Analog computer

An analog computer or analogue computer is a type of computer 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 symbolically and by discrete values of both time and amplitude.

Analog computers can have a very wide range of complexity. Slide rules and nomograms are the simplest, while naval gunfire control computers and large hybrid digital/analog computers were among the most complicated.[1] Systems for process control and protective relays used analog computation to perform control and protective functions.

Analog computers were widely used in scientific and industrial applications even after the advent of digital computers, because at the time they were typically much faster, but they started to become obsolete as early as the 1950s and 1960s, although remained in use in some specific applications, such as aircraft flight simulators, the flight computer in aircraft, and for teaching control systems in universities. More complex applications, such as aircraft flight simulators and synthetic aperture radar, remained the domain of analog computing (and hybrid computing) well into the 1980s, since digital computers were insufficient for the task.[2]

A page from the Bombardier's Information File (BIF) that describes the components and controls of the Norden bombsight. The Norden bombsight was a highly sophisticated optical/mechanical analog computer used by the United States Army Air Force during World War II, the Korean War, and the Vietnam War to aid the pilot of a bomber aircraft in dropping bombs accurately.
PACE-TR-10 analog computer - National Cryptologic Museum - DSC07908
TR-10 desktop analog computer of the late 1960s and early 70s

Timeline of analog computers


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.

NAMA Machine d'Anticythère 1
The Antikythera mechanism, dating between 150 and 100 BC, was an early analog computer.

The Antikythera mechanism was an orrery and is believed to be an early mechanical analog computer, according to Derek J. de Solla Price.[3] It was designed to calculate astronomical positions. It was discovered in 1901 in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, and has been dated to circa 100 BC during the Hellenistic period of Greece. Devices of a level of complexity comparable to that of the Antikythera mechanism would not reappear until a thousand years later.

Many mechanical aids to calculation and measurement were constructed for astronomical and navigation use. The planisphere was first described by Ptolemy in the 2nd century AD. The astrolabe was invented in the Hellenistic world in either the 1st or 2nd centuries BC and is often attributed to Hipparchus. A combination of the planisphere and dioptra, the astrolabe was effectively an analog computer capable of working out several different kinds of problems in spherical astronomy. An astrolabe incorporating a mechanical calendar computer[4][5] and gear-wheels was invented by Abi Bakr of Isfahan, Persia in 1235.[6] Abū Rayhān al-Bīrūnī invented the first mechanical geared lunisolar calendar astrolabe,[7] an early fixed-wired knowledge processing machine[8] with a gear train and gear-wheels,[9] circa 1000 AD. The castle clock, a hydropowered mechanical astronomical clock invented by Al-Jazari in 1206, was the first programmable analog computer.[10][11][12]

The sector, 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 planimeter was a manual instrument to calculate the area of a closed figure by tracing over it with a mechanical linkage.

Sliderule 2005
A slide rule. The sliding central slip is set to 1.3, the cursor to 2.0 and points to the multiplied result of 2.6.

The slide rule was invented around 1620–1630, shortly after the publication of the concept of the logarithm. 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 functions such as logarithms and exponentials, circular and hyperbolic trigonometry and other functions. Aviation is one of the few fields where slide rules are still in widespread use, particularly for solving time–distance problems in light aircraft.

Mathematician and engineer Giovanni Plana devised a Perpetual Calendar machine which, though a system of pulleys and cylinders and over, could predict the perpetual calendar for every year from 0 AD to 4000 AD, keeping track of leap years and varying day length.

The tide-predicting machine invented by Sir William Thomson in 1872 was of great utility to navigation in shallow waters. It used a system of pulleys and wires to automatically calculate predicted tide levels for a set period at a particular location.

The differential analyser, a mechanical analog computer designed to solve differential equations by integration, used wheel-and-disc mechanisms to perform the integration. In 1876 James Thomson had already discussed the possible construction of such calculators, but he had been stymied by the limited output torque of the ball-and-disk integrators.[13] In a differential analyzer, the output of one integrator drove the input of the next integrator, or a graphing output. The torque amplifier was the advance that allowed these machines to work. Starting in the 1920s, Vannevar Bush and others developed mechanical differential analyzers.

