Reactance

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

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.

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

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:

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

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 $R_{\mathrm{ac}}$ 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.

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

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

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 $L_{\mathrm{inc}}=\mathbf{d}\lambda /\mathbf{d}i$ 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

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.

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.

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 $R_{\mathrm{ESR}}$ and equivalent series inductance $L_{\mathrm{ESL}}$ in series with the intended capacitance. A large leakage resistance $R_{\mathrm{leak}}$ lies in parallel with the capacitance for a model that also describes slow DC discharge. The approximate terminal impedance is

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, $R_{\mathrm{ESR}}$ 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

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

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

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 $v(t)=\sqrt{2}V\cos (\omega t+\varphi )$ is represented by $\tilde{V}=V{e}^{j\varphi }$. 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

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

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.

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.

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 $i=C\mathbf{d}v/\mathbf{d}t$ for a capacitor gives

The stored electric-field energy is consequently

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.

Substitution of $v=L\mathbf{d}i/\mathbf{d}t$ for an inductor gives

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

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.

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

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.

“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

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.

Inductive reactance is

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.

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:

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.

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

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

$d$ 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

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

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 $v(t)=A\cos (\omega t+\varphi )$ and a delay represented by $v(t-\Delta t)$,

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,

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,

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 $R_{\mathrm{ac}}+j\omega L$. 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.