Bézier Curves

Mean Time Between Failures (MTBF) is a reliability metric representing the predicted elapsed time between inherent failures of a repairable system. For systems with a constant failure rate [latex]\lambda[/latex], MTBF is its reciprocal: [latex]MTBF = 1/\lambda[/latex]. This simple relationship is fundamental in reliability engineering for predicting system uptime and planning maintenance schedules, assuming failures follow an exponential distribution.

For repairable systems, Mean Time Between Failures (MTBF) is the sum of Mean Time To Failure (MTTF) and Mean Time To Repair (MTTR). The formula is [latex]MTBF = MTTF + MTTR[/latex]. MTTF represents the average operational time before a failure, while MTTR is the average time taken to repair the system. This distinction is critical for understanding system availability.

For a system of components in series, where any single failure causes system failure, the overall failure rate is the sum of individual rates. The system's MTBF is the reciprocal of this sum: [latex]MTBF_{system} = (\sum_{i=1}^{n} \lambda_i)^{-1} = (1/MTBF_1 + 1/MTBF_2 + ... + 1/MTBF_n)^{-1}[/latex]. This implies the system's MTBF is always less than the lowest individual component's MTBF.

Physics of Failure (PoF) is a reliability engineering approach that uses knowledge of materials science and physics to understand and model the root-cause mechanisms of failure. Instead of relying purely on statistical data from past failures, it focuses on predicting failure by analyzing the physical processes (e.g., fatigue, corrosion, creep) that lead to degradation and breakdown.

The ISO 5725 standard defines accuracy as the combination of trueness and precision. Trueness is the closeness of the mean of a large series of measurements to the accepted reference value, quantifying systematic error or bias. Precision is the closeness of agreement among a set of results, quantifying random error. Thus, accuracy requires both high trueness and high precision.

The Deborah number is a dimensionless quantity in rheology, used to characterize the fluidity of materials. It is the ratio of the relaxation time, which is an intrinsic property of the material, to the characteristic time scale of the experiment or observation. The formula is [latex]De = \frac{t_c}{t_p}[/latex], where [latex]t_c[/latex] is the relaxation time and [latex]t_p[/latex] is the observation time.

Repeatability assesses precision under identical conditions, while reproducibility assesses precision when one or more conditions change, such as the operator, instrument, or location. Reproducibility variance is always greater than or equal to repeatability variance. This distinction is crucial for understanding measurement variability, especially in inter-laboratory comparisons where conditions are inherently different.

Cleanrooms utilize two primary airflow principles to control contamination. Turbulent (or non-unidirectional) flow involves mixed air streams, suitable for less stringent classes (ISO 6-9). Laminar (or unidirectional) flow uses parallel, constant-velocity air streams to sweep particles out of the environment, essential for high-purity applications like ISO 1-5, preventing cross-contamination and ensuring rapid particle removal.

CMOS (Complementary Metal-Oxide-Semiconductor) is the dominant technology for constructing integrated circuits. It uses complementary pairs of p-type and n-type MOSFETs to build logic gates. Its primary advantage is very low static power consumption, as one transistor in the pair is always off during steady state, resulting in minimal current flow except during switching transitions.

Molecular electronics explores using individual molecules or nanoscale molecular collections as fundamental electronic components. This approach aims to build circuits at the ultimate limit of miniaturization, far beyond traditional silicon-based technology. Key components include molecular wires, switches, and rectifiers, leveraging quantum mechanical properties like electron tunneling through molecular orbitals for their function.

A Single-Electron Transistor (SET) is a switching device that uses controlled electron tunneling to manipulate the flow of single electrons. It consists of a quantum dot (the 'island') coupled to source and drain leads via tunnel junctions, and capacitively coupled to a gate electrode. Its operation relies on the Coulomb blockade effect, enabling extreme sensitivity and low power consumption.

General relativity predicts that accelerating masses generate ripples in the fabric of spacetime called gravitational waves. These waves propagate outwards at the speed of light, carrying energy as gravitational radiation. They are created by cataclysmic cosmic events like the merger of black holes or neutron stars, causing spacetime to stretch and squeeze as they pass.