Standard circuit measurements
Measuring a resistance means reading one number. Measuring an impedance means reading two —magnitude and phase— and also asking at what frequency, because the answer changes. A 100 nF capacitor is not worth the same at 1 kHz as at 10 MHz: at some point it stops behaving like a capacitor at all.
01What it means to measure an impedance
In DC, Ohm’s law is enough. In AC the current can lead or lag the voltage, and a complex number is needed to describe the whole picture: the impedance. All of this was developed in the vector study of alternating current; here the subject is how to measure it.
R = 1 kΩ in series with C = 100 nF:
| Frequency | XC = 1/2πfC | |Z| | Phase | Behaves as |
|---|---|---|---|---|
| 100 Hz | 15,915 Ω | 15,947 Ω | −86.4° | Almost a pure capacitor |
| 1 kHz | 1,592 Ω | 1,880 Ω | −57.9° | A mix of the two |
| 10 kHz | 159 Ω | 1,013 Ω | −9.0° | Almost a pure resistor |
The same pair of components, three different impedances. That is why no impedance measurement means anything without the frequency at which it was made, and why LCR bridges let you choose it.
02Direct methods
The impedance is driven by the generator, and you measure the voltage across it and the current through it (as the drop across a small standard resistor) and divide. This gives the magnitude.
For the phase you need a two-channel oscilloscope: you measure the shift between the two waves.
Channel 1 on the generator, channel 2 on the standard resistor. The amplitude ratio gives the magnitude and the horizontal offset gives the phase:
The unknown is replaced by a decade box, adjusted until the circuit behaves the same. It is slow but very accurate, because the standard and the unknown see exactly the same circuit.
The unknown is placed in an LC tank with a known standard and you look for the resonant frequency. The value follows from that. It is the classic method for RF inductors and the one the Q meter uses.
A ×1 probe adds about 100 pF in parallel with the point being measured. In an audio circuit you won’t notice; in an RF circuit, or when measuring a 100 pF capacitor, the instrument changes what it is trying to measure. The ×10 probe lowers that capacitance to about 10 pF and raises the resistance to 10 MΩ: always use ×10 to measure impedances, compensated with the trimmer against the instrument’s calibration square wave.
03Measuring bridges
They are the most accurate family of instruments there is, for a conceptual reason: they don’t measure, they compare. The result does not depend on the accuracy of the detector or on the supply voltage, only on that of the standards. The detector only has to tell “there is a signal” from “there is no signal.”
| Bridge | What it is for | Detail |
|---|---|---|
| Wheatstone | Resistances in DC | The father of them all. It is used to measure strain gauges and temperature sensors with parts-per-million accuracy. |
| Kelvin | Very low resistances (mΩ) | Four-wire: it cancels the resistance of the leads and contacts, which would otherwise dominate the measurement. |
| Maxwell | Inductors with low Q (Q < 10) | Compares the inductance against a standard capacitor, which is much more stable than a standard inductor. Lx = R2·R3·C1. |
| Hay | Inductors with high Q (Q > 10) | A variant of the Maxwell bridge with the capacitor and its resistor in series instead of in parallel. |
| Schering | Capacitors and insulation | Measures the loss tangent of a dielectric. It is the instrument used to test high-voltage cables and insulators. |
| Wien | Frequency | It balances at a single frequency: it serves to measure it and, above all, to generate it in the oscillator that bears its name (see oscillators). |
With R2 = 1 kΩ, R3 = 4.7 kΩ and C1 = 10 nF, at balance:
- Lx = R2·R3·C1 = 1000 × 4700 × 10−8 = 47 mH
- If in addition R1 turned out to be 22 kΩ: Rx = R2·R3/R1 = 4.7×106/22,000 = 213.6 Ω
- Q at 1 kHz = 2πfL/Rx = 6283 × 0.047 / 213.6 = 1.38
A Q this low confirms that the right bridge was chosen: for a high-Q inductor you would have to use the Hay bridge.
Because at the balance point the detector reads zero, and for that it does not need to be calibrated: only sensitive. All of the accuracy rests on the three standards, which are precision resistors and capacitors, stable and inexpensive components. It is the same principle as a balance scale versus a spring scale.
