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611969
An Active FET Receiver Front End Mixer
Kenneth Voyce
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Recommended Citation
Approved by:
by
Kenneth G. Voyce
A Thesis Submitted
in
Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
in
Electrical Engineering
Prof.
Walton F. Walker
(Thesis Advisor)
Prof.
Name Illegible
Prof.
Name Illegible
Prof.
W. F. Walker
(Department Head)
DEPARTMENT OF ELECTRICAL ENGINEERING
COLLEGE OF APPLIED SCIENCE
ROCHESTER INSTITUTE OF TECHNOLOGY
In
_{many}
recent receiver_{designs,}
the
frontend
mixerdetermines
the
receiver's_{sensitivity}
andits
_{susceptibility}
to
distortion
from
large input
signals.Two
balanced
FET
mixers aredeveloped for
suchan
_{application;}
a common source configuration and a common gateconfiguration.
Each
exhibits a range of110
dB
between
the
availableS+N
input
power_{necessary for}
10 dD
_{sensitivity}
andthe
availableinput
power of a
twotone
signal which causesintertnodulation
productsdown
40 dB from
the
desired
signals*An
expressionfor
the
conversiontransconductance
of ajunction
FET
used as a mixeris
first derived from
the device
transfer
characteristic equation*
The
advantages of abalanced
mixer configuration
arethen
discussed*
A
noise equivalent circuit whichdescribes
the
sources of noisein
each ofthe
proposed mixersis
developed*
This
leads
to
the
calculation of noise
factor
and_{hence,}
_{sensitivity}
ofthe
commonsource mixer as a
function
ofR
,the
generatorresistance*
The dependence
ofdistortion
onR
for
the
common source mi'xeris
then
found
and plotted*A
comparison ofthis
plot andthe
_{sensitivity}
vs.
R
plot showsthat
the
maximumdynamic
range resultsfrom
a choices
of
R
equalto
200
ohms.
The dynamic
range ofthe
common gate mixeris
_{investigated}
andfound
to
be
equalto
that
ofthe
common source mixer.The
_{sensitivity,}
The
effects of_{frequency}
dependent
terminations
onthe
mixer portsare examined and
found
to
be
quiteimportant
to
mixer performance*The
bias
problemis
solved with provision madefor
the
variationin
parameters
from
device
to
device.
The
methods of_{testing}
the
mixer arethen discussed
andtest
results are shown
to
agree_{excellently}
withthe
results predicted_{by}
the
theoretical
analysis.A
brief
description
ofFET
operation and an observation onthe
TABLE OF
CONTENTS
Page
ABSTRACT
i
LIST
OF
FIGURES
vLIST OF
TABLES
viiLIST
OF SYMBOLS
viiiI.
INTRODUCTION
2
II.
THE FET AS A MIXER
5
Development
ofthe
Junction FET
5
Calculation
ofConversion Transconductance
5
Balanced Mixers
10
III.
NOISE ANALYSIS
12
Sensitivity
andNoise
Factor.
12
Sources
ofNoise
in FET's
15
Noise
Equivalent Circuit
17
Noise Factor Calculation
19
Sensitivity
vs.R
for
Common Source Mixer
23
Sensitivity
ofCommon Gate Mixer
24
Summary
ofNoise
Analysis
28
IV.
LARGE
SIGNAL HANDLING CAPABILITY
30
Intermodulation
andCrossmodulation Distortion
Relationship
30
Balanced
Mixer Distortion
36
Page
Compari
son.
39
Bandwidth
Check
40
VI.
FREQUENCY
DEPENDENT TERMINATIONS
41
VII.
BIASING
AND CIRCUIT
COMPONENTS
43
Component
_{Sensitivity}
50
VIII.
EXPERIMENTAL
PROCEDURE AND RESULTS
53
*
Results
53
Di
scussion58
IX.
CONCLUSIONS
64
APPENDIX
1
66
Brief
Description
ofFET Operation
66
APPENDIX II
70
An Observation
onthe
Calculation
ofNoise Factor
70
Figure
Page
1
Common Source FET Balanced Mixer
3
2
Common
Gate FET Balanced Mixer
4
3
Simple
FET
Mixer
6
4
Noise Equivalent Circuit
ofFET
18
5
Balanced
Mixer
Noise Equivalent Circuit
19
6
Section
ofNoise
Equivalent
Circuit
20
7
Input
_{Conductance,}
_{g}
23
s
8
Dynamic Range
ofCommon Source Balanced Mixer
25
9
FET
Equivalent Circuit
26
10
FET
Equivalent
Circuit
withShorted Output
26
11
I.M.
