worst-case-analysis

name	worst-case-analysis
description	Worst-case analysis (WCA) — extreme value tolerance analysis (EVTA/worst-case stackup), root sum square (RSS) vs. worst-case tolerance, Monte Carlo simulation for non-normal distributions, electronic WCA (voltage/current/temperature extremes, parameter drift over life), manufacturing and assembly tolerance extremes, reliability prediction under worst-case conditions, conservative bounding (1σ/2σ/3σ process capability), and MIL-STD-1543/NASA best practices for safety-critical design.
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Worst-Case Analysis (WCA) — Complete Skill

WCA Fundamentals

Purpose and Application

Worst-case analysis (WCA): determine if a system meets requirements when all independent variables simultaneously take their worst possible values Required for: safety-critical systems; NASA/DoD Class A/B hardware; human-rated spaceflight; medical devices; defense electronics

When WCA is mandatory: Flight-critical systems (loss of crew/mission risk) Single-point failures with catastrophic consequence Systems where failure data insufficient for statistical approach

WCA vs. RSS: Worst-case (EVTA): guarantees performance over all combinations; most conservative; no failures below specification guaranteed RSS: 99.73% (3σ) coverage typically; some assemblies may fail; acceptable for non-safety-critical

Analysis Types

Mechanical WCA: dimension and tolerance stackup with all tolerances at extreme values → check if assembly fits/functions Electrical/electronic WCA: parameters at worst-case over temperature, radiation, aging, supply voltage extremes → performance within limits? Thermal WCA: maximum temperature with highest power dissipation + worst-case environment → within thermal limit? Reliability WCA: failure rates at worst-case temperature, vibration, voltage stress → MTBF acceptable?

Mechanical Worst-Case Stackup

Extreme Value Tolerance Analysis (EVTA)

Stack gap equation: G = Σ aᵢ × tᵢ [G = resultant gap; aᵢ = ±1 sensitivity coefficient; tᵢ = component nominal + tolerance]

Worst-case gap: G_max = Σ|tᵢ| + Σ|nominal contributions| [all positive tolerances stack positively] G_min = −Σ|tᵢ| + Σ|nominal contributions| [all positive tolerances stack negatively]

Requirement: G_min ≥ G_required (no interference at minimum; clearance always positive)

Example: Assembly of 3 parts in series; nominals: L₁ = 50±0.1, L₂ = 30±0.05, L₃ = 20±0.1 mm; slot = 100±0.2 mm Nominal gap = 100 − (50+30+20) = 0 mm Worst-case max gap = 0.2 + 0.1 + 0.05 + 0.1 = +0.45 mm Worst-case min gap = −(0.2 + 0.1 + 0.05 + 0.1) = −0.45 mm Result: interference possible! Redesign or tighten tolerances

Comparison: RSS gap: G_rss = √(0.2² + 0.1² + 0.05² + 0.1²) = √(0.04+0.01+0.0025+0.01) = √0.0625 = 0.25 mm (3σ) RSS predicts 99.73% of assemblies within ±0.25 mm → some interference still possible

Sensitivity Coefficients

Aᵢ = ∂G/∂xᵢ: how gap changes with each parameter Measured via analytical differentiation or FEA sensitivity study For linear stackup: aᵢ = ±1 (chain dimension: +1 if closing, −1 if opening) For nonlinear (angular stackup): aᵢ = ∂f/∂xᵢ evaluated at nominal conditions

Electronic Worst-Case Analysis

Parameter Extremes

Four corners of electronic WCA:

1. High temperature (T_max) + high supply voltage (V_max) + end-of-life drift (EOL_high)
2. High temperature + low supply voltage + EOL_high
3. Low temperature (T_min) + high supply voltage + EOL_low (initial)
4. Low temperature + low supply voltage + EOL_low

All four must pass specification

Component Parameter Drift

Initial tolerance: manufacturing variation (T_initial = ±X% at room temperature) Temperature coefficient (TempCo): TempCo = ΔP/P₀ / ΔT [ppm/°C or %/°C] Radiation drift (space): parameter change per krad(Si) total dose Aging/EOL drift: stability over operating life; manufacturer data or MIL-PRF specification Combined worst-case parameter: P_max = P_nominal × (1 + tolerance_initial + |TempCo| × ΔT + drift_EOL + radiation_drift) P_min = P_nominal × (1 − tolerance_initial − |TempCo| × ΔT − drift_EOL − radiation_drift)

Example (resistor in voltage divider): R_nominal = 10 kΩ; initial tolerance = 1%; TempCo = 50 ppm/°C; ΔT = −55 to +125°C (ΔT_max = 80°C from 25°C) Aging drift = 0.5% R_max = 10,000 × (1 + 0.01 + 50e-6 × 80 + 0.005) = 10,000 × 1.019 = 10,190 Ω (WC high) R_min = 10,000 × (1 − 0.01 − 0.004 − 0.005) = 10,000 × 0.981 = 9,810 Ω (WC low)

Circuit Performance at Extremes

Voltage divider example: V_out = V_in × R2 / (R1 + R2) WC V_out_high: V_in_max, R2_max, R1_min → V_out_max = V_in_max × R2_max / (R1_min + R2_max) WC V_out_low: V_in_min, R2_min, R1_max → V_out_min = V_in_min × R2_min / (R1_max + R2_min)

