Inverting summing amplifiers are well known. However, attached is a circuit with two versions. The first gives an output from the amplifier that is equal to the average value of three input signals. The second gives an output equal to the sum. The topology can be extended to higher inputs, but I cannot recall it being documented in text books on op amps. Can it be called as a non-inverting summing amplifier ?
As a first remark, both the circuits cannot be regarded as voltage amplifiers, since they do not show a high impedance at inputs e1, e2 and e3. Indeed, also the so-called "inverting amplifier" is not a voltage amplifier for the same reason. As a matter of fact, both the circuits include a passive resistive summing network (Millman network) and a canonical non-inverting voltage amplifier.
Moreover, both the topologies can be regarded as special cases of the generalized summing opamp circuit ("The general opamp circuit" in the following link), which is largely documented in textbooks.
http://www.nhn.ou.edu/~bumm/ELAB/Lect_Notes/Op_Amps_v1_2_2.html
Yes, it can be called as a non inverting amplifier. Because input signal is in phase with output signal. Also input signal is given to non - inverting terminal of the Op - Amp .
First circuit is a type of non-inverting average circuit and second one can be called as non-inverting summer
In practise one needs to be careful with these type of things. If the source resistance/impedance of the signals to the inputs e1, e2 & e3 is not low and also equally matched compered to the three R's then the suming will not work as expected.
My goodness! How does such a rudimentary matter
become the subject for a "Research Gate"?? Surely
this is "Circuits 101"?
My previous answer was not very constructive, simply
because the original question didn't seem to be looking
for a better solution - only an endorsement of the one
presented (which might be called a "Bread and Butter"
solution; or perhaps a "First Instincts" solution.)
If the questioner is interested in the general challenge
of summing N analog voltages, for, say 2 < N < 10,
with essentially zero conductive loading, while affording
the option of having either an inverting or non-inverting
response - equally weighted - from any if the inputs, I
would be happy to show how, in these pages.
Barrie
It seems very intriguing... Let's try to guess the remedy - maybe a resistor connected between the op-amp output and the non-inverting input?
Cyril:
Obviously, the number of solutions to such a challenge
is very large, if we are permitted many active elements.
While my solutions are compact, accurate and efficient,
they must be implemented in an integrated form.
Barrie
Barrie Gilbert
Thank you for your offer to guide. I will enjoy and follow this puzzle and your solutions with the humble interest that only an old student can have.
Glen
Glen:
Lacking any (known) demand in the wider market,
there is no IC product that performs this nice feat.
However, the principle is very simple, and at least
one product from Analog Devices (of my design)
uses it. That product is the AD830 (attached) and
the principle is called "Active Feedback" or AFA.
Around 1974 (or was it 1774??) I wrote a paper
entitled "A High Performance Monolithic Multiplier
Using Active Feedback", IEEE J. of Solid-State
Circuits, vol. SC-9, pp. 364-373. It was the first
use of the principle, and has been widely used
since. One of my team members (Brunner) and
I wrote a short summary of the idea (attached)
in "The Active Feedback Amplifier - A Versatile
Analog Building Block" to explain it in the most
basic way.
You will see that it employs a pair of matched
transconductance cells whose current outputs
drive an output error amplifier (fast integrator).
Though each gm cell may be nonlinear, their
distortion terms cancel, due to the use of one
of them as a feedback element.
It will be apparent that any number of these gm
cells (within reason) may be used for the inputs
and the output will respond to the sum of all of
them.
There are a few caveats to note.
Whereas in the basic AFA only a single pair of'
gm cells is used, and cancellation can then be
near-perfect, the same cannot be said when a
larger number of input stages are used, unless
they are independently linear. However, that is
not hard to achieve through skilled design.
The second consideration concerns the number
of input channels to provide. One consideration
is the number of IC package pins. For example,
the common 16-pin format can accommodate 6
channels: 6 x 2 = 12 for their differential inputs
(an internal feedback path would be used, thus
saving 2 pins for the feedback gm) - plus 2 for
supplies, one for the output and another pin to
optionally make a single-pole LPF from the IC,
which totals 16 pins.
A third consideration is: Who "out there" in the
market would ever want to sum/difference six
concurrent analog sources? An IC development
is costly and needs to be justified on demand.
Of course, unused inputs could be grounded,
although the power consumption and the RMS-
summed noise would persist.
However, it may be noted that we already have
a solution at hand: the AD830 can sum/difference
N, non-loading channels simply by operating the
same number of this IC in parallel. (The output
of the k-th stage simply daisy-chains into the
output-connected gm input of the (k+1)th stage).
Finally, it should be noted that because the gm
cells of the AD830 are inherently matched to a
very close tolerance (specified as 0.1% - see
Data Sheet) one does not need all those ugly
old, source-loading resistors for the summing
function. And it is not necessary for the overall
block to be limited to a gain of unity (which is
simply the default condition): any gain can be
provided, ultimately limited only by noise.
Barrie
Barrie,
thank you for being so generous with your time
and your encouraging response.
Have read your attached articles , and thanks.
They have been informative.
'Active Feedback Amps' are indeed a superior approach
to Instrumented Amps.
I plan to investigate their application in my own developments,
which up to now have centered on
Phase Filtering in BandPass Filters
as applied to Radio CW Operations.
I use OpAmps everywhere, calculate equations,
and run Spice all the time, before protoboarding and hardwiring. .
Next phase will be on RF input
and Tayloe Phase-DeModulating to drive the audio filters.
I am using a Direct Conversion approach ( No I.F. stages )
to receive 14MHz signals.
This has traditionally been done with JFET transistors,
but I look forward to trying a Video OpAmp approach.
I recall cajoling a high-speed OpAmp
to successfully run 7 nano second rise times.
It was a very successful project then, in 1779 or was it 1979.
Now,just as I an looking into RF amps in my radio system ,
you have introduced me to todays Amps, such as the AD830.
Indeed, there will be a match among these chips.
... Thank you.
Just read your "Four Dees" and much more ... a really good read.
I have obtained your downloads and specs on similar IC amps.
I obtained Bumm's 'Notes on Op Amps', a very clear and basic text.
The next few months will be interesting,
as I consider more closely the IC analog chips I have to use.
I think that Sujit, the original poster,
should rexamine his application,
and let that 'need' direct his search for a 'solution',
which he apparently has in hand.
...
Sujit is asking for a more precise name.
..... A rose is a rose by any other name.
Glen
.
