High or low: what a bit costs

A wire does not hold a bit. A wire holds a voltage, and a voltage is a continuous quantity — 0.00 V, 0.43 V, 1.62 V, and every value in between, drifting with temperature, with what the neighbouring wire just did, with how much current the thing at the far end is drawing. Nothing in physics divides that line in two.

The division is imposed. We declare a band near ground to mean 0, a band near the supply to mean 1, and — this is the part that is easy to skip — we declare the space between them to mean nothing at all. A bit is that declaration. It is a decision about a voltage, not a property of one.

The decision is not free. It is paid for three times.

The first cost: headroom

If a gate promised only “I will output something below the threshold for a zero”, then a receiver reading exactly at the threshold would be right at the edge of being wrong, and the smallest disturbance would flip it. So the two bands are not adjacent. A gate must output more decisively than a gate is required to accept: outputs are pushed hard toward the rails, inputs are allowed to be sloppier. The gap between what is promised and what is required is the noise margin, and it is voltage deliberately left unused.

That asymmetry buys something remarkable: a gate emits a signal cleaner than the one it received. Noise picked up along a wire is squeezed back out at the next gate instead of accumulating. This is why a chain of ten thousand gates works at all. An analog chain of ten thousand stages would be mud by stage fifty.

The second cost: time

A gate does not change its output when its input changes. It changes it afterwards.

Every wire has capacitance — it is a conductor near other conductors, and that is all a capacitor is. To move a wire from 0 to 1 you must move charge onto it, and the transistor doing the moving delivers current at a finite rate. The voltage climbs an exponential curve toward the rail, and the gate downstream does not see a valid 1 until that curve has climbed past the accept threshold.

So delay is not a tax added to the logic. Delay is the logic, seen on a clock. And it composes the way the circuit is shaped: a signal that must pass through eight gates in a row waits for eight of those climbs, one after another, no matter how simple each gate is.

This is the quantity the labs in this handbook count. When Adding, gate by gate reports a depth in substeps, it is not counting work — it is counting the longest chain a value had to walk before the whole circuit stopped changing. Feed the adder zeros and the depth does not drop. Nothing was easier; the shape was the same.

The third cost: energy

Charging that capacitance takes energy, and roughly half of what leaves the supply is dissipated as heat in the transistor doing the charging rather than stored on the wire. Discharging spends the stored half as heat too. So each full swing of a wire costs about C·V² of energy, split across the two transitions — a cost paid per change, not per unit of time.

Which gives the central number of digital design: dynamic power grows with the capacitance you switch, with the switching rate, and with the square of the supply voltage. Halving the supply quarters this term. It also shrinks the noise margin and slows every one of those exponential climbs, which is the entire tension in the field compressed into one sentence.

A gate that is not switching, meanwhile, costs almost nothing. That is not obvious and it is not universal — it is a property of the complementary arrangement Wanlass and Sah published in 1963 (source), where a pull-up network and a pull-down network are wired so that in any settled state exactly one of them conducts. There is no path from supply to ground, so a gate holding an answer draws essentially no current. The title of that paper is Nanowatt logic, and the claim was as radical as it sounds: the logic families it displaced burned power continuously just to hold a value still. Wanlass wrote the same idea as a patent four years later (source), and it states the stand-by power problem more plainly than most textbooks do.

The next article opens those two networks up. This one only needs the consequence: the reason a modern chip can hold billions of bits without melting is that holding is nearly free, and it is changing that costs.

Why the algebra came first

It would be reasonable to assume the algebra was invented to describe the circuits. It was the other way round. Boole’s algebra of logic was almost a century old, and unemployed, when Claude Shannon — writing a master’s thesis at MIT, then publishing it in 1938 (source) — observed that a network of relay contacts and a Boolean expression are not merely analogous. They are the same object under two notations: contacts in series are a conjunction, contacts in parallel are a disjunction, and simplifying the expression is removing contacts from the circuit.

That is the hinge this entire handbook turns on. Because a circuit can be written as an expression, a circuit can be designed by manipulating expressions, checked by evaluating them, and composed without knowing what is inside the parts being composed. Every level above this one — gates, adders, the ALU, the datapath — exists because that translation is exact.

What the ISA does not say

Open the RISC-V ISA manual (source) and search for the word volt. It is not there. There is no threshold, no noise margin, no capacitance, no picosecond.

That absence is a deliberate engineering artefact, and it is the reason the rest of this handbook is possible. The instruction set is a contract that says what a machine must compute, phrased entirely in bit patterns and their meanings. Everything in this article sits below that contract. A RISC-V core built from relays would be slow, enormous, and correct.

So the layers are honest about what they know. This one knows about volts and knows nothing about instructions. The one above it knows about gates and nothing about volts. The lab lets you cross the boundary by hand: double-click a gate in Adding, gate by gate and the transistors are there, switching, with the same rule holding one level down.