Important In the future all pages covering analog computing, its techniques, technology, application etc. will slowly be moved to a new location at http://www.analogmuseum.org. This is the result of the growing size of the collection and documentation and the wish to have a central point in the web to gather information about analog computing. The following pages are becoming obsolete and will not be developed further!

Apart from my digital machines (cf. my computer room) I started to experiment with simple operational amplifier circuits back in school. When I got my first TELEFUNKEN RAT700 analog computer, this was the beginning of my collection of (mostly) electronic analog computers.

The following pages will eventually contain an introduction into the art of analog computing as well as detailed descriptions of the machines in my collection. As one can see, I am still working on these pages since it is quite time consuming to describe the machines and to write an introductory course. So currently there are only a couple of links to my machines and to some other interesting material concerning analog computing.

	This machine, the Telefunken RA 770, is, in my humble oppinion, the best and most fascinating electronic analog computer ever built by any company. With this machine a dream dating back to the days of my childhood finally came true.
	This is another one of my personal favourites: The first analog computer ever made by TELEFUNKEN. This machine is the one and only engineering model which was created from 1955 to 1956 by TELEFUNKEN (Ulm).
	The TELEFUNKEN RAT700 analog computer - this was the first of TELEFUNKEN's very successful table top analog computers.
	The TELEFUNKEN RA741 analog computer.
	The TELEFUNKEN RA742 analog computer - this was the last of TELEFUNKEN's very successful table top analog computers.
	The TELEFUNKEN OMS 811 dual beam oscilliscope. This oscilloscope was especially designed to be used in conjunction with TELEFUNKEN's transistorized analog computers. It features two channels with independent x- and y-deflection as well as synchronisation mechanisms for camera control, etc. Here you can find some notes about the restoration of such an oscilloscope.
	The TELEFUNKEN DEX102 "Digitalzusatz". This was a digital extension for the RA742 table top analog computer. Both devices together build a small hybrid computer.
	The Hitachi-240 analog computer. This system has 40 chopper stabilized operational amplifiers and a lot of special functions, but is quite a challenge to repair and maintain.
	The EAI TR-10 analog computer. This is the first transistorized table top analog computer built by EAI (1961).
	The EAI-2000 analog computer. This is one of the last analog computers built by this famous company. It is a rather small hybrid computer system featuring a built in digital prozessor with a control terminal which allows complete control of the digital part as well as the analog part of the machine.
	The Dornier DO-960 analog computer. This system is one of the most sophisticated analog computers ever built.
	The Dornier DO-80 analog computer. The smallest system built by the German manufacturer Dornier.
	The Solartron Analogue Tutor. A small tube based system intended primarily for educational purposes.
	Some General Dynamics analog computer modules which have been used in the control of a nuclear power plant.
	The MEDA 43 analog/hybrid computer. This machine was designed and built in the former czechoslovakia and found widespread use in the countries behind the iron curtain.

The following links point to introductory material covering the technology of analog computers, the art of programming these machines and their history.

• The slides for a talk with the title "Faszination Analogrechnen" which I gave at the University of Hamburg on February 6th, 2008, may be found here (in German, about 1.6 MB).
• Since I will talk about analog and hybrid computing at the VCFE 7.0 (2006) in Munich, I prepared some slides for this talk which may be found here: Analog and Hybrid Computing - an introduction to analog and hybrid computing.
• Workshop - Analog Computing - slides covering the solution of three typical problems on an analog computer in detail. These examples are 1) mass-spring-damper system, 2) Lynx/Rabbit ecosystem and a 3) bouncing ball in a box. These slides (about 700 kB) were prepared for a talk to be given at the VCFE 7.0 2006 in Munich.
• The following slides, Telefunken Analog and Hybrid Computers , give a short overview of Telefunken's developments in the area of analog and hybrid computing (about 4.5 MB). I created these slides for a talk to be given at the VCFE 7.0 2006 in Munich.
• My personal favourite link is the one to EAI's multimedia lecture Understanding the ANALOG/HYBRID Computer. This course gives a simple introduction to the principles of an analog computer with some examples. The original lecture consisted of eighty slides and a corresponding audio tape. Both have been merged into a podcast by my friend Christian Peters.
• Another interesting resource are the annotations for an ancient slide collection covering the TELEFUNKEN analog computers - Dia-Reihe Analogrechner, in German, 27 MB!
• A Practical Approach to Analog Computers by John D. Strong and George Hannauer gives a good introduction to analog computing using the then new EAI-231R tube based analog computer, one of the finest instruments ever made.
• A very comprehensive text is the quite famous Handbook of Analog Computation from EAI.
• Fundamentals of the Analog Computer by Dr. Howe, one of the original founders of ADI.
• Analog was not a Computer Trade Mark a wonderful introductory paper written by George Fox Lang, Data Physics Corporation, published in the August 2000 issue of "Sound and Vibration".
• Thanks to Mr. Knut Rothstein I am able to present the scans of the EAI-brochure Simulation by Hybrid Computer - the Unbeatable Competitive Edge which he generously donated to the collection.
• Joe Sousa maintains the Philbrick Archive - an incredibly valuable collection of materials about Philbrick's operational amplifiers etc. If you are looking for more information about the famous K2-W amplifier or the like this is the right place.

