Foundations — Static vs dynamic power dissipation

This page assumes you know nothing. We will build every letter, every squiggle, and every picture that the parent note Static vs dynamic power dissipation throws at you, in an order where each idea leans on the one before it.

1. Charge, current, voltage — the three words under everything

Before any power formula makes sense, you need the three most basic electrical quantities. They are not the same thing, and mixing them up is where most confusion starts.

WHY the topic needs these: Every power number in this chapter is built by combining exactly these three. In the picture, the red arrow is the current — the flow. The tank height is voltage — the push. The water itself is charge — the stuff. Keep this picture in your head; we reuse it constantly.

The one relationship you must feel in your bones:

2. What a capacitor is, and why $Q=CV$

Every wire and every transistor gate in a chip acts like a tiny capacitor: two conductors separated by an insulator that can hold charge.

The picture: two parallel plates. Pile charge $+Q$ on the top, $-Q$ on the bottom, and a voltage $V$ appears between them. A fatter capacitor (bigger $C$, wide bucket) needs more charge to reach the same voltage.

WHY the topic needs it: In the parent note, $C_L$ is the "load capacitance" — the total bucket of wire + next-gate inputs that your gate must fill (charge to $V_{DD}$) or empty (drain to $0$) every time the output flips. This is the thing dynamic power pays to move. See Capacitance in interconnects for where $C_L$ physically comes from.

3. Energy and power — and why they are different

These two words get used interchangeably in daily life. In this chapter they are strictly separate, and the parent note's most common mistake comes from confusing them.

WHY this split is the whole game: Charging a capacitor once costs a fixed energy ($\frac{1}{2}{C}_L{V}_{DD}^2$ joules). But power = that fixed energy × how many times per second you do it. That is exactly why the switching-power formula has an $f$ (frequency) in it and the energy formula does not.

4. The exponential $e^x$ — and why leakage uses it

Static (leakage) power is described with $e^{(\dots )}$. You cannot read the topic without knowing what this symbol does.

The picture: compare a straight line (add-the-same-amount) with the exponential curve (multiply-by-the-same-factor). The red exponential stays low, then explodes.

WHY the topic needs it: Subthreshold leakage current is $I_{sub}\propto e^{(V_{GS}-V_T)/(n{V}_{th})}.$ It says: shrink the threshold voltage $V_T$ a little, and leakage multiplies, not merely adds. That "explosion" shape is the entire reason leakage became a monster in tiny chips. See Threshold voltage and subthreshold conduction.

• $V_{GS}$ — voltage from gate to source of a transistor (the "on/off" control knob).
• $V_T$ — the threshold voltage: the gate voltage above which the transistor turns properly on.
• $V_{th}=\frac{kT}{q}\approx 26$ mV — the thermal voltage, a temperature-set natural scale ($k$ = Boltzmann's constant, $T$ = temperature in kelvin, $q$ = charge of one electron). It sets how sharp the exponential is.
• $n$ — a fudge factor slightly above 1 describing a non-ideal transistor.

5. The integral sign $\int$ — adding up tiny slices

The parent note derives energy with ${\int }_0^{V_{DD}}\dots dV$. Here is all you need to read it.

WHY the topic needs it: When a capacitor charges, the current isn't constant — it starts big and fades to zero. To get total energy you must add up the contribution of every tiny instant. That "add up infinitely many tiny bits" is precisely what $\int$ does. The clever trick in the derivation — swapping $\int (\dots )dt$ for $\int (\dots )dV$ — is just changing what you slice by: instead of slicing time, slice voltage from $0$ to $V_{DD}$, which is easier because the limits are exactly the start and end voltages.

6. The named voltages, currents, and factors

Now that the building blocks exist, here is the full symbol dictionary the parent note assumes.

WHY $\alpha$ specifically: real circuits don't toggle every clock. If they did, we'd overcount power hugely. $\alpha$ scales the maximum switching power down to the realistic average. It is a pure probability — a picture of "out of 100 ticks, how many flip this wire."

The picture: a clock ticking; the red marks show the few ticks where the node actually flipped. $\alpha$ = (red ticks) ÷ (all ticks). Here 3 flips out of 10 ticks means $\alpha =0.3$.

7. How it all feeds the topic

Everything on the left is built on this page; everything on the right is what the parent note derives. If any left-hand box feels shaky, reread its section above before moving on.

Equipment checklist

Test yourself — cover the right side and answer aloud.