Download Linear amplifier design and linear integrated circuits Hobler, William Joseph 1968

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Thesis and Dissertation Collection
1968
Linear amplifier design and linear integrated circuits
Hobler, William Joseph
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/12556
N PS ARCHIVE
1968
HOBLER, W.
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MNEAR AMPLIFIER DESIGN AND
LINEAR INTEGRATED CIRCUITS
WILUAM JOSEPH HQBUR. M.
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i
II
LINEAR AMPLIFIER DESIGN
AND
LINEAR INTEGRATED CIRCUITS
by
William Joseph Hobler, Jr.
Lieutenant Commander, United States Navy
B.S., U. S. Naval Academy, 1957
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOL
June 1968
ABSTRACT
g.
Through the utilization of the functional matrix description
of linear
integrated circuits, a design method is evolved from which the
terminating admittances
may be
specified.
These terminating admittances
are optimized with respect to power gain for a stable device and
optimized with respect to power gain for a specified stability level.
The total input and output conductances are obtained for use
response design.
in
which
a not
in
frequency
Finally the procedure is extended to include the case
optimum conductance
is forced
on the designer.
TABLE OF CONTENTS
Section
I.
II.
III.
Page
INTRODUCTION
DEVICE AND CIRCUIT STABILITY
15
Characterization
15
Definition of Stability
17
AMPLIFIERS USING INHERENTLY STABLE DEVICES
Example
IV.
9
AMPLIFIERS USING POTENTIALLY UNSTABLE DEVICES
21
26
31
Example
V.
THE TERMINAL PROPERTIES OF THE AMPLIFIER STAGE
44
VI.
NON -OPTIMUM LOAD AND SOURCE CONDUCTANCES
54
CONCLUSIONS
5 7
VII.
FOOTNOTES
77
BIBLIOGRAPHY
80
LIST
OF TABLES
TABLE
I.
PAGE
Comparison
of
Weights and Sizes of Conventional
and Microelectronic Circuits
II.
The Values
of
Gamma which Maximize Gain
11
as a
Function of the Stability Figure and
60
Transmission Angle
III.
IV.
V.
VI.
The Gain Factor, F,
Asa
Function of the Stability
Figure and Transmission Angle
62
R of
Gamma and
Theta
64
of
Gamma and
Theta
70
I
H
of P
and R
76
LIST
OF FIGURES
FIGURE
PAGE
1.
Terminated Two-Port
2.
Power
3.
Optimum Power Gain and Conductance Product as
in a
17
Terminated Two-Port
21
a Function of Stability Figure
4.
The Factors 1/r and
5.
The Value
of
Gamma
(1
- 1/r) 2 as a Function of
27
r
41
Required to Maximize Gain as
a Function of 6, for Several Values of the
42
Stability Figure
6.
The Gain Factor F as a Function
of 0, for Several
Values of the Stability Figure
7.
Cascode Amplifier with Optimum Terminations
43
52
CHAPTER
I
INTRODUCTION
The design
of
low power level circuitry can no longer be regarded
as an exercise in stable bias design, impedance matching and feedback
design.
As more and more devices appear on the market, the selection
of a device
which will be stable under the specified circuit operating
conditions becomes increasingly important and difficult.
active device densities which result from the industry
The high
move toward
miniaturization constrain the circuit designer to those designs which
are compatible with this packaging and the thermal variations inherent
The circuit designer must then select the
to these high densities.
proper device, configure that device to meet the performance specifications, prove the stability of the circuit and allow sufficient engineering time to permit proper packaging and thermal design.
Recently Linear Integrated Circuits (LICs) have become economically
competitive with other low power devices in the frequency range
through 130
for the
MHz. The
designer, therefore, must consider LICs
convenience of 135 db gain
amplifier with 70 to 90 db
in a flatpack or
common mode
rejection.
D-C
if
only
an instrument
The designer
of
military equipment can look toward increased reliability, decreased
maintenance and decreased logistic support requirements which result
from use of LICs.
The designer of consumer electronics may select
LICs because of the lower
initial circuit cost, easier
automation of
the manufacturing process and the decrease in assembly operations
which
all
reduce the manufacturing cost.
Although the introduction of LICs has increased the selection of
devices, LICs provide the designer with the following distinct
advantages:
1
.
Reduced size and weight
2
.
Reduced power drain
3.
Improved reliability
4
Lower cost
.
5.
Improved maintainability
6.
Improved stability
It is
the intention of this paper to justify briefly the first five of
these advantages with indications of future trends in an effort to
illuminate a broad range of system or product improvements accruing
from Linear Integrated Circuits.
The sixth advantage, stability,
is to
be just one portion of a fairly vigorous investigation of a circuit design
technique which
is
comprehensive enough to include stability
of the
design.
SYSTEM OR PRODUCT IMPROVEMENT
BY USE OF LINEAR INTEGRATED CIRCUITS
1.
Reduced Size and Weight.
The presently available size and
weight reduction are most conveniently displayed by comparison of
these parameters in Table
I.
10
TABLE
I
COMPARISON OF WEIGHTS AND SIZES
OF CONVENTIONAL AND MICROELECTRONIC CIRCUITS
Con-
Miniature
struction
Conven-
Weight
Integrated Circuit
Thin Film
1/4 x 1/4
x 0.025
substrata
tional
TO-5
Can
1/4 x 1/4
Flatpack
0.055
0.35
0.0055
0.000035
0.002
0.1
0.04
0.008
0.000016
0.008
Chip
in oz.
Volume
in
3
in. °
Note, although the basic chip size represents a reduction in
volume
of 10,000, the
package controls the volume and degrades this
advantage to a 10-20 to one reduction.
we may expect
Hallman predicts, "By 1975,
the overall size of avionics packaging to be approxi-
mately 1/30 the size of the equivalent 1965 package.
However, the
1980 package will occupy only 1/75 the volume of the equivalent 1965
functional package."
2
functional capabilities.
These size reductions presume no increase
However, increases
in
in capabilities are
indicated by a predicted 25 per cent increase in active device population
on board a U.S. Navy destroyer in the period 1965 to 1970.
of integrated circuits will reduce size
and weight by 17 per cent during
Thus, in both comparable systems and increased
the same period.
capability systems
Application
,
the size and weight of the electronics package will
decrease through integration,
11
2.
