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A circular outline does not imply uniform electrical heating. In a continuous resistive annulus driven between complete inner and outer electrodes, the same current crosses progressively larger circumferences. Current per unit width falls with radius, and electrical power per unit area falls with radius squared. This specific model explains why an acceptable terminal resistance can coexist with strongly unequal inner- and outer-rim generation.
Key design decisions
- Confirm radial current between complete rim electrodes before using the annular-sheet equation.
- Calculate local electrical generation separately from total power divided by the ring area.
- Review radius scaling and thermal boundaries independently; unchanged resistance does not mean unchanged temperature.
Identify the electrical annulus, not merely the ceramic outline
The model requires a continuous uniform resistive sheet between circular inner and outer boundaries. Each boundary is contacted around its complete circumference by an electrode whose voltage drop is negligible relative to the sheet drop. Current flows radially through the sheet. A serpentine resistor printed on a ring-shaped substrate does not meet these assumptions merely because the substrate has a central hole.
Use the actual layer and terminal map to decide whether the model applies. An interrupted ring, a narrow terminal tab feeding a resistive rim, or separated arc-shaped resistor tracks can produce angular current variation. Those cases need their own electrical boundary model. The calculation below is a defined design model; a new annular product construction still requires its material, electrode and manufacturing review.
Add the resistance of concentric differential bands
Let the sheet resistance be Rsq in ohms per square and the inner and outer radii be a and b. A thin band at radius r has radial length dr and current-path width 2 pi r. Its differential resistance is Rsq dr/(2 pi r). These bands are traversed in series, so integration produces a natural logarithm rather than a simple radial-length-to-one-width ratio.
The sheet resistance must represent the processed film at the evaluated condition. It incorporates resistivity and film thickness in the uniform-sheet approximation. Do not use the ceramic thickness as the resistive-film thickness. At this stage the formula describes the electrical sheet, not heat conduction through the ceramic or the overall assembly's operating-temperature capability.
R = Rsq ln(b/a)/(2 pi); K(r) = I/(2 pi r)
- R is terminal resistance in ohms and Rsq is uniform sheet resistance in ohms per square.
- a and b are inner and outer electrical radii in the same length unit, with b greater than a.
- K is sheet current per unit width in A/m when radius is in metres; I is total radial current in amperes.
Uniform isotropic sheet, complete equipotential rim electrodes, negligible contact and electrode resistance, no angular interruption and a fixed evaluated material state.
Derive the power density before predicting temperature
For the sheet model, local power per face area is Rsq times K squared. Substituting the radial sheet current gives pA(r) = I squared Rsq/(4 pi squared r squared). Consequently the inner-rim to outer-rim power-density ratio is (b/a) squared. This ratio follows from geometry even when the sheet itself is perfectly uniform.
The generated electrical heat still has to leave through the substrate, supports, contacting load and exposed surfaces. A fourfold generation ratio is not a fourfold temperature ratio. Strong radial spreading or a cooler inner boundary can moderate or reverse the measured temperature profile. Preserve both the electrical generation map and the thermal contact model rather than using a thermal image as a direct map of resistor power.
Calculate terminal power and two local values together
Assume a = 5 mm, b = 10 mm and Rsq = 100 ohms per square, with 2 V applied directly between ideal rim electrodes. The terminal resistance is approximately 11.032 ohms, current is 0.18129 A and total sheet power is 0.36259 W. These values are independent calculation inputs, not specifications of a released heater.
The inner-rim electrical generation is approximately 3.330 milliwatts per square millimetre, and the outer-rim value is 0.833 milliwatt per square millimetre. The annular-area average is approximately 1.539 milliwatts per square millimetre. Reporting only that average conceals both the inner concentration and the lower generation at the outside.
| Quantity | Calculated value | Meaning |
|---|---|---|
| Terminal resistance | 11.032 ohms | Complete radial path between ideal rim electrodes |
| Total power at 2 V | 0.36259 W | Integrated generation in the sheet |
| Inner-rim generation | 3.330 mW/mm² | Local value at the smaller circumference |
| Outer-rim generation | 0.833 mW/mm² | One quarter of the inner-rim value |
| Annular average | 1.539 mW/mm² | Total power divided by 235.62 mm²; not a local maximum |
Locate a boundary that divides electrical power equally
Because the concentric bands carry the same total current, each band's power is proportional to its differential resistance. The fraction of total power generated between a and an intermediate radius r is ln(r/a)/ln(b/a). Half of the power is therefore generated inside r = square root of ab, not generally inside the arithmetic midpoint of the radii.
For the example, that radius is approximately 7.071 mm. The inner region occupies only one third of the total annular area but generates half the power. This gives a useful regional check for a spatial model: integration over the inner and outer regions should reproduce the predicted shares before thermal coupling is added. Equal radial widths or equal areas are not automatically equal-power partitions.
Recognize resistance invariance when both radii scale
If both radii are doubled while sheet resistance remains unchanged, b/a and the calculated terminal resistance remain unchanged. At the same terminal voltage the total power also remains unchanged. However, the annular area increases fourfold and the local generation at corresponding scaled positions falls to one quarter of the previous value.
Thus a terminal-resistance match cannot establish that two geometrically scaled annuli have equivalent thermal behavior. Their perimeter lengths, thermal contacts and substrate dimensions also change. If the design instead seeks equal power per unit area at twice the linear scale, four times the total electrical power is required under this model, which means twice the terminal voltage at unchanged resistance. That is a new electrical and thermal operating point requiring its own review.
Check the mean-circumference shortcut against the logarithm
A tempting approximation replaces the changing circumference with its value at the mean radius. It gives R approximately equal to Rsq times (b - a) divided by 2 pi times (a + b)/2. For a narrow annulus the width variation is small and this can be a useful screening estimate.
For the 5-to-10 mm example the shortcut gives approximately 10.610 ohms instead of 11.032 ohms, an underestimate of about 3.82 percent. The shortcut also offers no local generation profile. Use the exact logarithmic expression when its assumptions apply; if the electrodes are incomplete or the sheet is nonuniform, a more exact arithmetic expression for the wrong boundary does not resolve that larger modeling error.
Verify electrode and sheet assumptions before thermal acceptance
Measure terminal resistance and inspect the actual inner and outer electrical boundaries, including interruptions and contact transitions. Compare voltage at multiple angular positions on the electrodes where the measurement arrangement permits it. A meaningful electrode drop indicates that the equipotential assumption requires refinement; one terminal measurement cannot reveal where that drop is generated.
Retain film uniformity observations and connect the electrical model to measured radial and angular temperatures under the installed load. Unexpected angular peaks require examination of feed geometry, sheet defects and local heat removal. The completed handoff should distinguish calculated electrical concentration from measured temperature performance, with the actual drawing and material state retained for both.
Provide the annular electrical and thermal boundaries
Send the real current path and the installed ring interfaces before applying the radial-sheet model.
- Inner and outer electrical radii, layer map, electrode coverage, interruptions and terminal feed locations.
- Processed sheet resistance, its temperature dependence and available film-uniformity observations.
- Applied terminal voltage/current, electrode drops and the required total and regional power allocation.
- Substrate construction, inner and outer contacts, face loading and angular mounting asymmetry.
- Scaled artwork, radial and angular temperature records and the proposed measurement acceptance regions.
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