Distortion
31
12
Distortion Model
ofMixer
33
13
U222
Characteristic
44
14
FET
_{Biasing}
Example
46
15
Bias
Circuit
47
16
U222
Characteristic
48
17
Improved Bias Circuit
_{49}
18
Input Transformer
52
19
Output
Transformer
_{52}
20
_{Sensitivity}
andNoise Factor
Test
54
Figure
Page
22
Crossmodulation
Test
56
23
Spurious Response Test
59
24
Spurious Response Data
61
25
Spurious
Response
Data
62
26
Spurious Response
Data
63
I
N Channel FET
67
II
FET Characteristics
68
III
FET Transfer Characteristic
69
IV
Input Noise Equivalent Circuit
70
V
TtooPort Noise Equivalent Circuit
71
LIST
OF
TABLES
Table
Page
I
Noise Factor Parameters
24
LIST OF SYMBOLS
V
FET drain
to
source voltageV^
FET
gateto
source voltageGo
Vp
FET
pinchoff voltageI
FET drain
currentI
_FET drain
current withVGS
_{0}
^
the FET
bias
voltageVGS/Vp
g
conversiontransconductance
mcG
conversion power gainc
R
generator or source resistanceF
noisefactor
S./N.
input
signalto
noise ratioS
/N
output signalto
noise ratio o o*i a specific
(S
+_{N}
)/N
o o o
I
.thermal
noise output current cmG.
input
conductance of common sourceFET
g..
input
conductance ofFET
in
either configurationI
shot noise ofthe
gateleakage
currentgs
F
optimum noisefactor
g.
source conductance which yieldsF
o o
Rg
bias
resistorVG_
dc bias
voltageVfi_
RM
ratio of amplitudeI.M.
productsto
amplitudes ofdesired
signalsV_
voltagefor
eachtone
of atwotone
signal which L.n.causes
R_
_{40 dB}
^xmod
voltage ofAM
signal which produces_{17.}
crossmodulationThe
radio_{frequency}
spectrumhas
become
quite crowded withsignals
_{ranging from}
those
which are_{barely}
detectable
to
those
whichoriginate
in
_{nearby high}
powertransmitters.
The
needfor
receiverswhich will
detect
a_{very}
weak signal without_{being}
distorted
_{by}
large
undesired signals
_{is,}
_{therefore,}
apparent.This
problem_{is especially}
prevalent
in
_{military}
communicationsbecause
several systems often mustoperate
from
within a small area.This
characteristic of a_{receiver,}
calledits
dynamic
range,
has
received much attention
in
recent receiverdesigns.
One
significantimprovement has
resultedfrom the
elimination ofR.F.
amplifiers withtheir
poor_{selectivity,}
and_{hence,}
_{susceptibility}
to
distortion from
large
undesired signals._{This,}
_{however,}
has
left
the
burden
ofsensitivity
and signal_{handling}
_{capability}
to
the
receiver'sfront
end mixer.
Wide
dynamic
range mixershave
_{lately}
been developed
_{utilizing}
low noise,
excellent squarelaw
_{devices,}
such as_{Schottky}
diodes
andfield
effecttransistors
(FET).
Presented
in this
paperis
such amixer which was
developed for
usein
the
2MHz
12
_{MHz}
band
_{with} _{an}_{i.f.}
frequency
of26.5
MHz.
The
performance oftwo
balanced
mixer configurations
willbe
analyzed and compared.
_{For convenience,}
full
schematics ofthe
two
R.F.
Input
Oo
o
ih
V
M
OOutput
I.F.
Common Source FET Balanced
Mixer
Figure
1
O
R.F.
Input
_{q}
#
I.F.
Output
Common Gate FET Balanced Mixer
Figure
2
Development
ofthe Junction FET
The junctiongate FET
wasfirst described
in
a paper_{by}
W.
_{Shockley}
in
1952.
Earlier
workhad been done
onfield
effectdevices in
the
1930's
_{by}
J.
E.
Lillienfeld
andin the
latter
1940' s_{by}
Shockley,
but
these devices
did
not exhibit good enough performanceto
be
practical.
_{During}
the
warthe
field
effectdevices
did
not receivemuch
_{attention,}
because
the
emphasis atthat
time
was onthe development
of
the
point contacttransistor
and other solid statedevices
whichapparently
were an outgrowth ofthe
_{early}
work_{by}
Shockley.
An
excellent approximationto
the
junction FET
transfer
char2
acteristic,
equation_{1,}
wasderived
_{by}
R.
D.
Middlebrook
, andit
is
this
expression which willbe
usedin the
_{following}
calculations.xd
has
f1
if
)*
m
For
abrief
review of_{FET operation,}
seeAppendix I.
Calculation
ofConversion
Transconductance
An
expressionfor
the
conversiontransconductance
of anFET
usedas a mixer can now
be derived
_{using}
equation1.
One
possible circuitis
shown
in Figure
3.
The
bias
mustfirst be
set at_{V_e/V_,}
_{^,}
wheref
is
Co
Vsuch
that the
input
signals will notdrive
the
device
into
saturation orr.f.
O
1t
TT
i.f,
6
B+
Simple FET Mixer
Sj(t)
_{AjV}
_{coscjjt}(2)
S2(t)
A2Vp
cos2t(3)
the
gateto
source voltage canbe
writtenVGS
"^VP
+A1VP
cosfc,it +A2VP
cos<2t(A)
Substituting
into
equation_{(1)}
yieldsVXDSS
"(1^+
Al
CS"lt +A2
cos"2t)2
2
(l<")
+_{2(W)}
(A.
cosco.t +A2
cosco_{t)}
2
+
_{(A.}
_{Cost*.t} +_{A}
cos cot)(5)
Examining
the
third
term
_{only}
2
I_o<_{(A.}
_{cosfcj.t)}
+2A.A2
cosw.t cosa*2t+
(A2
costo2t)2
(6)
Remembering
an_{identity}
from
_{trigonometry}
we can expressthe
secondterm
in
equation6
asIDSSrt'AlA2
[cos(t,,i
+_{w2^}
+cos((jJi
ma,2^tJ
^
amplitude
A.A.
