Build a small analogue computer as a hands-on instrument for solving and observing equations—not as a replacement for a PC. A modular circuit built from op-amps, resistors, capacitors and potentiometers can add, scale and integrate voltages to demonstrate dynamic systems. Start with arithmetic and a simple oscillator; attempt the Lorenz system only after each block works, because its products require calibrated analogue multipliers.
What an analogue computer does
An electronic analogue computer represents mathematical variables with electrical signals—usually voltages—and connects circuit blocks to reproduce relationships in a model. Resistors set sums and scale factors, op-amps provide addition and inversion, capacitors in feedback provide integration, potentiometers adjust coefficients, and multiplier circuits form products. A meter or oscilloscope lets you inspect the signals that represent the computation.
This is different from merely using analogue electronics. An amplifier, sensor or synthesizer may process analogue signals, but it is not automatically an analogue computer: the circuit must be configured to represent a mathematical system. Historically, analogue computers also used physical quantities such as water flow; the Phillips MONIAC is one example discussed in the 2011 Elektor project on analogue computing. The project article uses an electronic computer to demonstrate the Lorenz system.
Choose a manageable first project
A modular, patchable op-amp design is the most approachable architecture. It lets you test and reuse functional blocks, rather than debugging a large circuit all at once. Choose the scope before buying parts:
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| Build | What it computes | Best for | Main challenge |
|---|---|---|---|
| Arithmetic trainer | Inversion, scaling, addition and subtraction | Learning op-amp basics | Correct resistor ratios and signal polarity |
| Differential-equation demonstrator | First-order equations, coupled systems or an oscillator | Most first-time builders | Integrator drift, reset and signal-range control |
| Nonlinear computer | Systems with products, including a Lorenz-system demonstration | Builders comfortable with calibration | Multiplier scaling, offsets and sensitivity to initial conditions |
Do not begin with the Lorenz circuit. A simpler progression makes faults easier to isolate: first verify arithmetic, then an integrator, then a two-integrator oscillator, then a multiplier, and only then connect a nonlinear system.
Plan the modules, power and instruments
A useful modular computer has summing amplifiers, integrators with reset switches, coefficient controls, test points and buffered outputs. A direct Lorenz implementation also needs analogue multiplication. For a small project, bipolar rails such as ±12 V or ±15 V are common choices, but the selected op-amps and multiplier must support the chosen supply. Those rail values do not guarantee that an output can swing to ±10 V; output range depends on the device, load and operating conditions.
- Core components: op-amps selected for the actual supply and circuit requirements, resistors, capacitors, potentiometers, reset switches, patch points and hookup wire.
- For a direct Lorenz build: dedicated analogue multiplier ICs or a transistor-based analogue multiplier. An op-amp alone does not multiply two signals.
- Useful construction materials: a solderless breadboard for initial arithmetic and integrator tests; perfboard or a PCB for a larger, repeatable build.
- Minimum check equipment: a multimeter and a current-limited supply appropriate to the circuit. Confirm the rails and polarity before installing ICs.
- Strongly recommended for debugging: a two- or four-channel oscilloscope, which reveals ramp shape, clipping, oscillation and relationships between variables. A multimeter cannot show those waveforms. A signal source is useful for applying known test inputs.
- Optional: data-acquisition hardware for recording signals, or a circuit simulator for checking signs and scaling before wiring.
Choose a maximum intended variable voltage—such as ±5 V or ±10 V—before calculating resistor ratios or multiplier compensation. This is a design target, not a promise that the circuit can reach the rails. Keep supply-return currents out of sensitive signal paths, add local decoupling close to each IC, and use a clearly defined signal ground.
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Build and verify the arithmetic blocks
Inverting and summing amplifiers
An ideal inverting amplifier follows Vout = −(Rf/Rin)Vin. With several inputs, Vout = −Rf(V1/R1 + V2/R2 + …). This makes the circuit a basic equation-building block: each input contributes a signed, weighted term, and resistor ratios set the weights. Potentiometers can make a coefficient adjustable, but their printed scales are not calibrated mathematical values.
Assemble one amplifier first. Apply a known DC input, measure the output, and check both the expected gain and its sign. Confirm that the output remains inside the intended signal range. Add and test a voltage follower, difference stage and sign inverter as needed; verify each independently before patching blocks together.
Integrators
An ideal inverting integrator produces Vout(t) = −(1/RC)∫Vin(t) dt. Its input resistor and feedback capacitor establish the integration scale and polarity. In practice, offset voltage, bias current, capacitor leakage, noise and input DC errors accumulate over time. The output can drift to a supply rail even when the intended equation should remain stable.
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Include a reset switch across the feedback capacitor so runs begin from a repeatable state. A resistor in parallel with the capacitor can limit very slow drift, but it also changes the integrator’s behavior at low frequencies; treat it as part of the design rather than an invisible fix.
Build the dynamic system in stages
- Set the voltage convention and rails. Choose the intended variable-voltage range, then select op-amps and any multiplier that support the supply. Build the power section first, check rail polarity and voltage with a multimeter before inserting ICs, and use current limiting or a fuse.
- Verify one arithmetic block. Apply a known DC level and measure the output. Confirm the gain, polarity and headroom before adding other modules.
