LUMPED MODEL TRANSISTOR NOISE Á2ALÏSIS
DOXAN ILRAH.Li
KKYA
 THESIS
su4nitted to
OREGON STATE IJNIVE:RSITY
in
rti1 fulfillment of
the reçiirenients for the
deree of
akSTLR OF SCIENCE
June 1962
PFEOVED ¡
Profsor
of E1ectric1 Engineering
In Charge of Major
Head of Lepartient of Electrical Engineering
Chirmn
of School Graduate Comniitto.
Dean of Graduate School
Date thesis
i
Typed by Mary
resented
Jd&Lfl$
7
ACKNOVLEDOE4ENT
The work reported in this thesis was performed
under
tÎie
direction of J. C. Looney, Professor of
Eìectrical
n4neering at Oregon State University.
to
xress his appreciation
suggestions,
The author wishes
to Professor Looney for his
constructive criticism and advice, and also
extend.s his appreciation to Professor
L. N.
Stone,
Professor
h.
J. Oorth.iys of Electrical Yngineerin,
Professor
A.
'1.
Lonseth
of'
the
their valu bie encouragement;
and
athematics Department for
and of course to his wife.
TAULE OF CONT±TS
P
IN2RODUCTION .
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TRANSISTOhNOISEi'iODr.L,,...a..e....e.....
14
LiJI'4P;D
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4Qtij,S
NOISE THEO!
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;ITH LUMPiiD
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MOLiEL PARAMETERS
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CORRELATION bETWEEN LUNPED MODEL PARAMETERS
UANTITIES ,
WITHNEASURLBLE
LtJPEJ)
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PAFJETEBS
MODEL SM.LL SIuNAL
THE NOISE FIuURE
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COi4PARISON OF CALCULATED AND MEASURED DATA
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CO11PARISON WITH THE RESULTS OF CTHiRS
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XPERIMENTAL INVESTIGATION
CONCLUSION
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BIBLIOGRAPHY
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APPFDIX
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16
19
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30
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32
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38
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39
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LIST OF FIGEThÌ5
Pge
Figure
is
Lumped noae1
Genrtion
of a pnp oifí'usion transistor
currents
nd recombixiatlon
noise gener&tor and
:3.
TAie
4.
T1eChaniSß for
d'
betweentwojunetions
p
5.
Lumped pnp transistor
wlth
6.
Lumped pup
kir
e
itioúel
Transistor
:I4e
15.
16.
17.
18,
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4.
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9
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12
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p1ifier nOise
iaociel
7
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gonertors
.
a
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.
s
.
vs. log f
log
f
Pn vs. log
f
vS.
1ocy
a
e
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s
e
a
e
a
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a
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a
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a
a
:igra.n of instruiìenttiou
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20
e
a
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a
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23
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23
s
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s
28
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28
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28
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31
fiure
.
*
a
Cd.cu1ated nd neasured noise figure
function of 'E' for
R5
opt.
20
e
Calculated and reasured noise figire
function of R, for I ot. condition
*
.
e
.
mesured noise
15
a
oie
R8
.
e
Trpic1 spectrwn level ehractristic for G.
Type 1390-B Randor
Generttor .
Ca1cu1ted
function of
.
.
Correlation of noise ouree of an am1tfjer
into V and I voltage and current genemtore
F
12
15
trnsitor model with noise
sin1 y parbmeterB . . .
9
i3.
.
rbb
Equivalent ciicuit forai of Figure
i2.
.
type aterta1
L
i1.
.
................
7. Small
loe
.
.
condition
.
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e
e
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Ft.
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31.
a
a
a
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a
35
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36
a
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as a
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37
NOISES
Any undesired sound.
by extension, noise is
any unwanted disturbance within & useful
fre-
quency band, such as undesired electric waves
in eny transmission channel or device.
Such
disturbances when produced by other services
are called interference.
The Interntion'l Lictiorxry
of Physics and Electronics
MLYSIS
LUNPED MODEL TIi3SISTOR NOISE
IÏIIkOLJCTION
The
rocesses of electron-uiole-pair cretion,
recombination form the ba8is of tr.nsistor
accompanies these processes.
These
ction.
iotion, and
Noise unavoidably
rocsses are random,
so
related
noise is randoni in nature.
At room temperature
there is
a
electron pairs and on the averae an
electrons with holes.
upon
equa].
energy by a valence
in dyninio equilibrium with
conduction electrons and holes per unit
the
rete of recombination of
ir is
Forma tion of a hole-electron
the chance acquisition of
recombination is
continual formation of hole-
dependent
electron and
the population of
of volume fluctuating about
average value.
It is
desirable to use
s lumped
transistor noise because of its
physics of tne
transistor.
cnaracter of noise in
comare the results
model.
transstr model
to investigate
inherent simplicity with regard to
The object of this work is to study the
transistors tr laboratory
measurements and to
with those predicted by a simple lumped noise
2
LJ4PD
Serniconäuctor
having two
ìOIELS
kthd of carriere, holes
exhibit a more complicated mode of behavior th.n vacuum
and electrons,
tubes.
