Foundations — Voltage droop and decoupling capacitors

This page assumes you have seen none of the notation in the parent note. We build every letter, every symbol, and every picture from the ground up, in an order where each idea leans only on the ones before it. Once you finish this page, the parent note Voltage droop and decoupling capacitors will read like plain English.

0 · The two quantities everything is made of: charge and current

Before voltage, before capacitors, before anything, there are just electrons moving through wires.

The link between them is the single most important idea on this whole page:

$I=\frac{\Delta Q}{\Delta t}$

Read this out loud: current is how much charge ($\Delta Q$) moves in how much time ($\Delta t$). The triangle symbol $\Delta$ ("delta") just means "a change in" or "a small amount of" — it is the workhorse of this entire topic, so meet it now.

1 · Voltage $V$ — the "push"

A modern chip runs on a "rail" of about $1\text{V}$. That is the pressure its transistors expect.

The gap between the normal $1\text{V}$ and $V_{min}$ is your whole safety budget. Everything in this topic is about not spending that budget during a sudden current spike. (This budget is explored further in Clock timing margin and Vmin.)

2 · Resistance $R$ and Ohm's law — the "steady" obstacle

The rule connecting push, flow and obstacle is Ohm's law:

$V_R=IR$

3 · The star of the show: rate of change, written $\frac{di}{dt}$

Here is the single symbol most people trip over. Let's earn it slowly.

We already met $\Delta$ as "a change in." Now shrink the time window until it is as tiny as you can imagine — an instant. Written that way, $\frac{\Delta i}{\Delta t}$ becomes $\frac{di}{dt}$.

Look at the figure: same start and same end current, but the red curve gets there in a flash. Its slope (its $di/dt$) is enormous. Everything bad flows from that steep red line.

4 · Inductance $L$ — why wires hate sudden change

The law that makes $L$ the villain:

$V_L=L\frac{di}{dt}$

Read it: the voltage a wire "steals" is its inductance times how fast the current is changing. Notice — it does not depend on how big the current is, only on how fast it changes. That is the whole plot twist of the topic.

5 · Capacitance $C$ — the local bucket that saves the day

The defining relationship:

$Q=CV$

Plain words: the charge stored equals the bucket size times the pressure across it. A bigger bucket ($C$) holds more charge ($Q$) at the same pressure ($V$).

6 · Parasitics: $ESR$ and $ESL$ — no bucket is perfect

A real capacitor is not a pure bucket. It has flaws baked in by its physical body and leads:

7 · Impedance $Z$ and $\omega$ — one number for "difficulty at a given speed"

The parent note writes $Z_{PDN}$ and $Z_{cap}(\omega )$. Let's unpack both symbols.

The key mental model: inductance's opposition grows with $\omega$; capacitance's opposition shrinks with $\omega$. That tug-of-war is what makes a capacitor helpful at some speeds and useless at others — and it's fully unpacked in LC resonance and impedance. The $j$ you'll see in $Z_{cap}=ESR+j\omega ESL+\frac{1}{j\omega C}$ is just a bookkeeping symbol for "this opposition is out of step in timing with that one" — you can safely treat it as a label for now.

8 · Putting the words together: the PDN

Every symbol above lives somewhere in this chain: $R$ and $L$ in the wires, $C$ (with its $ESR$, $ESL$) in the caps, $V$ at the pins, $I$ and $di/dt$ set by the chip's appetite. The PDN's overall "difficulty dial" is $Z_{PDN}$, and droop is simply:

$\Delta V=-\Delta I\cdot Z_{PDN}$

Now every letter in that line has a home. (Full treatment in Power Delivery Network (PDN).)

How these foundations feed the topic

Equipment checklist

Cover the right side and see if you can recall each before revealing.