An 8-bit DAC produces an out voltage of 1 V for a digital input of 00110010. Determine the largest value of the output voltage from the DAC.

Explanation:

Digital to Analog Converter (DAC):

Definition: A Digital-to-Analog Converter (DAC) is an electronic device that converts digital signals (binary numbers) into analog voltages or currents. The resolution of a DAC determines the number of distinct output levels it can produce and is typically expressed in bits (e.g., 8-bit DAC).

Working Principle: The DAC takes a digital input, which is a binary number, and maps it to a corresponding analog voltage or current output. For an n-bit DAC, the number of possible output levels is \( 2^n \). The maximum output voltage (full-scale voltage) corresponds to the highest digital input value (all bits set to 1). The analog output voltage is proportional to the digital input value, and the relationship can be expressed as:

\( V_{out} = V_{ref} \times \frac{D}{2^n} \)

Where:

• \( V_{out} \): Analog output voltage
• \( V_{ref} \): Reference voltage (maximum output voltage of the DAC)
• \( D \): Decimal equivalent of the digital input
• \( n \): Resolution of the DAC in bits

Problem Statement:

An 8-bit DAC produces an output voltage of 1 V for a digital input of 00110010. Determine the largest value of the output voltage from the DAC.

Solution:

Given data:

• Resolution of the DAC (\( n \)) = 8 bits
• Output voltage for digital input of 00110010 = 1 V
• Digital input (binary) = 00110010

Step 1: Convert the binary input 00110010 to its decimal equivalent.

Binary: 00110010

Decimal: \( (0 \times 2^7) + (0 \times 2^6) + (1 \times 2^5) + (1 \times 2^4) + (0 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) \)

\( = 0 + 0 + 32 + 16 + 0 + 0 + 2 + 0 = 50 \)

Decimal equivalent = 50

Step 2: Determine the reference voltage (\( V_{ref} \)) of the DAC.

Using the formula for DAC output voltage:

\( V_{out} = V_{ref} \times \frac{D}{2^n} \)

Substitute \( V_{out} = 1 \), \( D = 50 \), and \( n = 8 \):

\( 1 = V_{ref} \times \frac{50}{256} \)

\( V_{ref} = \frac{1 \times 256}{50} \)

\( V_{ref} = 5.12 \, \text{V} \)

The reference voltage of the DAC is \( 5.12 \, \text{V} \).

Step 3: Calculate the largest output voltage from the DAC.

The largest output voltage corresponds to the maximum digital input value, which is \( 11111111 \) in binary or \( 255 \) in decimal.

Using the DAC output voltage formula:

\( V_{out,\text{max}} = V_{ref} \times \frac{D_{\text{max}}}{2^n} \)

Substitute \( V_{ref} = 5.12 \), \( D_{\text{max}} = 255 \), and \( n = 8 \):

\( V_{out,\text{max}} = 5.12 \times \frac{255}{256} \)

\( V_{out,\text{max}} = 5.12 \times 0.99609375 \)

\( V_{out,\text{max}} \approx 5.1 \, \text{V} \)

Conclusion: The largest output voltage from the 8-bit DAC is approximately \( 5.1 \, \text{V} \). Hence, the correct answer is Option 2.

Additional Information

To further analyze the other options:

Option 1: \( 12.75 \, \text{kV} \)

This option is incorrect. The reference voltage of the DAC was calculated to be \( 5.12 \, \text{V} \), and the maximum output voltage is \( 5.1 \, \text{V} \). A value of \( 12.75 \, \text{kV} \) is unrealistic and far beyond the calculated maximum output voltage.

Option 3: \( 255 \, \text{V} \)

This option is incorrect. While the maximum digital input value is \( 255 \), this does not imply that the output voltage is \( 255 \, \text{V} \). The output voltage depends on the reference voltage and the resolution of the DAC, and the calculated maximum output voltage is \( 5.1 \, \text{V} \).

Option 4: \( 20 \, \text{mV} \)

This option is incorrect. A value of \( 20 \, \text{mV} \) is much smaller than the calculated maximum output voltage of \( 5.1 \, \text{V} \). \( 20 \, \text{mV} \) might represent a low output voltage for a smaller digital input value but cannot be the maximum output voltage.

Conclusion:

Understanding the operation of DACs and their relationship between digital inputs, reference voltage, and output voltage is crucial for solving this problem. The correct answer is Option 2, \( 5.1 \, \text{V} \), which represents the largest output voltage from the given 8-bit DAC.