Happy’s Essential Skills: Design of Experiments

Estimated reading time: 9 minutes

I guess I was lucky to be exposed to engineering statistics early in my college education. I never took a statistics course from the math department; if I had I might have thought it to be boring. Instead, it came as part of the chemical engineering basics. Since there is no “higher math” in most statistics, it is a good introductory course for engineers and essential to analyze lab and experimental results that will be part of the science and engineering education. My first design of experiment was done by long-hand; then, we did it by punched cards, and finally, with our slide rules (guess that dates me!). It wasn’t until the HP PCB problem solving that I wrote a basic program to conduct my DOEs on an HP 2116 computer.

Critical to DOE was the type of variables. In production, qualitative factors can be more significant than quantitative factors. Important quantitative factors (variables) are usually controlled, but qualitative variables can change without notice. Qualitative factors include: time of the year, day of the week, production shifts, production line, individual workers or machines, supplier sources, maintenance frequency, and even source of water. If you remember in my second column, and Figure 6 contained therein, for DOE with “factors not all being quantitative,” “screening experiments” are called for, such as described by Plackett-Berman[1] and in Fractional Factorial[2] (center boxes in Figure 1). Other application areas are comparative, modeling and optimizing.

Screening experiments (also called fractional factorial) are test plan used for an initial scan of problems having a large number—usually six or more—of presumed independent variables. The purpose of such plans is to determine which variables have the largest effects on the dependent variables. Results show only main or first-order effects (interactions), only the sensitivity of Y to a significant change in X1, X2 or X3, etc. Generally, interaction and second-order effects are not detected in screening plans.

Once the independent variables have been reduced to four or less, full factorial experiments can be conducted to understand all interactions and if the responses are non-linear and linear equations can be developed. Further experimentation can be conducted as ‘evolutionary operations’ to discover optimum settings and performances.

In the HP PCB problems, indeed the causes of the problems were an interaction of Monday vs. Friday, Day Shift vs. Graveyard Shift, process tank #1 vs. #4, and chemical supplier source. It was the qualitative variables that were at the Root Cause! A “one-at-a-time” experimentation couldn’t duplicate the root cause.

Some Examples

The next three figures show four different PCB process DOE results. The first, in Figure 2, is an experiment to minimize shifting of innerlayers during multilayer lamination. The variable and levels were a full factorial design of three variables at two levels:

1. Vented panel borders: with venting and without venting

2. Tooling methods for layup: ¼-inch holes and four 1/8-inch slots-centerline

3. Lamination pressure: 294 PSI & 344 PSI

The results are the image shift in microns. The lowest shift was 76 μm using vented borders, ¼-inch peripheral holes and the higher pressure. Analysis shows that the tooling method has the most positive effect on shifting and interacts with panel venting (V).

Figure 2: An example of factorial design of experiments (DOE) in printed circuit manufacturing to minimize innerlayer shifting during lamination.

The second experiment, in Figure 3, uses optimizing photoresist exposure, developing and etching to provide the highest production yield. The variable and levels were a full factorial design of three variables at three levels (center point):

1. Exposure energy in mjoules: 70, 50 & 30

2. Developer speed in inches per minute: 45, 40 & 35

3. Etcher speed in inches per minute: 45, 40 & 35.

The variables were chosen with the center point being the current production process: 50 mjoules, 40 in/min developer and 40 in/min etcher. The highest yield was 95% using slower developer speed, lower exposure intensity, and the slower etcher. Analysis shows that the developer speed has the greatest effect on yield and interacts with etcher speed.

Figure 3: An example of factorial design of experiments (DOE) in printed circuit manufacturing to optimize yield in exposure, developing and etch.

The third experiment, in Figure 4, serves to find the highest hole quality in a multilayer board. The variables and levels were a full factorial design of four variables at three levels:

1. Drill methods: (-) resharpened 4-8 times (0) resharpened

2. Drill diameter: (-) 0.008” (0) .014” (+) 0.020”

3. Infeed rate: (-) xx in. per min. (0)  xx in. per min (+) xx in. per min

4. Construction: (-)Std. Foil-Lam (0) Thick-prepreg w/foil-Lam (+) Std. Core-Lam.

The results are the hole quality (rms roughness %) and max. innerlayer mushrooming in microns.

The best quality was 0 microns mushrooming and

The fourth experiment, shown in Figure 4, is to further find the highest hole quality and to look at drilling productivity. The variables and levels were a fractional factorial design of three variables at two levels:

1. Drill method:  (-) new drills (+) resharpened 6 times

2. Stack height: (-) 1 high (+) 3 high 

3. Panel venting dams: (-) no-flow dams  (+) full venting dams

The results are the hole quality (rms roughness %) and max. innerlayer mushrooming in microns.

The best quality was < 4% rms hole-wall roughness using a plane of new drill bits for stacks of 1-high  with any appropriate venting dams. Analysis shows that the old resharpened drills could be used with drill stacks 3-high and has a usable hole-wall roughness but it interacts with drill infeed rates.

Figure 4: Two more examples of DOE for hole quality in multilayer boards. Full factorial design on the left was conducted to optimize drilled hole quality. Fractional factorial DOE on the right further optimizes hole quality and production productivity.

Notice that this last experiment was a fractional factorial. The power of a scanning experiment using the fractional factorial methodology is that N number of variables can be reviewed with only N+2 experiments. This is useful to find main effects, but not interaction, while later experiments will provide examination of interactions and optimization.

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