Impedance Resistor⚠ unverified
Electrical / AC · Compute the impedance of a resistor
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| R | R | — | 1.0 | Resistance |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Z | — | Impedance, in ohms (Ohm) |
The science & history
Understanding the Parameters
-
Resistance $R$ — the same quantity as in DC Ohm's law; for an ideal resistor it is real and frequency‑independent.
-
$Z_R$ — a complex impedance that happens to have no imaginary part: magnitude $R$, angle $0^\circ$. This "in‑phase" behaviour is what makes a resistor the only one of the three basic elements that dissipates real power.
Derivation (Approaching a Proof)
Represent voltage and current as phasors ($e^{j\omega t}$ amplitudes). A resistor obeys Ohm's law instantaneously, $v(t) = R\,i(t)$, which carries directly to the phasors:
$$Z_R = \frac{V}{I} = R.$$
There is no derivative to introduce a factor of $j$ (as there is for $C$ and $L$), so the phase is $0^\circ$ at every frequency. Because voltage and current peak together, the average power $P = V_{rms}I_{rms}\cos 0^\circ = V_{rms}I_{rms}$ is fully real — the resistor converts electrical energy irreversibly to heat, whereas reactive elements merely store and return it.
History
The resistor's impedance is the trivial (real‑axis) case of Steinmetz's complex‑impedance framework (1893). Placing $R$ on the same footing as $j\omega L$ and $1/(j\omega C)$ is what lets a mixed R–L–C network be reduced by ordinary complex series/parallel arithmetic.
Related Concepts: Impedance Capacitor, Impedance Inductor, Impedance, Ohm's Law solve for current
Notes: Registry calculator impedance-resistor (unverified; the live card also mislabels the units as
dimensionless — should be ohms). Ideal resistor; real resistors gain slight parasitic reactance at high
frequency.