Analog computer

Encyclopedia

An

that uses the continuously-changeable aspects of physical phenomena such as electrical

, mechanical

, or hydraulic quantities to model

the problem being solved. In contrast, digital computers represent varying quantities incrementally, as their numerical values change.

Mechanical analog computers were very important in gun fire control in World War II and the Korean War; they were made in significant numbers. In particular, development of transistor

s made electronic analog computers practical, and before digital computers had developed sufficiently, they were commonly used in science and industry.

Analog computers can have a very wide range of complexity. Slide rule

s and nomographs are the simplest, while naval gunfire control computers and large hybrid digital/analog computers were among the most complicated.

Running an electronic analog computer, assuming a satisfactory setup, started with the computer held with some variables fixed at their initial values. Moving a switch released the holds and permitted the problem to run. In some instances, the computer could, after a certain running time interval, repeatedly return to the initial-conditions state to reset the problem, and run it again.

s and dashpot

s (viscous-fluid dampers), and electrical components, such as capacitor

s, inductor

s, and resistor

s is striking in terms of mathematics. They can be modeled using equations that are of essentially the same form.

However, the difference between these systems is what makes analog computing useful. If one considers a simple mass-spring system, constructing the physical system would require making or modifying the springs and masses. This would be followed by attaching them to each other and an appropriate anchor, collecting test equipment with the appropriate input range, and finally, taking measurements. In more complicated cases, such as suspensions for racing cars, experimental construction, modification, and testing is not so simple nor inexpensive.

The electrical equivalent can be constructed with a few operational amplifier

s (Op amps) and some passive linear components; all measurements can be taken directly with an oscilloscope

. In the circuit, the (simulated) 'stiffness of the spring', for instance, can be changed by adjusting a potentiometer

. The electrical system is an analogy to the physical system, hence the name, but it is less expensive to construct, generally safer, and typically much easier to modify.

As well, an electronic circuit can typically operate at higher frequencies than the system being simulated. This allows the simulation to run faster than real time (which could, in some instances, be hours, weeks, or longer). Experienced users of electronic analog computers said that they offered a comparatively intimate control and understanding of the problem, relative to digital simulations.

The drawback of the mechanical-electrical analogy is that electronics are limited by the range over which the variables may vary. This is called dynamic range

. They are also limited by noise levels. Floating-point digital calculations have comparatively-huge dynamic range (good modern handheld scientific/engineering calculators have exponents of 500).

These electric circuits can also easily perform a wide variety of simulations. For example, voltage

can simulate water pressure and electric current

can simulate rate of flow in terms of cubic metres per second (in fact, given the proper scale factors, all that is required would be a stable resistor, in that case). Given flow rate and accumulated volume of liquid, a simple integrator provides the latter; both variables are voltages. In practice, current was rarely used in electronic analog computers, because voltage is much easier to work with.

Analog computers are especially well-suited to representing situations described by differential equations. Occasionally, they were used when a differential equation proved very difficult to solve by traditional means.

An electronic digital system uses two voltage levels to represent binary numbers. In many cases, the binary numbers are simply codes that correspond, for instance, to brightness of primary colors, or letters of the alphabet (or other symbols). The manipulation of these binary numbers is how digital computers work. The electronic analog computer, however, manipulates electrical voltages that are proportional to the magnitudes of quantities in the problem being solved.

Accuracy of an analog computer is limited by its computing elements as well as quality of the internal power and electrical interconnections. The precision of the analog computer readout was limited chiefly by the precision of the readout equipment used, generally three or four significant figures. Precision of a digital computer is limited by the word size; arbitrary-precision arithmetic

, while relatively slow, provides any practical degree of precision that might be needed.

, in which an analog output is converted into digits

. The information then can be sent into a standard digital computer for further computation. Because of their ease of use and because of technological breakthroughs in digital computers in the early 70s, the analog-digital hybrids were replacing the analog-only systems.

Hybrid computers are used to obtain a very accurate

but not very mathematically precise

'seed' value, using an analog computer front-end, which value is then fed into a digital computer, using an iterative process to achieve the final desired degree of precision. With a three or four digit precision, highly-accurate numerical seed, the total computation time necessary to reach the desired precision is dramatically reduced, since many fewer digital iterations are required (and the analog computer reaches its result almost instantaneously). Or, for example, the analog computer might be used to solve a non-analytic differential equation problem for use at some stage of an overall computation (where precision is not very important). In any case, the hybrid computer is usually substantially faster than a digital computer, but can supply a far more precise computation than an analog computer. It is useful for real-time

applications requiring such a combination (e.g., a high frequency phased-array radar or a weather system computation).

Typical electronic analog computers contain anywhere from a few to a hundred or more operational amplifier

s ("op amps"), named because they perform mathematical operations. Op amps are a particular type of feedback amplifier with very high gain and stable input (low and stable offset). They are always used with precision feedback components that, in operation, all but cancel out the currents arriving from input components. The majority of op amps in a representative setup are summing amplifiers, which add and subtract analog voltages, providing the result at their output jacks. As well, op amps with capacitor feedback are usually included in a setup; they integrate the sum of their inputs with respect to time.

Integrating with respect to another variable is the nearly-exclusive province of mechanical analog integrators; it is almost never done in electronic analog computers. However, given that a problem solution does not change with time, time can serve as one of the variables.

Other computing elements include analog multipliers, nonlinear function generators, and analog comparators.

Inductors were never used in typical electronic analog computers, because their departure from ideal behavior is too great for computing of any great accuracy. Analog computer setups that at first would seem to require inductors can be rearranged and redefined to use capacitors. Capacitors and resistors, on the other hand, can be made much closer to ideal than inductors, which is why they constitute the majority of passive computing components.

The use of electrical properties in analog computers means that calculations are normally performed in real time

(or faster), at a speed determined mostly by the frequency response of the operational amplifiers and other computing elements. In the history of electronic analog computers, there were some special high-speed types.

Nonlinear functions and calculations can be constructed to a limited precision (three or four digits) by designing function generator

s — special circuits of various combinations of resistors and diodes to provide the nonlinearity. Typically, as the input voltage increases, progressively more diodes conduct.

When compensated for temperature, the forward voltage drop of a transistor's base-emitter junction can provide a usably-accurate logarithmic or exponential function. Op amps scale the output voltage so that it is usable with the rest of the computer.

Any physical process which models some computation can be interpreted as an analog computer. Some examples, invented for the purpose of illustrating the concept of analog computation, include using a bundle of spaghetti

as a model of

; a board, a set of nails, and a rubber band as a model of finding the

Most practical mechanical analog computers of any significant complexity used rotating shafts to carry variables from one mechanism to another. Cables and pulleys were used in a Fourier synthesizer, a tide-predicting machine

, which summed the individual harmonic components. Another category, not nearly as well known, used rotating shafts only for input and output, with precision racks and pinions. The racks were connected to linkages that performed the computation. At least one US Naval sonar fire control computer of the later 1950s, made by Librascope, was of this type, as was the principal computer in the Mk. 56 Gun Fire Control System.