Reduced Power Drain.
contends that 50 per cent
Whitelock
to five per cent of the presently required power will be required for
comparable systems when integrated circuitry
is
used.
These figures
are supported by a reduction from 21 watts to 3.9 watts due to the use
PCM
of LICs in a
Multicoder by Burns and Foulke.
power requirements
for a conventional versus
Similarly, the
an integrated circuit
FM
Telemetry Encoder was reduced by half.
3.
Improved Reliability.
Hallman states, "Even so, we may
expect that our principal reliability problem in 1980 will be one of
devising ways to measure and/or predict mean-time-between-failure
rates in excess of 1000 hours.
This problem exists today in an
Ultra -Reliable Transceiver under evaluation by the
this program
U.S. Navy.
In
two six channel UHF transceivers have been produced with
the design goal of optimum performance for two years with no maintenance,
Q
with a confidence interval of 98 per cent.
between-failures of 17,520 hours.
reliability,
maximum
This
is a
mean-time-
"To approach this degree of
an all-solid-state transceiver was mandatory along with
utilization of microelectronic techniques and redundant and
Q
adaptive circuitry."
4.
Lower Cost.
Perhaps the most telling argument to support the
lower cost contention comes from the highly profit-motivated consumer
electronics industry.
H. H. Scott described
its
decision to use
integrated circuitry to replace discrete components as based purely on
economic considerations.
However,
12
this lower cost is reflected
"
throughout the spectrum of system
During initial construction,
life.
the integrated circuit can replace numbers of discrete components, thus
reducing the production operations required, saving labor costs.
The
increased reliability increases the mean-time-between-repair causing
a decrease in
circuitry
down time and
may be included
down time and/or
factor.
In fact,
in
repair cost.
some
reduced volume where reduced
of the
training or salary of repair personnel is a determining
it
may become economical
(costing $1.50) rather than troubleshoot
Improved Maintainability.
5.
Additionally, trouble shooting
to discard a 135
db amplifier
it.
"As the effective failure-
or
trouble-free life of subassemblies increases, the economic advantage
of simply replacing defective
subassemblies and discarding the faulty
module becomes more attractive.
Also, the skill levels required of
maintenance personnel are reduced considerably.
summary, the benefits that are derived from microelectronic
In
techniques - reliability, size, weight, maitainability
consumption
- are real.
,
cost and power
"For all these reasons, circuit and system
design must assume greater importance and must be applied with greater
care.
The time and money spent on optimizing circuit design will
become
a worthwhile investment,
.
.
One
result os such effort could
be a more thorough and complete characterization of circuits than
have ever had in the past and
in a
systematic way.
"
12
we
Such complete
characterization of circuits has been available to the circuit designer
for
some years.
Systematic methods of applying these characterizations
13
and describing circuit operation
literature is to be
is
also available.
If
current applications
used as an indicator, these techniques have not been
broadly applied to Linear Integrated Circuits.
14
CHAPTER
II
DEVICE AND CIRCUIT STABILITY
On
design
of the
most common and most exasperating difficulties
the problem of oscillation.
is
some form
in circuit
Oscillation is always caused by
of feedback from the output to the input.
Such feedback can
be a result of the circuit interconnections either wired in or through
stray capacitance and inductive coupling.
Feedback may also occur
through reverse transmission within the active device.
Oscillation due
to the reverse transmission will be referred to as device-caused
1
o
The existence of internal feedback or reverse trans-
oscillation
.
mission
expressed by the fact that
is
if
the device is described by the
"z", "y", "h", or "g" matrices that element having the subscript 12 is
different from zero.
Most active devices presently available have
sufficient internal feedback to cause oscillation
and operated
shown
at the circuits resonant frequency. \^
that Linear Integrated Circuits
it
is
It
can and will be
possess sufficient internal feed-
back to support device-caused oscillation.
oscillate
when properly terminated
Since the circuit
may
proper to begin an investigation of LIC application with
a discussion of stability.
1.
Characterization.
Any discussion
with the characterization of the device.
of active
devices must begin
The characterization most suit-
able for the study of the performance of an LIC
is a
functional character-
ization from which the designer determines the potential of the circuit
15
including stability, sensitivity, gain, and optimum load and source
impedances.
In characterization of LICs,
separately characterize each component.
to reach every point in the circuit.
is
If
it
is not practical to
Initially,
it
is not
possible
an equivalent circuit formulation
attempted, the stray capacitance and lead inductance form complex
pi-networks around the active circuits which severely compound the
characterization.
Additionally, since a diffused resistor is, in fact,
a distributed element, its equivalent circuit is a
and the equivalent circuit becomes untenable.
R-C transmission
line
To avoid the difficulties
imposed by the above consideration, the functional two-port matrix
1
c
characterization is used for LICs.
The familiar two-port characterizations are the "h", "g", "z" and
"y" parameters.
the device
.
Any
These parameters implicitly but completely characterize
set of two-port parameters
may be calculated from any
other set so that the set of parameters actually used is determined by
the designer.
Since the device may be parameterized in any of four
ways and an analysis must produce
identical physical results for any
set of parameters, analysis utilizing
any parameter set
is
providing the chosen parameters are consistently applied.
equally valid
Therefore, an
analysis using one set of parameters provides a general series of design
relationships.
17
Here
it is
well to note that the following analysis
will produce certain "invariant" properties.
This use of the term
invarient implies invariant in form under the transformations between
the
common two-port parameters.
16
CHAPTER
II
DEVICE AND CIRCUIT STABILITY
On
design
most common and most exasperating difficulties
of the
problem of oscillation.
is the
some form
in circuit
Oscillation is always caused
of feedback from the output to the input.
by-
Such feedback can
be a result of the circuit interconnections either wired in or through
stray capacitance and inductive coupling.
Feedback may also occur
through reverse transmission within the active device.
Oscillation due
to the reverse transmission will be referred to as device-caused
l
3
The existence of internal feedback or reverse trans-
oscillation
.
mission
expressed by the fact that
"z"
,
is
"y"
,
if
the device is described by the
"h", or "g" matrices that element having the subscript 12 is
different from zero.
Most active devices presently available have
sufficient internal feedback to cause oscillation
and operated
shown
at the circuits resonant frequency. \^
that Linear Integrated Circuits
it
is
It
can and will be
possess sufficient internal feed-
back to support device-caused oscillation.
oscillate
when properly terminated
Since the circuit may
proper to begin an investigation of LIC application with
a discussion of stability.
