analog computer for the atari 8bit computer magazine see analog computing a page from the bombardiers information file bif that describes the components and controls of the norden bombsight the norden bombsight was a highly sophisticated opticalmechanical 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 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 as their numerical values change as an analog computer does not use discrete values but rather continuous values processes cannot be reliably repeated with exact equivalence as they can with turing machines unlike digital signal processing analog computers do not suffer from the quantization noise but are limited by analog noise analog computers were widely used in scientific and industrial applications where digital computers of the time lacked sufficient performance 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 digitalanalog computers were among the most complicated systems for process control and protective relays used analog computation to perform control and protective functions the advent of digital computing made simple analog computers obsolete as early as the 1950s and 1960s although analog computers remained in use in some specific applications like the flight computer in aircraft and for teaching control systems in universities more complex applications such as synthetic aperture radar remained the domain of analog computing well into the 1980s since digital computers were insufficient for the task setup setting up an analog computer required scale factors to be chosen along with initial conditionsthat is starting values another essential was creating the required network of interconnections between computing elements sometimes it was necessary to rethink the structure of the problem so that the computer would function satisfactorily no variables could be allowed to exceed the computers 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 initialconditions state to reset the problem and run it again timeline of analog computers see also history of computing hardware precursors see also timeline of computing hardware before 1950 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 southpointing chariot invented in ancient china during the first millennium bc can be considered the earliest analog computer it was a mechanicalgeared wheeled vehicle used to discern the southern cardinal direction the antikythera mechanism dating between 150 and 100 bc was an early analog computer the antikythera mechanism was an orrery and is claimed to be an early mechanical analog computer according to derek j de solla price 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 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 a star chart invented by ab rayn albrn in the early 11th century 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 a slide rule the slide rule was invented around 16201630 shortly after the publication of the concept of the logarithm it is a handoperated 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 timedistance problems in light aircraft the tidepredicting 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 wheelanddisc 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 ballanddisk integrators 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 at the lewis flight propulsion laboratory circa 1949 heathkit ec1 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 ones 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 firecontrol 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 these were essentially scale models of the electrical properties of the fullsize 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 hlzer built a fully electronic analog computer at peenemnde army research center 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 peiod 19301945 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 worked out into the deltar an analogue computer supporting the closure of estuaries in the south west 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 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 generalpurpose analyzer facility for the solution of field problems developed there by gilbert d mccann charles h wilts and bart locanthi educational analog computers illustrated the principles of analog calculation the heathkit ec1 a 199 educational analog computer was made by the heath company usa c 1960 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 limited and a simple slide rule was more accurate however the unit did demonstrate the basic principle in industrial process control thousands of 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 vacuumtube and solidstate devices to emulation of analog controllers by microprocessors electronic analog computers polish analog computer akat1 the similarity between linear mechanical components such as springs and dashpots viscousfluid 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 massspring 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 mechanicalelectrical 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 floatingpoint digital calculations have a comparatively huge dynamic range 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 analog computers are especially wellsuited to representing situations described by differential equations occasionally they were used when a differential equation proved very difficult to solve by traditional means 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 arbitraryprecision arithmetic while relatively slow provides any practical degree of precision that might be needed many small computers dedicated to specific computations are still part of industrial regulation equipment but from the 1950s to the 1970s generalpurpose 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 8800 analog computer solid state operational amplifiers 64 integrators 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 in the usa nasa huntsville houston martin marietta orlando lockheed westinghouse hughes aircraft in europe cea french atomic energy commission matra aerospatiale bac british aircraft corporation analogdigital 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 setup initiating multiple analog runs and automatically feeding and collecting data the digital computer may also participate to the calculation itself using analogtodigital and digitaltoanalog 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 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 100000 simulations runs for each certification of the automatic landing systems of airbus and concorde aircraft 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 wasnt 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 implementations 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 