I would like to generate curves that rise and fall exponentially.
These curves simulate a phenomenon unrelated to electronics, but the
best simulation would involve plotting the voltage induced by a magnet
being moved at some linear velocity past a coil in a circuit that
contains a capacitor. The plotted curve represents the transient
voltage generated by induction that decays over a time across R and C.
Although one could use a voltage source and switch, or a pulse
generator, I specifically need to model individual events that involve
a magnet moved past a coil.
The circuit typically would look like this:
------------
| __|__
| | |
| C1 R1
| | |
magnet [F] L1 ------
| |
| |
| |
------------
The voltage transient is given by an equation with "F" being the flux,
"L" the coil inductance and "x" representing time. It would be
measured across "R1". The decay over time "x" is given by an equation
applicable to the parallel circuit with component values "R" and "C"
and the induced voltage.
The generation of the transient takes place in a short time,
milliseconds, typically. The decay can extend over several seconds
for the type of curve I would like to see. If the above were a real
circuit, C would discharge rapidly because of the low resistance of
the coil. I assume that it is possible to offer a function that can
be plotted that uses the peak voltage generated as the voltage that
then decays via RC. In a "real" circuit I would perhaps insert a
diode to preclude the discharge through "L".
The solution I need is a single function that plots the exponential
rise of the induced voltage and the exponential decay of that voltage
via RC in one curve. It would be of the form f(x)=
I could then select component values to change the shape of the
transient curve from x=0 to some other value of x where "x" is time.
The answer would consist of the function as described, that can be
demonstrated using any plotting program that can accommodate multiple
variables. Offering a link to an online or downloadable plotting
program that was used to validate the proposed solution prior to
submission would demonstrate that the task described here was indeed
answered. |
Request for Question Clarification by
hedgie-ga
on
16 Mar 2006 21:20 PST
Cereb
There some issues of terminology, which I would like to sort out before we
get to solving the 'the system'. The system could be mechanical, electric,
or some abstract time series in economics. For all kinds of 'systems'
people often ask the 'if' questions. e.g.:
If we increase the toll on a bridge (for example),
how will that affect to volume of the traffic across that bridge?
Mathematical formulation of these situations was provided by the Control Theory
http://en.wikipedia.org/wiki/Control_theory
The variable we control (e.g toll on the bridge) is called 'forcing
function' and the
variable which we then measure is the 'response of the system'. Often
the forcing function is a step : we increase the toll, and we see that
after same time the system approaches new equilibrium :
Step response
http://en.wikipedia.org/wiki/Step_response
Approach to the new equilibrium can often be approximated by a single
exponential described by a single parameter (exponent) called:
relaxation time
http://en.wikipedia.org/wiki/Relaxation_time
It is an important property of the 'Linear Systems' that once we know
the response of the system to 'step function', we can easily
calculate response to any forcing function.
For example, to increase 'the toll for a month by $5, and then put it
back to original level' (so that forcing function is a 'square pulse'
will make the system to respond by slowly lowering the volume of the
traffic, and then, after a month, traffic will slowly start returning
to original level.
The response to such a pulse function is obtained as superposition of
the two responses:
1) the response to initial step up, and then
2) the response to step down (shifted in time, by one month).
Hoping to find dome nice applet illustrating that, I did this search:
search terms: linear dynamic system applet
and picked these links:
--------------------------\\
The behavior of a linear, continuous-time, time-invariant system with
input signalx(t) and output signal y(t)
http://www.jhu.edu/signals/convolve/index.html
The following applet was created for students to investigate the basic
properties of linear systems via a simple mass, spring and Damper
system.
http://users.ece.gatech.edu:80/~bonnie/book1/applets/suspension/MSDdemo.htm
---------------------//
None is ideal, but they do illustrate a notion of a linear system.
If we can agree on the above terms, the question arises, which using
these terms, can be formulated as follows:
What is the forcing function for the system we want to solve?
We are talking about a motion of a magnet, magnet which causes
variable magnetic field, field which comes and goes. That looks like a
pulse function.
The pulse function is inducing the voltage in the coil L1.
OK. We know that 'to solve the linear system' we just need to find a
response to a step function, but we did not really defined what that
forcing function is.
Usually, in the circuits theory, forcing function is selected as a
voltage over the input terminals.
Then response can be current at any conductor, or voltage across any component.
So, to make question more exactly defined, I would like to have the
forcing function better defined.