- Test one integrator. Apply a square wave and observe a ramp or triangle-like output. Reverse the input polarity to confirm the ramp reverses. Operate the reset switch and check that it gives a reproducible starting condition.
- Construct a two-integrator oscillator. Check whether its amplitude and frequency change predictably as you adjust coefficients. This tests dynamic feedback before nonlinear terms are introduced.
- Add and calibrate the multiplier. Tie one input to zero to measure output offset. Apply known voltages to both inputs, measure the output, and determine the scale factor. Include that factor in the surrounding equation and resistor network.
- Patch a simple differential equation first. Check that its equilibrium or oscillation matches the expected qualitative behavior before attempting a nonlinear system.
- Document and calibrate the whole computer. Record offsets, gain errors, saturation limits and reset behavior. Make a wiring diagram or patch table that maps each mathematical term to its circuit block.
Translate the Lorenz system into circuit blocks
The Lorenz equations are dx/dt = σ(y − x), dy/dt = x(ρ − z) − y and dz/dt = xy − βz. Each state variable is an integrator output. The weighted terms that make up a derivative are added in a summing amplifier, and that result is fed into the corresponding integrator. Potentiometers or resistor ratios set σ, ρ and β.
The nonlinear terms require multiplication. One multiplier forms xy; another is needed for xz if the second equation is built as ρx − xz − y. Equivalently, the product can be formed as x(ρ − z) by subtracting first and then multiplying. A multiplier’s scale constant is part of the equation: if its output is kVxVy, the surrounding circuit must compensate for k and for the chosen representation of variables as voltages.
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Reset the integrators, set initial conditions, and monitor all state outputs when running the system. A reset switch alone may not set precise mathematical initial values; trim controls or input offsets can help, but the goal of a basic build is usually a qualitative demonstration. Chaotic motion is sensitive to initial conditions, and component tolerances and drift make an exact match to a numerical trajectory unrealistic without careful calibration.
The 2011 Elektor article, “The Chaos Machine: Analogue Computing Rediscovered,” is a historical reference for the Lorenz-computer idea. It should not be mistaken for a verified modern bill of materials or a currently maintained construction manual.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Choose construction style for the finished machine
| Method | Good fit | Trade-offs |
|---|---|---|
| Solderless breadboard | Early arithmetic blocks and integrator experiments | Fast to change, but loose contacts, parasitic capacitance, long wiring and poor repeatability can disrupt sensitive integrators. |
| Perfboard | A one-off prototype after the circuit works | More permanent without PCB design, but routing and grounding are harder to inspect and revise. |
| PCB | A repeatable instrument or several matching modules | Can reduce wiring errors, but requires layout work and a board mistake can affect the whole build. |
A practical compromise is to use a breadboard for the arithmetic and integrator stages, then move the complete, tested circuit to perfboard or a PCB. Modular front-panel patching makes equations easier to inspect, but it adds connectors and wiring that must be kept short and clearly labeled.
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Troubleshoot by checking the circuit, not just the equation
If an integrator runs to a rail
Remove its input and reset the capacitor. Verify the supply rails, feedback connection, capacitor value and input polarity. Then apply a small known signal and check the ramp direction and time scale before reconnecting the wider circuit. A drifting output can reflect offset or bias current rather than a wrong equation.
If a multiplier gives the wrong result
First measure its zero-input offset, then test it with known input voltages. A plausible-looking output can still be wrong by a constant scale factor; calculate and compensate for the measured factor in the equation network.
If outputs clip, oscillate or behave erratically
- Check whether the signal range exceeds the op-amp’s actual output swing; supply voltage alone does not determine usable output range.
- Inspect local decoupling, grounding and feedback wiring. Keep sensitive integrator and multiplier returns separate from high-current supply paths.
- Shorten long breadboard connections and check the op-amp output with an oscilloscope before assuming the mathematical model is unstable.
- Check for floating inputs, incorrect capacitor or resistor values, loose contacts and excessive input amplitude.
Stop the run if an integrator rails or oscillates uncontrollably. Disconnect the patch, verify the module by itself and reintroduce connections one at a time.
What the project teaches—and what it cannot replace
A physical computer makes feedback and differential equations visible as changing voltages. It is a useful way to learn op-amp circuits, signal scaling, measurement, dynamic systems and the ways real components depart from ideal equations. Its computation is real-time only within the bandwidth and signal limits of the chosen circuit.
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1Clear out junk files and repair common Windows errors2Fix the driver behind crashes, sound loss and screen glitches3Repair Windows errors before they cause bigger problemsFor most practical work, digital simulation is more flexible, repeatable and convenient. A simulator can help check signs, gains and expected behavior before construction, but it will not reveal loose breadboard contacts or grounding errors. A microcontroller with converters can imitate an analogue system, but that is a hybrid or digital implementation—not a purely analogue computer. Vintage trainers offer historical interest but can bring obsolete parts, incomplete documentation and electrical-safety concerns.
For a broader look at the historical concept, the All About Circuits discussion of analogue-computer construction points to the Internet Archive “anacomp” collection. Use old designs as references, not as assurance that parts, supply assumptions or safety practices remain suitable today.
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