R2R and ƩΔ D/A converters: architectures and performance

The technology in Digital to Analog converters, DACs then, has evolved a lot, thanks to better and better technologies and more stable components with better tolerances. In this article we try in a very simple and discursive way to tell how it works and the steps forward so far

In the beginning, music was a live-only event

The recording of sounds, voice, noises, and so on is a desire that for centuries has unleashed human ingenuity aimed at indelibly storing what was happening in nature or in social events with the intent of then replaying them to those who were not present at the time of the recording. In the field of music it was a necessity from the first moment since any sound event represents a precise and unrepeatable instant of a sequence of notes, sometimes not even written down, perhaps in front of a set of people who later could describe the event but not report it in its essence. Only a number of musicians were eventually able to try to repeat the event, and in earlier times this was available to few or could happen only in certain places and for particular moments; in past centuries folk festivals were also a rare event. Symphonic performances, for example, took place only if a wealthy gentleman could pay for the necessary number of orchestral players with all the attendant expenses for the performance; it is no coincidence that composers such as Liszt or Busoni, for example, made transcriptions from symphonic music to piano-only music one of the major incomes of their coffers

Yes that’s right, we find transcriptions of Beethoven’s or Bach’s symphonies that could be performed with just the piano in such a way as to give an idea to a wider audience far from city environments what it was all about and above all not to miss how much superlative composers were able to do. The first concrete example of a device capable of making audio recordings belongs to the 1800s, and in particular we can mention Thomas Edison’s Phonograph (1877) or Emile Berliner’s actual Gramophone (1887). The next century is filled with refinements of these technologies that would produce today’s vinyls but also the development of other media and other technologies such as that of magnetic tapes. In the second half of the twentieth century, audio recording and its trade became one of the most profitable markets on the planet, and the need to reduce space and to be able to enjoy audio entertainment on the move became one of the reasons for the research of large manufacturers such as Philips and Sony, which applied as a solution to these needs the digital representation of the signal, or rather commercialized from 1982 onwards the audio stored in the Compact Disc format that would from then on revolutionize the way audio content was enjoyed.

The binary representation of numbers

Let us now ask what the digital representation of a musical signal is and how a sequence of numbers transforms into an analog sound understandable to our ears that interpret stimuli only of an analog nature. It all stems from some mathematical studies that established how an electrical signal, in this case audio, can under a number of precise conditions-maximum electrical amplitude, maximum frequency value present, periodicity in time of its signals-be transformed into a very specific sequence of numbers and thus be stored in such a way in memory of an electronic nature, of any kind whether volatile or permanent such as precisely a CD or a computer Hard Disk. It is necessary at this point, however, to see how in electronics numerical values are “written,” we say represented. An electronic memory is made up of billions of small cells that may or may not be charged so they can be described as a 1 if they are charged or a 0 if they are discharged.

Is it then possible to represent any number through only 0 or 1? Certainly, in that case we say we are doing a binary representation and the elementary cells from before are called bits, let’s see how it works. The mathematical representation of a number depends on the “base” we have chosen, typically we choose base 10 (this is no accident since we have 10 fingers and started counting from there) and then we write a multi-digit number where the first digit on the right represents units from 0 to 9, the second the tens from 10 to 90, the third the hundreds from 100 to 900, and the fourth the thousands from 1000 to 9000. In this way all we are doing is placing according to a certain sequence the values from 0 to 9 (10 values then) of the units (corresponds to the digit of the units multiplied by 10 ⁰ which is worth 1) or of the tens (corresponds to the tens digit multiplied by 10¹ which is worth 10) or hundreds (corresponds to the figure of units multiplied by 10² which is worth 100) and so on. When we talk about numbers in electronics, hence bits, on the other hand, we cannot use base 10 because there are not 10 fingers but only 2 possible values: 0 or 1 corresponding to whether the memory cell is charged or not. Also in this way all numbers can be represented, however, more digits will be needed because following the same reasoning as before the first digit on the right will be worth only 0 or 1×1 (2 ⁰=1), the second only 0 or 1×2 (2¹=2), the third only 0 or 1×4 (2²=4), the fourth only 0 or 1×4 (2³=8) and so on. To give a (I hope) clarifying example let us take the value 12 that is the sum of 1 ten and 2 units; according to what we are used to employing this representation starting from the left it is written 2×10 (10¹=10) added to 2×1 (10⁰=1) which is worth precisely 12. On the other hand, if we want to think of it the electronic way for the value 12, we have to write the following representation 1100 which corresponds to 1×8 (2 ³=8), added to 1×4 (2 ²=4), added to 0x2 (2 ¹=2) finally added to 0x1 (2 ⁰=1) which makes precisely 12. From here we can understand how the famous digital samples of an audio signal are nothing but numerical values corresponding to an electrical signal amplitude, typically a voltage in volts but also a current in amperes, which, however, are written with a binary representation and thus sequences of 1s and 0s. When it comes to CDs each sample is represented with 16 bits which by now we have learned corresponds to a sequence of 16 zeros or ones depending on the value, the same applies to high-definition, HD files where there are 24 bits.