• Have you ever wondered who a rendezvous in space is performed when two space craft shall be coupled together like the CM/LEM or a Space Shuttle and the ISS etc.? If so, you might want to have a look at a simple orbital rendezvous simulation I programmed this weekend (12/13-JAN-2008).
• A couple of days ago I reimplemented the vehicle simulation listed below on my EAI 580 analog computer using no special external equipment like the four channel oscilloscope multiplexer, etc. This was possible since the EAI 580 features quite a few electronic analog switches which can be employed to implement a display multiplexer on the computer itself. This new simulation is more realistic than the old one below and shows the car frame, the wheels and the road itself in motion. Read more about this simulation here.
• On September 22nd, my friend Dr. Karina Schreiber (a mathematician, too) came for a visit and we had lots of fun creating some artwork using an analog computer. Some of the pictures we created can be seen here.
• On December 1st, 2007, I finished a more complex program on the Telefunken RA 770 analog computer to display a so called Joukowski airofoil with lines of airflow around this profile. The first video clip (about 8 MB!) shows a single line of airflow around this airofoil under manual control while the second video clip (about 2 MB) shows the automatic generation of a family of 16 curves using the very same program which is described in more detail here.
• Last weekend I developed a small analog computer program to display rotating three dimensional figures on an oscilloscope screen - this program together with a short movie of a rotating spiral can be found here.
• Simulating a (simplified) car suspension system - this page contains links to three AVI-files showing the overall setup of a complex simulation run on an analog computer as well as showing the real time simulation output.
• Programming a bouncing ball in a box - this link leads to a page showing the overall program, setup and an AVI-file showing the bouncing ball. There also is a set of slides describing the simulation in detail.
• Watch two rotating circles spinning. This film was taken from the display of the OMS 811 dual beam oscilloscope connected to the RA742 analog computer.
• This short film shows an ink plotter in action which plots a quite beautiful curve generated by the RA742 analog computer.
• The following film shows a single rotating circle generated by the EAI-2000 analog computer.

• On 29-MAY-2006 my wife and I had the pleasure to have Prof. Giloi with his wife and Prof. Lauber with his wife as our guests. Some photos of this event which was very significant for me can be found here.
• Mr. Bruce Baker, a former employee of Martin-Marietta and analog computing kindly scanned some pictures and papers related with his work on aerospace simulation using analog computers.
• On 29-APR-2007 my wife and I were invited by Prof. Dr. Meyer-Brötz, the father of the first transistorized analog computers made by Telefunken. His first two machines were the RA 800 and the RAT 700, developed in 1959/1960. Prof. Dr. Meyer-Brötz left the area of analog computing in 1966 when he realized that digital computers would eventually supersede analog computers in terms of computing power. His interests shifted to electronic character recognition and the like in the following years. It was a wonderful experience and a gift to meet him and his wife in person. He has lots of memories of the time of his analog computer developments at Telefunken and is a truly wise man. This picture was taken by his wife at the end of our visit. He sent it to us with this short handwritten note.

