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Let me start by saying that I am not an electrical engineer but a mathematician, who just started to dive into the world of analog computing using for the moment Simulink to simulate an analog computer virtually.

Consider the first-order differential equation

$$ \frac{dy(t)}{dt} = c - Fy(t), \quad y(0)=y_0 $$

with explicitly given constant \$c\$. The coefficient \$F\$ is given implicitly by the following integral

$$ c=\int_a^b f(x) dx, $$ where \$f(x)\$ is a computable function that depends the spatial coordinate \$x\$ (but not on \$t\$) and the integration bounds \$a < b\$ are fixed.

I already know how to compute the above differential equations with analog computers using integrators given that \$c\$ and \$F\$ are known constants. Assume that I am not really interested in the transient behaviour of the differential equation but in the steady-state solution

$$ y(t)=c/F. $$

My question is, how can I perform the integration of \$f(x)\$ in the finite bounds \$a<b\$ on a (virtual) analog computer?

Matthias Möller
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  • Welcom to EE.SE. This site uses [MathJAX](https://electronics.meta.stackexchange.com/questions/5565/most-common-mathjax-uses-in-electrical-engineering) which enables you to write properly formatted formulas, if you wish. Use `\$` as start and end tag for in-line formulas and `$$` for formulas on their own line. – Transistor Nov 09 '17 at 10:59
  • What is the problem with using an integrator to integrate? Also, as per Transistor suggestion, I edited the post to fix the math ;) – Vladimir Cravero Nov 09 '17 at 11:00
  • Thanks for bringing up MathJAX and for editing the post to fix the math ;) – Matthias Möller Nov 09 '17 at 15:26
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    Yes you can, it's called an integrator you can build these with op amps. – Voltage Spike Nov 09 '17 at 16:12
  • \$f(x)\$ being a spacial function means its values are some quantity depending on space. The question is: how do you want to sense that quantity in order to process (integerate) it by analog computation? What is this quantity depending on space? (mechanical thickness, electrical conductivity, optical reflectivity,...?) – Curd Nov 09 '17 at 16:34
  • Let's assume that \$f(x)=sin(x)\$ and that \$a=\pi\$ and \$b=2\pi\$. My naive assumption was that it should be possible to integrate a sine signal over the interval \$[\pi,2\pi]\$ (do switch-on/switch-offs exist?) and 'store' the value somehow. Is this possible with analog computing? – Matthias Möller Nov 09 '17 at 17:12
  • Why do you call \$x\$ *spatial* coordinate? if \$x\$ is space it must have a unit (e.g. meters) and also \$a\$ and \$b\$ can not be plain numbers. Although you say you are a mathematician: if you insist to build/use/simulate an analog computer (which can be anything) you must get more concrete, i.e. physical. – Curd Nov 09 '17 at 22:00
  • @laptop2d: but it seems OP does not want to integrate over time; the common OpAmp circuits I assume you are are referring to integrate over time. – Curd Nov 09 '17 at 22:03
  • I am not aware of any analog signals that exist outside of a time-domain. – Voltage Spike Nov 09 '17 at 22:22
  • @laptop2d: The question is not whether they "exists outside of a time-domain", but whether the **interesting** signal is a function of time or a function of space. E.g. variable reflectivity across a sheet of paper (printed picture, writing...) is a function of space, not of time (OP emphasized that dependency of the function is not on time). If e.g. mean reflectivity of the whole sheet or of one line is interesting, you need to integrate over space, not time. – Curd Nov 10 '17 at 08:23
  • Let me be more specific. The final application that I have in mind are _partial_ differential equations. A common practice is to apply some spatial discretisation method to turn the problem into a system of ordinary differential equations of the form given in the original post. The finite element method is one of these discretisation methods, whereby the \$F\$ is a (constant) coefficient that is computed as the integral of a function over space. Anyway, thank you so much for all your help. – Matthias Möller Nov 10 '17 at 11:20

2 Answers2

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In your particular case this makes little sense for an analog computer, as analog computers are at their best when dealing with time-dependent functions. It only makes sense as an educational problem. In analog computers time represents itself, everything else must be explicitly "coded."

You have basically two options:

  • Convert your space-based problem into a time-based problem, and start/stop the integration at the proper time points.

Note that any other function that depends on time would have problems with this transformation, so it might not be applicable to your example.

  • Discretize your space-based problem into an array of equations that are still applicable in the time-domain.

The second approach is a common one, and is just the application of Newton-Rampson style methodologies to the space domain.

The problem with this second approach is that it can lead to unstable analog implementations, as long-chains of feedbacks are sure to introduce dependencies on second-order analog effects (e.g., higher-frequency poles).

Analog computers are at their best when you let physics represent themselves, if you have to rely on discretization or other types of discontinuous approximations you are likely to run into problems.

Edgar Brown
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How can I perform the integration of f(x) on the finite bounds a to b on a (virtual) analog computer?

This is, strictly speaking, impossible since any virtual analog computer is implemented on a digital computer and is therefore in discrete time. HOWEVER you can simply use the integration block available in Simulink, which should approximate the analog behavior you are studying to an acceptable degree of precision. If you need a truly analog result then you need to actually use a real analog computer or calculate the response yourself using continuous time transfer functions.

user135797
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