As Director of Jesting at the NW Labs, here in Orygun,
I (or is it Dr. Leif?) have a lot more tricks up my sleeves
and beneath my multi-coloured patchwork of yellow and
red diamonds, I'm eager for someone to ask about them.
.
There is in me a heavy regret that the decades-old arts
of analog circuit design are today becoming, inexorably,
bit by bit, and byte by marching byte, buried in the deep
umbra of the Dark Side.
-
I don't want to turn this cross-roads into an advertizing
bazaar; so, with great restraint, I won't list any more of
the charming ICs that I suspect might be of great value
to you. Nevertheless, please let me know if I can help
through conventional channels.
.
Barrie
Sujit K. Biswas
* You specified use of voltage inputs,
but you will find that there are variations depending on how much i(input) current moves into the (+) input on the OpAmp.
* Going with your Voltage perspective,
I ran this through manual number calculations
then through Spice ( see attached ):
* Applying some variation in numbers, it is apparent that
if R1=R2=R3
then the circuit #2 will Average ( with gain 1+R9/R10).
* However, if R1R2R3
then the circuit #2 does something different,
and is not useful for Averaging the inputs.
* Perhaps there is not a unique name for every circuit in the world.
* Perhaps this could be called a 'non-inverting Combiner' .
* Don't let the lack of a proper name stop your work.
Give it a simple generic name+date ,
and move on to greater things in your project.
Glen
www.GeoCities.WS/glene77is
Friends of the Analog:
.
The original function - however implemented - cannot
be called "averaging" unless it is also equipped with a
low-pass filter of appropriate form. The acquisition of
an average in which the "wait time" is unlimited is, of
course, very easy; the LP filtering needed to provide
a faster "running" average is more complicated.
'
It's apparent that the suggested (very straightforward
and unremarkable) circuit aims to perform simply the
summation of several inputs, while maintaining the
unity gain. As noted, a limitation of this circuit is that
it presumes the signal sources are already at a very
low impedance (including their HF output impedance
of course). But IF that really is the case, N inputs can
just be added in a passive resistor matrix followed by
a non-loading buffer having the required gain. If one
objects that "this will cause interactions in the driving
sources", then the first assumption (driving from very
low impedances) is already violated.
.
It will be noted that the input offset voltage and input
noise of the op-amp will always be multiplied up in a
straightforward and simply calculated way.
.
There are much better ways to add/subtract several
voltage sources, and the preceding dialogue on this
matter prompts me to write a generously extended
article for ADI's "Analog Dialog" magazine, showing
the versatility of the AD830 in numerous such tasks.
'
Barrie
.
Its great to see the long discussions going on, but all I wanted to know if there was a given name to the circuits.
Nothing is ideal, yet we assign names like "summing amplifier", "differential amplifier" etc.
'
It is sometimes useful to ascribe a name to a circuit topology
when it is clear that (a) the form is novel (b) it is non-obvious
(c) it is likely to be used many times over in one's work and
used and written about by other in professional publications
for years to come. Many examples come to mind, but I can't
see how the circuit you presented meets these criteria.
.
Sujit,
Long before the days of personal computers and internet,
( back when we had to do day-long calculations
and build-ups on ProtoBoards ),
I had done a series of experiments in physiology,
monitoring stress reactions in humans.
There were many months of puzzling responses,
not that I could not get useful measurements,
but puzzling in the sense that there seemed to be
no common basis for the measurements.
I could adjust the linear sensing circuit for each experiment,
but could not make an across the board comparison,
without tedious calculated interpolations.
* I had a circuit-with-no-name, but it worked, so far.
But, in my mind, not far enough.
* My mentor suggested that the circuit might be more useful if it contained a Log-Amp to convert my linear sense data into a proportional output for graphing and comparisons.
* My circuit-with-no-name was an obsession
until I was introduced to non-linear circuitry.
* Over this history of development I went back to school
at a different college, taking with me my circuit,
then named a Logrithmic-Delta-Ohm-Meter.
* To me that circuit was a badge of merit,
having turned a simple circuit into something very useful
for use in our department at the university,
* To me, it proved that I could resolve some difficult issues.
So, that said, Sujit, I encourage you to continue developing
circuits of much greater complexity and usefulness
to further your research
... possibly giving one a meritorious name of your own choosing.
I have attached a hand-written note from that period.
My circuit-with-no-name...
Our clients were mostly Cerebral-palsy, and were unable to provide good information. This circuit automatically differentiated between base-line-drift and target signals, eliminating the 'white coat effect' and the stress that comes by being 'wired-up'. This circuit collected data from finger-arteries by piezo pickup for heart monitoring. This circuit could aid in 'listening' to muscle contractions in response to stimuli using EKG cups. With various sensors we could monitor results without depending on the client's verbal responses.
So, good luck.
Glen
.
Glen:
-
There are perhaps four different name-tags we give
(or are later given by others) to our electronic children:
(1) That of the inventor; (2) One describing the circuit
topology; (3) or the specific function; (4) Code names.
.
All of them serve to capture the definitive essence of
the form, and (2) and (3) overlap considerably.; and in
some cases "great names" are attached to cells of
of very modest complexity. In this category, we have,
for example, the Eccles-Jordan Multivibrator, or the
Brokaw Band-Gap, the Darlington Follower and the
like. These names arise in the fullness of time for a
good reason: They have proven their worth and are
in some significant way unique.
.
The last category is Code Names. There often refer
special forms that a team or a group or a company
use a lot, and the name of the form in an asset in
communicating a novel and unique topology that is
used countless times within the Company although
initially little known beyond the four wall, and often
never. I can cite many examples of names that I've
given to my "children" of this sort. Code names in
many cases are an attempt to immediately create
the desired image in the mind of the adopter even
though they started out as merely a codification:
Orchid, X-AMP, Spider, Kermit, V-mirror and Tail
Chaser are some of mine.
.
So if one truly believes that "Here is a novel and
truly valuable form, that I myself expect to make
extensive use of for a long time to come", then it
is pretty harmless to just make up a name that
collapses the concept into something pithy.
.
On the other hand, the well-seasoned designer
invents or adapts useful cell-level topologies in
the course of everyday work. It's just what we do,
and we would never think to give them a name.
The fact that they "haven't been documented in
any text-book" indicates that they were exactly
what we needed at the moment for our design
but unlikely to be of much use to anybody else.
.
Barrie
Barry,
Right On !
