---
title: AC Fundamentals
module: Alternating Current
moduleNumber: 9
lessonNumber: 1
order: 901
summary: >
  A wall socket delivers a voltage that averages to zero over each cycle, yet it still
  heats a filament and runs a motor. The resolution is that dissipation follows the
  mean of the square, not the mean, so we define the root-mean-square value that makes
  an alternating source the equal of a DC one for resistive heating. We show a
  sinusoid's RMS is its peak divided by $\sqrt2$, work out the average power an ideal
  resistor draws when its current stays in phase with the applied voltage, and separate
  the peak, average, and RMS descriptions that a single number cannot combine.
topics: [Alternating Current]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 29 — Alternating Current; §29-1 Alternating Current in a Resistor"
---

## A sinusoidal source and its reference convention

An alternating source changes polarity periodically. A sinusoidal source voltage is
written

$$
v(t)=V_0\cos(\omega t+\phi_v),
$$

where $V_0$ is peak voltage, $\omega=2\pi f$ is angular frequency, $f$ is
ordinary frequency, and $\phi_v$ is a chosen phase reference. The phase reference
does not alter the physical waveform. It specifies where the time coordinate is
called zero.

Peak value, peak-to-peak value, and root-mean-square value are different quantities.
A zero-centred sine wave has

$$
V_{\rm pp}=2V_0,
\qquad
\overline{v}=0.
$$

The zero average voltage does not imply zero electrical energy transfer. A resistor
heats on both half cycles because its instantaneous power contains the product of
voltage and current, whose sign remains positive for an ideal passive resistor.

$$
% caption: One period of a sinusoidal source $v(t)=V_0\cos(\omega t+\phi_v)$. The
% amplitude $V_0$ is the peak, the peak-to-peak span is $2V_0$, and the period
% $T=1/f$ fixes the frequency; the signed average over a full period is zero.
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  ({\x},{1.60+1.00*sin(120*(\x-1.05))});
\draw[<->,black] (1.80,3.00)--(4.80,3.00);
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\node[above,black!70,fill=white,inner sep=1pt] at (1.55,2.60) {peak};
\node[below,black!70,fill=white,inner sep=1pt] at (3.30,.60) {negative peak};
\end{tikzpicture}
$$

The positive reference terminal of a source and the sign of its displayed voltage
must be recorded together. Reversing meter leads changes the measured waveform from
$v(t)$ to $-v(t)$, equivalent to a phase shift of $\pi$. This sign reversal
does not change peak magnitude or RMS value, but it does matter when two waveforms
are compared for phase or when instantaneous power is calculated.

## Ideal resistor response

Use the passive sign convention for a resistor: current $i(t)$ enters the terminal
labelled positive for $v(t)$. Ohm's law then applies at every instant:

$$
v(t)=Ri(t).
$$

An applied sinusoidal voltage gives

$$
i(t)=\frac{V_0}{R}\cos(\omega t+\phi_v)
=I_0\cos(\omega t+\phi_v),
\qquad
I_0=\frac{V_0}{R}.
$$

Voltage and current have the same phase in an ideal resistor. Their zero crossings,
positive peaks, and negative peaks occur at the same times. A real resistor can have
small lead inductance or parasitic capacitance at high frequency, but the ideal
resistor model assumes those effects are negligible over the stated frequency range.

$$
% caption: Ideal resistor driven by a sinusoidal source under the passive sign
% convention. Current enters the terminal marked $+$, so the instantaneous product
% $v(t)i(t)$ is the power absorbed; Ohm's law $v(t)=Ri(t)$ holds at every instant.
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$$

With ideal resistor response, a time plot of current can be obtained from the voltage
plot by changing only vertical scale. The ratio of corresponding samples is $1/R$.
A phase offset in measured resistor data signals a reference error, a time delay in
the instruments, a nonideal component, or an element other than a pure resistor in
the measured path.

$$
% caption: Voltage and current of an ideal resistor stay in phase. Both cross zero
% and reach their extrema at the same instants (dashed lines); their peak ratio is
% the resistance $R=V_0/I_0$, so a measured phase offset signals a nonideal element.
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  ({\x},{1.60+.58*sin(120*(\x-.95))});
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\node[right,black,fill=white,inner sep=1pt] at (4.78,2.18) {current};
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$$

## RMS values and equivalent heating

The RMS value is defined by the square average over one full period:

$$
V_{\rm rms}=
\sqrt{\frac{1}{T}\int_0^T v^2(t)\d t},
\qquad
I_{\rm rms}=
\sqrt{\frac{1}{T}\int_0^T i^2(t)\d t}.
$$

A sinusoidal waveform has average $\cos^2$ value $1/2$ over one period.
Therefore

$$
V_{\rm rms}=\frac{V_0}{\sqrt2},
\qquad
I_{\rm rms}=\frac{I_0}{\sqrt2}.
$$

RMS voltage is the DC voltage that would produce the same average heating in the
same resistor. A labelled $120\ \mathrm V$ household supply denotes RMS voltage,
not peak voltage. Its ideal sinusoidal peak is approximately
$120\sqrt2\ \mathrm V$, subject to the actual supply waveform and measurement
standard.

