Calculating 8-bit DAC Output Voltage

This question asks us to find the output voltage of an 8-bit Digital-to-Analog Converter (DAC) given its resolution and a specific digital input code. Understanding the relationship between the digital input, the DAC's resolution, and the analog output is key.

An 8-bit DAC means the digital input is represented by 8 bits, ranging from (00000000)₂ to (11111111)₂.

The resolution of the DAC is given as 20 mV/LSB. LSB stands for Least Significant Bit. This resolution value tells us how much the analog output voltage changes for each increment of the digital input code, specifically for a change in the LSB.

The output voltage (V₀) of a DAC is typically calculated by multiplying the decimal equivalent of the digital input code by the resolution (step size) per LSB.

Digital Input to Decimal Conversion

The given digital input is (10000000)₂. We need to convert this binary number to its decimal equivalent. For an 8-bit binary number \((b_7 b_6 b_5 b_4 b_3 b_2 b_1 b_0)_2\), the decimal value is calculated as:

\( \text{Decimal Value} = b_7 \times 2^7 + b_6 \times 2^6 + b_5 \times 2^5 + b_4 \times 2^4 + b_3 \times 2^3 + b_2 \times 2^2 + b_1 \times 2^1 + b_0 \times 2^0 \)

For (10000000)₂:

• \(b_7 = 1\)
• \(b_6 = 0\)
• \(b_5 = 0\)
• \(b_4 = 0\)
• \(b_3 = 0\)
• \(b_2 = 0\)
• \(b_1 = 0\)
• \(b_0 = 0\)

So, the decimal value is:

\( \text{Decimal Value} = 1 \times 2^7 + 0 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \)

\( \text{Decimal Value} = 1 \times 128 + 0 + 0 + 0 + 0 + 0 + 0 + 0 \)

\( \text{Decimal Value} = 128 \)

The digital input (10000000)₂ corresponds to a decimal value of 128.

Calculating the Output Voltage V₀

Now, we can calculate the output voltage V₀ using the decimal equivalent of the input code and the given resolution:

\( V_0 = \text{Decimal Value of Input} \times \text{Resolution per LSB} \)

Given resolution is 20 mV/LSB, which is equal to 0.020 V/LSB.

\( V_0 = 128 \times 20 \, \text{mV} \)

\( V_0 = 128 \times 0.02 \, \text{V} \)

Performing the multiplication:

\( V_0 = 2.56 \, \text{V} \)

Thus, the output voltage V₀ for the input (10000000)₂ is 2.56 V.

Let's review the options provided:

1. 256 V
2. 2560 V
3. 2.56 V
4. 25.6 V

Our calculated output voltage V₀ is 2.56 V, which matches Option 3.

Summary of 8-bit DAC Calculation

To find the output voltage of an 8-bit DAC:

• Convert the binary input code to its decimal equivalent.
• Multiply the decimal value by the DAC's resolution per LSB.

In this case, the 8-bit DAC with 20 mV/LSB resolution and (10000000)₂ input results in an output voltage of 2.56 V.