Analysis is the name of the game ... functionality is a key-word.
In your responses, I see your easy way with analysis
and some of what I missed over the years in your summaries.
Mentoring and a team approach
with sharing of ideas / problems
can be so much more productive than working alone.
I can only appreciate at a distance
what your team must have been like over the years.
I gravitate towards (3) a functional name and tack-on (4) codes and sequences. Have three years of explorations on PartSIM, under prefixes like AFX or P2S or OOAD or CALF
( CALF went into small-time production ).
Quick naming helps me move on ,
and see where my mind has been..
I do hope that young Sujit will adapt useful topologies
that function well together ,
into larger groups for even greater functionalilty.
Pithy Named or Numbered,
personal or known-in-the-world,
our work should move forward and be productive.
No telling what good things will happen
... that seems to be part of the game.
One of our projects in 1980
was the prototype of the No-Hands Computer Interface
that Stephen Hawking used for a dozen years.
I still have the proto-type in a box, here.
Barry, It is delightful to read your 'loaded' responses.
Glen
Barry,
I do hope that young Sujit will benefit
from these exchanges
... that is the point of this forum.
Perhaps, this topic will rest at ease now.
Glen
.
Paolo:
.
In the link you provided, there is this comment:
.
Op Amp Golden Rules [ ]
1) The op amp has infinite open-loop gain
.
But in the remaining notes there is not a word
about the internal "inertia" of the op amp. I use
this word rather than the more ambiguous term
"gain-bandwidth", in part because the time-
domain response of modern op amps is very
complex and the simple "dominant-pole" idea
does not, in general, describe their behavior
of these amplifiers; and -- often much more
importantly -- the naive "small-signal model"
fails to countenance, in every case, the real
life of an amplifier, which can appear to be
benign using a small-signal approach, but
(often, and even usually) drastically different
when viewed under transient conditions.
'
In this connection, many of the topologies in
the linked notes - particularly the integrators
and the (unwise-to-use) differentiators - will
NOT behave in the nicely-analytic way that
is shown, simply because the internal poles
and zeroes - the inertia - invariably play a
HUGE role in determining the time-domain
response.
.
This note is getting too long, so I'll present
my "Golden Rule #1" in a follow-up note.
.
Barrie
'
"Golden Rule #1" for an idealized op amp:
'
1) The op amp is an integrator
'
Corollary: It is characterized completely
by its unity-gain frequency.
.
Ideally, the operational amplifier is a simple
dominant-pole-stabilized integrator, having
very high (but usually unimportant) DC gain.
If we stick to the notion of true ideality, the
DC gain of an integrator will be "infinite", so
automatically satisfying the classic assertion
that "an op amp has infinite open-loop gain".
.
As an example, some years ago I designed
a special (and relatively simple) amplifier in
BiCMOS whose DC gain is about 250 million
(168dB), yet it is a single-stage amplifier,
having a clean -20dB/decade response all
the way down to its 0dB frequency (of about
200 Mhz) and continuing beyond that point.
In order to view its "DC" gain in simulation
the frequency sweep starts at nanohertz !
It uses no "tricky" (non-robust) methods -
such as a critical dependence on certain
matching conditions - and it's in production,
'
The ideal op-amp/integrator is modeled
as (1) a differential-input gm cell (VCCS)
driving (2) a grounded capacitor, C,
whose consequential voltage is buffered
by (3) a X1 cell (VCVS). That's all there
is to it. The unity-gain angular frequency
of this op-amp/integrator is the inverse of
its (only) time-constant, T = gm/C. Thus
VOUT(t) = 1/T integral VIN(t)
.
So here's a problem for you to think about:
Assume an ideal op amp with no inertial
constraints (that is, "infinite bandwidth",
as tacitly assumed in the provided linked
notes) is used to implement a current-to-
voltage converter (which we might call a
transimpedance stage were it not for the
fact that this zero-inertia element has no
no "impedance" attributes, so the circuit
form is that of a transresistance stage).
.
In order to set up this mode of operation,
we apply a (pure) current to the inverting
input node, and ground the non-inverting
node. Finally we add a real resistor on a
real PC board to set the transresistance
where we want it. Big hint: this is where
all the troubles arise, and it is the key to
solving the following problem:
'
What is the maximum permissible gain
of this op amp, used in this particular
way, before it bursts into song?
.
If you have a cosh in your solution you've
probably got the correct numerical answer,
Keep in mind that I'm assuming the usual
model equation for gain, that is, Ao/(1+sT)
except that here T = 0. Your quest is to
determine the maximum value of Ao below
which the system is unconditionally stable
and above which Bad Things happen,
.
The lesson to be taken away from my little
teaser is that "infinite" - or even simply the
very high bandwidth of many real-world
op amps - is not always a desirable case.
.
Barrie
dear Barrie,
Thank you for your comments! In my previous answer - I must confess - I gave a link to the first document (or maybe it is the second...) I found on the Internet showing something with its own name, which actually covers the suggested circuit as a special case, i.e. to say that one more name is not needed... A matter of names..., Even though philosopher Plato maintains that names come before real things, circuits performing useful functions in the best way considering real world constraints are surely more of interest in Engineering... And now that the discussion has become much more interesting - thanks to your contribution -, my original answer and my link seem to be really poor. Nonetheless, even the notes I use in my lectures (for undergraduate automotive engineering students), except they are more verbose, are perhaps not very different... I am looking forward to see your "Golden rules"!
PS: I have written this before your last answer. It seems really interesting,... I guess the problem is related with the parasitics of a real resistor and of a real board...
'
Paolo:
'
First a small - and probably unnecessary - self-
correct: I meant (of course) to write T = C/gm.
.
And, yes, it is the resistor that is troublesome,
and a reminder that its artifacts are frequently
overlooked in teaching, perhaps as a "merely
practical" matter. In most modern simulators
the parasitic capacitances of an (IC) resistor
are modeled rather simply, dividing the total
capacitance CTOT into CTOT/4 at each end
and CTOT/2 at the mid-point, The additional
simplification is that the (substrate) node to
which the capacitors are returned is a zero-
voltage plane. (Good luck with that one!).
.
But since you are "on to me" regarding that
resistor, I will help a little by adding that the
solution I have in mind depends on the use
of a continuous resistance and capacitance
(that is, not a lumped model) the product of
which is just CTOTRTOT. And to help you to
more quickly find the (shocking) answer for
AoMAX, I will open the door a little wider and
add that this value of time-constant is quite
irrelevant.