Propagation delay WCA: t_pd_max = sum of all gate worst-case delays (at V_supply_min, T_max, EOL) t_pd_min = sum of all gate best-case delays (at V_supply_max, T_min, fresh) Setup/hold margin: setup margin = t_clock_period − t_pd_max − t_setup ≥ 0 (always)

Power supply ripple and noise: V_supply = V_nominal ± V_ripple ± V_noise (WC = V_nominal − V_ripple_max − V_noise_max for V_min) Must exceed V_supply_min for all components at all conditions

Operational Amplifier WCA

OPAMP gain error (inverting configuration): G = −R_f / R_in [nominal; WC: both at extremes simultaneously] V_out_error = V_os + (I_b × R_f) + (G × V_in_offset) V_os = offset voltage WC: at T_max; EOL drift; V_os_WC = V_os_initial × (1 + T_drift + EOL_drift) I_b = input bias current WC: I_b_WC = I_b_max × (1 + TempCo × ΔT)

Total output error budget: E_total = E_offset + E_gain + E_noise + E_nonlinearity ≤ E_allowable

Thermal WCA

Maximum junction temperature: T_j_max = T_ambient_max + P_dissipation_max × θ_ja [θ_ja = junction-to-ambient thermal resistance] T_ambient_max: worst-case operating environment (e.g., 85°C for automotive; −40 to +125°C; MIL-STD-810 Hot) P_dissipation_max: component at maximum current + maximum supply voltage + minimum efficiency

Derating: Junction temperature derating per MIL-HDBK-217: maintain T_j < 0.7 × T_j_max_rated Screen components for T_j_derated < T_j_calculated_WC

Structural WCA

Mechanical worst-case failure analysis: σ_max_WC = (F_max × SF) / A_min [F_max = maximum load; SF = safety factor; A_min = minimum cross-section at tightest tolerance] σ_max_WC ≤ F_tu_min / SF_material [F_tu_min = minimum material ultimate strength from MIL-HDBK-5J Basis A]

MIL-HDBK-5J Material Basis: Basis A: value exceeded by 99% of population with 95% confidence (extremely conservative) Basis B: value exceeded by 90% with 95% confidence Safety-critical: Basis A; general structure: Basis B; WCA for dimensions requires A

Force WCA for mechanisms: Drive train WCA: T_output_min = T_motor_min × G_ratio_min × η_gearbox_min − T_friction_max − T_backlash_max Must T_output_min ≥ T_required_max for all conditions

Monte Carlo Simulation (WCA with Distributions)

Monte Carlo WCA: sample parameter distributions → propagate → find probability of failure Steps:

1. Define parameter distributions (uniform for WC tolerance; normal for process; Weibull for wear)
2. Sample N_MC = 10,000–100,000 iterations
3. Calculate output G(x₁...xₙ) at each sample
4. P(failure) = fraction with G < G_min_required
5. 3σ claim: if P(fail) < 0.00135 for each tail → equivalent to 3σ normal

Advantage over EVTA: accounts for distribution shape; identifies which parameters dominate (Sobol sensitivity) Disadvantage: does not guarantee zero failures (probabilistic, not deterministic)

Combined approach: WCA first → if fails, determine if Monte Carlo with known distributions shows adequate reliability

NASA/DoD WCA Requirements

NASA-STD-5005 (structural): ultimate factor of safety 2.0 (vs. CS-25/FAR 1.5) for crewed vehicles MIL-STD-1543: WCA requirements for launch vehicle power, signal, data systems JPL Design Principles DP-7: electronics WCA procedure at part/assembly/system level NASA-HDBK-4007: WCA for flight systems (electrical)

WCA Report contents: Block diagram; analysis conditions (temperature range, voltage extremes, EOL); parameter tables (nominal, tolerance, TempCo, EOL); four-corner analysis table; pass/fail vs. specification; margin summary (positive margin required ≥ 0% for safety-critical; ≥ 10% for robustness)

Standards and References

Standard	Scope
NASA-HDBK-4007	Flight system WCA
MIL-STD-1543	WCA for launch vehicles
MIL-HDBK-5J / MMPDS	Material basis A/B data
MIL-HDBK-217F	Reliability prediction (failure rates)
JPL D-8545	Parts, materials, and processes (PMP) control
AIAA S-120	Aerospace electrical system WCA

Output

Provide: system description (application: spacecraft/aircraft/medical device/industrial; failure consequence: catastrophic/critical/marginal; WCA required: yes/no; why), analysis type (mechanical stackup/electronic WCA/thermal WCA/structural WCA), mechanical stackup (if applicable: dimensions, nominals, tolerances; sensitivity coefficients aᵢ; EVTA G_max and G_min; requirement G_min ≥ 0; pass/fail; comparison with RSS margin), electronic WCA (four corners: (T_max, V_max, EOL_high) / (T_min, V_min, EOL_low) / other combos; parameter tables with nominal, initial tol, TempCo, EOL drift; output performance at each corner; pass/fail vs. specification), margin summary (margin = (value − spec) / spec × 100% for each corner; minimum margin [%]; all must be ≥ 0%), thermal (T_j_WC [°C] = T_ambient_WC + P_WC × θ_ja; vs. T_j_max_rated [°C]; derating factor [%]), structural (F_WC [N]; A_min [mm²]; σ_WC [MPa]; vs. F_tu_min Basis A [MPa]; margin [%]), Monte Carlo (if used: N_MC; distribution types; P(fail) = [value]; equivalent sigma level), and applicable standard (NASA-HDBK-4007; MIL-STD-1543; MMPDS Basis A/B; MIL-HDBK-217F).