Calculating your PCB Complexity & First Pass Yields
There are four things I know about PCB yields—mathematically!
PCB yields are not normally distributed. They are what is called GAMMA Distributed.
Defects that produce low yields may be systemic and repeatable, others may be just random—but you can test for this.
You can predict yields if you take the time to collect the data.
Yields are such wonderful things! I wonder why we don’t write about them more. They are the dividing phenomena that often times determine whether we make a profit or not. They certainly control our efficiency and throughput. Most customers acknowledge high yields with competence, control, and capability for the fabricator.
So why don’t you read more about PCB yields? Personally, I think it’s because we have been very poorly schooled in statistics. I don’t mean the boring statistics like the kind I had in college, with endless equations and proofs, where only the professor seemed to care about the endless stream of numbers. No, I’m talking about statistics that are fun and revealing and tools every engineer needs to embrace. The kind of statistics that allow the numbers (yields) to reveal certain truths.
My love of the numbers (and statistics) came from three sources:
I took one of the E.I. DuPont de Nemours Professional Development Courses in “Design Of Experiments”
I read a magazine article in Chemical Engineering in 1970 (like this one) written by a 3M engineer about “Statistical Analysis for Multivariable Systems” that introduced me to Plackett and Burman’s  factorial design
The only statistics book I enjoyed reading (and learning) was “Conversational Statistics” by Harry Roberts of the University of Chicago 
The critical parameter in the calculation of First Pass Yield (FPY) was the metric ‘complexity index’. This is a dimensionless parameter that is, what I call a pseudo-independent variable; it replaces several important independent variables that would be complicated if taken individually. This variable is invented, but it follows several pseudo-independent variables which are essential for modern engineering, the Dimensionless Numbers. An example that you might remember is The Reynolds Number (a measure of inertia force divided by viscous force), used in fluid flow calculations…but...back to yields…
The First Pass Yield (FPY) is computed from a board’s design characteristic. This is the yield of production before any repair or rework. PCB yield data is a Gamma Distribution. This is only common-sense, as you can have a typically high-yielding board have some bad production runs, the resulting mean and standard deviation would reflect the lower yield data, but on the “ + “ side, you cannot have greater than 100% yield. Thus, the normal mean and standard deviation does introduce some errors which we will ignore. If you have the ability to calculate and insert a gamma-distribution average, by all means do it.
Defining a PCB
The basic truth about printed circuit boards, substrates and hybrid circuits is that design factors have a cumulative effect on manufacturing yield. These factors all affect producibility. Specifications may be selected which individually do not adversely affect yields, but, cumulatively, can dramatically reduce yield and profitability. A simple algorithm is available that collects these factors into a single metric called the Complexity Index (CI). It is given in equation 1:
There is no standard for the Complexity Index. It is whatever fits your data the best. You need to experiment to find what definition of the CI equation gives the highest r2 value (goodness of fit).
FPY is affected primarily by the physical attributes of the board; these can be quantified by Parametric Panels (PCQR2). But random effects also affect FPY like handling, operator training, equipment maintenance, production shifts, and unexpected events. These show up in the high variability of the yield predictions.
First Pass Yield Calculations
The first pass yield equation is derived from the Weibel probability failure equations. This equation is of a more general form of the equation typically used to predict ASIC yields by defect density and provided here as equation 2:
To determine the constants A and B in equation 2, a fabricator must characterize his manufacturing process by selecting the number of printed circuits currently being produced and having various complexity indexes, hopefully some low, medium and high. The first pass yield (at electrical test without repair) of these printed circuits for several production runs are recorded.
Any statistical software program—such as EXCEL—that has a model-based regression analysis can now determine A and B from this model:
Constant A determines the slope at the inflection of the yield curve and constant B determines the x-axis location of the inflection. The confidence level in yield is predictability determined by the statistical margin of error in the calculation of A and B. Figure 1 shows the characteristic Yield versus PCB Complexity and the influence of the coefficients A and B.
Figure 1. The First Pass Yield is reflected in the overall yields in production. Improving the FPY will improve all board yields (move the curve to the RIGHT) and DFM feedback and Tooling helps simplify a board that moves its Complexity to the LEFT
Plackett, R.L. and Burman, J.P., The Design of Optimum Multifactorial Experiments, Biometrika, Vol.33-part 1, April 1946, pp 305-325
Harry V. Roberts, Conversational Statistics, University Business Series-University of Chicago, December 1974, Provided by Hewlett-Packard and Stanford University, # HP 5951-5606