Modern era

Analog Computing Machine GPN-2000-000354
Analog computing machine at the Lewis Flight Propulsion Laboratory circa 1949.
Heathkit Analog Computer
Heathkit EC-1 educational analog computer

The Dumaresq was a mechanical calculating device invented around 1902 by Lieutenant John Dumaresq of the Royal Navy. It was an analog computer which related vital variables of the fire control problem to the movement of one's own ship and that of a target ship. It was often used with other devices, such as a Vickers range clock to generate range and deflection data so the gun sights of the ship could be continuously set. A number of versions of the Dumaresq were produced of increasing complexity as development proceeded.

By 1912 Arthur Pollen had developed an electrically driven mechanical analog computer for fire-control systems, based on the differential analyser. It was used by the Imperial Russian Navy in World War I.

Starting in 1929, AC network analyzers were constructed to solve calculation problems related to electrical power systems that were too large to solve with numerical methods at the time.[14] These were essentially scale models of the electrical properties of the full-size system. Since network analyzers could handle problems too large for analytic methods or hand computation, they were also used to solve problems in nuclear physics and in the design of structures. More than 50 large network analyzers were built by the end of the 1950s.

World War II era gun directors, gun data computers, and bomb sights used mechanical analog computers. In 1942 Helmut Hölzer built a fully electronic analog computer at Peenemünde Army Research Center[15][16][17] as an embedded control system (mixing device) to calculate V-2 rocket trajectories from the accelerations and orientations (measured by gyroscopes) and to stabilize and guide the missile.[18][19] Mechanical analog computers were very important in gun fire control in World War II, The Korean War and well past the Vietnam War; they were made in significant numbers.

In the period 1930–1945 in the Netherlands Johan van Veen developed an analogue computer to calculate and predict tidal currents when the geometry of the channels are changed. Around 1950 this idea was developed into the Deltar, an analogue computer supporting the closure of estuaries in the southwest of the Netherlands (the Delta Works).

The FERMIAC was an analog computer invented by physicist Enrico Fermi in 1947 to aid in his studies of neutron transport.[20] Project Cyclone was an analog computer developed by Reeves in 1950 for the analysis and design of dynamic systems.[21] 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.[22] The MONIAC Computer was a hydraulic model of a national economy first unveiled in 1949.[23]

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 Locanthi.[24][25]

Educational analog computers illustrated the principles of analog calculation. The Heathkit EC-1, a $199 educational analog computer, was made by the Heath Company, US c. 1960.[26] It was programmed using patch cords that connected nine operational amplifiers and other components.[27] General Electric also marketed an "educational" analog computer kit of a simple design in the early 1960s consisting of a two transistor tone generators 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 limited and a simple slide rule was more accurate; however, the unit did demonstrate the basic principle.

Analog computer designs were published in electronics magazines. One example is the PE Analogue Computer, published in Practical Electronics in the September 1978 edition. Another more modern hybrid computer design was published in Everyday Practical Electronics in 2002.[28] An example described in the EPE Hybrid Computer was the flight of a VTOL aircraft like the Harrier jump jet.[28] The altitude and speed of the aircraft were calculated by the analog part of the computer and sent to a PC via a digital microprocessor, to be displayed on the PC screen.

In industrial process control, analog loop controllers were used to automatically regulate temperature, flow, pressure, or other process conditions. The technology of these controllers ranged from purely mechanical integrators, through vacuum-tube and solid-state devices, to emulation of analog controllers by microprocessors.

Electronic analog computers

Polish analog computer AKAT-1 (1959)
Analogrechner HW-in-Loop Ausschnitt
EAI 8800 Analog computing system used for hardware-in-the-loop simulation of a Claas tractor (1986)

The similarity between linear mechanical components, such as springs and dashpots (viscous-fluid dampers), and electrical components, such as capacitors, inductors, and resistors is striking in terms of mathematics. They can be modeled using equations of 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 both complicated and expensive.

The electrical equivalent can be constructed with a few operational amplifiers (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 the parameters of a capacitor. 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 due to the fixed supply voltage. Therefore each problem has to be scaled to its parameters and dimensions, e. g. the expected magnitudes of the velocity and the position of a spring pendulum. Improperly scaled problems might suffer from higher noise levels. Floating-point digital calculations have a huge dynamic range but might also suffer from imprecision if tiny differences of huge values lead to numerical instability.

