---
title: Reactance
module: Alternating Current
moduleNumber: 9
lessonNumber: 2
order: 902
summary: >
  A resistor obeys Ohm's law instant by instant, but a capacitor responds to how fast
  its voltage changes and an inductor to how fast its current changes. Under a steady
  sinusoid that rate-dependence collapses to a fixed quarter-cycle phase shift and a
  frequency-dependent amplitude ratio, the reactance. We derive $X_C=1/(\omega C)$ and
  $X_L=\omega L$, adopt phasors to turn the defining derivatives into multiplication by
  $j\omega$ so a single complex impedance carries amplitude and phase together, and
  track the energy an ideal reactance stores and returns without dissipating it. Real
  windings and dielectrics add loss, leakage, and self-resonance that bound where the
  ideal formulas hold.
topics: [Alternating Current]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 29 — Alternating-Current Circuits; §29-2 Alternating Current in Inductors and Capacitors"
---

## Reactive Elements

Capacitors and inductors respond to a changing electrical state. A capacitor relates
its current to the rate at which its terminal voltage changes; an inductor relates its
terminal voltage to the rate at which its current changes. Neither relation has the
instant-by-instant form of Ohm's law. That distinction produces a phase difference
under sinusoidal conditions and permits energy to move back and forth between source
and component without becoming heat in an ideal element.

With the passive sign convention, current enters the terminal marked positive for
voltage. The defining relations are

$$
i_C(t)=C\frac{\d v_C}{\d t},
\qquad
v_L(t)=L\frac{\d i_L}{\d t}.
$$

The symbols $C$ and $L$ denote capacitance and inductance. Capacitance has units of
farads, with one farad equal to one coulomb per volt. Inductance has units of henries,
with one henry equal to one volt-second per ampere. The signs carry physics: reversing
either reference direction reverses one measured waveform and changes the numerical
phase assigned to it. A circuit sketch, probe polarity, and current-arrow direction
should therefore accompany a phase measurement.

A sine wave retains its frequency under differentiation, which advances phase by a
quarter cycle. Integration retains frequency and delays phase by a quarter
cycle. These calculus facts explain the familiar capacitor and inductor phase rules;
the rules are consequences of the constitutive equations rather than independent
mnemonics. They also explain why a frequency value is essential. The same component
has a different current-to-voltage ratio at a different drive frequency.

$$
% caption: Terminal conventions for the two reactive elements. In each drawing the labelled current enters the terminal marked positive, fixing the sign in the constitutive relations $i_C=C\,\d v_C/\d t$ and $v_L=L\,\d i_L/\d t$ before any sine-wave phase comparison is made.
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## Component Nonidealities

An inductor contains a winding, an insulating structure, and often a magnetic core.
Its DC winding resistance, its AC winding resistance, core loss, and interturn
capacitance all enter a terminal measurement. Over a limited low-to-middle frequency
range, the familiar approximation

$$
Z_L\approx R_{\rm ac}+j\omega L
$$

is often adequate. The resistance in that expression is the AC resistance under the
actual current distribution, temperature, and frequency. It need not equal a
two-wire room-temperature resistance measurement. A broader small-signal model places
a core-loss branch and a parasitic capacitance across the winding branch:

$$
Z_{L,\rm real}\approx
\left[\left(R_{\rm ac}+j\omega L\right)\parallel R_{\rm core}\right]
\parallel\frac{1}{j\omega C_{\rm par}}.
$$

The symbols name mechanisms, not universal constants. Core loss changes
with flux amplitude and waveform. Parasitic capacitance is distributed among turns,
the core, shielding, and nearby conductors. A model fitted to a low-amplitude sweep
may therefore fail under a large DC bias or a high-current ripple condition.

Winding loss begins with conductor resistance. A winding resistance measured at
reference temperature $T_0$ has the first correction

$$
R(T)\approx R(T_0)\left[1+\alpha\left(T-T_0\right)\right],
$$

where $\alpha$ is the conductor temperature coefficient over the selected range.
Copper's value near room temperature is about $0.00393\ \mathrm{K}^{-1}$. A coil that
warms from DC current can thus change the loss seen by a superposed AC ripple. At
higher frequency, skin effect concentrates current near the conductor surface and
proximity effect redistributes it in response to neighboring turns. Both effects
raise $R_{\rm ac}$ above the DC value. Fine stranded wire, foil geometry, turn spacing,
and winding arrangement alter that rise; a catalogue DC-resistance entry cannot
substitute for an AC-loss curve.

$$
P_{\rm cu}=I_{\rm rms}^2R_{\rm ac}.
$$

The winding current contributes to core temperature, insulation temperature, and
terminal temperature through different thermal paths. A small package can meet a
current rating in moving air and exceed its allowed rise in a closed enclosure. The
measurement should record the duration before temperature is sampled, because copper
loss changes as the winding warms and a core may lag the winding thermally.

