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Inverted R-2R DAC output and advantages

Short answer

In the inverted R-2R ladder, each 2R leg current is I/2^k, and bits switch currents to virtual ground or ground. The output is Vo=−(Rf/R)Vref(b1/2+b2/4+...+bn/2^n)=−(Rf/R)Vref·D/2^n. Advantages: only two resistor values, easy expansion, constant currents, high speed, and good accuracy.

Derive the expression for output voltage in inverted R-2R ladder type Digital to Analog converter (DAC). What are the advantages of R-2R type DAC?

An inverted (current-mode) R-2R ladder DAC uses only two resistor values, R and 2R. Each 2R leg is switched either to the virtual ground of the op-amp (bit = 1) or to real ground (bit = 0). Since both positions are at 0 V, the ladder currents never change.

 Vref     R        R        R
  o--+--/\/\--+--/\/\--+--/\/\--+-----+
     |        |        |        |     |
     2R       2R       2R       2R    2R
     | I/2    | I/4    | I/8    | I/16 |
    [S1]     [S2]     [S3]     [S4]  GND
   (MSB)                     (LSB)
 bit = 1: switch to Iout bus
 bit = 0: switch to GND

             Rf
   Iout  +--/\/\--+
   bus --+--(-)   |
              >---+--- Vo
      GND ---(+)

Derivation (n-bit)

  1. Looking right from any node, the resistance is 2R in parallel with 2R = R, plus R in series gives 2R again. So the total resistance seen by Vref is R, and
I = Vref / R
  1. At each node the current splits equally between the 2R leg and the rest of the ladder (both are 2R to ground potential). So the leg currents are
I1 = I/2, I2 = I/4, I3 = I/8, ... In = I/2ⁿ
  1. Bit bk = 1 sends Ik to the summing bus (virtual ground); bk = 0 sends it to ground.
Iout = (Vref/R) [b1/2 + b2/4 + b3/8 + ... + bn/2ⁿ]
  1. The op-amp converts this current to voltage:
Vo = -Rf · Iout
   = -(Rf/R) · Vref · [b1·2⁻¹ + b2·2⁻² + ... + bn·2⁻ⁿ]
   = -(Rf/R) · Vref · D/2ⁿ

where D is the decimal value of the input (b1 = MSB). With Rf = R, 4-bit, Vref = 10 V and input 1010: Vo = −10 × 10/16 = −6.25 V.

Advantages of the R-2R type DAC

  • Only two resistor values (R and 2R), easy to make and match accurately in ICs, regardless of the number of bits.
  • Easily expanded to more bits by adding R-2R sections.
  • Constant current in each leg and constant load on Vref (inverted type), since switches only move current between two 0 V points.
  • High speed and fewer glitches: node voltages do not change, so stray capacitances are not charged and discharged.
  • Good accuracy and temperature tracking; resistor values stay in a practical range (unlike the weighted-resistor DAC).

✓ Answer checked by Claude Opus 5.5 (high effort)

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