The processes of ionization of atoms to provide holes and
electrons as well as the
rcouìbine,
rocesses by which holes and electron
ieducin, both carrier
.opulation with each event
re
new,
Luniped models due
to J.
G.
Linvill (8, 9, p,
to-one correspondence to the physic&l
semiconductor,
devices,
the lumped nodels
The models used
47) bear a one-
'rocesses encountered in the
apøroximatin the distributed
ordinarily
exhibit relationshiis between
current and excess density of carriers.
A síngle aoiel is then
applicable both for snall signals and large sigials.
The behavior of a transistor is simply described in tenas of
the behmvior of the two
junctions, since the tranlstor is essentially
Lut a
two diodes sharing a common region.
transistor
differently from the two diodes placed in the sa'e
Fire
Lekmaves
quite
enveloLe.
i is a simplified lumped model developed Ìy J. G.
Lirmvill (8, 9, p. 47)
is represented
for transistors.
In the model, each junction
y a rectangular box snd serves to transform the
voltage V applied across it to an excess ainority carrier density
just inside the
base
re4on.
The property of the junction can be
expressed by Ecuations (i) snd (2).
Fee = Pn
g V
-
/
(1)
3
ec
= P
(exp
qV
- i)
(2)
kT
Wkaere
xce8s censity at e;iitter.
?ee
Excess density at collector.
Pn
Minority carrier density in
q
Electron charge.
T
Junction tenperture in ¿bolute degrees in K°.
k
type base.
oltzman Constant.
ApplIed forward voltae across the junction.
V
As can be seen, the
bas
4
leed.
The symbols
iodeI has synaaetry about the axis of the
hei,
kTee
r preeent to
.
a
first-order approx-
imtion the distributed generation-recombinatlon echanisa of ±iinority
carriera near the collector and emitter junction respectively. They
are
Ve
cllcd
coìibinance e1eenta which
ha"e
the analogy of conductance,
can define cobinance as
as conductances
C
This analo
I
V
p
is true if the reference for expressing excess
density is selected in &ccordnee with the Equation (lb).
11d
represents to a
nechanism of
first
iInority carriers
Sc end S0
order p;iroximation the diffusion
between the two junctions.
are the symbols which represent, to a
first
order
approximation, the storage iaechanisn of ninority carriers near the
collector and emitter jiuiction, respectively.
to the capacitors of an electric circuit.
They are analogous
We can see that
p
r°
V
(r
Fec
SCI
Tc
I
j
Fisura
(i)
Lumpd modJ.of
tTJni.sor.
.
PNP di-Fu3ion
C
I
by
cn
corbin.nce elernent
The
ties
jÇ!ct
S
expressed in ieasurable
be
connecting the coflector to the
(,
current rneasuring device
that
O.
y
9). Since
bse tLrouh
VC = O,
a low
qUiXti-
ïnpedce
Ecuation (2) shows
forward biasing the eniitter with a current
short circuit collector current can
rnesured, so
be
(3a)
e
It will
later that
be ahown
=
(3b)
ee
sii:irí
explained in Ejuation (h2)
a
y;Hd
1eg..
i:-
I
kid
(lid
1d
f'
+ 1ec
In the case where ec
col?n in EQuations (i) nd (2), I
lector leakage current can be found by esurin the reverse- biased
collector current with the íaitter o'en
I co
P
=
4
(-ci
H
ee +-
(1ee
and will Le
hd li 4- k:i
cc
ee j ec)
-
Pn (ilci Hee
+
to
(
, 9)
±
Sidlarly
leo
eu1
11d
±
'1ec
+
Hee
k)
()
Hi.)
These equations were derived ±rorn E;qutions (37) and (33).
The
storance elements are deteriineá froi the alpha cutoff
frequencies
constant,
and
ec
f1
constant,
as followsz
so
When
collector
volte is
s.
i
2TTfN
Hec±
_________
H+
(7)
11d
Sc
211 f0(1 -
(8)
Hd
We can solve Equations (3), (4), (5), (6), (7),
a
()
to get for
np transistor
(leo (i -
Hee
N5)
(9)
Pfl(i_o(NcXI)
(i
Hec = -
-0i)
(io)
Pn (1 - 0NiXI)
EcoO(I
'eo'N
(u)
n
= -
'eo
P
217
o(N (i
-°N
(12)
oi)
leo
Sc
P
27f
(13)
ixI (i
<N 0<i)
7
itOISE THEORY WITH LUMPED MODEL PÁFAMETERS
difference of the aeneration and recombination currents.
ibriun no net current flows, which means
eual
mist be
current.
is
the net recombination currerLt
In a lump of n material
that genertion
In
the
eiil-
current
to and in the opposite direction from the recoinbination
The average
recombination nd generation
currents, (1, 3,
10, p. 46-47)
qPV
/7.
(14)
q P
q
electron charge
V
Volume
Pn
equilibrium hole density in n
:c
terial
mean hole life ti:e in n materi1
p
almost independent of hole density (P s
'g
depends on hole concentration, so it can be
T
.q(P+
1e)
e
bUt 'r
modified to the fox
V
(15)
where
excess hole censity.
We
can
rapresent these
connected between minority
two currents as current generEtors
¿d
majority carrìer lines.