$$
% caption: RMS from the squared waveform. Squaring $v(t)$ gives a nonnegative trace
% whose period average, the mean square, is half the peak square for a sinusoid;
% $V_{\rm rms}=\sqrt{\langle v^2\rangle}=V_0/\sqrt2$.
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\node[right,black!70] at (5.78,1.70) {mean square};
\end{tikzpicture}
$$

The average of a waveform and its RMS value answer different questions. Average
voltage tracks signed DC content. RMS voltage tracks the mean square relevant to
resistive heating. A waveform can have zero average and substantial RMS magnitude.
An AC-coupled meter may reject the DC component before calculating RMS; a true-RMS
meter may include both components depending on its specified measurement mode.
Record the instrument's mode before comparing a displayed value with a theoretical
RMS expression.

## Instantaneous and average resistor power

The instantaneous power absorbed by a resistor is

$$
p(t)=v(t)i(t)=\frac{v^2(t)}{R}=i^2(t)R.
$$

For sinusoidal voltage and current in phase,

$$
p(t)=V_0I_0\cos^2(\omega t+\phi_v)
=\frac{V_0I_0}{2}
\left[1+\cos\!\left(2\omega t+2\phi_v\right)\right].
$$

Power oscillates at twice the source frequency and never becomes negative for an
ideal resistor under the passive sign convention. Its average is

$$
P_{\rm avg}=V_{\rm rms}I_{\rm rms}
=\frac{V_{\rm rms}^2}{R}
=I_{\rm rms}^2R.
$$

The energy converted to internal energy in an interval from $t_1$ to $t_2$ is
$W=\int_{t_1}^{t_2}p(t)\d t$. Over an integer number of complete periods,
$W=P_{\rm avg}\Delta t$. Over a fraction of a period, the oscillating power term
must be retained; replacing it immediately with average power changes the predicted
short-time energy.

$$
% caption: Instantaneous power in a resistor over one cycle,
% $p(t)=V_0 I_0\cos^2(\omega t+\phi_v)$. It never goes negative and oscillates at
% twice the source frequency about its average $P_{\rm avg}=V_0 I_0/2=V_{\rm rms}I_{\rm rms}$.
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$$

## Waveform measurement and reporting

An oscilloscope records voltage difference between its probe reference and its tip.
To infer resistor current from voltage, measure voltage across the resistor with a
stated polarity and divide by the measured resistance within the frequency range
where the component is approximately ideal. A current probe has its own phase delay,
bandwidth, sensitivity, and orientation convention. Comparing voltage and current
channels requires a common trigger and matched time bases.

Sample many points per period. A coarse sample interval can shift an apparent peak,
underestimate RMS value, and disguise phase offset. The sampling frequency must
exceed twice the highest frequency content for reconstruction in principle; practical
phase and peak measurements need a substantially higher rate and an anti-alias
filter. A clipped amplifier or limited probe range distorts the waveform and changes
its RMS and heating prediction.

State peak or RMS explicitly, give frequency and resistance, record phase convention,
and state the measurement bandwidth.

> **Worked example (RMS current and power in a resistor).** A sinusoidal source with
> $V_{\rm rms}=6.00\ \mathrm V$ drives $R=120\ \Omega$. Find the RMS current, the
> average power, and the peak current.
>
> Ohm's law holds instantaneously, so it holds for RMS values:
> $$
> I_{\rm rms}=\frac{V_{\rm rms}}{R}=\frac{6.00\ \mathrm V}{120\ \Omega}
> =50.0\ \mathrm{mA}.
> $$
> The average power dissipated is
> $$
> P_{\rm avg}=V_{\rm rms}I_{\rm rms}=\frac{V_{\rm rms}^2}{R}
> =\frac{(6.00\ \mathrm V)^2}{120\ \Omega}=0.300\ \mathrm W.
> $$
> The peak current is $I_0=\sqrt2\,I_{\rm rms}=70.7\ \mathrm{mA}$. Reporting
> $50.0\ \mathrm{mA}$ as a peak would understate the instantaneous current extrema;
> reporting $70.7\ \mathrm{mA}$ as RMS would overstate resistor heating by a factor
> of two.

## RMS of arbitrary waveforms and resistor data reduction

Root-mean-square voltage is defined by heating equivalence, not by a sinusoidal
shortcut. For any periodic voltage $v(t)$ with period $T$, the RMS value is

$$
V_{\rm rms}=\sqrt{\frac{1}{T}\int_0^T v^2(t)\d t}.
$$

The square makes positive and negative voltage intervals contribute equally to
resistor heating. Averaging $v(t)$ itself can give zero for a symmetric alternating
waveform even while the resistor dissipates energy continuously. The square-root at
the end restores voltage units. A DC source with the same RMS voltage produces the
same average power in an ideal resistor as the periodic waveform.

