Introduction to analog computing
Introduction to analog computing
The simulation and models
Since it began studying the phenomena and to design systems based on a scientific have always used the models to predict before the realization of the result of an experiment or the behavior of a system.
A template is nothing more than a system that evolves into the system similar way we want to study, and that we can control more easily compared to the real system.
The technical study of a system / experiment by observing the patterns of behavior is called Simulation
Models can be classified according to the nature of the model:
 physical models which in turn are graded in
 Scale models the nature of which is exactly that of the model to be studied but are more suited to the size of the study purpose, a plastic for modeling a building, a pilot plant for modeling a large chemical plant etc. This type of model has the advantage of being able to immediately perform simulation, but has the disadvantage linked to scale phenomena, that is, those phenomena that occur in a manner not proportional to the size.
 analog models are physical models based on physical phenomena other than those who want to study but we can identify variables that evolve in parallel to those of the system to be studied. It remains the difficulty of characterizing the similarity or analogy between the original system and the system model that is, the correspondence between the variables of the two systems and the choice of model parameters.
 mathematical models ie constituted by a set of equations / inequations that analytically describe the evolution of the variables of the system studied, Also these are classified based on how they are used
 analytical Models They are used by the direct and exact solution of the equations, providing direct result of the system, or they are used to establish a correspondence between models belonging to different physical domains and establish analogy use to build analog models.
 numerical models which include the calculation mode to generate a numerical solution (usually approximated) the analytical model to run on a digital computer the numerical model starts from the analytical (usually, even if there are alternative approaches) and it includes the development of a solution algorithm that translates into a calculation program
The Simulation
As already mentioned consists in the experimentation carried out on models of the system and repeated by varying the conditions that influence the behavior and the possible applied input that interest in the practical application of the system. In analog simulation such experiments are made on a similar model in the numerical simulation by implementing the solution algorithm.
The analog computer
The analog computer is any system that brings together components, that appropriately configured enabling to build a analog model a large number of systems, because this is easier using electronic components, almost all of the analog computer is based on these technologies. Since then the majority of the models that it is interesting to simulate models are described (mathematical model) by differential equations, an analog computer is (or she was in the early days of their introduction) also named differential analyzer. To better understand the use of the models and the analog simulation we design a simple physical example.
Model of the suspension of a motor vehicle (1/4 of a motor vehicle)
1 Mechanical suspension Model
In the physical model represented on z_{r} k represents the vertical profile of the road_{t} the elastic constant of the tire, k_{s} that of c suspension_{s} is the coefficient of friction damper m_{s} the pot of (a quarter) m of the vehicle_{in} the mass of unsprung parts relating to the wheel z_{s} e x_{in} are the vertical displacements of the body and the wheel, respectively,. The second law is written for the wheel (mass times acceleration equals the sum of the forces acting)
Where v_{s }, v_{in }, a_{s }, a_{in } They are velocity and acceleration of the vertical displacements of the body respectively and the wheel. Recall that the speed and the acceleration are respectively the derivatives with respect to time of the displacement and the speed then the two equations written constitute a system of equations differential
Where it used the classical notation of physics to indicate the dot the derivative with respect to time of a variable, and so on for the successive derivatives.
, etc
The equation that we wrote is a system of two differential equations of second order with two unknowns whose solution can be calculated by knowing the value of the parameters, the trend of the function forcing with_{r }(t) the initial conditions with_{s}(0), v_{s}(0), with_{in}(0), v_{in}(0)
The system is a linear system as appearing in the equations only the derivation operations, multiplication by a constant and the algebraic sum, such a system can be adequately simulated on an analog computer that has supplements, adders, inverters and multipliers by a constant.
Definition of an "analogue program"
As seen in the excellent article by Amilcare supplements, adders and inverters can be made with operational with feedback and the multipliers by a constant, variable (excuse the pun) by potentiometers. In fact, in order to avoid too many problems and the operational calibrations are mounted fixed gain (classically 1,1,10) and if by examples must multiply by 3,75 it is done by means of an amplifier for 10 and a divider realized with a potentiometer, which lowers the output for multiplying 0.375, on this page from a manual of EAI (excuse the quality) the correspondence between circuits, symbols of the analog calculation and mathematical operations (for the moment only operations linear).
