The use of statistical techniques for production line monitoring and control (SPC) is an established concept. Creative use of these techniques is also valuable when adapted to the design and specification of magnetic components. This paper introduces a technique for developing meaningful, cost-effective specification by predicting estimates of process capability from a minimum amount of sample data. A toroidal inductor design example is described.
Statistical techniques can be applied to the design process, providing off-line quality control similar to the on-line quality control typically provided by SPC. Creating designs that are not sensitive to manufacturing variation plays a crucial role in determining attainable production quality levels. The use of statistical techniques to analyze designs and specifications at the earliest point of the design process leads directly to higher quality and lower cost.
Only a brief working knowledge of statistics is required, and it is not necessary for the designer to become a statistician in order to make use of appropriate design tools.
Predicting process capability and production distributions from a sample toroidal Inductor design is possible using certain techniques. Therefore, reasonable specifications for both functional and physical parameters of the designed part can be determined.
Only variable parameters are considered as opposed to attribute parameters. Attribute parameters are such things as the legibility of part marking and lead solderability. It is important, however, that attribute characteristics not be ignored because it is feasible to have as many defectives for attributes as for variable parameters. Attribute parameter analysis requires production line data. Only variable parameters are appropriate to consider at the design stage.
Statistics can be easily used to check and review component designs and tolerances. The following brief overview provides most of the tools required for such analysis.
Two types of measures may be used to define a random distribution: measures of central tendency and dispersion.
The most common and useful measure of central tendency is the arithmetic mean, which is defined as:
(1)
[x is used to denote sample average, µ is used for population mean]
The most often used measure of dispersion is the standard deviation (or its square, the variance). The standard deviation, σ, is defined as the average distance from the mean:
(2)
where i = 1 to n
[s denotes sample standard deviation, σ is used for the population.]
Most random data in practice are observed to follow a normal (Gaussian) distribution pattern. A normal distribution is a continuous, symmetrical pattern described by:
(3)
where –∞ < x < ∞
It is assumed that parameters of interest follow normally distributed patterns. This has been observed in practice to be a good assumption. Further, it can be shown according to the Central Limit Theorem that any group of numbers that represent sums or averages will be normally distributed regardless of whether or not the distribution from which they were drawn is normal. This holds true for magnetic component design because most parameters of interest are functions of a combination of several factors.
Knowing or assuming that data will follow a standard type of distribution greatly facilitates analysis. For example, for normal distributions, eq. (3) can be used to calculate the percentage of parts expected to fall within a certain number of standard deviations from the mean value.
Table 1. Percent Within a Given Distance from the Mean | |
Expect 68.27% 95.45% 99.73% 99.9937% 99.99932% 99.9999425% 99.9999998% |
Within ±1σ ±2σ ±3σ ±4σ ±4.5σ ±5σ ±6σ |
Table 2. PPM Defectives for Various Specification Limits | |
Specification Limit ±1σ ±2σ ±3σ ±4σ ±4.5σ ±5σ ±6σ |
Expected ppm 317,310 45,500 2700 63 6.8 0.58 0.002 |
Table 3. Desired Inductor Parameters |
Functional: Inductance = 225 µH ±20% DC Resistance = 0.100 Ohms maximum Physical: PC board Area = 1.5″ × 1.5″ max board. Height from PC board = 0.675″ maximum Mounting = header with 0.015″ minimum standoff from PC board. Quality: Parts must be shipped at 1,000 PPM defectives maximum. |
Table 4. Statistical Sample Data | ||||
x | s | est σ | ||
L (µH) | Lot 1 2 |
236.3 236.6 |
2.3 2.9 |
2.3 3.1 |
R (mΩ) | Lot 1 2 |
98.1 97.7 |
0.82 1.32 |
0.82 1.4 |
L (in) | Lot 1 2 |
1.12 1.12 |
0.02 0.02 |
0.02 0.021 |
W (in) | Lot 1 2 |
1.14 1.14 |
0.01 0.003 |
0.01 0.003 |
H (in) | Lot 1 2 |
0.61 0.595 |
0.02 0.006 |
0.02 0.006 |
Table 5. Expected Defect Levels | |
Parameter Inductance Resistance Length Width Height Total |
Expected ppm 0 16400 0 0 1200 17600 |
Table 6. Expected Defect Levels | |
Parameter Inductance Resistance Length Width Height Total |
Expected ppm 0 0 0 0 25 25 |