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A corner can contribute less resistance than its centerline length suggests because current divides among paths with different radii. The size of that difference must be stated against a specific baseline. This calculation uses an ideal circular strip to obtain a transparent analytical comparison, then asks how much the approximation changes the complete resistor. It does not provide a universal factor for right-angle artwork, a preferred bend shape, or an allowable current rating.
Key design decisions
- Define the corner region and its contact boundaries before comparing square counts.
- Compare both the corner-only discrepancy and its fraction of complete resistance.
- Use the analytical sector as a benchmark for a model with the actual printed outline.
Specify the circular region being calculated
Consider a conducting annular sector with inner radius a, outer radius b and angular extent theta. Its thickness and sheet resistance are uniform. The radial faces at the two angular ends are ideal equipotential contacts; the inner and outer arcs are electrically insulating boundaries. Current therefore runs around the arc. Use radians for theta and the same length unit for both radii.
These contacts are deliberate mathematical boundaries. When a sector joins ordinary straight arms, the junction cross-sections may not be equipotential in the same way. Additional spreading can then alter the integrated resistance. Label the sector result as an ideal benchmark and keep the arm-to-bend transition outside its claim. A chamfered elbow, a square corner and a U-turn with changing width are different geometries requiring their own solution.
Integrate the conductance of parallel arc strips
Split the annulus into thin strips at radius r. Each has path length theta times r and width dr, giving resistance Rs theta r/dr. The strips share the same endpoint voltage, so their conductances add. Integrating dr divided by Rs theta r from a to b produces a logarithm, rather than an arithmetic average of the path lengths.
The result contains no absolute radius scale if b/a stays constant. Enlarging both radii and width by the same factor leaves the ideal resistance unchanged. A manufactured thick film need not preserve that scale invariance because edge rounding, thickness and material formation may change relative to the feature size. Treat the invariance as a property of the stated mathematical sheet, not as evidence that two printed sizes are interchangeable.
Gbend = ln(b/a)/(Rs theta); Rbend = Rs theta/ln(b/a); Rcenter = Rs theta(a+b)/[2(b−a)].
- Gbend: integrated bend conductance in siemens.
- Rbend: sector resistance in ohms; Rs: uniform sheet resistance in ohms per square.
- a and b: positive inner and outer radii, with b greater than a.
- theta: bend angle in radians; Rcenter: centerline-length straight-strip estimate.
Uniform isotropic ohmic sheet, radial equipotential end contacts and no current through the circular edges; contacts, straight arms and self-heating excluded.
Use one centerline baseline for the subtraction
For a constant-width circular strip, width is b minus a and the centerline radius is their arithmetic mean. Dividing centerline arc length by width gives the familiar straight-strip estimate. Define the signed discrepancy as Rbend minus Rcenter. For this ideal sector it is negative, because the shorter inner strips carry more current than a uniform centerline representation allows.
A different partition can move resistance between the bend and its arms without changing the whole pattern. Before comparing a coupon-derived factor with this result, match the entry and exit planes. Also establish whether a reported coefficient is the full bend contribution or only a correction to an already-counted centerline path. Adding a full bend resistance to a baseline that already contains the bend counts the same region twice.
Work through a sixty-degree corner
Take a hypothetical sheet resistance of 120 Ω per square, a = 0.40 mm, b = 1.00 mm and theta = π/3. The radius ratio is 2.5, so ln(b/a) is approximately 0.916291. The calculated bend is 137.14 Ω. The 0.60-millimeter-wide centerline path has length 0.733038 mm, giving 146.61 Ω under the straight approximation.
The signed difference is approximately −9.46 Ω, or −6.46% relative to the centerline estimate. That percentage describes this isolated ideal corner. It is not the error of an entire resistor unless the resistor consists solely of that sector. Rounding ln(2.5) early would introduce avoidable numerical error, so keep adequate intermediate precision and round the final values to the usefulness of the geometric inputs.
| Term | Calculation | Value |
|---|---|---|
| Bend width | 1.00 − 0.40 | 0.60 mm |
| Radius ratio | 1.00 / 0.40 | 2.50 |
| Integrated bend squares | (π/3) / ln(2.5) | 1.14287 |
| Centerline squares | (π/3) × 0.70 / 0.60 | 1.22173 |
| Difference in resistance | 120 × (1.142866 − 1.221730) | −9.46 Ω |
Calculate how the straight body dilutes corner error
Suppose the corner is part of a hypothetical resistor containing 1000 Ω of straight body and 50 Ω of independently established terminal contribution. The integrated-sector prediction is 1187.14 Ω, while the centerline treatment predicts 1196.61 Ω. The centerline overestimate is about 0.80% relative to the corrected whole-resistor value, much smaller than the corner-only percentage.
For several separated identical bends, add their full resistance contributions once and compare against the same number of centerline regions. Additivity must be checked when turns are close enough to disturb one another's entry distribution. Do not multiply a local percentage by the number of bends; sum resistance differences in ohms and divide by the correct complete resistance. This bookkeeping reveals whether resolving the corner is necessary for the actual initial-value budget.
Separate integrated resistance from inner-edge concentration
The angular electric field in the sector varies inversely with radius. Its inner-edge value is therefore b/a times its outer-edge value under the ideal model. For the example that ratio is 2.5. This does not mean the temperature ratio is 2.5: local electrical loss and heat transport obey different relationships, and the substrate and attachment redistribute heat.
At a fixed sector voltage, local areal loss varies with the square of inverse radius for the uniform sheet. A resistance estimate can thus improve while a substantial inner-edge concentration remains. Neither the integrated result nor the centerline discrepancy establishes a permissible power density. Use the actual finite printed radius and a defined thermal installation when the engineering decision concerns damage or operating temperature.
Validate the arithmetic before modeling the printed bend
First reproduce the sector formula in an electrical field model using the specified radial contacts. Compare terminal current and integrated loss across a mesh sequence; the numerical resistance should approach the analytical value. This checks units, conductivity conversion, angle and boundary implementation without relying on an unknown material correction.
Then replace the ideal outline with measured geometry and connect the actual straight arms. Hold sheet properties fixed initially so the difference isolates shape and junction effects. Only afterward introduce measured thickness or sheet-response variation. Validate representative low-excitation specimens with recorded contact planes and dimensions. Keep the ideal benchmark, actual-outline prediction and measurement as separate columns, so agreement obtained by changing several inputs simultaneously is not mistaken for verification.
Diagnose a corner comparison that does not close
A factor near 57.3 can indicate degrees entered where radians were required. A large geometry-independent offset suggests contacts or arms were omitted. A resistance change after moving the reference planes signals a partition mismatch. Disagreement that grows when adjacent turns move closer questions the independent-corner assumption. These signatures identify which part of the calculation should be revisited.
A converged terminal resistance alongside a continually increasing pointwise peak at a mathematically sharp corner is a different issue: refine the physical radius definition before treating that peak as a real stress. Record the chosen comparison denominator, geometric boundaries and remaining model discrepancy with the drawing. The deliverable is the calculated contribution and its uncertainty within a defined outline, not a broadly transferable corner allowance.
Provide the corner comparison geometry
Include both the bend and the straight reference used to judge its contribution.
- Measured inner and outer outlines, bend angle, film thickness convention and the proposed entry and exit planes.
- Sheet-resistance basis, straight-arm dimensions, terminal contribution and processing state at measurement.
- Resistance target, allowed initial prediction error and whether the requested comparison is local or whole-pattern.
- Low-excitation coupon measurements, field-model boundary conditions, mesh checks and intended load/thermal installation.
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