1.
Characterization.
Any discussion
with the characterization of the device.
of active
devices must begin
The characterization most suit-
able for the study of the performance of an LIC is a functional characterization from which the designer determines the potential of the circuit
15
including stability, sensitivity, gain, and optimum load and source
impedances.
In characterization of LICs,
separately characterize each component.
to reach every point in the circuit.
is
If
it
is not practical to
Initially,
is not
it
possible
an equivalent circuit formulation
attempted, the stray capacitance and lead inductance form complex
pi-networks around the active circuits which severely compound the
Additionally, since a diffused resistor is, in fact,
characterization.
a distributed element, its equivalent circuit is a
and the equivalent circuit becomes untenable.
R-C transmission
line
To avoid the difficulties
imposed by the above consideration, the functional two-port matrix
characterization
is
used
for LICs.
The familiar two-port characterizations are the "h", "g", "z" and
"y" parameters.
the device
.
Any
These parameters implicitly but completely characterize
set of two-port parameters
may be calculated from any
other set so that the set of parameters actually used is determined by
the designer.
Since the device
may be parameterized
ways and an analysis must produce identical physical
set of parameters, analysis utilizing
in
any
of four
results for any
any parameter set
is
providing the chosen parameters are consistently applied.
equally valid
Therefore, an
analysis using one set of parameters provides a general series of design
relationships.
17
Here
it is
well to note that the following analysis
will produce certain "invariant" properties.
This use of the term
invarient implies invariant in form under the transformations between
the
common two-port parameters.
16
.
Although
way
to the
it
in
is true that
,
Mason's unilateral power gain U
which the terminals are paired
is invariant
in a standard transistor,
1 ft
since the substrate of an integrated circuit transistor is always connected
common
to the
terminal, any change in the configuration changes the
internal circuit.
gain.
19
This alteration of the internal circuit changes the power
Therefore, whenever the device under consideration is an LIC
the invariance deals with the use of consistent sets of parameters and
not with the shifting of terminal pairs
2.
Any useful active device may be
Definitions of Stability.
categorized, with respect to stability, as inherently stable or potentially
unstable.
The device
is
considered inherently stable
can cause oscillation.
of passive terminations
feedback within the active device
if
no combination
However,
is sufficient to
if
there can be found
ft.
1
\
i
1
FIGURE
1
Terminated Two-port
17
With
of y
s
is potentially
unstable.
L
20
.
some combination
y T which will cause oscillations, the active device
1«.
the internal
support device-caused
oscillation, the device is considered potentially unstable
reference to Figure 1,
if
and
Rollett introduced the invariant stability factors, k and K defined as
'
\%-J%\
Qu-1^
OuWtA^
A device
is stable if k
stable
K
if
>
1
and
^
Yi^xA
%i «*& 4z^=
1
and
G,,G 22
~
g,
,
g 22
~
^L.4 3-M,
Note that
if
no combination
and yr can cause K to become less than unity and G, iG,, ^L
device
is
inherently stable.
(2)
Similarly, a circuit is
0«
1
0.
(1)
of
ys
the
Bahrs casts these two expressions into
equivalent forms given below:
In this form the device is inherently stable
valid.
If
and
G,
-J
may be made
G 22
if
the above inequality is
are defined as above a potentially unstable device
inherently stable by neutralization or by loading the device
sufficiently with large source and load conductances.
of these
conductances and the active device
is inherently stable if
S«^ v >M/z(H&««)
where
M
and 6 are as defined
in
equations
18
The combination
(4).
(5)
Note that the parameters involved
functions of frequency.
in
measuring the stability are
Any particular device may be potentially
unstable over any portion of
its
statement that an active device
usable frequency range.
is potentially
unstable
Thus, the
is really
incomplete unless the frequency range over which this instability exists
is
included in the statement.
The expression
new
G-,-,G
22
22
(M/2)
(1
+ Cos
suggests the use of a
0)
stability factor for the terminated two-port.
K*
MO-t Co^e)
(6)
where
(7)
The terminated two-port will be stable
stability factor
if
K
^
1.
Similarly, an inherent
may be defined by
M(HQw©)
where
M
and
8 are defined
By this definition,
if
by
K,
(7)
^> 1
(8)
above.
no combination of passive source
and load terminations can support device-caused oscillations.
if
Conversely,
K^ is less than unity, a suitable pair of passive terminations exists
which can cause oscillation.
Therefore, the device is inherently stable
19
.
If Kj
2?
1
anc* potentially unstable
conductance
added to the ports
is
the actual stability factor
if
K^ <.
1
.
If
sufficient external
of a potentially unstable device, K
may be made
greater than unity and the circuit
stable.
The actual stability factor K
and of Bahrs
.
It is
port parameters
is identical
with the factor K of Stern
not invariant under the transformations of the two-
23
20
CHAPTER
III
AMPLIFIERS USING INHERENTLY STABLE DEVICES
For two-port networks which require power at the input port, as
LICs do,
power
it
is
more significant to measure performance
in the circuit.
in
terms of the
The ratio of power delivered to the load to the
power available from the source
defined as transducer gain
is
,
and
given the symbol Gp.
'
r
fos^aJJk. i****JL fe« uu
?*
fkwJl
(9)
?** CS
liKisl
y
Oil
Jll
V
k
'V w%
FIGURE
Power
in a
2
Terminated Two-Port
Unfortunately, transducer gain
is a
function of the source
impedance, the device parameters, and the load impedance.
It is,
therefore, convenient to consider another quantity which provides an
21
1
upper bound on transducer gain and
(—
is
given the symbol
Z.
i
I
easier to calculate.
Power gain
power supplied to load to the power into the two-port
is the ratio of the
and
is
24
G
.
yO
mini
The transducer gain may be expressed as
(ii)
For an inherently stable device,
to unity.
P.
/P ava ^^
always less than or equal
is
The maximum value of unity can be achieved by conjugately
matching y o to
y.
.
At a given frequency or in a narrow band of
XI
frequencies, this can always be done by using a suitable transformer
and a susceptive element.
Thus, the transducer gain
the power gain for an inherently stable device.
is
the
same as
The design problem
reduces to finding the load termination which will maximize the power
.
gain.
