| name | failure-rate-analysis |
| description | Failure rate analysis — constant/increasing/decreasing hazard, MTBF/MTTF/MTTR, exponential/Weibull failure models, MIL-HDBK-217, Telcordia SR-332, FIDES, electronic component reliability. |
| metadata | {"priority":7,"promptSignals":{"phrases":["failure rate","MTBF","failure rate analysis","MIL-HDBK-217","reliability prediction","hazard rate","exponential failure"],"minScore":3}} |
Failure Rate Analysis — Complete Skill
Reliability Fundamentals
Reliability function:
R(t) = P(T > t) = 1 - F(t)
Failure rate (hazard rate) function:
h(t) = f(t) / R(t) = -dR/dt / R(t) [failures/hour or failures/cycle]
MTTF (Mean Time To Failure — non-repairable):
MTTF = ∫₀^∞ R(t) dt
MTBF (Mean Time Between Failures — repairable systems):
MTBF = 1/λ for constant failure rate
MTTR (Mean Time To Repair):
MTTR = average time to restore function after failure
Availability:
A = MTBF / (MTBF + MTTR)
Failure Rate Models
Exponential (Constant Failure Rate)
Hazard rate: h(t) = λ = constant
Reliability: R(t) = exp(-λt)
MTTF = 1/λ
Valid for: electronic components during useful life (no wear-in, no wear-out)
R at t = MTTF: R = exp(-1) = 36.8% (NOT 50%)
Failure rate units:
1 FIT (Failure In Time) = 10⁻⁹ failures/hr = 1 failure per 10⁹ device-hours
1 %/1000hr = 10⁴ FIT
Weibull Distribution (Most General)
Hazard rate: h(t) = (β/η) × (t/η)^(β-1)
Reliability: R(t) = exp[-(t/η)^β]
β = shape parameter (Weibull slope):
β < 1: decreasing failure rate (infant mortality; burn-in needed)
β = 1: constant failure rate (exponential; random failures)
β > 1: increasing failure rate (wear-out; fatigue; corrosion)
β = 3.5: approximately normal distribution
η = scale parameter (characteristic life): R(η) = exp(-1) = 36.8% always
MTTF = η × Γ(1 + 1/β) [Γ = gamma function]
B_10 life (10% failure):
t_10 = η × (-ln(0.90))^(1/β) = η × (0.1054)^(1/β)
Lognormal Distribution
Appropriate for fatigue and crack growth failures; metal fatigue
f(t) = 1/(t σ_log √2π) × exp[-(ln(t) - μ_log)²/(2σ_log²)]
MTTF = exp(μ_log + σ_log²/2)
Bathtub Curve
Three phases:
-
Infant mortality (β < 1): manufacturing defects; decreasing h(t)
Duration: 0–1000 hours typical (electronics); shorter for screened components
Mitigation: burn-in testing; incoming inspection; ESS (Environmental Stress Screening)
-
Useful life (β ≈ 1): random failures; constant h(t)
Duration: component-specific design life
Prediction: MIL-HDBK-217, Telcordia; constant λ
-
Wear-out (β > 1): degradation mechanisms; increasing h(t)
Duration: begins at 80–90% of design life
Mitigation: preventive maintenance before wear-out; condition monitoring
Electronic Component Reliability Prediction
MIL-HDBK-217F Notice 2 (1995)
Component failure rate model:
λ_p = λ_b × Π_E × Π_Q × Π_T × Π_other
λ_b = base failure rate [failures/10⁶ hr]
Π_E = environment factor; Π_Q = quality factor; Π_T = temperature factor
Temperature factor (Arrhenius):
Π_T = exp[-E_a/k × (1/T_junction - 1/T_ref)] [E_a in eV; k = 8.617×10⁻⁵ eV/K]
E_a typical: 0.5–1.0 eV for silicon ICs; 0.3–0.7 eV for capacitors
10°C rule: doubling of life per 10°C decrease (approximately; if E_a = 0.7 eV, ΔT = 10°C → ~2.2× life at 55°C)
Quality factors Π_Q:
JANTXV: 0.7; JANTX: 1.0; JAN: 2.0; Lower: 3.0–5.0; Worst: 10+
Environment factors Π_E:
GB (ground benign): 1.0; GF (ground fixed): 3.0; GM (ground mobile): 16; AIF (airborne inhabited): 6; SF (space): 0.5
Telcordia SR-332 (Bellcore)
Updated method; based on measured field data; better for telecom components
Method 1: Predicted (component count); Method 2: Similar (field data comparison); Method 3: Test-based
More accurate than MIL-HDBK-217 for commercial telecom
FIDES Guide 2009/2022
European aerospace/defense standard; physical models + stress models
Temperature cycling, humidity, vibration, altitude as degradation drivers
Accounts for application profile (% time in each environment condition)
System Reliability (Series/Parallel)
Series system (all components must work):
R_sys = Π R_i [product of all reliabilities]
λ_sys = Σ λ_i [sum of all failure rates for exponential]
Parallel system (at least one must work):
R_sys = 1 - Π (1 - R_i)
k-of-n (majority voting):
R_sys = Σ_{i=k}^n C(n,i) × R^i × (1-R)^(n-i) [each component identical R]
Example: 2-of-3:
R_2/3 = 3R² - 2R³
Failure Mode Analysis
Failure Modes Effects Analysis (FMEA) per IEC 60812:
For each component: failure mode → effect on system → severity × frequency
Risk Priority Number (RPN):
RPN = Severity × Occurrence × Detectability [each 1–10 scale]
High RPN (> 100–200) → action required
Criticality Analysis (FMECA):
C_r = Σ (λ × α × β × t) for each failure mode
α = failure mode ratio; β = conditional probability mode causes mission failure
t = mission duration
Confidence Intervals for Failure Rate
From test data (no failures in T total test hours):
λ_max = χ²(2f+2, 1-C_L) / (2T) [upper bound at confidence level C_L]
f = number of failures; χ² = chi-squared distribution
Example: 0 failures in 10,000 hr → λ_max = χ²(2, 0.90) / 20,000 = 4.605/20,000 = 2.3 × 10⁻⁴/hr
Useful Component Failure Rates (FITs at 25°C)
| Component | Typical λ [FITs] |
|---|
| Si MOSFET (logic, room T) | 5–50 |
| Electrolytic capacitor | 50–500 |
| Film resistor | 2–10 |
| Ceramic capacitor | 1–5 |
| Oscillator (crystal) | 50–200 |
| PCB connector | 50–200 |
| Power IGBT (junction T = 100°C) | 200–1000 |
| Mechanical relay | 500–5000 |
| Lithium battery | 1000–5000 |
Output
Provide: failure rate λ [FIT or failures/hr], MTBF [hours], MTTF [hours], reliability R(t) at mission time t, Weibull β (failure mode type), B_10 life [hours], temperature derating recommendation (maximum T_junction), Π factors for dominant environment/quality levels, system reliability (series/parallel), confidence interval on λ from test data, applicable standard (MIL-HDBK-217F, Telcordia SR-332, IEC 60812 FMEA).