Magnetic material adds another operating limit. With $N$ turns and flux $\Phi$ linked
to the winding, the induced winding voltage follows

$$
v=N\frac{\d\Phi}{\d t}.
$$

The ideal relation between RMS voltage and peak flux in an approximately uniform core
under sinusoidal excitation is

$$
V_{\rm rms}=4.44fN\Phi_{\rm pk}.
$$

The voltage inducing core flux excludes any appreciable copper drop. Use the full
terminal voltage only when that drop is negligible. Lowering frequency at fixed applied
RMS voltage increases the required flux swing. A DC current offset shifts the magnetic
operating point. If the material approaches saturation, the incremental inductance
$L_{\rm inc}=\d\lambda/\d i$ declines, the current waveform becomes distorted, and the
ideal phase relation no longer describes the whole cycle. The winding can then draw
more ripple current than a constant-$L$ calculation predicts.

Core loss is commonly reported through material-specific curves or an empirical law
of the form

$$
p_{\rm core}\approx kf^aB_{\rm pk}^b.
$$

The constants and exponents depend on material, temperature, waveform, sample shape,
and measurement method. This expression supports interpolation only within the data
range used to determine it. A square-wave drive, a DC-biased waveform, or a different
air gap changes the loss problem. State whether the reported loss is per core, per
volume, or per mass before combining it with winding loss.

Parasitic capacitance eventually shunts turns and changes the measured high-frequency
impedance. An inductor can show an inductive rising region, a turning region, and then
a capacitive falling region as frequency rises. The turning frequency depends on coil
construction, core geometry, mounting metal, and measurement leads. It marks the
boundary of the single-inductor model. A high-frequency use needs impedance and phase
data beyond the intended band; an extrapolated low-frequency inductance is
insufficient.

Separate the stated ratings before selecting an inductor. Saturation-current rating
addresses a drop in inductance under DC bias; RMS-current or temperature-rise rating
addresses heating; voltage and frequency limits address flux and insulation stress.
Each rating constrains a distinct operating condition. A component may tolerate its
rated DC current yet overheat from a large AC ripple, or remain cool while losing
enough inductance under bias to miss a current-control target. A complete operating point includes DC current, ripple
amplitude, waveform, frequency, ambient temperature, cooling arrangement, and the
measurement bandwidth used to assess current peaks.

### Frequency windows and model boundaries

Reactance has a numerical meaning only after the frequency and the terminal quantity
have been stated.

> **Worked example (reactance across two frequency decades).** A $100\ \mathrm{nF}$
> capacitor and a $10\ \mathrm{mH}$ inductor are each driven at $1.00\ \mathrm{kHz}$
> and then at $100\ \mathrm{kHz}$. Find their reactances at both frequencies.
>
> At $1.00\ \mathrm{kHz}$,
> $$
> X_C=\frac{1}{2\pi(1.00\times10^3)(100\times10^{-9})}=1.59\ \mathrm{k\Omega},
> \qquad
> X_L=2\pi(1.00\times10^3)(10.0\times10^{-3})=62.8\ \Omega.
> $$
> Reactance scales as $f^{\pm1}$, so raising the frequency by a factor of $100$ divides
> $X_C$ by $100$ and multiplies $X_L$ by $100$:
> $$
> X_C(100\ \mathrm{kHz})=15.9\ \Omega,
> \qquad
> X_L(100\ \mathrm{kHz})=6.28\ \mathrm{k\Omega}.
> $$
> The two reactances swap roles across the sweep: the capacitor falls from
> $1.59\ \mathrm{k\Omega}$ to $15.9\ \Omega$, the inductor rises from $62.8\ \Omega$
> to $6.28\ \mathrm{k\Omega}$.

A capacitor selected to block a low-frequency signal can become a low-impedance
current path at the upper end of a sweep. An inductor that limits ripple at one
frequency can have little reactance at a much lower frequency. The source impedance,
lead resistance, and component ratings determine whether those calculated currents
can actually occur.

$$
% caption: Ideal reactance on logarithmic axes. The capacitor $X_C=1/(\omega C)$ falls one decade per decade of frequency; the inductor $X_L=\omega L$ rises one decade per decade. They cross where the two reactances are equal, a point set by the component values.
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The limiting cases require physical interpretation. The expression
$X_C=1/(\omega C)$ tends to infinity as frequency tends to zero, yet a voltage step
can produce a large finite charging current for a short interval. The expression
$X_L=\omega L$ tends to zero as frequency tends to zero, yet an inductor with
winding resistance settles to a finite DC current. A low-frequency sine test has a
finite period and needs a duration long compared with earlier transients before a
steady-state phasor describes the record.

Frequency selection must respect the lumped-element boundary. The source wavelength,
lead length, fixture dimensions, and probe ground lead introduce distributed delay
when their electrical length is no longer small. At that point a single terminal
voltage may differ across the physical component package or fixture, and a source
may launch a reflected waveform into a cable. The reactance formula remains a local
constitutive relation for the ideal element; the wiring needs a transmission-line or
distributed-network description. A short lead at audio frequency normally meets the
lumped assumption. The same lead can distort a measurement made with a rapid edge or
at radio frequency.

The log slopes provide a compact diagnostic during a frequency sweep. After taking
the magnitude of the measured impedance, a near-constant factor-of-ten decrease in
$|Z|$ for every factor-of-ten increase in $f$ supports a capacitive range. A
near-constant factor-of-ten increase supports an inductive range. A flat region points
to resistance, a measurement floor, or a parasitic limit. The phase record separates
these possibilities: a nearly capacitive region has phase near $-90^\circ$, a nearly
inductive region has phase near $+90^\circ$, and a resistive region has phase near
$0^\circ$ after the measurement reference has been corrected.

