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Equation 5 · Comparing the Main Approaches to Chiplets and Advanced Packaging

What does this equation mean?

ΔTj  =  P∑i=1NRth,i,\Delta T_j \;=\; P \sum_{i=1}^{N} R_{\mathrm{th},i},

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Inputs and operationsP sum_i=1^N R_th,i
Result or conditionΔ T_j
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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.

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ΔTj\Delta T_j

Symbol Δ T_j

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

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PP

Symbol P

the power dissipated at that tier and each Rth,iR_{\mathrm{th},i} is the thermal resistance of one layer already between that tier and the heat sink — die bulk, bonding interface, or an intervening active tier.

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ii

Symbol i

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

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Rth,iR_{\mathrm{th},i}

Symbol R_th,i

the thermal resistance of one layer already between that tier and the heat sink — die bulk, bonding interface, or an intervening active tier.

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=

=

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

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change

change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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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

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

The bill for that proximity arrives as heat, and it arrives in a specific, structural way rather than as a vague “3D runs hotter” intuition. A stack has exactly one face in contact with the package’s heat-removal path — typically the topmost die, under a lid or heat spreader — so the temperature rise at a given tier depends on everything the heat has to cross to reach that face: ΔTj  =  P∑i=1NRth,i\Delta T_j \;=\; P \sum_{i=1}^{N} R_{\mathrm{th},i}. where P is the power dissipated at that tier and each Rth,iR_{\mathrm{th},i} is the thermal resistance of one layer already between that tier and the heat sink — die bulk, bonding interface, or an intervening active tier. A tier two layers from the heat-removal face inherits the resistance of…
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The bill for that proximity arrives as heat, and it arrives in a specific, structural way rather than as a vague “3D runs hotter” intuition. A stack has exactly one face in contact with the package’s heat-removal path — typically the topmost die, under a lid or heat spreader — so the temperature rise at a given tier depends on everything the heat has to cross to reach that face: ΔTj  =  P∑i=1NRth,i\Delta T_j \;=\; P \sum_{i=1}^{N} R_{\mathrm{th},i}. where P is the power dissipated at that tier and each Rth,iR_{\mathrm{th},i} is the thermal resistance of one layer already between that tier and the heat sink — die bulk, bonding interface, or an intervening active tier. A tier two layers from the heat-removal face inherits the resistance of everything below it in a way a single die never does, which is why thermal co-design and power-tier placement, not raw stack height, are what determine whether a given 3D design is thermally viable.

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Sources cited in the article section

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