XXIV.
NOISE IN ELECTRON DEVICES
Prof. R. B. Adler
Prof. H. A. Haus
Prof. J. B. Wiesner
R. P. Rafuse
RESEARCH OBJECTIVES
While our work on the general problem of noise in linear networks came to a conclusion with the publication of Technology Press Research Monograph No. 2, "Circuit
Theory of Linear Noisy Networks," by H. A. Haus and R. B. Adler (published jointly
by the Technology Press of Massachusetts Institute of Technology and John Wiley and
Sons, Inc., New York, 1959), related special problems continue to emerge. These are
conveniently handled by the general methods developed in the monograph, and we shall
continue to present applications of the general theory to cases of current interest as
they arise.
R. B. Adler
A.
MEASUREMENT OF ABSOLUTE NOISE PERFORMANCE OF PARAMETRIC
AMPLIFIERS
Any measurement procedure that is used to determine the absolute noise performance of a negative-resistance amplifier must necessarily include the behavior of the
amplifier in cascade.
For practical considerations (bandwidth, nonlinearity, and so
on), one of the members of the cascade usually is chosen to be a unilateral, and inherently noisy, amplifier of a class whose normal performance is not as good as the
This situation is,
capabilities of the parametric amplifier.
in fact,
encountered if a
negative-resistance amplifier is added to an existing system in order to improve its
noise performance.
It should be noted that devices with negative-real input and/or output impedances,
as well as positive-real devices driven from sources with negative-real impedances,
are capable of being characterized and analyzed within the framework of exchangeable
A suitable model for a generalized unilat-
gain, noise figure, and noise measure (1).
eral noisy amplifier (2, 3) is shown in Fig. XXIV-1.
The excess exchangeable noise
figure for this representation is
(F-1) = (R /R
)+ G R
(1)
+ 2p(R G n)/2
where
This
research
was
supported
in
part by
Purchase
Order
Lincoln Laboratory, a center for research operated by M. I. T.,
the U.S. Air Force under Air Force Contract AF19(604)-5200.
205
DDL-B222
with
which is supported by
(XXIV.
NOISE IN ELECTRON DEVICES)
e i
n n
P=
2
_ _2.2
Figure XXIV-2 illustrates the behavior of (F-1) versus source resistance.
minimum magnitude excess noise figures are
The two
(F+ -1)
= 2(1+p)(RnGn)/2
(2)
(F -1)
= -2(1-p)(R G )/2
(3)
g,
o opt
(R/Gnll/2
for Rg opt = ±(R/G)/,
respectively.
Penfield has derived a simple model for the
signal-frequency behavior of a lossy varactor (4, 5). He also has shown that the best
performance is obtained with the diode series resistance as the total idler circuit load.
Figure XXIV-3 gives two models for the signal-frequency behavior of a lossy varactor
for which
m = Penfield modulation ratio (m
max
< 1/Tr)
Oc = diode cutoff frequency = 1/RsCmin
w. = idler frequency
Ws = signal frequency
R
s
C
R
= series loss
= average capacity
d
2
m
w.
_
1
2
w
c R
C s
m
2 2
C
c
W. O
G
R
22
s
2C2
s o
s
1+
M
e
(4)
s
22
C2
s o
2
m cc
-2 m
c R
s
W
.
s
1 s
(F-l)
Co T.
O
- (l1/G)
wi T 0
wi T
s
1
P
T
R
i
R
s
+Rdd
The best value of Me is obtained when Rd = Rd, max (strong pumping).
206
e n = 4kT o RnAf
NOISE
NOISELESS
NETWORK
NETWORK
AMPLIFIER
NOISY
Fig. XXIV-1.
[T]=
' =4kTGn Af
R
Any linear noisy two-port can
Characterization of a noisy amplifier.
be separated into a noiseless two-port preceded by a pair of correlated
noise generators.
(F- I)
C+
+
Rg
(F_- I)
Rg,opt+
Fig. XXIV-2.
