Resolution MCQ Quiz - Objective Question with Answer for Resolution - Download Free PDF

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.

• The smallest change in the input signal that can be detected by an instrument is referred to as its "resolution."
• The term resolution describes the finest detail that a device or system can detect or measure.
• It is a key parameter for systems that deal with digital signals, as it is directly linked to the quality or level of detail of the output.

Concept:

The resolution  for n bit DAC is given by:

\(R=\frac{V_{max} - V_{min}}{2^n -1}\)

Calculation:

\(0.01=\frac{10-0}{2^n-1}\)

\(2^n -1 =1000\)

\(2^n=1001\)

n=10 

Minimum number of bits in the output is 10

Concept:

• Analog-to-Digital Converters (ADCs) transform an analog voltage to a binary number (a series of 1’s and 0’s).
• Then eventually to a digital number (base 10) for reading on a meter, monitor, or chart.
• The ADC resolution depends upon the number of bits used to represent the digit number. 
• As the number of bits increases the resolution of an Analog to Digital Converter improves and the quantization error decreases.

Resolution for n – bit A/D converter  will be:

\(R= \frac{V_{FS} \ \times \ (2^ib^i)}{{{2^n} - 1}}\)

Where 

R = Resolution

VFS is reference voltage 'or' Full-scale voltage

n = number of bits

2ⁱbⁱ gives output voltage value.

Calculation:

Given:

V_FS = 5.8 V

n = 4

R = 05 V

\( 5= \frac{5.8 \ \times \ 2^ib^i}{{{2^{4}} - 1}}\)

\( 2^ib^i= \frac{5 \ \times \ 15}{{5.8}}\)

2ibi = 12.93 V ≈ 12 V

In bits, the output voltage will be 1100

Hence option (1) is the correct answer.

Important Points

The resolution of DAC is a change in analog voltage corresponding to the LSB bit increment at the input.

The resolution (R) is calculated as:

\( R= \frac{{{V_{FS}}}}{{{2^N} - 1}}\)

No. of levels = 2N – 1

Vr is reference voltage 'or' Full-scale voltage

Resolution: It is defined as the smallest change in the analog output voltage corresponding to a change of one bit in the digital output.

The percentage resolution (%R) of an n-bit DAC is:

\(\%R = \frac{1}{{{2^n} - 1}} \times 100\)

The resolution of an n-bit DAC with a range of output voltage from 0 to V is given by:

\(R = \frac{V}{{{2^n} - 1}}volts\)

Hence the difference between analog voltage represented by two adjacent digital codes of an analog to digital converter is called resolution.

Hence option (2) is the correct answer.

Important Points

Accuracy:

• The accuracy of the A/D converter determines how close the actual digital output is to the theoretically expected digital output for given analog input.
• In other words, the accuracy of the converter determines how many bits in the digital output code represent useful information about the input signal.

% Accuracy of a n bit ADC = (1 / 2n ) × 100

Concept:

• Analog-to-Digital Converters (ADCs) transform an analog voltage to a binary number (a series of 1’s and 0’s).
• Then eventually to a digital number (base 10) for reading on a meter, monitor, or chart.
• The ADC resolution depends upon the number of bits used to represent the digit number. 
• As the number of bits increases the resolution of an Analog to Digital Converter improves and the quantization error decreases.

Resolution for n – bit A/D converter  will be:

\(R= \frac{V_{FS} \ \times \ (2^ib^i)}{{{2^n} - 1}}\)

Where 

R = Resolution

VFS is reference voltage 'or' Full-scale voltage

n = number of bits

2ⁱbⁱ gives output voltage value.

Calculation:

Given:

V_FS = 5.8 V

n = 4

R = 05 V

\( 5= \frac{5.8 \ \times \ 2^ib^i}{{{2^{4}} - 1}}\)

\( 2^ib^i= \frac{5 \ \times \ 15}{{5.8}}\)

2ibi = 12.93 V ≈ 12 V

In bits, the output voltage will be 1100

Hence option (1) is the correct answer.

