Equation 14 · When the Substrate Became the Product
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol Y_pkg
kg is part of the quantity the equation computes from the expression on the right.
Symbol Y_assy
ssy is an input to the expression that computes the quantity on the left.
Symbol i
i appears in the bound of this product. The bound states where the repeated operation starts, ends, or which values it includes.
Symbol N
N appears in the bound of this product. The bound states where the repeated operation starts, ends, or which values it includes.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →Starting index or lower bound: i=1
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.
Ending index or upper bound: N
This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.
How to interpret it
Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
A multi-die package inherits a property that a single die does not have: its yield is a product, not a term. If a package contains dies with individual probabilities of being good, and the assembly itself succeeds with probability , then . Multiplication is brutal at scale. Eight components at 99 percent each land near 92 percent together before assembly loss is counted; the same eight at 95 percent land near 66 percent. And the loss is not proportional to the failing part — scrapping a package destroys every good die in it plus the assembly work. This is precisely why known-good-die testing dominates the discussion: the whole point is to move the …
Read the full surrounding passage
A multi-die package inherits a property that a single die does not have: its yield is a product, not a term. If a package contains dies with individual probabilities of being good, and the assembly itself succeeds with probability , then . Multiplication is brutal at scale. Eight components at 99 percent each land near 92 percent together before assembly loss is counted; the same eight at 95 percent land near 66 percent. And the loss is not proportional to the failing part — scrapping a package destroys every good die in it plus the assembly work. This is precisely why known-good-die testing dominates the discussion: the whole point is to move the terms as close to unity as possible before dies are committed to an assembly that cannot be undone.
Sources cited in the article section
- [2] Chiplet Actuary: A Quantitative Cost Model and Multi-Chiplet Architecture Exploration ↗
- [9] Opportunities and Challenges for 3D Systems and Their Design ↗
- [11] Heterogeneous Integration Roadmap ↗
These citations give research context. Read each source to check which claims it supports.
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