The
complete expressionfor
the
drain
currentis
nowID/IDSS
"(1"<*)2
+
_{2(1"f)}
(A1
COSOJit +A2
costo2t)
2
2
+
(A.
cosw.t)
+(A2
cosw_{t)}
+
AjA2
fcosCoaj
+ co2)t +cosfWj
 _{cc>2)t}j
(8)
If
_{S.(t)}
were alocal
oscillatorsignal,
L.O.,
of constant amplitude andif
S_(t)
were adesired
R.F.
signal*
the
output at adesired inter
mediate
_{frequency}
(cj.
uo6) would
be
XDi.f.
^DSS
A1A2C0S
(W1
w2)fc_{(9)}
As
expectedthe
i.f.
currentis
proportionalto
the
amplitudes ofboth
input
signals.The
conversion_{transconductance,}
a ,is defined
asthe
ratio ofthe
outputif.
currentto
the
desired
input
voltage.8roc
"rDi.f.
/
A2Vp
(10)
Substituting
for I
.f
from
equation9
*DSSA1
*mc
"^
<U>
Di.t.l
L
/j3)
<$*
G
4 g
R,
R
c
smc
X
2
. n(14)
s
Where R
is
the
source resistance. sMore
specifically,
aSiliconix
U222
FET
withthe
_{following}
parameters :
*
VP
_{8V}
*

_{130}
_{ma}Rs
_{100}
_{ohms}RL
IB2500
ohms(15)
when used as a mixer with a one volt
L.O.
,A.
1/8,
wouldfrom
equations
11
and_{14,}
have
a conversiontransconductance
ofg. 
_{2000}
_{micromhos}(16)
and a conversion gain of
Gc
4
or6
dB
_{(17)}
Balanced
Mixers
Several
improvements
canbe
made onthe
above mixer_{by}
_{utilizing}
a
balanced
configuration.Ttoo
possible circuits ofthis
type
are shownin Figures 1
and2.
Perhaps
the
most significantimprovement
is
the
cancellation ofnoise
from
the
L.O.
port whichincludes
voltages atthe r.f.,
i.f.,
andimage
frequencies.
Referring
to
Figures
1
and2
we seethat
the
i.f.
signalis
coupledout of
the
circuitthrough
a pushpulltransformer.
_{Therefore,}
the
i.f.
currents
_{resulting}
from
the
two
halves
ofthe
balanced
circuit mustbe
in
the
proper phase or cancellation willtake
place withinthe
trans
former*
This is
exactly
whathappens
to
the three
noise signal components
from
the
L.O.
port.Since the
r.f. noise signal andthe
L.O.
signal are
both
single ended_{inputs, they}
resultin
drain
currents atthe
i.f.
_{frequency}
which cancel.The
sameis
true
ofthe
image
_{frequency}
noise component.
The
i.f.
_{frequency}
component resultsin
i.f.
drain
currents without
_{being}
mixed.It
alsocancels,
though, because
the
input
is singleended,
whilethe
outputis
pushpull*This
cancellationmakes
_{it unnecessary}
to
addhigh
andlow
passfilters
atthe
L.O.
port,
which
is
indeed
an attractiveresult,
for
in
some cases practicalfilters
can notbe
realizedthat
will passthe
L.O.
signal and rejectthe
r.f.,
i.f.
andimage
frequencies.
In
this
same mannerthe
L.O.
signalis
isblated from
the
r.f.reaches
the
antenna.With
suitablefilters
atthe
r.f.port,
radiationat
the
L.O.
_{frequency}
ceasesto
be
a problem.The
r.f.
signalis,
of_{course,}
alsoisolated
from
the
L.O.
port.If
it
were_{not,}
filters
wouldhave
to
be
added which would preventdissipation
of r.f. power._{Similarly}
the
i.f.
signalis isolated
from
the L.O.
port againpreventing
powerloss.
The
advantages of abalanced
configuration,
therefore,
makeits
use
in front
end mixerdesigns very
desirable.
Indeed,
the
cancellationof
L.O.
noiseis
mostsignificant,
but in
orderto
optimizethe
noiseperformance of
the
completebalanced
circuit,
the
analysisin Section III
III.
NOISE ANALYSIS
Sensitivity
andNoise Factor
Before
the
noise performance ofthe
two
proposed circuits canbe
_{investigated,}
an_{understanding}
of noise_{factor,}
_{sensitivity,}
andthe
relationship between
the two
is
necessary*
The
noisefactor
of atwoport
networkis defined
asthe
input
signal
to
noise ratiodivided
_{by}
the
output signalto
noise ratio*Sj/N
Noise
factor
y
*

_{F}
_{(18)}
o o
It
_{is,}
therefore,
a measure ofthe
degree
ofdegradation
of signalto
noise ratio_{by}
the
twoport
network.For example,
atransistor
amplifier raises
the
powerlevel
ofthe
signal and noise atits
input,
and
in
additiondegrades
the
signalto
noise ratio_{by}
_{introducing}
noisewhich
is
createdin
or caused_{by}
the
circuititself.