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. An integrator can provide the total accumulated volume of liquid, using an input current proportional to the (possibly varying) flow rate.

Federpendel als Analogrechenschaltung
Analog circuit for the dynamics of a spring-mass system (without scaling factors)
Damped spring
Damped motion of a spring-mass system

Analog computers are especially well-suited to representing situations described by differential equations. Occasionally, they were used when a system of differential equations proved very difficult to solve by traditional means. As a simple example, the dynamics of a spring-mass system can be described by the equation , with as the vertical position of a mass , the damping coefficient, the spring constant and the gravity of Earth. For analog computing, the equation is to be programmed as . The equivalent analog circuit consists of two integrators for the state variables (speed) and (position), one inverter, and three potentiometers. The circuit has to consider that both integration and addition units invert the signal polarity.

The 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. The 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. However, in most cases the precision of an analog computer is absolutely sufficient given the uncertainty of the model characteristics and its technical parameters.

Many small computers dedicated to specific computations are still part of industrial regulation equipment, but from the 1950s to the 1970s, general-purpose analog computers were the only systems fast enough for real time simulation of dynamic systems, especially in the aircraft, military and aerospace field.

In the 1960s, the major manufacturer was Electronic Associates of Princeton, New Jersey, with its 231R Analog Computer (vacuum tubes, 20 integrators) and subsequently its EAI 8800 Analog Computer (solid state operational amplifiers, 64 integrators).[29] Its challenger was Applied Dynamics of Ann Arbor, Michigan.

Although the basic technology for analog computers is usually operational amplifiers (also called "continuous current amplifiers" because they have no low frequency limitation), in the 1960s an attempt was made in the French ANALAC computer to use an alternative technology: medium frequency carrier and non dissipative reversible circuits.

In the 1970s every big company and administration concerned with problems in dynamics had a big analog computing center, for example:

Analog–digital hybrids

Analog computing devices are fast, digital computing devices are more versatile and accurate, so the idea is to combine the two processes for the best efficiency. An example of such hybrid elementary device is the hybrid multiplier where one input is an analog signal, the other input is a digital signal and the output is analog. It acts as an analog potentiometer upgradable digitally. This kind of hybrid technique is mainly used for fast dedicated real time computation when computing time is very critical as signal processing for radars and generally for controllers in embedded systems.

In the early 1970s analog computer manufacturers tried to tie together their analog computer with a digital computer to get the advantages of the two techniques. In such systems, the digital computer controlled the analog computer, providing initial set-up, initiating multiple analog runs, and automatically feeding and collecting data. The digital computer may also participate to the calculation itself using analog-to-digital and digital-to-analog converters.

The largest manufacturer of hybrid computers was Electronics Associates. Their hybrid computer model 8900 was made of a digital computer and one or more analog consoles. These systems were mainly dedicated to large projects such as the Apollo program and Space Shuttle at NASA, or Ariane in Europe, especially during the integration step where at the beginning everything is simulated, and progressively real components replace their simulated part.[30]

Only one company was known as offering general commercial computing services on its hybrid computers, CISI of France, in the 1970s.

The best reference in this field is the 100,000 simulations runs for each certification of the automatic landing systems of Airbus and Concorde aircraft.[31]

After 1980, purely digital computers progressed more and more rapidly and were fast enough to compete with analog computers. One key to the speed of analog computers was their fully parallel computation, but this was also a limitation. The more equations required for a problem, the more analog components were needed, even when the problem wasn't time critical. "Programming" a problem meant interconnecting the analog operators; even with a removable wiring panel this was not very versatile. Today there are no more big hybrid computers, but only hybrid components.


Mechanical analog computers

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, 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[32] the fire control computer mechanisms. 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 trigonometric 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 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).

Electronic analog computers

Analogrechner Schaltbrett vorne
Switching board of EAI 8800 analog computer (front view)

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 amplifiers ("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.

Electrical elements such as inductors and capacitors used in electrical analog computers had to be carefully manufactured to reduce non-ideal effects. For example, in the construction of AC power network analyzers, one motive for using higher frequencies for the calculator (instead of the actual power frequency) was that higher-quality inductors could be more easily made. Many general-purpose analog computers avoided the use of inductors entirely, re-casting the problem in a form that could be solved using only resistive and capacitive elements, since high-quality capacitors are relatively easy to make.

