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Sizing a printed resistor from its target value is an inverse problem. The target includes everything between the specified sensing planes, while the rectangular body equation describes only one part of that path. Subtracting a measured end contribution before choosing the body can change the required aspect ratio appreciably. This calculation uses a supplied, applicable correction; it does not extract that correction from coupons or choose a resistor paste. Its output is a dimensioned candidate with a traceable arithmetic check.
Key design decisions
- Choose either resistance-valued ends or an equivalent-length correction, and use that convention consistently.
- Solve for the fired body before converting its dimensions into artwork.
- Reject a numerical solution when the resulting geometry falls outside the correction's characterized range.
Assign the target to two sensing planes
Mark the exact terminals across which the required resistance applies. Conductor extensions between the resistor and the customer's terminals may contribute to that value. If the correction study measured inside those extensions, either add their independently established resistance or move the target boundary. An unexplained terminal convention can invalidate an otherwise correct inverse calculation.
Use the resistance required at the selected manufacturing stage. For a resistor that will later be adjusted upward, the pre-trim target is a separately justified input. Do not derive it by subtracting an arbitrary percentage from the final nominal value. The calculation cannot determine trim capability, permitted remaining neck, or post-trim stability from geometry alone. Record temperature and excitation alongside the stage-specific target so that the supplied sheet resistance and end terms describe the same state.
Keep ohmic ends distinct from extra length
One model describes the combined ends as a signed resistance b. Another uses a total equivalent length dL. They are related through b = Rs dL/W for a uniform body of width W. This relationship means that keeping b constant while changing width is a different assumption from keeping dL constant. A spreadsheet must not silently switch between those assumptions.
The sign belongs to the declared length origin. A positive b adds resistance relative to the rectangular baseline; a negative fitted b reduces it. Negative b does not mean a passive contact generates energy. It may represent a geometric region counted in L that conducts better than the assumed body. Never convert both b and dL into additive terms in the same prediction, because they may describe the same end contribution twice.
Solve the rectangle before rounding dimensions
With resistance-valued ends, remove their combined contribution from the target and divide the remaining value by the sheet resistance. Multiplying the resulting dimensionless ratio by the chosen fired width gives fired body length. The calculation requires positive sheet resistance and a positive body allocation; a nonpositive allocation indicates an inconsistent target or a model used outside its domain.
For an equivalent-length model, first calculate the ideal length and then subtract the signed length correction. Treat a width-dependent end function as b(W), evaluating it at each candidate width. If the supplied correction also depends on length, solve the full equation iteratively and verify uniqueness across the permitted geometry interval instead of using the explicit shortcut.
a = (Rt − b)/Rs; L = W a. Equivalently, L = W Rt/Rs − dL when b = Rs dL/W.
- Rt: stage-specific resistance target in ohms.
- Rs: sheet resistance of the applicable processed film in ohms per square.
- b: combined signed end contribution in ohms at the selected width.
- a: dimensionless body aspect ratio L/W; L, W and dL use one length unit.
Uniform untrimmed straight body; matched processing and sensing conditions; no separate corner term; end correction valid at the solved dimensions.
Calculate a length and check it in the forward direction
Consider an illustrative target Rt of 2400 Ω, sheet resistance Rs of 600 Ω per square, combined b of 120 Ω and fired width W of 0.75 mm. The body receives 2280 Ω, so a equals 3.80 and L equals 2.85 mm. Substitution gives 600 × 2.85/0.75 + 120 = 2400 Ω. Using four squares without removing the ends would instead predict 2520 Ω.
The equivalent total length correction is 120 × 0.75/600 = 0.15 mm. The second formulation gives 0.75 × 2400/600 − 0.15 = 2.85 mm, confirming that the conventions agree here. If a drawing rounds the body to 2.90 mm, the forward prediction becomes 2440 Ω, a 40 Ω shift. These numbers illustrate design arithmetic and are neither sample measurements nor a dimensional capability statement.
| Quantity | Operation | Result |
|---|---|---|
| Resistance assigned to body | 2400 − 120 | 2280 Ω |
| Required body squares | 2280 / 600 | 3.80 |
| Fired length | 0.75 × 3.80 | 2.85 mm |
| Total equivalent end length | 120 × 0.75 / 600 | 0.15 mm |
| Forward result after 2.90 mm rounding | 600 × 2.90 / 0.75 + 120 | 2440 Ω |
Recalculate when the correction sign or width changes
Keeping the example's target and sheet resistance but using b = −120 Ω gives 4.20 body squares and a length of 3.15 mm. Taking the absolute value of b would choose the wrong direction and miss the target by twice the end contribution. Retain the sign in the input record, worksheet and drawing review.
Now increase width to 1.00 mm. If a verified resistance-valued b remains 120 Ω, the length becomes 3.80 mm. If instead the applicable model preserves dL = 0.15 mm, length becomes 3.85 mm and the corresponding b becomes 90 Ω. Neither assumption can be selected merely because it gives a convenient dimension. Request a width-specific correction or bound the uncertainty until matching evidence resolves the choice.
Translate coefficient uncertainty into length allowance
For the explicit model, the length sensitivity to the end intercept is −W/Rs. In the example this is −0.00125 mm per ohm. An illustrative 24 Ω uncertainty in b therefore corresponds to 0.030 mm in calculated length before other components are included. This indicates whether refining the end estimate would materially improve sizing.
Width uncertainty, sheet-response uncertainty and target-setting uncertainty enter separately. If b and Rs came from one fitted length series, retain their covariance when propagating uncertainty. Their errors may compensate or reinforce in the inverse length. Use bounded corner combinations for drawing tolerances and an uncertainty model for measurement estimates; the two answer different questions. Neither should be represented as extra decimal places on a nominal length.
Validate the solved dimension against withheld specimens
Check the calculation against coupons near the proposed length and width that were not used to establish the end correction. Measure their actual body dimensions, resistance and observation planes. Predict resistance using the frozen coefficients before viewing each result. This prevents an end factor being retuned until every test appears to agree.
Include a nearby shorter and longer body and, when scaling width, a second width family. Record signed prediction errors against dimensions, panel position and processing state. Agreement at one target can conceal compensating errors between body and ends. Verification must also address the proposed rectangle's load and physical construction separately; a correct low-power resistance prediction does not establish its usable voltage or thermal envelope.
Recognize solutions that should be rejected
A negative solved length, a sign reversal after changing units, or a large jump after switching correction conventions indicates a calculation or domain problem. A consistent resistance offset after rounding points toward the dimensional translation. Errors that grow on short bodies suggest interaction between ends, while a width-specific bias questions the assumed scaling of the correction.
Issue the resulting candidate with its fired length, width, target stage, signed end basis and forward prediction. Keep the mapping from fired dimensions to screen artwork as a separately controlled process transformation. Recalculate after changes in terminal geometry or the coordinate definition. The useful result is a defensible dimension to prototype, including what uncertainty remains, rather than an unqualified instruction to print the ideal rectangle.
Send the inverse-sizing inputs
A sizing review needs the target and the exact correction convention together.
- Required resistance at the specified pre-trim or finished stage, sensing planes, temperature and test excitation.
- Applicable sheet-resistance estimate and signed end coefficients with units, fitted range, uncertainty and covariance if available.
- Chosen fired width, permitted length envelope, conductor overlaps and inspectable length-origin drawing.
- Available dimensional and electrical coupon data, artwork mapping, rounding rule and acceptance allocation.
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