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A two-polarity resistance measurement removes a constant offset, but an offset that changes between readings remains in the result. This matters when a printed resistor produces only a small voltage and the fixture is still thermally evolving. Three appropriately timed readings can separate a locally linear offset drift from a stable resistive signal. The cancellation depends on the signed currents, actual observation times and electrical settling; simply averaging three resistance displays does not implement the method.
Key design decisions
- Retain raw signed voltage, signed current and observation time for every reading; a final resistance number cannot expose a timing error.
- Use time interpolation when the middle observation is not halfway between the other two.
- Check residual drift curvature and settling by changing the sequence timing while keeping the physical sense boundary fixed.
Separate the resistance term from a changing offset
Write each observation as Vj = Ij R + E(tj), where current is signed and E is the additive voltage at that observation time. Keep the voltage-lead orientation fixed while reversing source current. The intended resistance R must remain sufficiently stable during the short sequence. A contact interruption, nonlinear interface or changing specimen temperature can violate this model even when the arithmetic is correct.
For equally spaced observations using positive, negative and positive current of equal magnitude I, assume the offset is locally linear in time. The combination V1 − 2V2 + V3 cancels both its constant term and its linear slope. The resistive contribution remains 4IR. This is a calculation on voltages, not the mean of three unsigned readings or three instrument-computed resistance values.
R3 = (V1 − 2V2 + V3)/(4I)
- V1, V2, V3: signed sense voltages in volts, with fixed sense-lead orientation.
- I: positive current magnitude in amperes for a +I, −I, +I sequence.
- R3: three-reading resistance estimate in ohms.
Observation times are equally spaced, current magnitudes match, resistance is stable, and offset is locally linear. Each voltage is recorded after the relevant electrical settling.
Reproduce a case where two readings leave a drift error
Assume a 0.050-ohm resistor and a current magnitude of 0.010 ampere. The resistive voltage magnitude is 500 microvolts. At times zero, one and two seconds, let the additive offset be 20, 24 and 28 microvolts. The signed observed voltages are therefore 520, −476 and 528 microvolts.
Using only the first positive-negative pair gives (520 − (−476)) microvolts divided by 0.020 ampere, or 0.0498 ohm. The three-reading combination gives (520 − 2 × (−476) + 528) microvolts divided by 0.040 ampere, or 0.0500 ohm. The difference is caused entirely by the assumed changing offset. Compare the original voltage sequence before assigning such a method-dependent difference to the printed film.
Interpolate the outer observations to the middle timestamp
A triggered instrument may not produce equally spaced observation centers. Let w = (t3 − t2)/(t3 − t1). The weighted outer voltage wV1 + (1 − w)V3 represents the outer-polarity observation interpolated to the middle time when its offset is linear. Subtract V2, then divide by the corresponding difference in signed currents.
Keep the preceding resistor and current but move the third observation to three seconds. A four-microvolt-per-second offset slope gives voltages 520, −476 and 532 microvolts. The equal-spacing formula incorrectly returns 0.0501 ohm. With w = 2/3, the interpolated outer voltage is 524 microvolts; subtracting −476 and dividing by 0.020 ampere returns 0.0500 ohm. A known timing asymmetry is therefore correctable within the stated linear model.
R = [wV1 + (1 − w)V3 − V2]/[wI1 + (1 − w)I3 − I2]; w = (t3 − t2)/(t3 − t1)
- tj: effective observation time in seconds; t1 < t2 < t3.
- Ij: measured signed current in amperes associated with Vj.
- w: dimensionless interpolation weight; the denominator is the effective current difference, not a voltage.
Current and voltage observations describe the same settled state, R is stable across the sequence, and E(t) is linear over the observation interval. A small denominator makes the estimate poorly conditioned.
Use the actual polarity amplitudes in the denominator
Retain the unequal-time example but assume the three currents are +10.02, −9.98 and +10.00 milliamperes. With the same 0.050-ohm resistance and offset history, the voltages become 521, −475 and 532 microvolts. The weighted voltage difference is approximately 999.667 microvolts and the weighted current difference is 19.99333 milliamperes. Their ratio is again 0.0500 ohm.
Dividing that voltage difference by a nominal 20 milliamperes would instead give approximately 0.0499833 ohm. Offset cancellation does not calibrate the source current. Associate each voltage aperture with the corresponding current state, and account for the uncertainty of that current measurement. A compliance-limited or unsettled current is a reason to investigate the sequence, not a reason to force the nominal current into the denominator.