In
orderto
comparethe
noise performance of_{individual circuits,}
a measure ofthis
degradation is required,
hence
the
noisefactor
parameter*
The
noisefactor
of a network can alsobe
thought
of asthe
ratioof
_{(1)}
the
total
noise power atthe
outputto
_{(2)}
the
noise power atthe
output whichis
due
_{solely to}
noise atthe
input*
total
noise power outputnoise power output
due
to
input
noise(19)
N
N
F
" "This
form is
more convenientfor
noisefactor
calculations.The
noisefactor
oftwo
cascaded stages canbe found
using
the
3
following
equation:
F
1
FT
"Fl
+ V"(21>
where
F_
is
the
total
noise_{factor,}
F.
is the
noisefactor
ofthe
first
stage,
F
is the
noisefactor
ofthe
secondstage,
andG.
is
the
powergain of
the
first
stage*The
_{sensitivity}
of a receiveris
a measure of_{its ability}
to
detect
small
desired
signals*The
inherent
noise ofthe
receiverlimits
this
ability because
a small signal will notbe
discernable if
its
poweris
not comparable
to
the
noise powerin the
signalpath*
Because
the
_{sensitivity}
andtotal
noisefactor
of a receiver areboth
measures ofits
noise_{performance,}
_{they}
mustbe
related*A
oneto
one
relationship exists,
however,
only
afterthe
bandwidth
and_{operating}
temperature
ofthe
receiverhave been
specified*Sensitivity
is
_{usually}
specified asthe
minimum available powerfrom
a source whichis
_{necessary}
to
obtain a given signal plus noiseto
noise ratio at
the
receiver output*With
a ratio of ct specifiedV2
*
(22)
o^
_{*}
1
substituting into
equation_{18,}
L 
_{F}
<
1)
(24)
Wi
Since S.
/N.
is
a ratio of powers atthe
sameimpedance
level,
R
,it
11 s
can
be
written as a ratio ofthe
mean square voltages*2
S
2
*where e
is the
mean square open circuit signal voltage and4kTR
Af
3
is
the
mean squarethermal
noise voltagedue
to
the
resistance, R
.5
The
availableinput
poweris
now2
e~ 
_{kTAfF}
(<*
_{1)}
_{(26)}
s
which expresses
the
_{relationship between}
noise_{factor,}
_{F,}
andsensitivity.
In the
analysisthat
follows
it
is
desirable
to
find
the
_{sensitivity}
of a receiver which
incorporates
either oftwo
proposed mixers.To
accomplish
this
a noise equivalent circuitfor
the Siliconix U222 FET
will
first be developed.
This
will allow usto
calculatethe
noise^
23k
is
the
Boltzmann
constant1.38
x10
joules/degree Kelvin.
T is the
temperature
in
degrees
Kelvin.
factors
ofthe two
mixer configurations andthen
the
_{sensitivity}
ofa receiver
_{incorporating}
either ofthe
mixers canbe found from
equation26.
_{This is only}
validif
we assumethat
the
mixerdetermines
the
total
noise
factor
ofthe
receiver.Equation
21
showsthat
this
assumtionis
legitimate
if
the
mixeris
followed
_{by}
a_{lownoise,}
highgain
circuit.Sources
ofNoise
in FET's
Before
an equivalentcircuit,
whichdescribes FET
noiseperformance,
can
be
drawn,
the
sources of noisein FET's
mustbe
investigated
to
determine
the
extentto
which each will contributeto
the
total
noisepower.
The
_{primary}
source of noisein FET's is thermal
noise ofthe
4
channel.
Van der Ziel
has
shownthat the
current_{flowing}
in the
shortcircuited output of an
FET
whichis due
to
this
thermal
noise ofthe
channel canbe
expressed as*dn
'_{4kT*maxAf}
*(VV
(27)
That
it
shouldtake
this
form is
not_{surprising}
because
the
thermal
3
noise current
due
to
a conductance_{g is}
i
4kTgAf.
The
additionalfactor
in
the
FET thermal
noiseexpression,
Q(VGV_)
is
an_{arbitrary}
constant of value one or
less.
Its
valueis
determined
_{by}
the
choiceof
V
and_{Vn}
or_{bias,}
andis
_{approximately}
0.7
for
operationin
the
saturated
(current
source)
region,
g
is the
maximumtransconductance
max
at
the
particularbias
voltageV_.
The
term
g;_{Q(Vpvt^}
can^e
thermal
noise currentI.
anEquation 27
wasderived
through
consideration of noise voltagesproduced
_{along the}
channel which modulatethe
channel width andhence
produce an amplified noise voltage at
the
drain*
The
expressionis
theoretically
valid_{only for bias}
conditions
below
saturation(pinchoff)
but
Van
der
Ziel's
experiments showedthat
the
expressionis
_{"nearly}
correctin
the
saturated part ofthe
characteristic as_{long}
as
the
field
strengthin
the
cutoff part ofthe
channelis
nottoo
4
large".
This
expression_{will,}
_{therefore,}
be
usedto
evaluatethe
amplitude of channel
thermal
noisefor
the
U222
FET*
The
agreementbetween
_{theory}
and experiment willbe
shownto
be
excellentin
Section VIII.
Another
important
source of noisein FET's is
that
due
to
capacitively
coupled gate noise. 'At
_{moderatly high frequencies}
thermal
noise voltages_{along the}
channel willbe capacitively
coupledto
the
gate_{causing}
a currentto
flow in
the
shortcircuitedinput.