### Real capacitors: loss, leakage, and lead inductance

An ideal capacitor contains only $C$. A manufactured capacitor has conducting plates,
dielectric material, terminations, and leads. Over a stated frequency range, a first
model places an equivalent series resistance $R_{\rm ESR}$ and equivalent
series inductance $L_{\rm ESL}$ in series with the intended capacitance. A large
leakage resistance $R_{\rm leak}$ lies in parallel with the capacitance for a model
that also describes slow DC discharge. The approximate terminal impedance is

$$
Z_{C,\rm real}\approx R_{\rm ESR}+j\omega L_{\rm ESL}
+\left(\frac{1}{R_{\rm leak}}+j\omega C\right)^{-1}.
$$

No single three-element circuit describes every capacitor across every amplitude,
temperature, age, and frequency. The model separates low-frequency leakage, midband
loss, and high-frequency lead inductance. At low frequency, finite leakage changes the long-time current and makes the phase
less nearly capacitive. In the normal capacitive range, $R_{\rm ESR}$ produces heating
and shifts the phase toward zero. At high frequency, lead and package inductance
eventually dominate. The measured component then appears inductive even though its
intended function is capacitive.

Equivalent series resistance varies with frequency and temperature. A sinusoidal
branch current dissipates

$$
P_{\rm ESR}=I_{\rm rms}^2R_{\rm ESR}.
$$

> **Worked example (ripple heating in a capacitor's ESR).** A capacitor with
> $R_{\rm ESR}=0.080\ \Omega$ carries $I_{\rm rms}=2.50\ \mathrm A$ of ripple current.
> Find the power dissipated inside it.
> $$
> P_{\rm ESR}=I_{\rm rms}^2R_{\rm ESR}=(2.50\ \mathrm A)^2(0.080\ \Omega)
> =0.500\ \mathrm W.
> $$
> The ideal capacitance dissipates zero average power, yet this $0.500\ \mathrm W$ of
> real heating can exceed the component's thermal limit — which is why a ripple-current
> rating stands beside the capacitance and voltage ratings.

That heat can exceed the component's thermal limit even when the ideal capacitor has
zero average power. Ripple-current ratings therefore matter alongside capacitance and
voltage rating. The rating often assumes a
particular ambient temperature, airflow, frequency range, and permitted internal
temperature rise. A much lower series resistance in a different capacitor can reduce
heating but may come with a different capacitance change under bias or a different
voltage limit.

For $R_{\rm ESR}\ll X_C$, the series model gives

$$
\tan\delta\approx\frac{R_{\rm ESR}}{X_C}
=\omega C R_{\rm ESR}.
$$

Here $\delta$ measures the departure from an ideal negative-right-angle impedance.
State the series or parallel model and the frequency with any reported value.
A quoted dissipation factor at $120\ \mathrm{Hz}$ does not automatically predict loss
at $1\ \mathrm{MHz}$. In particular, a measurement system that treats every loss as
a constant resistor can fit one narrow sweep region while missing dielectric and
electrode processes elsewhere.

The frequency at the minimum is often approximated by

$$
f_{\rm self}\approx\frac{1}{2\pi\sqrt{L_{\rm ESL}C}}.
$$

The value identifies the onset of package-inductance effects. The minimum is broadened
and shifted by series loss, mounting geometry, dielectric behavior, and the test
fixture. A capacitor used
for high-frequency bypassing should have its impedance curve checked over the band
where the load creates current demand. The nominal capacitance alone cannot establish
that performance.

Leakage and dielectric absorption belong to slow-time behavior. After a charged
capacitor is briefly discharged, some dielectric systems develop a small recovered
terminal voltage. Bound-charge relaxation produces that recovery; it is not evidence
that charge crossed an ideal dielectric gap. A leakage test must state the charging
time, discharge interval, measurement input resistance, temperature, and whether the
meter is connected continuously. Without those conditions, two reported leakage
currents may describe different physical histories.

Capacitance itself can depend on voltage bias, temperature, frequency, and mechanical
stress. The size and direction of those changes depend strongly on dielectric family.
Film and some ceramic parts serve different accuracy and energy-density requirements;
an application needs the curve for the selected part rather than an assumption based
on the word capacitor. A bias-dependent capacitance changes the predicted reactance,
the ripple current, and the apparent phase. Measure the component at the operating
DC bias and AC amplitude whenever the tolerance budget is tight.

## Phasors and Impedance

A time trace makes phase visible, but it is awkward for repeated sinusoidal
calculations. A phasor replaces a sine wave at one angular frequency by a directed
complex quantity. Its length represents an amplitude and its angle represents phase
relative to a stated reference. For RMS phasors, a voltage waveform
$v(t)=\sqrt{2}V\cos(\omega t+\phi)$ is represented by $\widetilde V=V e^{j\phi}$.
The time dependence is understood to be common to every phasor in the calculation.
The symbol $j$ is used for the square root of negative one in electrical engineering
notation, leaving $i$ available for current.