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 generators—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 sorting numbers; 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 of finding the shortest path in a network. These are all described in Dewdney (1984).


A 1960 Newmark analogue computer, made up of five units. This computer was used to solve differential equations and is currently housed at the Cambridge Museum of Technology.

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:

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

In some analog computer designs, multiplication is much preferred to division. Division is carried out with a 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.


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


In the 1950s to 1970s, digital computers based on first vacuum tubes, transistors, integrated circuits and then micro-processors became more economical and precise. This led digital computers to largely replace analog computers. Even so, some research in analog computation is still being done. A few universities still use analog computers to teach control system theory. The American company Comdyna manufactured small analog computers.[33] At Indiana University Bloomington, Jonathan Mills has developed the Extended Analog Computer based on sampling voltages in a foam sheet.[34] At the Harvard Robotics Laboratory,[35] analog computation is a research topic. Lyric Semiconductor's error correction circuits use analog probabilistic signals. Slide rules are still popular among aircraft personnel.


With the development of very-large-scale integration (VLSI) technology, Yannis Tsividis' group at Columbia University has been revisiting analog/hybrid computers design in standard CMOS process. Two VLSI chips have been developed, an 80th-order analog computer (250 nm) by Glenn Cowan[36] in 2005[37] and a 4th-order hybrid computer (65 nm) developed by Ning Guo[38] in 2015,[39] both targeting at energy-efficient ODE/PDE applications. Glenn's chip contains 16 macros, in which there are 25 analog computing blocks, namely integrators, multipliers, fanouts, few nonlinear blocks. Ning's chip contains one macro block, in which there are 26 computing blocks including integrators, multipliers, fanouts, ADCs, SRAMs and DACs. Arbitrary nonlinear function generation is made possible by the ADC+SRAM+DAC chain, where the SRAM block stores the nonlinear function data. The experiments from the related publications revealed that VLSI analog/hybrid computers demonstrated about 1–2 orders magnitude of advantage in both solution time and energy while achieving accuracy within 5%, which points to the promise of using analog/hybrid computing techniques in the area of energy-efficient approximate computing. In 2016, a team of researchers developed a compiler to solve differential equations using analog circuits.[40]

Practical examples

X-15 Analog computer
X-15 simulator analog computer

These are examples of analog computers that have been constructed or practically used:

Analog (audio) synthesizers 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 US. It is now known as The Society for 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.[41]

