Bridges and impedance measurement

Measuring a resistor of a few kilohms to 1 % is easy with any multimeter. Measuring a 10 mΩ current shunt, a strain gauge whose resistance changes by a few hundredths of a percent, or the equivalent series resistance of an electrolytic capacitor at 100 kHz is not. This lesson covers the methods built for those jobs: the four-wire Kelvin connection that removes lead resistance, the Wheatstone bridge that turns tiny resistance changes into measurable voltages, and the LCR meter that measures capacitors and inductors as complex impedances at a chosen frequency.

Two Cedarline problems motivate the lesson. The LED driver's current-sense resistor turned out, in lesson 3, to dominate the uncertainty of the current measurement, so Amara wants its actual value. And the 470 µF bulk capacitor on the 24 V rail is a common failure point in outdoor controllers, where heat dries out electrolytic capacitors, so she wants Leo to measure the capacitance and ESR of new and aged samples.

Two-wire and four-wire resistance measurement

A multimeter in its ordinary two-wire resistance mode drives its test current through the same leads it uses to measure voltage. The voltage it reads therefore includes the drop across both leads and both contact resistances, typically 0.1 to 0.5 Ω in total. On a 10 kΩ resistor that is negligible. On a 2.2 Ω sense resistor it is a 5 to 20 % error, and on a 10 mΩ shunt it swamps the reading completely. Zeroing the meter with the probes shorted together helps, but contact resistance changes every time the probes are pressed.

The four-wire, or Kelvin, method separates the two jobs. One pair of leads, the force leads, carries the test current through the resistor. A second pair, the sense leads, connects to the resistor's terminals inside the force connections and measures only the voltage across the resistor itself. Because the voltmeter draws almost no current, the sense leads carry no current and drop no voltage, so their resistance does not matter. Bench meters with a 4W ohms function have four terminals for exactly this, and low-value resistors and shunts are often made with four terminals so the sense points are defined precisely.

The Wheatstone bridge

A Wheatstone bridge is four resistors in a diamond, driven by a voltage source across one diagonal, with a voltmeter or detector across the other. When the ratio of the resistors on one side equals the ratio on the other, the two midpoints sit at the same voltage and the detector reads zero; the bridge is balanced. If three resistors are known and one is adjustable, balancing the bridge measures the fourth by a ratio, independent of the source voltage and of the detector's calibration. That was how precision resistance measurements were made for a century, and the principle survives in many instruments.

Today bridges are used mostly off balance, as a way to measure small changes. Suppose all four resistors are equal at R and one of them, a sensor, changes to R + ΔR. For small changes the output is approximately V_s·ΔR/(4R). A strain gauge on a steel bracket might change by 0.1 % under load, giving only 2.5 mV from a 10 V bridge, but that 2.5 mV sits on a zero baseline rather than on top of 5 V, so it can be amplified a thousandfold without saturating anything. Using two or four active gauges in the bridge doubles or quadruples the output and cancels temperature effects that change all the gauges equally.

Impedance and the LCR meter

Capacitors and inductors are never pure. A real capacitor has an equivalent series resistance (ESR) from its leads, plates and electrolyte, an equivalent series inductance from its leads and construction, and a leakage resistance in parallel. A real inductor has winding resistance and capacitance between turns. Their impedances therefore depend on frequency, and measuring them means measuring a complex impedance Z = R + jX at a chosen test frequency, which is what an LCR meter does. It applies a small AC test signal, measures the current and voltage magnitudes and the phase between them, and computes the impedance, then reports it as the user chooses: capacitance and ESR, inductance and series resistance, dissipation factor D, quality factor Q, or the magnitude and phase directly.

An LCR meter lets you choose the equivalent circuit, series or parallel, and the two give different numbers for the same impedance. The convention is to use the series model for low impedances, such as large capacitors and small inductors, where the series resistance matters most, and the parallel model for high impedances, such as small capacitors and large inductors, where leakage dominates. For electrolytic capacitors, ESR at 100 kHz and capacitance at 120 Hz, the ripple frequency of a full-wave rectifier on a 60 Hz supply, are the standard figures on data sheets. LCR meters also use four-terminal connections, often called Kelvin clips, so that lead impedance does not corrupt low-impedance readings, and they let you perform open and short corrections to subtract the fixture.