• Applied Dynamics International, ADI for short, made wonderful analog computers from 1957. Although they have stopped their analog computing business their web site still has two real treasures about analog computing:
  • Fundamentals of the Analog Computer by Dr. Howe, one of the original founders of ADI.
  • Analog Computers in Academia and Industry by Dr. Howe, too.
• Doug Coward's online Analog Computer Museum. He has wonderful and quite complete pictures of the various models from different manufacturers. Further more there is a great introduction into the art of analog computing.
• My friend Hans Kulk, a composer, is not only enthusiastic about analog computers as I am, he also uses his machines to create and compose music.

Maybe you think, as I constantly do, about building your own analog computer. If so, here are some topics which should be considered before attempting such an endeavor:

• If you want to build just a small scale demonstration model you can easily choose off the shelf operational amplifiers, 1% resistors and some selected off the shelf capacitors to build your computer. Using relays for controlling the integrators is perfectly fine for such a computer since you most probably just want to have a way to select between the three basic modes of operation: Initial condition (called "Pause" in German), compute ("Rechnen") and Hold ("Halt").
• If you think about building something more sophisticated as I do, you should at least take the following points into account:
  • Operational amplifiers:

    Gain: This is by far the most complicated and important point when you intend to build an analog computer suitable for doing real work. Since the assumption of an inifite gain does not hold for real world operational amplifiers computing elements based on these devices will have some inherent errors due to the finite gain. These errors correlate with the gain and grow smaller as the gain gets larger. Professional operational amplifiers used in precision analog computers like the Telefunken RA741 and other machines have DC gains of about 1000000000 (10 to the power of nine)! Gains like this are not easily obtained using modern integrated circuit operational amplifiers.

    Drift and noise: Every real world operational amplifier is subject to errors due to drift effects (which correlate with temperature and other factors) and noise. Noise is indeed a problem but not as much as drift is since noise tends to cancel out in calculations in the long run. Drift is a more severe problem since even tiniest drift effects will accumulate in the integrators of a computer setup. The most effective and widely used scheme to keep drift effects negligibly small is to use two amplifiers: One AC coupled main amplifier, since AC coupled amplifiers do not expose drift effects, and a chopper stabilized auto zero amplifier. The raw idea is described in the description of my Telefunken RA741. Noteworthy at this point is that the chopper stabilized amplifier (I would suggest using something like MAXIM's MAX430 for the auto zero amplifier) will sample the voltage at the summing point of the main amplifier which should be zero all the time (synthetic ground). Drift effects in the main amplifier will result in a non-null voltage at the summing point which will be amplified by the auto zero amplifier and fed back into the main amplifier's non-inverting input thus canceling out the drift effect. So building a suitable operational amplifier for a precision analog computer will at least require the use of two operational amplifiers - I would suggest using an OPA27 or the like for the main amplifier and the already mentioned MAX430 as the auto zero amplifier.

    Overload indication: Every analog computer requires some means of overload indication which can be used to stop a calculation as soon as an operational amplifier either

    • exceeds the value of +/- 1.1 machine units at its output or
    • is overloaded in a way that requires a higher output current than the amplifier can readily deliver.
    The first topic can easily accomplished by using two comparators which will generate an output signal as soon as the output voltage of the main amplifier goes out of bound. The second topic is not as easily accomplished as this. Measuring the voltage drop over a series resistor in the output lead of the main amplifier is generally a bad idea since this resistor will introduce additional errors in the calculations. The best method to detect this kind of overloading is to measure the voltage at the summing junction of the main amplifier. This voltage will be zero at all times under normal (non-overload) conditions since the auto zero amplifier will always generate a proper correctional signal. Only when the main main amplifier will be overloaded with respect to its maximum output current, it will not longer be able to maintain a ground potential at its summing junction through the feedback impedance of the overall amplifier setup. So a deviation from zero at this point may be savely used as a sign of current overload.

