---
title: AC Power
module: Alternating Current
moduleNumber: 9
lessonNumber: 4
order: 904
summary: >
  Multiply an AC load's RMS voltage by its RMS current and you get an answer in
  volt-amperes that the wiring must carry, but not in general the watts the load
  consumes. The phase between voltage and current splits that product into a part that
  does net work and a part that merely sloshes energy back and forth. We derive the
  average power $P=V_{\rm rms}I_{\rm rms}\cos\phi$, package amplitude and phase into
  complex power $S=P+jQ$ so that real, reactive, and apparent power form one right
  triangle, and see why a harmonic-rich current forces the time-domain definition
  $P=\langle vi\rangle$ in place of a single phase angle.
topics: [Alternating Current]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 29 — Alternating-Current Circuits; §§29-1, 29-6"
---

## Signed instantaneous power

Electrical power begins with terminal references. Mark a positive voltage terminal
and define positive current as entering that terminal. The instantaneous power

$$
p(t)=v(t)i(t)
$$

is then positive when the two-terminal element absorbs energy and negative when it
returns energy through its terminals. The sign belongs to the selected reference
directions. Reversing a voltage probe or a current-sensor orientation changes the
algebraic sign of the measured power without changing the physical circuit.

The energy transferred during an interval is the signed integral

$$
W_{t_1\to t_2}=\int_{t_1}^{t_2}v(t)i(t)\d t.
$$

Positive and negative portions of a power trace therefore have different meanings.
Positive area records energy sent into the selected element. Negative area records
energy returned from that element to the rest of the circuit. A resistor under the
passive sign convention has nonnegative instantaneous power because $p=i^2R$.
An ideal storage element can have either sign during a cycle while its stored energy
remains nonnegative. A measurement record must retain the sign; replacing every power
sample by its magnitude erases energy return and inflates the calculated average.

$$
% caption: Passive terminal convention for a two-terminal AC load. Current enters the terminal marked positive, so the product of terminal voltage and current is positive when the load absorbs energy from the source.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick] (0.8,0.8)--(0.8,1.38);
\draw[black,thick] (0.8,2.02)--(0.8,2.6);
\draw[black,thick] (0.8,1.7) circle (0.32);
\node[left,black] at (0.44,1.7) {source};
\draw[black,thick] (0.8,2.6)--(4.0,2.6);
\draw[black,thick] (0.8,0.8)--(4.0,0.8);
\draw[black,thick] (4.0,2.6)--(4.0,2.1);
\draw[black,thick] (4.0,1.3)--(4.0,0.8);
\draw[draw=black,thick] (3.4,1.3) rectangle (4.6,2.1);
\node at (4.0,1.7) {load};
\draw[->,acc,thick] (2.35,2.6)--(3.25,2.6) node[midway,above] {current};
\node[above,black] at (4.0,2.62) {+};
\draw[<->,black,thick] (4.95,1.3)--(4.95,2.1) node[midway,right] {load voltage};
\end{tikzpicture}
$$

For sinusoidal voltage and current at one frequency, use RMS amplitudes and a stated
phase difference:

$$
v(t)=\sqrt2V\cos(\omega t),
\qquad
i(t)=\sqrt2I\cos(\omega t-\phi).
$$

The product identity gives

$$
p(t)=VI\cos\phi+VI\cos(2\omega t-\phi).
$$

The first term is constant. The second oscillates at twice the source frequency and
has zero mean over any integer number of periods. The average power is consequently

$$
P=\frac{1}{T}\int_{t_0}^{t_0+T}p(t)\d t=VI\cos\phi.
$$

The expression requires sinusoidal voltage and current at a common angular frequency.
It does not apply by inserting the RMS values of arbitrary distorted traces and a
single phase angle; a harmonic-rich load needs the time-domain average instead.

$$
% caption: In-phase voltage and current, with their instantaneous product below. The power stays nonnegative, oscillates at twice the line frequency, and has a positive average equal to the product of RMS voltage and RMS current.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.8,0.55)--(6.1,0.55) node[right] {time};
\draw[->,black] (0.8,0.3)--(0.8,3.35) node[above] {value};
\draw[black,dashed] (0.8,2.55)--(5.7,2.55);
\draw[acc,very thick] plot[domain=1.0:5.6,samples=200] ({\x},{2.55+0.55*sin(360*(\x-1.0)/2.3)});
\draw[black,very thick] plot[domain=1.0:5.6,samples=200] ({\x},{2.55+0.33*sin(360*(\x-1.0)/2.3)});
\draw[acc!75,very thick] plot[domain=1.0:5.6,samples=220] ({\x},{0.6+0.9*(sin(360*(\x-1.0)/2.3))^2});
\draw[black,dashed] (1.0,1.05)--(5.6,1.05) node[right,black] {average};
\node[acc] at (2.15,3.28) {voltage};
\node[black] at (3.05,2.02) {current};
\node[acc!80] at (4.35,1.9) {power};
\end{tikzpicture}
$$

The phase difference changes the power trace before it changes the RMS readings.
With current delayed by a quarter cycle, $\phi=90^\circ$, the constant term vanishes.
The trace alternates equally between positive and negative values. Energy enters the
load during one interval and returns during another, producing zero average power for
an ideal lossless branch. With a phase angle between zero and ninety degrees, the
trace has both signs but a positive average. The negative intervals are not errors;
they carry the reactive part of the energy exchange.

