Applying Statistical Techniques to the Design of Custom Magnetics

Figure 2. Toroid Core Dimensions

The nominal inductance factor (AL) this core is 900 µH per 100 turns. The required turn count for 500 µH is:
 

               (4)

The wire size must be chosen to meet the dc resistance requirement:
 

               (5)

             (6)

Core inner circumference = 1.79″
Winding = wire diameter × turns = 0.0281″ × 50 = 1.41″

(1.41″/1.79″) × 360° = 284° of the core ID is required for the winding. This means that the winding should fit comfortably into a single layer. Choose a plastic header for mounting that is 0.060″ thick plus 0.015″ standoffs.

At this point the initial component design has been created, which consists of 50 turns of 22 AWG wire wound on a 1.06″ diameter core. Figure 3 shows the expected appearance of the coil.

Figure 3. Wound Inductor

The next step in the analysis is to review the component design to determine the critical parameters.

A parameter is designated as critical if it is likely to cause a defective part. This is the case if the parameter that is out of tolerance causes the part to be not usable, and if there is some finite likelihood that the parameter will be out of tolerance.

A parameter is likely to be out of tolerance if either the standard deviation is large with respect to the specification limits, or the distribution is skewed too closely to a specification limit.

The reason for focusing attention on critical parameters is cost. There are costs associated with each manufactured defective inductor. There are also inspection costs for each inductor manufactured. Those costs increase with the number of parameters that require inspection. Quality costs are lowest when the combination of cost due to defectives and inspection cost is minimized.

For the toroid design example it is assumed that if the inductance, resistance, length, height, or width are out of tolerance, then the part is a non-usable (defective) item This satisfies the first requirement of a critical parameter. Further, it is assumed that these are the only parameters that can cause a defect. (Attribute parameters will not be considered.)

It is possible to imagine manufacturing or material variations that cause the part to be out of tolerance for inductance, resistance, or any of the three physical parameters. Therefore it is not possible yet to eliminate any of these from being considered critical and we identity five expected critical parameters:

In the previous section it was observed that a trial run produced 100 pieces with only one being defective (one piece being 100.3 mOhm as shown in Figure 5). This defect information alone does not indicate whether or not there are likely to be defectives when larger numbers of the parts are built. It should be noted that a quality level of 1000 ppm corresponds to a percentage defectives of 0.1%. Even a 100 piece sample without defectives alone may not be sufficient to conclude that the design meets 1000 ppm.

There are several capability indices commonly used as measures of process capability including CP, CPU, k and CPK. For a simple straightforward estimator of inductor parameters, CP is sufficient. CP is the allowable process spread divided by the actual process spread. It is most common to use six times the standard deviation as the measure of the actual spread. The 6σ rule can be varied In order to achieve designs of differing acceptable outgoing quality level (AOQL).
 

               (7)
 

It can be seen from equation 7 that it is desirable for CP to be greater than one so that the parts will fall within the required specification.

For the following example a straightforward use of the sample mean and standard deviation alone is a powerful analysis tool. However, for most parts destined for production, it is most beneficial to use one of the common process capability indices. This allows the same terminology to carry over from design to production.

To determine the proper definition of process spread we must refer to the acceptable quality levels. In this example the requirement states that the expected quality is to be 1000 ppm or less. For simplicity, it is assumed that the five critical parameters are the only parameters that can cause defectives, and each one of these will be allowed 1000/5 = 200 ppm each.

Calculations similar to those for Table 4 show that the allowed specification for 200 ppm must represent a ±3.7σ or more. In other words, we will use ±3.7σ to represent the actual process spread and when comparing CP to 1.

Inductance:
CP = (270 – 180) / 3.7 × 2.3 = 10.6 >> 1

The other four parameters are one-sided specifications. Using 3.7σ yields an expected ppm of 100. In this case, the difference between the mean and the specification limit is compared to σ.

Resistance:
(0.100 – 0.098) / 0.00082 = 2.4 (<3.7)

Length:
(1.50 – 1.12) / 0.02 = 19

Width:
(1.50 – 1.14) / 0.01 = 36

Height:
(0.675 – 0.61) / 0.02 = 3.25 (<3.7)

It is seen from the above that length, width and inductance are significantly greater than 3.7 standard deviations away from the specification limits. Resistance and height are not within acceptable limits, however, and can be expected to create quality problems. Table 5 shows a summary of the expected quality level based on the analysis so far.
 

Table 5. Expected Defect Levels
Parameter Inductance Resistance Length Width Height Total	Expected ppm 0 16400 0 0 1200 17600

In other words, we can expect at least 17,600 ppm defects. This may seem a bit surprising since the 100 piece sample lot had only one defect and it was barely over the limit! This design must be corrected for the two questionable parameters.

It is important to note that observing a trial run of parts without defects is not sufficient to predict that there will be no defects in production.

Redesign

The resistance can be decreased by changing the wire size from #22 to #21. The difference in the two wire sizes is 0.003 Ω/ft. Therefore, a change to #21 should lower the mean resistance by 18.7 mΩ, but we would expect the same standard deviation. The new mean of 79.41 mΩ is now 25 standard deviations away from the maximum limit of 100 mΩ. This now yields an expected defect level of 0 ppm.

The increase in wire diameter due to the wire size change is not enough to affect the length and width dimensions or the inductance parameter that were all well within specification, however the height is still a problem.

