situated between bil and e. For good transistors the surface leakage current is negligibly small, and furthermore it appears in practice that for an appreciable class of transistors the real part of the collector admittance is quite small, so that it can be omittedI9). So the representation of fig. 1.14 is sufficiently accurate to represent a considerable class of transistors, both alloy and diffused-base transistors.

In addition to the influence of the collector represented by the admittance Ceb-re,l some other effects of the collector action need to be discussed20). The carrier transit time in the collector depletion layer causes a phase difference at high frequencies between the entering and leaving carriers. Calculations show that this effect can be disregarded here because the measurements th at will be discussed later were carried out at frequencies below 15 Mc/s. Avalanche multi-plication in the depletion layer can be neglected when the d.c. voltage is suf-ficiently small. Finally, multiplication effects in the collector body are also neglected as the minority-carrier density is assumed to be very low.

(30)

1.4.2. Admittanee matrix in common-emitter configuration

We first discuss the current-amplification factors a and a' and the input admittance in coIl).mon base. Obviously these quantities deviate from their va lues for the intrinsic transistor, owing to the presence ofthe emitter depletion-layer capacitance Ceb. It is easily seen that

wh ere

a=

~

a'

1

+

e' I- at

+

e and Ye

=

Ye.i(1

+

e),

Ye

=

Ye.i

+

jwCeb and e= -jwCeb - .

Ye,i

By analogy with the intrinsic transistor the frequency

Ir

is defined as 1 1 1

f1 = ao,/crx' = - = - -.

2n re(Cd

+

Ceb) 2nreCe

(1.73)

(1.74)

Now fcrx' is that frequency at which la'l

=

1-

ao'

V

i:-

When the expressions (1.73) are inserted in (1.46), (1.51) and (~.53), respectively, a set of approximate expressions is obtained for these important quantities ofthe practical transistor.

It can be shown, however, that these expressions can be brought into the same form as (1.46), (1.51) and (1.53), provided that the constants are modified in the following way21):

n

+

1 lP, n

+

1

K

=

Kt, lP= and M

=

Mt, (1.75)

Kin

+

1 Kin

+

1 Mtn

+

1 where

The practical approximations are valid whenf .;;

Ir

:

exp (- jlPw/wc) a

=

ao ; Wc

=

KWl, 1

+

jw/wc (l.46a) , , 1 - jlPw/wc a

=

ao - - ---,---,- -1

+

jaO'W/Wl (1.5Ia) . and Ytb "'" Ye =

~

+

jw Ce =

~

( 1

+

j

~

);

Wl = - . re M re MWl reCe (1.53a)

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The admittance matrix can be determined from the equivalent circuit of fig. 1.14. The complete expressions become rather complicated and therefore suita-bIe approximations are introduced. These approximations have to be valid feir transistors of practical interest, e.g. transistors that yield a stabie power gain greater than unity in a practical circuit. This means that they must have a suf-ficiently high value of the gain-stability factor.

At medium and high frequencies the influence of the feedback parameters

!-tee and !-tbc is negligibly smalI. Then the equivalent circuit can be simplified to that of fig. 1.15, and a calculation shows that

I

Yte

I

=

laYe-Ye{ 1

+

Rb2Ye(1-a)

}

I

Yre IYe{ 1

+

Rb2Ye(1- a)}1 (1. 76)

• (i) . (i)

IJ

.!4

-

Ze C

RbT h" Rb2 0'

C(i~

(~')

t~C (Y..-

(IXia)

Ceb

v..t

b.

=

y. .

e

. e;l-J-é,j

ai a

be

ii')

(e

'

l) I(:e

T . C

e e e e Jé =

rc;t

JW eb

/nfrinsic transistor

Fig. 1.15. Simplified equivalent·circuit representation of the practical transistor in common-emitter connection.

Hence a high value of the gain-stability factor requires that

Now because of the properties that laYel

=

laiYe,11 "'" ao/re (cf. eq. (1.57)) and that

l

a'

l

=

Wl/ W wh en W > Wert.' (cf. eq. (1.51)), this condition reads:

Ce

Ce,b ~-

----1

+

Rb2/re (1. 77)

Henceforth it is assumed that (1.77) is fulfilled. Then the admittance matrix for medium and high frequencies is found to be

Ye

Ye

=

'

-1

+

RbYe(l- a) , (1.78)

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For low frequencies the matrix has to be extended by the addition of terms containing /-lee and /-lbc to the second column. The complete expressions of the matrix elements are given in appendix A, where they are further evaluated in terms of frequency in order to enable a confrontation with experiments (cf. chapter 4). The input admittance is particularly of interest when noise is concern-ed. It immediately follows fr om (1.78) that

(I-a) Ye

Yle

=

.

1

+

RbYe(l-a) (1.79)

The second term in the denominator is usually small but ought not to be neglect-ed. A graphical representation of Yte, assuming that eq. (1.53a) holds, is shown in fig. 1.16 where the lirnjting values at low and high frequencies are indicated. The actual behaviour of Yte deviates somewhat at high frequencies.

jOJq,

Î

-~,

1 1

li'b~r, li'b

Fig. 1.16. Complex diagram of the approximated input admittance of the practical transistor in cornrnon-emitter connection (cf. eq. (1.79)).