$$
% caption: Quadrature voltage and current, with their product below. Equal positive and negative lobes give zero net area over a complete cycle, so the average power vanishes although both RMS values are nonzero.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.8,0.55)--(6.1,0.55) node[right] {time};
\draw[->,black] (0.8,0.3)--(0.8,3.35) node[above] {value};
\draw[black,dashed] (0.8,2.55)--(5.7,2.55);
\draw[acc,very thick] plot[domain=1.0:5.6,samples=200] ({\x},{2.55+0.55*sin(360*(\x-1.0)/2.3)});
\draw[black,very thick] plot[domain=1.0:5.6,samples=200] ({\x},{2.55+0.4*cos(360*(\x-1.0)/2.3)});
\draw[black,dashed] (0.8,1.15)--(5.7,1.15);
\draw[acc!75,very thick] plot[domain=1.0:5.6,samples=240] ({\x},{1.15+0.5*sin(720*(\x-1.0)/2.3)});
\node[acc] at (1.75,3.3) {voltage};
\node[black] at (5.35,3.2) {current};
\node[acc!80] at (4.75,1.95) {power};
\end{tikzpicture}
$$

An average must state its interval. The integral over an integer number of settled
periods removes the double-frequency term exactly. An arbitrary partial-cycle window
can have a nonzero mean even for a lossless ideal element. A record that starts at one
phase and ends at another samples unequal positive and negative energy lobes. For a
finite acquisition, either use a whole-number cycle count, use a window synchronized
to the line, or report the exact integration window and its phase relation to the
source.

## RMS values and average-power windows

RMS values give a heating-equivalent amplitude for a waveform over a stated interval:

$$
V_{\rm rms}=\sqrt{\frac{1}{T}\int_{t_0}^{t_0+T}v^2(t)\d t},
\qquad
I_{\rm rms}=\sqrt{\frac{1}{T}\int_{t_0}^{t_0+T}i^2(t)\d t}.
$$

A sine wave has RMS amplitude equal to peak amplitude divided by $\sqrt2$. A general
periodic waveform requires evaluation from the squared trace. The
average of the signed waveform can be zero while the RMS value remains large. A DC
offset contributes to RMS because its square contributes to heating. Subtracting the
mean before the square computes the AC-only RMS value, which answers a different
question and should be labelled accordingly.

The average real power for any measured waveform is defined directly by

$$
P=\frac{1}{T}\int_{t_0}^{t_0+T}v(t)i(t)\d t.
$$

The time-domain definition remains valid for distorted traces, DC offsets, clipped current, and
multiple harmonics. It does not require a phase angle. A time-domain power analyzer
implements this operation with samples, calibration factors, and a finite window.
The quality of the result depends on synchronized voltage and current channels, enough
bandwidth for the waveform, and a window long enough to represent the periodic state.

$$
% caption: RMS as a heating-equivalent amplitude. Squaring the waveform makes both polarities contribute, and the mean of the squared trace over one period is the square of the RMS value.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.8,0.55)--(6.1,0.55) node[right] {time};
\draw[->,black] (0.8,0.3)--(0.8,3.4) node[above] {value};
\draw[black,dashed] (0.8,2.55)--(5.7,2.55);
\draw[acc,very thick] plot[domain=1.0:5.6,samples=240] ({\x},{2.55+0.5*sin(360*(\x-1.0)/2.3)+0.16*sin(3*360*(\x-1.0)/2.3)});
\draw[black,very thick] plot[domain=1.0:5.6,samples=260] ({\x},{0.6+3.0*(0.5*sin(360*(\x-1.0)/2.3)+0.16*sin(3*360*(\x-1.0)/2.3))^2});
\draw[black,dashed] (1.0,1.02)--(5.6,1.02) node[right,black] {mean square};
\node[acc] at (2.0,3.32) {waveform};
\node[black] at (5.1,1.75) {squared};
\end{tikzpicture}
$$

RMS meters have specified waveform, crest-factor, bandwidth, and coupling limits.
A meter calibrated for sine waves may display a value inferred from average rectified
signal rather than the true root-mean-square value for a distorted load. A true-RMS
meter still has a maximum crest factor and a finite bandwidth; narrow current spikes
can exceed its capability while contributing substantially to heating. Check the
instrument manual for its stated limits before treating a displayed RMS number as an
input to a power calculation.

Window choice also affects numerical RMS. A record with many samples but a noninteger
number of periods can weight the waveform unevenly. A slowly changing amplitude or
frequency can make a one-period calculation unrepresentative. Use a stable line
reference when available, average several settled cycles, and compare consecutive
windows. A disagreement between windows indicates drift, beating, insufficient record
length, or a waveform that is not periodic over the selected interval.

## Sinusoidal power factor

For single-frequency sinusoidal voltage and current, the ratio

$$
\mathrm{PF}=\frac{P}{VI}=\cos\phi
$$

is the power factor. Here $V$ and $I$ are RMS magnitudes and $\phi$ is the voltage
phase minus the current phase under the stated reference convention. Unity power
factor occurs when voltage and current are in phase. A phase difference reduces the
average real power carried by fixed RMS voltage and current magnitudes. State whether
current is leading or lagging; the numerical cosine alone does not identify the sign
of reactive exchange.

$$
% caption: Voltage and current phasors with current as the horizontal reference. The voltage projection along the current sets real power; the perpendicular projection sets reactive power, and the angle between them is the phase.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.7,0.7)--(6.0,0.7) node[right] {current axis};
\draw[->,black] (1.1,0.4)--(1.1,3.1) node[above] {quadrature axis};
\draw[->,black,very thick] (1.1,0.7)--(2.9,0.7) node[midway,below] {current};
\draw[->,acc,very thick] (1.1,0.7)--(4.3,2.05) node[above right,acc] {voltage};
\draw[dashed,black] (4.3,2.05)--(4.3,0.7);
\draw[black] (1.65,0.7) arc (0:23:0.55);
\node[black] at (2.2,0.98) {phase};
\node[below,black] at (4.3,0.66) {in-phase part};
\node[right,black] at (4.37,1.38) {quadrature part};
\end{tikzpicture}
$$

Power factor is a ratio of averages and magnitudes, not a statement about how much
current a load draws. A low-power-factor load can draw substantial RMS current while
converting relatively little average energy into heat or mechanical work. The current
still causes conductor heating through $I^2R$, voltage drop in source wiring, and
capacity demand in equipment that carries the branch. A correct report gives real
power, RMS voltage, RMS current, power factor, and leading or lagging state together.

