1. C. Pronk 1 Analog computing © Kees Pronk http://vitrinemuseum.ewi.tudelft.nl

2. C. Pronk 2 Digital vs. Analog Digital equipment works in discrete time steps, and, in discrete value steps. Analog equipment works with continuous time, and, with continous values.

3. C. Pronk 3 Early examples Early examples of analog computing mechanisms are: The slide rule An Astrolab (such as this one in Franeker)Franeker

4. C. Pronk 4 Example system Here is a typical system consisting of a Spring (k), a Mass (m) and a Damping (c). The system is excited by some force and will move in the y-direction. We want to calculate the y-value in real-time! y

5. C. Pronk 5 Some physics: For the mass it holds that: F m = m. a = m. ÿ For the spring it holds that: F s = m. y For the damper it holds that: F d = d. ý For the whole system it holds that: F m + F s + F d = 0 or m. ÿ + d. ý + s. y = 0 Rewriting: ÿ = (- 1/m). (d. ý + s. y)

6. C. Pronk 6 Physics - 2 ÿ = (- 1/m). (d. ý + s. y) Given the second derivative ÿ, we can calculate ý by using an integrator: ÿ ý and in a similar way we can calculate y from ý ÿ ý y

7. C. Pronk 7 Physics - 3 The complete schematic: ÿ ý y s d - (d ý + sy) 1 / m 1 / m - (d ý + sy)

8. C. Pronk 8 Some history Analog computing was being used before digital computers became powerful enough to do calculations in real-time. Analog computers are based upon the mathematical equivalence of electrical circuits with phenomena which need to be studied such as Spring/mass/damper systems Control systems (closed loop systems e.g. pilot in the loop) Systems of differential equations (e.g. economic systems)

9. C. Pronk 9 Amplifier with gain - A RfRf A R1R1 in out Under the assumption that the amplifier gain A is very large (and some other electronics assumptions), the amplification of this circuit is V out /V in = - R f / R 1 Note: such an amplifier inverts the input signal (a positive input signal becomes a negative output signal).

10. C. Pronk 10 Summing amplifier with gain -A RfRf A R1R1 in 1 out The amplification of this circuit is V out / V in1 = - R f / R 1 and V out / V in2 = -R f / R 2, or, V out = V in1. – (R f / R 1 ) + V in2. – (R f / R 2 ) R2R2 in 2

11. C. Pronk 11 Integrator circuit C A R in out V out = - V in. (t / RC) This cicuit functions as an integrator. The output voltage is a time-integral of the input voltage. Such a circuit converts e.g. speed into distance, or, acceleration into speed. An integrator "smoothes" variations in the input voltage.

12. C. Pronk 12 Differentiator circuit C A R in out V out = -RC. (d V in / dt) This circuit functions as an differentiator. The output voltage is the time differential of the input voltage. Such a circuit converts e.g. distance into speed, or, speed into acceleration. A differentiator amplifies noise and high frequencies and is therefore seldomly used in this pure configuration.

13. C. Pronk 13 Other circuit elements Apart from amplifiers, integrators and differentiators there exist other elements such as: logarithmic amplifiers non-inverting amplifiers comparators switching elements.

14. C. Pronk 14 The EAI 680 analog computer Such an analog computer has been used for research and labs in the aerospace faculty at TUDelft in the "Stabilisation and Control" group.

15. C. Pronk 15 EAI computers Electronic Associates Inc. was an important manufacturer of analog computers. Later on, analog computers were combined with digital computers to form so-called hybrid computers. E.g. EAI 640 (digital) + EAI 680 (analog) EAI 690 (hybrid computer). Still later on, when digital computers became fast enough, analog computers were phased out.

16. C. Pronk 16 From the EAI-680 Chopper relay Note the large (precision) capacitor in the integrator. Integrator / amplifier

17. C. Pronk 17 From the EAI-680 Switching unit

18. C. Pronk 18 Some electronic facts In the beginning the accuacy of analog dc amplifiers was not sufficient for dependable computing. The following requirements could not be met: infinite open loop gain infinite imput impedance zero output resistance zero temperature drift Therefore, the input voltage was chopped into ac; ac amplified and rectified again in so- called chopper-amplifiers

19. C. Pronk 19 Programming an analog computer An analog computer is programmed by connecting the various input and output ports by wires. To easily set-up an experiment, patch panels were provided. A patch-panel was programmed for a particular experiment. Patch panels could be easily exchanged. The connections of the amplifiers and other components are "behind" the patch panel.

20. C. Pronk 20 Patch panel from the EAI-680

21. C. Pronk 21 Modern developments The same ideas of composing a circuit from building blocks can be found in tools such as Simulink and Matlab. See http://www.mathworks.com/ Example:

22. C. Pronk 22 More information? http://www.analogmuseum.org/ http://www.vaxman.de/analog_computing/analog_computing.html http://en.wikipedia.org/wiki/Analog_computer http://technikum29.de/en/computer/analog http://www.play-hookey.com/analog/setting_coefficient.html

23. C. Pronk 23 This museum The 'vitrinemuseum' shows early computer hardware as used for various labs at Delft University of Technology. Have a look at http://vitrinemuseum.ewi.tudelft.nl http://vitrinemuseum.ewi.tudelft.nl