1) One choice would be to put input terminals in series with the
inductor L1 , like this
|---------IIIIII---------0 0---------------|
L1 ~ input
and apply a voltage to the input terminals . That would produce the
current in the L1, similar to moving a magnet close;
Then we remove the voltage, and the circuit eventually comes to the
rest, same was as after the magnet was removed.
Or
2) In reference to the picture on the right top of this page
http://www.sparkmuseum.com/INDUCT.HTM
we could have two coils. What they call 'high voltage output' would
be connected to the rest of the circuit (where L1 s now) and the
battery would represent the applied voltage across the 'input
terminals'.
Switching the K on and after few seconds off, would be a voltage pulse.
We would have additional two parameters in this case; we would have
L1 (as now) but also
L0 (inductance of the primary)
and L12 - the cross inductance between the primary and secondary coil.
Which one (or perhaps still another) interpretation of your question
you want to use?
Hedgie
|
Clarification of Question by
cereb-ga
on
17 Mar 2006 19:40 PST
I was hoping to hear from you, Hedgie. Thank you for responding. I am
certain that you can resolve my unusual task and that it is largely a
matter of communication which, I trust, we can overcome and fairly
rapidly, probably now.
I carefully read your comments and attempted to understand those as
well as the linked-to information. You certainly have invested effort
to tutor me.
The closest I could come to selecting from the options you recommend
would be the two-coil version. If I used the example of a typical
transformer having primary and secondary windings, I could see one
winding (not the L1) driven by a battery and creating a field. I
could make sure that "my" circuit has no voltage anywhere. Then, by
turning the energized winding off with a switch would induce a voltage
and current in my L1 coil. The collapsing field in the formerly
energized coil at would be a time-varying field simulating a moving
magnet and RC would be off and running. EXCEPT that in a real
circuit, the charge on C would dissipate through the coil before RC
ever got off to a start, because the intended rise time of the induced
voltage typically would milliseconds and the fall time seconds to
yield my curve.
I think it is best if I describe the intended use of what we are
dealing with here. It deals with human responses much as the
Weber-Lechner law does, but I model something else.
Basically I model a "motivational signal" that registers in the human
mind, brought on by some event, which then fades by something similar
to "forgetfulness".
For example, if I gave an employee a one dollar raise, that would
generate some mental signal. But if I gave the employee twothousand
Dollars for Christmas, that would yield a much larger signal. In
either case, the employee will experience a decay in that signal and
return to normal almost, as prior to either signal or reward. The
total amount I gave will be the same for 2000 hours worked, however.
So I take 2000 as the flux of a magnet - the flow of money in this
case. A rapid flux change would create a steep pulse, dribbling the
amount out over time, a small pulse (the signal in the employee will
say "it's just a Dollar"). In either case, the signal, stored in
memory C decreases because C is leaking charge through R, causes C "to
forget".
I can use C to model some human capacity or ability to register, and
use R to model "forgetfulness", typically (though not quite). Taken
together, I would have a function V(t) =" inductive term and RC term"
which I could plot, and by assigning values to LCR and the "event
time" and magnetic flux for event value or significance, I can then
develop my scenarios and illustrated them with curves.
In the above example I could use a pulse, because it takes about the
same time to advise an employee of the hourly rate or the bonus, and
let pulse height model the value. But there are other cases where it
would not work as nicely.The moving magnet gives me greater latitude
in modeling signal generation scenarios.
I hope this helps, Hedgie. I was reluctant to use the "Science"
category for this not-so-scientific model, at least at this stage.
|
OK cereb,
Let's take it small in steps -
meaning: do not consider this to be the answer.
It is the opening step of a dialog.
Do not worry too much about the the length of the dialog.
Obviously, it has to be finite;
(This is one example of one 'too long' dialog:
http://answers.google.com/answers/threadview?id=500730
)
As you see, before the dialog gets too long, I do usually say that we have to
come to closure in few more installments.
The reason I want to go in small steps is to make sure we are communicating.
The problem with the magnet
______________________________
Waving the magnet around the coil, is a legitimate 'forcing function'
or 'stimulus'. We are able to model electro-mechanical systems, such as
electric motors, where physical motion and induced EMF interact. The problem
with your original description was that it was too undefined: The induced of EMF
depends on speed, and the speed-profile of the motion, which determines
the shape of voltage induced in L1 was not well defined.