They
ìndeendent, random, ipu1sive in n turc and exhibit full shot
effect (Van aer Ziel) (Lt, p
47),
re
Since
fluctution current due to th
of electric charges passes through th
coruculir structure
seilconductor materid, an
equilibriura condition can be restored by supplying a
noise current
generator to each of thei in parallel with
'r a
This additional noise current is noniially
the d-c current I and so the superposition
rents can
:.e
applied to this case.
eral
Fiure
cornpared with
rinciple of noise
Now 'r and
I
as an exact d-c current without any
acconipaxying
the terninaïs (see
'g generators.
cui'-
can &e rgarded
randoni part at
2).
'r
ig =
'g
-F-
±
nr
(16)
iflg
Two theorems from the theory of fluctuation
in temperature-lImited
tube currents
Ll1 be useful. tiere.
The first theorem concerns a curreiit I
consisting of many short
current pulses of ecual charge content
Q
tiaie.+
It states;
the time
istributed randomly in
At frequencies low compared to the
reciprocal of
durtion of one pulse,
f1uctuation in
the current
I
&re
described by
U-n:) : 2 Q2
Li f
(17)
where
+
*
complex conjugate
(____)
mean value averaged o7er a time auch
longer than
The
chrgn content
is equal
to
fi
-
dt integrated over one pulse.
Lr-
]lr+
L
Lg
gun.
)
Gnrrion
curr-1n1s.
cgnd Rcornbrìafion
I',
* average nuniber of pulses
If the pulses are unidirectional
I
cn
EQution (17)
so
(i j*)
The secon
theDre
unit tiie.
N0 is equal to the direct current
written as
he
2
Ç,
or
.
f
I
(]3)
st tes wuen several currents I
are flowing
into a coxnon terilhinal x, but fron several circuit branches
XL
(n = 1, 2,
totd current
.....), the
I
consists of many short
unidirectioni1 current ?uìses of eaual charge content q distrii'uted
If
randomly in time.
is to pass
a given pulse
throuch but one of
tne circuit brncììes xn, nd the prob..bility of passage throuh a
given branch
aì
is to be the sane for all pulses, then, at frequen-
cies low coiip&red to
time
reci. focal of time duration of ono puise,
fluctuation in the currents I
'V
(19)
.
express our noise generators in the form,
(i
'
(i
i*
nr
nr'
1*
ng
= 2 q
tr
(20)
'2qIgLf.
(21)
ng'
These equations are derived
where
described by,
j *) = 2 q
X
(j
Then we can
i
for "spot" or narrow-bad noise currente
f is the incremental bandwith.
e
can combine Fcuatioz
and (21) into one equation
(i
ana from
(
i
)
E.
=
nr
jr
(ing
j)
2 q
(Ir
+
1g)
Af
(22)
nations (14) and (15)
i)
y
= 2 q
(
q
±
e)v)
Af
(23)
(
2))
u
!
q
c.
P) A
f
p
2q1(2P-trom the 1uped
ioui (°,
(64)
n. h7)
(25)
so
i)
(in
= 2 q
r
(2
P-i-
This noise generator nd H. are
There
and
and
shown in Figure
re two kinds of carriers
electrons, aìd there are
carriers
(26)
e)
resent in
two rnechanisms
diffusion of carriers,
The
a region in which
seiniconth.ictors,
holes and electrons in the
virtue of their thermal
their ensity ts uniform
place an iraaginary plane inside an count the holes
passing through
it per unit
would find on the average
passing in one
tie
tht
holes
drïft of
of current flow,
seniconcuctor rnbterLl re in rndoni tìotiorx
velocities. In
3.
in each of the
we
can
or electrons
directions.
e
the net current .s zero , the currents
direction being equal to the currents assing in the
other for either holes or electrons,
in which tne censit3r of carriers
Now
if
we
consider
are not equal, there
two regions
i certainly
net flow of carriers from the region of hih density to the region
of low density. The net flo is :roportional to the gr&dient of
density
Therefore, as in Figure
and the
(i
l)th region
4,
due to
the hole current b.twen the
diffusion is
(i)th
a
.1
Figur. (3)
Th Noise
and Hr
Gnrator
p___ ___
1gur
(4-)
Dif-fusion
°F
mzchanisni
p±ypmi1
bczFwan
+VVO
jund-ìon5.
13
ipd (i, i
[p(i)
-j--
1)
-p(i±
=
A
Âqt
i)]
-
(27)
&d (9, p. 47),
hpd=
A q D0
(26)
L\x
then
pd
(i, i
-j-
i)
Hpd
=
{2(i)
- p(i +
'pd(i) - 1pd(i
-j--
pd consista of -wo rardom indpendent
seen
1)
(29a)
i)
(29b)
impulse
currents as can be
from Equation (2%,b) and should exhibIt full shot effect
(12, p. 47). Giving the
same
resonind as Equation (16)
we
can
express the diffusion current as
'd
-F-
(30)
1nd
then
i
(i,
I ± i)
= 'dt
-ndi
- 1d(t
+ i')
-
(31)
+
1)
Using the vacuu tube theory used in the derivation of
(23
(1) we can express the diffusion
Equations
current noise
generator as,
(-a
ici)
2 q (ia(i)±
2 q (lid
1d'
P(j) ±
+
:i)
e(i
-i-
i))
L
f
(32)
1h.