/G
n
Exchangeable excess noise figure. The two branches corresponding
to ±R are particularly convenient when cascading. When the output
g
impedance of the preceding stage is negative, its exchangeable gain
is also negative, and a definite contribution to over-all noise-figure
results.
Rd
Rs
Co
-00OIR
Fig. XXIV-3.
Signal-frequency varactor models.
A parallel equivalent circuit for the
Penfield model is derived.
207
(XXIV.
NOISE IN ELECTRON DEVICES)
NEGATIVE - RESISTANCE
AMPLIFIER
L
Rs
-G0
IDEAL
en
yNI
i
IDEAL
SECOND STAGE
Fig.
XXIV-4.
Figure XXIV-4
Model of a negative-resistance amplifier in cascade
with a noisy second stage. A signal-frequency model
of a parametric amplifier is cascaded with a generalized noisy second stage. Two separate over-all noise
figure expressions are necessary for the two regions.
presents a typical signal-frequency equivalent circuit for a "two-
tank" parametric amplifier.
The behavior of the cascade noise figure versus Rg is
-1
characterized by two regions for R
(F12-1)G
= OR
o, opt
and for R
<
/N
2
g
(N2G,
\o,
max)
max
~ For
R
g
> 1/N
2
G
o,max
+ (F/R ) + A
(7)
g
Go, max
(F12-1)G
2R
=
+ (/R g) + X
(8)
g
o, max
where
(FZ-1)
(F 1 2 -1) = (F 1 -1) +
e, 1
It is
apparent from the expression for noise measure
G
M
=M
el2
el
+
AM
of any cascade (1),
-1
(9)
- 1
e\G
e12
that if Me, 1 is optimized by making Rd = Rd, max' then Me,
is optimized by making
12
AMe a minimum and Ge, 1a maximum. This may be achieved in practice by pumping the
varactor as hard as possible (achieving m max), adjusting the turns ratio of the first
max
transformer, N, in such a way that N2R
(Go,
)- (G
oo), and simultaneously
adjusting the second transformer turns ratio 'yN - oo, with the result that the impedance
seen by the second stage is Rg, opt (from Fig. XXIV-2).
The over-all exchangeable
gain will be infinite in the limit, and Me, 12 reduces to (F
12
The calculated values of 0,
F,
A,
-1).
4, 0£, and X necessary to predict the cascade
208
(XXIV.
NOISE IN ELECTRON DEVICES)
noise figure for non-optimum conditions are:
22
(1-p
S=
Gn
n
P
y
1
aR
To
\
+
y
T.
1 N2 aR
o
Y2
R
yR
il
. T oN 2
1+
T.
s T
W.
n
R
s/
aR
a
y
T.i a(R )2
s+
o
T
2
1
o 2y R
si
n
42
n
n
4N2Rw
n
2
(coT.)
s i
1
2 Rn W. T 0
4y R n
/
I
(G n/R n)1
n2
oT
P
I
RG
2R3
s
2
2 4\
- 2a R
s
s
W. Ti
1
o/
2
S
T.
s
1+P
. T
1 o
i To [yN]
(10)
Rn
s
(11)
P
i
1i
o [y N]2R
s
n
FaRZ 2pR ( ns )/J
1 aR 2
1/
+
(G /R
Co
P
+ --
(12)
1P
Pi
s
WiT o (yN) 2 R
where
2
W2
coC
so
l + W2C 2 R 2
SOS
These values are calculated under the constraint that G
is always adjusted (by varying
the pump amplitude) so that the over-all noise figure is a minimum for any Rg (provided
that G o, max is not reached).
The other parameters
2
(13)
n
Co T.
2= s -
+
N2G
o.
T
+
2
2 N4R G
n o
1
o
max
C T
P
(2
2
G
+G
max
X = 2p(R G )1/2 - 2(yN)
n
+ aR N 2 - aR2N2G
max
1 oT
s
- 2p(R Gn ) 1/2 N 2 G
nn
s
Oma
x
(14)
max
(15)
R G
max
209
y=co
12
2
10
_A
-p
i/N 2 Go,max
0
Fig. XXIV-5.
