How are the CPU power and temperature caculated/estimated?

Question

1. From Wikipedia

   The power consumed by a CPU, is approximately proportional to CPU frequency, and to the square of the CPU voltage: $$ P = C V^2 f $$ (where C is capacitance, f is frequency and V is voltage).

   I wonder how that is derived from basic circuit theory?

   How is a CPU modeled as a circuit? Why is it modeled as a capacitance, how about a mixture of resistance, capacitance, and inductance?

   Is the above formula for $P$ related to that the energy/work of a capacitance is $$ W = \frac{C V^2}{2}? $$

   Do we have to distinguish between AC and DC circuits here?

2. From another source, the temperature of a CPU is estimated as a constant factor $$ \text{Processor Temperature} = ( \text{C/W Value} \times \text{Overclocked Wattage}) + \text{Case Temperature} $$ where, if I understand correctly, $\text{Overclocked Wattage}$ is the $P$ in my first formula, and $\text{C/W Value}$ is the constant factor multiplied to $P$.

   I wonder why we can model the temperature as a linear function of $P$? Specifically, why is there a constant factor $\text{C/W Value}$?

3. In practice, I have encountered two cases.

   When I scale down the CPU frequency, the CPU temperature decreases. If the CPU frequency is $f$ in my first part (is it?), then the first formula explains this case well.

   But there is another case that I cannot find explanation from the above parts. When I am running a heavy program, if I use another program called cpulimit in Linux to limit the percentage of CPU usage to for example $50\%$ for the program's process (originally there is no limitation, i.e. CPU usage percentage can be 100% for the program), the CPU temperature can also go down. How will you explain this?

   I posted my questions on http://superuser.com/questions/432377/whats-more-harmful-to-a-cpu-high-load-or-high-temperature, but replies (especially the one by Dennis) there don't seem convincing.

Thanks and regards! ?

Answer 1

Re question 1: A processor is obviously a very complex object, but it's made of from basic structures called logic gates. A logic gate consumes power mainly when it's changing state, and the frequency with which it changes state will probably be, on average, proportional to the clock frequency. To work out the work done whenever the gate changes state you can model it as a capacitor with some effective capacitance, $C_g$, and you get:

$$W = \frac{1}{2}C_gV^2$$

and the power is the work per state change times the number of state changes per second, so:

$$P_g \propto C_gV^2f$$

If you add up all the logic gates in the processor you can define an effective total capacitance, $C$, that will be the sum of all the gate capacitances, $C_g$, so:

$$P \propto CV^2f$$

You'd have to establish the constant of proportionality by experiment.

Re question 2: presumably the CPU is connected to a heatsink, and the equation is just saying that the heat flow into the heatsink (i.e. out of the CPU) is proportional to the temperature difference between the CPU and the (presumably roughly constant) temperature of the heatsink. This seems a reasonable approximation, but it is only an approximation.

Re question 3: there are a couple of possible mechanisms at work. Modern CPUs scale their clock frequencies depending on load, so by only loading at 50% the CPU may be running below it's maximum clock speed. I must admit I don't know how the clock scaling works in modern CPUs and the chaps at Stack Overflow or Superuser would probably know more about this.

The other possibility depends on what the CPU does in the 50% of the time it's not running your program. At the beginning of this answer I said that the frequency with which the logic gates change state will probably be, on average, proportional to the clock frequency. However the constant of proportionality will probably depend on what the CPU is doing. A CPU that is idling may be flipping fewer logic gates per second that the same CPU when it's crunching numbers, so an idling CPU will use less power. That explains why the power usage and hence temperature falls when you limit the CPU used by your program.

(I'm assuming it's not a dual core CPU and 50% means only using one core!)