From a physical quantity to x(t): constant & time-varying signals

In this course, an electrical time-domain signal is a voltage, current or other measurable quantity represented as a function of time. We may write a generic signal as

The letter x names the modeled quantity; the parenthesis (t) states that the value may depend on time. For a voltage and a current we use

The function is a representation. The physical voltage still exists between two terminals with a chosen reference direction; the mathematical function records the value attributed to that voltage as time changes.

In x(t), time is the independent variable. A statement such as v(1 ms)=5 V means: at the time coordinate 1 millisecond, the modeled voltage value is 5 volts.

The two axes therefore carry different physical meanings and units:

Writing only “the signal equals 5” discards both the quantity and the unit. In engineering, a graph without labeled axes and units is an incomplete quantitative statement.

ADR-0018 and EF-01 establish the house notation used throughout AoN:

This is not cosmetic typography. It prevents a later equation from silently confusing a constant operating-point value with a waveform whose value changes with time.

A quantity is constant on an interval if its value is the same for every time in that interval:

For a 5 V ideal source held constant over the observation interval:

A time-varying quantity is simply one for which at least two times in the interval have different values:

Time-varying does not imply sinusoidal, periodic or alternating in sign. A ramp from 1 V to 4 V is time-varying while remaining positive. A single pulse is time-varying but not periodic. Noise is time-varying without being a deterministic sinusoid.

AoN uses uppercase symbols for the constant/DC quantities of elementary circuit analysis. Later, “DC component” will have a more specific signal-processing meaning related to an average or zero-frequency component; the two contexts will be stated explicitly rather than conflated.

A compact mathematical description of Figure 1.1 is

From the model we can read v(1 ms)=5 V and v(3 ms)=2 V. We can also ask interval questions: during the interval from 0 to 2 ms the value is constant even though the complete signal is not constant over all time.

This distinction between a local interval and the whole domain will matter later when we discuss steady state, transients, windows of observation and sampled records.

The vertical jumps in Figure 1.1 mean the idealized mathematical value changes with zero transition time. That is often a useful abstraction, but no real electrical system has unlimited bandwidth.

For an ideal step from 0 V to 5 V at t=0, the model can later be written compactly using a unit-step function. At this introductory stage, the piecewise definition is clearer because it exposes the interval assumptions directly.

A real source, interconnect and measurement instrument will round the transition over a finite time. Therefore an infinitely sharp corner in a drawing should be read as “the transition is much faster than the time scale of interest,” not as evidence of infinite physical slew rate.

The notation x(t) describes a continuous-time model: in the mathematics, a value is defined for every time in its domain. A digital instrument instead returns a finite set of observations at particular times or after finite integration intervals.

If samples are taken at times t₀, t₁, …, the recorded dataset is better thought of as

The approximation sign matters: the instrument has bandwidth, loading, aperture/integration behavior, noise and uncertainty. MS-01 already taught that the measuring instrument becomes part of the experiment.

We will introduce formal sampling notation later. For now, keep three objects distinct: the physical quantity, the mathematical time function, and the finite data record produced by an instrument.

This table is deliberately ahead of the formulas. Its purpose is to prevent a common conceptual mistake: “variable,” “periodic,” “sinusoidal,” “AC” and “sign-changing” are not interchangeable words.

Example A — constant operating point. A regulated rail is modeled as V=3.30 V over a 10 s experiment. Under the stated model there is no time argument because the quantity is taken as constant on that interval. If ripple becomes relevant, the model must change to v(t).

Example B — linear ramp.

Then v(0)=1 V, v(0.5 s)=2.5 V and v(1 s)=4 V. The coefficient 3 V/s has units because multiplying it by seconds must produce volts.

Example C — same value, different signal. Two waveforms can both satisfy v(1 ms)=5 V and yet be completely different elsewhere. One sample never identifies an entire time function.

1. Explain in one sentence the difference among a physical voltage, the function v(t), and an oscilloscope data record.
2. For the Figure 1.1 model, find v(−1 ms), v(1.5 ms) and v(4 ms).
3. Is a ramp from 2 V to 8 V over 3 s time-varying? periodic? required to change sign?
4. A current is modeled as I=2 mA for an entire test interval. Which AoN symbol style is appropriate? What would change if measurable ripple were included?
5. For v(t)=2 V+(4 V/s)t on 0≤t≤0.5 s, find v(0), v(0.25 s) and v(0.5 s).
6. Why is an ideal vertical step not a claim that a real source has infinite bandwidth?
7. Can two different waveforms have the same value at one instant? Give a concrete example.
8. A digital instrument returns 1000 samples. Does this by itself prove that the underlying physical signal is defined only at 1000 instants? Explain.

Chapter 02 will build amplitude, offset and period only after this time-domain vocabulary is stable.