Foundations — DRAM refresh and charge leakage

Before you can read the parent note, you must be able to look at each symbol — $Q$, $C$, $V$, $R$, $\tau$, $e^{-t/\tau }$, $t_{REF}$ — and see a picture, not just letters. This page builds every one of them from nothing, in the order they depend on each other.

1. Charge — the stuff that IS the bit

The picture: imagine a bucket. The water level is what we can measure; the amount of water is the charge. More electrons piled on the node = more charge = a fuller bucket.

Why the topic needs it: the entire bit is "is there charge here or not?" — charge present = logic 1, charge absent = logic 0. If you can't picture charge, you can't picture a stored bit.

2. Voltage — how "hard" the charge pushes

Why two words for what seems like one thing? Charge and voltage are different: a wide bucket and a narrow bucket can hold the same water height (voltage) but very different amounts of water (charge). What links them is the width of the bucket — and that width is called capacitance, next.

Why the topic needs it: the sense amplifier can't count electrons; it can only feel the voltage on the bitline. So the "1 vs 0" decision is made on $V$, and the whole decay derivation tracks $V(t)$.

3. Capacitance and $Q=CV$ — the bucket's width

The picture: in the figure below, two buckets at the same water height (same $V$) hold different water (different $Q$) because one is wider (bigger $C$).

Why the topic needs it: this one equation is the bridge between the thing we store ($Q$) and the thing we measure ($V$). Every leakage calculation swaps between them using $Q=CV$.

4. Current — the rate the bucket drains

The notation $\frac{dQ}{dt}$. The little "$d$" pair means "a tiny change in $Q$ over a tiny change in time $t$" — the instantaneous slope of the charge-vs-time graph. If charge is leaving, $Q$ is shrinking, so its change is negative. That is exactly why the parent writes: $I=-\frac{dQ}{dt}$ The minus sign just says "the current out equals the rate at which stored charge falls."

Why the topic needs it: leakage is a current. To find how long a bit survives, we must connect "how fast it drains" ($I$) to "how much is left" ($Q$, hence $V$).

5. Resistance and Ohm's law — how narrow the leak is

Why the topic needs it: the parent models every leakage path (subthreshold, junction, dielectric) as one equivalent resistor $R_{leak}$ from the storage node to ground. That single simplification is what makes the decay solvable.

6. The exponential $e^{-t/\tau }$ — why decay is never a straight line

The picture: the curve below starts at $V_0$, passes through $0.37V_0$ at $t=\tau$, and flattens toward zero. The horizontal line $V_{th}$ (the sense threshold) is where the bit becomes unreadable.

7. The threshold $V_{th}$ and retention time $t_{ret}$

The notation $\ln$. $\ln$ (natural logarithm) is the inverse question to $e^{-t/\tau }$: "the exponential turned $t$ into a voltage ratio; $\ln$ turns the voltage ratio back into a time." It's the only tool that can undo an exponential, which is exactly why it appears the instant we solve for a time.

8. The refresh symbols — $t_{REF}$, $N$, $t_{RI}$

Why the topic needs it: this is the payoff — once you can compute $\tau$ and $t_{ret}$, you set $t_{REF}$ safely below it and divide by $N$ to schedule the refreshes.

Prerequisite map

This whole chain also underlies the comparison in SRAM vs DRAM — SRAM skips the leaky-bucket problem entirely by using a self-driving latch.

Equipment checklist

Test yourself — you are ready for the parent note only if every line below is instant.