.
Barrie
'
Fellow Travelers:
.
Regarding that AMAX problem, you are all invited
to flex your cerebral muscles, including you Sujit,
since you unwittingly opened this Pandora's Box
in the first place! And, as for a name, how about
"SummIt". It sounds a little like your name and it
also sounds like something at the pinnacle of all
that is possible, while also describing its function.
.
AMAX = ........
'
Another hint: An accurate, idealized solution can
be most readily found by starting with the phase
behavior of a distributed-RC transmission line -
no inductors - and look for the loss magnitude
where the phase lag is exactly 180 degs, since
that's precisely where the loop gets nasty, and
beyond which point you might as well go home
and sink into a soothing bubble bath. At least,
that's what I do, while catching up on reading
my stack of Journals; only to be disappointed
again by the noteworthy lack of pure invention
in the arena of basic analog-circuit topologies.
.
But.. to get the analytic solution - from which, of
course a numerical solution can be extracted to
1,000 decimal places - you obviously must treat
this as a mathematical challenge rather than as
a facet of circuit design. So throw my schematic
out with those lead-tainted Maggi noodles, and
ascend to the pure summit air of mathematics.
'
The analytic solution is so beautiful.
.
Barrie
dear Barrie:
I believe the solution can be obtained considering the resistor as an RC-distributed transmission line with a propagation constant
k=omega*sqrt(R0/(j*omega) * C0)= sqrt(-j omega R0 C0)= k0 - j*k0, (1)
being k0=sqrt((omega R0 C0)/2) where R0 and C0 are the per-unit-length resistance and capacitance of the "resistor", which are expressed as R/l and C/l, being R and C its total resistance and parasitic capacitance (towards the PCB ground) and l its physical length. The interesting thing is that the real part of (1), describing propagation, and the imaginary part, describing attenuation, have the same magnitude k0.
Then , the loop gain of the amplifier is
T= A0 * beta = A0 * V_minus /V_out (2)
being V_minus /V_out the voltage transfer along the feedback resistor/RC transmission line, from the opamp output back to the inverting input, which can be expressed, if I am not doing mistakes:
V_minus/V_out = exp(-j k l) * (1+Gamma_minus)/(1+Gamma_out) (3)
= 2*exp(-j k l)/(1+exp(2 j k l))
= 2*exp(-j k0 l)*exp(-k0 l)/(1+exp(2 j k_0 l)*exp(-2 k_0 l))
being Gamma_(minus)=1 the reflection coefficient at the inverting input (open circuit) and Gamma_out= exp(2 j k l) the reflection coefficient at the opamp output.
Considering (2), (very) "Bad Things" are expected to happen if |T|>1 when the phase shift is 180° (Bode Stability criterion, to give its name, or perhaps Barkhausen in-stability criterion, which is another name). A 180° degrees phase shift means k0*l=pi in (3) (a "half wavelength" line) and under such condition
V_minus /V_out =- 2*exp(-k0 l)/(1+exp(-2 k_0 l))
= -1 / cosh (k0 l)
= -1 / cosh (pi) (4)
Putting (4) into (2), |T|
Dear Sujit,
I'm surprised you did not find any reference on this circuit, because in Italy you can find it in any book of electronics for high school. See the attached link.
You can also seach by yourself "sommatore non invertente" in google.
regards
S.
https://www.google.com/search?q=sommatore+non+invertente&num=20&sa=X&biw=1280&bih=833&tbm=isch&tbo=u&source=univ&ved=0CB0QsARqFQoTCNPt6N7XicYCFUtzcgodQjQA6g
Simone,
Sujit's circuit, as given, can have un-equal Voltage inputs, and un-equal Resistance, so this Vout would be something linear like Vo = (1 + Rf / R) (v1 / v2 + R1 / R2 + vn / Rn) .... so what is a good name for this ? .
Given equal input resistances, the Vout would be Vo = (1 + Rf / R) / N (v1 / v2 + R1 / R2 + vn / Rn), which is a simple Average , so maybe named Summing Average Non-Inverting Amp.
Sujit was simply requesting a new name for this possibly new circuit, not offering a broad discussion.
Barrie is responding with an excellent discussion of the subject from his own background.
I have found this combination of grad project
and high-level coaching to be a wonderful discussion. Rock-On!
IMHO,
The name-of-the-game here is adding Dimensions (Thinking)
and Discovering more.than we thought existed .
All In all,
this thread does not go directly to resolving Sujit's minor problem,
but really expands the scope of intellectual involvement.
Glen
Paolo,
I liked your explanation of Negative Feedback in OpAmps
using the analogy of Driving a Car, with/without eyes open !
Glen
Dear Glenn,
altough English is not my native language, with average I intend more a time or statistical mean than a linear combination.
So non-inverting summing amplifier would be more appropriate.
Moreover, also the well know [inverting] summing amplifier (https://en.wikipedia.org/wiki/Operational_amplifier_applications) performs a linear combination (weighted sum) of the inputs, if the resistances are not equal, yet its name doesn't change for that.
regards
S.
P.S. in your formulas you swapped v2 whith R1
dear Glenn,
I actually borrowed and adapted the analogy of the car from my master, prof. Vincenzo Pozzolo. I am glad to acknowledge his extraordinary skills as a teacher in this discussion.
Indeed, I believe that that example is especially appropriate for future automotive engineers...
Dear Paolo and Barrie,
here attached you can find my solution to the proposed problem, accomplished by using the two port representation of a distributed resistence.
In particular I used the chain matrix of the resistor.
Furthermore I added feedback load effects to the amplifier and calculated the open-loop gain, assuming a zero output resistance for the amplifier.
Applying Bode stability criterion I got the same results as Paolo.
soon
S.
Dear all,
if we substitute the istantaneous OPAMP model A = Ao; with the Barrie's model : A(s) = 1 / sT, or A(w) = wt / jw, what will happen to our circuit?
You can se from the attached file, that wt max = 2.51 wo ; where wo = pi^2 / RC ;
In the reported example I used data of an integrated resistor insted of a discrete one, to be mounted in a PCB.
soon
S.
.
Friends of the Analog:
.
I can't find my earlier note containing a challenge
about the maximum permissible gain of an op amp
having a completely flat frequency response, that
is, infinite bandwidth, when connected as a I-to-V
converter - a pure transresistance - which, from a
PRACTICAL perspective requires the use of a real
resistor, of necessity modeled as a DISTRIBUTED
transmission line (this component being mounted
above a ground plane on a PC board).