See also


  1. ^ "Gears of war: When mechanical analog computers ruled the waves". 2014-03-18.
  2. ^ Johnston, Sean F. (2006). Holographic Visions: A History of New Science. OUP Oxford. p. 90. ISBN 978-0191513886.
  3. ^ The Antikythera Mechanism Research Project Archived 2008-04-28 at the Wayback Machine, The Antikythera Mechanism Research Project. Retrieved 1 July 2007.
  4. ^ Fuat Sezgin "Catalogue of the Exhibition of the Institute for the History of Arabic-Islamic Science (at the Johann Wolfgang Goethe University", Frankfurt, Germany) Frankfurt Book Fair 2004, p.35 & 38.
  5. ^ François Charette, Archaeology: High tech from Ancient Greece, Nature 444, 551–552(30 November 2006), doi:10.1038/444551a
  6. ^ Silvio A. Bedini, Francis R. Maddison (1966). "Mechanical Universe: The Astrarium of Giovanni de' Dondi", Transactions of the American Philosophical Society 56 (5), pp. 1–69.
  7. ^ D. De S. Price (1984). "A History of Calculating Machines", IEEE Micro 4 (1), pp. 22–52.
  8. ^ Tuncer Őren (2001). "Advances in Computer and Information Sciences: From Abacus to Holonic Agents", Turk J Elec Engin 9 (1), pp. 63–70 [64].
  9. ^ Donald Routledge Hill (1985). "Al-Biruni's mechanical calendar", Annals of Science 42, pp. 139–163.
  10. ^ "Episode 11: Ancient Robots", Ancient Discoveries, History Channel, retrieved 2008-09-06
  11. ^ Howard R. Turner (1997), Science in Medieval Islam: An Illustrated Introduction, p. 184, University of Texas Press, ISBN 0-292-78149-0
  12. ^ Donald Routledge Hill, "Mechanical Engineering in the Medieval Near East", Scientific American, May 1991, pp. 64–69 (cf. Donald Routledge Hill, Mechanical Engineering)
  13. ^ Ray Girvan, "The revealed grace of the mechanism: computing after Babbage" Archived November 3, 2012, at the Wayback Machine, Scientific Computing World, May/June 2003
  14. ^ Thomas Parke Hughes Networks of power: electrification in Western society, 1880–1930 JHU Press, 1993 ISBN 0-8018-4614-5 page 376
  15. ^ James E. Tomayko, Helmut Hoelzer's Fully Electronic Analog Computer; In: IEEE Annals of the History of Computing, Vol. 7, No. 3, pp. 227–240, July–Sept. 1985, doi:10.1109/MAHC.1985.10025
  16. ^ Neufeld, Michael J. (2013). The Rocket and the Reich: Peenemunde and the Coming of the Ballistic Missile Era. Smithsonian Institution. p. 138. ISBN 9781588344663.
  17. ^ Ulmann, Bernd (2013-07-22). Analog Computing. Walter de Gruyter. p. 38. ISBN 9783486755183.
  18. ^ Neufeld (2013), p. 106.
  19. ^ Tomayko, James E. (1 July 1985). "Helmut Hoelzer". IEEE Annals of the History of Computing. 7 (3): 227–240. doi:10.1109/MAHC.1985.10025.
  20. ^ Metropolis, N. "The Beginning of the Monte Carlo Method." Los Alamos Science, No. 15, p. 125
  21. ^ Small, J. S. "The analogue alternative: The electronic analogue computer in Britain and the USA, 1930–1975" Psychology Press, 2001, p. 90
  22. ^ Small, J. S. "The analogue alternative: The electronic analogue computer in Britain and the USA, 1930–1975" Psychology Press, 2001, p. 93
  23. ^ Bissell, C. (2007-02-01). "Historical perspectives – The Moniac A Hydromechanical Analog Computer of the 1950s" (PDF). IEEE Control Systems Magazine. 27 (1): 69–74. doi:10.1109/MCS.2007.284511. ISSN 1066-033X.
  24. ^ "History – Accounts".
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External links

Analog device

Analog device is usually a combination of both analog machine and analog media that can together measure, record, reproduce, or broadcast continuous information, for example, the almost infinite number of grades of transparency, voltage, resistance, rotation, or pressure. In theory, the continuous information (also analog signal) has an infinite number of possible values with the only limitation on resolution being the accuracy of the analog device.


The Monte Carlo trolley, or FERMIAC, was an analog computer invented by physicist Enrico Fermi to aid in his studies of neutron transport.

General purpose analog computer

The General Purpose Analog Computer (GPAC) is a mathematical model of analog computers first introduced in 1941 by Claude Shannon. This model consists of circuits where several basic units are interconnected in order to compute some function. The GPAC can be implemented in practice through the use of mechanical devices or analog electronics. Although analog computers have fallen almost into oblivion due to emergence of the digital computer, the GPAC has recently been studied as a way to provide evidence for the physical Church–Turing thesis. This is because the GPAC is also known to model a large class of dynamical systems defined with ordinary differential equations, which appear frequently in the context of physics. In particular it was shown in 2007 that (a deterministic variant of) the GPAC is equivalent, in computability terms, to Turing machines, thereby proving the physical Church–Turing thesis for the class of systems modelled by the GPAC.

This was recently strengthened to polynomial time equivalence.


HRS-100, ХРС-100, GVS-100 or ГВС-100, (see Ref.#1, #2, #3 and #4) (Serbian: Hibridni Računarski Sistem, Russian: Гибридная Вычислительная Система, English: Hybrid Computer System) was a third generation hybrid computer developed by Mihajlo Pupin Institute (Serbia, then SFR Yugoslavia) and engineers from USSR in the period from 1968. to 1971. Three systems HRS-100 were deployed in Academy of Sciences of USSR in Moscow and Novosibirsk ( Akademgorodok) in 1971 and 1978. More production was contemplated for use in Czechoslovakia and German Democratic Republic (DDR), but that was not realised.