Considering
the
source ofthis
noisecurrent,
one would suspectthat
it
would
be
_{partially}
correlated withthe
thermal
noise currentI.
Brunke
andVan
der
Ziel
showedthis to
be true,
however, they
alsoconcluded
that
the
correlation was so slightthat
it
couldbe
ignored
in
a noise analysis of anFET
circuit.
The
gate noise current wasshown
to
be
simply thermal
noisedue
to
the
device
input
conductanceGllr
T~
2

_{4kTG*.Af}
(28)
Here the
notation G*represents
the
input
conductance ofthe
device
connected
in
a common source configuration.This
mustbe
emphasizedbecause
the
_{capacitively}
coupled gate noisedoes
not change whenthe
device
is
connectedin
a common gate configuration._{Hence,}
the
term
s
G..
whichdescribes
this
noise,
will appearin
the
calculationsfor
the
noisefactor
ofboth
mixers.The
term g..,
however,
willbe
usedto
representthe
input
conductance of either configuration.Other
noise sourcesin FET's
are excess or1/f
noise and shotnoise of
the
gatecurrent.
The
1/f
noiseis
negligiblefor
operationat
frequencies
above audio andthe
shot noise ofthe
gate current whichcan
be
expressed asI
_{2ql}
_{A}
_{f}
_{is insignificant}
_{when} _{compared}_{to}
gs
_{g}
the
thermal
noisedue
to
g...The thermal
noise ofthe
channelis
_{essentially}
constantin
amplitude over a wide
_{frequency}
range,
and_{therefore,}
the
_{limiting}
factor for
noise performance athigh
frequencies
is
the
gain cutoff pointof
the
transistor.
Since
our_{frequency}
range ofinterest
is
below
this
cutoff point and abovethe
range where1/f
noise mustbe
_{considered,}
the
_{only}
two
sources of noise which mustbe described in
the
analysisare
thermal
noise ofthe
channel and_{capacitively}
coupled gatenoise.
Noise
Equivalent
Circuit
The
noise equivalent circuit of afield
effecttransistor
is
onewhich
includes
the
two
noise generatorsI.
andI
in
additionto
the
an gn
I
is
the
drain
to
gateleakage
current.
generators
and admittances whichdescribe
the
devices
twoport
parameters,If
the
yparameter
equivalent circuitfor
the
device
is chosen, the
generators
I.
andI
areincluded
as shownin Figure
4,
because
_{they}
represent
the
shortcircuit output noise current andthe
shortcircuitinput
noise
current
respectively.
O
1
j
0
)v
C
)Y12E2
Q>21E1
QXn
Y22
O
C
Figure
4
This
noise equivalent circuit can nowbe
usedto
calculatethe
noiseperformance of
_{any}
circuit which utilizes anFET.
It
is
now possibleto
obtain a noise equivalent circuitfor
the
balanced
mixerconfiguration
_{by}
_{replacing}
the FET's
ofFigures
1
and2
_{by}
the
equivalentcircuit
ofFigure
4.
The result,
whichis
shown1:1:1
T
^(Dp^iOvi1*,^^
T
1:1:1
El
Q)v
>8H
(D8.ncEl
(D1
dn
<
822
Figure
5
Noise Factor Calculation
Figure
5
can nowbe
usedto
derive
an expressionfor
the
noisefactor
ofthe
balanced
mixer circuit.It
has
been
shown_{previously}
in
this
sectionthat
noisefactor
canbe
calculated_{by}
_{finding}
the
total
mean square current
in
the
shorted output and_{dividing}
that
_{by}
the
portion
in
the
output whichis
due
to
the
source.Note that
in the
following
calculationit is
assumedthe
capacitance ofthe transformer
and
that
ofthe
devices has
no effect onthe
noisefactor.
This
assumption
is
validif
the
capacitive reactanceis
large
with respectto
_{1/g}
, orif the
capacitive effectis
neutralized asby
series Speaking,
for
example.This
willbe
examinedfurther
in Section V.
If
the
input
and outputtransformers
eachhave
turns
ratios of.
_{2}
_{2}
_{2}
_{2}
\a
"2<mc El
+Xdn
>
(29)
The
dependence
ofE.
onthe three
input
current generators must nowbe
found.
The
most straightforward
approachis
to
_{apply}
the
superposition principle.
The
part of_{E,}
whichdepends
on one ofthe
I
generators,
E,
,I
_{gn}
*"can
be
calculated
from
the
circuit
in
Figure
6.
v
48.
1:1:1
0i
gn
'11
'11
2
2
C
m gni8
<*V
+ 28ll)2(30)
There
aretwo
such generators andthe
voltagedue
to
both
is
therefore
2E,
2
ig
2C
(4gs
?_{2guV}
(31)
The
contribution
ofthe
source generatoris
found
in
the
same mannerto
be
2
r
(32)
rr^
ig
(4gs
?_{28nV}
Substituting
equations31
and32
into
equation29
yields7"
2
.2
XnT
"28mc
2r
2
ff1
41.
s(4g3
* 28U)2(4gs
*2gu)2
*
_{2}
I~H2 nd(33)
The
mean square currentdue
to
the
source generatoris
nowthe
secondterm
in
equation33
orM
_{(4gs}
41.
