IN11 · A capacitor crosses self-resonance

LCR & Impedance Analyzer

Why can the same capacitor look resistive or inductive at another frequency? A real capacitor also has loss and inductance in its leads and internal current path. As frequency rises, capacitive reactance falls and inductive reactance grows. Above self-resonance the inductive part wins. Calibration removes the test fixture’s contribution; the capacitor’s own parasitics remain in the corrected result.

Sweep synchronous voltage and current, acquire open/short/load standards and separate fixture errors from a component’s own parasitics.

Saved compensated impedance magnitude $1.53\,\Omega$

Saved compensated phase $-86.3^\circ$

Saved series equivalent resistance $0.0999\,\Omega$

Saved series equivalent capacitance $1.04\,\mu\mathrm{F}$

Measure through the fixture

Drag the turquoise carriage to change the real lead length. Acquire new standards before saving another sweep.

Component carrierFrequency

100\,\mathrm{kHz}

f/\mathrm{Hz}

Fixture lead length

20

\ell/\mathrm{cm}

Source RMS voltage

0.1

V_s/\mathrm{V}

DUT capacitance

1.00

C/\mu\mathrm{F}

DUT series inductance

100

L/\mathrm{nH}

DUT resistance

100

R/\Omega

A complete default calibration and detector sweep are saved.

Compensation, detector and standard preview

Next saved compensationLive physical carrierCapacitor ESR

0.1

R_s/\Omega

Relative voltage gain

1.02

g_V/g_I

Voltage-channel delay

20

\tau_V/\mathrm{ns}

Voltage quadrature noise bound

500

\epsilon_V/\mathrm{nV}

Current quadrature noise bound

1

\epsilon_I/\mathrm{nA}

Detector seed

184

s_0

Standard preview changes the actual physical connection. The calibration action independently acquires all four standard sweeps at the current plane; the check is excluded from the fit.

Model assumptions

Linear small-signal lumped circuits use RMS phasors with positive-time complex exponent. Voltage is high minus low at the instrument force/sense junction, before series fixture residuals; positive current enters the high terminal. A source with 10-ohm output resistance drives the fixture and DUT. A teaching series resistance/inductance followed by parallel leakage/capacitance describes the fixture. The DUT retains its own ESR and ESL. An ideal phase-sensitive sixteen-cycle integral supplies simultaneous complex voltage/current, followed by bounded quadrature noise and range flags; ADC sampling and detailed lock-in filters are outside this model. SO assumes matched channel gain/phase; SOL fits a declared one-port fractional map from raw standards. An independent check standard is withheld. Compensation, circuit fits and equivalent parameters read only saved data and known standard models. Calibration geometry and channel paths must match exactly. The sensitivity diagnostic is not an uncertainty certificate. Fixture and dielectric geometry are schematic; dielectric thickness is enlarged one hundred times. Distributed wave effects, material dispersion, temperature dependence and nonlinear excitation are omitted.

Raw and compensated frequency sweep

Saved default simulated detector sweep

$f/\mathrm{Hz},\qquad |Z|/\Omega$

$f/\mathrm{Hz},\qquad \arg Z/{}^\circ$

Copper: raw detector voltage divided by current. Teal: compensation from saved standards. Dashed white: separate ideal DUT reference. Vertical line: the saved-frequency cursor.

The default sweep and independent standard check are valid.

Equivalent parameters describe the selected saved complex point. Changing this display does not acquire new signals.

Standards, circuit fit, raw samples and limits

The check standard is not used in calibration

$f/\mathrm{Hz},\qquad \epsilon_Z/\%$

$\epsilon_{\max}=0.00188\%$

Explicit whole-sweep fit family

$R_s=0.100\,\Omega,\quad L_s=100\,\mathrm{nH},\quad C=1.00\,\mu\mathrm{F}$

The model is fitted only to compensated saved samples.

$f/\mathrm{Hz}$	$\Re V/\mathrm{V}$	$\Im V/\mathrm{V}$	$\Re I/\mathrm{A}$	$\Im I/\mathrm{A}$

Calibration stores independent open, short, 100-ohm load and withheld 10-ohm plus 50-nH check records. These are declared ideal standard models, with no assigned traceable uncertainty. Each stored RMS phasor is the analytic output of a sixteen-cycle synchronous integral plus bounded quadrature noise.

Force and sense meet at the instrument-side reference plane, before both fixture leads. They do not measure directly across the DUT. Moving the carriage changes actual lead geometry and invalidates old standards. SOL removes this declared fixture/channel map; DUT loss and inductance remain. A one-at-a-time noise-bound sensitivity diagnostic rejects unstable subtraction; its ten-percent threshold is a teaching guard, not a confidence interval.

The capacitor cutaway has two parallel dielectric gaps with prescribed relative permittivity 1000; gap thickness is enlarged one hundred times. Lead and coil shapes are schematic and do not solve electromagnetic fields. Phase motion and signal markers are slowed illustrations. The linear lumped model omits material dispersion, propagation, temperature and nonlinear drive.

Start with the essentials

Focus question: Why can the same capacitor look resistive or inductive at another frequency?
One-sentence intuition: Fixture correction removes the fixture; the component’s own resistance and inductance remain.

Core mathematical model

A voltage-to-current ratio with a phase

Z=\frac{V}{I},\qquad v(t)=\sqrt2\,\Re[V e^{j\omega t}]

Phasors use RMS amplitudes and a positive-time exponent. Voltage is high minus low at the instrument-side sense junction; positive current enters the high terminal.

The capacitor retains its own current-path inductance

Z_{\mathrm{DUT}}=R_s+j\left(\omega L_s-\frac{1}{\omega C}\right)

The series resistance dissipates energy. Capacitive reactance falls with frequency and inductive reactance rises, giving a self-resonance when they cancel.

The defined fixture has series and parallel residuals

Z_m=Z_s+\frac{1}{Y_p+1/Z_{\mathrm{DUT}}}

The voltage-sense junction lies before the series fixture residual. Open and short standards use the same physical DUT plane; a known load also identifies relative channel gain and phase.

Three standards determine a fractional correction

m=\frac{aZ+b}{cZ+1},\qquad Z=\frac{b-m}{mc-a}

The implementation fits the equivalent homogeneous equations directly to raw standard voltages and currents, so an ideal open can have exactly zero current. A separate check standard is excluded from the fit.

Common difficulties

An equivalent circuit is a description

Typical misconceptionA parallel capacitance display proves that the component contains that exact internal circuit.

Better mental modelAt one frequency several circuits can share the same complex impedance. A wider sweep tests a specified family, and a small residual remains conditional on that family.

Run the experiment

01
Cross self-resonance
Move the frequency dial through the saved sweep.
What to observe: The compensated reactance changes sign. The component still contains its own series inductance.
02
Move the physical fixture
Drag the turquoise DUT carriage farther from the instrument. Save raw data, then acquire new standards and save again.
What to observe: The old calibration becomes stale because the real lead geometry changed. Old records retain their original calibration.
03
Compare two standards with three
Open compensation settings and compare SO with SOL after introducing a voltage-channel delay.
What to observe: A known load supplies the missing channel information. The independently measured check standard reveals an inadequate correction.

LCR & Impedance Analyzer

Raw and compensated frequency sweep

The check standard is not used in calibration

Measure the voltage and current before naming an equivalent

Core mathematical model

A voltage-to-current ratio with a phase

The capacitor retains its own current-path inductance

The defined fixture has series and parallel residuals

Three standards determine a fractional correction

Common difficulties

An equivalent circuit is a description

Run the experiment

Cross self-resonance

Move the physical fixture

Compare two standards with three

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