04At high frequency nothing is what it claims to be
This is the point that causes the most surprises in the workshop. A real component is not its symbol: a capacitor carries inductance in its leads and resistance in its plates; an inductor carries capacitance between turns. At low frequency that doesn’t matter. At high frequency it dominates.
It is C in series with ESR (equivalent series resistance) and ESL (inductance of the leads and plates). Above self-resonance it stops filtering.
A 100 µF electrolytic with 20 nH of ESL resonates at 112 kHz: above that it no longer works as a filter. A 100 nF ceramic with 5 nH resonates at 7.1 MHz. That is why a switching power supply uses both in parallel: each covers the band where the other fails.
It is L in series with the wire resistance and in parallel with the distributed capacitance between turns. Above its self-resonance it behaves like a capacitor.
In addition, the wire resistance grows with frequency because of the skin effect: current flows only along the surface. In copper at 1 MHz the effective depth is barely 66 µm. Hence the Litz wire in RF inductors, and the fact that a thick conductor doesn’t help as much as you might expect.
It is the ratio of the energy stored to the energy dissipated per cycle. For an inductor, Q = XL/R = 2πfL/R. A high Q means low loss: a well-made air-core inductor has a Q of 100 to 300; one with a saturated ferrite core, less than 10. In a filter the Q determines how sharp the curve is, and in an oscillator, the frequency stability.
An aged electrolytic keeps its capacitance and yet no longer filters: what has risen is its ESR. An ordinary capacitance meter passes it as good. Only an ESR meter —which injects about 100 kHz and measures the resistive part— detects it, and it can do so with the component still soldered on the circuit board. It is the instrument that saves the most time when repairing switching power supplies.
| Instrument | What it provides | Typical range |
|---|---|---|
| Multimeter with C function | Capacitance only, at low frequency | nF to mF, no ESR or phase |
| ESR meter | Series resistance at ~100 kHz | 0.01 to 100 Ω, in-circuit |
| Benchtop LCR bridge | L, C, R, Q, D and phase at the chosen frequency | 100 Hz to 100 kHz or more |
| Q meter | Q and L by resonance | RF: 50 kHz to 50 MHz |
| Vector network analyzer (VNA) | Magnitude and phase versus frequency, full sweep | From kHz to GHz. Affordable pocket-sized models exist today. |
| SWR meter | Standing wave ratio: how far the load departs from 50 Ω | Antennas and transmission lines |
Impedance measurement on antennas and lines is the ground of high-frequency measurements, which are taken up again in Telecommunications, where the goal becomes matching: making the load equal 50 Ω so that no energy comes back along the line.
05Frequency measurement
The counting method with a timebase was already developed in digital measuring instruments. Two methods remain that are still useful and that belong to this topic:
Oscilloscope in X-Y mode: the unknown signal on one axis and a reference on the other. The figure holds still only if the frequency ratio is a simple ratio of integers, and that ratio is read by counting the tangent points with the horizontal edge and with the vertical edge:
In addition, with the two frequencies equal, the figure directly shows the phase difference: a straight line if they are in phase, a circle if they are 90° apart.
A known LC circuit is tuned until the response is a maximum. It is the method of RF instruments —the grid-dip meter— and it also works the other way around: to measure an unknown inductor or capacitor using a known frequency.
06In the lab
Build R = 1 kΩ with C = 100 nF and drive it from the generator at 100 Hz, 1 kHz and 10 kHz. Use the oscilloscope to measure the total voltage and the drop across R (which gives the current), calculate |Z| and measure the phase shift from the offset between the two waves. Compare with the table in section 1. Discrepancies larger than 5 % are usually due to the capacitor’s tolerance, which is worth measuring separately.
Build the bridge with three known resistors and a variable decade box; use the multimeter on its millivolt range as the detector. Measure several unknown resistors and compare with the direct reading of the ohmmeter. Then repeat with the supply at half the voltage: the result does not change, and that is precisely the point of the method.
With a generator and oscilloscope, sweep the frequency across a 100 nF ceramic capacitor with a small series resistor, and record the voltage across the capacitor. It falls to a minimum and then rises again. That frequency is the self-resonance. Repeat with a 100 µF electrolytic: the resonance appears much earlier. This is the experimental justification for putting the two in parallel.