The definition applies to clipped, pulsed, asymmetric, and offset waveforms. It also
shows why peak value alone is insufficient. Two waveforms can have the same peak but
different RMS values because they spend different fractions of a cycle near that peak.
A waveform with a DC offset has RMS value containing both the offset and alternating
parts; removing the mean before squaring computes RMS of the alternating component
only, which answers a different measurement question.

A resistor has instantaneous power $p(t)=v^2(t)/R=i^2(t)R$. The energy converted to
thermal energy over an interval from $t_1$ to $t_2$ is

$$
W=\int_{t_1}^{t_2}\frac{v^2(t)}{R}\d t.
$$

Over an integer number of periods, $W=P_{\rm avg}\Delta t$ and
$P_{\rm avg}=V_{\rm rms}^2/R$. Over a partial cycle, the energy need not equal the
average power times an arbitrary fraction of a period. A waveform with narrow high
voltage peaks deposits much of its energy during those peaks. Partial-cycle energy
calculations therefore require the time-resolved voltage record or an analytic waveform
model; one reported RMS value does not determine energy over a partial cycle.

Measurement instruments impose their own limits. A true-RMS meter estimates the
square-and-average operation over a stated bandwidth and crest-factor range. A meter
calibrated only for sinusoidal waveforms can display an accurate RMS value for a sine
wave but a biased value for a pulsed waveform. Oscilloscope data allow direct numerical
integration, provided the sample rate resolves the fastest voltage changes, the input
range avoids clipping, and the probe attenuation and ground reference are calibrated.

Bandwidth limits matter twice. Insufficient analogue bandwidth rounds fast waveform
edges before digitization, reducing the measured squared voltage. Insufficient sample
rate aliases high-frequency content into lower-frequency samples. Both effects can
produce a plausible-looking trace with an incorrect RMS value. Record the instrument
bandwidth, sample interval, record length, trigger condition, probe factor, and any
digital filtering before reducing waveform data to RMS or energy.

> **Worked example (cross-checking voltage and current channels).** A source drives a
> measured resistor $R=120.0\ \Omega$. A calibrated scope record over ten complete
> cycles gives $V_{\rm rms}=6.00\ \mathrm V$ on the voltage channel and
> $I_{\rm rms}=49.9\ \mathrm{mA}$ on the current channel. Check the two channels
> against Ohm's law.
>
> The current predicted from the voltage channel is
> $$
> I_{\rm rms}=\frac{V_{\rm rms}}{R}=\frac{6.00\ \mathrm V}{120.0\ \Omega}
> =50.0\ \mathrm{mA},
> $$
> which matches the measured $49.9\ \mathrm{mA}$ to the stated precision. The two
> independent power estimates agree as well:
> $$
> \frac{V_{\rm rms}^2}{R}=\frac{(6.00)^2}{120.0}=0.300\ \mathrm W,
> \qquad
> I_{\rm rms}^2R=(0.0499)^2(120.0)=0.299\ \mathrm W.
> $$

Selecting a recorded partial interval of $2.50\ \mathrm{ms}$ and numerically summing
$v_j^2\Delta t/R$ over its samples gives $0.748\ \mathrm{mJ}$, close to
$P_{\rm avg}\Delta t=0.750\ \mathrm{mJ}$. That agreement holds only because the chosen
interval spans an integer number of periods; a noninteger interval requires the
sample-by-sample sum and can differ appreciably from the average-power estimate.

Uncertainty propagation should follow the reduction path. Resistor tolerance,
voltage-channel scale, current-channel scale, sample timing, and waveform clipping all
affect the result. Compare the independent voltage-based and current-based power
estimates; a persistent difference can indicate probe phase error, an incorrect resistor
value, or a channel calibration offset. Preserve the raw record and the integration
window so that RMS and thermal-energy claims can be reproduced.

| reduction | quantity retained | independent diagnostic |
|---|---|---|
| full-record heating | $V_{\rm rms}^2/R$, $I_{\rm rms}^2R$, and $\langle vi\rangle$ | the three estimates agree for a calibrated resistive load |
| finite-window energy | $\sum_j v_j^2\Delta t/R$ over listed sample indices | refine the sample interval and compare with a whole-cycle estimate only when applicable |
| DC content | $V_0$ and the AC-only RMS component | distinguish DC coupling from an AC-coupled meter display |
| crest response | peak value, RMS value, and instrument crest limit | inspect raw peaks before accepting an in-range display |
| phase response | signed $v(t)i(t)$ samples on a common time base | a systematic phase residual identifies channel delay or reactance |

Arbitrary-waveform RMS calculations should state whether the reported value includes
DC content. A source waveform with mean voltage $V_0$ and zero-mean alternating part
$v_{\rm ac}(t)$ has full RMS value
$V_{\rm rms}^2=V_0^2+V_{{\rm ac,rms}}^2$ when the alternating part averages to zero.
The resistor heats from both terms. An instrument set to AC coupling removes the mean
before measuring, so its display cannot be substituted directly into a full heating
calculation without restoring the DC contribution.

Crest factor is the ratio of peak magnitude to RMS magnitude. Narrow pulses can have
a large crest factor even when their RMS value is modest. A meter with a restricted
crest-factor specification can overload internally or return a biased result before its
displayed range is exceeded. Inspect the raw trace for short peaks and compare peak,
RMS, and instrument limits rather than assuming that an in-range RMS display is valid.