A passive sinusoidal load has nonnegative average real power and power-factor magnitude
between zero and one. Under the common convention, reactive power is positive for
lagging current and negative for leading current. Generator or source conventions can
reverse signs, so label the terminal
reference and whether a reported value describes absorbed or delivered power.

$$
% caption: Power factor as a function of phase magnitude. The cosine falls from one when voltage and current are in phase to zero at a quarter-cycle separation; leading and lagging cases share this magnitude but carry opposite reactive sign.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.7,0.6)--(6.1,0.6) node[right] {phase};
\draw[->,black] (0.7,0.6)--(0.7,3.05) node[above] {power factor};
\draw[acc,very thick] plot[domain=0:90,samples=120] ({0.7+5.0*\x/90},{0.6+2.1*cos(\x)});
\draw[dashed,black] (0.7,2.7)--(5.7,2.7);
\node[left,black] at (0.66,2.7) {1};
\node[below,black] at (0.7,0.55) {0};
\node[below,black] at (5.7,0.55) {quarter cycle};
\node[acc] at (1.7,2.35) {in phase};
\end{tikzpicture}
$$

## Complex power and power triangles

Complex power packages the amplitude and phase data of one sinusoidal frequency into
one quantity. With RMS phasors and the passive sign convention,

$$
\widetilde S=\widetilde V\widetilde I^{\ast}=P+jQ.
$$

The real component $P$ is average real power in watts. The imaginary component $Q$
is reactive power in volt-ampere reactive units. The magnitude

$$
|\widetilde S|=VI
$$

is apparent power in volt-amperes. The three quantities have different roles in a
measurement. Real power predicts net energy transferred over time. Apparent power
sets the product of RMS voltage and current that conductors, switches, and much of the
distribution equipment must carry. Reactive power records the quadrature component
for a sinusoidal record and its sign depends on the stated convention.

$$
P=VI\cos\phi,
\qquad
Q=VI\sin\phi,
\qquad
|\widetilde S|^2=P^2+Q^2.
$$

The conjugate on current is essential. If voltage has phase $\phi_V$ and current has
phase $\phi_I$, then the complex-power angle is $\phi_V-\phi_I$, the voltage-current
phase used in the real-power formula. Omitting the conjugate changes the angle to a
sum of phases and makes the result depend on an arbitrary time reference. Complex
power is therefore reference-invariant only when the voltage and current phasors are
formed with one common time convention.

$$
% caption: Complex-power triangle for an absorbing sinusoidal load. The horizontal leg is real power, the vertical leg is reactive power under the stated sign convention, and the hypotenuse is apparent power.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.7,0.65)--(5.9,0.65) node[right] {real power};
\draw[->,black] (1.1,0.4)--(1.1,3.05) node[above] {reactive power};
\draw[->,black,very thick] (1.1,0.65)--(4.5,0.65) node[midway,below] {P};
\draw[->,black,very thick] (4.5,0.65)--(4.5,2.35) node[midway,right] {Q};
\draw[->,acc,very thick] (1.1,0.65)--(4.5,2.35);
\node[acc] at (2.35,1.75) {apparent S};
\draw[black] (1.6,0.65) arc (0:27:0.5);
\node[black] at (1.98,0.86) {phase};
\end{tikzpicture}
$$

In a lagging-current load, voltage leads current and $Q$ is positive under the common
load convention. In a leading-current load, voltage lags current and $Q$ is negative.
The real-power sign still follows energy direction at the selected port.
An absorbing load can have positive $P$ with either positive or negative $Q$. A source
delivering energy has negative $P$ under the same passive convention. Some utility
and generator displays use a different sign convention, so values from separate
instruments should not be combined until their sign definitions are aligned.

$$
% caption: Reactive-power sign under a load convention. Lagging current places the complex-power point above the real axis; leading current places it below. Both retain positive real power because the load absorbs average energy.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.7,1.6)--(6.0,1.6) node[right] {real power};
\draw[->,black] (3.3,0.4)--(3.3,2.85) node[above] {reactive power};
\draw[->,acc,very thick] (3.3,1.6)--(5.25,2.45) node[above,acc] {lagging};
\draw[->,black,very thick] (3.3,1.6)--(5.25,0.75) node[below,black] {leading};
\draw[dashed,black] (5.25,2.45)--(5.25,0.75);
\node[right,black] at (5.35,1.2) {same P};
\end{tikzpicture}
$$

The power triangle provides several consistency checks. Its horizontal component
cannot exceed the apparent-power magnitude for a passive sinusoidal load. A reported
power factor magnitude greater than one signals a calibration, scaling, or sign error.
If $P$, $Q$, and $S$ are separately displayed, square the values and test the triangle
relation within the instrument resolution. A disagreement can also arise when the
displayed quantities use different averaging windows or when waveform distortion makes
the simple single-frequency phasor construction inapplicable.