Gather electrolytic capacitors taken from old equipment. Measure them first with the capacitance meter —almost all of them will read fine— and then with the ESR meter. Rank them by ESR and compare them with a new one of the same value. It is the lab that most quickly convinces you that capacitance alone is not enough to judge a capacitor.
Oscilloscope in X-Y mode with two generators. Adjust the second one until you get a stationary ellipse (1:1 ratio), then look for 2:1 and 3:2 and sketch each figure. With a 1:1 ratio, vary the phase and watch how the ellipse goes from a line to a circle and back.
07Common mistakes
| Symptom | Usual cause |
|---|---|
| The measured impedance does not match the calculated one | They were compared at different frequencies, or the capacitor’s tolerance (±20 % in electrolytics) explains the difference by itself. |
| The value changes when you bring your hand close | High impedance and body capacitance. Shield the circuit and use short leads. |
| A capacitor measures fine but the equipment still fails | High ESR. The capacitance meter does not see it; you need an ESR meter. |
| The bridge never reaches balance | The second adjustment is missing: in AC you must balance magnitude and phase, iterating between the two knobs. |
| When measuring very low resistances, the value does not repeat | Resistance of leads and contacts. A four-wire (Kelvin) measurement is needed. |
| The inductor reads a different inductance on each instrument | Each one measures it at its own frequency. With a ferrite core, moreover, the value depends on the signal level. |
| The capacitor “behaves like an inductor” | It is being operated above its self-resonance. That is correct: choose a different type of capacitor. |
| The oscilloscope probe distorts the signal | A ×1 probe, or a ×10 probe that is not compensated. Compensate it with the instrument’s own square wave. |
08Self-assessment
Why is an impedance measurement meaningless without stating the frequency?
Because reactance depends on frequency: XC = 1/2πfC falls as f rises, and XL = 2πfL rises. The same series R-C circuit can behave as a pure capacitor at 100 Hz and almost as a pure resistor at 10 kHz.
R = 1 kΩ and C = 100 nF in series at 1 kHz: what are the magnitude and phase?
XC = 1/(2π·1000·100 n) = 1592 Ω. |Z| = √(1000² + 1592²) = 1880 Ω. φ = −arctan(1592/1000) = −57.9°: the current leads the voltage.
Why is a bridge more accurate than a direct measurement?
Because it compares against standards instead of measuring. At balance the detector reads zero, so it does not need to be calibrated nor the source to be stable: all of the accuracy lies in the standard components.
In a Maxwell bridge with R2 = 2.2 kΩ, R3 = 10 kΩ and C1 = 4.7 nF, what is the inductance?
Lx = R2·R3·C1 = 2200 × 10,000 × 4.7×10−9 = 0.1034 H, that is, about 103 mH.
What is the self-resonance of a capacitor and why does it matter?
It is the frequency at which its capacitance resonates with the parasitic inductance of its leads. Below it, it behaves as a capacitor; above it, as an inductor, and it stops filtering. That is why power supplies use electrolytics and ceramics in parallel.
What does an ESR meter measure, and why is a capacitance meter not enough?
It measures the capacitor’s series resistance at about 100 kHz. An aged electrolytic keeps its capacitance but its ESR increases: the capacitance meter passes it as good and the circuit fails anyway.
What does it mean for an inductor to have Q = 150?
That at the measurement frequency its reactance is 150 times greater than its resistance: it stores 150 times more energy than it dissipates per cycle. It is a good-quality inductor, suited to selective filters and stable oscillators.
Why, at high frequency, does a thick conductor not lower the resistance as much as you would expect?
Because of the skin effect: current concentrates at the surface. In copper at 1 MHz the effective depth is about 66 µm, so the center of the conductor barely conducts. Hence Litz wire and tubular conductors in RF.
In X-Y mode a stationary figure appears that touches the horizontal edge 3 times and the vertical edge 2 times. What is the ratio between the frequencies?
fy/fx = nx/ny = 3/2. If the horizontal reference is 1 kHz, the vertical signal is 1.5 kHz.
Why is it better to measure with a ×10 probe rather than ×1?
Because the ×1 adds about 100 pF and 1 MΩ to the circuit, and that alters what you want to measure. The ×10 drops to about 10 pF and rises to 10 MΩ, at the cost of dividing the signal by ten. It must be compensated with the oscilloscope’s calibration square wave.