Electronic Measurements II · 120 h · Topic 1 of 10

Measuring device parameters

A 10 μF electrolytic capacitor measured at 1 MHz is no longer a capacitor: it is above its self-resonance and the LCR meter sees it as an inductance.

LCR meter Series and parallel model Dissipation factor Self-resonance Curve tracer

01The LCR meter

The bridges of Electronic Measurements I are balanced by hand. A modern LCR meter does the same automatically with a self-balancing bridge: it applies a sinusoidal voltage of known frequency to the component, takes the other terminal to a virtual ground with an amplifier, measures the current with a range resistor and calculates the complex impedance with two synchronous detectors, in phase and in quadrature:

\[ Z = \frac{V}{I} = R + jX \] From \( R \) and \( X \) the instrument deduces the value the user asks for according to the chosen model: series or parallel capacitance or inductance, and the dissipation factor \( D = R/|X| \) or the quality factor \( Q = 1/D \).

A real component is not ideal, and the number the instrument shows depends on the model and on the test frequency. For the same component, the series and parallel models give different values:

\[ C_p = \frac{C_s}{1 + D^2} \] With small losses, \( D \ll 1 \), the two models coincide. By convention the series model is used for low impedances (large capacitors, small inductors) and the parallel model for high impedances.

02The real capacitor

Besides its capacitance, a capacitor has an equivalent series resistance (ESR), from the electrodes, the dielectric and the electrolyte, and a series inductance (ESL), from its leads and its construction. Its impedance falls with frequency down to the self-resonance, where it equals only the ESR, and above that it grows like that of an inductor:

\[ Z(f) = ESR + j\left(2\pi f\, ESL - \frac{1}{2\pi f C}\right) \] \[ f_r = \frac{1}{2\pi \sqrt{ESL \cdot C}} \] A 10 μF electrolytic with 10 nH of ESL resonates at 503 kHz. Above that, it does not decouple: that is why small ceramic capacitors are placed in parallel with the large ones on boards.

Lab · what an LCR meter reads on a real capacitor

The impedance of a capacitor with ESR and ESL as a function of frequency. The dot marks the meter’s test frequency; the data are what it would show in the series and parallel models.

Nominal capacitance =

ESR =

ESL =

03Connection and compensation

The cables and the fixture add their own impedance. For low impedances, four terminals are used, with the current and the voltage on separate pairs, as in the Kelvin method; for high impedances, shields that take the stray capacitances to ground. Before measuring, compensation is performed: one measurement with the terminals open and another shorted, which the instrument subtracts. Without it, a 1 pF capacitor or a 10 mΩ resistor are read with large errors.

04Active devices: the curve tracer

For transistors and diodes, the curve tracer applies a stepped voltage to the base or gate and a sweep to the collector or drain, and draws the family of output curves. From them one reads the current gain, the output resistance, the saturation voltage and the breakdown voltage. Semiconductor parameter analyzers, with source-measure units (SMUs), do the same with picoampere precision and make it possible to measure leakage currents and junction capacitances. The models being measured are those of the bipolar transistor and the MOSFET, in Electronic Devices.

05In the laboratory

Activity 1 · Sweeping an electrolytic

Measure an electrolytic capacitor with an LCR meter at all its test frequencies, in series model, and estimate the ESR and the ESL. Compare with the datasheet.

Activity 2 · Compensation

Measure a 10 pF capacitor with and without open-circuit compensation, and a 0.1 Ω resistor with and without short-circuit compensation.

Activity 3 · Transistor curves

Use a curve tracer to obtain the output family of a bipolar transistor and calculate β and the Early voltage.

06Common mistakes

Measuring a capacitor without noting the test frequency. The value changes with it, and near resonance it stops making sense.
Forgetting the compensation. The cables can add more than the component.
Confusing the series model with the parallel one. With large losses, they give very different values.
Measuring in circuit. Components in parallel change the reading.

07Self-assessment

Where does a 100 nF capacitor with 5 nH of ESL resonate?

\( 1/(2\pi\sqrt{5\cdot10^{-9} \cdot 10^{-7}}) \approx 7.1 \) MHz.

A meter reads C_s = 1 μF with D = 0.5. What would it read in parallel?

\( 1/(1 + 0.25) = 0.8 \) μF.

Why is four-terminal measurement used for low resistances?

So that the drop in the current leads does not enter the voltage measurement.

What does a curve tracer measure?

The families of output curves of an active device, sweeping the output voltage for several values of the input.

08Further reading

Keysight Technologies. Impedance Measurement Handbook. 6th ed., Keysight, 2016. The reference guide on LCR meters, models, fixtures and compensation.
Albert D. Helfrick and William D. Cooper. Modern Electronic Instrumentation and Measurement Techniques. Prentice Hall, 1990. Spanish edition: “Instrumentación electrónica moderna y técnicas de medición”, Prentice Hall Hispanoamericana, 1991. Bridges and component measuring instruments.
Clyde F. Coombs Jr. (ed.). Electronic Instrument Handbook. 3rd ed., McGraw-Hill, 1999. Chapters by instrument, among them the impedance meter and the curve tracer.

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