Digital to Analog Converter

At the highest level, the DAC is a black box that takes in a reference voltage, $V_{\text{ref}}$, and scales it by both a proportionality constant, $|k|$, and the binary value, $B$, to produce an analog output signal, such that:

$$x=k\cdot V_{\text{ref}}\cdot B$$

where

• $x$ is the output
• $k$ is a proportionality constant
• $V_{\text{ref}}$ is the reference voltage
• $B$ is the binary word.

We will use $|k|=\frac{1}{2^n}$. This means that the maximum output voltage will be:

$$V_{\text{max}}=k{V}_{\text{ref}}(2^n-1)$$

therefore it will always be smaller than $V_{\text{ref}}$.

The binary value $B$ can be provided to the DAC in either serial or parallel form; either way, a parallel port is used to provide a connection to the system bus, thus providing a path for the CPU to set the binary value by writing to an interface register .

Generic block diagram of a DAC:

Quantization and LSB

Analog signals are continuous, digital signals are discrete. As such, for $n$ digital values, we can only produce outputs at $n$ points along the analog range, ideally at equally spaced intervals. Those intervals, or steps between outputs, are 1 LSB each.

For example, if we have $V_{\text{ref}}=5\text{V}$, $n=2$, $k=\frac{1}{2^n}$, the only analog outputs that can be produced are:

$$x=\frac{1}{2^2}\cdot 5\text{V}\cdot B=1.25B$$

where $B$ is an integer between 1 and 3. While $V_{\text{ref}}$ was $5\text{V}$, the maximum output that can be produced is $3.75\text{V}$, which is exactly $1$ LSB less than $V_{\text{ref}}$. For the converters considered in this course, $V_{\text{out max}}$ will always be 1 LSB less than of $V_{\text{ref}}$.