Power-factor correction changes the reactive component seen by an upstream source.
With fixed real power in a sinusoidal load, lowering $|Q|$ lowers apparent power and
the required RMS current at fixed RMS voltage. The correction method is outside this
lesson; the accounting is not. Any added branch changes measured $P$, $Q$, and current
through wiring. Verify the result with simultaneous voltage-current records rather
than assuming a nominal correction value has reached the operating circuit.

$$
% caption: Power-factor correction at fixed real power. Reducing the reactive leg shortens the apparent-power hypotenuse at the same real-power coordinate, lowering the RMS current drawn from a fixed-voltage source.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.7,0.65)--(5.9,0.65) node[right] {real power};
\draw[->,black] (1.1,0.4)--(1.1,3.05) node[above] {reactive power};
\draw[->,black,very thick] (1.1,0.65)--(4.6,2.55);
\draw[->,acc,very thick] (1.1,0.65)--(4.6,1.45);
\draw[dashed,black] (4.6,0.65)--(4.6,2.55);
\node[black] at (2.9,2.25) {large S};
\node[acc] at (3.5,0.92) {small S};
\node[below,black] at (4.6,0.62) {same P};
\end{tikzpicture}
$$

Complex power applies to a sinusoidal steady-state phasor at one frequency. A single
pair of phasors cannot represent a waveform containing significant harmonic content,
switching edges, or nonperiodic transients. For those records, calculate real power
from the time-domain product or resolve the waveform into harmonics and calculate
compatible components. A meter may display apparent power as total RMS voltage times
total RMS current for a distorted waveform; that value need not satisfy the sinusoidal
triangle relation with any single $Q$.

The distinction matters when comparing instruments. A basic meter may estimate phase
from the voltage and current fundamentals, then display $VI\cos\phi$. A power analyzer
may multiply broadband sampled records and report total real power. Both can be valid
measurements of different quantities. The display labels, bandwidth, harmonic mode,
and averaging interval determine which interpretation applies.

## Synchronized voltage-current measurements

A wattmeter measures power by obtaining a voltage signal and a current signal with
known scale, polarity, timing, and bandwidth, then forming their product. Traditional
electrodynamic instruments implement the product through interacting voltage and
current coils. Digital power analyzers obtain sampled channels and perform the
multiplication numerically. The implementations differ, but each must establish the
same four facts: which terminals define the voltage, which conductor defines positive
current, which time reference aligns the channels, and which frequency content the
measurement path passes.

For sampled records $v[n]$ and $i[n]$ with sample interval $\Delta t$, an estimate
over $N$ samples is

$$
P_{\rm est}=\frac{1}{N}\sum_{n=0}^{N-1}v[n]i[n].
$$

If the records cover duration $N\Delta t$, the corresponding energy estimate is

$$
W_{\rm est}=\Delta t\sum_{n=0}^{N-1}v[n]i[n].
$$

The voltage and current samples must describe the same physical instant. Delaying one
channel by a few samples shifts its product with the other channel. The error can be
small for a resistive load and large for a low-power-factor load. A high sample rate
does not repair an unknown fixed channel delay; it only resolves the delay more
finely once the channels have been calibrated.

The voltage connection should span the same terminals used in the power definition.
Measuring upstream of a long lead includes lead drop and any intervening load. Measuring
across a load while sensing current in a neighboring branch combines unrelated
quantities. Draw a two-terminal boundary around the object whose absorbed power is
reported. Place the voltage sense points on that boundary and put the current sensor
in the one conductor crossing it. In a multiwire system, the choice of return path
and the number of voltage channels becomes part of the measurement definition.

Current can be measured with a calibrated shunt, a current transformer, a Hall probe,
or a dedicated current input. A shunt senses voltage proportional to current and has a
known polarity, but it adds series resistance and can heat. A clamp probe avoids
opening the conductor but has sensitivity, offset, phase, and bandwidth specifications.
A current transformer requires appropriate loading and is intended for alternating
current. Each sensor has an orientation mark. Reversing it turns absorbed real power
into delivered real power in the record. Establish orientation with a known resistive
load before measuring a reactive or distorted load.

Voltage probes introduce their own constraints. A grounded oscilloscope often ties
multiple probe reference clips together and to protective earth. Connecting two clips
to different floating circuit nodes can short those nodes through the instrument.
Measure a floating voltage with a rated differential probe, an isolated front end, or
a topology that keeps all ordinary probe references at one approved common node. The
probe voltage rating and common-mode range apply to the actual waveform, including
transients, rather than only the displayed RMS value.

The instrument bandwidth must exceed the frequency content that materially contributes
to the power product. A bandwidth too low rounds voltage or current peaks and can
underestimate RMS current, real power, or harmonic content. A sampling rate too low
aliases high-frequency content into lower-frequency artifacts. Apply a known analog
bandwidth limit before sampling when the sensor or analyzer lacks enough rate for the
full waveform. The cutoff and the residual spectrum should be reported because they
define which power quantity was measured.

For sinusoidal records, a time skew $\Delta t$ appears as a phase error
$\delta=\omega\Delta t$ under a fixed sign convention. If the true phase is $\phi$,
the measured average power can become

$$
P_{\rm meas}=VI\cos(\phi+\delta).
$$

Near unity power factor, a small phase error changes power modestly. Near quadrature,
the same error can create a large fractional error because the true real power is
small compared with $VI$. A low-power-factor measurement therefore needs a more
careful phase calibration than a resistive-load measurement with the same voltage and
current scales.

Calibrate gain and phase as a pair. Apply the same stable sinusoidal signal to both
complete channel paths, including cables, probes, attenuator settings, and current
sensor range. The measured amplitude ratio establishes relative gain; the measured
phase establishes channel skew. A known resistive reference provides a second check:
its voltage and current should be in phase within the expected reference impedance
and measurement uncertainty. Keep this calibration record with the power data. A
later probe replacement, range change, or software bandwidth setting can invalidate
the earlier correction.

Instrument uncertainty has several components. Voltage and current gain errors scale
the product. Phase error changes the product according to the operating power factor.
Offset and noise can bias a small signal. Finite ADC resolution can quantize narrow
current pulses. Sensor heating can change a shunt resistance. Current-probe position
can change coupling to nearby conductors. A credible result separates these terms
instead of attaching one unexplained percentage to a displayed watt value.