TRANSISTOR NOISE MODE.
Fiure
5
shows the intrinsic, lumped,
pnp trnsistor nìodel
(3, 9, p. 47) to which the extrinsic base resistance has been added
to inprove the accuracy of the nodel. All the noise iray be accounted
for
associating a noise current generator with each of the con-
L
ductces &pearing
in the lwiiped iiocel and associating a thermal
noise voltage generator with the extrinsic resistive eieiient as
Fiure
shown in
e
6.
on write the corresponing equations for the n3ise generators
as
(me
e) = 2 q Lïr Hee (2 P
) = 2 q
(d ')
(Cflb
eb)
Lf 1ec
2 q
s 4
11d
T
Af
+ see)
(33)
eec)
(4.)
(2
(2 Pn
rbb
S
4
ee
+
fec)
(5)
(3ö)
fv
Figur(5)
rvcC
D
Hd
Lumd
pnp +ransÌsÖr
if1-ì
rnodczl
rbb.
¿nd
çv
C
I
Rd
H
gur-
H
() Lunip4
wi-h noi
modal
rìar
or-s.
16
CORRELATION BETWEEN LU4PTD NODEL Pi\RAMETERS
WITH MEASURkI4E QUANTITIES
In the lumped model of Figure
we can write the following
5
relationshi3a between the d-c values of the excess carrier density
and current
vrt
±
'E = (Hee
IC = - Hd
les
s
lid)
ee - hd
ee
+
(,
p. 47),
9,
(rLec#
(37)
ec
lid)
(33)
ec
The above equations express the f&ct that for the lurnped model
the f1oT of carriers ecross the bese region is a linear
carrier
ensity near the junctions.
shm,
function of
Unuer the conditions that the
collector volt&.ge is held zero, Ecuations (i) and (2) show that
becomes zero, and we can
the collector I
ec
rite the ratio of the currents coming out
to th&t injected into the emitter I
as in
quation
(39),
C
(exp.
1'ee
(txp.
(i)
i)
(2)
qV
iT
.;
The
i)
IÇT
eC
-
VE
(V0 -
3)
transistor
:lE_________
dee
Hd
car.
be
cXN
.
ooerated in the reverse direction with
current injectea at the collector holding V
O.
We can
define,
kki
-
.-
IC
In the
(39)
saine
(Vj,
=
o)
:
=
Heo+
(40)
d
manner we can evaluate
'CO eraitter leakage currenta,
c'K
I
collector
ie!ae
ctrreit and
r7
-
=
±
(Hee
1ec
p.S,
±
'EO =
By
(eec
H2 e
He
H
ee
-
Ha)
+
±
-i--
Hd
H)
--
11d
ll
(41)
Ha
Re c
+
insoection of EcuLtion (39), (40), (41), (2)
saze equation founo by Moll
cx
(4')
s
H
we
(4,),
i
=
()
it_o
From
the above ecuations
=
C)
we
can derive
1E0
H
-
can find the
?
(1
Pn
(i
-
1) - -Pn
(î -
.a__
oc1)
i]
-
o
O(
r
'
"ee
:- --
.
o(i)
-
ri (1
O(:
%f)
o( LI1À,
H----- P (1 - O(N °( i)
u
(44)
°<N
(1
- oc i)
(45)
O(
-
P
(1
I
leo
-°N
(46)
Accoring to Shock1ey' theory (1g) for th low injection
levels (under noriia]. oprating condition8) in a pnp transistor
<
VC
I
qV0j
Therefore
then,
O
»
it will
kT
(47)
s
he a good ap3roxiniation
to write,
18
'ec
-
which will be used 1ter,
(4e)
i»
LUNPED MODEL &4ALL SIGNLL PiRiiiETiLFS
In oraer to derive noise figure foiiu1as
in
model parameters, it is necessary to
lent circuit for the
iodel.
lumped
e!nitter orientation are used
of Equations (I7) and (4),
under nonul operatin
they
re not zero.
than
Stili
The y
shown in
l2 and y
conditions.
sml1
prieters
lixaped
sia1
euiva-
for the
Figure 7. ¡s
e
common-
conse:uence
would be practically zero
But vrious experiments showed
Early (3) tried to explain this by
the modula-
tion of the thickness
voltu'e.
s
evelor a
texs of
of the bese layer due to the
collector ac
l2 and Y22 are of a smaller order of maiitude
ii and y
an9
my
Le
neglected in practical eases.
Their
contribution to noise is thus negligible. Lxpressin the
ecuivalent
circuit variables in te1ns of the
variables in the lumped model,
neglecting the noise generators, as shown
in Figure 8, we find,
Il
-
1'ee
=-P
(Hee
kT
+
j
1JS)
___
(Hee
j
Ve
(49)
hence,
vi = _ Ve
(
50)
so
Il = F
T
exp
(c Ve
(hee
\kT
q
=
y11 a
12
-
ee 'd
exp
(
j
WSe) Vl
Ve
T
ee
Se)
(
51)
(52)
)
(7)
I1V1
y Framars.
