---
title: "R-2R ladder DAC versus weighted resistor"
subject: "Instrumentation I"
chapter: 5
key: "semester_4/instrumentation_i:r-2r-ladder-dac"
source: https://www.bctengineer.com/q/semester_4/instrumentation_i/r-2r-ladder-dac
site: BCT Engineering
---

# R-2R ladder DAC versus weighted resistor

Instrumentation I · Chapter 5: Analog-Digital and Digital-Analog Conversion

- Asked in: year not given, 8 marks
- Video: [R-2R Ladder DAC](https://www.youtube.com/watch?v=PiGKTNV1ijg) — AK ECE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_i/r-2r-ladder-dac
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_i/chapter-5

## Question

Explain the method of D/A conversion using R-2R ladder network. Why do we prefer to use this DAC over WRN type?

## Answer

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)
```text
 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)
| Point | R-2R ladder | Weighted resistor |
|---|---|---|
| Resistor values | Only two (R, 2R) | n values (R, 2R, 4R, … 2ⁿ⁻¹R) |
| Value range | Ratio 2:1 | Ratio 2ⁿ⁻¹ : 1 (e.g. 128:1 for 8 bit) |
| Accuracy and matching | Easy; only ratios matter | Hard for the large resistor values |
| IC fabrication | Easy | Difficult |
| Cost and stability | Better | Poorer; large resistors drift |
| Resolution possible | High (12 to 16 bits) | Limited to about 4 to 6 bits |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Looking from any node of the R-2R ladder toward the LSB end, the resistance is:
   A. R
   B. 4R
   C. Different at each node
   D. 2R
   Answer: D. The 2R terminator in parallel with 2R gives R, plus series R gives 2R, repeating at every node. (Key point: Resistance seen)

2. The output of an n-bit R-2R ladder DAC is:
   A. Vref · D / 2^n
   B. Vref · D · 2^n
   C. Vref · D / n
   D. Vref · 2^n / D
   Answer: A. Bit k contributes Vref/2^k, so Vout = Vref(b1/2 + b2/4 + ...) = Vref D/2^n. (Key point: Output expression)

3. For a 4-bit R-2R DAC with Vref = 5 V, the input 1010 gives:
   A. 3.125 V
   B. 1.5625 V
   C. 2.5 V
   D. 4.6875 V
   Answer: A. D = 10, Vout = 5 × 10/16 = 3.125 V. (Key point: Numerical example)

4. Why is R-2R preferred over the weighted-resistor network?
   A. It uses n different resistor values
   B. It needs only two resistor values (R and 2R), so ratios are easy to match and fabricate
   C. It uses very large resistors
   D. It is limited to 4 bits
   Answer: B. WRN needs a resistor ratio of 2^(n-1):1, which is hard to match; R-2R needs only a 2:1 ratio. (Key point: Advantage over WRN)

## Related in other subjects

- [Bipolar DAC; weighted resistor DAC drawbacks](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Abipolar-dac-weighted-resistor) — Advanced Electronics, chapter 3, asked 2×
- [Non-inverted R-2R ladder DAC output](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Anon-inverted-r2r-dac) — Advanced Electronics, chapter 3, asked 4×
- [4-bit inverted R-2R DAC and why better](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Ainverted-r2r-4bit-better) — Advanced Electronics, chapter 3
- [Current-mode R-2R preference; bipolar DAC](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Acurrent-mode-r2r-bipolar-dac) — Advanced Electronics, chapter 3