Analog Computers

English translation

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GAU’s Analog Computer: Multiplies, Divides, Squares, and Extracts Square Roots

This document is an English translation of the original Italian article “Il calcolatore analogico di GAU: moltiplica, divide, eleva al quadrato ed estrae radici,” published on a hobbyist website circa 2005–2010, itself based on an article by “GAU” originally published in the Italian electronics magazine Sperimentare, July 1968, under the title “Archimede, calcolatore elettronico.”

Introduction

The slide rule has always been the distinctive symbol of every true engineer, much as the drum is for the shaman. Over time, desktop computers, pocket calculators, and PCs arrived on desks, yet the initiatory fascination of the slide rule remained unsurpassed.

The author recently had occasion to read an article in an old magazine — guarded by an acquaintance more jealously than a Leonardo manuscript — describing an analog computer. The magazine was the July 1968 issue of Sperimentare, and the article was titled “Archimede, calcolatore elettronico” (Archimedes, electronic calculator).

Some online research was carried out, but beyond numerous articles on analog computation simulated with electronic components, nothing as simple as the original could be found. The circuit is considered worth republishing because it is extremely interesting. The documentation includes a photo of the original schematic taken with a mobile phone while the stern librarian was distracted.

The idea is emphatically not the author’s own; respectful credit is given to the author of the article, whose name is unfortunately recorded only by the initials “GAU,” for the ingenious solutions adopted in a circuit of rare elegance in its simplicity.

What GAU’s Analog Computer Does

With a 9-volt battery, three potentiometers, one microammeter, and little else, the device performs addition, subtraction, multiplication, division, squaring, and square-root extraction on numbers ranging from zero to ten with a precision of one decimal place. It has three graduated dials for setting the numbers on which to perform calculations and for reading the result.

The cornerstone of the circuit is the independence of the calculation results from the battery voltage: operation is possible anywhere from 1.5 V to 18 V without difficulty.

In the construction described here, the switch that enables addition and subtraction was eliminated — operations not present on slide rules — because those operations require an auxiliary battery with a voltage exactly equal to the main one; results would not be consistent over time. Instead, the ability to increase the precision of division operations was added for cases where the quotient is greater than 10: without this feature the computer would have used only one-tenth of the useful travel of one potentiometer. This is a small “retrofit” that marginally improves the original design.

Operations on All Numbers

GAU’s computer works on numbers between 0 and 10 with two significant figures.

What if one needs to multiply 2,456 by 435,469?

The trick is to express the factors in exponential (scientific) notation and calculate the normalised fractional part (mantissa) and the exponent (characteristic) of the result separately.

For example:

  • 2.786 × 432,469 = ?

    • 2.786 = 2.8 × 10³; 432,469 = 4.3 × 10⁵
    • 2.8 × 4.3 × 10^(3+5) = 12 × 10⁸ = 1,200,000,000 (exact: 1,204,858,634)

  • 75,826 ÷ 471 = ?

    • 75,826 = 76 × 10³; 471 = 4.7 × 10²
    • 76 ÷ 4.7 × 10^(3−2) = 16 × 10¹ = 160 (exact: 160.989)

  • 6,521² = ?

    • 6,521 = 6.5 × 10³
    • 6.5 × 6.5 × 10^(3×2) = 42 × 10⁶ = 42,000,000 (exact: 42,523,441)

  • √264,247 = ?

    • The number must be written with an even exponent
    • 264,247 = 26 × 10⁴
    • √(26 × 10⁴) = √26 × 10^(4/2) = 5.1 × 10² = 510 (exact: 514.05)

Precision of Results

The theoretical precision of calculations performed with numbers rounded to two significant figures is on average 1.2% (in 10% of cases it drops to 2.5%). This applies to values above 10% of the reading scale. Using the first 10% of the scale, precision drops considerably — which is why textbooks recommend taking measurements near full scale.