Bound what a three-reading sequence cannot cancel
Add an offset curvature of two microvolts per second squared to the earlier linear drift: E(t) = 20 + 4t + 2t² microvolts. At times zero, one and two seconds, the offsets are 20, 26 and 36 microvolts. The observed voltages become 520, −474 and 536 microvolts, and the equal-spacing estimate becomes 0.0501 ohm. Linear cancellation leaves the curved component behind.
For E(t) containing b t² and equal spacing h, the resulting resistance error is b h²/(2I). In this assumed case, reducing h from one second to half a second reduces that term from 100 to 25 microohms. This quadratic timing test can help identify curvature, but other errors may change with sequence speed. It does not authorize arbitrarily fast measurements without checking electrical settling.
Find a timing interval between settling and thermal change
After reversal, source response, fixture capacitance and the measurement aperture can contribute transient voltage. As a separate hypothetical model, a 100-microvolt transient decaying with a 20-millisecond time constant needs about 92.1 milliseconds to fall to one microvolt. That estimate comes from solving 100 exp(−d/0.020) = 1 with d in seconds.
Determine the actual settling behavior with retained voltage and current records rather than borrowing this example. The useful observation interval must be long enough for electrical settling yet short enough for the offset to remain approximately linear and the resistor state stable. Observation time should represent the aperture center or another consistently defined effective time. Trigger time alone may not describe when an integrating voltmeter formed its value.
Do not mistake cancellation for zero uncertainty
With equal independent voltage noise of standard deviation sV on each observation, the equal-spacing estimate has a voltage-noise contribution sqrt(6)sV/(4I). For an assumed sV of 0.5 microvolt and I of 0.010 ampere, this is approximately 30.6 microohms. The corresponding two-reading expression gives 35.4 microohms under the same per-reading noise assumption.
Those numbers are not a fair speed comparison unless aperture, total elapsed time and instrument processing are also matched. Overlapping three-reading estimates share observations and are correlated. Do not count every overlapping output as an independent repeat when estimating uncertainty. Add current calibration, timing, residual drift and specimen stability contributions separately rather than reporting the cancellation calculation as a complete measurement uncertainty.
Use deliberate changes to test the estimator assumptions
Repeat a short set of sequences while changing one timing or excitation feature at a time. Keep the specimen, Kelvin sense coordinates and handling state unchanged. Retain an original record before any cleaning or remounting so a later improvement can be assigned to the correct change.
| Observation | Controlled comparison | Interpretation to investigate |
|---|---|---|
| Result shifts with middle-sample timing | Use measured timestamps and interpolation weights | Unequal spacing may be leaking linear offset drift |
| Result changes strongly as spacing is shortened | Inspect settling records and repeat at longer dwell | Electrical transients may not have decayed |
| Residual scales approximately with spacing squared | Compare several settled sequence intervals | Curved offset drift is a possible contribution |
| Nominal-current and measured-current calculations disagree | Retain signed actual currents and compliance flags | Source amplitude or synchronization may dominate |
| Results change after reversing the entire sequence | Compare +−+ and −+− with consistent sign treatment | State changes, nonlinear contacts or asymmetric settling need review |
Report a method that another laboratory can reproduce
Store raw voltages, signed currents, effective times, integration apertures, source delays and the exact estimator. Include the resistance temperature state and the evidence that the selected sequence is settled. Report whether estimates overlap and how repeatability was calculated. This makes the value reproducible without implying that a particular instrument mode guarantees correctness.
Keep this temporal correction separate from the physical sensing boundary. A perfectly corrected voltage can still include an unintended terminal segment, and a stable film can still heat during the sequence. For low-ohmic printed-resistor review, provide both the drawing and the time-resolved data. The combined record distinguishes what was measured from how additive drift was removed.
Provide the raw low-ohmic measurement sequence
Send the observations needed to review drift cancellation without losing the physical resistor boundary.
- Resistance target, reference temperature and drawing with the established force and sense contacts.
- Raw signed voltages, signed actual currents, effective timestamps and instrument range or compliance records.
- Integration aperture, reversal dwell, trigger sequence, initialization and estimator expression.
- Timing-variation and polarity-order comparisons with sample temperature and handling state.
- Repeatability data identifying overlapping estimates and the required uncertainty allocation.
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