An 8-bit Digital-to-Analog converter (DAC) using two identical 4-bit DACs with equal reference voltage is shown in Figure. If b₀ represents LSB, b₇ MSB and the op-amp is ideal, to obtain correct analog values corresponding to an 8-bit DAC at the output V₀, what should be the value of resistor R ?

Explanation:

8-bit Digital-to-Analog Converter (DAC) using Two 4-bit DACs

Problem Understanding: In the given question, an 8-bit DAC is implemented using two identical 4-bit DACs with equal reference voltage. The binary inputs b₇ to b₀ are represented, where b₇ is the Most Significant Bit (MSB) and b₀ is the Least Significant Bit (LSB). To achieve the correct analog output voltage (V₀) corresponding to an 8-bit DAC, we need to determine the value of the resistor R in the circuit. The operational amplifier (op-amp) is assumed to be ideal.

Working Principle:

To understand the operation of this circuit, note the following:

• The 8-bit input is divided into two 4-bit groups: the higher 4 bits (b₇ to b₄) and the lower 4 bits (b₃ to b₀).
• Each 4-bit group is fed into one of the two identical 4-bit DACs.
• The higher 4 bits produce an analog output proportional to their binary value, scaled to the range of the higher nibble.
• The lower 4 bits produce an analog output proportional to their binary value, scaled to the range of the lower nibble.
• The outputs of the two DACs are combined using a resistor network and an operational amplifier to produce a final output voltage V₀ that corresponds to the 8-bit binary input.

Analysis:

The higher nibble (b₇ to b₄) represents the most significant part of the binary input. Its contribution to the output voltage should dominate over the lower nibble (b₃ to b₀). To achieve this, the output of the DAC handling the higher nibble is scaled by a factor of 16 (since 2⁴ = 16) relative to the output of the DAC handling the lower nibble.

The resistor R in the circuit is responsible for ensuring this scaling. Specifically, the resistor network and the ideal op-amp configure the circuit such that:

• The output voltage of the higher nibble DAC is multiplied by 16.
• The output voltage of the lower nibble DAC is added directly.

Mathematical Derivation:

Let the outputs of the two 4-bit DACs be V_H (higher nibble) and V_L (lower nibble), respectively. The final output voltage V₀ is given by:

V₀ = 16 × V_H + V_L

Both DACs have the same reference voltage and identical resistor networks. Therefore, the scaling factor of 16 is achieved by appropriately choosing the value of resistor R. For a standard 4-bit DAC, the output voltage is proportional to the binary input (b₃ to b₀ or b₇ to b₄) scaled by the reference voltage and the DAC’s resolution.

The resistor R is chosen such that the higher nibble’s output voltage is effectively scaled by 16 relative to the lower nibble’s output. This is achieved by using a resistor value of 1 kΩ.

Correct Answer: The value of R should be 1 kΩ.

Additional Information

To further understand the reasoning, let’s analyze why the other options are incorrect:

Option 1: 8 kΩ

This value is too high. If R were 8 kΩ, the scaling factor would not properly match the requirement of multiplying the higher nibble’s output by 16. This would result in incorrect analog values at the output.

Option 2: 0.25 kΩ

This value is too low. A resistor value of 0.25 kΩ would result in an incorrect scaling factor, leading to a mismatch between the contributions of the higher and lower nibbles.

Option 4: 0.5 kΩ

Similar to Option 2, this value is also too low. The scaling factor would not correctly match the requirement, resulting in incorrect output voltages.

Option 5: Not provided in the question

As the correct scaling requires R = 1 kΩ, any other value not listed in the options would also fail to produce the correct output.

Conclusion:

The resistor R plays a crucial role in ensuring the correct scaling of the higher nibble’s output relative to the lower nibble’s output in the 8-bit DAC circuit. By choosing R = 1 kΩ, the circuit achieves the desired scaling, resulting in accurate analog values corresponding to the 8-bit binary inputs. This understanding is essential in designing such circuits to ensure proper functionality and accuracy.