25
The power gain of the device
admittance and
is
expressible as
9
is
independent of the source
Ft
&^, - 4,^vr4^o)i /wl J 2
After
some manipulation equation
= 1<fr
^.2.
M
(12)
may be
rewritten as
%L
(13)
\ + B^ - -MS uG«®-MBM>><*t*
g.. (<9 Z
22
Now
by defining a normalized transducer gain
power gain G
pN such
*
<?r--
and a normalized
that
^
CLn-J.
G rm
-
J
^J
G^
^
spm
(14)
Hi
so that
<?<
PN
<?<
Since
-7
%»7%x
|y 21 /y 12
yields a
is a
|
constant of the device, a maximum value of
maximum value
of
G
or
p
G r Now
(15)
G PN
from equations (13) and (15)
the normalized power gain is
GPI1
Kf5xi-<3W>
Z
J16)
Since B
b
b
22 - 22 + L occurs only in the second term of the denominator
the value of B
22 which maximizes
B
n
or the
optimum value
G PN
O
of B
,
is
obviously
(17)
B 97
opt
B 21
(18)
oft
23
Thus the optimum output susceptance
is
independent of the load
into equation (16) and rearranging
Substitution of B 99
zz op
conductance.
yields
Now
d>;
if
^
an inherent stability figure
£jbl2l2: _
M
is
—
is
(20)
defined as
where
M
then equation
(19)
(19)
defined as 0^ where
Go©
and a partial stability figure
r
———
~ p—
^f>N "
(21)
may be
written as
The inherent stability figure was defined as a function of the device
parameters whereas the partial stability figure
parameters and the load conductance.
is a
Equation
function of the device
(22)
represents the
normalized power gain for any load conductance for the optimum load
susceptance.
The optimum load conductance may be calculated from
24
that partial stability figure which maximizes
equation
(22)
G pN
Differentiating
.
and equating the result to zero yields
(#-*£>&+' 7°
(23)
From the definitions of the two stability figures,
it
is
obvious that
P
is
always greater than
stability figure is
^V
If
~~
J2L
p opt
*' +
this value of
of
,
optimum partial
where
~^'
substituted into the expression of
is
maximum value
#•, thus the solution for the
Gp^ G p ,
,
,
T
G p,
the
T
results
o
I
(25)
The value of normalized power gain
Since for
plotted in Figure 3.
is
unbounded
at
0"
=
1, the
all
for various values of 0. is
values of
device
is
n
at a value of
^opt
By expression
g
g
(M/2)
(1
(3)
G n^ ,w
T
less than unity the power
potentially unstable.
greater than unity the gain is finite for all
value at
0.
j2L
For
0.
going through a maximum
.
Bahrs had defined the stability relation
+ Cos8) which
may be
tkV±- C^&>\
M
25
written as
27
(26)
then the condition for inherent stability may just as validly be written
as
{
1,
)>
since
i
A
for potential instability
manner
2
gjl g 22
-Cos6.
Similarly, the condition
M
may be
\1-
written as 0*
I
n an analogous
a stability figure for the terminated two-port is defined as
4-
-G^e
s
{27)
At this point the problem of amplifier design is completed since
the load susceptance
may be determined from equation
conductance may be determined from
JZL
p opt
and equation
input admittance need only be conjugately matched.
load
may be reduced by recognizing
Therefore,
r
G p ^ and
= G22
and the load
Then the
(21).
The computation
that equations (20)
and the optimum value of conductance gL
M
(18)
,
(21)
and
(24)
~ g 22 yields
/-
equation
(28)
are functions of the inherent stability
(28)
o
figure and
may be
plotted as functions of 0^, as is done in figure 3.
ILLUSTRATIVE EXAMPLE
To illustrate the design procedure using an inherently stable
device, an
RCA
differential amplifier,
operating frequency of 10.7 mega
CA
3028, was selected and an
Hz used as center frequency.
26
For
8
_
0*3
^
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s a
c
a
this condition the specification sheet lists the following admittance
parameters.
29
y 22 = .04 +
0002 mmhos
Y21 ~ ~^ 7 +
mmhos
=
y-,9
= .01 -
1.
Calculate intermediate but individually useful values.
.
5
+
mmhos
y
j
.
5
j
.
a.
M
b.
9 = arg(y
= «37 x 10
= Yi2Y2l
y
-6
j
.
-j •
23
5
mmhos
9
mhos^
= 178.0°
)
12 21
2
.
(1)
CosO =
(2)
SinO = .03403
-
.99942
Calculate the inherent stability figure
0.
(20)
yirJuAj^J2^ jJroMlA—
« 1.0074
3
.
From figure
a..
b.
4.
G pN
3
enter with
0-
and read the following
= .935
o
g, ,g T
11 L
/M
Q
= .03
,
Calculate the optimum load terminations
a.
g
L
= (M/g
).
3
03 = 2.22 x 10"
o
28
mhos
b.
-
,
Since
fc>
L
=
o
G 22
M &*^ &
opt
I
"*V
^a
5,
" k?9 from equation (18)
(29)
Calculate the optimum source terminations.
straight forward calculation
Lathi, by a
shows that the conjugate match
at the
input terminals for a device terminated with the optimum admittance is
as follows:
30
(30)
2 ?ll
6.
5
(3D
_
Calculate the stage power gain,
^=£.»ir-«£2£i
s
(14)
in.
The design problem
source addmittance of y
is
thus complete for the stated conditions.
= .27 75 -
j e
A
4874 milli mhos and a load
admittance y T = .0222 - j.23 milli mhos should yield a maximum stage
power gain
of
3459.5 or 35,4 db.
have been stopped
In this real world this
at the calculation of 0.
.
Figure
the sensitivity of the stage gain to changes in
29
0.
,
3
design should
amply illustrates
when
is
i
close
to unity.
It
does not take too much experience
in the electronics field
to conceive of parameter drifts which would shift this device into the
potentially unstable category.
An immediate attempt to design a less
sensitive and more stable circuit would be the addition of some
conductance
While
it is
power gain
at the input
true that
is
and output ports producing larger
g^
increased,
maximum
it
is
not true that the
for a specific level of stability
arbitrary procedure.
may be achieved by
A more rigorous and satisfying procedure
and g^o.
this
is
obtained by treating this device as a potentially unstable device,
selecting a circuit stability figure
section.
