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(C.22) So if fez) is analytic in some domain D, its components u and v are harmonic functions (Le., solutions of Laplace's equation) in D. But do not despair. Despite the path dependence of limits such as (C.18), it is still possible to define a quaternion derivative, to have a quaternion analog of the Cauchy-Riemann equations, and quaternion functions satisfying this analog will be solutions of Laplace's equation in four dimensions [146]. Given a unitlO quaternion n = no + inl + jn2 + kn3, we can define the directional quaternion derivative of :F in the direction of n, 0:F/ on, as we define a directional derivative in vectorial calculus,

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The PCR has a natural interpretation: (1 PCR)100 is just the percentage of the speci cations width used by the process. Thus, the vane-opening process uses approximately (1 1.55)100 64.5% of the speci cations width. Figure 16-14(a) shows a process for which the PCR exceeds unity. Since the process natural tolerance limits lie inside the specifications, very few defective or nonconforming units will be produced. If PCR 1, as shown in Fig. 16-14(b), more nonconforming units result. In fact, for a normally distributed process, if PCR 1, the fraction

3 (a)

n) - :F(Q).

Figure 16-14 Process fallout and the process capability ratio (PCR).


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nonconforming is 0.27%, or 2700 parts per million. Finally, when the PCR is less than unity, as in Fig. 16-14(c), the process is very yield-sensitive and a large number of nonconforming units will be produced. The definition of the PCR given in Equation 16-19 implicitly assumes that the process is centered at the nominal dimension. If the process is running off-center, its actual capability will be less than indicated by the PCR. It is convenient to think of PCR as a measure of potential capability, that is, capability with a centered process. If the process is not centered, a measure of actual capability is often used. This ratio, called PCRk , is defined below.

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PCRk PCRk min c USL 3 , LSL 3 d (16-21)

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In vectorial calculus, such a directional derivative is given by the scalar product of the gradient with the unit vector that defines the given direction. Here, too, a quaternion gradient can be introduced in an analogous way. Viewing F(Q) as F(qo, ql, q2, q3), where Q qO+ql i+q2j+q3k, we can expand:F( Q+ n) = :F(qo+ no, ql nl, q2+ n2, q3 + n3) in a Taylor series around:F(Q) = :F(qO, q}, q2, q3) and rewrite (C.23) as

In effect, PCRk is a one-sided process capability ratio that is calculated relative to the speci cation limit nearest to the process mean. For the vane-opening process, we nd that the

estimate of the process capability ratio PCRk is ^ PCR k min c min c USL 3 x x , LSL d 3 1.06, 33.19 20 312.152 2.04 d 1.06

~:F(Q) = lim :F(Q+ n)-:F(Q) on -+0 o 0F o = lim {nooo:F +n 1 :F +n2 :F +n3 +O( 2)} 0 Xl 0 X2 0 X3 -+0 Xo =S(nO):F(Q), (C.24)

40 33.19 312.152

= 1.

Note that if PCR PCRk, the process is centered at the nominal dimension. Since ^ 1.06 for the vane-opening process and ^ 1.55, the process is obviously runPCR PCR k ning off-center, as was rst noted in Figs. 16-14 and 16-17. This off-center operation was ultimately traced to an oversized wax tool. Changing the tooling resulted in a substantial improvement in the process (Montgomery, 2001). The fractions of nonconforming output (or fallout) below the lower speci cation limit and above the upper speci cation limit are often of interest. Suppose that the output from a normally distributed process in statistical control is denoted as X. The fractions are determined from P1X LSL2 P1Z 1LSL 2 2 P1X USL2 P1Z 1USL 2 2

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For an electronic manufacturing process a current has speci cations of 100 10 milliamperes. The process mean and standard deviation are 107.0 and 1.5, respectively. The process mean is nearer to the USL. Consequently, PCR 1110 902 16 1.52 2.22 and PCRk 1110 1072 13 1.52 0.67

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