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The R2R digital-to-analogue converters: a few clarifications!

By Abigail Church |

Most digital audio files use PCM (Pulse Code Modulation) encoding, which has also always been used on CDs.

This encoding is always characterized by a number of bits and a sampling frequency, for example 16 bits at 44.1 kHz for CDs.

The sampling frequency corresponds to the number of times per seconds that the amplitude of the analogue signals can be measured, i.e. 44,100 times for a CD. Signal samples are therefore regularly taken, and what happens between two samples is ignored.

The number of bits help translate the captured (or sampled) signal amplitude every second into binary language (0 or 1). With 16 bits, it’s 65,536 (216) values which can be encoded, and there’s no intermediary value. This operation is called quantization and the difference between two consecutive values is called a quantization step.

Any captured signal value will therefore be encoded to the closest value among these 65 536 values, the difference between the real value of the analog signal and its quantized value being called a quantization error.

For a long time, CD players have used digital-to-analog converters which processed the digital audio signals as they were encoded using R2R-type resistor networks. An example of this type of converter is pictured on the diagram below where only three bits are represented for clarity.

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When the three bits are at 1, the VO output voltage is defined by the following formula:

Vo = (Vref / 23) x [ (20 x b0) + (21 x b1) + (22 x b2)] x [1 + (RF / R1)]

[1 + (RF / R1)] represents the buffer stage gain.

For the accuracy of the conversion, it is thus imperative that every R resistor is of rigorously identical values, as well as the 2R resistors that need to be exactly double the value of R, which was difficult for manufacturers and it implied high manufacturing costs, as any difference in value caused a conversion error, which was even more sensitive since the concerned bit is an MSB or the most significant bit (high or very high amplitude signal encoding), and this hasn’t improved with the emergence of converters working on 24-bits, which offer distinctly more important dynamics than 16-bit models.

Today, only a few rare 24-bit R2R converter models are available, and they remain expensive. Few digital-to-analog converter manufacturers use them, as the Delta/Sigma-type models—a process working similarly to DSD, which they can also decode—have supplanted them.

However, cards using precision resistors and logic circuits under the guidance of powerful programmed circuits have emerged a few years ago and carry out R2R-type conversions.


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