A segmented reactor limb has several gaps, each affecting local fringing and the combined magnetic reluctance. Checking only the sum can accept a narrow gap paired with a wide one. Checking only each gap can miss a systematic spacer error across the whole limb. [1]
For a gap-dominated, linear screening model with equal effective area A, L ≈ μ₀N²A/gΣ, where gΣ is total gap length. Here μ₀ is free-space permeability in H/m, N is turn count, A is effective area in m², total gap length is in metres and L is in henries. The model neglects core reluctance, fringing and saturation; it is useful for initial sensitivity, not final reactor design.
Two tolerance questions need two records
| Constraint | Measurement | What it prevents |
|---|---|---|
| Total gap | Sum of loaded gap lengths | Overall reluctance and inductance drift |
| Individual gap | Local minimum, mean and tilt | Excessive local field or magnetic pull |
| Spacer batch | Thickness and compression correlation | Shared bias hidden by independent-error assumptions |
| Assembly state | Preload and temperature during measurement | Comparing incompatible dimensions |
Do not assume independent spacer variation
For random gap errors, Var(gΣ) = ΣVar(gi) + 2ΣCov(gi, gj) over distinct pairs. Positive batch correlation increases the sum’s uncertainty. If all spacers carry the same thickness bias, it accumulates directly rather than cancelling statistically.
Separate systematic batch offset, individual scatter and loaded compression. Record spacer lot by gap position. Use the actual correlation evidence or a bounded worst case where it is unknown; a root-sum-square result needs an independence basis.
Combine standard uncertainties with covariance where effects are correlated; repeatability alone does not include all systematic contributions. [2]
Include shared spacer-batch error
Assumptions: ten loaded gaps total 20 mm; each error has standard deviation 0.10 mm. Independence gives total σ = 0.316 mm; pairwise correlation 0.5 gives σ = √[10×0.10² + 90×0.5×0.10²] = 0.742 mm. Under an assumed two-σ design band and L proportional to 1/gΣ, maximum inductance rises are 3.27% and 8.01%, respectively. A +5% ceiling passes only the independent model. Hold that independence assumption without evidence. Separately, measured 1.6 and 2.4 mm gaps can retain their nominal 4 mm sum while the first violates an assumed 1.8 mm local minimum. Both records are required. [1] [2]
References
- H. B. Duc, T. P. Minh, T. P. Anh and V. D. Quoc, A Novel Approach for the Modeling of Electromagnetic Forces in Air-Gap Shunt Reactors, Engineering, Technology & Applied Science Research 12, 8223–8227, 2022. DOI: 10.48084/etasr.4692.
- B. N. Taylor and C. E. Kuyatt, NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, 1994, sections 5 and 6.