## Analogue Computers

Introduction

Electronic analogue computers come at the end of a chain of development that includes a wide variety of devices such as slide-rules, nomograms, and planimeters. In contrast to these simpler devices, they are more flexible, in that they are not designed for the solution of a single problem, but can be "programmed" to solve many different problems. Also, they include time as an important factor – i.e. they are designed for solving dynamic, rather than static, problems. Before looking at the electronic form, we should mention other methods that have been used.

Suppose we take a vertical cylinder with a small leak at the bottom, and fill it with water. The pressure across the leak will be proportional to the depth of the water, and the rate at which water flows through the leak will be proportional to the pressure. This means that the rate at which water leaks out of the cylinder is proportional to the amount of water present. This is precisely the same behaviour as the decay of a radionuclide (popularly referred to as a "radioactive isotope") – i.e. the rate of decay is proportional to the amount of the nuclide present at any instant. Therefore we can use a leaky cylinder as an "analogue" for the process of radioactive decay. We can do more than that: if the water that leaks out of the cylinder is caught in a second cylinder, then that represents the amount of the decay product accumulating over time. In addition, if this "daughter" nuclide is also radioactive, its decay can be represented by a leak in the second cylinder, and so on. Therefore we have a simple analogue computer for showing the time-variation of the different members of a radioactive decay series.

By using the volume of water to represent money, quite complicated analogues have been constructed for economic models.

Numerical values are represented by physical movement in the slide-rule, but research machines have been designed in which the rotation of a shaft is used. Mathematical functions can be represented by the profile of a cam. Several units can be coupled together with pulleys and belts, and quite complicated problems can be solved. If the linkage is done via cogwheels, then it can be difficult to decide whether we have an analogue or a digital computer.

The type of electronic analogue computer formerly used in research laboratories consisted of a wall-sized panel divided into a large number of rectangular modules, each having a number of sockets and lamps. The sockets were joined up with a tangle of different-coloured leads, and the results appeared as the flashing of the lamps and readings on voltmeters. In more recent machines, the layout of the circuit is much clearer, and very close to the way that the circuit is drawn out on paper. In the 1930s, a machine for doing calculations would have been called a calculator, while the term "computer" meant either a person who did calculations, or an analogue computer. In his stories about robots, Isaac Asimov recognised that they would be controlled by some sort of computer, and he gave them "brains" in the form of analogue computers involving the circulation of electrons and positrons in a sponge-like metallic structure. If instead he had described electrons and holes (i.e. "bubbles" in the sea of electrons) circulating in silicon or germanium, then he would have been even more prophetic. Analogue computers were still of practical importance in the 1960s, and many of the calculations behind the Apollo moon expeditions were carried out on this type of machine.

The decreasing cost of integrated circuits and the wider availability of digital computers have tended to overshadow analogue machines, in parallel with the way in which pocket calculators displaced slide-rules. The techniques underlying analogue computation, however, are still relevant, and we can learn about these by simulating an analogue computer on a digital machine. Analogue signal processing is of continuing interest in all branches of science and engineering. The programming technique known as Neural Networks has also a great similarity to the setting-up of an analogue computer. These notes will describe the properties of a "traditional" analogue computer, and will make additional comments relevant to a digital simulation, which has been constructed to go with this account, plus some problems that can be solved using the simulated machine.

The computer represents dynamically-varying quantities as potentials (voltages), often limited to a range such as –10 to +10 V. It contains components referred to as "passive" (potentiometers) and "active" (summers, integrators and multipliers).

Potentiometers

Each potentiometer has one end at earth potential, while the other end is where the input voltage is applied. The output is taken from a slider, and can vary continuously from zero up to the applied voltage. The simulated computer contains up to 120 potentiometers, each considered to be permanently connected to one of the input terminals of a summer or integrator (see below). The potentiometer values are set by inserting the appropriate factor in the cell provided. In a schematic diagram, a potentiometer is shown as a small circle with two leads, and is represented by the mathematical formula Vout = kVin, where k is the constant of the potentiometer. In a real analogue computer, the allowed range is 0 ≤ k ≤ 1, but in the simulation, any value (positive or negative) will be possible.