This is illustrated in these two links:
Induced e.m.f. = depends on speed
http://courses.science.fau.edu/~rjordan/rev_notes/28.2.htm
shape of e.m.f. pulse induced in coil by a magnet
http://www.data-harvest.co.uk/datalogging/easysense_fa.html
Also, such a situation may require a non-linear model if the distance varies
in a wide range. Again, we can solve non-linear models, but here, in what I
take to be a conceptual stage, it is better to make the model as simple as
possible (but 'not more simple than that :-)
http://www.quotationspage.com/quote/3430.html
So, following that adage, let's start really simply - we can add componets later:
Description of the task
_______________________
You do have a stimulus (or forcing function) which is the
'rate of pay as a function of time'.
We consider two rectanglual pulses with same area of 2000 units.
One is narrow and tall (the Xmass bonus), other is wide and low (a raise).
We will call them Rx(t) and Rr(t).
You have 'a worker' (what a control engineer would call 'a plant')
which produces output P(t) (measured in some output units).
The 'worker' is modelled by a linear system K, (kernel)
so that the output (of production) P depends on rate of pay:
P(t) = Sum K(t - p) R(p)
Sum is over p ,ranging from infinite past up to present = t.
meaning: the current rate of production P(t) depends not just on the current
rate of pay R(t), but also on the
last week R(t-1),
and week before that R(t-2)... etc
and the 'memory of the past' fades at the rate controlled by some
'time constant' tau= R *C .
We know that K can be represented by a circuit (mechanical or electrical)
but whatever circuit we choose, we want to be able, for the given K
and the stimulus R we choose, to obtain a plot of P(t).
In addition, we want to represent K by some simple circuit with small
number of components, so we can tweek the values, obtain a modified K,
and perhaps give some interpretation to their function in terms of the
psychology of the worker.
Questions (RFC)
1) First question: Is this, as described above, the task?
An ideal tool for this would be Scilab (a free simulation paltform) which can
produce plots like this:
http://www.scilab.org/doc/demos_html/node179.html
as described here
http://www.scilab.org/
Commercial programs like
http://www.wolfram.com/
ttp://www.math.ufl.edu/help/matlab-tutorial/
or free version of matlab,
http://www.octave.org/
would also do the job.
What job? Job b) using the Kernel K to calculate Response P and plot it
and a) Calculating the K from a diagram of the circuit.
It is a bit more than a plotting program, but not much more.
2) Question two is, do you have some such program installed,
or do you want to install it?
It is not necessary to install one, but if you want to work interactively:
change the component and get a response, interactively, it would
make the task easier.
Alternative would be to start with the plotting applet, and some simple
Kernel, someting like K(d) = e + g* expt( r * d)
and get sample responses to different stimuli ( Rx and Rr ..)
P.S. r = 1/tau can be complex number - in which case we get an oscilatory
response - indication of 'inertia' or inductor in the circuit.
3) Are you familiar with complex numbers, stuff like
http://hyperphysics.phy-astr.gsu.edu/Hbase/cmplx.html
http://planetmath.org/encyclopedia/ComplexExponentialFunction.html
http://www.mathworks.com/access/helpdesk/help/toolbox/dspblks/complexexponential.html
Do you want to use or avoid that terminology?
Hedgie |
Request for Answer Clarification by
cereb-ga
on
18 Mar 2006 12:07 PST
Thorough job, Hedgie, Thank you again.
I made curve generation my principle goal, and without having yet
generated such curve I trust that I can do so with the links etc. you
gave to me. I will download the simulation software and see what I
can do with it to generate a plot shape of the type I wish to
generate, and will then be ready in case more interaction is
indicated.
I knew that the magnet approach would be cumbersome to actually
generate the desired curve on a scope, for example. The "real" issue
for me is this: I have invested 20 years of work related to
human/social/societal behavior. It has been rather novel work,
seminal in parts, and at age 73 I feel the pressure to present it so
it can be understood by many people who search for credible solutions
to seemingly insurmountable obstacles to change the way we manage the
affairs within and among societies.
Since we have only "effective objectivity through collective
subjectivity" to rely on, such as laws, treaties and such, my aim was
to find a way to overcome the constraints of humans being able to be
objective about humans, and I searched for an objective frame of
reference to overcome these constraints. In that I suceeded,
presumptuous as that may sound.
I use phenomenology for explaining the method I employed. Social
scientists could use some of my findings employing expert systems and
AI tools to simulate policy formulation based upon the core model for
individual behavior and its use for a societal core model that uses
the individual core model. But I am more interested in making my work
plausible to many people who have valid concerns for our future, even
make it credible and mainly actionable. So I felt the need to model
much of what I found.
One of the most consequential Findings of my work relates to our
individual as well as collective behavior of "growth" that applies in
economics, in governance and the exponentially rising expansion or
even race for scientific progress. The curve I am after now is but
a start, subsequent graphical illustrations will further illustrate
what drives this obsession with "growth", which in turn creates the
many problems we face everywhere that derive from this obsession.