Online, there is a remarkably-clear illustrated reference (OP 1140) that describes World War II mechanical analog fire control computer mechanisms. Lacking access to OP 1140, a text description of many important mechanisms follows.

For adding and subtracting, precision miter-gear differentials were in common use in some computers; the Ford Instrument Mark I Fire Control Computer

contained about 160 of them.

Integration with respect to another variable was done by a rotating disc driven by one variable. Output came from a pickoff device (such as a wheel) positioned at a radius on the disc proportional to the second variable. (A carrier with a pair of steel balls supported by small rollers worked especially well. A roller, its axis parallel to the disc's surface, provided the output. It was held against the pair of balls by a spring.)

Arbitrary functions of one variable were provided by cams, with gearing to convert follower movement to shaft rotation.

Functions of two variables were provided by three-dimensional cams. In one good design, one of the variables rotated the cam. A hemispherical follower moved its carrier on a pivot axis parallel to that of the cam's rotating axis. Pivoting motion was the output. The second variable moved the follower along the axis of the cam. One practical application was ballistics in gunnery.

Coordinate conversion from polar to rectangular was done by a mechanical resolver (called a "component solver" in US Navy fire control computers). Two discs on a common axis positioned a sliding block with pin (stubby shaft) on it. One disc was a face cam, and a follower on the block in the face cam's groove set the radius. The other disc, closer to the pin, contained a straight slot in which the block moved. The input angle rotated the latter disc (the face cam disc, for an unchanging radius, rotated with the other (angle) disc; a differential and a few gears did this correction).

Referring to the mechanism's frame, the location of the pin corresponded to the tip of the vector represented by the angle and magnitude inputs. Mounted on that pin was a square block.

Rectilinear-coordinate outputs (both sine and cosine, typically) came from two slotted plates, each slot fitting on the block just mentioned. The plates moved in straight lines, the movement of one plate at right angles to that of the other. The slots were at right angles to the direction of movement. Each plate, by itself, was like a Scotch yoke, known to steam engine enthusiasts.

During World War II, a similar mechanism converted rectilinear to polar coordinates, but it was not particularly successful and was eliminated in a significant redesign (USN, Mk. 1 to Mk. 1A).

Multiplication was done by mechanisms based on the geometry of similar right triangles. Using the trig. terms for a right triangle, specifically opposite, adjacent, and hypotenuse, the adjacent side was fixed by construction. One variable changed the magnitude of the opposite side. In many cases, this variable changed sign; the hypotenuse could coincide with the adjacent side (a zero input), or move beyond the adjacent side, representing a sign change.

Typically, a pinion-operated rack moving parallel to the (trig.-defined) opposite side would position a slide with a slot coincident with the hypotenuse. A pivot on the rack let the slide's angle change freely. At the other end of the slide (the angle, in trig, terms), a block on a pin fixed to the frame defined the vertex between the hypotenuse and the adjacent side.

At any distance along the adjacent side, a line perpendicular to it intersects the hypotenuse at a particular point. The distance between that point and the adjacent side is some fraction that is the product of

The second input variable in this type of multiplier positions a slotted plate perpendicular to the adjacent side. That slot contains a block, and that block's position in its slot is determined by another block right next to it. The latter slides along the hypotenuse, so the two blocks are positioned at a distance from the (trig.) adjacent side by an amount proportional to the product.

To provide the product as an output, a third element, another slotted plate, also moves parallel to the (trig.) opposite side of the theoretical triangle. As usual, the slot is perpendicular to the direction of movement. A block in its slot, pivoted to the hypotenuse block positions it.

A special type of integrator, used at a point where only moderate accuracy was needed, was based on a steel ball, instead of a disc. It had two inputs, one to rotate the ball, and the other to define the angle of the ball's rotating axis. That axis was always in a plane that contained the axes of two movement-pickoff rollers, quite similar to the mechanism of a rolling-ball computer mouse (in this mechanism, the pickoff rollers were roughly the same diameter as the ball). The pickoff roller axes were at right angles.

A pair of rollers "above" and "below" the pickoff plane were mounted in rotating holders that were geared together. That gearing was driven by the angle input, and established the rotating axis of the ball. The other input rotated the "bottom" roller to make the ball rotate.

Essentially, the whole mechanism, called a component integrator, was a variable-speed drive with one motion input and two outputs, as well as an angle input. The angle input varied the ratio (and direction) of coupling between the "motion" input and the outputs according to the sine and cosine of the input angle.

Although they were did not accomplish any computation, electromechanical position servos were essential in mechanical analog computers of the "rotating-shaft" type for providing operating torque to the inputs of subsequent computing mechanisms, as well as driving output data-transmission devices such as large torque-transmitter synchros in naval computers.

Other non-computational mechanisms included internal odometer-style counters with interpolating drum dials for indicating internal variables, and mechanical multi-turn limit stops.

Considering that accurately-controlled rotational speed in analog fire-control computers was a basic element of their accuracy, there was a motor with its average speed controlled by a balance wheel, hairspring, jeweled-bearing differential, a twin-lobe cam, and spring-loaded contacts (ship's AC power frequency was not necessarily accurate, nor dependable enough, when these computers were designed).

Key hydraulic components might include pipes, valves and containers.

Key mechanical components might include rotating shafts for carrying data within the computer, miter-gear differentials, disc/ball/roller integrators, cams (2-D and 3-D), mechanical resolvers and multipliers, and torque servos.

Key electrical/electronic components might include:

The core mathematical operations used in an electric analog computer are:

Differentiation with respect to time is not frequently used, and in practice is avoided by redefining the problem when possible. It corresponds in the frequency domain to a high-pass filter, which means that high-frequency noise is amplified; differentiation also risks instability.

is composed of four basic components: DC and AC magnitudes, frequency, and phase. The real limits of range on these characteristics limit analog computers. Some of these limits include the operational amplifier offset, finite gain, and frequency response, noise floor

, non-linearities, temperature coefficient

, and parasitic effects

within semiconductor devices. For commercially available electronic components, ranges of these aspects of input and output signals are always figures of merit.

. The American company Comdyna manufactures small analog computers. At Indiana University Bloomington, Jonathan Mills has developed the Extended Analog Computer based on sampling voltages in a foam sheet. At the Harvard Robotics Laboratory, analog computation is a research topic. Lyric Semiconductor's error correction circuits use analog probabilistic signals.

Analog (audio) synthesizer

s can also be viewed as a form of analog computer, and their technology was originally based in part on electronic analog computer technology. The ARP 2600's Ring Modulator was actually a moderate-accuracy analog multiplier.

The Simulation Council (or Simulations Council) was an association of analog computer users in USA. It is now known as the The Society of Modeling and Simulation International. The Simulation Council newsletters from 1952 to 1963 are available online and show the concerns and technologies at the time, and the common use of analog computers for missilry.

s (because they operate on the set of real number

s). Digital computers, by contrast, must first quantize

the signal into a finite number of values, and so can only work with the rational number

set (or, with an approximation of irrational numbers).