Repeated whole-cycle records estimate short-term repeatability. Change the source
level, current-sensor range, or integration window one at a time when locating a
discrepancy. A power value that changes after moving the voltage probe can indicate lead
drop. A value that changes only with current-probe orientation indicates a sign or
coupling issue. A value that drifts over minutes can indicate thermal change in the
load or sensor. The recorded diagnostic steps belong beside the final uncertainty.

## Distortion, harmonics, and true power factor

Many practical current records are not sinusoidal. Rectifier inputs, switching power
supplies, electronic drives, saturated magnetic circuits, and pulsed loads can draw
current concentrated near selected portions of the voltage cycle. RMS current then
includes harmonic components that do not share one phase angle with the voltage
fundamental. The time-domain definition remains the starting point:

$$
P=\frac{1}{T}\int_{t_0}^{t_0+T}v(t)i(t)\d t,
\qquad
\mathrm{PF}_{\rm true}=\frac{P}{V_{\rm rms}I_{\rm rms}}.
$$

True power factor combines displacement and distortion. It relates total RMS voltage
and total RMS current to measured real power over the stated interval. A display that
reports only the fundamental voltage-current phase gives a different quantity when the
waveform contains substantial harmonics.

Write periodic voltage and current as harmonic sums with a common fundamental angular
frequency:

$$
v(t)=V_0+\sum_{n=1}^{\infty}\sqrt2V_n\cos(n\omega_1t+\alpha_n),
$$

$$
i(t)=I_0+\sum_{n=1}^{\infty}\sqrt2I_n\cos(n\omega_1t+\beta_n).
$$

Over a complete fundamental period, products of different harmonic orders average to
zero. The real power becomes

$$
P=V_0I_0+\sum_{n=1}^{\infty}V_nI_n\cos(\alpha_n-\beta_n).
$$

The RMS magnitudes include squared harmonic terms:

$$
V_{\rm rms}^2=V_0^2+\sum_{n=1}^{\infty}V_n^2,
\qquad
I_{\rm rms}^2=I_0^2+\sum_{n=1}^{\infty}I_n^2.
$$

The harmonic formula requires a periodic record with a common fundamental frequency
and a complete fundamental-period average. Time-domain multiplication handles the
same situation without explicitly extracting harmonics and remains preferable for
waveforms with changing frequency or transient content.

$$
% caption: Nearly sinusoidal voltage with a pulsed current. Narrow current spikes carry modest average power yet raise the RMS current and lower the true power factor; the signed product must be averaged from simultaneous samples.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.7,1.5)--(6.1,1.5) node[right] {time};
\draw[->,black] (0.85,0.4)--(0.85,2.9) node[above] {value};
\draw[acc,very thick] plot[domain=1.0:5.8,samples=220] ({\x},{1.5+0.85*sin(360*(\x-1.0)/2.4)});
\draw[black,very thick] (1.0,1.5)--(1.45,1.5)--(1.6,2.35)--(1.75,1.5)--(3.85,1.5)--(4.0,2.3)--(4.15,1.5)--(5.8,1.5);
\node[acc] at (5.15,2.45) {voltage};
\node[black] at (2.7,2.05) {current pulse};
\end{tikzpicture}
$$

With nearly sinusoidal supply voltage, let $V=V_1$ and separate the current into its
fundamental RMS value $I_1$ and higher harmonics. The real power is then

$$
P=V_1I_1\cos\phi_1.
$$

The true power factor can be written

$$
\mathrm{PF}_{\rm true}=
\left(\frac{I_1}{I_{\rm rms}}\right)\cos\phi_1.
$$

The first factor is the current distortion factor. The second is the displacement
power factor. Harmonic current lowers the first factor even when the current
fundamental is in phase with the supply voltage. A phase-only meter can therefore
report a displacement factor close to one while a broadband power analyzer reports a
lower true power factor and a larger RMS current than a fundamental-only calculation.

Harmonic spectra locate the components responsible for distortion. A magnitude plot
shows RMS amplitude by harmonic order; a phase plot shows relative timing for each
component. Both are needed to reconstruct real-power contributions. A large harmonic
current magnitude does not determine its power contribution without the matching
voltage harmonic and their phase difference. In a distribution system with an
approximately sinusoidal voltage, high-order current harmonics often increase RMS
current and heating while contributing little real power. The exact result follows the
measured voltage spectrum, not a general slogan about harmonics.

$$
% caption: Example harmonic-current spectrum. The fundamental carries the main current, while higher odd harmonics add to total RMS current; matching voltage harmonics and their phases are still needed to assign real power.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.7,0.6)--(6.1,0.6) node[right] {harmonic order};
\draw[->,black] (0.7,0.6)--(0.7,3.05) node[above] {current RMS};
\draw[acc,very thick] (1.4,0.6)--(1.4,2.7);
\draw[black,very thick] (2.6,0.6)--(2.6,1.5);
\draw[black,very thick] (3.8,0.6)--(3.8,1.05);
\draw[black,very thick] (5.0,0.6)--(5.0,0.85);
\node[below,black] at (1.4,0.55) {1};
\node[below,black] at (2.6,0.55) {3};
\node[below,black] at (3.8,0.55) {5};
\node[below,black] at (5.0,0.55) {7};
\node[acc] at (1.4,2.9) {fundamental};
\node[black] at (3.55,1.75) {harmonics};
\end{tikzpicture}
$$

The term total harmonic distortion is often used to summarize higher harmonic RMS
content relative to a reference component. For current with no DC term,

$$
\mathrm{THD}_I=
\frac{\sqrt{I_{\rm rms}^2-I_1^2}}{I_1}.
$$

The ratio summarizes waveform shape, but it does not give real power by
itself. Two currents can have equal current THD and different true power factors
because their fundamental phases, voltage distortion, or harmonic phase relationships
differ. A THD report should identify the bandwidth, the harmonic cutoff, the reference
fundamental, the window length, and whether the analyzer removed DC before computing
the ratio.