Multiplication by $j$ rotates a phasor by positive ninety degrees; multiplication by
$-j$ rotates it by negative ninety degrees. The capacitor relation and inductor
relation therefore become

$$
\widetilde I_C=j\omega C\widetilde V_C,
\qquad
\widetilde V_L=j\omega L\widetilde I_L.
$$

Solving each relation for voltage divided by current defines the complex impedances

$$
Z_C=\frac{\widetilde V_C}{\widetilde I_C}=-\frac{j}{\omega C},
\qquad
Z_L=\frac{\widetilde V_L}{\widetilde I_L}=j\omega L.
$$

The magnitude of either impedance is the corresponding reactance. The sign of its
imaginary part carries phase information that a positive reactance number alone does
not contain. A capacitor has negative imaginary impedance, consistent with current
leading voltage. An inductor has positive imaginary impedance, consistent with voltage
leading current. Treating $X_C$ and $X_L$ as ordinary positive resistances erases this
phase information and leads to incorrect circuit reductions.

$$
% caption: Phasors capture the quarter-cycle phase difference without a full time trace. For the capacitor the current phasor is rotated a right angle ahead of the voltage; for the inductor the voltage phasor is rotated a right angle ahead of the current.
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Phasors are not a second physical voltage. They are bookkeeping devices valid for
linear circuits in sinusoidal steady state at one frequency. A square wave contains
many frequencies, and a switching transient is not described by a single phasor.
Individual Fourier components can each be treated with a phasor, but the final
time-domain waveform must then be reconstructed from all of them. Similarly, a
nonlinear inductor near saturation does not retain one fixed $L$, so its current may
not remain sinusoidal even under sinusoidal drive.

The impedance form has a dimensional check. Both $j\omega L$ and
$-j/(\omega C)$ have units of ohms. The algebraic sign is a phase sign, not a claim
that a passive component has negative energy or negative heating. In an ideal reactive
element, the average conversion to heat is zero because voltage and current are
quadrature signals. The component still may carry substantial RMS current and may
have substantial voltage across its terminals. Ratings based on current, voltage,
dielectric stress, winding temperature, and magnetic flux can be exceeded even when
the ideal average power is zero.

$$
% caption: The impedance plane separates capacitive from inductive response. Equal reactance magnitudes sit on opposite sides of the real axis: the inductor at $+j\omega L$, the capacitor at $-j/(\omega C)$, because the two elements rotate phase in opposite senses.
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When a resistor is present with a reactive component, the real and imaginary pieces
remain separate. A series resistor and inductor, for example, have
$Z=R+j\omega L$; a series resistor and capacitor have $Z=R-j/(\omega C)$. The
impedance magnitude determines the ratio of RMS voltage magnitude to RMS current
magnitude, while its angle gives the voltage-current phase difference. Those facts
extend directly to networks of several components. A single-component measurement
must still establish whether the real device follows its ideal impedance over the
reported frequency, bias, and amplitude range.

## Energy and Reactance

Instantaneous power into any two-terminal component is $p(t)=v(t)i(t)$ under the
passive sign convention. Substituting $i=C\,\d v/\d t$ for a capacitor gives

$$
p_C=Cv_C\frac{\d v_C}{\d t}
=\frac{\d}{\d t}\!\left(\frac{1}{2}Cv_C^2\right).
$$

The stored electric-field energy is consequently

$$
U_C=\frac{1}{2}Cv_C^2.
$$

Positive instantaneous power increases stored energy. Negative instantaneous power
means the capacitor is returning energy through its terminals. Under a sinusoidal
voltage, this exchange happens twice per period: field energy rises from zero to a
maximum, falls back to zero, then rises again. Energy itself is nonnegative, whereas
the direction of its transfer may reverse. That difference prevents a common error in
which a negative instantaneous power segment is described as negative stored energy.

$$
% caption: Capacitor energy exchange over a cycle. Terminal power alternates in sign about zero while the stored electric-field energy $\tfrac12 Cv_C^2$ rises and falls between zero and a positive maximum at twice the drive frequency.
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Substitution of $v=L\,\d i/\d t$ for an inductor gives

$$
p_L=Li_L\frac{\d i_L}{\d t}
=\frac{\d}{\d t}\!\left(\frac{1}{2}Li_L^2\right),
\qquad
U_L=\frac{1}{2}Li_L^2.
$$

The interpretation parallels the capacitor but the energy occupies the magnetic field
associated with the current. An ideal inductor absorbs energy while current magnitude
increases and returns energy while current magnitude decreases. Its stored energy is
largest at the positive and negative current peaks, because the square removes the
sign of current. It is zero when the ideal inductor current is zero. The energy
formula is valid only while the stated inductance represents the magnetic state. A
core driven into saturation requires an energy calculation from the measured flux or
current-dependent inductance. A constant small-signal value does not describe that
state.

The average ideal reactive power over any whole number of periods is zero, not because
power is absent, but because equal positive and negative energy transfers cancel.
This cancellation depends on a complete averaging interval. Over a fraction of a
cycle, an ideal capacitor or inductor can absorb or return nonzero net energy. In a
measurement, an integration window that starts and ends at arbitrary phase can report
a nonzero average even with a lossless model. The interval must therefore be stated,
or the record must contain an integer number of settled cycles, when average power is
used to estimate component loss.

Reactive energy transfer can still burden a source and wiring. A large capacitor at
high frequency may draw high RMS current with little ideal average power. A large
inductor at low reactance can do the same. Current causes copper loss in leads,
switches, and source resistance; voltage can stress insulation and probe inputs. The
statement that an ideal component dissipates no average power is local to that ideal
element. It does not eliminate losses elsewhere in the real circuit.