These idealized analog computers may

(caused by the quantum movements of ions). Ambient noise may be severely reduced – but never to zero – by using cryogenically

cooled parametric amplifiers

. Moreover, given unlimited time and memory, the (ideal) digital computer may also solve real number problems.

Other types of computers:

People associated with analog computer development:

Analog computation:

**analog computer**is a form of computerComputer

A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

that uses the continuously-changeable aspects of physical phenomena such as electrical

Electrical network

An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...

, mechanical

Mechanics

Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, or hydraulic quantities to model

Scientific modelling

Scientific modelling is the process of generating abstract, conceptual, graphical and/or mathematical models. Science offers a growing collection of methods, techniques and theory about all kinds of specialized scientific modelling...

the problem being solved. In contrast, digital computers represent varying quantities incrementally, as their numerical values change.

Mechanical analog computers were very important in gun fire control in World War II and the Korean War; they were made in significant numbers. In particular, development of transistor

Transistor

A transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...

s made electronic analog computers practical, and before digital computers had developed sufficiently, they were commonly used in science and industry.

Analog computers can have a very wide range of complexity. Slide rule

Slide rule

The slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.Slide rules come in a...

s and nomographs are the simplest, while naval gunfire control computers and large hybrid digital/analog computers were among the most complicated.

## Setup

Setting up an analog computer required scale factors to be chosen, along with initial conditions—that is, starting values. Another essential was creating the required network of interconnections between computing elements. Sometimes it was necessary to re-think the structure of the problem so that the computer would function satisfactorily. No variables could be allowed to exceed the computer's limits, and differentiation was to be avoided, typically by rearranging the "network" of interconnects, using integrators in a different sense.Running an electronic analog computer, assuming a satisfactory setup, started with the computer held with some variables fixed at their initial values. Moving a switch released the holds and permitted the problem to run. In some instances, the computer could, after a certain running time interval, repeatedly return to the initial-conditions state to reset the problem, and run it again.

### Precursors

This is a list of examples of early computation devices which are considered to be precursors of the modern computers. Some of them may even have been dubbed as 'computers' by the press, although they may fail to fit the modern definitions.- The Antikythera mechanismAntikythera mechanismThe Antikythera mechanism is an ancient mechanical computer designed to calculate astronomical positions. It was recovered in 1900–1901 from the Antikythera wreck. Its significance and complexity were not understood until decades later. Its time of construction is now estimated between 150 and 100...

is believed to be the earliest known mechanical analog "computer" by Derek J. de Solla PriceDerek J. de Solla PriceDerek John de Solla Price was a physicist, historian of science, and information scientist,credited as the father of scientometrics.-Biography:...

. It was designed to calculate astronomical positions. It was discovered in 1901 in the Antikythera wreckAntikythera wreckThe Antikythera wreck is a shipwreck from the 1st or 2nd century BC. It was discovered by sponge divers off Point Glyphadia on the Greek island Antikythera in the early 1900s...

off the Greek island of AntikytheraAntikytheraAntikythera or Anticythera is a Greek island lying on the edge of the Aegean Sea, between Crete and Peloponnese. Since the 2011 local government reform it is part of the municipality of Kythira island....

, between Kythera and CreteCreteCrete is the largest and most populous of the Greek islands, the fifth largest island in the Mediterranean Sea, and one of the thirteen administrative regions of Greece. It forms a significant part of the economy and cultural heritage of Greece while retaining its own local cultural traits...

, and has been dated to*circa*100 BC. Devices of a level of complexity comparable to that of the Antikythera mechanism would not reappear until a thousand years later. - The astrolabeAstrolabeAn astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...

was invented in the Hellenistic worldHellenistic civilizationHellenistic civilization represents the zenith of Greek influence in the ancient world from 323 BCE to about 146 BCE...

in either the 1st or 2nd centuries BC and is often attributed to HipparchusHipparchusHipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...

. A combination of the planispherePlanisphereA planisphere is a star chart analog computing instrument in the form of two adjustable disks that rotate on a common pivot. It can be adjusted to display the visible stars for any time and date. It is an instrument to assist in learning how to recognize stars and constellations...

and dioptraDioptraA dioptra is a classical astronomical and surveying instrument, dating from the 3rd century BCE. The dioptra was a sighting tube or, alternatively, a rod with a sight at both ends, attached to a stand...

, the astrolabe was effectively an analog computer capable of working out several different kinds of problems in spherical astronomySpherical astronomySpherical astronomy or positional astronomy is the branch of astronomy that is used to determine the location of objects on the celestial sphere, as seen at a particular date, time, and location on the Earth. It relies on the mathematical methods of spherical geometry and the measurements of...

. An astrolabe incorporating a mechanical calendarCalendarA calendar is a system of organizing days for social, religious, commercial, or administrative purposes. This is done by giving names to periods of time, typically days, weeks, months, and years. The name given to each day is known as a date. Periods in a calendar are usually, though not...

computer and gearGearA gear is a rotating machine part having cut teeth, or cogs, which mesh with another toothed part in order to transmit torque. Two or more gears working in tandem are called a transmission and can produce a mechanical advantage through a gear ratio and thus may be considered a simple machine....

-wheels was invented by Abi Bakr of Isfahan in 1235. - Abū Rayhān al-Bīrūnī invented the first mechanical geared lunisolar calendarLunisolar calendarA lunisolar calendar is a calendar in many cultures whose date indicates both the moon phase and the time of the solar year. If the solar year is defined as a tropical year then a lunisolar calendar will give an indication of the season; if it is taken as a sidereal year then the calendar will...

astrolabe, an early fixed-wireWireA wire is a single, usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads and to carry electricity and telecommunications signals. Wire is commonly formed by drawing the metal through a hole in a die or draw plate. Standard sizes are determined by various...

d knowledge processing machineMachineA machine manages power to accomplish a task, examples include, a mechanical system, a computing system, an electronic system, and a molecular machine. In common usage, the meaning is that of a device having parts that perform or assist in performing any type of work...

with a gear trainGear trainA gear train is formed by mounting gears on a frame so that the teeth of the gears engage. Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, this provides a smooth transmission of rotation from one gear to the next.The transmission of...

and gear-wheels,*circa*1000 AD. - The PlanispherePlanisphereA planisphere is a star chart analog computing instrument in the form of two adjustable disks that rotate on a common pivot. It can be adjusted to display the visible stars for any time and date. It is an instrument to assist in learning how to recognize stars and constellations...

was a star chartStar chartA star chart is a map of the night sky. Astronomers divide these into grids to use them more easily. They are used to identify and locate astronomical objects such as stars, constellations and galaxies. They have been used for human navigation since time immemorial...

astrolabe invented by Abū Rayhān al-Bīrūnī in the early 11th century. - The sectorSector (instrument)The sector, also known as a proportional compass or military compass, was a major calculating instrument in use from the end of the sixteenth century until the nineteenth century. It is an instrument consisting of two rulers of equal length which are joined by a hinge. A number of scales are...

