One formula

Dynamic power is P = alpha x C x V squared x f. Every symbol below is a stated EXAMPLE parameter, never a real-process figure.

Alpha is the fraction of capacitance toggling per cycle. C is the switched capacitance, V the supply, and f the clock rate.

A worked EXAMPLE

Substitute step by step and keep the units. The calc block below is the reference the program rechecks.

EXAMPLE parameters (not real figures): alpha = 0.2, C = 50 pF, V = 0.9 V, f = 1 GHz
V^2 = 0.9 x 0.9 = 0.81
C x f = 50e-12 x 1e9 = 0.05
P = 0.2 x 0.05 x 0.81 = 0.2 x 0.0405 = 0.0081 W = 8.1 mW
half voltage (V = 0.45 V): V^2 = 0.2025
P = 0.2 x 0.05 x 0.2025 = 0.002025 W = 2.025 mW = 0.25 x 8.1 mW
gated activity (alpha = 0.05): P = 0.05 x 0.05 x 0.81 = 0.002025 W = 2.025 mW
double frequency (f = 2 GHz): C x f = 0.1, P = 0.2 x 0.1 x 0.81 = 0.0162 W = 16.2 mW

What each lever does

Voltage is quadratic: halving it quarters power, before timing is considered. Activity and frequency are linear: halving either halves power.

Capacitance follows the netlist: smaller cells and fewer toggled nets lower C. That is where synthesis earns its savings, and results vary by design.

A runnable checker

The program below recomputes every hand-worked number from the same EXAMPLE parameters. Its five self-checks pin the base case, the voltage, activity, and frequency variants, and the quarter-power ratio.

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import sys

def dyn_power(alpha, c_f, v, f_hz):
    return alpha * c_f * v * v * f_hz

C = 50e-12
checks = [
    (dyn_power(0.2, C, 0.9, 1e9), 0.0081),
    (dyn_power(0.2, C, 0.45, 1e9), 0.002025),
    (dyn_power(0.05, C, 0.9, 1e9), 0.002025),
    (dyn_power(0.2, C, 0.9, 2e9), 0.0162),
    (dyn_power(0.2, C, 0.45, 1e9) / dyn_power(0.2, C, 0.9, 1e9), 0.25),
]

for i in range(len(checks)):
    got, want = checks[i]
    if abs(got - want) > 1e-9:
        print("FAIL: check " + str(i + 1) + " got " + str(got) + " want " + str(want))
        sys.exit(1)

print("PASS: dynamic power agrees with hand arithmetic on 5 checks")

Related topics

• Next: Leakage and Vt Mixing

• See also: Clock Gating and Low Power