Analog computing

[gary36]

I was looking for a all analog computation of the reactor point kinetic equations(studied in under grad course) using opamps

The equation is described as

I referred to the article https://analogparadigm.com/downloads/alpaca_30.pdf. But could not quite get , as to how the circuit represents the differential equation.

 

[danadakk]

How to Solve Differential Equations Using Op Amps - dummies

The op amp circuit can solve mathematical equations fast, including calculus problems such as differential equations. To solve a differential equation by findin

www.dummies.com

Integrator and Differentiator shown :

Operational Amplifiers (Basic Circuits) Equations

Learn Operational Amplifiers (Basic Circuits) equations. We'll also share about amplifier circuits and formulas for Operational Amplifiers' configurations.

www.circuitbread.com

https://www.analogmuseum.org/library/heathkit_ec1_operation_manual.pdf

Regards, Dana.

 

[BradtheRad]

Not sure if it solves a differential equation by being a differential detector. The op amp contains a long-tail pair which acts as the differential detector. An unchanging current is shared by two networks resembling class A amplifiers. If the left-hand column draws less Amperes, this allows the right-hand column to increase the amount of Amperes it draws. And vice-versa and conversely. The resulting network is not too different from a Wheatstone bridge. And the Wheatstone bridge concept is frequently extended to the input network.

To determine slope (rising or falling) of a signal, I've seen a capacitor integrator at one input of the op amp causes the output to go high or low. The signal is split so it goes unchanged to one input, and a slightly delayed copy goes to the other input. If the delayed copy is greater than the initial signal, then the output goes one direction. If the delayed copy is less than the initial signal, the output goes the other way.

 

[dick_freebird]

I don't think the hand drawn circuit performs the equation.
Equation has Vi as "output", circuit has Vi as input. Maybe
if you put the whole thing inside a feedback loop, that
would have an output at Vi which truly did what the text
indicates.

 

[danadakk]

The links in post #2 just ref material.

 

[BradtheRad]

With some experimentation I found the proper amount of gain for my setup. Incoming signal reaches peak amplitude briefly. At the same moment output is at zero thus indicating slope of incoming signal and performing differential function.

A lissajous plot shows the relationship. Incoming signal is a sine wave (pretty much) therefore the differential looks almost like the cosine.

To be strictly correct a second op amp is needed to invert polarity of the result.

 

[gary36]

I was trying to understand the method to solve the second equation(#1), dc/dt=beta/L* n - lamda*C. The problem with this equation is that outputs do go unbounded due to presence of integrator, eventually saturating with the input being very small. Am I correct?

 

[BradtheRad]

Yes, I had to fiddle with component values to keep the output within bounds.
The output tended to pin to the supply rails. (My input signal is Falstad's simulator ANTenna model whose amplitude varies all over the place. I took the screenshot while amplitude was large and getting larger.)

I guess it seems strange to put an integrating capacitor to solve a differential equation. There may be another topology which is more straightforward.

 

[gary36]

can you suggest a workaround

 

[BradtheRad]

A series capacitor is known as performing a differentiating function. If built into an electronic circuit, values should be chosen to suit the frequency and detail of input signal.

 

[danadakk]

Some differentiator considerations :

https://neurophysics.ucsd.edu/courses/physics_120/op_amp_differentiator_ckt.pdf

https://www.ti.com/lit/an/sboa276c/sboa276c.pdf?ts=1727929323002

Regards, Dana.

 

[gary36]

I am afraid Dana, the link provided by you did not solve my problem. I understand the method to implement differentiator. But the query was related to solving the equation dc/dt= beta/L *n - lamda*c. This equation , when implemented using opamps results in output saturation.

 

[danadakk]

A Pressurized Water Reactor can go prompt critical if injected into
coolant system is a large volume of cold water, which causes
neutron moderation factor to skyrocket. I was not trained in Reactor
Physics in detail (had not had advanced math at that point in my career),
just generalities, but think these equations can describe both stable and
saturating solutions. The bigger Neutron lifetime is the greater Neutron
production results as a result of greater probability of a fission which is
a multiplying factor in Neutron fission rate hence Neutron density.
PWRs use T (as well as rod position and other factors) of coolant to
moderate, eg. negative feedback. So as you pointed out when solution
is looked at the integrator can integrate into the rail, and then because
the OpAmp is a bounded circuit it can saturate if there is no negative
feedback over time to limit it.

So all that implies (the equations) is the various factors that affect
fission levels, and stable or unstable results can occur.

Maybe use this to get at a simulation ....? https://www.symbolab.com/solver/ordinary-differential-equation-calculator

Or use Matlab to sim..... or just solve the ODE and examine limits.

So in short solving the diff equation should allow you to see either an exponentially
growing, exponentially decaying, or stable output. The later two allowing a OpAmp integrator
to function properly, the first case saturate. Of course this all is dependent of scaling
in the integrator to permit those conditions, eg., decay and stable.

 

[gary36]

I did a matlab simulation and observed that due to absence of feedback the output from integrator keeps increasing without control. Can we perhaps reset the integrator so that cycle starts all over again?

 

[FvM]

If the real process has stable equilibrium, a correct model should be also stable (without requiring periodical reset). Appropriate initial conditions may be necessary.

 

[danadakk]

But if I look at the first equation, the neutron density is a reflection of initial source, current
density, and growth or DECAY term (second equation, conditions governed). The growth phase,
with smaller decay, will saturate a finite integrator.

 

[gary36]

Yes, second equation results in saturation over time. Hence thinking of reset. I was referring to the document (See the attachment). Any final thoughts on this article.

 

[BradtheRad]

Another approach to solving for slope immediately by passing source through an inductor. Several waveforms are applied: triangle, square, sine.

--- Updated Oct 5, 2024 ---

1) First simulation is a triangle wave whose slope is unchanging as the wave rises, then as the wave drops slope is unchanged but reversed polarity.

2) Middle simulation is a square wave whose slope is zero most of the time except for indefinite large spikes/glitches at transitions.

3) Sine wave slope turns out to be similar waveform but advanced 90 degrees.

 

[gary36]

Hi BradtheRad

Could not quite get what you were trying to convey. We were discussing about computation of inverse point kinetic equation

 

[danadakk]

Yes, second equation results in saturation over time. Hence thinking of reset. I was referring to the document (See the attachment). Any final thoughts on this article.

My only thought is various G values in the structure, eg setting parameter values, will
determine if stimulant reactor is stable, decaying, or unbounded. Not sure what else to say here.

Page 19 shows what happens with precursors population density and resultant decay
for parameters chosen in design.

Design given its age makes one wonder what todays solution looks like using micro.
One could take the the system, use sample data approach, and perform the various
operations on the data.

Regards, Dana.