R-2R Ladder D/A Converters

Introduction to D/A Converters

A Digital-to-Analog Converter (DAC), or C/A converter, maps a digital input word with a fixed length of $N$ bits to an analog output, usually a voltage or current. A binary-weighted resistor DAC is easy to describe, but its resistor values grow by powers of two. For an 8-bit version, the nominal range from the smallest to the largest weighted resistor is $2^7 : 1 = 128:1$, which makes accurate matching difficult.

The R-2R ladder DAC addresses the matching problem with a repeating network that uses only two nominal resistor values, $R$ and $2R$. Their 2:1 ratio is easier to match on an integrated circuit than a series of widely separated values.

Architecture and Principle of Operation

The R-2R ladder DAC consists of a repeating resistor network connected to switches controlled by the input word. Depending on the implementation, the ladder is used in voltage mode or current mode, and its output can feed an operational amplifier (Op-Amp).

Binary weighting in the ladder

For the standard switch and termination arrangement, the Thévenin equivalent seen at each ladder section has the same resistance. The reference current therefore divides by two at each section, producing binary-weighted currents that can be summed by the amplifier.

1. Binary-weighted currents: Looking from a node toward the less significant bits, the ideal equivalent resistance is $2R$. This repeated load gives each successive branch half the weight of the preceding branch. The Most Significant Bit (MSB) has the largest contribution and the Least Significant Bit (LSB) the smallest. In the current-mode arrangement described here, the ideal input impedance seen by the reference remains constant, so the binary division does not depend on the code.
2. Switching and summing: Each input bit controls a switch. In the current-mode arrangement, a 1 steers its weighted current to the amplifier summing node, while a 0 steers it to the reference return or ground defined by the circuit.
3. Amplifier output: In a current-output implementation, the op-amp sums the selected currents and converts them to a proportional voltage. An inverting feedback configuration gives a negative output for a positive reference. A second inverting stage with a gain of -1 can restore positive polarity.

In a real converter, resistor mismatch, switch resistance, reference variation, amplifier offset, and finite settling time limit linearity and speed. Code changes can also produce a transient glitch while several switches change state.

Calculation example: 4-bit R-2R DAC

Consider 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

For the ideal inverting topology used in this example, with the feedback scaling matched to the ladder, the output voltage is:

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

Here $N$ is the number of bits and $D$ is the decimal value of the input word. The sign follows the inverting amplifier configuration.

Step-by-step calculation

1. Convert the 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:

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