1.4.3. Admittanee matrix in common-base configuration

The admütance-matrix elements of this configuration can be obtained from those of the common-emitter configuration with the aid of the inverse relations of (1.66) to (1.69). Now the requirement IY/b/Yrbl ~ 1 leads to the condition

(1.80) Assuming that this condition is fulfilled, the admittance matrix for medium and high frequencies reads:

(1.81) Ye

Yb

=

---,----1

+

RbYe(l-a) a

At low frequencies terms containing /-lee and /-lbc must be added to the second column, but this will not be considered here. An examination of (1.81) shows that negative feedback occurs across the outer part of the base resistance. This

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is unlike the behaviour of the common-emitter configuration wh ere positive feedback occurs across the inner part of the base resistance.

The behaviour of the input admittance Yib is more complicated than th at of Yie, because it strongly depends on the base resistance. As an illustration Yib is graphically represented in fig. 1.17 for two different cases, viz. Rb/re ~ 1 and ao' > Rb/re ~ 1. Note th at the sign ofthe imaginary part of Yib is opposite to that of Yie when Rb > re. This becomes important wh en noise is concerned, to be discussed in chapter 3. \ \ \ \ \ \ "-Wf !jwCib I I

t

i IR <Al 1-11«1 I re a' > Rb __ 1 o re

Fig. 1.17. Complex diagram of the approximated input admittance of the practical transistor in common-base connection at two different values of the ratio of the base resistance to the emitter differential resistance. The applied approximation (cf. eq. (J .81» is not allowed beyond

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2. NOISE THEORY OF THE INTRINSIC TRANSISTOR

2.1. Introduction

Before dealing with transistor noise some basic definitions and concepts of noise theory are briefly treated (see references 22 and 23). Some fundamental noise processes are tben considered, after which tbe fundamental noise sources of transistors are investigated.

The brief description of some basic definitions and concepts of the theory of noise does not pretend to be either rigorous or complete, but only serves to justify some methods applied in the next section and in chapter 3.

The quantity th at represents some property of a physical process that varies in a quite irregular way is called a random (or stochastic) variabIe. For instance the indication of a sensitive voltmeter, connected to the terminals of aresistor in thermodynamic equilibrium, varies in a quite irregular way and may be used to represent the voltage fluctuations due to the tbermal agitation of the free e1ectrons in the resistor.

A sample function of a random process represents the values of the random variabie observed during a certain interval of time.

Let y(t) represent some property of a random process. The statistical ave rage (or expectation value) E{y(t) }of y(t) at the instant t is obtained from simultane-ously recorded sample functions of a large number of identical systems. The statistical average of the product of the random variabie at two different, but fixed, instants of time tI and t2 is called the autocorrelation function q>y(tl, t2) of the random process.

When the statistical average of tbe random variabIe is independent of time and the autocorrelation function only depends on the time difference r

=

tl-t2 the process is a stationary random process:

cpy(r)

=

E{y(t) y*(t-r)}. (2.1)

Another way of averaging a random variabie is to determine its mean value during a long time interval T, measured on one system. Ergodic random proc-esses have the property that the time average equals the statistical average and the time-autocorrelation function equals the statistical autocorrelation function

T

y(t )= lim

~

f

y(t) dt = E{y(t)}

T-+oo

(2.2)

o and

T

cpy(r)

=

lim

~fy(t)y*(t-r)dt

=

E{y(t)y*(t- r)}.

T-+oo •

(2.3)

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Henceforth it is assumed that we are dealing with ergodic stationary random processes.

An important procedure for determining the stochastic properties of a ran-dom process is the frequency decomposition of the observed fluctuations. The following method is usually applied. Let y(t) be a sample function of the consid-ered stationary random process (-00 ~ t ~ (0). Define the truncated sample

function YT(t) in the following way:

I

Y(t)inO ~ t ~ T,

YT(t)

=

o

everywhere else.

It is assumed that the Fourier transform of YT(t) exists when YT(t) is defined in any interval 0 ~ t ~ T:

YT(t) = J AT(f) expjwt dj (w = 2nl), (2.4)

- 0 0

where

AT(f) = J YT(t) exp- jwt dt.

- 00

The correlation function rpy(r) for a given r of the random process can now be expressed in terms ofthe Fourier components AT(f):

T 00

rpy(r) = lim

~JYT(tjYT*(t

-

'f)

dt = lim

~JAT(f)AT*

(f)expjwrdf (2.5)

T~oo T~oo

o - 0 0

The spectral density Sy(f) of the fluctuations at the frequency j is usually defined as follows*):

(2.6)

Now it follows from (2.5) and (2.6) th at the autocorrelation function and the spectral density are each other's Fourier transform:

*) Strictly speaking the definition should be read:

10

+

tLlI

Sy (10) = lim }I lim

~

f

AT (I) AT* (I) dj, (2.6a)

Ll/~O T~oo/o-tL1f

because AT(f) is a random variabIe and the varianee of

~

AT(f) AT*(f) does not necessarily vanish in the limit T = 00.

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