The ideal equations assume a lumped component: terminal dimensions and connecting
wires are small enough that one voltage and one current describe the element at the
frequency of interest. That approximation can fail at radio frequency, along long
leads, or during a fast transient. It applies when the stated frequency range, layout,
and measurement bandwidth support it. The ideal model is a limited description whose
accuracy must be checked against the intended measurement.

### Capacitor current and capacitive reactance

Apply a voltage $v_C(t)=V_0\cos(\omega t)$ to an ideal capacitor. Differentiation
gives

$$
i_C(t)=-\omega C V_0\sin(\omega t)
=\omega C V_0\cos\!\left(\omega t+\frac{\pi}{2}\right).
$$

Current reaches its positive peak one quarter period before voltage reaches its
positive peak. In words, capacitor current leads capacitor voltage by ninety degrees.
Equally, capacitor voltage lags its current by ninety degrees. Both statements name
the same pair of traces; the reference signal determines the wording. At voltage
extrema, the capacitor charge is
then largest in magnitude, but its rate of change is zero, so the current is zero. At
zero voltage, charge passes through zero while changing most rapidly, so current has
its largest magnitude.

$$
% caption: Ideal capacitor voltage and current. The current leads the voltage by a quarter period: it reaches its positive maximum before the voltage does, and it is zero at the voltage extrema where the plate charge momentarily stops changing.
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\draw[acc,very thick] plot[domain=0.95:5.80,samples=220] ({\x},{1.55+0.92*cos(360*(\x-2.35)/2.80)});
\draw[black,very thick] plot[domain=0.95:5.80,samples=220] ({\x},{1.55+0.92*cos(360*(\x-2.35)/2.80+90)});
\node[acc,fill=white,inner sep=1pt] at (2.85,2.75) {voltage};
\node[black,fill=white,inner sep=1pt] at (1.55,2.75) {current};
\draw[<->,black] (1.65,0.55)--(2.35,0.55) node[midway,below] {quarter period};
\end{tikzpicture}
$$

The ratio of voltage amplitude to current amplitude is called capacitive reactance:

$$
X_C=\frac{1}{\omega C},
\qquad
I_0=\frac{V_0}{X_C},
\qquad
I_{C,\rm rms}=\frac{V_{C,\rm rms}}{X_C}.
$$

Reactance is measured in ohms because it relates voltage and current amplitudes, but
it is not a resistance. A larger $X_C$ means less current for a fixed sinusoidal
voltage amplitude. Doubling frequency or capacitance halves $X_C$, so a capacitor
passes more alternating current at higher frequency. At zero frequency, the ideal
formula tends to infinite reactance: after the finite charging interval of a DC step,
the ideal capacitor carries no steady DC current.

$$
% caption: Capacitive reactance $X_C=1/(\omega C)$ falls as frequency rises. For a fixed capacitance, doubling the frequency halves the voltage-to-current amplitude ratio, so a capacitor passes more current at higher frequency.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.70,0.55)--(6.00,0.55) node[right] {frequency};
\draw[->,black] (0.70,0.55)--(0.70,2.95) node[above] {reactance};
\draw[acc,very thick] plot[domain=0.95:5.75,samples=220] ({\x},{0.45+1.95/\x});
\draw[dashed,black] (2.10,0.55)--(2.10,1.38)--(0.70,1.38);
\draw[dashed,black] (4.20,0.55)--(4.20,0.91)--(0.70,0.91);
\node[below] at (2.10,0.55) {lower f};
\node[below] at (4.20,0.55) {higher f};
\end{tikzpicture}
$$

“Passes AC” describes conduction current in the wires; charge does not cross the
dielectric gap. Charge accumulates on one plate while charge of opposite sign is
removed from the other. The electric field and changing plate charge establish the
terminal relation. During the following half cycle the process reverses. A current
probe placed in one lead records the same branch current that charges and discharges the
plates, even though ideal dielectric material does not conduct through its thickness.

In laboratory work, calculate $X_C$ at the actual measured frequency rather than a
nominal source setting. A source frequency error directly changes the predicted
current. If voltage is measured across a capacitor that has a series resistor or a
nonzero source resistance, use the capacitor terminal voltage, not automatically the
generator display. The generator display may refer to an open-circuit amplitude; the
capacitor current then changes the loaded output. A simultaneous voltage and current
record tests both the loading assumption and the expected quarter-cycle phase.

### Inductor voltage and inductive reactance

Prescribe $i_L(t)=I_0\cos(\omega t)$ for an ideal inductor. Differentiation gives

$$
v_L(t)=-\omega L I_0\sin(\omega t)
=\omega L I_0\cos\!\left(\omega t+\frac{\pi}{2}\right).
$$

Inductor voltage leads inductor current by ninety degrees, or current lags voltage by
ninety degrees. The physical interpretation uses the changing magnetic field. A large
rate of current change requires a large terminal voltage. At maximum current, the
current slope is zero and ideal inductor voltage is zero. When current crosses zero,
its slope is largest in magnitude and inductor voltage is at an extremum. These points
offer a quick sign check when comparing an oscilloscope record with the stated current
reference.

