Weighted Resistor D/A Converters

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 with a fixed length of n bits into a continuous analog value.

This function is the inverse of an Analog-to-Digital Converter (ADC). The circuit has n digital inputs and one analog output, and its output voltage is determined by the input word and the converter's transfer relationship.

Output Voltage Relation for the Shown Circuit

For the idealized inverting circuit shown and its resistor ratios, the output voltage $U_{OUT}$ as a function of the digital word is described by the following relation (for Natural Binary Code, NBC):

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

$U_{REF}$ is the stable reference voltage, measured in volts. $U_{OUT}$ has the same voltage unit.
$a_1, a_2, \dots, a_n \in \{0,1\}$ are the bits of the input word. In this notation, $a_1$ is the MSB and $a_n$ is the LSB.
$n$ is the number of input bits.
The shown relation assumes a feedback resistor of $R/2$ , as in the schematic.

Architecture of the Weighted Resistor DAC

The weighted resistor DAC implements D/A conversion with a summing circuit built around an operational amplifier. Its resistor network uses values weighted according to powers of two, so each input bit contributes a binary-scaled current.

Principle of Operation

The circuit uses an inverting 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 $R$ , measured in ohms. 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 current values are expressed in amperes. 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. A second inverting amplifier stage with a gain of -1 can be added in cascade to obtain a positive output voltage.

Advantages and Limitations

Advantages

Simplicity: The circuit is easy to analyze and directly illustrates the D/A conversion principle.
Speed: In this simplified circuit, conversion speed depends mainly on the op-amp settling time.

Disadvantages

Large Resistor Value Spread: The resistor range 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, it is $2^{15}:1$ , or 32,768:1.
Manufacturing Difficulty: Fabricating precise resistors over this range on a single integrated circuit is difficult and costly. DAC accuracy depends directly on matching the resistor ratios.
High Cost and Limited Accuracy: At larger word lengths, manufacturing this converter is costly and maintaining accuracy is difficult. Resistor mismatch and noise reduce precision.

The resistor spread and matching requirements limit pure weighted-resistor DACs at higher resolutions. They have largely been superseded by more practical architectures such as the R-2R Ladder DAC, which uses only two resistor values in a 2:1 ratio.