+IaVa
1
+Y22
Small
c'nal
Ii
p
r
-
1
vi
ylvI
I
Figure
(8)
Euivaln
crcuÌt o
urz(7).
21
so,
(qV"\
q
eed
1
y,'
.1
ICT
Ve
-
exp
LT
(qv'
T
I
.
)
-"
22
THE NOIgE FIGURE
The re8ultin.-; nodei shown in Figure 9 represents
e
common
eriitter pnp transistor amplifier with resistive source and
with three independent
stiot
noise voltage generator.
of source resistance R
The genera]. noïse equation
i
nd one
'.
therrnal
a function
for a narrow-band is,
B
N.
F
noise current generators
1od
-
-t-
C
()
.....
s
This is a
re.ctive
:enerl ecuation
escribing any amplifier.
is held constant
coiraonent of the source
and
If tue
the resistive
coxnonent is varied, the variEtion of noise factor should follow
this law.
The noise performance of the
characterized by the parameters
which is our case.
a
.iii4ifier can therefore
be
B and C if the source is resistive
If the source is partly reictive, another texìi is
necessary which is beyond the scope of this paper.
This representation is advantageous because the amplification
properties of the aïnclifiers do not appear in the noise factor
calculation.
Over an incremental frequency band
f,
be represented by a noise free arnalifier
genertor and a
noise
as shown in Figure 10.
noisy amplifier
Figure 9 shows
a
can
with a noise voltage
current generator at the input, V
nected at the input of the amplifier.
factor is (2),
a
source
and
resistance con-
The forraula for the noise
23
ur(S)
rnplifr
Tran5Isfor
NÖì5cz
mocZI
-r
__
-4-
I
NO5
_1i'
FIERJVa
1-REE
In
y
MPLÌÌE
(k)
4igur(iC) CorrczLhcn
oj
rnpli1uzr nfrû
nid c.urriznt
rioì
öurcs
Vn and
1En
/oa
noIscL g(Zfl(ZrâFÖrS.
an
24
F=1+
(vv) ± Il)
I) +
For frequencies at which
()
(v
O
is a» roximateiy constant,
usually negligible (see Equation
T R
4
)
I
is
so that
v*)
(V
±
=
r
Af
De fining
neq
(v
v)
4 kT
Af
(57)
Eu&tion (45) becomes
F
i
*
Assuming no
find
V
aIt
correlation between the noise genertors,
frozu Fip.ire lOb with the
I
transistor in
emitter orientation as suggested tr Becing (2)
is shown in the
appendix.
V
aiid
I
v)
= 2 q
Af
{
(I
I)
= 2
c;
1f f
rhb+ re
I
(v
2
In)
»rounded
The derivtion
.
2
Z'bb4
IeJ
_ 'e
1
2
(5:)
(c)
ej
1l
(
cn
in terras of measurable
quantities are,
(v
we
f
C
(Tbb
+ re)
21
-qi.fIerbb
1e can assume that the
cutoff current
I
is
collector contribution
ìuch greater than the emitter
(ti)
to the collector-
contribution
25
nd I
e
is larg' £nough to allow the aìroxim'tion
cx'
°
1+
For our soec:Lfic purposes,
T
re
; DK2
f
e
cl
and
CO
inäeende;t of current
:i.
=
C,
L
+
p
o that l
and treating -- ¿nd ±.2 &s
(3
le
putting the values of 1quation
noie
e C8fl write the
F
fEetor
2]
[
59)
:ith
(rbb
+
,
tka11
(6o)
,
Now by
quantities.
i)
(
putting
into Equation
(
55),
mesur:b1e civantities as,
re) _
'e
r
J
q
4
(
Y
'co
1e
rbb+re
[
2 kT h8
O(
L
qR0
2}
I
e
rbb
J
I
+2kT
(<12'ej
j
or
F+0+![
4
[_
i
4-
2
Etj
L
+
+
I-
+++)
L
1e
j
2RsrL
e
L0+
!)
+
(
I
rbb2
i
[o(
::J
(3
I
(o3)
and
1cc
O(close to unity Equation (3) tekee the
forai,
F
.
±
(
f2)
+
+
r
rbb2
/
___
f'
s
2 re
\
(3
fCO2)
For our purposes, there is negligIble error in using
Equations
(62)
and (64).
If we
put the values of 1;uation (51) into Lutions
(82), (83) and rewrite Eauations (84),
for the
assume
noise factor F, and
Z5R5--
(5),
(86),
(80),
use iquation
(1),
(55)
the source impedance is complex
ji5, we canget,
2
2 q
±
rbb
I
2 q 'e
\
l
F
rbb
s I
4 kT
is
Equation ( 65)
Strut
( 6)
the same ss that obtained by Guggenbuhi and
whIch checks our
results with the other investigators.
Now it is necessar- ic develop m1nimu
secific case in order
we
will find tie
Taking the
o =
optimum oper&tion Qoints.
our
First
ootimurn R5.
partial ceriv tive of Lquation
R8 and equating to
d'
a h
to find the
for
noise figure
zero
-
we
:1.
h.
kT
2
( 55)
with
rspect
to
get,
-
_____
(T
(y n v}'
n
iT
T*
1/2
2
rb
IC
re
rbb4 IC
(
(66)
ICT:
ic(V
And by
putting Equation (66) into Eouation
'e
(2) we
get,
1/2
b
=
112_
'e)
{
[(
'e
rbb6J
j
C
r
- 'e rbb
re)
(
(
(67)
.
