R-2R ladder DAC versus weighted resistor

A DAC converts an n-bit digital word into an analog voltage or current. The R-2R ladder DAC uses a network with only two resistor values, R and 2R.

Circuit (4-bit, voltage mode)

 b1 (MSB)   b2        b3        b4 (LSB)
   |         |         |         |
  [2R]      [2R]      [2R]      [2R]
   |         |         |         |
 Vout o--[R]--o--[R]--o--[R]--o----[2R]--gnd
 (each switch connects its 2R leg to Vref if bit = 1, or to gnd if 0)

Each node has a 2R leg to its bit switch, adjacent nodes are joined by series resistors R, and the LSB end is terminated with 2R to ground.

Working

Looking from any node towards the LSB end, the resistance is always 2R (the 2R terminator in parallel with a 2R leg gives R, plus a series R gives 2R, and so on). Hence at each node the signal divides equally into two halves, and the effect of each bit is halved at each step away from the output.

By Thevenin's theorem and superposition, the contribution of bit k (counted from the MSB as b1) to the output is Vref/2ᵏ. Therefore:

Vout = Vref (b1/2 + b2/4 + b3/8 + … + bn/2ⁿ) = Vref · D / 2ⁿ

where D is the decimal value of the binary word.

Example (4-bit, Vref = 5 V): the word 1010 gives D = 10, so Vout = 5 × 10/16 = 3.125 V. For 1111, Vout = 5 × 15/16 = 4.6875 V. One LSB step = 5/16 = 0.3125 V.

(A current-mode version feeds an op-amp, giving Vo = −Vref·(Rf/R)·D/2ⁿ.)

Why R-2R is preferred to the weighted-resistor network (WRN)