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• #### Appendix J: List of Abbreviations 101-102

E

Dennard Scaling and Implications

The following description was taken from the 2010 National Research Council Computer Science and Telecommunications Board (CSTB) report The Future of Computing Performance: Game Over or Next Level?1

“In a classic 1974 paper, reprinted in Appendix D, Robert Dennard et al. showed that the MOS transistor has a set of very convenient scaling properties.10 The scaling properties are shown in Table 3.1, taken from that paper. If all the voltages in a MOS device are scaled down with the physical dimensions, the operation of the device scales in a particularly favorable way. The gates clearly become smaller because linear dimensions are scaled. That scaling also causes gates to become faster with lower energy per transition. If all dimensions and voltages are scaled by the scaling factor κ (κ has typically been 1.4), after scaling the gates become (1/κ)2 their previous size, and κ2 more gates can be placed on a chip of roughly the same size and cost as before. The delay of the gate also decreases by 1/κ, and, most importantly, the energy dissipated each time the gate switches decreases by (1/κ)3. To understand why the energy drops so rapidly, note that the energy that the gate dissipates is proportional to the energy that is stored at the output of the gate. That energy is proportional to a quantity called capacitance11 and the square of the supply voltage. The load capacitance of the wiring decreases by 1/κ because the smaller gates make all the wires shorter and capacitance is proportional to length. Therefore, the power requirements per unit of space on the chip (mm2), or energy per second per mm2, remain constant:

 Power = (number of gates)(CLoad/gate)(Clock Rate)(Vsupply2) Power density = NgCloadFclkVdd2 Ng = CMOS gates per unit area Cload = capacitive load per CMOS gate Fclk = clock frequency Vdd = supply voltage Power density = (κ2)(1/κ)(κ)(1/κ)2 = 1

That the power density (power requirements per unit space on the chip, even when each unit space contains many, many more gates) can remain constant across generations of CMOS scaling has been a critical property underlying progress in microprocessors and in ICs in general. In every technology generation, ICs can double in complexity and increase in clock frequency while consuming the same power and not increasing in cost. Given that description of classic CMOS scaling, one would expect the power of processors to have remained constant since the CMOS transition, but this has not been the case. During the late 1980s and early 1990s, supply voltages were stuck at 5 V for system reasons. So power density would have been expected to increase as technology scaled from 2 mm to 0.5 mm. However, until recently supply voltage has scaled with technology, but power densities continued to increase.”

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1NRC, 2011, The Future of Computing Performance: Game Over or Next Level?, Washington, D.C.: The National Academies Press (available online at http://www.nap.edu/catalog.php?record_id=12980).

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