, a calculating instrument used for solving problems in proportion, trigonometry, multiplication and division, and for various functions, such as squares and cube roots, was developed in the late 16th Century and found application in gunnery, surveying and navigation.

- The slide ruleSlide ruleThe slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.Slide rules come in a...

was invented around 1620–1630, shortly after the publication of the concept of the logarithmLogarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

. It is a hand-operated analog computer for doing multiplication and division. As slide rule development progressed, added scales provided reciprocals, squares and square roots, cubes and cube roots, as well as transcendental functionTranscendental functionA transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...

s such as logarithms and exponentials, circular and hyperbolic trigonometry and other functionsFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. - The tide-predicting machineTide-predicting machineA tide-predicting machine was a special-purpose mechanical analog computer of the late 19th and early 20th centuries, constructed and set up to predict the ebb and flow of sea tides and the irregular variations in their heights – which change in mixtures of rhythms, that never repeat...

conceived by Sir William ThomsonWilliam Thomson, 1st Baron KelvinWilliam Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...

was of great utility to navigation in shallow waters. - The differential analyserDifferential analyserThe differential analyser is a mechanical analogue computer designed to solve differential equations by integration, using wheel-and-disc mechanisms to perform the integration...

, a mechanical analog computer designed to solve differential equations by integration, using wheel-and-disc mechanisms to perform the integration. Invented in 1876 by James ThomsonJames Thomson (engineer)right|300px|James Thomson was an engineer and physicist whose reputation is substantial though it is overshadowed by that of his younger brother William Thomson .-Biography:Born in Belfast, he grew up mostly in Glasgow...

, they were first built in the 1920s and 1930s. Extensions and enhancements were the basis of some parts of mechanical analog gun fire control computers. - By 1912 Arthur PollenArthur PollenArthur Joseph Hungerford Pollen was a writer on naval affairs in the early 1900s who recognised the need for a computer-based fire-control system...

had developed an electrically driven mechanical analog computer for fire-control systemFire-control systemA fire-control system is a number of components working together, usually a gun data computer, a director, and radar, which is designed to assist a weapon system in hitting its target. It performs the same task as a human gunner firing a weapon, but attempts to do so faster and more...

s, based on the differential analyser. It was used by the Imperial Russian NavyImperial Russian NavyThe Imperial Russian Navy refers to the Tsarist fleets prior to the February Revolution.-First Romanovs:Under Tsar Mikhail Feodorovich, construction of the first three-masted ship, actually built within Russia, was completed in 1636. It was built in Balakhna by Danish shipbuilders from Holstein...

in World War IWorld War IWorld War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...

. - In 1929 the first AC transient network analyzerNetwork analyzer (electrical)A network analyzer is an instrument that measures the network parameters of electrical networks. Today, network analyzers commonly measure s–parameters because reflection and transmission of electrical networks are easy to measure at high frequencies, but there are other network parameter...

was built by General ElectricGeneral ElectricGeneral Electric Company , or GE, is an American multinational conglomerate corporation incorporated in Schenectady, New York and headquartered in Fairfield, Connecticut, United States...

at MIT. It was used for solving problems in electric power transmission that were too large to solve with numerical methods at the time.

### Modern era

- World War IIWorld War IIWorld War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...

era gun directorsDirector (military)A director, also called an auxiliary predictor, is a mechanical or electronic computer that continuously calculates trigonometric firing solutions for use against a moving target, and transmits targeting data to direct the weapon firing crew....

, gun data computerGun Data ComputerThe gun data computer is a series of artillery computers used by the U.S. Army, for coastal artillery, field artillery, and antiaircraft artillery applications...

s, and bomb sights used mechanical analog computers. - The FERMIACFERMIACThe Monte Carlo trolley, or FERMIAC, was an analog computer invented by physicist Enrico Fermi to aid in his studies of neutron transport.- Operation :...

was an analog computer invented by physicist Enrico Fermi in 1947 to aid in his studies of neutron transport. - Project Cyclone was an analog computer developed by Reeves in 1950 for the analysis and design of dynamic systems.
- Project Typhoon was an analog computer developed by RCA in 1952. It consisted of over 4000 electron tubes and used 100 dials and 6000 plug-in connectors to program.
- The MONIAC ComputerMONIAC ComputerThe MONIAC also known as the Phillips Hydraulic Computer and the Financephalograph, was created in 1949 by the New Zealand economist Bill Phillips to model the national economic processes of the United Kingdom, while Phillips was a student at the London School of Economics , The MONIAC was an...

was a hydraulic model of a national economy first unveiled in 1949. - Computer Engineering Associates was spun out of Caltech in 1950 to provide commercial services using the "Direct Analogy Electric Analog Computer" ("the largest and most impressive general-purpose analyzer facility for the solution of field problems") developed there by Gilbert D. McCann, Charles H. Wilts, and Bart LocanthiBart N. LocanthiBartholomew Nicholas Locanthi II was an audio engineer and leading expert in the US pro-audio industry in the 1970s and 1980s....

. - HeathkitHeathkitHeathkits were products of the Heath Company, Benton Harbor, Michigan. Their products included electronic test equipment, high fidelity home audio equipment, television receivers, amateur radio equipment, electronic ignition conversion modules for early model cars with point style ignitions, and...

EC-1, a $199 educational analog computer was made by the Heath Company, USA c. 1960. It was programmed using patch cords that connected nine operational amplifiers and other components - General Electric also marketed an "educational" analog computer kit of a simple design in the early 1960s consisting of a two transistor tone generator and three potentiometers wired such that the frequency of the oscillator was nulled when the potentiometer dials were positioned by hand to satisfy an equation. The relative resistance of the potentiometer was then equivalent to the formula of the equation being solved. Multiplication or division could be performed depending on which dials were considered inputs and which was the output. Accuracy and resolution was, of course, extremely limited and a simple slide rule was more accurate, however, the unit did demonstrate the basic principle.

## Electronic analog computers

The similarity between linear mechanical components, such as springSpring (device)

A spring is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication...

s and dashpot

Dashpot

A dashpot is a mechanical device, a damper which resists motion via viscous friction. The resulting force is proportional to the velocity, but acts in the opposite direction, slowing the motion and absorbing energy. It is commonly used in conjunction with a spring...

s (viscous-fluid dampers), and electrical components, such as capacitor

Capacitor

A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

s, inductor

Inductor

An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...

s, and resistor

Resistor

A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...

s is striking in terms of mathematics. They can be modeled using equations that are of essentially the same form.

However, the difference between these systems is what makes analog computing useful. If one considers a simple mass-spring system, constructing the physical system would require making or modifying the springs and masses. This would be followed by attaching them to each other and an appropriate anchor, collecting test equipment with the appropriate input range, and finally, taking measurements. In more complicated cases, such as suspensions for racing cars, experimental construction, modification, and testing is not so simple nor inexpensive.