HRS-100 was invented and developed to study the dynamical systems in real and accelerated scale time and for efficient solving of wide array of scientific tasks at the institutes of the A.S. of USSR (in the fields: Aerospace-nautics, Energetics, Control engineering, Microelectronics, Telecommunications, Bio-medical investigations, Chemical industry etc.).

Helmut Hölzer

Helmut Hoelzer was a Nazi Germany V-2 rocket engineer who was brought to the United States under Operation Paperclip. Hoelzer was the inventor and constructor of the world's first electronic analog computer.

Hybrid 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 and numerical operations, while the analog component often serves as a solver of differential equations and other mathematically complex equations. The first desktop hybrid computing system was the Hycomp 250, released by Packard Bell in 1961. Another early example was the HYDAC 2400, an integrated hybrid computer released by EAI in 1963. In the 1980s, Marconi Space and Defense Systems Limited (under Peggy Hodges) developed their "Starglow Hybrid Computer", which consisted of three EAI 8812 analog computers linked to an EAI 8100 digital computer, the latter also being linked to an SEL 3200 digital computer. Late in the 20th century, hybrids dwindled with the increasing capabilities of digital computers including digital signal processors.In general, analog computers are extraordinarily fast, since they are able to solve most mathematically complex equations at the rate at which a signal traverses the circuit, which is generally an appreciable fraction of the speed of light. On the other hand, the precision of analog computers is not good; they are limited to three, or at most, four digits of precision.

Digital computers can be built to take the solution of equations to almost unlimited precision, but quite slowly compared to analog computers. Generally, complex mathematical equations are approximated using iterative methods which take huge numbers of iterations, depending on how good the initial "guess" at the final value is and how much precision is desired. (This initial guess is known as the numerical "seed".) For many real-time operations in the 20th century, such digital calculations were too slow to be of much use (e.g., for very high frequency phased array radars or for weather calculations), but the precision of an analog computer is insufficient.

Hybrid computers can be used to obtain a very good but relatively imprecise 'seed' value, using an analog computer front-end, which is then fed into a digital computer iterative process to achieve the final desired degree of precision. With a three or four digit, highly accurate numerical seed, the total digital computation time to reach the desired precision is dramatically reduced, since many fewer iterations are required. One of the main technical problems to be overcome in hybrid computers is minimizing digital-computer noise in analog computing elements and grounding systems.

Consider that the nervous system in animals is a form of hybrid computer. Signals pass across the synapses from one nerve cell to the next as discrete (digital) packets of chemicals, which are then summed within the nerve cell in an analog fashion by building an electro-chemical potential until its threshold is reached, whereupon it discharges and sends out a series of digital packets to the next nerve cell. The advantages are at least threefold: noise within the system is minimized (and tends not to be additive), no common grounding system is required, and there is minimal degradation of the signal even if there are substantial differences in activity of the cells along a path (only the signal delays tend to vary). The individual nerve cells are analogous to analog computers; the synapses are analogous to digital computers.

Hybrid computers should be distinguished from hybrid systems. The latter may be no more than a digital computer equipped with an analog-to-digital converter at the input and/or a digital-to-analog converter at the output, to convert analog signals for ordinary digital signal processing, and conversely, e.g., for driving physical control systems, such as servomechanisms.

John Whitney (animator)

John Hales Whitney, Sr. was an American animator, composer and inventor, widely considered to be one of the fathers of computer animation.

Kerrison Predictor

The 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. Such devices had been used on ships for gunnery control for some time, and versions such as the Vickers Predictor were available for larger anti-aircraft guns intended to be used against high-altitude bombers, but the Kerrison's electromechanical analog computer was the first to be fast enough to be used in the demanding high-speed low-altitude role, which involved very short engagement times and high angular rates.

The design was also adopted for use in the United States, where it was produced by Singer Corporation as the M5 Antiaircraft Director.


