Introduction to D/A Converters

A Digital-to-Analog Converter (DAC), or C/A converter, is an electronic circuit whose task is to convert a digital signal (typically a binary word of a fixed length, N-bits) into an analog signal (usually a voltage or current). While the weighted resistor DAC is a simple concept, it suffers from a major drawback: it requires a very wide range of precision resistor values. For an 8-bit converter, the resistor values would need to span a ratio of $2^8 : 1 = 256:1$, which is very difficult to manufacture accurately.

The R-2R Ladder DAC elegantly solves this problem. It is a highly popular and practical architecture that achieves high precision using only two resistor values: $R$ and $2R$, making it much easier to fabricate accurately, especially in integrated circuits.

Architecture and Principle of Operation

The R-2R Ladder DAC consists of a repeating network of resistors arranged in a "ladder" configuration. This network is connected to a set of switches controlled by the input digital word, and its output is typically fed into an Operational Amplifier (Op-Amp).

How It Works: The Magic of the Ladder

The genius of the R-2R ladder lies in a property derived from Thévenin's theorem. At each "rung" of the ladder, the current from the reference voltage $U_{ref}$ splits exactly in half. This creates binary-weighted currents that can be summed by the op-amp.

1. Binary Weighted Currents: Looking from any node in the ladder towards the less significant bits, the equivalent resistance is always $2R$. This consistent impedance causes the current to divide perfectly at each node. The current associated with the Most Significant Bit (MSB) has the largest impact, and the current from the Least Significant Bit (LSB) has the smallest, with each bit's contribution being exactly half of its more significant neighbor.
2. Switching and Summing: Each bit of the digital input word controls a switch. If a bit is '1', its corresponding weighted current is directed to the summing input of the op-amp. If the bit is '0', the current is shunted to ground.
3. Op-Amp Output: The op-amp is configured as a summing amplifier. It sums all the currents directed to it and converts this sum into a proportional output voltage. Because it's typically an inverting configuration, the output voltage is negative. To get a positive output, a second inverting op-amp stage with a gain of -1 can be added.

Calculation Example: 4-Bit R-2R DAC

Let's calculate the output voltage for a 4-bit R-2R DAC with the following parameters:

• Reference Voltage: $U_{ref} = 10.24 \text{ V}$
• Digital Input Word (MSB to LSB): $[a_4, a_3, a_2, a_1] = [1, 0, 1, 1]$

Formula for Output Voltage

The output voltage of an inverting R-2R DAC is given by the formula:

$U_{out} = -U_{ref} \cdot \frac{D}{2^N}$

Where $N$ is the number of bits, and $D$ is the decimal equivalent of the binary input word.

Step-by-Step Calculation

1. Convert Binary Input to Decimal (D):The binary word is $[1, 0, 1, 1]$.$D = (1 \cdot 2^3) + (0 \cdot 2^2) + (1 \cdot 2^1) + (1 \cdot 2^0) = 8 + 0 + 2 + 1 = 11$.
2. Calculate the Denominator ($2^N$):The converter is 4-bit, so $N = 4$.$2^N = 2^4 = 16$.
3. Calculate the Output Voltage:Plug the values into the formula:

   $U_{out} = -10.24 \text{ V} \cdot \frac{11}{16} = -10.24 \text{ V} \cdot 0.6875 = -7.04 \text{ V}$