The electrical equivalent can be constructed with a few operational amplifier

Operational amplifier

An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...

s (Op amps) and some passive linear components; all measurements can be taken directly with an oscilloscope

Oscilloscope

An oscilloscope is a type of electronic test instrument that allows observation of constantly varying signal voltages, usually as a two-dimensional graph of one or more electrical potential differences using the vertical or 'Y' axis, plotted as a function of time,...

. In the circuit, the (simulated) 'stiffness of the spring', for instance, can be changed by adjusting a potentiometer

Potentiometer

A potentiometer , informally, a pot, is a three-terminal resistor with a sliding contact that forms an adjustable voltage divider. If only two terminals are used , it acts as a variable resistor or rheostat. Potentiometers are commonly used to control electrical devices such as volume controls on...

. The electrical system is an analogy to the physical system, hence the name, but it is less expensive to construct, generally safer, and typically much easier to modify.

As well, an electronic circuit can typically operate at higher frequencies than the system being simulated. This allows the simulation to run faster than real time (which could, in some instances, be hours, weeks, or longer). Experienced users of electronic analog computers said that they offered a comparatively intimate control and understanding of the problem, relative to digital simulations.

The drawback of the mechanical-electrical analogy is that electronics are limited by the range over which the variables may vary. This is called dynamic range

Dynamic range

Dynamic range, abbreviated DR or DNR, is the ratio between the largest and smallest possible values of a changeable quantity, such as in sound and light. It is measured as a ratio, or as a base-10 or base-2 logarithmic value.-Dynamic range and human perception:The human senses of sight and...

. They are also limited by noise levels. Floating-point digital calculations have comparatively-huge dynamic range (good modern handheld scientific/engineering calculators have exponents of 500).

These electric circuits can also easily perform a wide variety of simulations. For example, voltage

Voltage

Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

can simulate water pressure and electric current

Electric current

Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

can simulate rate of flow in terms of cubic metres per second (in fact, given the proper scale factors, all that is required would be a stable resistor, in that case). Given flow rate and accumulated volume of liquid, a simple integrator provides the latter; both variables are voltages. In practice, current was rarely used in electronic analog computers, because voltage is much easier to work with.

Analog computers are especially well-suited to representing situations described by differential equations. Occasionally, they were used when a differential equation proved very difficult to solve by traditional means.

An electronic digital system uses two voltage levels to represent binary numbers. In many cases, the binary numbers are simply codes that correspond, for instance, to brightness of primary colors, or letters of the alphabet (or other symbols). The manipulation of these binary numbers is how digital computers work. The electronic analog computer, however, manipulates electrical voltages that are proportional to the magnitudes of quantities in the problem being solved.

Accuracy of an analog computer is limited by its computing elements as well as quality of the internal power and electrical interconnections. The precision of the analog computer readout was limited chiefly by the precision of the readout equipment used, generally three or four significant figures. Precision of a digital computer is limited by the word size; arbitrary-precision arithmetic

Arbitrary-precision arithmetic

In computer science, arbitrary-precision arithmetic indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most ALU hardware, which typically...

, while relatively slow, provides any practical degree of precision that might be needed.

## Analog-digital hybrid computers

There is an intermediate device, a 'hybrid' computerHybrid computer

Hybrid computers are computers that exhibit features of analog computers and digital computers. The digital component normally serves as the controller and provides logical operations, while the analog component normally serves as a solver of differential equations.In general, analog computers are...

, in which an analog output is converted into digits

Numerical digit

A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...

. The information then can be sent into a standard digital computer for further computation. Because of their ease of use and because of technological breakthroughs in digital computers in the early 70s, the analog-digital hybrids were replacing the analog-only systems.

Hybrid computers are used to obtain a very accurate

ACCURATE

ACCURATE was established in 2005 by a group of computer scientists, psychologists and policy experts to address problems with electronic voting...

but not very mathematically precise

Accuracy and precision

In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...

'seed' value, using an analog computer front-end, which value is then fed into a digital computer, using an iterative process to achieve the final desired degree of precision. With a three or four digit precision, highly-accurate numerical seed, the total computation time necessary to reach the desired precision is dramatically reduced, since many fewer digital iterations are required (and the analog computer reaches its result almost instantaneously). Or, for example, the analog computer might be used to solve a non-analytic differential equation problem for use at some stage of an overall computation (where precision is not very important). In any case, the hybrid computer is usually substantially faster than a digital computer, but can supply a far more precise computation than an analog computer. It is useful for real-time

Real-time computing

In computer science, real-time computing , or reactive computing, is the study of hardware and software systems that are subject to a "real-time constraint"— e.g. operational deadlines from event to system response. Real-time programs must guarantee response within strict time constraints...

applications requiring such a combination (e.g., a high frequency phased-array radar or a weather system computation).

## Mechanisms

Electronic analog computers typically have front panels with numerous jacks (single-contact sockets) that permit patch cords (flexible wires with plugs at both ends) to create the interconnections which define the problem setup. In addition, there are precision high-resolution potentiometers (variable resistors) for setting up (and, when needed, varying) scale factors. In addition, there is likely to be a zero-center analog pointer-type meter for modest-accuracy voltage measurement. Stable, accurate voltage sources provide known magnitudes.Typical electronic analog computers contain anywhere from a few to a hundred or more operational amplifier

Operational amplifier

An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...

s ("op amps"), named because they perform mathematical operations. Op amps are a particular type of feedback amplifier with very high gain and stable input (low and stable offset). They are always used with precision feedback components that, in operation, all but cancel out the currents arriving from input components. The majority of op amps in a representative setup are summing amplifiers, which add and subtract analog voltages, providing the result at their output jacks. As well, op amps with capacitor feedback are usually included in a setup; they integrate the sum of their inputs with respect to time.

Integrating with respect to another variable is the nearly-exclusive province of mechanical analog integrators; it is almost never done in electronic analog computers. However, given that a problem solution does not change with time, time can serve as one of the variables.

Other computing elements include analog multipliers, nonlinear function generators, and analog comparators.

Inductors were never used in typical electronic analog computers, because their departure from ideal behavior is too great for computing of any great accuracy. Analog computer setups that at first would seem to require inductors can be rearranged and redefined to use capacitors. Capacitors and resistors, on the other hand, can be made much closer to ideal than inductors, which is why they constitute the majority of passive computing components.

The use of electrical properties in analog computers means that calculations are normally performed in real time

Real-time computing

In computer science, real-time computing , or reactive computing, is the study of hardware and software systems that are subject to a "real-time constraint"— e.g. operational deadlines from event to system response. Real-time programs must guarantee response within strict time constraints...

(or faster), at a speed determined mostly by the frequency response of the operational amplifiers and other computing elements. In the history of electronic analog computers, there were some special high-speed types.

Nonlinear functions and calculations can be constructed to a limited precision (three or four digits) by designing function generator

Function generator

A function generator is a piece of electronic test equipment or software used to generate different types of electrical waveforms over a wide range of frequencies. These waveforms can be either repetitive or single-shot, in which case some kind of triggering source is required...

s — special circuits of various combinations of resistors and diodes to provide the nonlinearity. Typically, as the input voltage increases, progressively more diodes conduct.