?2gn)'
The
noisefactor
is
nowfound
_{by}
_{dividing}
equation34
into
equation33
loJ
4Id2(8Q
+ St,/,)2p
_{1}
+JBL
? nd?
_{(35)}
2Is
V.
V
2
Substituting
equations_{27,}
28
andI
4kTg
Af
yieldss s
Gll
48x
Q(W(gs
+p
._{i}
_{+} ii +S_J2
5
ULt
_{(36)}
^g
%x
8
_{s}and
_{setting}
R
*C
D
(37)
nc
2
"me
The
noisefactor
ofthe
balanced
mixeris
now:Gn
UKr.
_{<8}
+_{811/9)}
p
_{1}
+ li _{+} J5
!i&_
(38)
2gs
8s
The
_{sensitivity}
as afunction
of noisefactor
was_{previously}
found
to
be
2
e
^
_{kTAf}
_{F}
U
_{D}
(39)
sr2
^f
_{kTAf}
fc1)
s,s
"11
,
4Rnc<8s
+8U/2)<
2gs
gs
(40)
which
expresses
_{sensitivity}
as afunction
ofthe
source_{admittance,}
gs,
where_{g}
is
as shownin
Figure
7.
8.
O
'g
O
J
Figure
7
Sensitivity
Vs.
R
for
Common Source Mixer
When the
parameters of equation_{(40)}
are assignedthe
values ofTable
I,
the
_{sensitivity}
ofthe
common source mixer canbe
plotted vs.Table I
^x
10,000
micromhos^mc
2,000
micromhosQ<VV
0.7
Gn
21.7
micromhos8n
21.7
micromhosk
1.38
X
IO"23
T
290K
Af
3,000
Hz
oi.
10
Note that the
_{sensitivity}
is
114dBm
at anR
of200
ohms whichcorrespondes
to
a noisefactor
of34.7.
Sensitivity
ofCommon Gate Mixer
Sensitivity
vs.R
ofthe
common gate mixer can alsobe found
from
equation40.
However,
it is
_{first necessary}
to
derive
anexpression
for
g.
.,the
input
conductance of a common gateFET,
andG
11 c
the
conversion gain ofthe
common gate mixer.The
equivalent circuitfor
a common gateFET
is
shownin
Figure
9.
For
our purposethe
outputis
assumed a short circuitfor
the
R.F.
signals,
as shownin
Figure
10.
Z,
is
nowderived
_{by}
_{first writing}
an expression
for
V
S

o^gs
4
9
(*
_{J}
f 9=:
8;
_{DS}
'SG
D
O
oFigure
9
O
D
I
_{g V}
V
 2m
_{sfi}
(41)
V8
_{8DS+JcSG}
Vsg
(gm
+gDS
+^CSG>
"Xs
(42)
and
_{now,}
in
^
+8DS
+ja>CSG
(43)
gll
8m
(44)
As
shownin
Section
VII
the
devices
arebiased
at_{g}
 _{g..} _{10,000}
micromhos or
R..
_{100}
ohms.
The
conversion gain ofthe
common gate singleended mixeris
found
in the
same manner asG
wasfound for
the
common source mixerin
c
Section II.
The
singleended common source mixer wasfound
to
have
a conversion gain of
6
_{dB,}
equation17.
If
the
device
were now switchedto
the
common gate
_{configuration^}
would not_{change,}
but
the
input
voltagewould
_{drop}
because
ofthe
higher input
conductance ofthe
common gateconfiguration
(equation 44).
The
availableinput
power would not changethough,
whilethe
output_{power,}
whichdepends
onthe
input
voltage,
would
drop.
At R
1/gs
100 ohms,
for
example,
the
sourceis
matchedto
the
100
ohminput
impedance
ofthe
common gateFET.
The
input
voltagehigh impedance
of a common sourcedevice.
The
conversion gain_{is,}
therefore,
6 dB less
than that
ofthe
common source mixer or0
dB*
The
conversion
gain of abalanced
common gate mixeris the
sameas
that
for
the
singleended mixerif the
source andload
impedances
are adjusted
properly.
The
gain_{is, therefore,}
0 dB
withR
equal200
ohms.
This is
nowthe
maximum possible conversion gainfor
this
mixer
because
the
sourceimpedance
is
matched._{Increasing}
ordecreasing
R
will resultin
alower
conversion gain.5
The
noisefactor
at values ofR
otherthan
200
ohmsis
therefore
s
of
little
_{importance,}
because
equation21
showsthat
the total
noisefactor
oftwo
cascaded stages_{increases rapidly}
asthe
loss
in
the
first
stage
increases.
Another
reason_{for considering}
the
noisefactor for
R
200
ohmsis
that
the
maximumdynamic
range occursfor
this
value.This
willbe
shownin Section V.
The sensitivity
ofthe
common gatebalanced
mixer atR
_{1/gs}
s
200
ohmsis
nowfound
from
equation40
_{by}
_{substituting}
from
Table
I
except with g.. 
_{10,000}
_{micromhos.}
_{The}
sensitivity is
108dBm
corresponding
to
a noisefactor
of_{139.}
Summary
ofNoise Analysis.
The
sources of noisein FET's
weredescribed
and an equivalentcircuit
for
the
device
which_{quantitatively}
representedthese
sourceswas
developed.
