What is sheet resistance?
Sheet resistance is the resistance of a thin film, expressed in Ω/□ (ohms per square). It is the natural unit for any conductive layer that is thin compared to its lateral dimensions, including evaporated metal films, sputtered ITO, screen-printed silver paste, inkjet-printed carbon, and any other printed or deposited conductor.
Unlike bulk resistivity (measured in Ω·m), which depends on the thickness of the material, sheet resistance already has the thickness folded in. The relationship is:
R_s = ρ / t where ρ is the bulk resistivity and t is the film thickness.
This is why ohms per square is so useful in printed electronics. The print thickness varies with screen mesh, ink viscosity, squeegee pressure, and curing. Rather than measure resistivity and thickness separately, we measure sheet resistance directly. It tells us everything we need to know about the printed layer's electrical performance in one number.
The "per square" name comes from a quirky property: a square of conductive film, regardless of size, has the same resistance corner-to-corner. A 1 mm square and a 100 mm square of the same film both measure the same end-to-end resistance, because doubling the length doubles resistance while doubling the width halves it. The two effects cancel.
Calculating trace resistance
For a rectangular printed trace of length L and width W, the total resistance is:
R = R_s × (L / W)
The ratio L/W is dimensionless and is called the number of squares. A trace that is 50 mm long and 2 mm wide has 25 squares. If the ink has a sheet resistance of 0.05 Ω/□, the trace resistance is 0.05 × 25 = 1.25 Ω.
This simple relationship is the foundation of printed heater design, antenna trace design, sensor layout, and electrode patterning. By choosing the right ink (which sets R_s) and the right geometry (which sets L/W), we can target any resistance value within practical limits.
Typical sheet resistance values
The table below gives realistic starting ranges for common printed inks at standard print thicknesses. Actual values depend on the specific ink formulation, print process, and curing conditions.
| Material | Typical sheet resistance | Common applications |
|---|---|---|
| Silver (Ag) ink | 10 to 100 mΩ/□ | Antennas, low-resistance traces, bus lines, heaters |
| Silver/Silver Chloride (Ag/AgCl) | 50 to 500 mΩ/□ | ECG, EEG, and other bio-potential electrodes |
| Copper (Cu) ink | 30 to 200 mΩ/□ | Cost-sensitive antennas and traces (Suryudey is increasingly using copper as a partial silver replacement) |
| Carbon ink (conductive) | 10 to 100 Ω/□ | Heaters, stimulation electrodes, flexible sensors |
| Carbon ink (resistive) | 100 to 1000 Ω/□ | Resistive heaters, force/strain sensing, current limiting |
| Silver nanowire (AgNW) | 5 to 50 Ω/□ | Transparent antennas, transparent heaters, transparent electrodes |
| PEDOT:PSS | 50 to 1000 Ω/□ | Flexible biosensors, transparent stretchable electrodes |
Why four-probe measurement?
The challenge with measuring sheet resistance is contact resistance. When you place two probes on a film and pass current through them, the meter sees three resistances in series: the film resistance you actually want, plus the contact resistance at each probe-film interface. For a low-resistance film like silver, the contact resistance can easily exceed the film resistance, completely corrupting the measurement.
The four-probe method (also called the four-point probe or Kelvin method) solves this elegantly. Four collinear probes are placed on the film. Current is forced through the outer two probes. Voltage is measured across the inner two probes using a high-impedance voltmeter that draws essentially no current.
Because the voltmeter draws no current, the voltage drop across the contact resistance of the inner two probes is essentially zero. The voltage reading is purely from the film between the two inner probes. Dividing this voltage by the forced current gives a resistance that depends only on the film, not on the quality of the electrical contacts.
For a large enough film with collinear probes spaced equally, the sheet resistance is calculated as:
R_s = (π / ln 2) × (V / I) ≈ 4.532 × (V / I)
For finite samples and other probe geometries (such as the Van der Pauw four-contact method), correction factors are applied. The Van der Pauw method is particularly useful for arbitrarily shaped samples and is the international standard for semiconductor sheet resistance measurement.
How Suryudey uses four-probe measurement
At Suryudey, every printed run is characterised with four-probe sheet resistance measurements. This is part of our quality control process, not an afterthought.
The reason: every batch of conductive ink, every screen mesh, every curing profile, and every substrate batch can shift the sheet resistance. A 20% variation in R_s means a 20% variation in heater resistance, which means a 20% variation in power dissipation at a fixed voltage. For a heated jacket, that translates to noticeable warmth differences from one production batch to the next. For an antenna, it shifts the resonant frequency. For a sensor, it shifts the baseline.
By measuring sheet resistance on every print run, we catch process drift early. If R_s starts climbing across batches, that's a signal that the ink is settling, the screen is loading, or the curing oven needs recalibration. The four-probe measurement gives us a precise, repeatable number that we can track over time and use to keep our printed electronics within specification.
This rigour is one of the reasons we can deliver printed heaters, antennas, and electrodes with consistent performance from prototype to production. It is also why we prefer to control the entire manufacturing process in-house. Outsourced printing introduces variables we cannot measure.