Accounting for potentiometer non-linearity from loading, scale misalignment, travel tolerance, pivot play, and reading approximation, and applying a fourfold safety factor (as is customary in bridge engineering), overall precision is still close to 5%. For electronics calculations — apart from rare exceptions — precision better than 5% is unnecessary for most components which have a nominal tolerance of 10% (e.g., E12-series resistors and capacitors).

The Circuit

The heart of the circuit consists of three potentiometers and a microammeter. All other components are minor accessories.

Compared to the original schematic, the following additions were made: a 10 K multiturn trimmer, a 100 K resistor, and switch S1 which, in the “open” position, multiplies the P1 reading scale by 10. The auxiliary battery for addition and subtraction and its associated switch were removed.

How It Works

To multiply 8 by 5, using a 10-volt battery for clarity:

The battery powers the linear wire-wound potentiometer P1, graduated in 10 divisions and set to 80% of its travel; at its output: 10 × 0.8 = 8 V, which then powers P2. P2, also graduated in 10 divisions, is set to 50% of its travel; at its output: 8 × 0.5 = 4 V. Ignoring decimal points, this voltage equals the product of the positions of P1 and P2.

P3 is also connected across the battery. Its output is proportional to cursor position. With its scale divided into 100 graduations, setting it to “40” yields 4 V at its output.

When the two voltages are equal, no current flows through the centre-zero microammeter. With the meter nulled:

PotentiometerScalePositionDivision ratioOutput voltage
P10–1080.810 × 0.8 = 8 V
P20–1050.58 × 0.5 = 4 V
P30–100400.410 × 0.4 = 4 V

Reading the scales: 8 × 5 = 40.

This result holds for any battery voltage. The operator sets the factors on P1 and P2, nulls the meter by adjusting P3, and reads the result on P3. P1 and P2 “perform the multiplication”; after nulling, P3 “reads the result.”

The Division Retrofit

If the quotient is less than 10 no problem arises, but in about half of all cases the quotient exceeds 10. When this happens, to obtain a quotient below 10 one must place P3 (the dividend) in the first 10% of its scale — the low-precision zone.

With the retrofit, a resistor is inserted in series with P3 to reduce the voltage across it to 90% of the battery voltage. The multiturn trimmer Tr (10 K) and resistor R (100 K) provide this. The theoretical value of (Tr + R) equals 9 × P1, but the required value is somewhat higher and must be set during calibration.

The P3 scale is “expanded” so that the former position “7.4” now corresponds to “74”; accordingly the P1 (quotient) scale is also multiplied by 10. The operator sets the dividend on P3 at “74”, the divisor on P2 at “5.5”, and with S1 in the “open” DIV×10 position reads the quotient “13” on P1 (interpreting the scale as multiplied by 10).

Example: 74,212 ÷ 552 = ?

  • 74,212 = 74 × 10³; 552 = 5.5 × 10²
  • 74 ÷ 5.5 × 10^(3−2) = 13 × 10¹ = 130

Practical Considerations

  • The battery voltage, whatever its value and even if it varies with time, does not affect results: this is the keystone of GAU’s computer.
  • The potentiometers must be wire-wound. Their nominal linearity is ±0.25%.
  • The resistance of P2 must be at least 10 times that of P1, so that the loading non-linearity of P1 caused by P2 is 2.5%.
  • The resistance value of the potentiometers does not affect results.
  • With P1 = 1 K, P2 = 10 K, P3 = 10 K, and a 9 V battery, total current draw is approximately 11 mA — less than an LED.
  • Because a “zero-current measurement” is performed, the meter need not be high-quality or even linear; it only needs to be sufficiently sensitive.
  • The meter is protected by a 33 K series resistor limiting current to approximately 300 µA in the worst case; two diodes further protect it by clamping to ±0.7 V.
  • The pushbutton P short-circuits the 33 K resistor, enabling a precise zero using maximum meter sensitivity — this was standard practice for decade bridges.