# Kelvin vs two-wire, quarter-bridge output, and capacitor ESR from an LCR-style measurement
import math
R_true, r_lead = 2.2183, 0.12
print(f"two-wire reads {R_true + 2*r_lead:.4f} ohm, four-wire reads {R_true:.4f} ohm")

def bridge(vs, r1, r2, r3, r4):
    return vs * (r3 / (r3 + r4) - r2 / (r1 + r2))

for strain_pct in (0.01, 0.05, 0.12, 0.5):
    rg = 350 * (1 + strain_pct / 100)
    exact = bridge(5.0, 350, 350, rg, 350)
    approx = 5.0 / 4 * strain_pct / 100
    print(f"dR/R {strain_pct:5.2f} %: exact {exact*1e3:7.4f} mV, approx {approx*1e3:7.4f} mV")

# LCR measurement of a 470 uF electrolytic: |Z| and phase at two frequencies
for label, f, zmag, theta_deg in (("new, 120 Hz", 120, 2.832, -88.6), ("new, 100 kHz", 100e3, 0.0420, -2.0),
                                  ("aged, 120 Hz", 120, 4.06, -84.3), ("aged, 100 kHz", 100e3, 0.380, -1.0)):
    th = math.radians(theta_deg)
    Rs, Xs = zmag * math.cos(th), zmag * math.sin(th)
    w = 2 * math.pi * f
    if f < 1000:
        Cs = -1 / (w * Xs)
        print(f"{label:14s}: ESR {Rs*1e3:7.1f} mohm, Cs {Cs*1e6:7.1f} uF, D = {Rs/abs(Xs):.3f}")
    else:   # reactance is a tiny fraction of |Z|, so |Z| is essentially the ESR
        print(f"{label:14s}: ESR {Rs*1e3:7.1f} mohm, reactance only {abs(Xs)*1e3:.1f} mohm")

two-wire reads 2.4583 ohm, four-wire reads 2.2183 ohm
dR/R  0.01 %: exact  0.1250 mV, approx  0.1250 mV
dR/R  0.05 %: exact  0.6248 mV, approx  0.6250 mV
dR/R  0.12 %: exact  1.4991 mV, approx  1.5000 mV
dR/R  0.50 %: exact  6.2344 mV, approx  6.2500 mV
new, 120 Hz   : ESR    69.2 mohm, Cs   468.5 uF, D = 0.024
new, 100 kHz  : ESR    42.0 mohm, reactance only 1.5 mohm
aged, 120 Hz  : ESR   403.2 mohm, Cs   328.3 uF, D = 0.100
aged, 100 kHz : ESR   379.9 mohm, reactance only 6.6 mohm

The capacitor results tell a story Amara has seen before. At 120 Hz the new capacitor measures about 468 µF with a dissipation factor of 0.024; the aged one, taken from a controller that spent eight summers in a sun-baked cabinet, measures about 330 µF with D = 0.10. At 100 kHz the impedance is almost purely resistive, because the capacitive reactance has fallen far below the ESR, so the magnitude there is essentially the ESR: about 42 mΩ new and 380 mΩ aged. The aged capacitor's ESR is nine times higher. With 1 A of ripple current, it would dissipate about 0.38 W instead of 0.04 W, heating itself and drying out faster. ESR rising faster than capacitance falls is the typical signature of electrolytic wear-out, which is why ESR meters are standard tools for repairing power supplies.

Self-resonance and test conditions

Every capacitor has some series inductance, so above a certain frequency it stops behaving as a capacitor at all. The self-resonant frequency, where the capacitive and inductive reactances cancel, is 1/(2π√(LC)); above it, the part is an inductor. A 100 nF ceramic decoupling capacitor with 1 nH of inductance resonates near 16 MHz; a 470 µF electrolytic with 20 nH resonates in the tens of kilohertz. Measurements must therefore state their frequency, and a capacitor's value at 1 kHz says little about its behaviour at 10 MHz. Ceramic capacitors add another trap: the class II dielectrics used for high values lose a large fraction of their capacitance as DC bias voltage rises, so a 10 µF part on a 5 V rail may behave like 4 µF. An LCR meter with a DC bias option measures that directly.