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A resistor turn must redirect current inside the space left by pads, adjacent tracks and the substrate boundary. Choosing its shape is a different decision from assigning an empirical corner correction to total resistance. The useful comparison asks how a rounded bend, a chamfer or a locally widened transition distributes current, which dimensions the printing route can reproduce, and what must move elsewhere to preserve the required resistance. Begin with the physical inside edge and the available envelope. A visually smooth outer outline says little about the shortest current path along the inside of the turn.
Key design decisions
- Rank candidate bends by the smallest reproducible inside radius relative to the local track width.
- Compare electrical concentration and occupied space before adjusting straight length to recover resistance.
- Confirm the selected outline on printed coupons with matched terminals and a controlled thermal attachment.
Reserve the entire turn envelope
Mark where the two straight arms enter the turn, their widths and the required change in direction. Include nearby conductor potentials, ceramic edges, coating openings and the region reserved for resistance adjustment. A larger radius can improve the distribution within a bend while reducing the gap to a neighboring feature. Evaluate both boundaries together; the outside edge cannot expand into an electrical or mechanical keepout simply because the inside edge looks better.
Separate movable features from fixed interfaces. Terminal locations may be contractual while a straight segment can move or lengthen. When several options fit, retain the same entry and exit planes so their resistance and regional loss can be compared. Changing the arm length at the same time changes the baseline and makes a claimed bend improvement harder to interpret. Keep an annotated outline for each candidate rather than relying on the names rounded or chamfered.
Use an annular sector to screen radius sensitivity
An ideal annular sheet provides a useful analytical comparison for a constant-width curved turn. Its inner paths are shorter than its outer paths, so they carry more sheet current under the same voltage between radial end faces. The model is deliberately limited: it excludes the junction to straight arms, print thickness gradients and termination effects. It therefore ranks a geometric tendency without predicting the finished product's peak temperature.
Normalize the inner-edge sheet current by current divided by track width. This makes the comparison independent of the chosen test current and uniform sheet resistance. The resulting concentration factor approaches one when the inside radius is large compared with width. It grows as the inner radius shrinks. Use the smallest physically plausible printed radius in a sensitivity case; the mathematical zero-radius limit is not a manufacturable specification.
Cinner = Kinner/(I/W) = W/[ri ln((ri + W)/ri)]; Rsector = Rs φ/ln((ri + W)/ri)
- Cinner is the dimensionless inner-edge sheet-current concentration factor.
- Kinner is sheet current in amperes per unit width; I is total turn current.
- W and ri are track width and inside radius in identical length units.
- Rs is uniform sheet resistance and φ is the turn angle in radians.
An isotropic annular sheet has equipotential radial ends, insulating circular edges, uniform sheet resistance and no straight-arm junction disturbance. This is an analytical screening geometry.
Compare two radii at the same track width
Consider a hypothetical quarter-turn with a width of one millimeter. An inside radius of 0.5 millimeter gives Cinner = 1/[0.5 ln(3)] = 1.820. Increasing the inside radius to two millimeters gives Cinner = 1/[2 ln(1.5)] = 1.233. At equal current and equal sheet resistance, inner-edge areal generation relative to the uniform straight-strip reference scales with the square of this factor: approximately 3.31 and 1.52 respectively.
The larger radius also changes resistance and footprint. With an illustrative sheet resistance of 100 ohms per square, the quarter-sector resistances are about 143.0 and 387.4 ohms. The second bend follows a longer path; it cannot replace the first without reviewing the remaining resistor length. Neither calculated generation ratio is a temperature ratio. Heat spreading, contact beneath the turn and the adjoining arms still determine the temperature field.
Choose a shape for its specific constraint
A circular fillet offers a continuous change of direction. A chamfer substitutes short straight boundaries and changes where the current must turn; it does not automatically remove all local field concentrations. Local widening provides additional conducting area but can relocate a constriction into the transition back to the original width. Inspect those transitions as part of the candidate rather than treating them as harmless connectors.