Theoretical behavior of cascade noise figure versus R
g
and y.
2.0
1.5
1.0 L
0
Fig. XXIV-6.
100
200
300
Excess noise figure for an experimental second stage; (F 2 -1) versus
source resistance was measured for a typical 108-mc receiver. The
values of R n , G n, and p are calculated from the measured values, and
the curve is drawn from these values.
(F 12 -I )
2.0
oMEASURED
0
1.0 -
I'
\
R:I/N2
G
R/ 2 o,m ax
100
Fig. XXIV-7.
200
300
Theoretical and experimental behavior of over-all excess noise
figure. The theoretical noise-figure performance for an experimental cascade is compared with the measured performance.
210
NOISE IN ELECTRON DEVICES)
(XXIV.
hold for values of Rg < (N2G
-1
max)1
Illustrated in Fig. XXIV-5 is the behavior of (F
12
-1) versus y and Rg.
it is necessary to know the extent to which the optimization procedure
tical reasons,
These relations
should be continued for a given varactor and a given second stage.
(Eqs.
For prac-
10-15) are all that are necessary.
Table XXIV-1.
Numerical values of A,
elements characterizing
fier, and those derived
values for the six basic
for two values of y.
0, r, X, £2, and (D. From the necessary
an experimental negative-resistance amplifrom the second stage, a set of numerical
parameters are derived. These are shown
y=l
y=2
= 65.7
= 1.4 X 10
i = 252.8
= 4.5 X 10-2
r
-2
0 = 5. 3 X 10-3
= -1. 28
£ = 3.0 X 10
A = 0. 14
r
- 2
X = -5.33
= 4.5 X 102
0 = 1.32 X 10 -
3
A = 0. 14
Figures XXIV-6 and XXIV-7 and Table XXIV-1 contain data on a specific experimental cascade.
R
It is apparent that (F12-1)best is nearly achieved by a combination of
= 100 ohms (with N 2
g
=
50) and y = 2.
It is interesting to note that, if the signal cir-
cuit is loaded too heavily (N small), a much poorer performance results.
The agree-
ment between theory and experiment indicates that the models chosen are adequate; the
optimization procedure is evident from the curves of Fig. XXIV-7.
In conclusion, it can be shown that, as y becomes very large, r,
0, and A reduce to
o T.R
s
p 1
.T N 2
(16)
10
co T.
(17)
N2
i
W. T
1
0
(aR
aR)
w. T
1
(18)
s
0
These values result in
oT
(Fs
F12
op t
o.1 T 0 0.
OT
P
1
T0
41/2
(aR2-a2
R4
-a saR
for R
o-1/2
2)] 2
- N2(a-aZ R
s/
I
g, opt
(19)
R. P. Rafuse
211
(XXIV.
NOISE IN ELECTRON DEVICES)
References
1. H. A. Haus and R. B. Adler, Circuit Theory uf Linear Noisy Networks (The
Technology Press, Cambridge, Mass., and John Wiley and Sons, Inc., New York, 1950).
2. A.
networks,
3.
H.
G. Th. Becking, H. Groendijk, and K. S. Knol, The noise factor of 4-terminal
Philips Research Reports 10, 349-357 (1955).
Rothe and W. Dahlke,
Theory of noisy four-poles,
Proc. IRE 44,
811-817
(1956).
4. H. A. Haus and P. Penfield, Jr., On the noise performance of parametric amplifiers, Internal Memorandum No. 19, Energy Conversion Group, Department of Electrical Engineering, M.I.T., Aug. 11, 1959 (unpublished).
5. P. Penfield, Jr., Interpretation of some varactor amplifier noise formulas,
Internal Memorandum, Microwave Associates, Inc., Burlington, Massachusetts, Sept. 1,
1959 (unpublished).
212