J
cuatian (7) is used to calculate the lowest part of the curve
which is shown below for specific pnp transistor (2N1128).
The derived noise figures were for snot noise
fiure
and the
re1tion between Lverage noise fiure and spot noise figure is
.;=
F(f)G(f)df
(68)
J(f)af
where
(7),
rnsicer
G(f)
am
of
he amplifier
Soot Noise Factor
F(f)
Average noise Factor,
In our region it can be seen from Figures 11
Equations (o5),
(66)
j2
and 13, and
that F(f) is independent of frequency so can be
reoved from the integral
aL. n
nd
f_ç_
--
JG553(f
Figures 11, 11 and
of frecuency.
(
i/i) noise
L
Lower
1,
tbn
show that transistor noise is 1ndeendent
f1 (around
coin&nt which is
ut
a
few
kca.
cicusea
region) the flicker
in this paper.
Tk
origin of this low freo;ency noise is related to the surface pro-
orties of the device (12).
Above f
(aprrodxnte region of f
cutoff frequency), noise
currents generated near the collector junction ure
collected und
1gu ni
F
Figur(()
vs.
LogÇ
(3
vs.
LóÇ
Pn
vs.
LoÇ.
Corn mon
zmiftr
currznI
Furci2)
Nbs
PwroufptJ
-Pn
J-
(t3)
29
ubeuent1y conducted
currents sources
signì
termin1.
re 3.istributed throughout the
is injected
collected noise
to the collector
t tAie
currents
with frequency above
f.
Since th
bse
noise
region and the
mitter-hse junction, the rtio of the
to the collected signal currents increases
In other worts, the power gain of
raidly with frequency then does
in the transistor base region.
tr.nsistor falls off
the noiEe
more
outut power jenerated
30
EXPERL'IENThJJ INVESTIGATION
methods available to determine noise fctor,
There are v&riou
the one usine
a noise generator
This method makes it
sirni.icity.
resonab1e eccuracy using
generator,
for reference is
VTVM,
20
keps.
selected for its
ossib1e to ueasure noise with
ibortory equipnent. Noise
lowoass filter, re1ted block digria
common
are shown in Figure 14.
In the experiment, the average noise figure is detenined by
using a
G.
Tyoe 1393-B Rindoui-Noie Generator with a band-width
.
of 20 keos. which has
a
constant noise spectrum in
te
s;ecific
region of our interest as shown in Figure 15.
The low pass filter is utilized to exclude the noise ;;enerated
in the unwanted
band.
of the
trxisistor mp1ifier
and at the same time
to narrow the VrVM's pass band.
The
in
common
in the
transistor under test is
bi&ed from
a battery power supply
emitter node, and operEted for small-sidn&1s current gain
common
emitter
mode.
Ail the connections nade in shielded
and coaxial cables &re used to prevent pickups.
503 Scooe
is
used to
Tecktronix type
control the noise wave shapes.
Ñ
O
i
S
TO
VO LTA(
E
____ DÌYDE
S -I
TOR
HTRJNS
UÑDE
I
T3TI
I
i
LOW
PA
VTVM
iLTER
I
SC-OPE
Ft'gur. (14)
Nock digr-im
o +hz fnsfHiman-
rrê
I
a
20K.
io
gJral5)
IOOK
Typical spach-um !cva/ airc-çt-ishc -sor
Randorn
NJd
6.ryp IO-B
Gn.ratör.
32
COMPARISON OF CLCTJLJTED
particular
saust be
The
bse spreaöing resistance
Radio frequency
A
(ii).
Terwn
trnsitor,
certain paraietera rbb, 1co3
measured for substitution into Equation (ô2).
For a
916
ND ME.SURED DATA
rbb
bridge using
detezined
rbb was
eaaured
with G.R. Typ*
the circuitry suggested
by ¡rieasiring the
h1h frequency
input impedance ¿md tking the real pirt as rbb which is
that gives rise to therm1 noise. For the auecific
an
avere value
ech
For
the part
transistor used
of 375 Q. was found.
tranaistor,
O was ieasured usine the setup in
Lab. O.S.U. 70523, and I
Corn.
tr
and!
Corn.
vere ueasured using the 8etup
Lab. O.S.U. 70553.
Figures 16 and
obtained arounc the
17 show
that
good agreeuent with aeasurement wa
inimuxa noise figure.
is the independent variable with a-c bies
at 0,995
when R5
xna
for
The
source resistance
current
F5
held constant
a 2N1128 PN? Gemaniurn Alloy Junction Transistor.
is incresed, the cifference
between Ftheo
Fep also
and
increased.
An R
Fiure
16.
opt. of 500Q. is obtained for specific
Fj
c&se shown in
.92, which is ver close to c.icu1ated values
obtained.
Frora Equation (66) R8 opt. was found
to be 517
Il
,
which la
again very close.