When compensated for temperature, the forward voltage drop of a transistor's base-emitter junction can provide a usably-accurate logarithmic or exponential function. Op amps scale the output voltage so that it is usable with the rest of the computer.

Any physical process which models some computation can be interpreted as an analog computer. Some examples, invented for the purpose of illustrating the concept of analog computation, include using a bundle of spaghetti

Spaghetti

Spaghetti is a long, thin, cylindrical pasta of Italian origin. Spaghetti is made of semolina or flour and water. Italian dried spaghetti is made from durum wheat semolina, but outside of Italy it may be made with other kinds of flour...

as a model of

*sorting numbers*Spaghetti sort

Spaghetti sort is a linear-time, analog algorithm for sorting a sequence of items, introduced by Alexander Dewdney in his Scientific American column.-Algorithm:...

; a board, a set of nails, and a rubber band as a model of finding the

*convex hull*

of a set of points; and strings tied together as a model ofConvex hull

In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....

of a set of points

*finding the shortest path in a network*. These are all described*in*A.K. Dewdney (see citation below).### Mechanical analog computer mechanisms

While a wide variety of mechanisms have been developed throughout history, some stand out because of their theoretical importance, or because they were manufactured in significant quantities.Most practical mechanical analog computers of any significant complexity used rotating shafts to carry variables from one mechanism to another. Cables and pulleys were used in a Fourier synthesizer, a tide-predicting machine

Tide-predicting machine

A tide-predicting machine was a special-purpose mechanical analog computer of the late 19th and early 20th centuries, constructed and set up to predict the ebb and flow of sea tides and the irregular variations in their heights – which change in mixtures of rhythms, that never repeat...

, which summed the individual harmonic components. Another category, not nearly as well known, used rotating shafts only for input and output, with precision racks and pinions. The racks were connected to linkages that performed the computation. At least one US Naval sonar fire control computer of the later 1950s, made by Librascope, was of this type, as was the principal computer in the Mk. 56 Gun Fire Control System.

Online, there is a remarkably-clear illustrated reference (OP 1140) that describes World War II mechanical analog fire control computer mechanisms. Lacking access to OP 1140, a text description of many important mechanisms follows.

For adding and subtracting, precision miter-gear differentials were in common use in some computers; the Ford Instrument Mark I Fire Control Computer

Mark I Fire Control Computer

The Mark 1, and later the Mark 1A, Fire Control Computer was a component of the Mark 37 Gun Fire Control System deployed by the United States Navy during World War II and up to 1969. It was used on a variety of ships, ranging from destroyers to battleships . The Mark 37 system used tachymetric...

contained about 160 of them.

Integration with respect to another variable was done by a rotating disc driven by one variable. Output came from a pickoff device (such as a wheel) positioned at a radius on the disc proportional to the second variable. (A carrier with a pair of steel balls supported by small rollers worked especially well. A roller, its axis parallel to the disc's surface, provided the output. It was held against the pair of balls by a spring.)

Arbitrary functions of one variable were provided by cams, with gearing to convert follower movement to shaft rotation.

Functions of two variables were provided by three-dimensional cams. In one good design, one of the variables rotated the cam. A hemispherical follower moved its carrier on a pivot axis parallel to that of the cam's rotating axis. Pivoting motion was the output. The second variable moved the follower along the axis of the cam. One practical application was ballistics in gunnery.

Coordinate conversion from polar to rectangular was done by a mechanical resolver (called a "component solver" in US Navy fire control computers). Two discs on a common axis positioned a sliding block with pin (stubby shaft) on it. One disc was a face cam, and a follower on the block in the face cam's groove set the radius. The other disc, closer to the pin, contained a straight slot in which the block moved. The input angle rotated the latter disc (the face cam disc, for an unchanging radius, rotated with the other (angle) disc; a differential and a few gears did this correction).

Referring to the mechanism's frame, the location of the pin corresponded to the tip of the vector represented by the angle and magnitude inputs. Mounted on that pin was a square block.

Rectilinear-coordinate outputs (both sine and cosine, typically) came from two slotted plates, each slot fitting on the block just mentioned. The plates moved in straight lines, the movement of one plate at right angles to that of the other. The slots were at right angles to the direction of movement. Each plate, by itself, was like a Scotch yoke, known to steam engine enthusiasts.

During World War II, a similar mechanism converted rectilinear to polar coordinates, but it was not particularly successful and was eliminated in a significant redesign (USN, Mk. 1 to Mk. 1A).

Multiplication was done by mechanisms based on the geometry of similar right triangles. Using the trig. terms for a right triangle, specifically opposite, adjacent, and hypotenuse, the adjacent side was fixed by construction. One variable changed the magnitude of the opposite side. In many cases, this variable changed sign; the hypotenuse could coincide with the adjacent side (a zero input), or move beyond the adjacent side, representing a sign change.

Typically, a pinion-operated rack moving parallel to the (trig.-defined) opposite side would position a slide with a slot coincident with the hypotenuse. A pivot on the rack let the slide's angle change freely. At the other end of the slide (the angle, in trig, terms), a block on a pin fixed to the frame defined the vertex between the hypotenuse and the adjacent side.

At any distance along the adjacent side, a line perpendicular to it intersects the hypotenuse at a particular point. The distance between that point and the adjacent side is some fraction that is the product of

*1*the distance from the vertex, and*2*the magnitude of the opposite side.The second input variable in this type of multiplier positions a slotted plate perpendicular to the adjacent side. That slot contains a block, and that block's position in its slot is determined by another block right next to it. The latter slides along the hypotenuse, so the two blocks are positioned at a distance from the (trig.) adjacent side by an amount proportional to the product.

To provide the product as an output, a third element, another slotted plate, also moves parallel to the (trig.) opposite side of the theoretical triangle. As usual, the slot is perpendicular to the direction of movement. A block in its slot, pivoted to the hypotenuse block positions it.

A special type of integrator, used at a point where only moderate accuracy was needed, was based on a steel ball, instead of a disc. It had two inputs, one to rotate the ball, and the other to define the angle of the ball's rotating axis. That axis was always in a plane that contained the axes of two movement-pickoff rollers, quite similar to the mechanism of a rolling-ball computer mouse (in this mechanism, the pickoff rollers were roughly the same diameter as the ball). The pickoff roller axes were at right angles.

A pair of rollers "above" and "below" the pickoff plane were mounted in rotating holders that were geared together. That gearing was driven by the angle input, and established the rotating axis of the ball. The other input rotated the "bottom" roller to make the ball rotate.

Essentially, the whole mechanism, called a component integrator, was a variable-speed drive with one motion input and two outputs, as well as an angle input. The angle input varied the ratio (and direction) of coupling between the "motion" input and the outputs according to the sine and cosine of the input angle.

Although they were did not accomplish any computation, electromechanical position servos were essential in mechanical analog computers of the "rotating-shaft" type for providing operating torque to the inputs of subsequent computing mechanisms, as well as driving output data-transmission devices such as large torque-transmitter synchros in naval computers.