Write an analog program is equivalent to plot a graph in which the computer components are interconnected in a timely manner, all components of the inputs / outputs available are given on suitable jacks (classically outlets for bananas from 4mm) on the front panel of the computer (patch panel) by means of flexible wires with the abovementioned bananas (generally it is possible to overlay multiple sockets to create circuit nodes) it is possible to implement the circuit, and then the interconnect program.
Returning to our example we reflect that single supplements are available only, do we have to use two for an equation of the second order, the easiest way is to rewrite the model as a system of differential equations of the first order system, we use the technique to introduce the state variables:
We define x_{1}= z_{s}, x_{2}= v_{s}, x_{3}= z_{in}, x_{4}= v_{in}, the equations become:
We define the coefficients
the program will be shown in the figure where the symbols used are a little different
2 Symbols “Analog Computer Design Tool” per LTSPICE
These symbols are used in an attractive package written by Arthur Glazar that runs under the application of Linear Technology LTSpice whose description can be found at this link https://www.edn.com/design/designtools/developmentkits/4376478/Analogcomputerdesigntool and which can be downloaded at this page https://www.edn.com/design/designtools/developmentkits/4376478/Analogcomputerdesigntool.
In practice this is a tool that allows numerical simulation of a analog program, I have used only employing it as a graphical tool for editing analogue programs, not to resort to the method used at the time of the "true" analog computation (pencil, bracket and stencil for symbols)
The differences between the package symbols and those standards are as follows
 Use of multipliers for both multiplication between signals that for the multiplication by a constant (this should be avoided in the real analog computation eg the obvious difference in cost and accuracy between a potentiometer and a multiplier circuit)
 The package integrators and adders are all noninverting as opposed to the standard convention providing for them invertenti and it is derived directly from the circuit embodiment.
 Adders have all input multipliers X1 (are not present in the inputs X10 e X100) in as much because of the point) there is no limitation, present in real calculators including coefficients with values in the range 01 (obtained with potentiometers).
 The presence of a shunt block that in an analog computer never mind due to the inherent instability and Feature noise amplification of such a circuit.
In any case, the pattern obtained by using the symbols of the toolbox is as follows and are obvious amendments to it by mapping it on a real analog computer. In the diagram the symbols A to E represent constants, the outputs of the integrators are the state variables x_{1}, x_{2}, x_{3}, x_{4}, and any supplement there is an additional input that is biased with initial value of the corresponding variable needed to solve the equation, l’input z_{r} It represents the road profile.
3 analog of the suspension Model
4 Symbols of analogue operators and realization circuit (Documentation E.A.I.)
On a true analog computer simulation it was made by identifying the number and type of components required cablandoli according to the scheme on the patch panel by setting initial conditions and parameters to the correct values by applying the wanted input via an appropriate function generator and observing the oscilloscope dedicated the trend of the variables of interest. It is easy to understand that, in the example that we are seeing we could be cyclically repeat the simulation by means of the adjusting knob of the potentiometers the coefficients under the control of the designer (for example, the elastic constant of the springs and the constant of the shock dissipation) until you get the answer that it was believed more comfortable.
Additional components for the analog calculation
Visas components so far are the basic components that are used to simulate linear systems There are also addons there can be used to introduce nonlinear phenomena in the simulation, an analog calculator was the most "powerful" how many more of these components and the more sophisticated onsite, Here we see a nonexhaustive list:
 Multipliers / Dividers between signals, We have already seen them but we only used to perform the multiplication by a constant around the circuit embodiments, for multiplication you can use different techniques
 Servomoltiplicatori using a servo potentiometer
 Multipliers based on the detection of the deflection imposed on the radius of an oscilloscope by the application of the two signals respectively to the plates for electrostatic deflection and to the yoke for the electromagnetic deflection.
 Circuits multipliers time division in which a signal is sampled in PWM whose duty cycle depends on the instantaneous value of the other signal at the output of a lowpass is obtained by a value proportional to the product
 Circuits parabolic output amplifiers that implement the formula
 Circuits logritmici / antilogaritmici that implement the formula
 Arbitrary function generators, the most common of them using a bank of adjustable resistors and diodes through which approximates the function with a broken, classically in 10 O 20 segments, the panel containing two of these generators is drawn in the figure

Other nonlinear blocks that implement
 Limitation or saturation
 dead zone
 Straightening
 Hysteresis
 Relay

sine and cosine functions
Other functions present were those that were used to generate the inputs to be applied to the system, (function generators x(t) which, however, they could be replaced by a sawtooth generator whose output was applied to a function generator and those necessary to observe the results, oscilloscopes both traditional electrodynamic big screen, both analog and digital voltmeters (remember that we are in the years 60 and a digital voltmeter is the most common tool) and analog plotter.