$$
% caption: Ideal inductor voltage and current. The voltage leads the current by a quarter period, because terminal voltage is proportional to the current slope: it peaks where the current is rising fastest and is zero at the current extrema.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.55,1.55)--(6.05,1.55) node[right] {time};
\draw[->,black] (0.80,0.30)--(0.80,2.95) node[above] {level};
\draw[acc,very thick] plot[domain=0.95:5.80,samples=220] ({\x},{1.55+0.92*cos(360*(\x-2.35)/2.80+90)});
\draw[black,very thick] plot[domain=0.95:5.80,samples=220] ({\x},{1.55+0.92*cos(360*(\x-2.35)/2.80)});
\node[acc,fill=white,inner sep=1pt] at (1.55,2.75) {voltage};
\node[black,fill=white,inner sep=1pt] at (2.85,2.75) {current};
\draw[<->,black] (1.65,0.55)--(2.35,0.55) node[midway,below] {quarter period};
\end{tikzpicture}
$$

Inductive reactance is

$$
X_L=\omega L,
\qquad
I_0=\frac{V_0}{X_L},
\qquad
I_{L,\rm rms}=\frac{V_{L,\rm rms}}{X_L}.
$$

It rises in direct proportion to frequency. An ideal inductor therefore carries less
sinusoidal current from a fixed-voltage source as frequency rises. At zero frequency,
the ideal expression gives zero reactance. That result describes the inductor after
the transient has settled: a lossless ideal coil is a short circuit for steady DC.
Real winding resistance and source resistance prevent unlimited current in an actual
DC experiment, and magnetic saturation can make the inductance depend on current.

$$
% caption: Inductive reactance $X_L=\omega L$ rises in direct proportion to frequency. The straight line through the origin on ordinary axes distinguishes an inductor from a capacitor, whose reactance falls over the same range.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.70,0.55)--(6.00,0.55) node[right] {frequency};
\draw[->,black] (0.70,0.55)--(0.70,2.95) node[above] {reactance};
\draw[acc,very thick] (0.70,0.55)--(5.75,2.70);
\draw[dashed,black] (2.10,0.55)--(2.10,1.15)--(0.70,1.15);
\draw[dashed,black] (4.20,0.55)--(4.20,2.04)--(0.70,2.04);
\node[below] at (2.10,0.55) {lower f};
\node[below] at (4.20,0.55) {higher f};
\end{tikzpicture}
$$

The voltage across an inductor includes any voltage across winding resistance only if
the measurement terminals include that resistance. Data sheets may quote inductance
for a small test signal and a specified DC bias. A coil driven near its current limit
can have a smaller incremental inductance because the magnetic core approaches
saturation. A calculation using a single catalogue $L$ then predicts too much
reactance and too little current. The relevant quantity is the inductance measured at
the drive amplitude, bias, temperature, and frequency of the application.

## Frequency-Response Measurement

Measure a reactive component as a two-terminal device. A sine source, a known series
sense resistor $R_s$, and the device under test form a simple arrangement. Record the
complex voltage across the sense resistor and the complex voltage across the device
with polarities marked on the drawing. The sense-resistor voltage provides branch
current; the device voltage divided by that current gives the terminal impedance:

$$
\widetilde I=\frac{\widetilde V_R}{R_s},
\qquad
\widetilde Z_D=R_s\frac{\widetilde V_D}{\widetilde V_R}.
$$

The ratio is valid even when the source has a nonzero output resistance, provided the
two measured voltages refer to the stated component terminals and the current has no
unmeasured parallel branch. It avoids treating the generator display as the device
voltage. It also produces magnitude and phase from the same data record, which makes
model checks more direct than separate amplitude and timing measurements.

$$
% caption: Series-sense measurement of a reactive device. The voltage across the known sense resistor gives the branch current, and the device voltage divided by that current gives the complex terminal impedance $\widetilde Z_D=R_s\,\widetilde V_D/\widetilde V_R$ at the marked polarity.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% source branch (left)
\draw[thick] (0.90,0.60)--(0.90,2.40);
\draw[thick,fill=white] (0.90,1.50) circle (0.42);
\node at (0.90,1.50) {AC};
% top wire: source to sense R
\draw[thick] (0.90,2.40)--(2.10,2.40);
\draw[thick,fill=black!6,draw=black] (2.10,2.12) rectangle (3.20,2.68);
\node at (2.65,2.40) {sense R};
\draw[thick] (3.20,2.40)--(4.05,2.40);
% DUT
\draw[thick,fill=acc!10,draw=acc] (4.05,2.02) rectangle (5.25,2.78);
\node[acc] at (4.65,2.40) {DUT};
% return path
\draw[thick] (5.25,2.40)--(6.05,2.40)--(6.05,0.60)--(0.90,0.60);
% current arrow
\draw[->,black,thick] (1.10,2.62)--(1.90,2.62) node[midway,above,black] {current};
% voltage spans
\draw[<->,acc,thick] (2.10,1.50)--(3.20,1.50);
\node[acc,below] at (2.65,1.48) {sense V};
\draw[<->,acc,thick] (4.05,1.50)--(5.25,1.50);
\node[acc,below] at (4.65,1.48) {device V};
\end{tikzpicture}
$$

The sense resistor needs a stated value at the measurement frequency. A wire-wound
power resistor can introduce inductance; a large resistor can add appreciable thermal
noise or reduce device voltage; a very small resistor can make the current signal too
small for accurate phase extraction. The voltage ratio obeys

$$
\frac{|\widetilde V_R|}{|\widetilde V_D|}
=\frac{R_s}{|\widetilde Z_D|}.
$$

A sense resistance of the same order as the expected impedance gives comparable
voltage magnitudes and generally avoids extreme ratio uncertainty. It also divides
the source voltage. If the device must be tested near its operating voltage, increase
source amplitude only after checking source-current, resistor-power, device-voltage,
and device-current limits. A single fixed $R_s$ rarely covers several decades of
impedance. Change it between overlapping sweep ranges, record each value, and compare
the overlap region for consistency.