The appropriate shortlist depends on available area and process reproduction. Do not select the option with the lowest simulated point value if it requires a radius or taper that the actual printed edge cannot preserve. The technical table identifies what each alternative must demonstrate before it can be retained. A straight reroute may be preferable when extra area exists, but it must still satisfy the resistance and terminal arrangement.
| Outline | Useful comparison | Feature requiring attention |
|---|---|---|
| Continuous circular turn | Inside-radius sensitivity at fixed width | Space consumed by the outside radius |
| Chamfered corner | Regional loss around every new junction | Two smaller direction changes replacing one corner |
| Locally widened turn | Concentration through both width transitions | Necking where the wide region rejoins an arm |
| Distributed gradual bend | Loss and resistance over the complete curve | Reproduction of a long changing boundary |
| Straight reroute | Resistance recovery and terminal access | Extra length and relocated thermal surroundings |
Evaluate the edge that survives printing
Inspect fired inner radii and widths on multiple examples, including different orientations relative to the print direction. A nominal fillet can be partly filled by paste spreading, while a nominal chamfer may round during transfer. Record the actual minimum neck and the profile approaching each arm. Use scaled optical observations and, where thickness is relevant, a suitable profile measurement; surface color cannot supply film thickness.
Apply the observed dimensional envelope to each candidate. One practical sensitivity comparison combines a smaller inside radius with a narrower adjoining neck, because those departures may occur together. Do not combine unrelated worst cases without explaining their physical plausibility. Conversely, do not use only the smoothest specimen as the model geometry. The decision needs a shape whose intended benefit survives ordinary dimensional variation in the reviewed process.
Validate the shortlist without hiding drive changes
Print the shortlisted turns beside a straight witness using the same material stack and processing history. Keep arm reference planes and measurement terminals consistent. First measure low-power resistance and geometry. Then compare powered behavior at matched current to examine geometric concentration, followed by the actual application drive condition. If total resistance differs, a constant-voltage comparison also changes current and can reverse an apparent thermal ranking.
Resolve the bend adequately with the selected thermal method and document its spatial averaging. Repeat the comparison after a controlled remounting so an interface gap does not decide the artwork choice. A numerical model should converge in terminal resistance and integrated turn loss; local field outputs additionally require resolved finite edge dimensions. The validation result should identify a preferred candidate, the remaining uncertainty and the tested geometry envelope.
Read failure locations before enlarging every radius
A narrow hot streak following an inside edge across several specimens is consistent with electrical concentration. A single damaged turn among otherwise identical ones points first toward a local void, scratch or thin region. A hot patch that follows a mounting location after specimens are exchanged suggests heat-removal variation. Record these locations against the same drawing coordinates before cleaning, coating removal or sectioning changes the evidence.
An apparently improved bend can still fail at its exit taper, or show excessive resistance variation because the inside radius is poorly reproduced. Track those signatures separately from total-value rejection. If the thermal maximum moves after a change, determine whether the new location is acceptable instead of declaring success from the disappearance of the original peak. A bend should be selected by the complete current path it leaves behind.
Document the selected outline and resistance recovery
The final design record should state the chosen bend dimensions, the physical radius convention, entry and exit coordinates, adjacent gaps and the permitted straight-length adjustment. Include the lower-bound radius case used in analysis and the inspection method that can distinguish an acceptable turn from a filled or necked one. Keep trim access away from an already sensitive transition unless that combined geometry has been evaluated.
Repeat the comparison when track width, turn angle, material system or local attachment changes enough to invalidate the evidence. A familiar radius in millimeters does not carry the same meaning after width doubles. The portable result is the defined geometry and its verified comparison, including radius-to-width sensitivity. Power and endurance requirements remain questions for the complete resistor and its installation.
Send the bend alternatives
Provide the geometric choices and the current-turn problem they must solve.
- Entry and exit planes, turn angle, local width, candidate inside radii, neighboring gaps and fixed keepouts.
- Required resistance, movable straight length, trim region and operating current or voltage waveform.
- Named resistor stack with scaled fired-edge observations, dimensional variation and available thickness evidence.
- Substrate attachment, matched-drive thermal comparisons, observation resolution and the failure locations of concern.
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