3.5 Energy Delay Squared Product

Martin, Nystroem and Penzes proposed $ET^{2}$ as a special case of the $ET^{n}$ metric that is voltage independent [64]. They proved mathematically that an $ET^{n}$ optimal design is optimal irrespective of the value of $n$. There are a few caveats to this result. It applies only when the circuit is operating within its normal range, i.e., supply voltage is not close to the threshold voltage or to the velocity saturated region of a transistor. The intuition behind their formulation is that two circuits with different supply voltages, power consumptions and performance may be compared by voltage/frequency scaling the systems until either their supply voltage or their frequency matches. Then the system with the better power consumption or performance may be picked. Unfortunately, if the initial difference in performance is too large as in the case of a 2.4 GHz Pentium 4 and a 400 MHz XScale described in Chapter 10, the scaled voltage will be outside the operating range. For example, if the Pentium operating at 1.6 volts has 10 times the performance of the XScale, to equalize their performance the Pentium's voltage needs to be to be scaled down to approximately 0.16 volts. This assumes that operating frequency scales linearly with supply voltage, an approximation that applies only in an extremely narrow voltage range. The new supply voltage of 0.16 volts is bound to be smaller than the threshold voltage of the $0.13\mu$ CMOS process in which the Pentium is fabricated. So the Pentium will not operate correctly at that voltage. Since the scaled supply voltage is not within the normal voltage range, the metric equivalent optimality promised by Martin et al. will not apply.

Results presented in Chapter 10 use $E$, $ET$ and $ET^{2}$ as metrics. The choice of $E$ gives an advantage to systems like the XScale processor that stress energy efficiency over performance. The choice of $ET$ favors systems like the perception processor that value both performance and energy efficiency. $ET^{2}$ favors high performance processors like the Pentium whose design allocates a large expenditure of energy in return for small improvements in performance. Since the range of supply voltage required to equalize the performance of the XScale and Pentium systems is outside the operating range for transistors in the $0.13\mu$ technology in which the Pentium 4 is implemented, this dissertation uses $ET^{2}$ merely as a metric that stresses performance over energy savings. No claims are made about metric equivalent optimality of the circuits for values of $n$ other than two.