The test fixture contributes series lead resistance and inductance, shunt capacitance,
and contact resistance. Keep the current path short and bring the voltage-sense points
to the device terminals rather than to a distant breadboard rail. A four-terminal-pair
instrument uses separate force and sense paths for this reason. With ordinary bench
equipment, the equivalent improvement is a compact fixture, a defined return path,
and a drawing that marks each probe-tip contact on the circuit.

An oscilloscope measures a voltage difference between a probe tip and its reference
lead. On many grounded instruments, the reference clips of several channels are
electrically common and connected to protective earth. Connecting them to two
different floating circuit nodes can short part of the circuit through the instrument.
Use a differential probe, an isolated measurement front end, or a channel-math method
with a shared reference when the topology requires a floating device voltage. The
probe voltage rating, common-mode range, and source isolation must be checked before
connection. The impedance formula is only as trustworthy as the actual circuit left
after the probes have been attached.

At each frequency, allow the source and instrument ranges to settle, then acquire an
integer number of periods when possible. The record should contain enough samples per
cycle to resolve the sine shape and enough cycles to average random noise. Triggering
from the source can stabilize the display, but phase still belongs to the measured
signals, not to a decorative trigger marker. A source with harmonic distortion can
create a phase estimate that depends on the extraction method. Record the residuals
of a sine fit or inspect the spectrum before calling a waveform sinusoidal.

Fit each sampled waveform to

$$
v[n]=a\cos(\omega t_n)+b\sin(\omega t_n)+d,
\qquad
A=\sqrt{a^2+b^2},
\qquad
\phi=\atanTwo(-b,a).
$$

$d$ is the fitted DC offset. The amplitude $A$ is a peak value; divide by
$\sqrt 2$ only after a sinusoidal fit has been justified. Fitting both quadratures
uses all samples and is less sensitive to a coarse peak location than reading a single
time difference between crossings. A zero-crossing measurement remains valuable as a
quick independent check, especially when the waveform is clean and the phase is far
from a crossing ambiguity.

Construct the device phasor ratio with the same amplitude convention for both channels.
Peak, peak-to-peak, and RMS scales cancel in the ratio only if both channels use the
same convention and remain linear. The current phasor inherits the polarity of the
sense resistor. Reversing that polarity adds $180^\circ$ to the reported current
phase, which changes an apparent capacitor into an apparent inductor if the reference
is not documented. Keep a sign table in the notebook: source direction, sense-resistor
polarity, device-voltage polarity, channel assignment, and software phase convention.

Sweep frequency on a logarithmic grid when the expected feature spans decades. At each
point, retain raw voltage records or the fitted coefficients together with the final
impedance magnitude. The complex record permits a later correction for a known channel
delay, a reassessment of probe loading, or a different equivalent-circuit fit. A
single magnitude curve cannot recover whether a departure arose from series loss,
shunt leakage, lead inductance, or an inverted voltage reference.

## Calibration and Model Choice

Calibration covers the complete measurement arrangement. Keep the cables, probe
settings, sense resistor, contact geometry, source level, and
frequency grid in place while measuring reference conditions. An open holder exposes
shunt capacitance and input leakage. A short holder exposes series lead impedance and
contact resistance. A known low-inductance resistor checks the magnitude scale and
phase reference in the middle of the intended impedance range. These records determine
whether the apparent component behavior exceeds the apparatus background.

Correct complex quantities rather than magnitudes. If a lead contribution is known
to be series over a narrow range, then

$$
\widetilde Z_{\rm corrected}
=\widetilde Z_{\rm measured}-\widetilde Z_{\rm series}
$$

is appropriate. If an open-holder contribution is known to be shunt, perform the
correction in admittance instead:

$$
\widetilde Y_{\rm corrected}
=\widetilde Y_{\rm measured}-\widetilde Y_{\rm shunt},
\qquad
\widetilde Y=\frac{1}{\widetilde Z}.
$$

Subtracting a short-holder magnitude from a device magnitude is generally invalid,
because phase determines whether two complex quantities add or cancel. Neither simple
correction repairs a holder whose parasitics are distributed or whose contacts shift
between samples. In that case, reduce lead length, use a better-defined holder, or
fit a network model that includes the measured parasitics. Keep the uncorrected record
as well as the corrected result so that the correction can be audited later.