Sampling and bandwidth choices can create false harmonic content. A sample rate below
twice the highest retained frequency aliases higher components into lower bins. A
noninteger record length relative to the fundamental causes spectral leakage, spreading
one harmonic across neighboring bins. A taper can reduce leakage for spectral display,
but it changes amplitude calibration unless the analyzer corrects for the window. For
real-power integration, retain a synchronized time-domain path and verify its result
against a harmonic sum only after both use compatible bandwidth and averaging rules.

Current crest factor is another instrument constraint. It is the ratio of current
peak to current RMS. A sharply pulsed current can have a high crest factor even when
its RMS value lies inside a meter's ordinary range. The input amplifier, current probe,
and ADC must accommodate the peak without clipping. Clipping a small fraction of a
current pulse can reduce calculated RMS current, alter harmonic content, and bias real
power. Check the raw maximum sample, overload indicators, and the specified crest
factor at the selected range.

Distortion also changes the meaning of reactive quantities. A single $Q=VI\sin\phi$
describes one sinusoidal frequency. Harmonic systems can report per-harmonic reactive
components, total nonactive power under a named standard, or only true power and
apparent power. Do not add a fundamental reactive display to a broadband apparent
power value and expect the sinusoidal power triangle to close. State the analyzer mode
and the mathematical definition supplied by that mode.

## Energy integration, meters, and reportable results

Energy is accumulated signed power. For a continuous record,

$$
E(t_1\to t_2)=\int_{t_1}^{t_2}p(t)\d t.
$$

For sampled records, sum the calibrated instantaneous products. A power trace may
vary rapidly within each line cycle while the energy trace changes smoothly on a much
longer time scale. The energy unit joule follows directly from watt-second. Electrical
energy reports often use kilowatt-hour:

$$
1\ \mathrm{kWh}=3.60\times10^6\ \mathrm J.
$$

A power analyzer or energy meter must maintain a calibrated clock as well as
calibrated voltage and current scales. A small constant power error accumulates linearly
with elapsed time. A clock error changes the duration multiplier. A meter reset,
rollover, missing sample interval, or time-zone conversion can create a discontinuity
in a long energy record even when the instantaneous power calculation was correct.

Energy meters can be configured to record import, export, or net energy according to
their installation and tariff arrangement. Under the passive sign convention at a
customer load, imported real power is positive. A local source can return power toward
the supply, producing negative real power at that boundary. A meter configured with
the opposite current orientation will reverse those labels. The meter's register names
and the service agreement define how signed intervals are accumulated and displayed.
Do not infer billing treatment from a power-factor sign or from a generic meter icon.

An energy bill can include more than accumulated real energy, depending on the local
tariff and customer class. Commonly separated quantities include real-energy use over
a billing interval, maximum or interval-average demand, and terms related to reactive
or apparent-power use. Their definitions belong to the actual tariff. Physics defines
the measured quantities but does not specify a universal billing formula. When
interpreting a bill, match each line item to the meter channel, averaging interval,
unit, and sign convention before comparing it with a laboratory power reading.

Demand is average power over a prescribed interval rather than the maximum single
sample. Over an interval of duration $\tau$,

$$
P_{\rm interval}=\frac{1}{\tau}\int_{t}^{t+\tau}p(t')\d t'.
$$

The interval may be fixed in clock time, rolling, or derived from another metering
rule. A short high-power event can raise an interval average even if the total energy
is modest. Conversely, two loads with equal energy use can have different maximum
interval demand because one concentrates its operation. A laboratory recorder should
state the integration duration before using the word demand.

> **Worked example (power triangle and energy for a lagging load).** A sinusoidal load
> measures $V=120\ \mathrm V$, $I=5.00\ \mathrm A$, and power factor $0.800$ lagging.
> Find the apparent, real, and reactive power, and the energy delivered over
> $3.50\ \mathrm h$ of steady operation.
>
> The apparent power is the RMS product, and real power is that scaled by the power
> factor:
> $$
> S=VI=(120\ \mathrm V)(5.00\ \mathrm A)=600\ \mathrm{VA},
> \qquad
> P=S\cos\phi=(600\ \mathrm{VA})(0.800)=480\ \mathrm W.
> $$
> The reactive magnitude closes the power triangle:
> $$
> |Q|=\sqrt{S^2-P^2}=\sqrt{(600\ \mathrm{VA})^2-(480\ \mathrm W)^2}=360\ \mathrm{var},
> $$
> positive because the current lags. Energy uses real power, not apparent power:
> $$
> E=P\,\Delta t=(0.480\ \mathrm{kW})(3.50\ \mathrm h)=1.68\ \mathrm{kWh}.
> $$
> Using $600\ \mathrm{VA}$ in place of $480\ \mathrm W$ here would overstate the
> transferred energy by twenty-five percent.

In a distorted case, begin with the measured time-domain real power rather than a
phase-only reconstruction.

$$
% caption: The worked operating point in the power plane. A 480 W real component and 360 var reactive component give a 600 VA apparent-power magnitude, drawn to scale; energy over time uses the real component alone.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.7,0.65)--(5.9,0.65) node[right] {watts};
\draw[->,black] (1.1,0.4)--(1.1,3.15) node[above] {vars};
\draw[->,black,very thick] (1.1,0.65)--(3.9,0.65) node[midway,below] {480 W};
\draw[->,black,very thick] (3.9,0.65)--(3.9,2.75) node[midway,right] {360 var};
\draw[->,acc,very thick] (1.1,0.65)--(3.9,2.75);
\node[acc] at (2.05,1.95) {600 VA};
\end{tikzpicture}
$$

Calibration converts raw samples to physical units. Let the calibrated channels be
$v=g_v(v_m-o_v)$ and $i=g_i(i_m-o_i)$, where $g_v$ and $g_i$ are gain factors and
$o_v$, $o_i$ are offsets in raw units. The calibrated real-power estimate is then

$$
P=\frac{1}{N}\sum_{n=0}^{N-1}
g_v\left(v_m[n]-o_v\right)
g_i\left(i_m[n]-o_i\right).
$$

Offset removal matters most for small signals and long averages. A DC voltage offset
times a DC current offset creates a false constant-power term. An offset times a real
waveform can create an additional error when the waveform has a DC component or an
uneven window. Measure channel offsets with inputs in the specified zero condition,
record the temperature and range, and repeat the check after a long high-current run.

