Processor Thermal Power Dissipation (Computer & Electrical Engineering) Solution

In modern computer microchip architecture, the local dynamic thermal power dissipation $P$ (in milliwatts, $\text{mW}$) of a processing core transistor sub-grid depends on its clock frequency $f$ (in gigahertz, $\text{GHz}$) and logic supply voltage $V$ (in volts, $\text{V}$), governed by:

To handle a demanding computational workload spike, an embedded Dynamic Voltage and Frequency Scaling (DVFS) microcontroller algorithm adjusts both operational parameters concurrently over run-time $t$ (in milliseconds, $\text{ms}$):

• $f(t) = 2 + \ln(t + 1)$
• $V(t) = 1.0 + 0.05t^2$

1. Compute the operating frequency $f(1)$ and voltage $V(1)$ at $t = 1\text{ ms}$.
2. State the multivariable chain rule formula for the total time derivative $\frac{dP}{dt}$.
3. Calculate the instantaneous rate of change of power dissipation $\frac{dP}{dt}$ at $t = 1\text{ ms}$.
4. Explain how modern thermal management firmware uses the partial sensitivities $\frac{\partial P}{\partial f}$ and $\frac{\partial P}{\partial V}$ to prevent processor thermal throttling.

Step-by-Step Solution:

1. Evaluate Operating State at $t = 1\text{ ms}$:

   • Frequency: $f(1) = 2 + \ln(1 + 1) = 2 + \ln(2) \approx 2 + 0.69315 = 2.69315\text{ GHz}$
   • Voltage: $V(1) = 1.0 + 0.05(1)^2 = 1.05\text{ V}$
   • Baseline Power: $P = 12(2.69315)(1.05)^2 \approx 12(2.69315)(1.1025) \approx 35.63\text{ mW}$

2. State Governing Multivariable Chain Rule Formula: Since $P$ depends on intermediate parameters $f$ and $V$, which are parameterized by run-time $t$:

   $$\frac{dP}{dt} = \frac{\partial P}{\partial f}\frac{df}{dt} + \frac{\partial P}{\partial V}\frac{dV}{dt}$$
   (Defined in Multivariable Chain Rule.)

3. Compute Partial and Parametric Derivatives:

   • Partial Derivatives:
     $$\frac{\partial P}{\partial f} = 12V^2$$
     $$\frac{\partial P}{\partial V} = 24fV$$
   • Time Derivatives:
     $$\frac{df}{dt} = \frac{d}{dt}[2 + \ln(t + 1)] = \frac{1}{t + 1}$$
     $$\frac{dV}{dt} = \frac{d}{dt}[1.0 + 0.05t^2] = 0.10t$$

4. Evaluate Individual Derivatives at $t = 1\text{ ms}$:

   • $\frac{\partial P}{\partial f} = 12(1.05)^2 = 12(1.1025) = 13.23\text{ mW/GHz}$
   • $\frac{\partial P}{\partial V} = 24(2.69315)(1.05) \approx 67.867\text{ mW/V}$
   • $\frac{df}{dt} = \frac{1}{1 + 1} = 0.50\text{ GHz/ms}$
   • $\frac{dV}{dt} = 0.10(1) = 0.10\text{ V/ms}$

5. Assemble Total Rate of Change:

   $$\frac{dP}{dt} = (13.23)(0.50) + (67.867)(0.10)$$
   $$\frac{dP}{dt} = 6.615 + 6.787 = 13.40\text{ mW/ms}$$

6. Firmware Sensitivity Analysis: Notice that $\frac{\partial P}{\partial V} = 67.87\text{ mW/V}$ is more than 5 times larger than $\frac{\partial P}{\partial f} = 13.23\text{ mW/GHz}$. Because dynamic power scales quadratically with voltage ($V^2$) but only linearly with frequency ($f$), DVFS algorithms prioritize reducing voltage first during thermal emergencies to maximize temperature drops while minimizing clock frequency penalties.