  • Resistors and capacitors: A precision analog computer needs precision parts for its calculation resistors and capacitors. Resistors with a precision of at least 0.1 percent are required. Really professional analog computers use 0.01 or even 0.005 percent resistors which are very (very!) expensive devices. The same holds true for the capacitors used in integrators and storage cells. These capacitors should have a precision well below 1 percent and should be adjustable (by paralleling a trimmer, etc.). Some precision analog computers have all of these parts mounted in a temperature controlled oven to ensure constant environmental conditions.
  • Integrator control: Every integrator in an analog computer normally uses two relays to control the run mode of the integrator (initial condition, run, hold). In cheap (or early) analog computers relays were used to implement these switches. Relays have some fine properties as the nearly infinite resistance in the open state and the very low resistance in the closed state. Apart from these two advantages, relays have three major drawbacks:
    • They need a significant amount of time to operate,
    • even relays from the same batch will have a slightly different timing and
    • relays bounce when being switched.
    The first problem is not too difficult at all - if fast repetition times of the computer are desired, the known relay setup times can simply be taken into account. The third problem can be overcome by using realys with mercury wetted contacts. The seconds problem is the main problem - even slightly different times at which the various integrators in a computer setup will be switched from one run mode to another will introduce subtle errors into calculations which are very difficult to analyse and compensate for.
  • Electronic switches: A modern precision analog computer would most probably make use of electronic switches for controlling integrators and storage, etc. In contrast to relays electronic switches feature negligible switching delays and they do not bounce at all. Unfortunately there are two problems with electronic switches, too: In the on-state they have a resistance far from being zero and in the off-state they have a finite resistance. Both effects will introduce additional problems and will require a redesign of integrator and storage circuits. In some cases it will be better to use three electronic switches instead of two relays to clamp certain signals, etc. A good introduction into this area may be found in the TELEFUNKEN ZEITUNG, Jahrgang 29, 1966, Heft 1 in an article written by Dr. A. Kley who was a leading figure in the field of electronic switch design and integrator control.
  • Additional modes: Apart from the traditional modes of a simple analog computer (see above) some more complicated modes will be required in a large scale precision analog computer:
    • Potentiometer setup: This mode will load all coefficient potentiometers as if they were wired in an active calculation to allow a precise setup of individual potentiometers. Further more each potentiometer should be easily selectable for readout on a compensation analog voltmeter or a precision digital voltmeter (having a resolution of 4.5 digits at least).
    • Static check: This mode will put the computer in a near-run mode but will switch on special jacks on the patch board which can be used to introduce synthetic variables into a program. The near-run mode differs from a real run mode in a way that all integrators are still in hold mode and their input summing junctions are grounded (thus the term "static" check - the calculation is static and time will not influence the state of any device in the computer).
    • Balance check: This mode will place all operational amplifiers into a special mode featuring a feedback impedance which will result in a very high amplification (normally about 1000). In addition to this the inputs of all these amplifiers are grounded so that at their output only the amplified imbalance of the amplifier can be measured. Measuring the output voltage of each amplifier they can be balanced to zero be means of setting their balance potentiometers.
    • Repetitive operation: Many (interesting) calculations require repetitive operation of the computer in which initial condition setup and run are performed repetitively. Repetitive operation is useful for example to display more or less flicker free pictures on an oscilloscope and allowing the user to change parameters of the calculation by adjusting potentiometers and see the effects of those changes immediately.
    • Iterative operation: In iterative operation there will be at least two groups of integrators - a "normal" group and a "complementary" group. While one group is in run mode, the other group is in hold mode and vice versa (a bit oversimplified). This mode is ideally suited for optimization tasks and other iterative processes to be mapped to the analog computer.
  • Multipliers: Apart from the most simple computational elements as summers and integrators an analog computer requires devices to perform multiplication as well. Electronic (precision) multiplication is a difficult task and some different techniques are normally applied:
    • Servo multipliers: At the heart of a servo multiplier are some ten turn potentiometers mounted on a common shaft which is connected to a servo motor. An operational amplifier is used to implement a feedback control circuit which allows to set the rotational angle of the common shaft depending on some input value (the multiplier). One of the potentiometers is used to close the feedback loop so this potentiometer is not available for multiplication. All other potentiometers located on the common shaft can be fed with multiplicands which will be automatically multiplied by means of the rotational shaft angle which is controlled by the multiplier. Precision servo multipliers have very low errors (down to 0.1 percent) and they allow the instant multiplication of a variety of multiplicands with a single multiplier which often comes in handy in many calculations. Their drawbacks are the very delicate hardware and their inherent slow speed (limited by the maximum rotational speed allowed for by the ten turn potentiometers and the gear box).
    • Time division multipliers: These multipliers also allow the multiplication of several multiplicands with a single multiplier. The idea is to use the multiplier to control the pulse with of a steady stream of pulses while the multiplicands are used to control the amplitude of such a pulse sequence. So multiplier and multiplicand modulate the X- and Y-dimension of rectangular areas which correspond to the desired products. These products will be generated by applying a low pass filter to the pulse outputs of the multiplier. This kind of multiplier is much faster than a servo multiplier while still allowing the calculation of several products in parallel. The pulse width modulation introduces some ripple and noise and the low pass filter still limits the bandwidth of multipliers like these.
    • Parabola multipliers: Parabola multipliers make use of the fact that (x + y) ** 2 - (x - y) ** 2 equals 4 * x * y. The only thing such a multiplier needs are two parabola function generators which can be quite easily built as diode function generators. The disadvantages of this technique are that only one product x * y can be generated at a time and the precision is somewhat lower due to the use of diode based function generators which approximate functions by polygons.
  • Function generators: Most calculations require the generation of more or less arbitrary functions. Apart from some quite arcane devices making use of rare physical effects, most function generators are based on biased diodes to approximate functions as polygons. Most professional analog computers featured variable diode function generators as well as fixed function generators (sine, cosine, log, exp, etc.).
  • Curve followers: Nearly every large analog computer features some special devices like curve followers and so called photo formers. A curve follower consists of a XY-plotter with a pickup coil mounted instead of a pen and a specially prepare sheet of paper on which the desired function has been drawn with conductive ink or laid out with fine wire and fixated with some tape on the paper. A high frequency generator is used to inject a signal into this function wire while the pickup coil acts as an antenna and is used to form a feedback loop with the aid of an operational amplifier. So whenever the X-position of the plotter's carriage changes, the Y-position will follow the function painted on the paper with conductive ink. The advantage of such a setup is the ease with which a function can be generated. Drawbacks are the inherently slow speed and the low precision.
  • Photo formers: A photo former works quite like a curve follower but is based on an oscilloscope, an opaque mask, a photomultiplier and a feedback control loop. The desired function is cut out of the opaque mask which is then placed over the oscilloscope screen in a way that the beam is blocked by the mask when it is below the function value at a particular X-position. If it is above the function value at some X-value it can shine through. This is picked up by a photomultiplier tube (a modern implementation might use a PIN diode) which in turn controls a feedback loop. This feedback loop tries to balance the oscilloscope beam always just at the edge of the opaque mask. The input to this device is an X-position signal which directly controls the X-position of the beam while the output is the Y-position voltage generated by the feedback loop to keep the dot on the edge of the mask.

