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Design 4-bit R-2R DAC; unipolar vs bipolar

Short answer

A 4-bit R-2R ladder DAC outputs Vo = Vref (b3/2 + b2/4 + b1/8 + b0/16) = Vref·D/16. Unipolar DAC outputs 0 to +FS (straight binary), while bipolar DAC outputs -FS to +FS using offset binary, requiring an internal offset of half full scale.

Design a 4 bit voltage mode R-2R ladder type digital to analog converter and derive the expression for its analog output. Also differentiate between unipolar and bipolar DACs.

A 4-bit voltage-mode R-2R ladder DAC uses only two resistor values, R and 2R. Each 2R leg is switched to Vref (bit 1) or ground (bit 0). The output is taken from the MSB end through a buffer.

Design choices

  • Resistors: R = 10 kΩ, 2R = 20 kΩ (1 % metal film, matched)
  • Reference: Vref = 5 V
  • Switches: CMOS analog switches (for example CD4066) controlled by b3 … b0
  • Output buffer: op-amp voltage follower (for example 741 or TL081) so the load does not disturb the ladder (ladder output resistance = R = 10 kΩ)
 Vo<-+---10k---+---10k---+---10k---+
     |         |         |         |
    20k       20k       20k       20k      20k
     |         |         |         |        |
    b3        b2        b1        b0       GND
  (MSB)                         (LSB)
  bit=1 -> Vref (5 V), bit=0 -> GND

  Vo --> (+) op-amp follower --> Vout = Vo

Derivation of output

Looking towards the LSB from any node, the resistance is 2R. So each node splits the current equally, and each bit's effect at the output is half that of the previous bit. By superposition:

b3 alone -> Vref/2
b2 alone -> Vref/4
b1 alone -> Vref/8
b0 alone -> Vref/16

Vo = Vref (b3/2 + b2/4 + b1/8 + b0/16)
   = Vref · D / 16      (D = 8b3 + 4b2 + 2b1 + b0)

With Vref = 5 V:

  • Resolution = 5/16 = 0.3125 V
  • Full scale (1111) = 5 × 15/16 = 4.6875 V
  • Example: 1010 gives 5 × 10/16 = 3.125 V

For a larger output, use a non-inverting amplifier: Vout = (1 + Rf/R1)·Vref·D/16.

Unipolar vs bipolar DAC

PointUnipolar DACBipolar DAC
Output range0 to +FS (one polarity)−FS to +FS (both polarities)
Zero output atCode 0000Mid-code 1000
CodingStraight binaryOffset binary / 2's complement
Extra circuitNoneOffset current/voltage of ½ FS
FormulaVref·D/2ⁿVref(2D/2ⁿ − 1)
UsePositive-only signalsAC signals, audio, servo control

✓ Answer checked by Claude Opus 5.5 (high effort)

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