TCR Sample Sets: Separate Different Paste Series Before Pooling the Result

Measurement purpose

Keep TCR population and pair identity visible through data aggregation.

Specimens and conditions

Material grouping

Series, grade, lot and completed process state retained.

Comparable definition

Same stated normalization and temperature interval, or differences reported.

Equipment and records required

• Measurement: Identified low-power resistance and actual temperature observations.
• Data analysis: Specimen-level records, group labels and explicit statistical weights.

Method sequence

1. Group

   Define material and process populations before examining pooled statistics.

   Record: Specimen grouping ledger.

2. Compare

   Calculate per-specimen curves and group summaries under stated conditions.

   Record: Normalized curves and within-group variation.

3. Decide

   Apply individual, group or pair requirements at their proper level.

   Record: Requirement-specific disposition.

Decision and uncertainty

A pooled mean cannot substitute for individual conformity or actual pair tracking.

Sampling composition, repeated observations and shared measurement errors affect interpretation differently.

Resistor design and qualification data owners.

Traceable outputs

Identify each material and process population

Record the full resistor paste product and grade, supplier lot, print geometry, termination system and firing history represented by each specimen. A broad commercial series name can include distinct sheet-resistance grades and processing requirements. Do not merge those records merely because the samples came from one shipment or have the same room-temperature resistance after trimming.

Also retain whether testing occurred before or after trim, glass protection and assembly. Those states answer different qualification questions. If a study deliberately compares several states, name them as separate groups and preserve the pairing where the same physical resistor is measured more than once. Repeated readings of one resistor are not additional independently produced resistors.

Put the coefficients on a comparable definition

For an endpoint coefficient normalized to the initial resistance, divide the resistance change by that initial resistance and by the temperature difference. State both endpoint temperatures and the normalization temperature. Reporting only ppm per kelvin can hide different temperature intervals, especially when resistance curves are not straight over the full operating range.

Retain the raw resistance and temperature pairs so normalization can be checked. Do not average resistance values from unrelated specimens first and then assume the resulting coefficient is the simple average of their normalized coefficients. Higher-resistance specimens receive different weighting in that pooled-resistance calculation. Choose the statistical summary only after deciding whether it represents specimens, circuits, lots or production quantities.

See how opposing responses create a misleading zero

Consider equal numbers of two hypothetical resistor groups tested over the same 50 kelvin rise. Every specimen in group A has a linear coefficient of positive 100 ppm per kelvin and every specimen in group B has negative 100 ppm per kelvin. An equally weighted mean of the individual coefficients is zero, but no individual resistor in this example has zero TCR.

Each A resistor increases by 0.5 percent and each B resistor decreases by 0.5 percent over the stated interval. If the actual requirement were an illustrative individual limit of plus or minus 60 ppm per kelvin, both groups would fail that requirement even though the combined mean looks ideal. The limit is invented solely to demonstrate the decision error; it is not a material rating or a ChipSimple capability claim.

Do not let the sampling mix decide the engineering conclusion

The example's 90-to-10 mixture has a coefficient mean of positive 80 ppm per kelvin. A later 10-to-90 mixture gives negative 80 ppm per kelvin, even though the underlying groups are unchanged. A chart of the pooled means could falsely suggest a large process shift. Report group membership and sample counts beside each summary so composition changes remain visible.

Decide whether groups are intentionally sampled in proportion to production or oversampled to inspect a less common but important construction. Neither choice is inherently wrong. They answer different questions. If a combined estimate is needed, state its weighting and its target population. Never silently apply production weights to a comparison experiment and then use its mean as proof that every group conforms.

Keep individual curves and within-group variation

Plot normalized resistance against actual temperature for each specimen, with series and lot identity visible. Preserve intermediate points where curvature matters and the return to the initial condition. An endpoint coefficient alone can hide differences within the interval, while a changed return value may indicate retained movement or a measurement problem rather than reversible temperature response.

Within each group, retain specimen-level results, spread, counts and unusual observations. Do not report a standard deviation calculated from repeated meter readings as though it measured production variation. If one sample differs strongly, investigate its identity and history before exclusion. A transcription correction or invalid measurement can be justified, but removing a real unfavorable specimen because it enlarges the spread cannot support acceptance.

Use the actual paired elements for a ratio requirement

Absolute TCR and relative tracking are different requirements. A report that pools unrelated individual coefficients does not establish tracking of the actual resistor pair used in a divider or network. Preserve the pairing identity and compare its responses under the required common-temperature condition. Detailed ratio and thermal-gradient design belongs to the linked matched-pair analysis, not to a pooled material average.

In the two-group example, placing an A element in the numerator and a B element in the denominator gives a normalized ratio of 1.005 divided by 0.995 after the 50 kelvin rise. That is approximately 1.01005, or a 1.005 percent ratio increase, despite the zero pooled mean. This calculation assumes the two linear responses and equal actual temperature changes. It does not describe any supplied resistor series.

Check whether series and measurement conditions are confounded

If every A specimen is measured first and every B specimen later, an instrument or temperature drift can resemble a series difference. Plan repeated reference checks or an appropriate interleaved sequence while preserving stabilization and specimen handling. Record actual temperatures and reading times instead of assigning every resistor the nominal chamber setting.

Likewise, a series tested in one fixture position should not be compared uncritically with another series tested only in a different position. Separate those factors through a practical controlled comparison. Keep uncertainty analysis distinct from observed population spread: common errors can shift groups together, while fixture-specific conditions can create an apparent difference. The linked covariance-aware TCR method addresses the numerical uncertainty calculation.

Release a group-specific conclusion that matches the requirement

The final table should identify each controlled group, coefficient definition, specimen count, temperature interval, process state and acceptance basis. Include the individual results and the reason for any pooled summary. Where a requirement applies to a circuit pair, add the pair record rather than substituting the group mean. Where data are insufficient for a group, retain that gap explicitly in the technical review.

For a thick-film resistor enquiry, provide the material alternatives and the circuit's actual absolute or ratio-temperature requirement. ChipSimple can review the relevant construction by drawing. The next action might be additional within-series samples, a corrected measurement sequence or a different material pairing; a favorable all-sample average alone is not a basis for transferring one group's evidence to another.