.
Thus, the model for the op amp is NOT that of an
integrator in this case. (Indeed, I proposed this
challenge to show how useless is the idea of an
op amp having "infinite bandwidth" as sometimes
stated - or tacitly assumed - in text books).
-
The responses seem to have gone of the rails.
But in order not to keep this "gate" open longer
than necessary, I will here the give the answer
'
The amplifier (just a simple VCVS) cannot have
a (flat) gain of more than a pitiable 11.591953...
which, in its analytic robes is just cosh(pi).
.
Barrie
11.59 = 21.3 dB.
I'm glad to ear that our (Paolo and mine) solutions are correct.
S.
I will correct my last post where I used data of an integrated resistor, not a PCB one, as should be done.
.
Friends:
.
Although we have moved very far from the original
enquiry, these exchanges have got us thinking - in
a useful way - about how to optimally visualize the
"good old op amp", an element so often treated in
ways that might detract a student at first exposure
from its simple, essential nature.
.
As long as we're thinking about its problematical
behavior using the notion of infinite bandwidth,
I must note that the "capacitively-loaded resistor"
does not have to be represented as an infinitely-
distributed line.The attached jpg shows a simpler
feedback network which not surprisingly will also
cause the system to oscillate; and of course, the
actual value of the time-constant(s) doesn't have
any bearing on the maximum permissible gain --
though it necessarily will alter the frequencies of
oscillation.
.
Here's what I found: Using first an AC analysis in
which the scalar gain parameter A is varied over
some range while the magnitude of the response
maxima are noted, the value of A resulting in the
maximum gain is (about) A = 20.9690082650087
and it is 220 dB -- equivalent to an impedance of
j1011 ohms, at a frequency of 3.272960203..MHz
'
However, in further simulation experiments, now in
the time domain, in which the circuit was hit with a
2 ps-wide 1 Amp stimulus, the critical value of gain
at which the oscillation magnitude neither grew nor
decayed, was found to be substantially greater, at
(roughly) 21.1719. The oscillation frequency at this
value of A was (about) 3.2847944982971781 MHz.
.
This discrepancy in this critical value of A is larger
than I'd expected. Note that as a matter of general
practice I typically set unusually tight convergence
tolerances. In these experiments chgtol = 1e-22 C,
abstol = 1e-15, vntol = 1 nV and reltol = 1n. I have
qualified the accuracy of this particular simulation
environment over many decades of its use and as
contributor to its capabilities.
.
So... I am puzzled by this anomaly. Can someone
out there "can explain it away"?
.
Barrie
Dear Barrie,
by preliminary hand calculations I obtain Amax = 27 and fo = 5.8489 MHz. I'll further investigate what I missing.
Anyway just from a first look at the circuit I noted that C4 and R1 are irrelevant if a VCVS is used. I guess they are there if Ri and Ro were added to the VCVS, but you can check if the simulations behave different without them.
My didn't, but I also got completel y different results from yours (A=29, fo=385 kHz).
I'll try with Cadence spectre when I'll be at work.
soon
S.
.
Simone:
.
Thanks for pointing out that error in the illustrative figure.
It arose in "tidying up" the drawing, during which the RC
network was accidentally mirrored (L-R reversed).
.
The simulations were carried out on the original circuit,
which was the "right way round". At least, I think that is
the case. When I'm less busy I will open up this can of
worms again and start from the beginning.
.
I note in passing that the characteristic frequency of the
network using R = 1 ohm and C = 1uF (the total values)
is 1/(2*pi*1*1u) = 159.155 kHz.
'
I had planned to make a table of lhe AMAX values, and
the corresponding frequencies of oscillation in a canonic
series of increasingly more distributed networks, but I'm
very busy working toward a tape-out this coming Friday
of two new ICs - and I'm still working on the circuits!
.
Barrie
Dear all,
since the discussion is gone far away from the original question, if you don't mind I would move the thread in the link below.
I tried to maintain Barrie's authorship of the original post, as clear as possible.
S.
https://www.researchgate.net/post/Discrepancy_in_critical_value_of_A_in_ideal_transimpedence_amplifier
the first circuit gives average as you say. the second is a summer also, but with gain of 3. Who says noninverting amplifiers should not have gain? Thus output of second amplifier is sum, as you say. You can also make it 10 times the sum if you want. Just ensure that amplifier saturation does not occur.
.
Fellow Travelers:
'
First I must say that the search for taxonomies
of simple circuits is a "Fool's Errand", Yet it is
one that often seems to be of more interest to
academia than to those of us in the real world
of product design. Perhaps the motivation is to
reach for fame as the "first" person to confer a
name and its acronym on another rudimentary
cell topology.
-
I enclose the word "first" because it frequently is
not that,in fact, Rather, the genuine originator of
some such topology happened to be working in
a "Real-World" environment and, because of the
regular, routine and habitual invention of circuit
noveltty (see below) that person might well coin
a useful name for each new cell simply in order to
converse with others in a team about it, by using
the shorthand reference it embodies.
.
For example I coined the term "current conveyor"
many decades ago, after considering the general
utility of a special sort of current mirror that could
operate on bipolarity input currents, "conveying"
them from a low-impedance node to a bipolarity
output at a high-swing high impedance node.
.
I used this newly-minted word in conversations
amongst my colleagues, but I didn't think of it
as anything more that a useful down-to-earth
description of how simple cells of this genre
actually behaved. But had I been a professor I
might well have writtten several erudite papers
about it, and several variants, leaving a trail of
arcane acronyms along the way. But I was in
the trenches, developing IC products - which
should never be confused with "circuits" - and
looking for ways to configure my elements in
the pursuit of increasingly more valuable real
products - not for journal papers in excess.
'
A second example: I coined the abbreviation
"PTAT", meaning - as is widely known today -
Proportional to Absolute Temperature, in the
mid-1960's. This time it did appear in a JSSC
paper of mine - but much later, in 1976 - and
even then merely as a tiny footnote. I viewed
it as conveying a useful concept, potentially
having widespread applicability in BJT circuit
design as well as a simple abbreviation, .
.
I have lost count of the number of concept-
loaded abbreviations - mostly for major cell
types - which are now in widespread use by
scores of designers within my Company -
and which appeared in various public forums,
where no record was taken,
'
Now to my appeal to "circuit researchers" in
academe. There is a rather crude American
expression for the thrust of this appeal: it is
'
GET REAL !
