Electronic – Why can the energy of local computing be written as \$f^2\$,not \$v^2\$

cpuenergyfrequencypower

I saw this formula in this book: Wireless Information and Power Transfer: Theory and Practice

And in the formula 14.8, it said:
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\$f_{i,n}\$ is the CPU frequency for the nth CPU cycle required for user i

I want to ask why can the energy be written as the square of frequency, \$f^2\$? the wiki said the energy should be \$CV^2\$, where \$C\$ is capacitance and \$V\$ is voltage. It didn't say that energy is equal to the square of frequency then times effective capacitance coefficient. Does anyone know about it?

Best Answer

We can divide the losses in FET logic (all processors are made in FET logic) in to categories:

  • static losses, i.e. leakage currents,
  • switching losses.

The reason is simple: Because they use FETs, the transistors don't need any current to flow through their gate to control the output. Therefore, the transistors use no current at all – aside from leakage.

When switching, however, the charge in the gate capacitor of a FET has to be changed – which means a current needs to flow. Since resistances are non-zero, with P=V·I and Ohm's law, it follows that P = I²·R.

To switch faster, i.e. to have a higher clock frequency, you need to have a higher current flowing in (simple: a gate capacitor exposed to a higher voltage charges faster, like every other capacitor; current is amount of charge per time) or out the gate capacitors every clock cycle. Therefore, I is (at least!) proportional to f, I = µ·f.

Therefore, P = I²·R = µ²·f²·R.

µ and R are material/structural constants of your semiconductor technology (this is a bit simplifying, but it doesn't really matter whether losses are purely ohmic or also have higher potences of the voltages involved here).

Therefore, P is proportional to the square of frequency, at the very least.

That's why the Pentium IV generation of processors, designed to clock incredibly fast, was extremely power-hungry.