Digital-to-Analog Converters

As we said earlier, the purpose of Digital-to-Analog converters is to translate a numerical value into an analog quantity such as a voltage of appropriate amplitude that can then be amplified, transmitted to a loudspeaker to be finally heard by our analog ears. So what a converter has to do is to take the 1s and 0s-the so-called bits-corresponding to the value to be converted, assign a weight to each bit that is worth at least twice or half as much as the adjacent bit (remember that in a binary representation this is the relative weight of the bits to each other, as one proceeds from right to left the bit is worth twice as much as the previous one, in a decimal representation the ratio is worth 10) transform them into a physical quantity and then add them together. Weighted transformation in an electronic circuit is as simple as it gets. Let’s look at Figure 1.

S0, S1, S2 and S3 are the first 4 bits of the binary value, if the bit S0 was worth 1 then the small diverter next to the word S0 should be connected to the VR generator (let’s assume 8 volts) if it was worth 0 then the diverter should be connected on the opposite side where there is a ground and thus 0 volts. All 4 branches have downstream of the diverter a resistor of different value but such that starting from the largest bit (the most significant) the value first is worth R, then it is worth its double so 2R, the next is worth twice the previous one i.e. 4R to get to the last one where the resistor doubles again and is worth 8R. If we consider the point downstream of all and these four resistors there we add up all the currents starting from the generator of the 8 volts VR, in the branch with R the current is worth I=V/R, in the branch with 2R the current is worth half and so on like that in the branch with 4R and in the 8R branch current is always smaller and worth half of the current before it. The small terminal circuit simply converts these currents into a voltage that the downstream part of the converter will interpret as an analog value of a binary number. Going back to the example we gave earlier i.e., the number 12 written in binary as 1100, the converter will do the following steps: it will analyze the binary word S3 S2 S1 S0, i.e., for the switches S3=1=8V/ S2=1=8V/ S1=0=0V/ S0=0=0V and then only the resistors R and 2R will have a current flowing through them. Let us make the assumption that as a resistor R we have chosen the value 1 then R=1 and 2R=2 from the formula we have seen we get 12 amperes which the next part of the little circuit will transform precisely into 12 volts. Child’s play. But, …there is always a but, to be truly accurate and exact in the transformation it is necessary to find a whole set of resistors that possesses a value exactly equal to that indicated in the scheme and this is first of all very expensive, the values that are required do not belong to the large-scale industrial achievements of the world’s resistor production, moreover the resistors with the highest value since they pass fractions of current while dissipating the rest in heat and thus turning into small stoves. But since the requirement to have the current halved each time comes from a mathematical formula someone got ingenious and designed the conversion circuit in Figure 2.

This circuit behaves exactly like the previous one with the special feature, however, that are present as weights of the various bits only resistors from value R and 2R, thus easy to locate and select for maximum accuracy, with the prerogative then that by choosing the right values even thermal dissipation can be contained within more than acceptable limits. In this so-called “ladder” architecture (ladder in English) the current coming from the bit with the least weight must flow through the entire ladder halving at each step thus respecting the double/half proportion required for binary encoding. Further evolution the R2R with inverted scale (Fig. 3). Thus, it seems that the best architecture for a D/A converter that requires great precision in converting signal amplitudes has been found, I emphasize that up to now we have not covered anything at all about temporal coherence with the Sampling Frequency, at the moment we do not care about that and we take the highest temporal precision for granted. Actually something more can be done by improving the interfacing with the downstream operational amplifier by reversing the scaling and connecting the active bits, those at 1, to the + input of the amplifier.

In this case you control even more accurately the current flowing through it but let us say clearly that the principle remains to diversify the paths according to the bits. As I said this is one of the earliest circuits designed and suffered from some of the shortcomings of the technology of the time, summarizing: it required extreme precision of resistor value, technologically robust construction to make resistor values constant to temperature change and especially to aging. The realization of a scaled D/A conversion circuit therefore turns out to be very expensive because of the technological choices and circuit implementation required for its maximum performance, and so it was that years ago the demands of market competitiveness, the strengthening of microelectronics technology with the consequent increase in processing speeds shifted the choice to a different conversion algorithm, I am essentially talking about Sigma-Delta (ƩΔ) architectures where the principle of oversampling is exploited to compare the conversion result with a ramp in voltage, a voltage of increasing amplitude, and arrive at the correct value by summation (Ʃ is the symbol for summation in mathematics) and cancellation of differences (Δ is the symbol for differences in mathematics) compared in multiple steps. This is a technique that requires processing and is easily accomplished with a microelectronics component, which can then be produced in millions of pieces and simplifies the user’s printed circuit board construction. Just to understand below you can see the architecture of its sibling the Analog-to-Digital converter.

The main advantage of this realization is that using oversampling lowers the noise floor and thus the realization is quiet with increased dynamics. For the curious, I can say that the noise energy spreads from the initial band to the oversampled band so the noise density decreases, and when you return to the audio band after low-pass filtering, the noise will have a lower value. The main disadvantage that oversampling requires clock generators, the internal clock of the circuits with which the operations are performed, at very high frequencies , several hundred kilohertz if not megahertz and it is very easy for these components to produce radio frequency interference that is difficult to shield and keep under control, So it is true that from the point of view of connections you simplify the circuit board but then you have to be very careful that the remaining connections do not disturb each other. It is this latter conversion technique that dominates the market today, but as always, the devil is in the hand, and the miniaturization of components combined with technologies developed in previous years especially for the military market has meant that even resistors can now be made with the accuracies and reliability necessary to reintroduce the scaled D/A converter architecture with great performance and satisfaction. It becomes a quality choice that only brands that aim to satisfy their customers with world-class performance can make. I’d say the race continues….