To improve the design for the overall height specification, a thinner mounting header can be used. A decrease from 0.060″ to 0.040″ thickness should decrease the mean to 0.59″, which would be 4.25 standard deviations from the allowed limit. This yields an expected defect rate of approximately 25 ppm. Table 6 shows the updated summary of expected defects.
 

Table 6. Expected Defect Levels
Parameter Inductance Resistance Length Width Height Total	Expected ppm 0 0 0 0 25 25

Review Critical Parameters

The toroid design example has shown that the application of a few basic statistical concepts is a very powerful tool for quickly identifying possible problem parameters. The techniques used can be more formalized by the use of standard indices for identifying process capability. Formalization provides a common language for design-to- production communication and a good frame of reference for identifying standard process capabilities as a large number of part numbers are brought to production over a long period of time.

Despite the power of a statistical tool, it is (like any other) only as good as the experience and common sense with which it is applied. Even in this fairly simple example, there may have been several factors overlooked in the extrapolation from sample data to predictions of production quality. The validity of the conclusions drawn regarding each of the critical parameters can be examined.

Inductance

The conclusion drawn regarding the percent of expected defects due to inductance did not consider possible lot-to-lot core variations. All pieces were made with cores from the same production lot.

The core inductance tolerance is typically +15%, −7.5%. Cores from three lots were examined, and it was found that although the standard deviation of each lot was fairly similar, the means varied substantially over almost the entire specified inductance range. Therefore the inductance variation will be much greater for long term production than for a single lot. Such parameters are not necessarily disastrous to the design, but must be taken into account.

In the example, the mean was already over the desired nominal inductance of 225 µH, however it was determined that almost the entire difference from nominal was due to the core permeability being over the nominal by the same amount. This was done by measuring a standard winding according to the core vendor catalog. Therefore the worst possible expected mean for the core permeability would be 15% over nominal. This would yield a mean inductance of 259 µH. Assuming the same standard deviation for this average inductance would yield an expected defect level of about 2 ppm; still well within acceptable limits.

Resistance

Experience has shown that coil resistance may vary by as much as 10% due to variations in wire diameter. If we increase the mean resistance expected by 10% we still have an expected defect level of approximately zero.

Length and Width

Both of these parameters were well within tolerance and are not likely to cause defects. The factor most likely to cause significant variation in these dimensions is crossed turns. Because no crossed turns were present in the sample run, none were predicted for production. This is a case where a short run did not give all possible information about possible defects. This can be a problem with sample runs and short productions runs. In this case the only way to obtain further information is to monitor any further production closely. The key to solving this type of problem is to close the feedback loop from production to design engineering so that the next time a similar part is designed, all probable sources of defects are accounted for at the time of design.

The length and width are both candidates for tighter specifications. If both length and width were tightened to 1.3″ max, for example, they would still not be likely to cause defects. Length would then be approximately 9σ and width would be 16σ from the allowed limit. Because circuit board space costs money, the tighter specification of the length and width dimensions would be recommended.

An interesting difference arises between the two dimensions. When comparing the two distributions, it is noticed that length is less on the average than width. This disparity is easily explained by the fact that since the coil does not cover the entire core, the width consists of the core plus two wire diameters, whereas the length only has the core and one wire diameter, as shown in Figure 3.

A question is created why the length would have a greater standard deviation. Upon further examination of the parts it is observed that the coil wires are not always formed identically into leads. In other words the first and last turn of the coil have greater variation than other turns. This is easy to understand but something that could easily be overlooked when specifying tolerances for this part. A simple observation of the two distributions highlights this situation.

Height

The initial analysis of the distribution for the height parameter showed a potential problem, but in the example case, one that was easy to fix. For the redesign, the average part was 4.25 standard deviations from the allowed limit. The height is a parameter that should be targeted for further investigation. A general rule of thumb is that the design should allow for a 1.5σ shift in distribution unless a greater variation is expected, such as in the case of the inductance tolerance. Since we had determined that we should be 3.7 from the limit, being 4.25 away only leaves a margin of 0.55σ shift before the acceptable quality level would be exceeded.

Conclusion

It is seen that applying a few simple statistical techniques to the custom inductor design process increases the effectiveness of analysis and reduces the risk of designing a component that will be a quality problem in production. Cost savings were achieved by saving circuit board area as well as by predicting potential problems with the resistance and the height parameters.

Bibliography

Ellis, Richard W. “Expand the Tolerance, Not the Spec” Quality, April 1987.
Hunter, J. Stuart “Statistics in Quality - Yesterday. Today and Tomorrow” Quality Anniversary Issue 1987.
Jones, Bradley “Beyond SPC: Designed Experiments for Quality Improvement” Quality Dec 1987.
Kackar, Raghu N. “Off-line Quality Control, Parameter Design, and the Taquchi Method” Journal of Quality Technology Vol 17, No 4, Oct 1985.
Kane, Victor E. Process Capability Indices” Journal of Quality Technology Vol 18, No1, Jan 1986.
Micrometals Catalog #4 tor EMI and Power Filters Micrometals, Inc 1982.
Norusis1 Marija J. Starter Stats by SPSS SPSS INC, 1987.
Ryan, Thomas A. Minitab Duxbury Press 1976.
Wadsworth, Harrison M. et al Modern Methods for Quality Control and Improvement John Wiley and Sons 1986.
Wilson, Karen W. “Five Factors in the Quality Issue” Purchasing Nov 5, 1987.

Author’s biography

Leonard Crane is Technical Marketing Manager for Coilcraft, Inc. and holds a BSEE degree from the University of Illinois and a Masters of Engineering Management from Northwestern University.