Channel delay is a frequent phase error. A recorded signal delayed by $\Delta t$
acquires a phase shift proportional to frequency. With the convention
$v(t)=A\cos(\omega t+\phi)$ and a delay represented by $v(t-\Delta t)$,

$$
\phi_{\rm delay}=-\omega\Delta t.
$$

The sign in a particular instrument chain should be established with a common signal
sent through both channels. Split one stable source signal into the two complete
measurement paths, measure the residual complex phase across the sweep, and store the
correction with its sign convention. A delay that appears harmless at $1\ \mathrm{kHz}$
can dominate a small phase angle at $1\ \mathrm{MHz}$. Cable substitutions, probe
attenuation settings, digital filters, and current-probe range changes can alter the
delay; repeat the reference measurement after any such change.

Magnitude uncertainty begins with the sense resistor and the two fitted amplitudes.
For independent small relative standard uncertainties,

$$
u_r\!\left(|Z_D|\right)\approx
\sqrt{
u_r(R_s)^2+
u_r\!\left(|V_D|\right)^2+
u_r\!\left(|V_R|\right)^2}.
$$

The expression omits correlations deliberately. A common voltage-gain error can
cancel partly in the ratio, while a shared noise source can correlate both channels.
When those effects are important, retain the complex fitted coefficients and propagate
the covariance through the ratio rather than adding every percentage specification in
quadrature. A large stated voltage accuracy does not necessarily produce a large
impedance-ratio uncertainty if the two channels share the same gain path; an
unaccounted phase mismatch can still dominate the interpretation.

For the impedance phase,

$$
\phi_Z=\phi_D-\phi_R,
\qquad
u^2(\phi_Z)=u^2(\phi_D)+u^2(\phi_R)
-2\cov(\phi_D,\phi_R).
$$

Use radians in this propagation. Phase uncertainty comes from sample noise, finite
record length, fit residuals, clock jitter, channel-delay correction, harmonic
distortion, and phase unwrapping. Near a voltage null, the phase of that channel is
poorly constrained even if its amplitude reading remains above zero. Select the
sense resistance and source level so that neither measured channel approaches the
noise floor or clipping limit across the claimed frequency range.

Repeat measurements after remounting the component when contact pressure or lead
placement might matter. Repeated records estimate repeatability, while a calibrated
sense resistor and reference device address traceability. These are distinct. A sweep
that repeats tightly can still be biased by a shared cable delay or a resistor value
measured at the wrong temperature. Conversely, a component with genuine temperature
drift can show broad repeated results even when the instrument is accurately
calibrated. Report both the environmental condition and the repeat strategy.

Phase unwrapping needs a declared branch convention. Display software often maps
phase to an interval such as $-180^\circ$ to $+180^\circ$. A smooth physical sweep
can cross that display boundary without a discontinuity in the complex impedance.
Unwrap only after checking successive complex points and preserving the raw wrapped
phase. A sudden $180^\circ$ jump can also signal an inverted probe, a saturated input,
or a failed contact; continuity alone does not prove that a numerical unwrap is
correct.

A laboratory result for a capacitor or inductor should identify enough conditions to
be reproduced and challenged:

- **Device state.** Record part identity, nominal value, mounting orientation, lead
  length, DC bias, temperature, and the elapsed powered time.
- **Excitation.** State sine frequency or sweep grid, source amplitude convention,
  source impedance, waveform distortion check, and the resulting device RMS voltage
  and current.
- **Measurement path.** Give sense-resistor value and calibration, probe types,
  bandwidth limits, common-reference or differential topology, cable arrangement,
  sampling rate, record length, and phasor-fitting convention.
- **Calibration record.** Include open, short, and known-resistor results or explain
  why their contribution is negligible relative to the reported device impedance.
- **Result.** Report complex impedance or magnitude and phase, uncertainty method,
  corrections applied in complex form, model range, and any observed departure from
  ideal $X_C$ or $X_L$ scaling.

The completed record distinguishes an ideal-law comparison from a component
characterization. At low frequency and low amplitude, the ideal formulas may predict
the data within uncertainty. At another bias, temperature, or frequency, a loss,
leakage, saturation, or parasitic model may be needed. The appropriate model is the
smallest one that accounts for the measured complex response over the stated operating
range, with the remaining residuals and limits left visible in the report.

### Model selection from complex data

Fit the simplest model only over the range where its assumptions hold. A capacitor
well below its high-frequency package limit and with negligible leakage keeps a
constant $C$, which predicts both $|Z|=1/(\omega C)$ and a phase near $-90^\circ$. A systematic
positive real part suggests series loss. A phase drifting toward zero at very low
frequency can indicate leakage or input loading. A high-frequency phase change toward
$+90^\circ$ points to lead inductance or test-holder inductance. Each observation
constrains the model differently; impedance magnitude alone cannot separate them.

An inductor model begins with measured winding resistance and low-amplitude
inductance. Compare the measured complex response against $R_{\rm ac}+j\omega L$.
An amplitude-dependent deviation calls for a bias or saturation check. A
temperature-dependent real part calls for an AC-resistance or core-loss check. A
high-frequency phase reversal calls for a turn-capacitance and mounting check. Report
these deviations separately from any single inductance extracted over a wide frequency
range. State the extraction band and the residual pattern.

Residuals deserve the same attention as fitted values. Plot the real and imaginary
parts of measured impedance after subtracting the selected model. Random residuals
with a scale consistent with the uncertainty estimate support the stated model range.
A smooth trend with frequency, drive amplitude, or temperature signals a missing
mechanism. Preserve raw complex data, corrections, and fitting choices so that a
later measurement at another operating point can test that mechanism directly.