Weighted Resistor D/A Converters

Converting digital words to analog signals using a weighted resistor network.

The Role of a D/A Converter

A Digital-to-Analog Converter (DAC or D/A) is a system that converts a digital signal-a sequence of bits representing a number-into an analog signal, such as a voltage or current. Its primary task is to translate a digital word of a fixed length (n-bits) into a continuous analog value.

This function is the inverse of an Analog-to-Digital Converter (ADC). Electronically, it is a circuit with n digital inputs and a single analog output. The output analog voltage depends on the input digital word according to a specific relationship.

General Output Voltage Formula

The relationship describing the output voltage $U_{OUT}$ as a function of the digital word is as follows (for Natural Binary Code, NBC):

$U_{OUT} = \pm U_{REF} \left( \frac{a_1}{2^1} + \frac{a_2}{2^2} + \frac{a_3}{2^3} + \dots + \frac{a_n}{2^n} \right)$

$U_{REF}$ is the stable reference voltage.
$a_1, a_2, \dots, a_n \in \{0,1\}$ are the bits of the input word. The notation varies; here, $a_n$ could be the MSB and $a_1$ the LSB.

Architecture of the Weighted Resistor DAC

The weighted resistor DAC is a classic architecture that achieves D/A conversion using a summing circuit built around an operational amplifier. Its key feature is a network of precision resistors whose values are weighted according to powers of two.

Schematic diagram of a weighted resistor DAC

Principle of Operation

The circuit operates based on the principle of a summing amplifier:

Digital Input and Switches: The n-bit digital input word (MSB to LSB) controls a set of switches. For each bit, the switch connects its corresponding resistor to either the reference voltage $U_{ref}$ (if the bit is '1') or to ground (if the bit is '0').
Weighted Resistors: Each resistor has a value that is a power-of-two multiple of a base resistance $R$ . The resistor for the Most Significant Bit (MSB) is $R$ , the next is $2R$ , then $4R$ , and so on, up to $2^{n-1}R$ for the Least Significant Bit (LSB).
Summing Amplifier: An is configured as a summing circuit. The currents flowing through the resistors connected to $U_{ref}$ are summed at the op-amp's inverting input. Because the resistors are weighted, each bit contributes a proportionally weighted amount of current to the sum. The MSB contributes the most current ( $I = U_{ref}/R$ ), while the LSB contributes the least ( $I = U_{ref}/(2^{n-1}R)$ ).
Output Voltage: The op-amp converts this total summed current into an output voltage, $U_{out}$ . Because it is an inverting summer, the output voltage is negative. To obtain a positive output voltage, a second inverting amplifier stage with a gain of -1 must be connected in cascade.

Advantages and Critical Disadvantages

Advantages

Simplicity: The concept is very straightforward and provides a clear illustration of the D/A conversion principle.
Speed: The conversion speed is generally high, limited primarily by the settling time of the op-amp.

Disadvantages

Large Resistor Value Spread: This is the most significant drawback. The range of resistor values grows exponentially with the number of bits. For an 8-bit DAC, the ratio between the largest and smallest resistor is $2^7:1$ or 128:1. For a 16-bit DAC, this ratio becomes a massive 32,768:1.
Manufacturing Difficulty: Fabricating a set of highly precise resistors over such a vast range on a single integrated circuit is extremely difficult and expensive. The accuracy of the DAC depends directly on the precise matching of these resistor ratios.
Low Production Cost and Accuracy: Despite its simple design, the cost of manufacturing this type of converter with high word lengths is high, and achieving accuracy is difficult. It is more susceptible to noise and less precise.

Due to these severe practical limitations, the pure weighted resistor DAC is rarely used for converters with high resolution (typically more than 4-6 bits). It has been largely superseded by more practical architectures like the R-2R Ladder DAC, which uses only two resistor values.