Understanding R-2R Digital to Analog Converters (DACs)

An R-2R Digital to Analog Converter (DAC) is a common type of DAC architecture that converts a digital binary input into an analog output voltage or current. It is characterized by its simple and scalable ladder network, which uses only two resistor values: R and 2R. This makes the design easy to implement and relatively accurate.

When an inverting op-amp is used at the output stage of an R-2R DAC, it takes the sum of currents produced by the ladder network and converts it into a proportional output voltage. The "inverting" nature of the op-amp means that the output voltage will have the opposite polarity (sign) compared to what would be expected from a non-inverting configuration, typically resulting in a negative voltage if the reference voltage is positive.

Analog Output Formula for R-2R DAC with Inverting Op-amp

For an N-bit R-2R DAC connected to an inverting op-amp, the analog output voltage \(V_{out}\) can be determined using the following formula:

\[ V_{out} = -V_{ref} \times \left( \frac{b_{N-1}}{2^1} + \frac{b_{N-2}}{2^2} + \dots + \frac{b_1}{2^{N-1}} + \frac{b_0}{2^N} \right) \]

A more simplified way to express this formula, especially useful for calculations, is:

\[ V_{out} = -V_{ref} \times \frac{\text{Decimal Equivalent of Binary Input Code}}{2^N} \]

• \(V_{ref}\) represents the reference voltage supplied to the DAC.
• \(b_{N-1} b_{N-2} \dots b_1 b_0\) is the N-bit binary input code, where \(b_{N-1}\) is the Most Significant Bit (MSB) and \(b_0\) is the Least Significant Bit (LSB).
• \(N\) signifies the number of bits the digital to analog converter can handle (e.g., 4-bit, 8-bit).

Calculating Analog Output for Input Code 1010

Let's apply the given values from the problem to calculate the analog output for the specified input code.

Given Parameters:

• Number of bits (N) = 4
• Reference voltage \(V_{ref}\) = 5V
• Binary input code = 1010

Step-by-Step Calculation of Analog Output:

1. Convert the binary input code to its decimal equivalent:
   The given binary input code is 1010. We convert this to decimal as follows:
   \( \text{Decimal Equivalent} = (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) \)
   \( = (1 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1) \)
   \( = 8 + 0 + 2 + 0 \)
   \( = 10 \)
2. Determine the denominator \(2^N\):
   Since the DAC is a 4-bit system, N = 4.
   \( 2^N = 2^4 = 16 \)
3. Apply the R-2R DAC formula for an inverting op-amp:
   Using the formula \( V_{out} = -V_{ref} \times \frac{\text{Decimal Equivalent}}{2^N} \):
   Substitute the values we found and the given reference voltage:
   \( V_{out} = -5V \times \frac{10}{16} \)
   \( V_{out} = -5V \times 0.625 \)
   \( V_{out} = -3.125 V \)

Final Analog Output

The calculated analog output for the input code 1010, given a 5V reference voltage and a 4-bit R-2R DAC with an inverting op-amp, is -3.125 V. The negative sign is a direct result of using the inverting op-amp configuration.