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Equation 3 · Chiplets and Advanced Packaging in Practice: An Advanced Technical Guide

What does this equation mean?

Ypackage≈∏i=1nyiY_{\text{package}} \approx \prod_{i=1}^{n} y_i

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Inputs and operations1^n y_i
Result or conditionY_package ≈ prod_i
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation gives an approximation: it relates the quantities while allowing an approximation. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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YpackageY_{\text{package}}

Symbol Y_package

YpY_package is part of the quantity the equation computes from the expression on the right.

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ii

Symbol i

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

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nn

Symbol n

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

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yiy_i

Symbol y_i

the independent per-die yield.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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≈

≈

Approximately equal to; the equality is not exact.

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subscript

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.

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superscript

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.

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i=1i=1

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.

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nn

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.

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How to interpret it

Its accuracy depends on the assumptions and range of use described in the article. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

The arithmetic behind that claim is a plain probability composition, and it is worth writing down because it is the one piece of real math in advanced packaging that most design reviews skip past. If each of n chiplets bonded into one package has an independent per-die yield yiy_i , the assembly’s final yield is approximately Ypackage≈∏i=1nyiY_{\text{package}} \approx \prod_{i=1}^{n} y_i. which is a first-order model — it assumes bonding introduces no additional defects and that die yields are independent, both of which are simplifications addressed separately below. Even under this optimistic assumption, four chiplets each yielding 95% compose to roughly 81% assembly yield, and a twenty-die stack at the same per-die yield falls to…
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The arithmetic behind that claim is a plain probability composition, and it is worth writing down because it is the one piece of real math in advanced packaging that most design reviews skip past. If each of n chiplets bonded into one package has an independent per-die yield yiy_i , the assembly’s final yield is approximately Ypackage≈∏i=1nyiY_{\text{package}} \approx \prod_{i=1}^{n} y_i. which is a first-order model — it assumes bonding introduces no additional defects and that die yields are independent, both of which are simplifications addressed separately below. Even under this optimistic assumption, four chiplets each yielding 95% compose to roughly 81% assembly yield, and a twenty-die stack at the same per-die yield falls to roughly 35% — the industry’s own framing of why an unscreened stack becomes uneconomical fast as die count rises [ 4 ] . The entire purpose of KGD testing is to push each yiy_i as close to 1 as pre-bond screening allows, since the model shows the product is unforgiving of even small per-die shortfalls once several dies are chained together.

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Sources cited in the surrounding passage

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