Other non-computational mechanisms included internal odometer-style counters with interpolating drum dials for indicating internal variables, and mechanical multi-turn limit stops.

Considering that accurately-controlled rotational speed in analog fire-control computers was a basic element of their accuracy, there was a motor with its average speed controlled by a balance wheel, hairspring, jeweled-bearing differential, a twin-lobe cam, and spring-loaded contacts (ship's AC power frequency was not necessarily accurate, nor dependable enough, when these computers were designed).

## Components

Analog computers often have a complicated framework, but they have, at their core, a set of key components which perform the calculations, which the operator manipulates through the computer's framework.Key hydraulic components might include pipes, valves and containers.

Key mechanical components might include rotating shafts for carrying data within the computer, miter-gear differentials, disc/ball/roller integrators, cams (2-D and 3-D), mechanical resolvers and multipliers, and torque servos.

Key electrical/electronic components might include:

- Precision resistors and capacitors
- operational amplifierOperational amplifierAn operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...

s - MultipliersAnalog multiplierIn electronics, an analog multiplier is a device which takes two analog signals and produces an output which is their product. Such circuits can be used to implement related functions such as squares , and square roots....
- potentiometerPotentiometerA potentiometer , informally, a pot, is a three-terminal resistor with a sliding contact that forms an adjustable voltage divider. If only two terminals are used , it acts as a variable resistor or rheostat. Potentiometers are commonly used to control electrical devices such as volume controls on...

s - fixed-function generatorFunction generatorA function generator is a piece of electronic test equipment or software used to generate different types of electrical waveforms over a wide range of frequencies. These waveforms can be either repetitive or single-shot, in which case some kind of triggering source is required...

s

The core mathematical operations used in an electric analog computer are:

- summationSummationSummation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,...
- integrationIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

with respect to time - inversionAdditive inverseIn mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....
- multiplicationMultiplicationMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
- exponentiationExponentiationExponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
- logarithmLogarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
- divisionDivision (mathematics)right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

, although multiplication is much preferred. Can be accomplished by placing the multiplier in the feedback path of an Operational Amplifier.

Differentiation with respect to time is not frequently used, and in practice is avoided by redefining the problem when possible. It corresponds in the frequency domain to a high-pass filter, which means that high-frequency noise is amplified; differentiation also risks instability.

## Limitations

In general, analog computers are limited by non-ideal effects. An analog signalAnalog signal

An analog or analogue signal is any continuous signal for which the time varying feature of the signal is a representation of some other time varying quantity, i.e., analogous to another time varying signal. It differs from a digital signal in terms of small fluctuations in the signal which are...

is composed of four basic components: DC and AC magnitudes, frequency, and phase. The real limits of range on these characteristics limit analog computers. Some of these limits include the operational amplifier offset, finite gain, and frequency response, noise floor

Noise floor

In signal theory, the noise floor is the measure of the signal created from the sum of all the noise sources and unwanted signals within a measurement system, where the noise is defined as any signal other than the one being monitored....

, non-linearities, temperature coefficient

Temperature coefficient

The temperature coefficient is the relative change of a physical property when the temperature is changed by 1 K.In the following formula, let R be the physical property to be measured and T be the temperature at which the property is measured. T0 is the reference temperature, and ΔT is the...

, and parasitic effects

Microelectronics

Microelectronics is a subfield of electronics. As the name suggests, microelectronics relates to the study and manufacture of very small electronic components. Usually, but not always, this means micrometre-scale or smaller,. These devices are made from semiconductors...

within semiconductor devices. For commercially available electronic components, ranges of these aspects of input and output signals are always figures of merit.

## Current research

Although digital computation is extremely popular, some research in analog computation is still being done. A few universities still use analog computers to teach control system theoryControl theory

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

. The American company Comdyna manufactures small analog computers. At Indiana University Bloomington, Jonathan Mills has developed the Extended Analog Computer based on sampling voltages in a foam sheet. At the Harvard Robotics Laboratory, analog computation is a research topic. Lyric Semiconductor's error correction circuits use analog probabilistic signals.

## Practical examples

These are examples of analog computers that have been constructed or practically used:- Antikythera mechanismAntikythera mechanismThe Antikythera mechanism is an ancient mechanical computer designed to calculate astronomical positions. It was recovered in 1900–1901 from the Antikythera wreck. Its significance and complexity were not understood until decades later. Its time of construction is now estimated between 150 and 100...
- astrolabeAstrolabeAn astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...
- differential analyzer
- DeltarDeltarThe Deltar was an analog computer, used from 1960 until 1984 in the design and implementation of the Delta Works....
- Kerrison PredictorKerrison PredictorThe Kerrison Predictor was one of the first fully automated anti-aircraft fire-control systems. The predictor could aim a gun at an aircraft based on simple inputs like the observed speed and the angle to the target...
- LibrascopeLibrascopeLibrascope was a Glendale, California division of General Precision Inc. founded in 1937 by Lewis W. Imm to improve aircraft load balancing, and acquired by General Precision in 1941....

, aircraft weight and balance computer - mechanical integrators, for example, the planimeterPlanimeterA planimeter is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.-Construction:There are several kinds of planimeters, but all operate in a similar way. The precise way in which they are constructed varies, with the main types of mechanical planimeter being...
- MONIAC ComputerMONIAC ComputerThe MONIAC also known as the Phillips Hydraulic Computer and the Financephalograph, was created in 1949 by the New Zealand economist Bill Phillips to model the national economic processes of the United Kingdom, while Phillips was a student at the London School of Economics , The MONIAC was an...

(hydraulic model of UK economy) - nomogramNomogramA nomogram, nomograph, or abac is a graphical calculating device developed by P.E. Elyasberg, a two-dimensional diagram designed to allow the approximate graphical computation of a function: it uses a coordinate system other than Cartesian coordinates...
- Norden bombsightNorden bombsightThe Norden bombsight was a tachometric bombsight used by the United States Army Air Forces and the United States Navy during World War II, and the United States Air Force in the Korean and the Vietnam Wars to aid the crew of bomber aircraft in dropping bombs accurately...
- RangekeeperRangekeeperRangekeepers were electromechanical fire control computers used primarily during the early part of the 20th century. They were sophisticated analog computers whose development reached its zenith following World War II, specifically the Computer Mk 47 in the Mk 68 Gun Fire Control system. During...

and related fire control computers - ScanimateScanimateScanimate is the name for an analog computer animation system developed from the late 1960s to the early 1980s.The Scanimate systems were used to produce much of the video-based animation seen on television between most of the 1970s and early 1980s in commercials, promotions, and show openings...
- slide ruleSlide ruleThe slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.Slide rules come in a...
- tide-predicting machineTide-predicting machineA tide-predicting machine was a special-purpose mechanical analog computer of the late 19th and early 20th centuries, constructed and set up to predict the ebb and flow of sea tides and the irregular variations in their heights – which change in mixtures of rhythms, that never repeat...
- Torpedo Data ComputerTorpedo Data ComputerThe Torpedo Data Computer was an early electromechanical analog computer used for torpedo fire-control on American submarines during World War II . Britain, Germany, and Japan also developed automated torpedo fire control equipment, but none were as advanced as US Navy's TDC...
- Leonardo Torres y QuevedoLeonardo Torres y QuevedoLeonardo Torres y Quevedo was a Spanish civil engineer and mathematician of the late nineteenth and early twentieth centuries.- Biography :Torres was born on 28 December 1852, on the Feast of the Holy Innocents, in Santa Cruz de Iguña, Molledo , Spain...

