Model and assumptions
DiaphragmCalc is a first-order analytical design tool. Results are analytical estimates and should be independently validated (FEA and test) before use in safety-critical or production designs. It is not FEA and not certified engineering software.
All calculations use SI units internally (m, Pa, kg/m³, V). The UI converts from µm, mm, kPa, bar, MPa, psi, GPa.
1. Geometry and boundary conditions
| Item | Assumption |
|---|---|
| Shapes | Square (side a) and circular (radius a; the UI takes diameter) |
| Edges | Fully clamped: zero deflection and zero slope |
| Plate | Homogeneous, isotropic, linear-elastic Kirchhoff thin plate |
| Load | Uniform differential pressure q on one face |
| Residual stress | None |
| Valid range | a/h ≥ ~10 (thin plate); w₀/h ≤ ~0.2 for linear results |
2. Equations
Flexural rigidity: D = E h³ / [12 (1 − ν²)]
| Quantity | Square (side a) | Circular (radius a) | Source |
|---|---|---|---|
| Centre deflection w₀ | 0.00126 q a⁴ / D | q a⁴ / (64 D) | Timoshenko & Woinowsky-Krieger (1959), Table 35; §63 |
| Edge stress σ_L (normal to edge, surface, mid-edge) | 0.3078 q (a/h)² | 0.75 q (a/h)² | same; square coefficient from M = 0.0513 q a² |
| Edge stress parallel to edge σ_T | ν σ_L | ν σ_L | zero curvature along a clamped edge |
| First resonance f₀ | 35.99 /(2π a²) √(D/ρh) | 10.21 /(2π a²) √(D/ρh) | Leissa (1969), NASA SP-160 |
Notes:
- The square-plate coefficients were tabulated for ν = 0.3. Their dependence on ν is weak (a few %), but for ν = 0.064 (Si ⟨110⟩) the numbers carry that extra uncertainty.
- Overpressure stress is σ_L × overpressure multiple (linear). At large overpressure the true bending stress grows sub-linearly while membrane stress is added; the linear value is a screening number only.
3. Large-deflection estimate
Circular plate (Timoshenko, energy method, ν = 0.3):
w₀/h + 0.488 (w₀/h)³ = q a⁴ / (64 D h)
Implemented as q a⁴/(E h⁴) = K₁ x + K₂ x³ with x = w/h, K₁ = 1/[c_w · 12(1 − ν²)] (c_w = linear deflection coefficient), K₂ = 0.488 K₁, solved by Newton iteration.
For the square plate the same K₂/K₁ = 0.488 ratio is assumed. This is an engineering estimate, not a published solution for the clamped square plate; treat square-plate nonlinearity as indicative only.
"Nonlinearity %" is the deflection nonlinearity (w_lin − w_NL)/w_lin at full scale. It is not the same as the terminal-based or best-fit-straight-line output nonlinearity on a datasheet, which is typically smaller and also includes stress and bridge nonlinearity.
4. Piezoresistive transduction
| Item | Assumption |
|---|---|
| Material | p-type Si, resistors along ⟨110⟩ on a (100) wafer |
| Coefficients | π₄₄ = 138.1 × 10⁻¹¹ Pa⁻¹ (Smith 1954), π_l = π₄₄/2, π_t = −π₄₄/2 (π₁₁, π₁₂ neglected) |
| Doping | Multiplier P(N) ≈ 1.0 / 0.97 / 0.85 / 0.60 / 0.30 at 10¹⁶ … 10²⁰ cm⁻³, 300 K (read from Kanda 1982; approximate) |
| Resistor size | Multiplier "stress averaging" ≤ 1 (1 = point resistor at the peak) |
| Placement | Radial resistor sees (σ_L, σ_T); tangential resistor sees (σ_T, σ_L) |
| Bridge | Full Wheatstone bridge, two resistors +x, two −y: V_out/V_in = (x − y)/(2 + x + y) |
| Temperature | Not modelled |
Material presets for E and ν are for the plate model only. The piezoresistive model is valid for p-type single-crystal silicon; the polysilicon preset uses the same coefficients and will over-predict sensitivity (polysilicon gauge factors are much lower).
5. Fluid loading
Added virtual mass incremental factor (Lamb 1920; Amabili & Kwak 1996):
β = Γ ρ_f a / (ρ h), Γ = 0.6689 (clamped circular plate, first mode, fluid on one side), f_wet = f₀ / √(1 + β)
Square plates use the equal-area radius a/√π. The model assumes an unbounded, incompressible fluid. Thin oil gaps, cavity compliance, isolation diaphragms and damping are not included.
6. Warnings raised
- w₀/h > 0.2: small-deflection results questionable.
- Safety factor below the user's target at the chosen overpressure.
- a/h < 10: thin-plate theory inaccurate.
7. What requires FEA
Anisotropic silicon stiffness; KOH sidewall/anchor compliance; bossed, corrugated or multilayer diaphragms; residual stress; resistor geometry and stress gradients; package, die-attach and thermal-mismatch stress; temperature coefficients; dynamic response with a real fluid cavity; burst prediction beyond linear bending.
References
- S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed., McGraw-Hill, 1959.
- A. W. Leissa, Vibration of Plates, NASA SP-160, 1969.
- C. S. Smith, "Piezoresistance effect in germanium and silicon", Phys. Rev. 94, 42 (1954).
- Y. Kanda, "A graphical representation of the piezoresistance coefficients in silicon", IEEE Trans. Electron Devices 29, 64 (1982).
- H. Lamb, "On the vibrations of an elastic plate in contact with water", Proc. R. Soc. A 98, 205 (1920).
- M. Amabili, M. K. Kwak, "Free vibrations of circular plates coupled with liquids: revising the Lamb problem", J. Fluids Struct. 10, 743 (1996).
- W. C. Young, R. G. Budynas, Roark's Formulas for Stress and Strain, 7th ed.