Signals and dynamic systems

• Identify input, state, output, disturbance, and time scale in a dynamic model.
• Calculate a first-order step response and relate its time constant to observed behavior.

• algebra-trig
• rates-integration

A signal is a quantity considered as it changes: screen brightness command versus time, measured motor speed, room temperature, or camera angle. A system maps input histories to output histories. Some systems respond nearly at once; others store energy or information, so the present output depends on state accumulated from the past.

A compact first-order model is τ dy/dt + y = Ku, where u(t) is input, y(t) output, K steady-state gain, and τ the time constant. For a constant step from zero to U at t = 0, y(t) = KU(1 - e^{-t/τ}). After one τ, the output reaches about 63.2% of its final change; after roughly 4τ it reaches about 98.2%. An RC voltage, a simplified thermal sensor, or a heavily damped speed response may fit this shape over a limited operating range.

State is the minimum remembered information needed to predict future behavior given future input. In a capacitor model, voltage is state; in a moving mechanism, position and velocity may both be needed. Disturbances enter outside the commanded input, while measurement noise corrupts what the controller sees. Deciding which variable is input or disturbance depends on the model boundary. This matters when choosing update rates: a sensor sampled much faster than its physical time constant may produce more data without revealing faster real dynamics.

A common misconception is that gain K alone describes response speed. Gain sets the eventual change per input; τ sets how quickly it approaches. Another is assuming any curved trace proves a first-order law. Dead time, saturation, multiple energy stores, friction, and changing gain can violate it. The optional RC transient simulation is a bounded ideal first-order model; it omits component tolerance, source resistance unless specified, leakage, and measurement loading. Use residuals and repeated tests before treating a fitted time constant as physical truth.

Problem

A simplified fan-speed system has K = 120 rpm/V and τ = 0.80 s. A command steps from 0 to 5.0 V. Predict speed at 0.80 s and 2.40 s.

Steps

1. Final model speed is KU = 120 rpm/V × 5.0 V = 600 rpm.
2. At t = τ: y = 600(1 - e⁻¹) = 379 rpm.
3. At t = 3τ: y = 600(1 - e⁻³) = 570 rpm.

Result

The model predicts about 379 rpm at 0.80 s and 570 rpm at 2.40 s.

Sanity check

Both values lie between the initial 0 and final 600 rpm and rise monotonically; 570 rpm is 95% of final at 3τ.

Two first-order systems have the same gain. System A has τ = 0.2 s and B has τ = 1.0 s. For the same step, which statement is correct?

A. A reaches its final value sooner B. B has five times the final value C. Both traces are identical D. A must overshoot

Your answer: ______________________________

Reasoning:

Goal

Estimate a first-order time constant and detect when the model is inadequate.

Equipment

• Low-voltage RC circuit or temperature sensor
• Microcontroller or multimeter with timed readings
• Stopwatch
• Known safe step input

Procedure

1. Predict the final output and 63% level from component values or an initial trial.
2. Apply a low-voltage step and record actual output at evenly spaced times without changing the circuit.
3. Estimate τ from the first crossing of 63% of the total change and compare predicted and actual traces.
4. Deliberate fault: add a second RC stage or insulating layer; diagnose model mismatch from a delayed or differently curved response rather than forcing one τ to fit all times.

What to observe

• Expected: a single-store system approaches a stable value without overshoot and crosses 63% near one τ.
• Measure: actual timestamps and outputs; label the theoretical curve separately and note meter/sample timing uncertainty.

Transfer challenge

Choose a reasonable logging interval for a slow environmental sensor and justify it relative to measured dynamics.

Cautions

• Use only low-voltage circuits within component and instrument ratings; discharge capacitors before rewiring.

Learner: ____________________ Date: ____________________

Evidence artifact or file: ____________________________________________

Notes: