When engineers simulate circuits with the intent of mass-production, the mindset changes from: “How do we hit targets,” to “How do we never miss?” In response, many circuit simulation frameworks like SPICE permit randomization to reveal how a circuit might misbehave when component true values deviate from their targeted (nominal) values.
Simulation end users might be tempted to implement this randomization using the popular Gaussian, or bell curve, distribution. This assumption can be completely wrong. Not because of physics, but because of how components are distributed in the real world.
First, a quick review of component tolerance is in order. In the electronics marketplace, components like resistors, capacitors, and inductors are sold with two primary figures of merit. The first is a nominal value, which expresses the target value of the device. The second is tolerance, which defines the minimum and maximum values any given device with that part number should be expected to have. Tolerance is usually expressed as a percentage:
X_{\min,\max} = X_{\text{nominal}} \left( 1 \pm \text{tolerance} \right)For example: 100 Ω ± 5% for resistive products where a multiplication of the 100 Ω by 5% is an implied one. The reason for deviation from nominal values can often be traced to production factors such as the quality of raw materials, equipment aging or calibration, operating procedure changes, and even subtle variations in plant temperature or humidity—among many other uncontrolled variables.
Some tolerances, especially those of bulk energy-storing capacitors, may be skewed (example: −20%, +80%). Regardless, any component falling outside the Xmin to Xmax range is deemed defective. If process capabilities naturally produce a Gaussian distribution, it would be reasonable to expect a bell curve that is truncated to exclude out-of-tolerance values.
In error modeling, it is prudent to run simulations that assume true component values follow some random distribution. But is a truncated Gaussian distribution the most reasonable choice?
BATCHING AND BINNING
Suppose a supplier producing components with nominal value Xnominal offers the product in a range of tolerances such as ±5%, ±2%, ±1%, ±0.5%, and so on. If end-of-line testing after production determines that a part that was expected to fall within a wide tolerance has actually landed within a tighter tolerance, it becomes advantageous to sell it as a tighter-tolerance device.
This process of reassigning parts to narrower categories goes by many names, such as skimming or culling, but we will refer to it as binning (i.e., regrouping like with like). Once binning has been applied to a batch of components, what remains in a given bin is the population of samples that fall within that tolerance and outside the next tighter tolerance.
When binning is applied aggressively, one can no longer reasonably expect a randomly selected part to closely match its nominal value—unless the part is already specified at the tightest tolerance available, or the supplier choses not to re-bin every single part that qualifies.
THE ACTUAL RANDOM DISTRIBUTION OF COMPONENTS
So what does the actual distribution of component values look like? It’s tough to say.
If a binning policy is aggressive and nearly all “lucky” samples are reassigned (ignoring detection errors), the resulting distribution may resemble a normal distribution with the center hollowed out. In this case, values near nominal may be underrepresented, making circuit simulations composed exclusively of nominal values surprisingly unrealistic.
If binning is applied less consistently—perhaps for logistical or market reasons—the distribution may instead appear bimodal, with two peaks on either side of nominal. In both cases, there should be little to no population outside the specified tolerance range, aside from undetected defects.
Without access to detailed information about a manufacturer’s process capabilities and binning policies at the time of production, it is impossible to know which distribution is most appropriate. Because of this uncertainty, datasheet tolerances should be interpreted as hard boundaries, and not as statements about statistical spread.
PRACTICAL IMPLICATIONS FOR SIMULATION
Given this uncertainty, what distribution should be used in simulation?
If I had to choose a general-purpose model, I would favor a uniform distribution within the specified tolerance range to estimate typical behavior. For components whose function is critical, I would extend that range slightly beyond the stated tolerance as a safety margin.