This
led
to the
noise equivalent circuit ofthe
balanced
mixer andto
the
calculation ofthe
noisefactor
as afunction
R
because
the
_{relationship}
of_{sensitivity}
and noisefactor
had been
previously
derived*
The
noise_{factor,}
and_{hence,}
sensitivity
ofthe
common gate mixer were
found
to
be
6 dB
worsethan
for
the
conmonsource mixer at a particular value of
R
(200
ohms)*The
lower limit
ofthe
dynamic
rangehas
nowbeen
established.The
upperlimit
orthe
large
signal_{handling}
_{capability}
willbe
IV.
LARGE SIGNAL HANDLING CAPABILITY
Intermodulation
andCrossmodulation Distortion
Relationship
When large
signals are present atthe
mixer_{input,}
distortion
is
produced
in the
circuit.
Another
measure ofthe
_{quality}
of a receiverfront
end mixer_{is,}
_{therefore,}
the
amount ofdistortion
producedfor
agiven amplitude of
input
signal.This
aspect of mixerdesign
will nowbe
examined and a quantitative result willbe
obtainedfor
the
FET
mixer.
The
two
types
ofdistortion
which mustbe
considered areinter
modulation_{distortion,}
I.M.
, and crossmodulationdistortion.
I.M.
is the
result oftwo
input
signals atfrequencies
f.
andf_
combining in
a nonlineardevice
to
give products at(kf.
i
_{If),}
wherek
and1
areintegers.
The
third
orderproducts,
(2f
f,)
_{and}2
l
(2f.
f9)
_{are} ofthe
most concernbecause
_{they}
can_{be very}
closeto
the
desired
signals,
f.
andf_
_{making it}
impossible
to
removethem
withselectivity.
An
example ofI.M.
in
atuned
amplifieris
shownin Figure
11.
The
amplifieris
driven
with alarge
twotone
signal_{causing}
I.M.
products at
(2f2
_{fy)}
_{and}(2fj
f2).
Crossmodulation
is defined
asthe transfer
ofthe
sidebands of alarge
A.M.
undesired signalto
adesired
signal.This
alsois
causedby
the
third
order curvature ofthe
device
transfer
characteristic.Two tone
Generator
\{
Spectrum
Analyzer
Tuned
Amplifier
withCenter
_{Frequency}
f
*
f
f.
f
_{f9}
1
o2
Twotone Input
*
f
(2fr2)
_{1}
_{o}
_{2}
(2f2l)
Amplifier
Output
withI.M.
Distortion
but
third
order curvature mustbe kept
at a minimum sothat
large
signals can
be
accepted withlittle I.M.
and crossmodulationdistortion
occurring.
In
a mixerthe
I.M.
and crossmodulationdistortion
productsactually
are not produced untilthe
desired
signalhas
passedthrough
the
mixertwice.
The
i.f.
_{frequency}
is
generated onthe
first
pass,
andfed back
to
the
_{input,}
through the
drain
to
gate capacitancefor
example.The
I.M.
products and crossmodulation atthe
i.f.
_{frequency}
arethen
g
generated on
the
second pass.This
suggeststhat
the
mixerbe
modeledas a perfect mixer which exhibits
the
proper second order_{curvature,}
followed
_{by}
a nonlinear
twoport
_{having}
the
properthird
order curvatureas
in
Figure
12.
An
expressionfor
_{the relationship}
between I.M.
andcrossmodulation
distortion
will nowbe derived
_{by}
_{closely}
_{examining}
the
non
_{linear}
_{twoport}
_{network.}The
transfer
characteristic ofthe
twoport
canbe
written as aTaylor
seriesI2K
_{c^j}
+ + + '_{(45)}
If
atwotone
signalis
now substitutedfor
v ,I.M.
distortion
willresult.
v. 
_{V}
_{cos}_{to}_{t}
+_{V}
cos uit
_{(46)}
Substituting
and_{retaining only}
those terms
withfrequencies in
the
I2<*c.(V
cos<*>t
+_{V}
_{cosu^t)}
+c
[
9/4
V3cosa>flt +
9/4
V3cos^t +
3/2
V3 cos(2wfl
o>b)t +3/2
V3 cos
(2b
**a)t]
(47)
L.O.
R.F.
Pure Second
Order Curvature
Non
_{linear}
Two
_{Port}
All Orders
ofCurvature
*
_{I.F.}
0H.
L2
KJ m **
_{u}
+
Non
_{linear}
+Vl
TwoPort
_{V2}
m
KJ
_{O}
The
amplitude of_{the desired signal,}
CJ
, atthe
second portis
now Si3
c.V, neglecting
c_{9/4 V}
relativeto
c.V.
This
is
justified
if
the
twoport
networkhas
muchless
third
order curvaturethan
it
does
first
order_{curvature,}
a_{property}
of most practicaldevices.
The
3
amplitude of
the
I.M.
product at(2
a*  co)
is
3cV
/2
_{and}the
ratioof
the
desired
signal amplitudeto
the
distortion
signal amplitudeis
2cRI.M.