Partial-cycle energy sets the electrical input to a thermal transient. Resistor
temperature does not respond instantaneously to the power trace because thermal mass and heat-loss
path integrate energy over much longer intervals. The electrical energy calculation
still uses the exact selected time window. Keep electrical conversion and thermal
temperature response separate: the former follows $v^2/R$ at each sample, while the
latter depends on heat capacity, mounting, airflow, and thermal contact.

Numerical integration should include a convergence check. Recompute the partial-cycle
energy after halving the sample interval or using a higher sample-rate record. If the
result changes beyond the voltage-scale uncertainty, the original record did not
resolve the waveform adequately. A trapezoidal sum can reduce integration error for
smooth records, but it cannot restore a clipped crest or high-frequency component
removed by analogue bandwidth. Retain the integration method and sample indices.

Channel agreement is a strong diagnostic for a resistive load. Voltage and current
should be in phase for an ideal resistor, and instantaneous power should remain
nonnegative apart from noise and probe offset. A systematic phase shift may arise from
probe delay, a current-sensor transfer function, or an unintended reactive element.
Check the phase relation before reducing $v(t)i(t)$ to average power; an RMS-only
comparison can omit a timing error that matters for partial-cycle energy.

The final report should identify waveform shape, mean value, RMS definition, crest
factor, source impedance, resistor value at measurement temperature, channel
calibration, bandwidth, sample rate, integration window, and independent power checks.
These details distinguish a resistor heating result from a sine-wave approximation and
make the RMS value meaningful for the actual source waveform.

Calibration should include a known sinusoidal reference and a non-sinusoidal check when
the intended source has sharp edges or pulses. The sinusoidal reference verifies scale
at one frequency; it does not test crest-factor handling or high-frequency bandwidth.
Use a reference resistor with a documented tolerance, compare voltage and current
channels, and repeat the reduction after changing the scope range. A result that shifts
with display range or probe attenuation identifies an instrument setting error rather
than a change in resistor heating.

## Peak, average, crest factor, and waveform uncertainty

Peak, average, and RMS values describe different aspects of a periodic waveform. The
peak magnitude is the largest instantaneous value. The ordinary average preserves sign
and can vanish for a symmetric alternating waveform. RMS is the heating-equivalent
quantity obtained by squaring before averaging. Confusing these values changes both
source ratings and resistor power estimates. A sine wave with peak value $V_p$ has
$V_{\rm rms}=V_p/\sqrt{2}$, but that conversion is not valid for a square wave, a
clipped wave, or a pulse train.

Crest factor is $C=V_{\rm peak}/V_{\rm rms}$. It measures how concentrated the
waveform is in time. A square wave has crest factor one; a sine wave has crest factor
$\sqrt{2}$; a narrow pulse train can have a much larger value. Instruments designed
for moderate crest factors can report a plausible RMS value while missing or clipping
short peaks. The peak limit of the input channel and the crest-factor limit of the RMS
algorithm must both exceed the waveform requirements.

$$
% caption: Peak, RMS, and average measure different things for a sinusoid. The peak
% is the largest instantaneous value, the RMS is $V_0/\sqrt2\approx0.71\,V_0$ (dashed
% level), and the signed average vanishes over a full period.
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Calculate RMS for a non-sinusoidal resistor voltage from samples over an integer
number of periods or from a record long enough to represent the repeating waveform.
With equally spaced samples, the discrete estimate is

$$
V_{\rm rms}\simeq\sqrt{\frac{1}{N}\sum_{j=1}^{N}v_j^2}.
$$

The corresponding average resistor power is $P\simeq V_{\rm rms}^2/R$. This estimate
weights each sample equally in time. If sampling is nonuniform, include the associated
time intervals; otherwise densely sampled portions of the trace receive too much
weight. A waveform record containing a noninteger number of cycles can still be used
if the start and end phases are accounted for, but the result then has window error
that should be reported.

Sampling validation begins with the fastest expected waveform feature. A sharply edged
pulse can contain significant components far above its repetition rate. Choose analogue
bandwidth and sample rate to resolve the rise time, then repeat the RMS reduction at a
higher sample rate or with a bandwidth limit changed in a controlled way. A stable RMS
estimate under those changes supports the selected acquisition settings. A changing
estimate indicates missing high-frequency content, clipping, or aliasing.

$$
% caption: Sampling a narrow pulse. Dense samples land on the pulse and recover its
% height, while sparse samples step over it and understate both the crest factor and
% the RMS value.
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\end{tikzpicture}
$$

Aliasing can be demonstrated with a validation record. Inject a known high-frequency
test signal or compare the source record against a higher-bandwidth reference channel.
If a component above the sampling limit appears at a false lower frequency in the
digitized trace, the RMS error can be either positive or negative depending on sample
phase and filtering. A low-pass filter placed before digitization must be documented;
it reduces aliasing but also removes physical high-frequency power that may matter for
resistor heating.