Important Points

The resolution of DAC is a change in analog voltage corresponding to the LSB bit increment at the input.

The resolution (R) is calculated as:

\( R= \frac{{{V_{FS}}}}{{{2^N} - 1}}\)

No. of levels = 2N – 1

Vr is reference voltage 'or' Full-scale voltage

Resolution:

Resolution of ADC is a change in analog voltage corresponding to a 1-bit increment.

Resolution is the number of bits per conversion cycle that the converter is capable of processing.

\(R=\frac{V_{range}}{2^n}\)

n = No. of bits of ADC

V_range = V_max - V_min

Analog output = Reslotion x Decimal equivalent to binary

V_o = R x D

Calculation:

V_range = V_max - V_min

V_max = 2.56 V

V_min = 0 V

⇒ V_range = 2.56 V

n = 8

Resolution is

\(R=\frac{V_{range}}{2^n}\)

⇒ \(R=\frac{2.56}{2^8}\)

∴ R = 0.01.

Output voltage V_o = 1 V

⇒ 1 = 0.01 x D

∴ D = 100

The binary value of 100 is

The binary equivalent of 100 is (011 001 00)₂

Concept:

The resolution of DAC is a change in analog voltage corresponding to LSB bit increment at the input.

The resolution (R) is calculated as:

\( R= \frac{{{V_{FS}}}}{{{2^n} - 1}}\)

No. of levels = 2n – 1

Vr is reference voltage 'or' Full-scale voltage

Calculation:

Given n = 10 bits

V_FS = 10.23 V

\(R= \frac{10.23}{{{2^{10}} - 1}}\)

R = 10mV

Explanation:

Resolution (R):

Resolution of Digital to Analog Converter(DAC) is a change in analog voltage corresponding to 1 LSB bit increment at the input.

\(R = \frac{V_r}{2^n-1}\)

Where

V_r = Reference voltage

n = number of bits

Resolution should be below 0.1% of the maximum value 

\(\frac{V_r}{2^n-1}≤\frac{0.1\ \times \ V_r}{100}\)

(2ⁿ - 1) ≥ 1000

2ⁿ  ≥   1001

n = 10 satisfies the above equation.

Hence n = 10.

Concept:

Resolution: It is defined as the smallest change in the analog output voltage corresponding to a change of one bit in the digital output.

The percentage resolution of an n-bit DAC is \(= \frac{1}{{{2^n} - 1}} \times 10\)

The resolution of an n-bit DAC with a range of output voltage from 0 to V is given by \(= \frac{{{V_{FS}}}}{{{2^n} - 1}}volts\)

The output of DAC for a given decimal equivalent of binary input is

\({V_{DAC}} = \frac{{{V_{FS}}}}{{{2^n} - 1}} \times \) (Decimal equivalent of binary input)

Calculation:          

Number of bits (n) = 10

V_FS = 1.023 V

Resolution \(= \frac{{{V_{FS}}}}{{{2^n} - 1}} = \frac{{1.023}}{{{2^{10}} - 1}} = \frac{{1.023}}{{1023}} = 1\;mV\)

Resolution is 1 mV; it indicates that if the binary input changes from one level to the next immediate level then the output voltage will change by 1 mV.

Initial Input of DAC = 1010101000

Changed Input of DAC = 1010101100

Change in Input = 0000000100 = 4

Change in output = 4 × 1 mV = 4 mV

Concept:

The resolution  for n bit DAC is given by:

\(R=\frac{V_{max} - V_{min}}{2^n -1}\)

Calculation:

\(0.01=\frac{10-0}{2^n-1}\)

\(2^n -1 =1000\)

\(2^n=1001\)

n=10 

Minimum number of bits in the output is 10

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.

• The smallest change in the input signal that can be detected by an instrument is referred to as its "resolution."
• The term resolution describes the finest detail that a device or system can detect or measure.
• It is a key parameter for systems that deal with digital signals, as it is directly linked to the quality or level of detail of the output.