As already stated elsewhere: I am an enthusiastic collector of old computing machinery, especially VAX-systems and analog computers. So I do NOT collect these items to make any profit! I will NOT sell any of my machines and I normally can NOT afford to pay for machines someone wants to give away. All of my money goes into preserving and maintaining these machines. Questions like "You have so many (analog) computers, give me one/some of your machines!" are an effrontery!

If you have a system you want to give away to a good home, I would love to take care of it. If you want to make money from your system, please do not ask me.

P.S.: I will really not give away any of my machines! Neither to students writing a thesis nor to anyone else. If I have two machines of a type I might be willing to swap but do not count on this. This is a private collection and I love my machines - maybe more than you can imagine. I will help where ever I can with my knowledge about these systems and about using, maintaining and programming analog computers. I will even scan drawings and handbooks to help you in getting your system running again.

Important: Today (04-FEB-2007) someone told me that he saved the remains of a Telefunken RA 741 analog computer from scrap as well as some parts from a RA 770, an EAI TR-48 and a Systron Donner system - and he told me that he could not take two EAI-380 systems since he had no room, so these were scrapped! I am in tears - honestly! It would have taken him a single phone call (0177 / 5633531) to tell me about the machines - I would have come with a suitable vehicle and would have taken care of the two computers! Please - if you know of a system looking for a good home, let me know! I will pay for all expenses - I will arrange and/or pay shipping, I will do everything I can to save analog computers from scrap! Do not let those systems be scrapped - they are our technological heritage and they should be preserved! Please help me in preserving this technology! (And please do not tell me about systems scrapped - I love machines! It hurts me to hear about such incidents!)