c. pronk 1 analog computing © kees pronk
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Analog computing
© Kees Pronk
http://vitrinemuseum.ewi.tudelft.nl
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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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Early examples
• Early examples of analog computing mechanisms are:
• The slide rule
• An Astrolab (such as this one in Franeker)
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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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Some physics:
• For the mass it holds that: Fm = m . a = m . ÿ
• For the spring it holds that: Fs = m . y
• For the damper it holds that: Fd = d. ý
• For the whole system it holds that: Fm + Fs + Fd = 0
• or
• m . ÿ + d . ý + s . y = 0
• Rewriting:
• ÿ = (- 1/m) . (d . ý + s . y)
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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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Physics - 3
• The complete schematic:
• ÿ ∫ ý ∫ y
s
d
- (d ý + sy)
1 / m1 / m - (d ý + sy)
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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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Amplifier with gain - A
Rf
A
R1inout
Under the assumption that the amplifier gain A is very large (and some other electronics assumptions), the amplification of this circuit is Vout/Vin = - Rf / R1
Note: such an amplifier inverts the input signal (a positive input signal becomes a negative output signal).
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Summing amplifier with gain -A
Rf
A
R1in1
out
The amplification of this circuit is Vout / Vin1 = - Rf / R1 and Vout / Vin2 = -Rf / R2,
or,
Vout = Vin1 . – (Rf / R1) + Vin2 . – (Rf / R2)
R2in2
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Integrator circuitC
A
Rinout
Vout = - Vin . (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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Differentiator circuit
C
A
R
inoutVout = -RC . (d Vin / 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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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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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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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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From the EAI-680
Chopper relayChopper relay
Note the large (precision) capacitor in the integrator.Note the large (precision) capacitor in the integrator.
Integrator / amplifierIntegrator / amplifier
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From the EAI-680
Switching unitSwitching unit
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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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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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Patch panel from the EAI-680
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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:
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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
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This museum
• The 'vitrinemuseum' shows early computer hardware as used for various labs at Delft University of Technology.
• Have a look athttp://vitrinemuseum.ewi.tudelft.nl