What we haven’t checked yet in our journey around (analog) electronics and digital electronics where the converter circuits from analog to digital and vice versa - for this we first want to explain the different terms around.

Analog signals are characterized by being continuous in time and continuous in value. The conversion process of analog signals is often referred to as quantization.

Digital signals, on the other hand, are signals that have undergone a form of quantization process, which means that instead of being continuous signals, they are discrete signals. Digital signals are both time-discrete and value-discrete.

The other way around is described as Analog-Digital-Converter. The fastest one, is a flash converter, a single IC which contains (2^n) -1 comparators where n stands for the bit width.

In addition to fast converter architectures such as the flash ADC and simple DAC structures like the R-2R ladder, there exists another very important class of converters that is optimized not for speed, but for precision and resolution: the sigma-delta converter.

Sigma-delta converters are widely used in modern electronic systems, especially in audio applications, sensor interfaces, and precision measurement equipment. Their main idea is to trade conversion speed for accuracy by using oversampling and digital signal processing.

Instead of converting an analog signal directly into a multi-bit digital word, a sigma-delta converter operates with a very high sampling frequency and a feedback loop. The analog input signal is continuously compared with a feedback signal derived from the converter output. The difference between these two signals is integrated over time. This integration process emphasizes the low-frequency components of the signal while averaging out high-frequency noise.

The name sigma-delta originates from the two operations used inside the converter loop. The sigma (Σ) represents the integration of the error signal over time, while the delta (Δ) represents the difference between the input signal and the feedback signal. Together, these operations form a closed-loop system that continuously corrects itself.

At the heart of the sigma-delta converter is a very simple quantizer, often implemented as a 1-bit comparator. This quantizer decides whether the integrator output is positive or negative and generates a corresponding digital output bit. The digital output is then converted back into an analog signal using a simple DAC and fed back to the input of the converter. This feedback mechanism forces the average value of the digital output to follow the analog input signal.

The direct output of a sigma-delta modulator is therefore not a conventional multi-bit binary number, but a fast stream of single bits, for example:

111011101111010111…​

The information is encoded in the density of ones within this bitstream. A higher density of ones corresponds to a higher input voltage, while a lower density corresponds to a lower input voltage. This type of representation is known as pulse-density modulation (PDM).

Sigma-delta converters operate with a sampling frequency that is much higher than the Nyquist frequency of the input signal. This technique is called oversampling. Oversampling reduces the quantization noise within the signal band and significantly relaxes the requirements for analog anti-aliasing filters.

Another key concept of sigma-delta converters is noise shaping. While quantization noise cannot be eliminated, the feedback loop pushes most of this noise toward higher frequencies, outside the frequency range of interest. A digital low-pass filter removes this high-frequency noise, and a decimator reduces the data rate to produce a high-resolution multi-bit digital output value.

Because of this principle, sigma-delta converters can achieve very high resolutions, often 16 to 24 bits, with relatively simple analog circuitry. However, this comes at the cost of conversion speed and latency introduced by digital filtering. For this reason, sigma-delta converters are not suitable for very high-frequency or fast transient signals, but they are ideal for precision applications.

(Not my own LT-Spice file but found here)

LTSpice file

One extraordinary simple example of a DAC is a resistor network R-2R-Ladder…​ It just consists of a simple resistor network and a simple single op-amp.