$$E_b \;\approx\; \tfrac{1}{2}\left(C_{\mathrm{fix}} + c_w \ell\right) V_{\mathrm{sw}}^{2}$$

Equation 5 · AI Hardware & Semiconductors

When the Substrate Became the Product

This equation gives an approximation: it relates the quantities while allowing an approximation.

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: $E_b \;\approx\; \tfrac{1}{2}\left(C_{\mathrm{fix}} + c_w \ell\right) V_{\mathrm{sw}}^{2}$. where $C_{\mathrm{fix}}$ collects driver, receiver, bump and pad capacitance, $c_w$ is capacitance per unit length of the conductor, $\ell$ is the reach, and $V_{\mathrm{sw}}$ 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,…

Meanings in this article

• $c_w$: capacitance per unit length of the conductor.
• $V_{\mathrm{sw}}^{2}$: the square of $V_s$w; the signalling swing.

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