Gain calibration should span the expected operating range. A voltage divider or probe
may be accurate at one amplitude and depart from its nominal ratio near range limits.
A shunt sensor can change resistance with temperature. A current clamp can have gain
and phase specifications that vary by range and frequency. Use traceable reference
sources or standards appropriate to the needed accuracy, then retain the calibration
date, conditions, and correction factors with the data.

With a nearly sinusoidal unity-power-factor reference, small independent relative gain
uncertainties give an approximate contribution

$$
\left(\frac{u(P)}{P}\right)^2
\approx
\left(\frac{u(g_v)}{g_v}\right)^2+
\left(\frac{u(g_i)}{g_i}\right)^2.
$$

Phase uncertainty adds a separate contribution. With a sinusoidal phase difference
$\phi$ and a small channel error $u(\delta)$ in radians, the local sensitivity is

$$
\frac{\partial P}{\partial\delta}=-VI\sin\phi.
$$

The phase term is small near unity power factor and becomes dominant near quadrature.
For distorted signals, evaluate uncertainty by propagating the calibrated sample
records, repeating complete acquisitions, or both. Treating a broadband trace as one
phasor discards uncertainty from harmonics, sampling, and waveform drift.

An audit-ready power report contains the measurement conditions and the result:

- **Boundary and signs.** Identify the load terminals, current direction, voltage
  polarity, source or load convention, and import or export definition.
- **Waveform and interval.** State RMS values, peak values when relevant, frequency,
  harmonic bandwidth, sampling rate, integration window, and whole-cycle treatment.
- **Instrument path.** List voltage probe, current sensor, ranges, scale factors,
  burden or inserted shunt, common-reference arrangement, and channel-delay result.
- **Calibrations.** Record gain references, phase reference, offset checks, timebase
  status, calibration date, and any correction applied to the raw channels.
- **Reported quantities.** Give real power, apparent power, power factor with leading
  or lagging label when meaningful, reactive definition or analyzer mode, energy,
  uncertainty method, and repeatability checks.

The report should separate direct observation from inference. A calibrated product of
simultaneous samples supports a real-power and energy result. A sinusoidal phasor fit
adds power-factor and complex-power interpretation when the waveform meets its scope.
A harmonic analysis adds spectral detail when its window and bandwidth are controlled.
Each layer remains traceable when the raw voltage-current records and calibration data
are preserved.

## Cross-checks and failure diagnosis

Begin a power measurement with a reference load whose behavior is known over the
measurement band. A stable resistor provides a direct check: voltage and current are
in phase, the signed average power is positive under the passive convention, and the
expected value is $V_{\rm rms}^2/R$ after allowing for lead and shunt resistance. A
negative result on this reference usually indicates a reversed current path or voltage
polarity. A nonzero phase on the reference can indicate channel delay, probe loading,
or a reference resistor that is no longer approximately resistive at the selected
frequency. Resolve these issues before interpreting a reactive or harmonic-rich load.

Perform a scale check at more than one point. A single gain check at a low voltage can
leave range-dependent error hidden near the operating voltage. A single current check
at a small current can leave sensor offset and clamp nonlinearity hidden near rated
current. Compare the analyzer's RMS voltage and current against independent calibrated
instruments within their common bandwidth. Then compare the analyzer's calculated
real power against an independent resistive reference. The three comparisons isolate
voltage scale, current scale, and product timing more effectively than one aggregate
watt reading.

Use the waveform itself as a diagnostic. Plot voltage, current, and instantaneous
power with common time coordinates. Look for clipped peaks, unexpected DC offset,
missing portions of a current pulse, beat envelopes, and trigger discontinuities.
Check the maximum raw sample against the input range. A display can show a plausible
RMS value while an unseen overload clips a few narrow peaks. Such clipping changes
RMS current and the power product in a way that an average-only display cannot distinguish.
For intermittent loads, retain several individual cycles as well as a long average;
the cycle-to-cycle variation may be the primary uncertainty source.

| validation check | quantity that should remain consistent | failure signature | next action |
|---|---|---|---|
| resistive reference | positive $V_{\rm rms}^2/R$ and near-zero phase | negative power or phase offset | reverse/check sensor polarity and channel delay |
| multilevel scale | voltage and current against independent standards | range-dependent residual | recalibrate gain, offset, or sensor burden |
| raw waveform | sample peak remains inside input range | clipped crest, missing pulse, beat, or DC offset | change range/bandwidth and reacquire |
| shifted whole-cycle window | real power stays within sampling uncertainty | start-phase dependence | synchronize or lengthen the record |

Window alignment needs an explicit check when the record is periodic. Shift the start
of a whole-cycle window by several fractions of a period and compare the resulting
real power. With a stable periodic waveform and an integer number of periods, the
result should remain within sampling and noise uncertainty. A systematic dependence
on start phase points to a noninteger cycle count, frequency drift, asynchronous
sampling, or a harmonic component not represented by the assumed period. Increase the
window duration or synchronize acquisition to a measured fundamental reference before
quoting a high-precision result.