"73
_{3c3V}
(48)
R_
now expressesthe
degree
ofI.M.
distortion
as afunction
ofthe
I.M.input
signal amplitude andthe
coefficients ofthe
Taylor
seriesrepresenting
the
device
transfer
characteristic.If
the
distortion
products aredown
40 dB
from the
desired
signal,
RT
M
_{100}
_{(49)}
I.M.
and
the
amplitude ofthe
desired
signals which producedthe
distortion
are
now,
cl
ri.M.
frsrs;
<*
9
Finding
an expressionfor
crossmodulationdistortion
canbe done
by
returning to
equation45
and_{substituting}
v
V.
costo.t +_{V.}
_{(1}
+ m cosa?_{t)}
cos_{to}
t
11^
n /(51)
which
is
the
sum of adesired
signal at <*>. and an undesiredA.M.
signal at
<*J.
_{Only}
those
terms
in the
_{vicinity}
of e*> _{will}_{be}
retained*
Ij^c.V.
cosw.t +(V.
coso)_{t)}
'$$VH
ll
+2m
cos<ot
+n
m
/2
cos2
oih)
(52)
3c.
Neglecting
#v
_{?}
relativeto
_{c.,}
as_{before,}
Ix(3c~
V
m cosfa>t
+_{c.)}
_{V,}
coswtZ
J
2
nil1
(53)
and
X2*C1
3^3
2
c,
2
COS CO
t
+1
n
V.costo
t
_{(54)}
Now,
for
17.
crossmodulation3(m)c.
i
V0
_{0.01}
c,
i.(55)
and
the
amplitude ofthe
undesired signal which_{is necessary}
to
producethe
crossmodulation
is
V
3000
(m)
Vxmod
"V3000 (m)
_{c3}
(56)
If the
undesired signal were307.
modulated,
m0*3
andVxmod
"\IWT3
(57)
from
50
and57
the
ratioV
.to
VT
is
xmod
I.M.
Cl
Vxmod
V^^3
VI.M.
_{/}
Cl
1.29
(58)
150
c3
In
shortthe
amplitude of a307.
A.M.
modulated signal_{necessary}
to
product17.
crossmodulationis
1*29
times
the
amplitude of eachtone
of a
twotone
signal which producesI.M.
productsdown
40 dB from
the
desired
signals.Balanced Mixer Distortion
VT
Mfor
the
FET
balanced
mixer wasexperimentally found
to
be
400
mv pertone
when measured acrossthe
complete_{secondary}
ofthe
input
transformer.
The
availableinput
power_{necessary for I.M.}
as a
function
ofR
, whereR
is the
real part ofthe
impedance
seens s
looking
into
the
_{secondary}
ofthe
input
transformer.
See Figure
7.
The
common
source mixerhas
a_{very high input}
_{impedance and,}
hence,
the
available power atthe
input
which willlead
to
a voltageof
V_
pertone
atthe
transformer
outputis
I.M.
(59)
with,
V
_{0.4}
_{v}I.M.
p
PjI.M.
_{Re}
(60)
A
plot of equation60
is
shownin
Figure
8.
A
plot of crossmodulationis
also shownin Figure
8.
The
availablepower
for 1%
crossmodulation shouldbe 2 dB
(1.29
in
dB)
higher
than the
power
in
onetone
ofthe twotone
signal._{However,}
sincethere
aretwotones
the
power ofthat
signalis
doubled
(3
dB)
andthe
crossmodulationplot ends
_{up}
1 dB below
the
I.M.
plot.The
common gate mixer which wehave
agreedin
Section
III
to
investigate
_{only}
atR
200
ohmshas
aninput impedance
which matchesR
V_
wtherefore,
will appearfrom
sourceto
source whenthe
S
I.M.
P
.2(vi")2
I.M.
200
(61)
With
_{VT}
w equalto
400
mvI.M.
P,
_{1.6}
_{mv}  _{+}_{2 dBm}
(62)
I.M.
Which
is
four
times the
power which causedthe
samedistortion
in
the
common source
mixer,
at anR
of200
ohms.s
The
large
signal_{handling}
_{capability}
of each configrationhas
nowV.
DYNAMIC RANGE
COMPARISON
Comparison
The
limits
of_{sensitivity}
and signal_{handling}
_{capability for}
the
common source mixer
have
nowbeen
established and are shownin
Figure
8.
The
maximum_{dynamic}
range occurs wherethe
slopes ofthe
two
plots arethe
same.
If
the
rangebetween
the
_{sensitivity}
andthe
I.M.
plotsis
considered,
wefind
that the
maximumis
110 dB
at anR
of arounds
200
ohms.
This
is
a_{very}
desirable
resultbecause
a wideband
input
trans
former
mustbe
_{built,}
andit
has
been
shown_{by}
Ruthroff
that
such atransformer
with atwo
to
one_{step up}
from
50
ohms(R
_{200}
ohms)
S
is
quite_{easily}
realizable._{The resulting}
_{sensitivity}
is
114dBm
or0.45
microvoltsinto
_{fifty}
ohms whichis
excellent._{PT}
.,is
4dBm,
I.M.which correspondes
to
atwotone
signal of100
mv pertone
into
_{fifty}
ohms.
The
dynamic
range ofthe
common gate mixerhas
alsobeen
established,
but only
at one particular value ofR
,200
ohms.However,
it
shouldnot
be
too
difficult
to
seethat
this
value ofR
alsoleads
to
_{very}
nearly
the
maximum possible_{dynamic range,}
for
this
mixer asit did
for
the
common source mixer.In Section III
the
_{sensitivity}
ofthe
common gate mixer atR
equal^