The
iffer'mce between
calculated and measured values
shown
in
3,
Figure 17 is due to the change in
which c&ued
R
opt. to
eniitter current is
for the specific tr&nsistor
he around 600 -Q
(Ex)
= 3.52
whicn
for
The
a
specific
is reason&tly close to
the theoretical value,
Figure 18 shows good agreement
between calcule ted
noise fiEures in the neighborhood of
F.
This
is unfortunute but not surprising
the nichel aire not considered
Figure 18
tht
not
at other values of
because the elements of
functions of 1L'
t
can be seen from
the elements have soue deendence on I
should be modified to
desired over
it
reflect this
a broader imnge.
nd mee sured
deoendence
if better
and the model
&greement
is
3h.
coMPrISoN WITH THE RESULTS OF OTHERS
(iuggenbuhl and Strutt
to
iiure
((a)
have noted experimental datE. similar
Their ueasured data show a similar departure f rom
18.
the
theoretical as a function of bias current and frequency.
W. H.
Fonger (5) uses another ap.roach to the problem and
virtually arrives &t the same noise-fi,ure
Ecuation (ó2).
noise
He
exresdon
as shown in
inicates that any one of the four uncorrelated
;enerators may be mtäe dominant as a function of bias point
and source termination.
s
noted before, Equation (62) can be written as Equation (ò)
which is the same equation Guggenlb1 and Strutt
Equation
(
of Beattie (1).
62)
(
)
used.
is :lso in agreement with the theoretical resulta
L
Figur.(6)
15
I
'4-
CALCULATED AND MEA'3URED NOI5E
N
flSURE OF
PNP
AU_Cri'
II
TRANSS-TOR. NO.1,AS A
S
'E 0FF.
RUNCTON O
I
I_13
_Io.c
vc.
ia
ir
/I/
¡O
\
\
S
Ql
\
8
\
ft
7
/
¡'lEA
N
5
r
'N
Z'N
4
3
-
uLAyD
a
.-1R5
.1000
30
200
7oOSOQ9
I
I
i
i
iii'ii
aoôo
I
.1
3000
i
(ahrn,$)
700
5O
i
i
i
i
i
EI1
Figur.
I
(iT)
AND MEASURED NOISE
1128 PNP JUNCTION
2N
c'&URE OF
ALLOY TRAN.SSTOI NOII, AS A
FUNCTION OF R-s, FOR 'E O.77Om.
CALCULJ\TED
j
j
I0
V=-IO.ov
/
\
/
/0
\
II
/
o
\
/
\
-Io
/
\
s
/
\
4
/
<I
z1z
MEAUQD
7+
\
/
GLL
4
/
/
/N
ULTEi
I
/
/
/'
/.
O
-.5--
-O-
-!)--
3
a
s-
°
I
1100
aoo
10
11000
-°
i
a000
I
i
R5
1.3000
(ohms)
oc
r
r:gur()
s
¡4
CALCULIATEE AiJD MEASUZD NOi.sE
F(GURE OF 2N 1128 PÑP JUNCTION
ALLOY TP,AN.S1STOR NO.1, AS A
RUHCT(ÖN oc r
500.Q.
,
-10.0v
'v'c
I-12
I-II
lo
\
\
s
\
o
8
z
z<V
'e
\\
T
\/'
A5URED
¡I
O
r
lL
r
r
r
/0
4
3
ZCALCULATP
a
-
1100
I
i400
«oo
2000
oc
I
i
i
I
I
I
'E (microampws)
3OOO
14-0001
CONCLUSION
Ce.lculation of noise figure for
transistor 1sed
on the lumped
noise
PN?
Gernniu Alloy junction
ioel
shown
in Figure
6
agrees
well with the eperiinentally nesurod vdues shown in Figures 16 :nd
theoretical reu1ts of Foncer (5), Ee&ttie (i), nd the
experimental results of Guggenl*ih]. nd Strutt () arc Liso th good
17.
The
greemct with this ode1.
?e
can conclude
that the
tr&nsistors aoie1's vLiidity hïs been ste.biished.
1uxped noice
39
131 xLIOGíAPk
1.
&na1yis of noise in seiiiIRE Transactions on Electronic Leviceß 6i13-
k3eattie, R. N.
conductors.
140.
1959.
2.
A luniped uode1
u. T, i. Groeidi2k &nd K.
factor of four tenninEl networks.
lo, 1955, p. 349.
t3ecking, A.
S.
Knol.
The
noie
Phillips Pesearch Teport
3.
Early, J. 1.
îffcLs of space-charge layer widening in
junction transistors.
Proceedings IRE 40:1401-1406.
Noveìiber 1952.
4.
Ebers, J. J.
5.
Fon:er,
6.
7.
nä J.
Lare-signEJ. behavior of junction
Proceedings IRE 42:17i1-1772.
1954.
trnsitors.
.
1-i.
Â. klaus.
p.
34J-405.
ii.
.
ioll.
Noise in trnsistors. In: L. L. Siu1iin and
Noise in electron devices. New York, i1ey, 1959.
nd 4. J. 0.
trutt. Theory anQ experlzftenta
on shot noise in serniconäuctor junction diodes cnd transistors.