Compare time-domain and phasor-domain quantities only within their common scope. A
sinusoidal voltage-current pair permits $P=VI\cos\phi$ and
$\widetilde S=\widetilde V\widetilde I^\ast$. The direct average of sampled products
should agree with that result after channel calibration. A discrepancy can indicate a
phase sign reversal, RMS-versus-peak confusion, a channel-skew error, or waveform
distortion. If harmonic content is visible, calculate real power from the sampled
product first. Then, if desired, compare it with a harmonic sum using matched
frequency bins and phase references.

Power-triangle checks apply only to sinusoidal quantities at one frequency. For a
sinusoidal record, verify that $S^2$ and $P^2+Q^2$ agree within uncertainty. Verify
that the power-factor magnitude equals $P/S$ and remains no greater than one. For a
distorted record, retain total real power, total RMS voltage, total RMS current, and
true power factor. Do not force a broadband analyzer display into a single-frequency
triangle unless the analyzer documentation defines an equivalent non-sinusoidal
quantity and the report names that definition.

Long energy records need reconciliation. Integrate the same calibrated power samples
into shorter blocks, then sum the block energies and compare with the full-record
integral. The results should match apart from deliberate rounding. Compare elapsed
time from the sample clock with elapsed time from an independent time reference. Log
instrument resets, configuration changes, power interruptions, and missing blocks.
An energy total without a data-completeness record cannot distinguish zero load from
unrecorded time.

Power-factor interpretation requires the operating point. A value of 0.80 at a small
current and the same value at a large current can have very different implications for
conductor heating and source capacity because apparent power scales with $VI$. Record
the simultaneous RMS voltage and current before comparing power factors across tests.
In a distorted load, record the fundamental displacement factor and the true power
factor separately when both are available. Their difference locates whether phase or
harmonics dominate the reduction in real power per RMS ampere.

The same discipline applies to energy and tariff interpretation. A laboratory energy
integral answers how much signed real energy crossed the selected boundary during its
record. A service meter may use a different boundary, a different clock interval, and
register-specific rules. Compare like quantities only after aligning sign, units,
interval, and meter configuration. The service contract sets the billing rule; the
waveform measurement provides physical evidence used to check a stated
quantity.

Preserve raw synchronized samples or phasors, calibration files, sensor orientation
photographs, a circuit-boundary drawing, instrument settings, timestamps, and analysis
code or formulas. Power measurement errors often originate in a changed probe range, a
moved clamp, or a forgotten time-base setting rather than in the algebra. The preserved
bundle lets a later analysis test those possibilities without reconstructing the
electrical setup from memory.

## A compact numerical reduction

> **Worked example (true power factor from synchronized records).** Synchronized,
> calibrated records give $V_{\rm rms}=120.0\ \mathrm V$ and $I_{\rm rms}=2.50\ \mathrm A$.
> Direct multiplication of the sampled records gives $P=240\ \mathrm W$, and a sine fit
> to the fundamental yields a current lag of $36.9^\circ$. Find the apparent power, true
> power factor, reactive magnitude, and the energy over $30\ \mathrm{min}$.
>
> The apparent power and true power factor follow from the RMS product:
> $$
> S=(120.0\ \mathrm V)(2.50\ \mathrm A)=300\ \mathrm{VA},
> \qquad
> \mathrm{PF}_{\rm true}=\frac{P}{S}=\frac{240\ \mathrm W}{300\ \mathrm{VA}}=0.800.
> $$
> The record is sinusoidal, so the displacement cosine agrees:
> $\cos(36.9^\circ)=0.800$. The reactive magnitude closes the triangle,
> $$
> |Q|=\sqrt{S^2-P^2}=\sqrt{(300\ \mathrm{VA})^2-(240\ \mathrm W)^2}=180\ \mathrm{var},
> $$
> positive under the lagging convention. Over $30\ \mathrm{min}$ of steady power,
> $E=(0.240\ \mathrm{kW})(0.500\ \mathrm h)=0.120\ \mathrm{kWh}$. Three checks hold at
> once: $P/S<1$, the sine-fit phase matches the product average, and the energy uses
> watts, not VA.

Now take the same RMS readings with a pulsed current waveform. Direct sample
multiplication may still give $240\ \mathrm W$, hence the same true power factor of
$0.800$, while a fundamental-only phase fit gives a displacement factor of $0.950$.
The gap signals current distortion: the current fundamental is only
$0.800/0.950=0.842$ of the total RMS current under a sinusoidal supply voltage, so a
one-angle power triangle no longer represents the waveform. Retain $P$, $V_{\rm rms}$,
$I_{\rm rms}$, true power factor, fundamental displacement factor, harmonic bandwidth,
and the time-domain integration method instead of one ambiguous reactive value.

A reliable power meter needs more than voltage and current scales. Phase calibration
determines whether the sinusoidal cross-check is meaningful. Bandwidth determines
whether pulsed current is retained. The integration window determines whether the
energy result represents the required interval. Each condition
changes a different part of the calculation and should remain visible beside the
final number.

Repeat the numerical reduction with the current reference deliberately reversed in a
controlled record. Voltage and RMS-current magnitudes remain unchanged, while the
signed real power, reactive sign, and imported or exported energy labels reverse. The
exercise verifies that the analysis preserves terminal convention rather than silently
taking an absolute value. Restore the documented sensor orientation before acquiring
the final record.

Compare a short whole-cycle calculation with a longer block calculation at the same
steady operating point. Their real-power means should agree within repeatability; the
longer block reduces random noise and exposes slow drift. Their energy totals should
scale with duration. A disagreement indicates a timebase issue, a nonstationary load,
or a window that does not contain an integer number of the relevant waveform periods.
These simple comparisons test the measurement chain before a power factor or energy
result is used for design, calibration, or billing interpretation.