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C. Pronk 1 Analog computing © Kees Pronk

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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.

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

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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

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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)

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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

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

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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)

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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).

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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

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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.

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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.

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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.

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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.

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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.

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

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C. Pronk 17 From the EAI-680 Switching unit

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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

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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.

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C. Pronk 20 Patch panel from the EAI-680

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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 Example:

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C. Pronk 22 More information?

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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

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