Multivac is the name of a fictional supercomputer appearing in several science fiction stories by American writer Isaac Asimov. According to his autobiography In Memory Yet Green, Asimov coined the name in imitation of UNIVAC, an early mainframe computer. Asimov had assumed the name "Univac" denoted a computer with a single vacuum tube (it actually is an acronym for "Universal Automatic Computer"), and on the basis that a computer with many such tubes would be more powerful, called his fictional computer "Multivac". His later short story "The Last Question", however, expands the AC suffix to be "analog computer".Like most of the technologies Asimov describes in his fiction, Multivac's exact specifications vary among appearances. In all cases, it is a government-run computer that answers questions, and is usually buried deep underground for security purposes. However, Asimov never settles on a particular size for the computer (except for mentioning it is very large) or the supporting facilities around it. In the short story "Franchise" it is described as half a mile long (c. 800 meters) and three stories high, at least as far as the general public knows, while "All the Troubles of the World" states it fills all of Washington D.C.. There are frequent mentions of corridors and people inside Multivac. Unlike the artificial intelligences portrayed in his Robot series, Multivac's interface is mechanized and impersonal, consisting of complex command consoles few humans can operate (with the exception of "Key Item").

Though the technology depended on bulky vacuum tubes, the concept – that all information could be contained on computer(s) and accessed from a domestic terminal – constitutes an early reference to the possibility of the Internet (as in "Anniversary").

In "The Last Question", Multivac is shown as having a life of many thousands of years, growing ever more enormous with each section of the story, which can explain its different reported sizes as occurring further down the internal timeline of the overarching story.

Network analyzer

Network analyzer may mean:

Packet analyzer, used on a computer data network

Network analyzer (AC power), an analog computer system used to study electrical power networks

Network analyzer (electrical), a type of electronic test equipment

Pulse computation

Pulse computation is a hybrid of digital and analog computation that uses aperiodic electrical spikes, as opposed to the periodic voltages in a digital computer or the continuously varying voltages in an analog computer. Pulse streams are unlocked, so they can arrive at arbitrary times and can be generated by analog processes, although each spike is allocated a binary value, as it would be in a digital computer.Pulse computation is primarily studied as part of the field of neural networks. The processing unit in such a network is called a "neuron".

Real 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. Within this theory, it is possible to prove interesting statements such as "The complement of the Mandelbrot set is only partially decidable."

These hypothetical computing machines can be viewed as idealised analog computers which operate on real numbers, whereas digital computers are limited to computable numbers. They may be further subdivided into differential and algebraic models (digital computers, in this context, should be thought of as topological, at least insofar as their operation on computable reals is concerned). Depending on the model chosen, this may enable real computers to solve problems that are inextricable on digital computers (For example, Hava Siegelmann's neural nets can have noncomputable real weights, making them able to compute nonrecursive languages.) or vice versa. (Claude Shannon's idealized analog computer can only solve algebraic differential equations, while a digital computer can solve some transcendental equations as well. However this comparison is not entirely fair since in Claude Shannon's idealized analog computer computations are immediately done; i.e. computation is done in real time. Shannon's model can be adapted to cope with this problem.)A canonical model of computation over the reals is Blum–Shub–Smale machine (BSS).

If real computation were physically realizable, one could use it to solve NP-complete problems, and even #P-complete problems, in polynomial time. Unlimited precision real numbers in the physical universe are prohibited by the holographic principle and the Bekenstein bound.


Scanimate is the name for an analog computer animation (video synthesizer) system developed from the late 1960s to the 1980s by Computer Image Corporation of Denver, Colorado.

The 8 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. One of the major advantages the Scanimate system had over film-based animation and computer animation was the ability to create animations in real time. The speed with which animation could be produced on the system because of this, as well as its range of possible effects, helped it to supersede film-based animation techniques for television graphics. By the mid-1980s, it was superseded by digital computer animation, which produced sharper images and more sophisticated 3D imagery.

Animations created on Scanimate and similar analog computer animation systems have a number of characteristic features that distinguish them from film-based animation: The motion is extremely fluid, using all 60 fields per second (in NTSC format video) or 50 fields (in PAL format video) rather than the 24 frames per second that film uses; the colors are much brighter and more saturated; and the images have a very "electronic" look that results from the direct manipulation of video signals through which the Scanimate produces the images.

Sibyl M. Rock

Sibyl Martha Rock (August 1, 1909 – November 17, 1981) was a pioneer in mass spectrometry and computing. Rock was a key person in Consolidated Engineering Corporation's (CEC) mass spectrometry team at a time when mass spectrometers were first being commercialized for use by researchers and scientists. Rock was instrumental in developing mathematical techniques for analyzing the results from mass spectrometers, in developing an analog computer with Clifford Berry for analysis of equations, and in sustaining an ongoing dialog between engineers and customers involved in development of both the mass spectrometer and an early digital computer, CEC's Datatron.