's Analogue Calculating Machines based on "fusee sans fin" - TorquetumTorquetumThe torquetum or turquet is a medieval astronomical instrument designed to take and convert measurements made in three sets of coordinates: Horizon, equatorial, and ecliptic...
- Water integratorWater integratorThe Water Integrator was an early analog computer built in the Soviet Union in 1928. It functioned by careful manipulation of water through a room full of interconnected pipes and pumps. The water level in various chambers represented stored numbers, and the rate of flow between them represented...
- Mechanical computerMechanical computerA mechanical computer is built from mechanical components such as levers and gears, rather than electronic components. The most common examples are adding machines and mechanical counters, which use the turning of gears to increment output displays...

Analog (audio) synthesizer

Analog synthesizer

An analog or analogue synthesizer is a synthesizer that uses analog circuits and analog computer techniques to generate sound electronically. The earliest analog synthesizers in the 1920s and 1930s such as the Trautonium were built with a variety of vacuum-tube and electro-mechanical technologies...

s can also be viewed as a form of analog computer, and their technology was originally based in part on electronic analog computer technology. The ARP 2600's Ring Modulator was actually a moderate-accuracy analog multiplier.

The Simulation Council (or Simulations Council) was an association of analog computer users in USA. It is now known as the The Society of Modeling and Simulation International. The Simulation Council newsletters from 1952 to 1963 are available online and show the concerns and technologies at the time, and the common use of analog computers for missilry.

## Real computers

Computer theorists often refer to idealized analog computers as real computerReal computation

In computability theory, the theory of real computation deals with hypothetical computing machines using infinite-precision real numbers. They are given this name because they operate on the set of real numbers...

s (because they operate on the set of real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s). Digital computers, by contrast, must first quantize

Quantization (signal processing)

Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The error introduced by...

the signal into a finite number of values, and so can only work with the rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

set (or, with an approximation of irrational numbers).

These idealized analog computers may

*in theory*solve problems that are intractable on digital computers; however as mentioned, in reality, analog computers are far from attaining this ideal, largely because of noise minimization problems.*In theory*, ambient noise is limited by quantum noiseQuantum noise

Quantum noise is uncertainty of some physical quantity due to its quantum origin.In the case of number of particles , the quantum noise is also called shot noise. Most optical communications use amplitude modulation...

(caused by the quantum movements of ions). Ambient noise may be severely reduced – but never to zero – by using cryogenically

Cryogenics

In physics, cryogenics is the study of the production of very low temperature and the behavior of materials at those temperatures. A person who studies elements under extremely cold temperature is called a cryogenicist. Rather than the relative temperature scales of Celsius and Fahrenheit,...

cooled parametric amplifiers

Parametric oscillator

A parametric oscillator is a harmonic oscillator whose parameters oscillate in time. For example, a well known parametric oscillator is a child pumping a swing by periodically standing and squatting to increase the size of the swing's oscillations. The varying of the parameters drives the system...

. Moreover, given unlimited time and memory, the (ideal) digital computer may also solve real number problems.

## See also

- Signal (electrical engineering)Signal (electrical engineering)In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity....
- Signal (computing)Signal (computing)A signal is a limited form of inter-process communication used in Unix, Unix-like, and other POSIX-compliant operating systems. Essentially it is an asynchronous notification sent to a process in order to notify it of an event that occurred. When a signal is sent to a process, the operating system...
- Differential equationDifferential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
- Dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
- Chaos theoryChaos theoryChaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
- Slide ruleSlide rule
- Analogical modelsAnalogical modelsAnalogical models are a method of representing a phenomenon of the world, often called the ‘target system’ by another, more understandable or analysable system. They are also called dynamical analogies.- Explanation :...
- Antikythera mechanismAntikythera mechanismThe Antikythera mechanism is an ancient mechanical computer designed to calculate astronomical positions. It was recovered in 1900–1901 from the Antikythera wreck. Its significance and complexity were not understood until decades later. Its time of construction is now estimated between 150 and 100...
- Field-programmable analog array

Other types of computers:

- DNA computer
- Molecular computer
- Quantum computerQuantum computerA quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from traditional computers based on transistors...
- Wetware computerWetware computerA wetware computer is an organic computer built from living neurons. , at the Georgia Institute of Technology, is the primary researcher driving the creation of these artificially constructed, but still organic brains...
- Digital computer

People associated with analog computer development:

- George A. PhilbrickGeorge A. PhilbrickGeorge A. Philbrick was responsible, through his company George A. Philbrick Researches, for the commercialization and wide adoption of operational amplifiers, a now-ubiquitous component of analog electronic systems, and the invention and commercialization of electronic analog computers based on...
- Leonardo Torres y QuevedoLeonardo Torres y QuevedoLeonardo Torres y Quevedo was a Spanish civil engineer and mathematician of the late nineteenth and early twentieth centuries.- Biography :Torres was born on 28 December 1852, on the Feast of the Holy Innocents, in Santa Cruz de Iguña, Molledo , Spain...

Analog computation:

- Analog electronics
- Analog VLSI
- Analog signal processingAnalog signal processingAnalog signal processing is any signal processing conducted on analog signals by analog means. "Analog" indicates something that is mathematically represented as a set of continuous values. This differs from "digital" which uses a series of discrete quantities to represent signal...
- Neural networkNeural networkThe term neural network was traditionally used to refer to a network or circuit of biological neurons. The modern usage of the term often refers to artificial neural networks, which are composed of artificial neurons or nodes...
- Continuous complexity theory
- Continuous computability theory
- Continuous algorithm
- Continuous-time control theoryControl theoryControl theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

## External links

- Large collection of electronic analog computers with lots of pictures and documentation
- Simulation of a car suspension system with an electronic analog computer
- Introduction to Analog-/Hybrid-Computing (pdf)
- Example programs for Analog Computers (pdf)
- Large collection of old analog and digital computers at Old Computer Museum
- A great disappearing act: the electronic analogue computer Chris Bissell, The Open University, Milton Keynes, UK Accessed February 2007
- German computer museum with still runnable analog computers
- Analog computer basics
- Lecture 20: Analog vs Digital
*(in a series of lectures on "History of computing and information technology")* - Analog computer trumps Turing model

- Jonathan W. Mills's Analog Notebook
- Indiana University Extended Analog Computer
- Harvard Robotics Laboratory Analog Computation
- The Enns Power Network Computer – an analog computer for the analysis of electric power systems (advertisement from 1955)

- Librascope Development Company – Type LC-1 WWII Navy PV-1 "Balance Computor"