30
and design as
in the following
CHAPTER
IV
AMPLIFIERS USING POTENTIALLY UNSTABLE DEVICES
The transducer gain
be expressed as
G™
,
of a
generalized terminated two port device may-
where
i
By defining
the transducer gain
may be
written as
^v>P>3
L
Optimization of the source and load susceptances involves finding
those values of b s and b» which minimize the denominator of equation
As a matter
-
of
convenience this denominator
fen <n-r B.i B
is written
n- a.)%-(Bn<; u
as
_+"Bri.ci,
31
(33)
r b)
(35)
The partial derivatives
of
D
with respect to
B-,
-,
and B22 ar© a s
follows
— -u^ W^-»-^„»
s-lftM^-BuBjt-^Six+Wiifii^CiiS,,-!,)^
-n
ll r^B, rb ll
-^
and
^
.
-l<*Wtt-B.,Bn -<l)B„+2CB,flrf<i*Bn.-t)^ u
=
at the
(36>
(37)
minimum D
g&
__
dJr
-
^
(38)
This yields
Gi.Srt-Mn-^B,, =^« fl n+ $n
Dividing equation
(40)
by equation
(39)
fi
"
-W<n
yields
—
(41)
^2/l
£?n
Equation
(41)
states that to maximize the gain of the stage the ratio of
susceptance to conductance
equal.
(4o)
This ratio
y ^
is to
at the input
be defined as
Bll
aiL=
B XX
sii
32
and output ports should be
,
that is
(42 )
Now
-
D
and
equation
(35)
KM
its partial
i
may be
+*
1
)
written
-A.fr
tttf^.^-Vj v
)
derivative with respect to
6
(43)
is
|| s-^faifctixxO-lfhA.)
(44)
This partial derivative must in turn be equal to zero at the minimum
of
D, which leads to the relation
^rJY
lS
ib*
However, the formulation
of a
-
O
(45)
<?//<$ llr
and b are
(4 6)
and from the definition of stability figure
,
(equation
(2 7)
(4 7)
equation number
^ +
l+
(
(45)
may be expressed as
^c^©!
^T&T©
<3
(48)
This cubic equation (48) is in a "normal" form whose roots are discussed
in a
number
of sources; see for
example Eurington.
33
31
Using equations
(46)
and
(4 7)
equation number
may be
(4 3)
rewritten as
which reduces to
(49)
-vD^+<W)-SA.e]*j
Examination of equation
reveals that
is in
if
positive value of
particularly the last squared term
(4 9)
the first two quadrants,
q
Further,
.
it
is
Q
and
~^
is
of both (48)
negative for
and
(4 9)
is
is
minimized by some
noted that the equation
minimized under the condition that the sign
the sign of sin9, that is,
D
of
positive for
in quadrants 3
and 4.
Q
is
(49) is
identical with
two quadrants
in the first
Therefore, examination
leads to
r(4j»)
-
V(<tj&) Investigation of equation
-)s(<tj*)
(so)
-U</>^)
(sl)
(48) results in
the conclusion that for
in the first quadrant the roots consist of one positive real root
imaginary roots.
and two
Since only physically realizable solutions are to be
considered, the positive root
is
the desired value of
(J
.
For
in
the second quadrant, the roots of (48) consist of one positive root and
two negative roots,
of
which the positive root will minimize D.
34
.
summary then,
In
and 180 degrees,
for a
given value of
D minimum may
and for
lying between
be found by solving equation
(48) for
the one positive real root and substituting this root into equation
For other values of 6 the relations (50) and
With
i^
this
minimum value
D
yield the desired D.
the transducer gain
may be
written as
_
T
which
of
(51) will
(4 9).
{52)
is
equation
r
^JH
Yoi/y^?
\
ti
mes
trie
normalized transducer gain
i
G TN
of
(14)
Ah
"
ft*
T
%y
"
(53)
Since the minimum value of
D
has been found the remaining factor
under control of the circuit designer
be maximized to give the
is the
maximum gain
product g s gL-
This product must
but this maximization is subject
to the stability criteria.
<J>
— -
-
Lo~$
©
M
(
27
)
that is, the product g s gL is restricted to the condition
*"
(54)
35
However, by the definitions
of equation (33)
,
g s and g L
may be written
as
and the product as
(55)
Finally by defining a dimensionless ratio of the total terminal
conductance to the device conductance G/g as
may be
the product
is a
maximum when
^tlh)
where
written as
14^ ^\^ U,l " )CA ^'
which
r
-_
C)A.|
{)
(57)
the partial derivatives are equal and zero.
^jjkj
a
,58)
cJAz-l
By the equation
(5 7)
g S g T is seen to be symmetrical with respect to
J_i
r,
J.
,
J.
and r~„, which, of course, means that the maximum will be reached
when these two
dropped from
r
ratios are equal.
Therefore, the subscript will be
and the product written as
(59)
36
.
and
W
5*
which results
>*
2„
9-
g^ = M
^
n
in the ratio r
(<*+
&+*)
(60)
as a function of the device constants and
the stability figure
H< e *&*&)
2
/L r
where
^ ^^
-
0^ is the inherent stability figure defined
(61)
by equation
The normalized transducer gain may now be written
equations
(59)
and
(20)
in terms of
(61).
---"bi-tfiM-'l
D
K
J
Ti
f™ /
^*^"^"^""^^^ ^—
t
l
^
1————
nirn,i__„___._
-sii
^TCg
fctftGHS)
[^^+QM©xi-^)-^©]+[^ f a^©).
>
sin©7
:
(62)
37
.
When
written in this form Gt.m is seen to consist of two groupings
of factors
."V
M.+ QJL6©
7/_
x
H
9m;
2.
!*, 3 L
F
y 21'
i2'
a^|^, x)
%)a*uL
F, the remaining factors in equation (62) is defined
is a function of the
<y
,
4,
below and
transmission admittances and the stability figure.
*>
^ Z(0 + Co*®)
[^
With
this
(<*
+
^»)(i+y2)-Cw©] +JVY0* 0>.«e-£*e]
z
(63)
background the normalized transducer gain maximum may-
be expressed as
F'l
5th-6
/
A
set of
z
+ &^e
Y rrt
JCx
C&4&
<$+ /^
<t>i
(64)
/
optimum terminating conductances may now be obtained
from equation
(5 6).