> **Worked example (RMS of a pulse train versus a sine assumption).** A pulse
> waveform drives a $100.0\ \Omega$ resistor. Calibrated samples over one repeated
> window give a mean-square voltage of $25.0\ \mathrm{V^2}$, with a maximum sampled
> voltage of $20.0\ \mathrm V$. Find the true RMS voltage, power, and crest factor,
> and compare with a sinusoidal shortcut.
>
> The RMS value comes from the mean square directly:
> $$
> V_{\rm rms}=\sqrt{25.0\ \mathrm{V^2}}=5.00\ \mathrm V,
> \qquad
> P=\frac{V_{\rm rms}^2}{R}=\frac{25.0\ \mathrm{V^2}}{100.0\ \Omega}=0.250\ \mathrm W.
> $$
> The crest factor is $C=V_{\rm peak}/V_{\rm rms}=20.0/5.00=4.00$. Assuming a sine
> shape and converting from the peak instead would give
> $V_{\rm rms}'=20.0/\sqrt2=14.1\ \mathrm V$ and $P'=(14.1)^2/100.0=2.00\ \mathrm W$
> — an eightfold power error. The waveform shape, not the arithmetic, decides which
> conversion is allowed: $V_{\rm rms}=V_{\rm peak}/\sqrt2$ holds only for a sinusoid.

The worked pulse reduction needs an uncertainty record tied to the sampled waveform.

- **Scale and timing terms.** For $P=V_{\mathrm{rms}}^2/R$, a small fractional
  voltage uncertainty contributes twice its fractional size to power; resistance
  tolerance contributes directly. Timing matters when the integration window omits
  whole periods or when peaks are narrow. Repeat records estimate random sample
  noise. Changed bandwidth or sample rate tests systematic resolution error. Averaging
  repeats cannot restore an unresolved peak.
- **Independent power check.** Compare numerical power with a thermal or
  current-channel result when available. A discrepancy that grows with crest factor
  points first to bandwidth, clipping, or meter-algorithm limits. A discrepancy that
  remains with a sine reference can indicate resistor value, probe scale, or channel
  calibration. Retain raw samples, settings, and the integration script.
- **Window definition.** A periodic record spanning an integer number of cycles has
  a start-independent RMS estimate. A record ending midway through a narrow pulse can
  omit a disproportionate fraction of squared voltage. Trigger on a repeatable
  waveform feature, estimate period from the record when frequency drifts, and carry
  the resulting window uncertainty.
- **Measurement loading.** Probe capacitance and resistance can change a fast
  source-resistor waveform; a current probe adds bandwidth, offset, and insertion
  impedance. Check the source voltage with and without the second channel connected
  when edges are fast. A changed trace is a circuit modification, not an uncertainty
  reduced by more averaging.
- **Aliasing and peak bound.** Acquire a control record at a higher sample rate,
  apply a known low-pass filter in reduction, and compare RMS and peak estimates. A
  disagreement beyond voltage-scale uncertainty identifies an analogue-bandwidth or
  sample-clock limit. If a narrow pulse top occupies only two samples, interpolate
  only with a stated waveform model; otherwise report a bound from the nearest
  samples and rise time. Peak uncertainty affects crest factor and can affect RMS at
  small duty fraction.
- **Final statement.** List voltage calibration, resistor tolerance, timing
  calibration, window choice, bandwidth, sample rate, clipping margin, probe loading,
  and peak-resolution bound. Mark each contribution as random or systematic and state
  the reference, repeat record, or acquisition change that constrained it.
## Spectral distortion, parasitics, and thermal calibration

An arbitrary periodic voltage can be represented as a sum of sinusoidal components.
The RMS value is related to the mean square of the complete waveform, so each resolved
spectral component contributes to heating. A waveform with DC value and distinct
sinusoidal components has squared RMS value equal to the sum of the squared DC value and
the squared RMS values of the components when the averaging interval contains complete
cycles. A power meter that attenuates high-frequency components therefore understates
the RMS voltage and resistor heating even if its low-frequency sinusoidal calibration
is accurate.

Distortion limits should be stated as bandwidth and waveform limits, not as a vague
claim that an instrument is “true RMS.” An RMS converter has a finite input bandwidth,
finite crest-factor range, and finite allowable peak. A waveform with harmonics beyond
the bandwidth reaches the converter with altered shape. A waveform with short peaks can
overload an internal stage even when the displayed RMS value is within range. Check the
source spectrum or rise time against the meter specification and verify the result with
a digitized record when high-frequency content affects the intended power result.

$$
% caption: Spectrum of a non-sinusoidal source. The fundamental and its harmonics
% each add to the mean square, so every component that reaches the resistor adds to
% the heating; a meter rolling off above its bandwidth (dashed) understates $V_{\rm rms}$.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.70,.55)--(6.05,.55) node[right] {frequency};
\draw[->,black] (.70,.55)--(.70,2.85) node[above] {amplitude};
\draw[acc,very thick] (1.50,.55)--(1.50,2.30);
\draw[acc,very thick] (2.65,.55)--(2.65,1.55);
\draw[acc,very thick] (3.80,.55)--(3.80,1.10);
\draw[acc,very thick] (4.95,.55)--(4.95,.82);
\draw[black,dashed] (5.55,.55)--(5.55,2.55);
\node[below] at (1.50,.55) {1st};
\node[below] at (2.65,.55) {2nd};
\node[below] at (3.80,.55) {3rd};
\node[below] at (4.95,.55) {4th};
\node[above,black!70] at (5.55,2.55) {meter limit};
\end{tikzpicture}
$$

