Equation 5 · Part 9 · From Origins to Frontier: A History of Chiplets and Advanced Packaging

Starting index or lower bound: i=1

Y_{\text{pkg}} = Y_{\text{assy}} \prod_{i=1}^{N} Y_i .

i=1

What this part means

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

Its job in the formula

i=1 appears in the bound of this product. The bound states where the repeated operation starts, ends, or which values it includes.

Full expressionStarting index or lower bound: i=1Article meaning

The passage around this formula

The arithmetic behind that reputation is straightforward and still governs every multi-die package built today. A packaged part’s yield is not the yield of its worst component; it is the product of every component’s yield together with the yield of the assembly step that joins them. Write the package yield $Y_{\text{pkg}}$ as a function of the assembly yield $Y_{\text{assy}}$ and the yield $Y_i$ of each of N individual die placed into the package: $Y_{\text{pkg}} = Y_{\text{assy}} \prod_{i=1}^{N} Y_i$ . Multiplication is unforgiving at scale. Every die added to a package is another factor less than one, so a package holding several die each individually manufactured at a very respectable yield can still fail at an uncomfortable rate…

Read this part in the article →

Learn the underlying idea

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

Open the illustrated sums and products: repeat an operation over an index guide →

Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.