1
that is
38
.
(65)
Similarly
2L=Vif
With these conductances, equation
susceptances
(42)
yields the remaining
32
(42)
=
*>
4
<3«-»3..K-
With
fc
»
b
(67)
this determination of the terminating admittances and the gain
of the amplifier the design is in
of the designer
of F,
L=V^O y -^V
q
equations.
,
essence complete.
The
effort required
has thus far not been reduced because the evaluation
the gain and conductances require solutions to transcendental
However,
this approach has separated the arguments of the
problem into grouping which can be conveniently computer programmed
and the results tabulated.
used by Lathi.
Initially
Q
was computed by Bahrs and
This paper has reformulated the work of these two into
39
a consistent notation.
Therefore,
has been recomputed and tabled
as a function of arg(y^2V2l)' anc* # the stability figure.
Similarly, F,
the gain factor has been tabled as a function of these two variables.
See Tables
II
is plotted in
and
III.
The other transcendental factor
Figure 4 as a function of the ratio
for several values of
although Table
II
steps
to illustrate the
r.
of interest
Figure
smooth behavior
through integral values and
5
plots
q
of this function,
in five degree
steps linear interpolation between any combination of values is a valid
operation.
Similarly, Figure 6 plots F for several values of
the validity of linear interpolation between values of F.
40
to support
41
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<>
H
CHAPTER V
THE TERMINAL PROPERTIES OF
THE AMPLIFIER STAGE
One
of
of the
more important considerations
bandpass characteristics.
The bandpass
of amplifier
is, of
design
is that
course, a function of
the total terminal admittance at the input and output terminals of the
stage.
The input admittance of a two port which
output by y
is
terminated at the
is
J-j
llV^L
to
(68)
which the input terminating admittance y
must be added to give the
total admittance at the input Y.
^ -'V1u-
i^ill
Qitfl
+ jlQ
44
(69)
Separation of the real and imaginary parts of
^7zx
(
l
-/-
2T
Y-
yields G-
and
B.
)
as
(70)
and
^«.( + * V
(71)
i
Considering
call
R
it
R(
first
Q
,
8)
Gj
.
By defining a function of
Q
,
and
8,
as
(y>^__
the input conductances
3-
i»
(72)
may be expressed as
-&*<*>*>
(73)
and
^
w
^/t^v
=
7//"
s„-
^ tt?w
5
^
'
•-
(74)
Similar arguments allow the output conductances to be expressed as
W
.
-
M
g tow
H
e>
fe- £
-,*:-
<5„
m
45
(75)
ca
—
The total stage terminal susceptances should be zero
maximum
c7//
o
gain.
If B^
—$
-
n
is to
be zero, then equation (71) is zero
—
2.1 (
/
at the point of
+ %
i
*
-o
)
(77)
^^
or7\(l+ f"r j y-
-O
(45)
The susceptance condition which yields zero total terminal susceptance
is, therefore, identical
with the condition of
maximum transducer
Under these conditions the terminal susceptance
is
known
gain.
to be zero.
However, under non-optimum conditions, the susceptance can most
conveniently be calculated by defining another function of
call
it I(
o
TCfye)
,
9),
q
and
0,
where
—
o^vu££ -^XG^S
/+ *^
(78)
Allowing the input and output susceptances are expressible as
Ov^n.
~
"il
(79)
B^
12
•
-
3
Ph -
—
y M I
^-^f-<^ZC6>eU&
c
'
46
r
(80)
.
tw
=
BcMrf =
_ M
b«-g^*,aj
(81)
^^-gjI^©)
These formulations
of
susceptance and conductance are convenient
virtue of the computer generated tables of R(
See Tables IV and V.
As with
^f
q
and F, R and
,
8)
I(
and
Q
,
I(
8)
q
,
by-
8)
are well
behaved functions so that linear interpolation between tabulated values
is a valid operation.
47
ILLUSTRATIVE EXAMPLE
The RCA integrated circuit CA 3028 may be used
The specification (op
figuration.
parameters at 10.7
y
n
=
6
(o.
MHz
cit.
.
=1.66 ( 75.1° mmhos
y 12 = (0.003 - j0.0) mmhos = .0003
y 21 =
("
~
18 )
J
mmhos
= (0.0006 + jO. 08)
y
=
arg
y 12 Y 21
I
(y
12
,
!
i
.
0°
.
mmhos
= 100.8 / -10.3°
mmhos
.08 / 90°
mmhos
mmhos =
= 0.0302 x 10~
y 21 ) = 9 = - 10.3°
Sine =-0.1788
Cos9 =0.984
2
/
Calculate the intermediate values
1.
M
p. 8) lists the
,
as
+ jl.6)-m mhos
The inherent stability figure
= 2Qllg 22 "
0.
Cose
M
3
- 2(0.6 x 10" )(0.6x 10~ 6 )
_
0#984
0.0302 x 10" 6
= - 0.960
3.
A POTENTIALLY UNSTABLE DEVICE
Choosing
0-3.0
a.
From Table
II
b.
From Table
III
c.
From Table IV
max = 0.05819
x\
F = 7.92925
R(
0,
48
8)
in a
= 0.99101
cascode con-
cascode y
Calculate the gain of the unit.
4.
=
From Figure
b.
/z.sr?r
4
- 1/r) 2
(1
= 0. 851
The normalized stage gain from equation
G
=
(64)
F(l-I) 2 =7.92925(0.851)
o
r
= 6.74
c.
q
Equation
-
(14)
yields gain
'i-
= 2.358 x 10 6
= 63.72 db
5.
a.
The terminating admittances are calculated
Source conductance, equation
(65)
49
b.
Source susceptance, equation
=
1(7,
ai
(67)
2
i.
oCo-ofsy-i.Llwo'
c.
Load conductance, equation
(66)
d.
Load susceptance, equation
(67)
*2-
6.
a.
The terminal conductances
Source conductance, equation
s 7,72 x/ar* 3 -
(74)
£g3£3 (o.?9/o/)
50
Load conductance, equation
b.
7
~^
-7^
Z^jrx/O
-
(76)
^'^0 2_V//3^
/ynkoA
This circuit was mechanized with some interesting results.
some
difficulty is encountered in synthesizing the admittance
6
6
y L = 7.12xl0~ - j80xl0"
.