.
When I was a young kid, I used to imagineer
my life as a budding mechanical engineer by
inventing Meccano models and mechanisms.
When I grew up - and decided on electronics
as a promising career - I quickly realized that
even the most elaborate objects I'd invented
using my modest collection of Meccano parts
were truly "tiny and tinny" contraptions which
paled miserably in comparison alongside the
magnificent and awe-inspiring mechanical
engineering marvels of the real world.
'
Accordingly, to my friends in certain corners
academia, I feel it must be said: Please stop
tinkering with the elementary things about
electronic circuits - the "Meccano" approach
to life - and use your precious energies and
time to look for Real World problems in the
realm of modern electronics, of which there
are plenty. It is encouraging to see that this
is how many universities in America and in
Asian countries are structured; and although
it is evident - from the many papers in such
journals as those of the IEEE - that the work
is in one way or another a little off the path
to reality, we must commend the teachers
and students of these scattered centres of
higher learning for having left behind them
the "Meccano phase" of their lives to focus
on matters of genuine - and often urgent -
importance in the REAL WORLD.
'
With respect,
'
Barrie
In practise one needs to be careful with these type of things. If the source resistance/impedance of the signals to the inputs e1, e2 & e3 is not low
What is a summing amplifier? It is possibly one where you can determine the weightage based on a resistor associated with THAT input. If that be so, a summer can only be an inverting summer as it meets these requirements. No doubt you can also get the sum of voltage using a non inverting amplifier but the charm is lost. If you decide to change the weightage by changing the resistor, the weightage of all others are also affected. so it is a kind of interactive summer where as the inverting summer is a non interactive summer. one can derive the equation for the output in terms of the inputs summed and their resistances, to get to know the noninverting summer (plus amplifier, if needed), better. Otherwise as unity gain amplifier, it averages the inputs as the SUM is not what the output is......
Very impressive answer... especially these "the charm is lost" and "interactive summer"... So circuitry may not be so ascetic (as it looks) but presented in a more artistic way...
.
Friends:
.
I do believe there is a place for art in design; and, of course,
I am not speaking here of the "art OF design". The first is an
emotional reaction to the intrinsic beauty of a circuit concept,
whether this refers to elegance, or perhaps poetic symmetry,
at either the schematic phase or in an IC layout. The second
sort of art - it is generally thought - has nothing to do with the
first. It is said (often in textbooks, and professional journals)
that circuit design is strictly a logical process. There can be
room for beauty or appearance: performance is everything.
.
But, these two aspects of design are not mutually exclusive!
.
The moment we lose touch with the beauty of our work, we
are on the road to becoming a product-designing robot and
to seeing our challenge as essentially that of optimization.
'
Where does passion fit into such a scenario? What place is
there for the life-long love of one's chosen vocation?
.
It's a matter of striking the right balance between these two
opposite views; a matter of merging beauty with function.
.
Who can not be aware of this?
.
Barrie
I find the circuit drawings from the 1920's and 1930's can be quite aesthetic.
In DxDesigner I have emulated this with special pass-through schematic symbols with rounded corners and hop-overs, as well as considering the contemporary style forms from the period for the components themselves.
It does add a certain satisfaction to the whole process, despite the extra effort to make it look nice. Example here:
http://www.audiophonics.com/images/schematics/susan-parker-zeus-system-1920s-style-audio-path-schematic-1a-950.gif
And yes, minimalist audio design is my passion.
A hybrid between the inverting and non-inverting configurations... a kind of a summing-subtracting circuit... It would be even more interesting, if you add more resistors (inputs) to the non-inverting input.
It is interesting to see if still there is a virtual ground at the inverting input... and whether the inverting inputs interact between each other...
Hi Cyril
I prefer to speak about the virtual short circuit (of inputs). I believe that it better reflects the substance function of the operational amplifier. Virtual ground it will be just if the non-inverting input is grounded .
"Virtual connection"? Yes, I think so. The virtual ground is a special case of this virtual connection when the non-inverting input is fixed (grounded)... and, as a result of the negative feedback, the inverting input is also fixed...
Now imagine the non-inverting circuit inputs are at constant voltage... the op-amp non-inverting input will be constant... and the op-amp inverting input will be constant as well. So it will serve as a virtual ground again. Only this will be movable (by the side of the non-inverting inputs) virtual ground.
So, as before, the non-inverting inputs are "interacting" while the inverting inputs are "non-interacting".
Maybe this Wikibooks story will be interesting for you:
https://en.wikibooks.org/wiki/Circuit_Idea/Simple_Op-amp_Summer_Design
I have dedicated it to the remarkable circuit designer Dieter Knollman; see the tallk page:
https://en.wikibooks.org/wiki/Talk:Circuit_Idea/Simple_Op-amp_Summer_Design
As you can see, the author was not very impressed by my interpretation ... but what to do, circuit designers are special people that do not like circuit fantasies:) BTW an amusing fact is that he has named his Plato formula and Daisy theorem with the names of his dogs:)
.
I continue to be amazed, and confused as to how this silly thread
can keep going on for SOOO.. LONG! And what has it to do with
CIRCUIT RESEARCH? Gosh, folks, the circuits being discussed
in this thread are RUDIMENTARY in the extreme. Is this REALLY
how circuit theorists spend their time? debating how many angels
dance on the head of a pin?
.
My advice: STOP NAMING THINGS and START INVENTING!
'
Barrie
.
Dear Sujit K. Biswas,
Let say that I have to put a caption to the Figure...
I personally would call "averaging" amplifier, actually "googling" for that I did find the same architecture.
For me, Non-inverting amplifiers are better to identify a category than a architecture. So as the output in not inverted, this circuit would be in the same category.
Non-inverting averaging amplifier, that is the name I would give to it.
You can see more here:
http://www.allaboutcircuits.com/textbook/semiconductors/chpt-8/averager-summer-circuits/
Kind Regards
Ps.: Dear Gilbert, from a big fan... I am curious, how you call this circuit if you really have to do it.
.
Diomadson:
.
Your name for this particular circuit topology
seems quite suitable, although a bit long. I
would probably not use this form, however,
unless the signal sources happen to have a
very low source resistance, or it is precisely
known because of two concerns (a) about
interaction of the sources and (b) ambiguity
in channel gains due to the resistance of the
sources,
.
There is a much better way of achieving the
averaging function, I will need to find some
old schematics to demonstrate how I would
do this IC form. I will attach them later.