5 Panel of the function generator
Substitutes modern analog computer.
Although thanks to the availability of lowcost integrated circuits and highprecision one might now easily build an analog computer with performance comparable to that of the analog computer used by NASA for solving mathematical problems associated with the APOLLO program (In this regard recommend you see the wonderful film The Right of Count) now such an activity may be just a fun hobby as the use, the power and simplicity of digital computers made it easy to perform the simulation of complex differential systems with these latter instruments. The fact remains that the reasoning connected with the analog calculation is a still widespread way and especially clearer to understand the structure of the system, It is for example the classic way of problem formulation in the field of automatic control systems. The fact remains that the transition from the differential model to the algorithm of numerical solution is an intricate operation and altogether unnatural.
Fortunately they were born of programming environments that automatically perform this translation for which the programming is made in fact by plotting a graphic scheme that closely resembles an analog program. I will mention here a couple, a paid and a free as examples of the wide range of existing solutions.
Matlab Simulink
The product package from Mathworks is a very complex system (it is expensive) software to be used for teaching, Research and production, in all fields and not only, It is generally available to anyone who has access to University computing resources or other Institutes of Education (many high schools) for which there is a licensing mechanism at reduced prices, or for students who buy it in the Student Version includes MATLAB, Simulink, and a dozen other Toolbox for a symbolic price.
Without dwelling here the description of the software I limit myself to show the simplicity of the environment simulation (Simulink precisely) showing the outline of the visa program before.
It is easy to recognize the mpping the same equations taking into account that in Simulink the symbol 1 / s is the integrator, the triangle the multiplier by a constant and the bead the algebraic sum (according to the signs + or  present on input) Simulink exist in a large amount of "springs" which represent the ways of generating the input, in this case it is chosen a step function. And various "sinks" to send the output signals.
6 The simulation program in Simulink
The program is run by clicking the run and the virtual oscilloscope display button (It opens by clicking on it) you can see the result of the simulation. In the figure the result with different parameter values
Scilab Xcos
The second that I want to mention is a software (semi) Matlab clone developed and made freely available by INRIA researchers, French public national institute for research in computer science and automatic, the software you download from this site without any special procedures http://www.scilab.org/en where you will also find a series of tutorials and books that explain the nature and use of Scilab in various contexts, Analog instrument with respect to Simulink is called Xcos. There are several differences between the two software, Obviously the first is highly professional and comprehensive, however, for what concerns the simulation of simple linear systems they behave in a perfectly identical (despite some differences on the graphic representation of the blocks. This is why I do not get bored with the written program (or better designed) per Scilab.
Conclusions
For the moment that's all, I realize that I was very (maybe too much) synthetic, but I wanted to avoid the risk of sliding. However, if the thing you were to stimulate, let me know, I am willing to investigate any related subject, historical notes on the use and implementation of analog machines (in this regard are interesting mechanical devices to achieve analogue operators) or still examples of modeling and simulation of systems a bit more complex, perhaps containing nonlinear components, or more information and examples of numerical simulation software such as Simulink.
It goes without saying that if someone wants to build “his” analog computer using modern electronics My advice is always available, In this regard I propose interssanti link: https://www.grappendorf.net/projects/analogcomputer/ http://www.analogmuseum.org/library/Webster1965.pdf https://www.clear.rice.edu/elec301/Projects99/anlgcomp/ http://www.analogmuseum.org/library/PEAC_small.pdf
Really very, very, very interesting. I also liked the four link at the bottom, especially the last one that shows the functional blocks of a possible “product”. Obviously it should be updated (also because it uses transient germanium…).
Of course the knowhow “freshen up”/integration would not trivial (at least for me, accustomed as they are to think in… “digital”…).
Currently a “chore” He has completely reset my already scarce free time, and I had to put “pause” both the coupon with FPGA that the new version of “retrocomputer” with Z80 (https://hackaday.io/project/19000a44icsz80homemadecomputeronbreadboard).
Surely if and when I decide to embark on a realization of this genre will ask advice and tips.
For the moment I can only thank you for this “treat” you gave us.
It is always a pleasure to deepen their knowledge, if he says those who had direct knowledge of the argument it is even more enjoyable.
Thank you for making us partakers of your esperinze.