2roceedinks IRE 45:839-43. 1957.
uuggentuihl,
IRE Standards on 4ethods of i4easuring NoiEe in Linear Twoorts.
?roceedins IRE 48:b0-6. 19&D.
8.
Linvill, J. u. Models of transistors &nd diodes,
1960 of Stanford University.
9.
Linvill, J.
u.
Luiiped .nociels of
Proceedins IPE, 46:1L41-1152.
10.
Shockley,
i1ectrons
D. Van Nostrand, 1950.
U. Ternn,
.nd
55].
tr5istors
ana diodes.
1958.
holes in semíconJxctors.
Ziel, A. Van lier.
New Ior,
p.
F. E.
lectronic and rado en:ineering.
York, McGraw-lili, 1955. 1078 p.
12.
C18ß Notea
.4th
Ed. New
Theory of shot noise in junction iiodea and
junction transistors. Proceedins IRE 4:l9-1646. 1955.
APPENDIX
Lrivation of the equations of
nd
Assuming no correlation between the noise generators, we can
find V
end.
from
Figure lOb, with the transistor in grounded
emitter orientation as suggested by ieoking (2).
Putting V2 = O
and i2 = O
we can solve,
v=-v1
111=_Il
(70)
hence,
V2
-
VC (1
- Ve
12 - 1nc
Yce Ve
(71)
(72a)
Ve - -
Ii-I--
I+
(72b)
y,
Putting Equations (72a) and (72b) into Ecuation (71) and for
V2 = O
,
i2 = O
u1e
Yc
I1__i,-f
\Yc I]
L
Ye
e 1nc
YO
(73)
+
Y0,
from,
VlVnb+
Ilrbb_VC+
V2
forV2O
,
12_O
(7i,a)
Vl_Vnb_l'V. +Ilrbb
(
74b)
Inc
Vl
_ Vflb4
(i
(1
-)
Yc
f
-
-i--
r
Ye)
me
rbb
ceJ
(74c)
hi
Putting !2!. - Ye
and
p
4. 1
V
- Vn
1
80
I
=
ce
- I
i
1e
n
which are
reasonable approximations,
nc
+ --
- V
Iflmne±
4
(r
i-ne
(75)
rbb
(76)
.
Then we csn find,
(v
V)
(Vnb
V)
4
(ne i8)
'bh2
2
i0)
__________
2
<
rbb
+
i
q
e
I
+ 2 Re [ifld
ee +
kT
(rbb+
i
(i
(v
I)
n)
=
ne
t8) rbb
q
(q_V\
kT
kT
(i
+
1_
(inc
(ne j8) "bb4
(rbb
exp.
icT
f2
R8 {(ind
+
j
Se)
[mj
(7e)
+
fqV8\
q
Pn
Re
(H
d)
J
J
(Hee _
j
u)
e)
42
1d
'-na
(*
ex
ee
-
(79)
Using Equations çi), (2), (:), (4o
(46)
(.i), (42),
pproximtions in Ecuations (47)
nci
Th-uatjons (33)
,
(
:4)
,
2 q L
ne
(6)
? 5)
(
f
"ee
i.r
(4e)
= 2 q
(d
14i) =
;:
s
f
J
2 kT
(h1ee
(so)
+
P±
2 q
eec)
Af
(u)
I
Pee±
Ar o [p
¿f
i.T
(Vl.b
(2
we can rcwrite
n
(
ec (2 F
Ar
q
(4), (5),
the form
2kTfe[
-2qfI
(i
,
(43),
exp
(kìeet
JSe
rbb
L3)
Pitt1ng tne iauation$ (so), (si), (2), and (33) into JL:uation
(79)
we can get,
(v
v)
4
f rbb
T
(Ree
+
j
±
- 2 q
Se)
f Re
T
rbb2
f I
J
1
exp
ee
ep
(ve)
r.b
J
4
Fn
L)Se)
'.3
fqV
q
2 kT
f rbb2
(H
kT
ee
+
i
W Se)]
)
+2kT/1frbb
I
2
rbb+
/q_Ve'
q
e
Pu
a2qf
kT
I
J
(Hee
8)
i
o
_rbb2I
(34)
I
Hence,
____
r
- 2
qLf
- 2 Re
T
f
A
exp
q
T)
+
(H ea
--
T
i
(85)
e
n
2qf
fqV\
exp
(q Ve
e]
f Re
-qL1fI
I
\tcd 2)
J2
: rbb
q
'
'C
2qAf
I)
_
f (
2 q
'e
2
(qV\
q
ea
j
i
1o2
+
i
5e)]
2kTAfrbb
ie
1
(Hee
C
W
íqV
-2kT Ar+ 2qíf
(e)
-2kTAf
44
Çr+
IqV\ (ll,
q
exp ___
i
)
Se)
-2qLfIr
(86)
Using the Ecuations (1.4), (45) and (46) and approximating in
our range of
ii
interest
1/re
(1
-t-
j
1/re
rbb+
(8'7)
2
rei
r2Ie
______
= 2 q
(II)2qf
IC
(of2
1ej
(89)
i
I
-2
qf 'e
l2
rbb
(88)
(rbb4 re)
(90)