I see Mr. Gilbert
that also you angels (on the needle) handcuffed. And a discussion on the simple basic circuit continues merrily .
Sincerely yours Josef Punčochář :-)))
I prefer the inverting summer for summing as it is non interactive. the 'gain" of other channels are not affected when I change the weightage resistor for the channel of interest. This is due to the low input resistance of the type of feedback used in inverting summers, as low as Rf/effective Av. this might be just a few ohms and hence we are actually converting the voltage to currents and adding the currents finally producing a voltage using Rf. the noninverting average/summer is way different. If the sum is what is the output, what is the amplification factor? no doubt you can also have amplification of more than 1, by suitably changing the feedback resistor. The summer does not allow changing the contribution from each channel, an advantage obtained with inverting summer, without also, the interaction between various channel gains.
I do not think that it is a good idea to forcibly close discussions about basic circuit ideas, because it is always possible someone who is tempted to think deeper, to enrich them with new viewpoints. Although there may be no direct practical and comercial use, it might be useful at least for didactic purposes.
For example, I would like to add some more speculations about the "philosophy" behind the resistive summing circuit (in addition to the brilliant V.s.V Mani's practical considerations). I will do this by following and commenting the evolution of the circuit from passive to active.
Generally speaking, the main problem of the passive resistive summer is the non-zero output voltage; its existence leads to the interaction of the input sources. If this voltage was zero, the input currents would depend only on the input voltages while, when it is not zero, the currents depend on the difference between the input voltages and the output voltage.
The paradox is that we want the output voltage to be not zero... and even to be as much as possible higher. The genius idea is to compensate (to zero) this "undesired" voltage and to use the compensating voltage as an output voltage.
For this purpose, we add another input (resistor) to the existing inputs of the summer where we apply a compensating voltage with an opposite polarity... and what is the most interesting, we use this input (of the passive summing circuit) voltage as an output (of the whole active circuit) voltage.
In this connection, the resistor of 100 k in the Josef's circuit diagram below, is not simply a "negative feedback resistor Rf" (as usually it is said); it is the same as the other four resistors (20 k, 10 k, 5 k). Thus we actually have a 5-resistor (input) passive summer that sums five voltages in total - four input voltages and one output voltage. From this perspective, this 4-input op-amp inverting summer consists of a 5-input passive summer and an op-amp.
I have developed this idea in the Wikibooks story below:
https://en.wikibooks.org/wiki/Circuit_Idea/Parallel_Voltage_Summer
Cyril,
This was didactic and a different perspective.
This will go through Spice and some thinking of the article.
Thank for presenting the Wiki Paper.
Hi everyone debaters
Once again I would like to point out that increasing the number of inputs always leads to a deterioration in noise ratios. Each inverted signal is amplified by means of "own" resistance ratio only. But the noise is amplified in proportion: the feedback resistance to parallel combination of resistances at the inverting input. Thus "noise gain" may be larger as "signal gain". It can be very unpleasant experience.
Now what?
:-)) Josef
I do not think Cyril.
Only for the noise (imagine them in a non-inverting input) will "work" all resistances (parallel) in the inverting input.
Josef
.
The noise of this particular circuit is readily calculated.
.
Most important in the application is the choice of the
value for the summing resistors, since, unless they are
very small - and that is strongly contraindicated if this
circuit is not to place a heavy load on the source(s) -
their thermal noise will probably be dominant.
.
Assuming the illustrative three-resistor input is used,
and neglecting (as we properly should) the separate
noise contributions of the sources to be summed, we
have an effective noise from these resistors equal to
their parallel sum. To this noise must be RMS-added
the input noise of the op amp, and in the case of an
of older op amp the noise current at the noninverting
input which operates on the effective value of the
parallel input resistance, which we can simply call 'R'.
.
Remember that the Noise Spectral Density (NSD) of a
resistor's thermal noise can be readily calculated as
129 pV per root Hertz times square root of R (at 300K).
Thus, a 1-kohm resistor generates 4.079 nV/Rt-Hz.
.
The total integrated noise of any individual source is
found by multiplying the NSD multiplied by the square
root of the effective power bandwidth of the channel.
This needs further definition, but for now, it must be
left for the reader to research.
.
Thus, our 1-kohm resistor generates an RMS Noise
Voltage of ~2 uV in a channel of 250 kHz bandwidth.
(This need not be a DC-based bandwidth. If we have
a channel of 50 KHz to 300 kHz, the noise would be
the same as a DC-250 kHz bandwidth - at least, if
the op amp's 'bandwidth' does not limit this number).
.
Without here including the op amp's own input noise,
which should be determined from its Data Sheet, we
must now add the extra noise due to the parallel sum
of the feedback resistors, calculated in the same way,
and RMS-added to these earlier noise sources. Now
we have a total input-referred NSD of, say, N.
.
Next, we must multiply the input-system NSD by the
closed-loop gain determined in part by the feedback
ratio and in part by the unity-gain frequency of the op
amp. Often in practice, the latter can be the dominant
factor, and cannot be ignored.
.
Suppose we set the feedback ratio to 0.1, for a gain
(we think) of 10. Now suppose that the op amp has
a unity-gain frequency of 5 MHz. Then its open-loop
gain at 250 kHz (assuming classical dominant-pole
HF stabilization) is 5 MHz / 0.25 MHz, that is, x20.
.
This number is not only a lot lower than one might
think an op amp provides, but that's all you get. (It
would be only x2 at 2.5 MHz, and of course x1 at
5 MHz). The DC open-loop gain of this op amp
may be 10 million, but to get up to that value the
operating frequency would have to about 5 MHz
divided by 10^7, that is, 0.5 Hz.
.
An OL gain of x10000 (80 dB) is needed to attain
a reduction in closed-gain of 0.1%, assuming we
are still hoping for a closed loop gain of ten. But
doing a similar calculation, we find that happens
at 500 Hz in this example.
.
This may be the place to point out that function of
a classical, dominant-pole-stabilized op amp is NOT
that of a voltage-gain amplifier with some 'bandwidth'.
.
THE OP AMP IS AN INTEGRATOR.
.
Think about it. Calculate the time-constant of this
integrator given the op amps' unity-gain frequency.
It's not rocket science.
'
This explanation could go on a lot longer, but it
points out that the bandwidth value used in the
noise calculation should be - not the unity-gain
frequency of the op amp - but rather its actual
gain at the signal frequency under closed-loop
conditions.