At sufficiently high frequency, a physical resistor is not an ideal frequency-
independent resistance. Lead inductance, body capacitance, mounting layout, and skin
effect can change the relation between voltage and current. The resistor then has a
frequency-dependent impedance, and $P=V_{\rm rms}^2/R$ using a catalog DC resistance
may not describe the measured source-resistor branch. A low-inductance resistor and a
short return path reduce this problem, but the measurement bandwidth and fixture
geometry still belong in the data record.

Frequency-domain validity can be checked from voltage and current records. For a
resistive branch, the voltage and current components should remain aligned in phase
over the spectral band that carries substantial power. A phase shift or a frequency-
dependent current-to-voltage ratio indicates a parasitic reactance or sensor transfer
error. Compute average power directly from the synchronized product $v(t)i(t)$ as a
cross-check; agreement with $V_{\rm rms}^2/R$ supports the ideal-resistor approximation
over the stated band.

Electrical cycles are usually much faster than resistor temperature changes. The
instantaneous electrical power can vary within every cycle while the resistor body
temperature responds to the average power over a thermal time constant set by heat
capacity and thermal resistance to its surroundings. A thermal measurement therefore
cannot validate partial-cycle energy directly. It can validate average power after a
long enough interval for temperature to approach the corresponding thermal response.

Thermal time constants also protect a resistor from brief power peaks only within
limits. A narrow high-power pulse deposits electrical energy immediately, but the
temperature rise spreads through the resistor body and mounting over a longer time.
Repeated pulses can accumulate heat if their average energy rate exceeds thermal loss.
Use the resistor's pulse-energy and average-power ratings rather than substituting one
for the other. The electrical RMS reduction establishes energy input; the thermal model
sets the allowable temperature response.

A calibration audit ties the RMS or power result to the physical acquisition chain.

| check | reference or perturbation | result retained |
|---|---|---|
| voltage scale | sine reference and probe attenuation | gain, offset, and uncertainty |
| current scale | transfer reference or known shunt | ratio, phase, and loading |
| time base | common fast edge or timing reference | skew and sample interval |
| spectral range | source spectrum and anti-alias setting | occupied band and excluded content |
| resistor state | measured value at operating temperature | resistance used in the reduction |

- **Traceability.** Verify voltage-probe attenuation, vertical scale, current-sensor
  transfer ratio, time base, and resistor value at measurement temperature. Use a
  sine reference for low-frequency gain, a known pulse or square reference for
  crest-factor and bandwidth handling, and a reference resistor to compare
  voltage-derived and current-derived power. Record the test result and uncertainty,
  not a bare instrument label.
- **One-change reduction.** Reprocess the same record after changing sample rate,
  analogue bandwidth, scope range, probe factor, or integration window one at a
  time. RMS value and average power should remain within propagated uncertainty when
  the change does not exclude physical waveform content. A systematic shift identifies
  the setting that limits the stated result. Retain raw waveforms, instrument settings,
  reference tests, spectral limits, fixture details, and the calculation version.
- **Common time interval.** The spectrum used to set a bandwidth decision must come
  from the same interval as the RMS reduction. A short record has limited frequency
  resolution, and a non-integer number of periods spreads one sinusoidal component
  across adjacent bins. That spreading leaves the time-domain mean square unchanged
  but can obscure a narrow component near an instrument limit. State record length,
  sample frequency, window function, and whether the transform measured amplitude or
  only identified occupied bands.
- **Connected frequency response.** A voltage probe adds capacitance, a current
  probe has a transfer function, and a shunt can add branch inductance. Their separate
  bandwidth specifications do not determine the response of the connected
  arrangement. Compare a low-capacitance probe or a short ground connection with the
  original setup for edge-rich waveforms, and retain the resulting change.
- **Aliasing.** Components above half the sample frequency can appear as lower
  frequencies and alter the computed mean square. Increase sample rate, change the
  anti-alias setting, and compare RMS and average power. Stable values bound
  sensitivity to that acquisition choice; they do not demonstrate that every
  high-frequency component was observed.

One waveform record can be reduced directly before any sinusoidal approximation:

$$
V_{\mathrm{rms}}=\sqrt{\frac{1}{N}\sum_{k=1}^{N}v_k^2},
\qquad
P=\frac{1}{N}\sum_{k=1}^{N}v_k i_k.
$$

Use the same calibrated sample interval for both sums. If voltage-only power is
reported, state the measured resistance and the evidence that phase shift and
frequency-dependent impedance are negligible over the occupied band. The direct
product remains the appropriate reduction whenever voltage and current records are
available at a common reference plane.