The parallel inductance required
=
L
c— = 185
g
resistance
=
is r
P
The
is
80 x 10 b x 6.28 x 10.7 x 10 b
P
Q
Initially-
~
.//
H
while the parallel
= 141 x 10 ohms.
7.12 x 10"6
inductance required for a parallel circuit
of the
is rather high.
similar but not as severe problem exists in the input circuitry.
problem
is
circumvented
at both parts of
A
The
using equivalent an series
circuits for a load and a parallel series circuit in the input to match
a 600
ohm signal generator.
The test
rig
was wired and
the source and load impedances
measured.
Several adjustments were required to obtain the required
paramters.
The final circuit used
is
shown
in Figure
All capacitance
"J.
units are microfarads, both inductances measured at 10.7
all resistor
values are listed as their nominal values.
51
MHz, and
With the device
Vt^f
^SShdH
&
@®®
fa*-
> 2/<
Figure 7
Cascode Amplifier with Optimum Terminations
removed, the
r
resistance from terminal
were 139 and 1020 ohms respectively.
2
and terminal
6 to
AC ground
The calculated optimum values
are 137 and 1105 ohms.
The voltage applied was lOmv peak.
A clipped output caused
decrease of the input to 500 microvolts peak that
is
353 x 10
a
volts
rms.
For a matched impedance, this voltage level results in an input
power
of
-8
0.18 x 10
watts.
these identical conditions.
Two
integrated circuits were tested under
The output voltages observed were 1.41
c:9
and 1.38 volts rms which yields a power of 3.97 x 10
3.7 x 10" 3 watts.
2. 11
x 10
2.36 x 10
.
The transducer gain P
in
/P out
is
-3
watts and
then 2.2 x 10 6 and
These values compare very favorably with the calculated
6
.
Granted that the circuit configuration
is impractical in form,
however, the results and verification of this design approach justify
the use of the configuration.
53
CHAPTER
VI
NON-OPTIMUM LOAD AND SOURCE CONDUCTANCES
The conductances determined by equations
(65)
and
(66)
may be
either not obtainable or not desirable by virtue of interstage network
The effect
considerations.
optimum values
of the departure of g
s
analyzed as follows:
is
The optimum value
of
G^
is
C&4
11
V
3m
'opt
and gr from the
V
<3>
<fo+ C&*e
(56)
and
1XX
However,
let the
Gil—
where p
G,
6^^ <% i C^ &
»T~
.
is a
p
actual value of
(56)
4/
^f//
^//
IP*"
measure
(83)
of the departure of
For a given value of
,
Gjn from the optimum value
the product G, -|G« must be constant,
?
opt
that is
54
V
Therefore
These relations result
in the values of g
1
v
(p
and g L as
(86)
I
4bH-> Iv>
But from equation (62)
(62)
G T1VT
ii\i
However,
W
"0
is
17777^ MfiJC^
t1(HG„&)
A measure
of gain
V 3"
degradation may be defined as
3>
H
(90)
l%>G*e
'J
mi
H=
,_,
u Ij_iMijLiMuiiimMM_jJM.-*n*tf~'~"~**'''
*--—
-—
—
-
Mimiiwi' in *
-
!
(91)
J
Equation
(91) is
symmetrical with respect to p or
\_,
that is, H(4) = H(p)
P
which shows that
H
is identical if
G92 = PG99
or
G ??
J
opt
=
! G 22
P
opt
The normalized transducer gain may now be written as
(92)
Table VI tabulates
H
as a function of
i
+
+
It
can be seen that when this ratio
gain
is
11
small.
For p =
T-.
c
5
^ 0and
_i
is
.
nnrxr
= .0025,
Cos8
the loss in gain is only (8 per cent.
is
an d p
Cos0
small the loss in transducer
+ COS8
+
j
design
CosB
tit
A great deal
0.
•!_!
r
r
possible
for small values of _i
+ CosO
+ Cose.
56
of flexibility in
.
.
CHAPTER
VII
CONCLUSION
This thesis
was motivated by
the introduction of linear integrated
circuits into the devices available to the circuit design engineer.
As
soon as an application procedure was defined, the thesis dropped
all
contact with any specific active device.
of this subject referred to
In fact, the original study
vacuum tubes and transistors as active devices
Linear integrated circuits, because of their complex structure, force
the circuit designer
away from any
particular device-oriented model to
the functional model of two port parameters.
performance using these parameters
The analysis of circuit
is of great
value for several
reasons
a.
A two
port parameter analysis is independent of the type of
active device used.
b.
Since all device response
is
compared within the same
analytical framework, the design engineer more rapidly assimilates
the response differences of each individual device.
c.
The use
of only one
design method facilitates application of
optimization techniques across the broad range of electronic
devices
d.
The use of two port parameters and this design technique
permit transforming from any given sen of parameters to any more
convenient parameter set.
57
Objectively one must note a lack of consistent schemes of
specifying devices used by the device manufacturers.
Within the same
industrial organization, technical specification vary in
scheme and
The vacuum tubes may use the transconductance and
completeness.
input/output capacitance; transistors, the h parameters; integrated
circuits
may
gain, etc.
,
utilize curves of the y parameters, curves of differential
or incomplete g parameters.
This proliferation of
specifications forces either a transformation of parameters or, for a
more thorough design, measurement of parameters.
The impact
discipline is,
I
of linear integrated circuits into the electronics
am
sure, beyond our furthest imagining.
of physical size resulting from LIC application
of
more functions per unit volume.
The reduction
makes feasible packaging
The increase
in available gain for
a given cost and volume encourages sacrifice of a greater percentage
of gain toward
improvement
any engineering trade
of linearity or stability or, for that matter,
off required
by a particular application.
As the
precision of these devices is increased, their application to small,
extremely fast analog computers of great accuracy
is sure to
emerge
as a real time problem solver of high speed technological which are
presently beyond the capability of our fastest digital computers and
the algorithms used therein.
In the research required
by this thesis, the lack of application of
optimum design techniques to solid state circuit design problems
more and more impressive.
is
The mean-time-between-failures of active
58
devices
is
now measurable
in
terms of decades.
The design
of
equipment utilizing these low failure rate components must optimize
performance over component response change with lifetime.
statistical nature of these response
The
changes forces the circuit designer
to consider statistical circuit analysis, that is, to optimize design.
