Resistor Variation: Separating Sheet Resistance from Geometry

Define the variation that the study is trying to explain

Specify whether the observed spread is among resistors on one panel, among panels in a printing run or among separate lots. Those are different populations. A run-level sheet-resistance movement can coexist with narrow within-panel variation, while local width changes can dominate only the smallest features. Pooling these levels can produce an apparently strong relationship that disappears when each run is examined separately.

Use untrimmed rectangles if the purpose is to diagnose the printed film. A laser cut introduces another current-path variable and a final value chosen by feedback, making the original geometry relationship harder to observe. If only trimmed parts are available, retain their pre-trim records or define a different model that includes cut shape. The attribution output should name a population and processing stage; it should not claim to isolate paste chemistry from every other influence.

Pair the resistance reading with the effective electrical dimensions

Assign a specimen identifier before testing and preserve it through imaging, dimensional inspection and electrical measurement. Measure length between the relevant electrical termination boundaries, not between the outer ends of a printed patch that overlaps metal. Measure width using a documented edge definition. Irregular edges, tapered deposits and end spreading may require several measurements or a more detailed current-path model.

Keep the physical specimen state fixed during the comparison. Temperature, humidity conditioning, applied test power and elapsed time after processing can alter resistance while the optical dimensions remain unchanged. Repeatedly measure a retained subset to estimate electrical and optical repeatability. A dimension reported with many decimal places is not necessarily accurate enough to explain small resistance shifts, especially when the visible edge does not coincide with the electrically active edge.

Separate multiplicative effects using measured square count

For an approximately rectangular body, define effective squares as length divided by width. After subtracting a justified series terminal contribution, estimate sheet resistance by dividing body resistance by those measured squares. Taking logarithms converts the multiplicative relationship into additive terms. This is useful for attribution because a geometric increase and a film decrease retain their opposite signs without a first-order percentage approximation.

Choose a reference rectangle and reference sheet resistance solely as normalization values. They do not have to equal the batch means, but they must remain fixed throughout the analysis. The inferred sheet resistance is an effective processed-film quantity. It includes thickness and material development as well as any unremoved interface or model error. Calling that residual a paste-lot change would claim more than these measurements establish.

Calculate one specimen before interpreting an entire distribution

Take a hypothetical reference with length 2.00 millimeters, width 0.50 millimeter and sheet resistance 1,000 ohms per square. Its four squares predict 4,000 ohms. A second specimen has effective dimensions 2.04 by 0.50 millimeter and a terminal-corrected reading of 4,202.4 ohms. Its measured square count is 4.08, so the inferred sheet resistance is 4,202.4 divided by 4.08, exactly 1,030 ohms per square.

The observed resistance increase is 5.06 percent. Geometry contributes a factor of 1.02 and sheet resistance contributes 1.03; multiplying them gives 1.0506. Adding two and three percent would omit the 0.06 percentage-point cross term. In logarithmic form, ln(1.0506) equals ln(1.02) plus ln(1.03), so the decomposition closes without that approximation. Using only nominal dimensions would label the entire resistance increase as sheet-resistance change. These assumed values demonstrate attribution; they are not measurements from a production lot.

Keep covariance when explaining the final resistance spread

Across a population, the variance of logarithmic resistance equals the variance of the sheet term plus the variance of the geometry term plus twice their covariance. A process that deposits a wider and thicker resistor can make geometric and film terms move together in a correlated way. Removing one variation source may therefore change the cancellation that previously narrowed the final resistance distribution.

For a hypothetical dataset, assume standard deviations of 0.020 in the sheet term and 0.015 in the geometric term, with correlation minus 0.60. The resistance-log variance is 0.020 squared plus 0.015 squared minus twice 0.60 times 0.020 times 0.015, giving 0.000265. Its standard deviation is approximately 0.0163, or 1.63 percent for small changes. Ignoring covariance would predict 2.50 percent. Neither result is an acceptance limit, and the correlation must come from paired data rather than a desired outcome.

The inferred sheet term is calculated from resistance minus the geometric term in logarithmic form. Dimensional measurement error therefore enters those two terms with opposite signs and can create artificial negative covariance. Use repeat imaging, electrical repeats and a measurement-error model to distinguish that arithmetic coupling from real process cancellation. A negative correlation in the derived columns alone does not prove that manufacturing variables compensate each other.

Use residual patterns to challenge the rectangular model

Compare the inferred sheet values by length, width, orientation, panel position and measurement session. A real material-process shift need not be spatially uniform, but a systematic dependence on geometry is a reason to test missing electrical effects. Do not force all samples onto one fitted line by assigning every discrepancy to measurement noise.

Validate attribution with a deliberate geometric contrast

Prepare a small matrix of repeated rectangles with independently varied length and width while keeping paste, terminations and thermal history comparable. Include more than two lengths if a terminal intercept is to be estimated; two points cannot reveal curvature. Distribute repeated geometries across the panel instead of placing every narrow specimen at one edge. Otherwise position and geometry become inseparable explanations.

Measure body resistance and processed dimensions, fit the declared model, then evaluate a retained geometry that was not used to set its coefficients. A prediction that succeeds only on the fitting specimens has weak value for changing artwork. Compare residuals with independently assessed measurement uncertainty and inspect failures. If the model fails for a geometry class, report that boundary and investigate the physical cause before using a refined statistical fit to recommend production adjustments.

Choose the process action from the attribution result

When measured square count explains most within-panel movement, review artwork transfer, edge definition and geometry control before changing the nominal sheet-resistance grade. When geometry-normalized values move together across a thermal or material change, examine the processed film and supporting witnesses. A residual alone cannot distinguish thickness, chemistry, firing history and substrate interaction; those require another controlled comparison.

Document how the proposed action affects covariance as well as the individual terms. Tightening width variation can improve dimensional conformity while unexpectedly changing final resistance spread if width previously offset another variable. Re-run the paired study after the adjustment using the same specimen definitions. The useful deliverable is an attributable change with a stated model boundary and a verified prediction, rather than a percentage allocation that treats correlated process variables as independently removable.