---
title: "Short note: Digital to analog conversion"
subject: "Digital Control System"
chapter: 1
key: "semester_6/digital_control_system:short-note-dac"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/short-note-dac
site: BCT Engineering
---

# Short note: Digital to analog conversion

Digital Control System · Chapter 1: Introduction to Discrete Time Control System

- Asked in: 2068 Jestha, 4 marks
- Video: [How Do DACs Work? - The Learning Circuit](https://www.youtube.com/watch?v=YAxrmoVtEtE) — element14 presents
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/digital_control_system/short-note-dac
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-1

## Question

Write a short note on digital to analog conversion.

## Short answer

A DAC converts an n-bit binary word to a proportional analog voltage using V_o = V_ref Σ(b_i/2^{i+1}) with resolution V_ref/2^n. Weighted-resistor DACs need a wide range of precise resistors, while R-2R ladder DACs use only R and 2R values and are most common. Specifications include resolution, accuracy, linearity, settling time, and monotonicity. The DAC output is held constant between updates by a zero-order hold, producing a staircase signal.

## Answer

A **digital-to-analog converter (DAC)** converts an n-bit binary word from the computer into a proportional analog voltage or current. In a digital control system it sits at the controller output, followed by a hold circuit and the actuator.

#### Output relation

$$
V_o = V_{ref}\left(\frac{b_{n-1}}{2} + \frac{b_{n-2}}{4} + \cdots + \frac{b_0}{2^n}\right)
$$

Resolution (one LSB) $= V_{ref}/2^n$.

**Example:** 4-bit DAC, $V_{ref} = 16$ V, input 1011: $V_o = 16(1/2 + 0 + 1/8 + 1/16) = 11$ V.

#### Common types

1. **Weighted-resistor DAC:** resistors $R, 2R, 4R, \dots, 2^{n-1}R$ feed an op-amp summer. It is simple but needs a wide range of precise resistor values.
2. **R-2R ladder DAC:** uses only two resistor values, $R$ in series and $2R$ in each leg; the contribution of each bit is halved at each node. It is the most widely used type.

```text
 Vo <--+--R--+--R--+--R--+-----+
       |     |     |     |     |
      2R    2R    2R    2R    2R
       |     |     |     |     |
      b3    b2    b1    b0    GND
 (each bit switch goes to Vref for 1 or to GND for 0;
  Vo is buffered by an op-amp)
```

#### Specifications

- **Resolution:** number of bits, or the smallest output step.
- **Accuracy and linearity:** deviation from the ideal straight line.
- **Settling time:** time to reach the final value within ±½ LSB.
- **Monotonicity:** the output must not fall when the code increases.

The DAC output is held constant between updates by a latch/hold (zero order hold), giving a staircase signal.

## Check yourself

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

1. For a 4-bit DAC with Vref = 16 V and input 1011, the output is:
   A. 13 V
   B. 11 V
   C. 5 V
   D. 8 V
   Answer: B. 16(1/2 + 0 + 1/8 + 1/16) = 11 V. (Key point: DAC output)

2. The resolution (one LSB) of an n-bit DAC is:
   A. Vref/n
   B. Vref x 2^n
   C. Vref/2^n
   D. Vref/2
   Answer: C. Smallest output step. (Key point: Resolution)

3. The main advantage of the R-2R ladder over the weighted-resistor DAC is:
   A. It needs resistors up to 2^(n-1)R
   B. It has no op-amp
   C. It gives higher bit count only
   D. Only two resistor values are needed
   Answer: D. Weighted-resistor needs a wide range of precise values. (Key point: R-2R ladder)

4. A monotonic DAC is one whose output:
   A. Does not fall when the input code increases
   B. Is always zero at code 0
   C. Equals the input code in volts
   D. Settles instantly
   Answer: A. A specification besides resolution, linearity and settling time. (Key point: Monotonicity)

## Related in Digital Control System

- [Advantages of digital over analog control](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adcs-advantages-over-analog) — Digital Control System, chapter 1, asked 3×
- [Data acquisition, conversion and distribution](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adata-acquisition-conversion-distribution) — Digital Control System, chapter 1, asked 9×
- [Digital control system block diagram and advantages](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adcs-block-diagram-advantages) — Digital Control System, chapter 1, asked 3×
- [Types of signals in digital control](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Asignal-types-in-dcs) — Digital Control System, chapter 1

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- [Analog/digital signals and digital advantages](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Adefine-analog-digital-advantages) — Digital Logic, chapter 1, asked 5×