- **Record selection.** Exclude startup, switching, or overload intervals only with
  a stated physical reason. Retain the excluded samples and their duration. A mean
  square formed from a quiet subset describes that subset, not necessarily the
  source's full duty cycle.
- **Scale conversion.** Apply gain and offset corrections before squaring samples.
  A small voltage offset can matter for low-level signals; a gain correction affects
  every squared sample. Current-channel delay must be corrected before forming the
  pointwise product, particularly when phase is near ninety degrees.
- **Cross-check.** Compare the direct product with a current-shunt calculation,
  voltage-only resistance calculation, or thermal energy measurement over a stated
  interval. Differences that depend on crest factor or sample rate identify waveform
  acquisition limits; differences that track temperature identify the resistor model
  or mounting.

A finite record with sample interval $\Delta t$ retains delivered energy as

$$
E=\sum_{k=1}^{N}v_k i_k\,\Delta t,
\qquad
P_{\mathrm{avg}}=\frac{E}{N\Delta t}.
$$

The equality between this average and the waveform power reduction holds only over
the same sample window. A long thermal observation can test $E$ after accounting
for cooling and the resistor's thermal mass; it cannot recover which individual
cycle carried a narrow peak. Recording electrical interval, ambient conditions, and
resistor temperature keeps the energy and thermal checks on the same physical duty
cycle.

Use a second record with a deliberately changed duty fraction when pulse heating is
important. At unchanged pulse amplitude, the electrical energy should scale with the
counted pulse number and integration interval. A thermal response that fails to follow
the corresponding average energy can indicate cooling changes, a temperature-dependent
resistance, or a measurement interval that omitted part of the duty cycle. The
comparison separates electrical waveform reduction from the later thermal model. A
resistor operating below its rated temperature often has a simple first-order thermal
response with time constant
$\tau_{\rm th}=R_{\rm th}C_{\rm th}$. Under approximately constant average power, the
rise tends toward $P R_{\rm th}$; after a step in power, the early response is slower
than the electrical change. Manufacturer derating and pulse limits remain applicable
to the resistor and its mounting. A board trace, heat sink, airflow, and enclosure
can change the effective thermal resistance substantially. Temperature readings require
their measurement location, air conditions, and elapsed time to be documented with
electrical power.

Pulse qualification also records pulse width and repetition period. A pulse far
shorter than $\tau_{\rm th}$ can raise the resistive film temperature before heat
spreads through the mounting, even when its average power is modest. Closely spaced
pulses accumulate from a raised initial temperature. Compare the peak electrical
energy per pulse with the component's stated pulse-energy limit, then use the
time-averaged record for the longer thermal response.

A complete calibration audit distinguishes correction from uncertainty. If a current
sensor has a measured gain offset, apply the stated correction to the waveform before
calculating power and carry the remaining uncertainty forward. If the sensor response
is unknown over a part of the occupied spectrum, report the excluded band or a bounded
sensitivity calculation rather than one percentage.
The same treatment applies to clipping, trigger jitter, and record truncation. Each
entry should identify the instrument or fixture, setting, reference used, date,
observed deviation, correction applied, uncertainty contribution, and the condition
under which the check is valid. That record lets a later measurement distinguish a
changed source waveform from a changed measurement system.

## Laboratory uncertainty, phase reference, and final reporting

An RMS or power result is incomplete until its uncertainty budget identifies the
measurement terms that control it. For power evaluated from sampled waveforms,
$P=N^{-1}\sum_{k=1}^{N}v_k i_k$, the dominant terms commonly include voltage gain,
current gain, channel offsets, timing mismatch, bandwidth, sample quantization,
resistor temperature coefficient, record selection, and repeatability. The list is
specific to the arrangement. A voltage-only calculation through a nominal resistance
does not have current-probe calibration as a direct term, but it does depend on the
resistance value at its operating temperature and on the validity of the resistive
model. A current-shunt calculation adds the shunt resistance, its heating, and any
loading caused by the shunt leads.

Separate random variation from systematic uncertainty before combining terms. Repeat
records under unchanged conditions measure noise, trigger placement variation, and
short-term source variation. Calibration limits, probe attenuation error, imperfect
bandwidth correction, and unobserved clipping are systematic until independently
bounded. Repeating the same biased measurement many times reduces neither an
unapplied gain correction nor a missing spectral component. A budget should therefore
state the estimate or correction, the uncertainty assigned to it, the basis for that
assignment, and whether the term is correlated with another term.

Sensitivity calculations make the budget testable. Reprocess the same record after
changing a voltage scale by its calibration limit, a current scale by its transfer
limit, or a timing offset by its bounded skew. The resulting change in average power
is the local sensitivity to that quantity. This method resolves interactions in a
non-sinusoidal waveform that a simple fractional rule can miss, including
between waveform shape, clipping margin, and instrument response. When voltage and
current scales were obtained from one common reference, their errors may move together;
the covariance must remain in the calculation. Adding every listed percentage in
quadrature without checking common causes can give an uncertainty that is too small.

