Digital circuits work because they use voltage ranges, not exact mathematical voltages.
That is what lets imperfect physical devices compose.
Ranges Instead of Exact Values
A circuit might treat inputs as:
- low enough to count as
0 - high enough to count as
1 - invalid or transitional in between
The exact thresholds depend on the technology and design.
The important mental model is that logical values are interpreted from physical ranges.
Noise Margin
Noise margin is the amount of unwanted voltage disturbance a signal can tolerate while still being interpreted correctly.
If a valid high output is far enough above the next circuit’s high-input threshold, small disturbances will not change the logical interpretation.
If a valid low output is far enough below the low-input threshold, small disturbances also remain safe.
This is why digital systems can be robust even though the physical world is noisy.
Signal Restoration
CMOS gates do not merely pass weak symbols forward.
When an input is recognized as low or high, the gate drives its output toward a fresh low or high range using supply and ground paths.
That means each gate can restore the signal level for the next stage.
Without restoration, small errors would accumulate quickly through many layers of logic.
Boundary
Digital abstraction is not the absence of analog behavior. It is an engineered use of analog behavior.
The durable model is:
Logic works at scale because circuits interpret voltage ranges with margins and regenerate outputs into valid ranges for the next circuit.