Construction

The computer is housed in a 15 × 10 × 5 cm plastic box. Assembly requires only a short insulated wire connecting the terminals of the potentiometers, meter, and switches mounted on the front panel. In the prototype, a sheet of pressed cardboard was used as a template for the final metal panel.

Dials and Their Calibration

P1 and P2 have a scale graduated from 0 to 10; P3 is graduated from 0 to 100. Dial artwork can be drawn with CAD software (the author used ProgeCAD 2009 Smart, available free for strictly non-commercial use).

Calibrating the P3 Scale

With S1 closed (DIV×1):

  1. Bring P2 to full scale (“10”).
  2. Set P1 to “0.5”.
  3. Adjust P3 to null the meter.
  4. With pushbutton P pressed, fine-adjust P3 to null the meter.
  5. Mark “5” on the P3 scale.
  6. Repeat steps 2–4 for all P1 values in steps of 0.5 from “1.0” to “10”, marking the corresponding P3 points from “10” to “100” in steps of 5.

The P3 scale is now calibrated at 20 points (0, 5, 10, 15 … 95, 100). Intermediate subdivisions in steps of 1 can be interpolated by hand.

Calibrating the P2 Scale

  1. Bring P1 to full scale (“10”).
  2. Set P3 to the previously marked “5”.
  3. Adjust P2 to null the meter.
  4. Fine-adjust with P pressed.
  5. Mark “0.5” on the P2 scale.
  6. Repeat for P3 values in steps of 5 from “10” to “100”, marking P2 from “1” to “10” in steps of 0.5.

The P2 scale is now calibrated at 20 points. Intermediate subdivisions in steps of 0.1 can be interpolated by hand.

Calibrating the DIV×10 Retrofit

  1. Place S1 in the “open” DIV×10 position.
  2. Using the calibrated scales, set P1 to “3”, P2 to “3”, P3 to “90”.
  3. Connect a digital voltmeter to the wiper of P2.
  4. Press P; read and note the exact voltage Vp2 (e.g., 0.760 V).
  5. Connect the voltmeter to the wiper of P3.
  6. Press P; read the voltage Vp3.
  7. With P pressed, adjust trimmer Tr until Vp3 exactly equals Vp2 (e.g., 0.760 V ± 0.01).
  8. Place S1 back to “closed” DIV×1.

The starting point 30 × 3 = 90 on already-calibrated scales guarantees a known equilibrium condition. The meter is not sensitive enough for this adjustment; the digital voltmeter is required.

How to Use the Calculator

Preparing the Data

All numbers must be expressed in exponential notation:

  • P1 and P2: “Units.Decimal” × 10^exponent (e.g., 4.3 × 10⁵)
  • P3: “Tens-Units” × 10^exponent (e.g., 33 × 10²)

Multiplication (S1 closed, DIV×1)

  1. Set the first factor on P1 (Units.Decimal).
  2. Set the second factor on P2 (Units.Decimal).
  3. Adjust P3 to null the meter.
  4. Press P; fine-adjust P3 to null precisely.
  5. Read the result on P3 (Tens-Units).
  6. Add the exponents.

Division (S1 closed initially)

  1. Set the dividend on P3 (Tens-Units).
  2. Set the divisor on P2 (Units.Decimal).
  3. Adjust P1 to null the meter.
  4. If the meter cannot be nulled, switch S1 to “open” DIV×10.
  5. Adjust P1 to null; press P and fine-adjust.
  6. Read the result on P1:
    • If S1 is DIV×1: read (Units.Decimal).
    • If S1 is DIV×10: read (Tens.Units).
  7. Always return S1 to closed (DIV×1).
  8. Subtract the exponents.

Squaring (S1 closed)

  1. Set the number on both P1 and P2 (Units.Decimal), keeping their positions as equal as possible.
  2. Adjust P3 to null; press P and fine-adjust.
  3. Read the result on P3 (Tens-Units).
  4. Multiply the exponent by 2.