.
This fact makes the output noise more difficult
to state, since we are getting now into factors
which are entirely application-specific, But the
principles remain firm, and perhaps a good
text book will come along in a little while......
'
Barrie
Maybe one partial solution to the problem, in the case of an AC summer, is the inclusion of decoupling capacitors at the inverting inputs? Then the DC noise will be not amplified and the DC loop gain will be maximal?
Cyril,
I think that the decoupling capacitors is application specific
( to be properly noted ) ,
and this topic started today as a consideration of principles.
I am game to consider both perspectives , one at a time.
In my own developments,
I use start Spice without decoupling caps,
to prove my concept,
then add decoupling et al , to improve the performance.
Real Life design is a combination of all there is. .
Respectfully
Josef,
I think these are not so big problems of the inverting configuration... It is designed mainly to sum, not to amplify... the amplification is an additional feature. For example, in the case of equal resistors, a 3-input inverting summer will amplify the noises at the non-inverting input with gain only of 4 times... and the loop gain will be 1/4 of the op-amp open loop gain... If we want more gain, we can connect an amplifier after the summer.
Cyril,
Perhaps you're right. But in any case it is not appropriate imprudent increasing the number of inputs. When working with small signals, each "noise dB" is welcome. And a reduction of 3 db frequency is evident.
Josef
Sujit, Cyril, Josef, Susan, especially Barrie, et al
This Question by Sujit may have set some record for the 73 responses.
And the posts continue to pierce the conceptual shell
of the summing amplifer.
* I did run variations of this through Spice,
and it gives results related to the way I set-up the schematics
... but no surprises.
* I reviewed Bumm,
and note that his presentation is based on the ideal OpAmp.
Must be this way
since the author cannot know all the possible application specifics.
* I enjoyed meeting my old friend OpAmp again
... but an inverting OA is still just an inverting OA.
The OA requires inventive application to become alive.
My experience has become real
... not so much with esoteric conceptualizing
... but with innovative applications,
such as, my little work in Phased Audio Filters
as applied in radio CW operations.
Glen
http://www.geocities.ws/glene77is/
.
Can we please move on to matters of substance?
I will not spend any more valuable time on this.
.
Dear Sujit K. Biswas sir,
Both are non inverting summing amplifier only. There are many other versions of non inverting summing amplifiers.
Observe carefully, both the circuits are providing the sum only but the difference lies in the gain provided by the circuit to each input. In first circuit, gain provided by the circuit is 1/3 to individual input whereas it is 1 in the second circuit.
The second Circuit is called the Averaging Amplifier.Refer Google search. Because of the nature of the feed back and all the resistances being equak it acts as Summing amplifier .
The first circuit becomes a non Inverting Summing Amplifier because of the Special Feed Back given.
The first circuit gives the average of the resistances acting as an averaging amplifier because of the Special feedback.
Both the circuits are non inverting.
To understand the two Circuits let us go a little bit deep into the Circuit Thery.
what we get using an opamp is the Transfer Impedance of the Circuit between the input and output Terminals .
In Laplace Transform Vo(s) /Vin(s) with initial conditions being zero.
On the Input side we have all the three resistances in Parallel.
Between the output and input terminals we have the feedback circuit.
In circuit 2 the feed back voltage (and also the Feedback Resistance) is zero.
In circuit 1 the feed back voltage is the voltage across the output and input terminals of the 3 resistances acting as a Potential divider (or its corresponding equivalent resistance).
Here the potential divider is connected to earth.
In actual opamp remember that what we have is Virtual earth. Hence
in circuit 2 the feedback voltage is zero and the resistance between output and input terminal is zero.
In both the circuits the feedback is Positive as can be observed being tapped from the positive terminal of the output.
Here in both the circuits we have all the three resistances equal.
Hence we have the results given.
P.S.
Fellow Engineers,
Reviewing this topic, 2022 August 29,
now seven years after it was posted by Dr. S. Biswas,
I find the circuit is like my "Voltage-Mixer" circuit.
Perhaps we should give Sujit Biswas circuit this new name.
The name "Voltage-Mixer" would not conjur up
images of other circuits.
...
In my opinion,
no matter however deep as we choose to dig
into the analysis of this circuit,
it will never resolve itself properly,
because we are trying to make a "Voltage-Mixer" circuit
speak to us
as if it were an "Inverting-Summing" circuit.
...
In perusing the writings that Barrie shared with me, (RIP),
and reading his line of thinking about circuit design,
I realize that Barrie was quick to divide a problem
and attribute/give new/appropriate names
along with some other theoretical perspectives.
...
Reviewing this long posting
has been a good experience for me,
I have enjoyed touching the thoughts of fellows from my past.
...
As Barrie suggested,
It is time to move on, in our thinking.
===
@Glen, I agree with you. "Voltage mixer" is a more figurative name than "voltage summer". Thanks for the heartfelt comment in which you reminded us of the unforgettable Barrie Gilbert...
Cyril ,
So good to read from you.
I see the handicap as too much reliance on textbook lessons, and not so great attention to "thinking" about the nature of the circuit. In one of Barrie's emails , he described his intuitive method this way: We must put on the robes of the electron and walk through the circuit.". We must "feel" the experience that the electron feels .
Textbook learning is only the beginning of the engineer's life. In Barrie's writings, you can read about "The Nature of Invention". We must be "one" with the electrons in the circuit design.
You can find a collection of Barrie's writings that he sent to me, on my website, on the "Alpha" page, just click on the photo of Barrie and his wife Alicia.
...
So, there is a great difference between
(1) "Summing" information
versus
(2) "Mixing" signals to generate intermodulations.
...
Perhaps this is a thing which cannot be "taught",
It is an idea that the apprentice must "think" through for himself.
... I am just an old apprentice,
caught tinkering in the Master's workshop.
to Cyril,
Hard to believe there are 8 years of comments to this simple question !
But ... Life is interesting.
Two months before Barrie Gilbert passed,
he emailed to me that
Life is for Exploring everything that comes our way,
Life is for Analyzing to the limits of our talent.
Life is for Appreciating these experiences !
... over those years, he was unforgettable.
...
Cyril,
I still find your writings to be interesting, sparking my imagination !
...
I am a very old apprentice now,
I was caught tinkering in the Master's workshop,
and given good advice.
===
- Badges
- Science topic
- Similar topics
- Mathematical Sciences
- Control Theory
- Control Systems