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How do modern oscilloscopes capture high-speed waveforms at billions of samples per second? They compare an incoming analog voltage to every possible threshold level simultaneously in a single clock cycle.
To perform this instant conversion, an nn-bit Flash Analogue-to-Digital Converter (ADC) relies on a potential divider chain of 2n2^n identical resistors of equal resistance RR, known as a reference voltage ladder.

The Ladder Architecture

To resolve an analog input into 2n2^n digital states, the architecture requires 2n−12^n - 1 op-amp comparators connected in parallel to test the signal against every tap voltage simultaneously.
To provide these thresholds, the circuit uses a potential divider chain of 2n2^n identical resistors of equal resistance RR connected across VrefV_{\text{ref}} to ground.
How do we derive the exact voltage at each comparator tap node?

Deriving the Node Voltages

In 1827, Georg Ohm demonstrated that current through a conductor is proportional to the potential difference across it. Because identical resistors in series share the same current, each resistor drops an identical slice of voltage.
The total resistance of a ladder with N=2nN = 2^n matched resistors is Rtotal=2n⋅RR_{\text{total}} = 2^n \cdot R.
Applying Ohm's law gives the steady ladder current: I=VrefRtotal=Vref2nRI = \frac{V_{\text{ref}}}{R_{\text{total}}} = \frac{V_{\text{ref}}}{2^n R}
The voltage at tap kk (measured across kk resistors from 0 V0\text{ V} ground) is the product of current II and cumulative resistance k⋅Rk \cdot R: Vk=I⋅(k⋅R)=(Vref2nR)⋅(k⋅R)=(k2n)VrefV_k = I \cdot (k \cdot R) = \left(\frac{V_{\text{ref}}}{2^n R}\right) \cdot (k \cdot R) = \left(\frac{k}{2^n}\right) V_{\text{ref}}