Square Root (S1 closed; exponent must be even)

Note: 234,534 must be written as 23 × 10⁴, NOT as 2.3 × 10⁵.

  1. Set the number on P3 (Tens-Units).
  2. Adjust both P1 and P2 to the same position to null the meter (Units.Decimal), keeping them as equal as possible.
  3. Press P; fine-adjust P1 and P2 together.
  4. Read the result on P1 (Units.Decimal).
  5. Divide the exponent by 2.

First Computations with GAU’s Computer

With the prototype scales divided only every 0.5 for P1/P2 and every 5 for P3, intermediate values were estimated by eye. After an evening of multiplication, division, squaring, and root calculations, results fell within 4–6% precision. When assistants performed the same operations, precision improved to 2–3%, confirming the importance of careful knob positioning and accurate scale reading. These results are consistent with the 5% theoretical prediction. Completing all 100 subdivisions on the scales should bring precision below 2%.

Theory: A Few More Formulae

The output voltage of a potentiometer Ra + Rb loaded by resistance Rc is:

$$V_{out} = \frac{R_b \cdot R_c}{R_a \cdot R_c + R_b \cdot R_c + R_a \cdot R_b} \times V_{batt}$$

When Rc is much larger than Ra and Rb:

$$V_{out} = \frac{R_b}{R_a + R_b} \times V_{batt} = K \times V_{batt}$$

At P1 output: V1 = V_batt × K1, which drives P2. At P2 output: V2 = V1 × K2 = V_batt × K1 × K2. At P3 output: V3 = V_batt × K3.

The circuit is an extreme simplification of the potentiometric measurement method: voltages at two nodes are equated by verifying that no current flows between them.

At the null condition V3 = V2, therefore:

$$V_{batt} \times K1 \times K2 = V_{batt} \times K3$$

Dividing both sides by V_batt (which is non-zero):

$$K1 \times K2 = K3$$

Because the potentiometers are linear and their output voltages correspond to cursor position:

Position(P1) × Position(P2) = Position(P3)

The computer has performed multiplication.

Furthermore, since K1 × K2 = K3, it follows that K1 = K3 / K2. Setting P3 and P2, nulling the bridge, and reading P1 gives the result of division:

Position(P1) = Position(P3) / Position(P2)

Possible Extensions

  • Operational amplifier voltage followers at the output of each potentiometer.
  • A ganged potentiometer with op-amps for squaring and square-root operations.
  • Multiturn potentiometers with precision counting dials.
  • Replacement of the meter by an op-amp comparator with two LEDs.
  • Scales pre-calculated to read directly Xc = 1/(2π f C) and Xl = 2π f L.

Parts List

QtyComponent
1Wire-wound linear potentiometer 1 K, 5 W
2Wire-wound linear potentiometers 10 K, 5 W
1Centre-zero microammeter 100 µA
2Diode 1N4148
1Resistor 33 K, ¼ W
1Resistor 100 K, ¼ W
1Multiturn trimmer 10 K
1Pushbutton
2Switches
19 V battery
3Knobs with pointer
1Enclosure

Estimated Costs (circa 2008)

ItemCost
Three potentiometers€18
Microammeter€10
Pushbutton, switches, knobs€2
Diodes, resistors, trimmer€2
Enclosure€3
Total€35

The Original 1968 Schematic

The photo of the original schematic shows the switch at lower left that places an auxiliary battery (divided by P2) in series with P1 for addition and subtraction. An alternative use of a non-centre-zero microammeter with a germanium diode bridge was also suggested — it was 1968!

References

Conclusion

One might object: was all this effort worth it, given that a calculator performing eight-digit arithmetic costs a couple of euros? True — but the same is said of the sextant now that GPS exists, and of Morse code now that mobile phones reach even the Ngorongoro Crater. GAU’s analog computer may be used only occasionally, as one might use a celestial compass or an air-bubble sextant, but computing a square root with three potentiometers is deeply satisfying.