tidepredicting 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 the fire control computer mechanisms for adding and subtracting precision mitergear 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 discs 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 threedimensional 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 cams 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 cams 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 mechanisms 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 rectilinearcoordinate 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 pinionoperated rack moving parallel to the trigdefined opposite side would position a slide with a slot coincident with the hypotenuse a pivot on the rack let the slides 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 blocks 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 balls rotating axis that axis was always in a plane that contained the axes of two movementpickoff rollers quite similar to the mechanism of a rollingball 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 variablespeed 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 rotatingshaft type for providing operating torque to the inputs of subsequent computing mechanisms as well as driving output datatransmission devices such as large torquetransmitter synchros in naval computers other noncomputational mechanisms included internal odometerstyle counters with interpolating drum dials for indicating internal variables and mechanical multiturn limit stops considering that accurately controlled rotational speed in analog firecontrol computers was a basic element of their accuracy there was a motor with its average speed controlled by a balance wheel hairspring jeweledbearing differential a twinlobe cam and springloaded contacts ships ac power frequency was not necessarily accurate nor dependable enough when these computers were designed electronic analog computers electronic analog computers typically have front panels with numerous jacks singlecontact 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 highresolution potentiometers variable resistors for setting up and when needed varying scale factors in addition there is likely to be a zerocenter analog pointertype meter for modestaccuracy 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 nonideal 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 higherquality inductors could be more easily made many generalpurpose analog computers avoided the use of inductors entirely recasting the problem in a form that could be solved using only resistive and capacitive elements since highquality 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 highspeed types nonlinear functions and calculations can be constructed to a limited precision three or four digits by designing function generatorsspecial 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 transistors baseemitter 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 components this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed march 2013 learn how and when to remove this template message 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 computers 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 discballroller integrators cams 2d and 3d mechanical resolvers and multipliers and torque servos key electricalelectronic components might include precision resistors and capacitors operational amplifiers multipliers potentiometers fixedfunction generators the core mathematical operations used in an electric analog computer are addition integration with respect to time inversion multiplication exponentiation logarithm division 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 highpass filter which means that highfrequency noise is amplified differentiation also risks instability limitations this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed april 2012 learn how and when to remove this template message in general analog computers are limited by nonideal 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 nonlinearities 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 decline in the 1950s to 1970s digital computers based on first vacuum tubes transistors integrated circuits and then microprocessors 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 resurgence in vlsi technology with the development of verylargescale integration vlsi technology yannis tsividis group at columbia university has been revisiting analoghybrid computers design in standard cmos process two vlsi chips have been developed an 80thorder analog computer 250nm by glenn cowan practical examples x15 simulator analog computer these are examples of analog computers that have been constructed or practically used boeing b29 superfortress central fire control system deltar kerrison predictor leonardo torres y quevedos analogue calculating machines based on fusee sans fin librascope aircraft weight and balance computer mechanical computer mechanical integrators for example the planimeter nomogram norden bombsight rangekeeper and related fire control computers scanimate torpedo data computer torquetum water integrator 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 2600s ring modulator was actually a moderateaccuracy analog multiplier the simulation council or simulations council was an association of analog computer users in usa 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 wikimedia commons has media related to analog computers analog neural network analogical models chaos theory differential equation dynamical system fieldprogrammable analog array general purpose analog computer lotfernrohr 7 series of ww ii german bombsights signal electrical engineering voskhod spacecraft globus imp navigation instrument xywriter notes ak dewdney on the spaghetti computer and other analog gadgets for problem solving scientific american 25061926 june 1984 reprinted in the armchair universe by ak dewdney published by wh freeman company 1988 isbn0716719398 universiteit van amsterdam computer museum 2007 analog computers jackson albert s analog computation london new york mcgrawhill 1960 oclc230146450 birunis eightgeared lunisolar calendar in archaeology high tech from ancient greece franois charette nature 444 55155230 november 2006 doi101038444551a the first computers large collection of electronic analog computers with lots of pictures documentation and samples of implementations some in german 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 analog computer trumps turing model jonathan w millss analog notebook 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 lc1 wwii navy pv1 balance computor authority control bnf cb119443710 data gnd 41423379 lccn sh85004769 ndl 00560244 