Statistical or six-sigma design requires the establishment of strong relationships between product defects and yields, reliability, cycle time, inventory, schedule and cost. To ensure robustness, reliability, first-pass success and higher yields-all of which translate into increased bottom-line profitability-it is extremely important to include six-sigma statistical methods in the overall design process.

This article discusses parametric design-for-yield (DFY) or statistical design methodologies and provides examples of the impact of using statistical design on actual designs.

What is parametric yield and how is it measured?

To achieve six-sigma quality, the product must first be designed so that the manufacturing processes are capable of yielding 99.99966 percent in the product. For example, assume the electrical design of an amplifier requires that the gain be greater than 12 dB. Both the manufacturing processes and the design must be capable of yielding 999,996.6 out of 1 million amplifiers that meet their requirements and only 3.4 out of the million amplifiers that do not meet their requirements, i.e., gain less than 12 dB.

So, the six-sigma design process translates into perfecting the design process so that 99.99966 percent of the products made are defect free. As a result, the six-sigma design process will also decrease design and manufacturing cycle time; decrease inventory, scrap and rework; and increase profitability.

We can see that achieving six-sigma quality depends on two design factors. The first factor is the customer requirements. Achieving six-sigma quality will be more difficult with tougher performance requirements. Using the amplifier design from the example above, higher yields will be achieved for the amp if the minimum requirement is 10 dB.

The second design factor is the anticipation of manufacturing process variations in the design process. If the design is very sensitive to component values, the gain distribution will be wider as component values vary during the manufacturing process. As a result, more amplifiers will fall out of the performance limits and the end-product yield will decrease. On the other hand, if the amplifier is designed so that the output response is insensitive to component variations, gain variation will be minimized and yield will be maximized. Once this is completed, the sensitive elements remaining can be tightly controlled. This can be accomplished by either using tighter-tolerance parts in the design or by monitoring the sensitive elements carefully during the manufacturing process.

To link the manufacturing control to design quality, six-sigma design uses the process capability indices (Cp and Cpk).

Process capability (Cp) is a measure of how closely and consistently the process can meet customer requirements. The customer requirements are indicated by the target value, the upper spec limit (USL) and the lower spec limit (LSL). This window is called the spec width.

I mentioned that the process is in control if all outputs lie within the upper and lower control limits. This process window is called the process width. If the UCL and LCL also happen to be the customer's target window (the USL and LSL), then Cp=1 using the following equation:

Cp = specification width/process width = (USL-LSL)/ plus/minus 3 sigma

What does this mean? If Cp = 1, the overall design is achieving three-sigma quality. In order to achieve six-sigma quality, Cp must equal 2.0, or in other words, if the spec width is twice the process width, we should meet this requirement 99.99966 percent of the time (see Fig. 1).

Statistical design methods

To simulate the process variation in circuit design, there are three preferred methods: corner, Monte Carlo and design of experiments/response surface methodology (DOE/RSM).

Corner analysis gives incorrect answers so often that it has no ability to predict accurate Cp and Cpk values. Monte Carlo can predict Cp and Cpk, but the simulation cost is very high due to the number of simulations (over 100) required to accurately predict the tails of the distribution.

Design of experiments (DOE) and response surface methodology (RSM) is a well-know statistical practice. The benefits of this method include process parameter sensitivity to circuit performance, reduction in the number of simulations to achieve accurate results, polynomial model creation with error compared with simulations, and performance correlation and sensitivity analysis.

Additional analysis techniques using Monte Carlo simulations can produce correlation and sensitivity results. The computing of the performance correlation from a 300-sample Monte Carlo analysis is shown in Fig. 2. Benefits of the correlation analysis include what's strongly or weakly related, and positively or negatively related performance criteria. It is easy to identify the very strong (.9593) positive correlation between the rising slew rate (SlewR) and the falling slew rate (Slew F).

Mark Rencher is president of Pivotal Enterprises (Gilbert,AZ) and Jack Sifri is with Agilent EEsof EDA (Westlake Village, Calif.)

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