Similarly, as modern society
becomes more and more dependent on
reliable accurate operation of a variety of electronic devices,
becomes more and more important
be optimized.
it
that the performance of this equipment
The electronic circuit and electronic equipment
designer, therefore, must learn and utilize the optimum design
techniques of the control systems engineer.
59
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76
FOOTNOTES
G. J. Veth, "Microelectronics in Space, " Proceedings of the Third
Conference on the Navy Microelectronics Program (Washington, D. C:
Office of Naval Research, 1965), p. 72.
2
L. B. Hallman, Jr.
"The Impact of Integrated Electronics on
Aerospace Avionic Subsystems, " IEEE Transactions on Aerospace and
Electronic Systems AES 1 no. 2 (October, 1965), p. 153.
,
,
3
C. Whitelock, "Microelectronics Program - Past, Present, and
Future " Proceedings of the Third Conference on the Navy Microelectronics
Program (Washington, D. C: Office of Naval Research, 1965), p. 75.
L.
,
4
Ibid.
,
p. 88.
Burns and K. W. Foulke, "An Advanced Microelectronic
PCM Multicoder," IEEE Transactions on Communication Theory COM 14
no. 2 (April, 1966), pp. 162-169.
W.
L.
,
6
Crecche, R. Rachal, and W. F. Liist, "Molecularized
Telemetry Encoder, " IEEE Transactions on Communication Theory
COM 14 no. 2 (April, 1966), pp. 169-177.
J.
R.
FM
,
7
Hallman, op.
cit.
,
p.
154.
D. G. Brown, "Ultra-Reliable Transceivers " Proceedings of the
Third Conference on the Navy Microelectronics Program (Washington,
D. C: Office of Naval Research, 1965), p. 43.
,
9
Ibid.
Anon., "Pop Op Amp,
"
Electronics
,
Vol. 40 no. 25 (December 11,
1967), p. 50.
Hallman, op.
cit.
,
p.
157.
12
B. H. Andrews, Keynote Address, Proceedings of the Third
Conference on the Navy Microelectronics Program (Washington, D.C.:
1965)
,
pp. vii-xi.
13
G. S. Bahrs, "Stable Amplifiers Employing Potentially Unstable
Transistors," 1957 IRE WESCON Convention Record Pt. 2 (1957), p. 185.
,
14
A. P. Stern, "Stability and Power Gain of Tuned Transistor
Amplifiers " Proceedings of the IRE Vol. 45 no. 3 (March, 1957), p. 335.
,
.
77
G. Linvill and L. G. Schimpf, "The Design of Tetrode
Transistor Amplifiers, " Bell System Tech Tournal, Vol. 35 no. 4 (July,
J.
1956), p. 814.
1
c
J. J. Robertson, "Tuned Amplifier Design with an Emitter-Coupled
Integrated R.F. Amplifier," Motorola Semi -conductor Products Inc
Application Note Integrated Circuit, AN 203 (n.d.) p, 3.
,
.
#
17
1 ft
Linvill and
Schimpf _op._cit.
,
p. 815
,
„
Mason, "Power Gain in Feedback Amplifier,"
on Circuit Theory, CT 1 no. 2 (June, 1954), pp. 20-25.
S.
J.
IRE Transactions
l^D. k. Lynn, Charles S. Meyer, and D. J. Hamilton (editors),
Analysis and Design cf Integrat ed Circuits (Motorola, Inc. New York:
McGraw-Hill, 1967),
20
Bahrs
p. 375.
_op. cit.
,
21
J. M. Rollett, "Stability and Power Gain Invariants of Linear
Two-ports," IRE Transactions on Circuit Theory, CT 9 no. 1 (March,
1962), p. 30.
22 Bahrs, op
23
B. P.
.
cit,
,
p.
186.
Lathi, "Optimal
Design
of
Multi-Stage Tuned-Transistor
Amplifiers Considering Gain, Stability and Sensitivity," System
Techniques Laboratory, Stanford Electro nics Laboratories, Technical
Report No. 755-3 (July 11, 1960), pp. 20-21.
24
Gibbons and J. G. Linvill, Transistors and Active Circuits
(New York: McGraw-Hill, 1961), pp. 233-235.
J.
F.
25 Lathi, op
2
fi
.
cit,
pp. 26-27.
,
Gibbons and Linvill,
op.,
cit,
,
p.
236.
97
Bahrs, op. cit.
2fi
B.
P.
,
p.
186.
Lathi, "Optimal Design of Multi-Stage Tuned-Transistor
Amplifiers Considering Gain, Stability and Sensitivity," System
Techniques Labo ratory, Stanfo rd Electronics Laboratories, Technical
Report No. 755-3 (July 11, 1960), pp. 28-33.
^Radio Corporation of America, RCA Linear Integrated Circuits
RF Amplifier CA 3028 (File No. 242. Harrison, New Jersey: Radio
Corporation of America, Electronic Components and Devices, 1967), p.
30
31
Lathi, op. cit.
,
pp. 110-113.
Ph.D., Handbook of Mathematical Tables and
McGraw-Hill, 1965), pp. 11-12.
R. S. Burington,
Formulas (New
York:
32
G. S. Bahrs "Amplifiers Employing Potentially Unstable
Elements," Stanford Electronics Laboratories Technical Report No. 105
(May 7, 1956), pp. 53-77.
,
,
79
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,
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Tuned Transistor Amplifiers "
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,
Vol. 40 no. 25 (December 11,
INITIAL DISTRIBUTION LIST
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Naval Postgraduate School
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Naval Ships Systems Command
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4.
Professor William M. Bauer
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Naval Postgraduate School
Monterey, California 93940
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LCDRWm.
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3015 Windish Street
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REPORT SECURITY CLASSIFICATION
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AND LINEAR INTEGRATED CIRCUITS
LINEAR AMPLIFIER DESIGN
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AUTHOR1SI
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HOBLER, William
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Through the utilization of the functional matrix description of linear
integrated circuits, a design method is evolved from which the terminating
admittances may be specified. These terminating admittances are optimized
with respect to power gain for a stable device and optimized with respect to
power gain for a specified stability level. The total input and output
conductances are obtained for use in frequency response design. Finally
the procedure is extended to include the case in which a not optimum
conductance is forced on the designer.
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LINEAR AMPLIFIERS
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