Tennis for Two

Tennis for Two (also known as Computer Tennis) is a sports video game, which simulates a game of tennis, and was one of the first games developed in the early history of video games. American physicist William Higinbotham designed the game in 1958 for display at the Brookhaven National Laboratory's annual public exhibition after learning that the government research institution's Donner Model 30 analog computer could simulate trajectories with wind resistance. He designed the game, displayed on an oscilloscope and played with two custom aluminum controllers, in a few hours, after which he and technician Robert V. Dvorak built it over three weeks. The game's visuals show a representation of a tennis court viewed from the side, and players adjust the angle of their shots with a knob on their controller and try to hit the ball over the net by pressing a button.

The game was very popular during the three-day exhibition, with players lining up to see the game, especially high school students. It was shown again the following year with a larger oscilloscope screen and a more complicated design that could simulate different gravity levels. It was then dismantled and largely forgotten until the late 1970s when Higinbotham testified in court about the game during lawsuits between Magnavox and Ralph H. Baer over video game patents. Since then, it has been celebrated as one of the earliest video games, and Brookhaven has made recreations of the original device. Under some definitions Tennis for Two is considered the first video game, as while it did not include any technological innovations over prior games, it was the first computer game to be created purely as an entertainment product rather than for academic research or commercial technology promotion.


The torquetum or turquet is a medieval astronomical instrument designed to take and convert measurements made in three sets of coordinates: Horizon, equatorial, and ecliptic. It is said to be a combination of Ptolemy's astrolabon and the plane astrolabe. In a sense, the torquetum is an analog computer.

Type 279 radar

The Type 279 radar was a British naval early-warning radar developed during World War II from the Type 79 metric early-warning set. It initially had separate transmitting and receiving antennas that were later combined in the Type 279M to single-antenna operation. This set also had a secondary surface-search mode with surface and aerial gunnery capability and used a Precision Ranging Panel, which passed accurate radar ranges directly to the HACS table (analog computer).

Voltage-controlled filter

A voltage-controlled filter (VCF) is an electronic filter whose operating characteristics (primarily cutoff frequency) can be set by an input control voltage. Voltage controlled filters are widely used in analogue music synthesizers.

A music synthesizer VCF allows its cutoff frequency, and sometimes its Q factor (resonance at the cutoff frequency), to be continuously varied. The filter outputs often include a lowpass response, and sometimes highpass, bandpass or notch responses.

Some musical VCFs offer a variable slope which determines the rate of attenuation outside the bandpass, often at 6dB/octave, 12dB/octave, 18dB/octave or 24dB/octave (one-, two-, three- and four-pole filters, respectively).

In modular analog synthesizers, VCFs receive signal input from signal sources, including oscillators and noise, or the output of other processors. By varying the cutoff frequency, the filter passes or attenuates partials of the input signal.

In some popular electronic music styles, "filter sweeps" have become a common effect. These sweeps are created by varying the cutoff frequency of the VCF (sometimes very slowly). Controlling the cutoff by means of a transient voltage control, such as an envelope generator, especially with relatively fast attack settings, may simulate the attack transients of natural or acoustic instruments.

Historically, musical VCFs have included variable feedback which creates a response peak (Q) at the cutoff frequency. This peak can be quite prominent, and when the filter's frequency is swept by a control, partials present in the input signal resonate. Some filters are designed to provide enough feedback to go into self-oscillation, and it can serve as a sine-wave source.

ARP Instruments made a multifunction voltage-controlled filter module capable of stable operation at a Q over 100; it could be shock-excited to ring like a vibraphone bar. Q was voltage-controllable, in part by a panel-mounted control. Its internal circuit was a classic analog computer state variable "loop", which provided outputs in quadrature.

A VCF is an example of an active non-linear filter. The characteristic musical sound of a particular VCF depends on both its linear (small-signal) frequency response and its non-linear response to larger amplitude inputs.

William Higinbotham

William Higinbotham (October 25, 1910 – November 10, 1994) was an American physicist. A member of the team that developed the first nuclear bomb, he later became a leader in the nonproliferation movement. He also has a place in the history of video games for his 1958 creation of Tennis for Two, the first interactive analog computer game and one of the first electronic games to use a graphical display.


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