The uncertainty budget also needs a stated confidence convention. A standard
uncertainty can be reported with its coverage factor, or an expanded interval can be
reported with the intended coverage and the assumptions used to obtain it. Resolution
is not an uncertainty by itself: a digitizer reading in fine increments can still have
an uncertain scale. Conversely, a coarse display may be adequate if the underlying
record, range, and calibration support the required interval. Rounding belongs at the
end of the reduction. More displayed digits than the uncertainty supports give a false
impression of agreement between instruments.

Phase reference deserves its own reconciliation procedure whenever voltage and current
come from different channels or instruments. A current probe may introduce delay;
separate recorders may have different time bases; and trigger timestamps need not mean
that samples represent the same physical instant. In a sinusoid, an uncorrected delay
$\Delta t$ changes the inferred phase by $2\pi f\Delta t$. Near a ninety-degree phase
relation, even a small error can dominate the real-power result because the average
product is sensitive to the difference between two nearly cancelling positive and
negative contributions.

$$
% caption: A fixed channel delay shifts the current record relative to the voltage.
% A delay $\Delta t$ becomes a phase error $2\pi f\,\Delta t$, which biases the
% average of $v(t)i(t)$ most severely near a quarter-cycle offset.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.55,1.55)--(6.05,1.55) node[right] {time};
\draw[->,black] (.78,.35)--(.78,3.05) node[above] {signal};
\draw[acc,very thick] plot[domain=1.00:5.30,samples=200]
  ({\x},{1.55+.85*sin(150*(\x-1.00))});
\draw[black,very thick] plot[domain=1.45:5.75,samples=200]
  ({\x},{1.55+.85*sin(150*(\x-1.45))});
\draw[<->,black] (1.00,.85)--(1.45,.85);
\node[below,black!70,fill=white,inner sep=1pt] at (1.23,.90) {delay};
\node[acc,fill=white,inner sep=1pt] at (1.60,2.70) {voltage};
\node[black,fill=white,inner sep=1pt] at (4.45,2.70) {current};
\end{tikzpicture}
$$

Reconcile polarity first. A reversed current probe changes the sign of average power
and cannot be repaired by a timing adjustment. A low-frequency resistive reference,
with voltage and current expected to be aligned, provides a simple sign and delay
check. A shared calibration source or a fast edge applied to both channels gives a
second check on channel skew. The same probe tips, cable lengths, bandwidth settings,
coupling modes, and acquisition path used for the power record should be used during
that check. A deskew value measured on a different path is not automatically valid for
the final arrangement.

For broad spectra, one scalar time shift may be inadequate. Compare the measured
current-to-voltage phase over the occupied band against a reference branch or a
characterized sensor transfer function. Frequency-dependent phase error requires a
frequency-dependent correction, a narrower stated bandwidth, or a measurement method
with known matched response. The direct product $v(t)i(t)$ is meaningful only after
each trace has been mapped to the same physical reference plane and time axis. A
power reading should identify where that voltage was measured: across the resistor
body, at a fixture terminal, or at the source output. Lead loss can make these values
different even when the current is common.

Select the resistor model from evidence rather than habit. Begin with the simplest
model consistent with the intended result: a real, temperature-adjusted resistance for
low-frequency heating measurements. Compare its prediction with measured voltage,
current, phase, and average power across the source amplitude and frequency range.
Residuals that are smaller than the measurement uncertainty do not justify a more
elaborate circuit. Systematic phase slope with frequency, a frequency-dependent
magnitude ratio, or a temperature-linked resistance change identifies the missing
behavior that the next model must represent.

A series inductance can represent lead and construction effects when the observed
positive phase slope is reproducible. A parallel capacitance can be appropriate when
the high-frequency current rises in a way that a series-only model cannot reproduce.
A temperature coefficient belongs when the electrical record and controlled thermal
change support it. The candidate parameters should be fitted over the stated range,
then tested on a different frequency, amplitude, or time record. Fitting every detail
of one noisy trace produces a model that may have little predictive value. The final
model is the smallest one that meets the measurement purpose within the uncertainty
budget; outside its validated range, the report should state that the model was not
tested rather than extending it by implication.

Final reporting checks connect the number to the physical experiment. Preserve the raw
or losslessly exported voltage and current arrays, sample interval, record start and
length, channel labels, probe factors, bandwidth limits, coupling settings, and the
calculation version. State the definition used for RMS, whether DC was included, the
integration interval for average power, the physical reference plane, and every
correction applied before the product was averaged. Include the resistor part or
measured value, its temperature or temperature estimate, fixture description, source
waveform, and the frequency or spectral range actually covered.

The final result should give a value, unit, uncertainty interval, coverage convention,
and explicit validity conditions. Those conditions include amplitude range, crest
factor, frequency content, resistor temperature, and any excluded transient interval.
Cross-check the reported power against energy delivered over a longer record, against
the voltage-only calculation where the resistor model is valid, and against the
expected temperature response where thermal equilibrium was reached. Disagreement is
evidence to investigate, not a value to average away. A report that retains these
checks can be recalculated later and can show whether a difference between trials is
electrical, thermal, or instrumental.