Can the Line Hold the Spec? Cp and Cpk

A circuit board's copper traces should be 100 µm wide, give or take 15. Whether the line can hold that depends on two things: how spread out it is (Cp compares the spec width to the process width) and whether it's centered (Cpk measures the distance to the nearer limit). A process can be tight but off-center, and then it still fails parts.

How it works

Process capability

In one sentence

For a normal process with mean $\mu$ and standard deviation $\sigma$ and spec limits $\text{LSL} < \text{USL}$ , $C_p = \frac{\text{USL}-\text{LSL}}{6\sigma}$ and $C_{pk} = \min\left(\frac{\text{USL}-\mu}{3\sigma}, \frac{\mu-\text{LSL}}{3\sigma}\right)$ . The nonconforming fraction is $\Phi\left(\frac{\text{LSL}-\mu}{\sigma}\right) + 1 - \Phi\left(\frac{\text{USL}-\mu}{\sigma}\right)$ .

General formula

{\htmlData{sym=0}{C_p}} = \frac{{\htmlData{sym=2}{\text{USL}}} - {\htmlData{sym=3}{\text{LSL}}}}{6{\htmlData{sym=5}{\sigma}}}, \qquad {\htmlData{sym=1}{C_{pk}}} = \min\!\left(\frac{{\htmlData{sym=2}{\text{USL}}} - {\htmlData{sym=4}{\mu}}}{3{\htmlData{sym=5}{\sigma}}},\ \frac{{\htmlData{sym=4}{\mu}} - {\htmlData{sym=3}{\text{LSL}}}}{3{\htmlData{sym=5}{\sigma}}}\right)

Spread and centering are different problems with different fixes. Cp tells you whether the process could fit the spec at all; Cpk tells you whether it actually does, where it's running now.

What the symbols mean

$C_p$
Potential capability
Spec width divided by the process width 6σ, ignoring centering.
$C_{pk}$
Actual capability
Distance from the mean to the nearer limit, in units of 3σ.
$\text{USL}$
Upper spec limit
The widest acceptable trace: 115 µm.
$\text{LSL}$
Lower spec limit
The narrowest acceptable trace: 85 µm.
$\mu$
Process mean
The average trace width the line produces.
$\sigma$
Process standard deviation
How much trace widths vary.

A process with standard deviation $\sigma$ spreads most of its output (99.73%) over $6\sigma$ ; $C_p$ compares the spec width to that.
$C_{pk}$ measures the distance from the mean to the nearer spec limit in units of $3\sigma$ , so it drops when the process drifts off center.
The fraction out of spec is the normal area outside the limits, $1 - [\Phi(z_{U}) - \Phi(z_{L})]$ , quoted in parts per million.
$C_p = C_{pk} = 1$ means about 2,700 ppm out of spec; "six sigma" ( $C_p = 2$ ) means limits $6\sigma$ away and essentially no defects.

Learn the math

Process Capability: Cp and CpkEng Stats · Lesson 29, Statistical Process Control

Can the Line Hold the Spec? Cp and Cpk