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Equation 5 · When the Substrate Became the Product

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Eb  ≈  12(Cfix+cwℓ)Vsw2,E_b \;\approx\; \tfrac{1}{2}\left(C_{\mathrm{fix}} + c_w \ell\right) V_{\mathrm{sw}}^{2},

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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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EbE_b

Symbol E_b

EbE_b is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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CfixC_{\mathrm{fix}}

Symbol C_fix

CfC_fix is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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cwc_w

Symbol c_w

capacitance per unit length of the conductor.

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Vsw2V_{\mathrm{sw}}^{2}

Symbol V_sw^2

the square of VsV_sw; the signalling swing.

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≈

≈

Approximately equal to; the equality is not exact.

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addition

addition

Add the term after the plus sign to the term or group before it.

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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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How to interpret it

Its accuracy depends on the assumptions and range of use described in the article.

What the article says around this equation

Here is the physical core. To first order, a short parallel die-to-die link is a driven capacitive wire, and the energy spent per bit is set by the capacitance that must be charged and discharged: Eb  ≈  12(Cfix+cwℓ)Vsw2E_b \;\approx\; \tfrac{1}{2}\left(C_{\mathrm{fix}} + c_w \ell\right) V_{\mathrm{sw}}^{2}. where CfixC_{\mathrm{fix}} collects driver, receiver, bump and pad capacitance, cwc_w is capacitance per unit length of the conductor, ℓ\ell is the reach, and VswV_{\mathrm{sw}} is the signalling swing. Two assumptions are doing the work and both deserve stating. First, this treats the link as a lumped capacitance rather than a transmission line — reasonable for a few millimetres of on-package wire at moderate rates, wrong once the link is long or fast enough to need equalisation,…
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Here is the physical core. To first order, a short parallel die-to-die link is a driven capacitive wire, and the energy spent per bit is set by the capacitance that must be charged and discharged: Eb  ≈  12(Cfix+cwℓ)Vsw2E_b \;\approx\; \tfrac{1}{2}\left(C_{\mathrm{fix}} + c_w \ell\right) V_{\mathrm{sw}}^{2}. where CfixC_{\mathrm{fix}} collects driver, receiver, bump and pad capacitance, cwc_w is capacitance per unit length of the conductor, ℓ\ell is the reach, and VswV_{\mathrm{sw}} is the signalling swing. Two assumptions are doing the work and both deserve stating. First, this treats the link as a lumped capacitance rather than a transmission line — reasonable for a few millimetres of on-package wire at moderate rates, wrong once the link is long or fast enough to need equalisation, retiming and serialisation, at which point the fixed circuit overhead dominates and the linear term stops being the whole story. Second, it ignores clocking and idle power, which for real links are not negligible.

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