I can use your approach to model that and other curves we may want to
deal with hereafter, because it may well serve those who want to model
things I describe in my work, and they would demand more rigor in my
reporting. I was hoping merely for something to illustrate to
non-professionals something they could use to better relate to these
graphs. Such approach would also more adequately reflect my
inabilities to offer more than "cute" analogies.
Certainly our present task would be completed by using your approach
to create scaleable graphs, and it would be completed upon generating
them. Maybe I will use my LCR/magnet example to just explain verbally
how such curves "could be" generated that way but because of the
modelling requirements for what I am describing, that your approach
was necessary.
This verbiage here may help to explain my desire for the LRC/magnet
model and it may help when we deal with the other curves related to
this "growth" obsession.
I think we are where we wanted to be, and I will now get the
simulation program and see what I can do with all your inputs. Above
all, I want to thank you for having stayed with this issue patiently
and constructively. If you wanted to create two curves to show
scalability using the simulation software link, that would be the
answer that closes this part. I would hope that we can then deal with
the use of that approach to create the other graphs I have not yet
described but that rely upon this one.
|
Clarification of Answer by
hedgie-ga
on
19 Mar 2006 22:52 PST
cereb-ga
http://answers.google.com/answers/threadview?id=705898
I = V / ( R1 + R2)
http://www.pa.uky.edu/~phy211/graph_applets/plot_graph.html
current
http://answers.google.com/answers/threadview?id=708116
I made curve generation my principle goal, and without having yet
generated such curve I trust that I can do so with the links etc. you
gave to me.
OK.Let's now move to the curve generation. Scilab software may become
very useful, particularly if you have experimental data to fit and will
need to introduce non-linear components. However, it is a handful and
there is a steep learning curve.
So, for now, we will plot simple solutions using the plotting applet.
We will use the simple kernel which we found in the previous question,
a solution which has a a) constant and b) one exponential with time
constant 1/l1= R1 *C1
Asymptotic regime
----------------
The constant is a0= 1/(R1 +R2) and it defines 'asymptotic behavior' aka
steady state: when the 'transients die away' and new (dynamic) equilibrium
is reached your circuit is
P(oo)= a0 * R(oo)
where oo is symbol for 'infinity' = long time after last change of R = rate of pay,
=long time after time 1/l1.
It says that 'rate of production' P is proportional to 'rate of pay' - which is
somewhat unrealistic, since workers productivity is limited, but not surprising
in a linear model. (Ur can be fixed by saying that R=log(pay) or so, but that is a
different issue).
Transient regime
----------------
In the time interval between change in pay R, and equilibrium, the
production P will oscillate around new level P(oo) like this
P(delta) = (a1/l1) * exp ( delta * l1) * sin( delta * omega1)
parameters of the curve a1 l1 omega1 comes from the kernel.
Actually, 'solving the circuit' means to express these parameters of the kernel
in terms of the values of components ( L1, R1, R2, C1 ...).
Let's plot the transient P(delta) or P(t) , which is P(t) shifted in time
(t is time, delta is time counted from the
moment t0 in which the change of pay was made) (we will set it to zero)
Let call the applet
http://www.pa.uky.edu/~phy211/graph_applets/plot_graph.html
and enter
((5*exp( -.03 *x) * sin (.1 *x ) )) +5 into function box
min x to be zero
max x to be 100
and press [plot the function] button
we get a transient, starting at P=5 , swinging up to P=8, down to 4, ...
to see settling we increase x.max
max x to be 100 and press [plot the function] button
To get further, we meed to find a 'step function' s(t,t0) = 0 if t<t0
and s(t,t0)=1 if t>t0
unfortunately, clicking on the Special Functions link produces error.
Meaning, we have to find a better plotting program.
Please, let me know if you have a preference
(but it has to be multi-platform, run not only on windows)
Hedgie
|
Request for Answer Clarification by
cereb-ga
on
20 Mar 2006 13:51 PST
The function to plot worked as you described. I made a typo in
entering the function by typing .3 instead of .03 and was rather
pleased to see the very shape of a curve I hoped to generate. It
basically looks like a damped version of the function you offered.
I agree, the Scilab software calls for an amount of learning I am not
prepared to get involved with, although I think it is an incredibly
powerful general engineering tool, and I am glad you steered me toward
it.
