Conductive Trace Resistance

The specimen is a 20 mm gauge length of trace held between two clamps on a stretchable substrate. Drag to move the right clamp; the strain is read against the dashed rest line. Lengths are to scale. The widths of the serpentine and the yarn are exaggerated so their structure can be seen, and the scale bar says by how much.

The moving dots are the current. A constant voltage sits across the clamps, so the electrons’ speed is proportional to 1/R: as the resistance climbs the current slows, and an open circuit stops it. The path they take is the real one for that structure — around a crack tip, along the meander, across a filament contact.

The chart plots R/R₀ against strain on a log scale, with all three structures for comparison and a faint trace of the path this particular sample has actually taken, loading and unloading. Where that path fails to return along the reference curve, the sample is damaged.

A stretchable printed conductor is not a continuous metal. It is a dense population of silver flakes in an elastomer binder, and it conducts because enough flakes overlap to form connected paths from one end to the other. Stretch the binder and the flakes, which are rigid, move apart. Paths break, the survivors get longer and more tortuous, and resistance rises roughly exponentially with strain before anything you could call a crack has appeared.

Past a threshold the film cracks from its edges. A crack is a hole in the network: the current must go round its tip through whatever conductor remains, and a few deep cracks can raise the resistance by orders of magnitude. Release the strain and the cracks close to hairlines but not to nothing, so a printed trace remembers its peak strain. That is the retained-damage figure in the readouts, and the reason a printed track that has been over-stretched once reads high forever after.

Copper extends by a few percent before it fails, so a straight copper trace on a stretchable substrate is not a stretchable interconnect. A serpentine one is: laminated onto TPU film, the meander opens like a spring, taking the substrate’s strain in bending while the copper’s length, and so its resistance, stays exactly what it was. Watch the readout while the meander straightens — it does not move.

Once the serpentine is straight the copper itself has to strain, and it does so with a gauge factor of about two: a 1% strain reads as a 2% resistance change, small enough to be nearly invisible on a log chart. Then it fractures. There is no gradual failure and no recovery: the trace goes from a few tens of milliohms to an open circuit, and a new sample is the only way back. The design lesson is that the meander’s amplitude and pitch set the strain budget, and the copper is only ever safe inside it.

A silver-plated nylon yarn is a bundle of filaments, each conductive along its length but insulated from its neighbours except where they physically touch. Its resistance is set as much by those contact points as by the silver. A little strain aligns the filaments and presses them together, which is why the resistance dips slightly first — a well-known and often unwelcome feature of yarn-based strain sensors.

With more strain the contacts begin to open, each at its own threshold, and every one that goes removes a path between filaments. The electrons have to run further along a single filament to find the next crossing, so resistance rises steadily and then steeply as the network thins. Unlike the printed film, this is reversible: relax the yarn and the contacts re-form, with some hysteresis, and the resistance comes back down. It is why conductive yarn tolerates stretch a printed trace cannot, and why its resistance is never quite stable.