You referred back to previous question where you suggested a Laplace
transform as well as to your "description of the task". Under the
title "transient regime" you offered the production equation stating
that the parameters of that equation come from the circuit components
and their values. And that is what I would like to see - the
correlation between curves and circuit elements and their values. So
far so good. The correlation itself I do not yet understand as yet.
You clearly understand what I am searching for, and I will therefore
follow your lead. I trust that in the end I will be able to follow
the steps you have already outlined, and others that will be needed,
to generate the curves. If you have a suitable plotting program, just
advise what it is, how it was generated, whether I have such program
or not, and I will accept that as the answer. Thank you for your
patience thus far.
Re other plotting programs - I have no preference. You suggested the
free version of matlab.
|
Clarification of Answer by
hedgie-ga
on
21 Mar 2006 19:20 PST
Octave - a free version of matlab is good, but like scilab, has more
complexity then we need. After looking around a bitm I picked
gnuplot. If the graphing software has the same capabilties, you can use that
of course. I will use gnuplot to visualise the response function,
and post in the next RFC. In the meantime, you may install it and work
through a tutorial.
Description
http://www-128.ibm.com/developerworks/library/l-gnuplot/
tutorials
http://www.gnuplot.info/help.html
examples
http://t16web.lanl.gov/Kawano/gnuplot/misc1-e.html
homepage - download -Current officially released version is gnuplot 4.0
http://www.gnuplot.info/
|
Clarification of Answer by
hedgie-ga
on
22 Mar 2006 05:20 PST
So,
I did entered the equations into gnuplot.
It seems to do what you want - a response to rectangular pulses.
I mean in a qualitative way. I used arbitrary numerical values.
I can post the pictures of plot, if you need that, but it
will be much better if you creeate them on your version of gnuplot,
so you can change the values and tweek the axes and labels.
What follows is a gnuplot script.
Meaning:
1) you put the text between lines ---------\\ ------------//
into a file called RCL.dem
and type
gnuplot RCL.dem
into a shell. I believe MS people call that 'command console' .
Then, at each prompt 'Hit return to continue'
you 'hit return' (it is, as you see, easy)
and script waks you thrue creation of stimulus pulses and responses
2) then you open the file RCL.dem in the editor and
either, modify values
a0, a1, lambda1 omega1, which describe the linear system
and
start end height , start2 height2, end2
which represent stimuli
(batch mode)
or
lift the command from the *.dem file and paste them into fnuplot console.
That should complete the portion
'how to plot response of the circuit'
of the question.
We will have to start thinking of the closure,
but please, do ask if these instrucyions are not clear.
The script
----------------------------------\\\
#
# $Id: RCL response of a simple linear system to a step function
#
# Requires data files "pulse.dat", kernel.dat from this directory,
# so change current working directory to this directory before running.
# gnuplot> set term <term-type>
# gnuplot> load 'RCL.dem'
#
set key left box
set title " Test availability of functions"
set samples 150
set zeroaxis
set xlabel "Date\nTime"
set yrange [-5:5]
set xrange [-1:50]
plot [-1:10] cos(x), 3*sgn(x), exp(-.5*x)
pause -1 "Hit return to continue"
set key right nobox
set title " Plot rectangular pulses : tall (bonus)"
set samples 100
plot 1+sgn(x)
# select pulse 1 == rise
start=3
end=33
height=1.5
#define pulse
pulse(x) =( sgn(x-start) - sgn(x-end))* height * .5
#plot it
set yrange [-.5:20]
set xrange [0:50]
plot pulse(x)
pause -1 "Hit return to continue"
set title " Plot both stimuli a) rise (pulse2) and b) bonus (pulse) "
set key left box
set samples 200
start2=3
end2=33
height2=5
pulse2(x)=( sgn(x-start2) - sgn(x-end2))* height2 *.5
start=3
end=5
height=10
set yrange [-.5:200]
set xrange [0:50]
set zeroaxis
plot pulse(x), pulse2(x)
pause -1 "Hit return to continue"
set title " Rise stimulus and Response to the 'rise' "
set zeroaxis
set yrange [-2:100]
R=0.5
a1=2
a0= 1/R
lambda1=.2
omega1=.9
kernel(x) = a0 + a1*exp(-lambda1*x)*cos(omega1*x)
response(x) = (x<0) ? 0 : kernel(x)
prod(x)= height*(1/R+response(x-start) - response(x-end))
plot prod(x)
pause -1 "Hit return to continue"
set title " Response to the 'bonus timulus' "
set yrange [-.5:50]
plot height2*(1/R+response(x-start2) - response(x-end2))
pause -1 "Hit return to exit"
reset
------------------------------------------///
Hedgie
|
