Data Converters MCQ Quiz - Objective Question with Answer for Data Converters - Download Free PDF

Sigma Delta Converters

Definition: Sigma Delta converters, also known as Delta-Sigma converters, are a class of analog-to-digital converters (ADC) that utilize oversampling techniques and noise shaping to achieve high resolution and accuracy. These converters are widely used in applications requiring precise measurements, such as audio processing, instrumentation, and sensor data acquisition.

Correct Option Analysis:

The correct option is:

Option 4: Pipe lining

Pipe lining is not a term associated with Sigma Delta converters. This term is typically used in the context of digital circuit design and computer architecture, where it refers to the process of dividing a computational task into smaller stages that are executed in parallel in a pipeline. In the case of Sigma Delta converters, the focus is on oversampling, decimation filtering, and noise shaping, which are integral to their operation. Pipe lining is unrelated to the principles or mechanisms of Sigma Delta converters.

The correct answer is option 1

Explanation:

1) Flash ADC

• Fastest ADC type.

• Uses a parallel bank of comparators to convert input voltage in a single step.

• Trade-off: very fast but limited resolution (typically 8 bits or less), because the number of comparators doubles with each bit of resolution.

2) Sigma-Delta ADC

• Very high resolution, but slow, because it oversamples and averages.

3) Successive Approximation ADC (SAR)

• Medium speed, medium resolution.

• Not as fast as Flash ADC, but higher resolution.

4) Dual-Slope ADC

• Very high resolution, but slowest.

• Typically used in digital multimeters.

The significance of 'resolution' in a Digital-to-Analog Converter (DAC) is that it defines the smallest change in analogue output that can be produced.

• Resolution in a DAC refers to the number of distinct analog output values that the DAC can produce. It is typically expressed in bits (e.g., 8-bit, 12-bit, 16-bit DAC).

• A higher resolution (more bits) means the DAC can produce a greater number of discrete analog output levels, resulting in smaller steps between successive levels.

• This "smallest change" in the analog output is often called the least significant bit (LSB) size or step size. It directly determines the precision with which the DAC can reproduce an analog signal from a digital input.

The other options are incorrect:

• Resolution does not directly set the power consumption.

• Resolution does not determine the maximum speed (conversion rate or settling time) of the DAC.

• Resolution does not limit the maximum input voltage; rather, the maximum analog output voltage (full scale) is determined by the reference voltage and the DAC's design, and the digital input code determines the fraction of that full scale.

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.

Explanation:

Analysis of the 4-bit Unipolar DAC and Comparator System:

The given problem involves a 4-bit unipolar DAC (Digital-to-Analog Converter) with a 10V reference voltage and a comparator whose threshold voltage is set at 7.5V. The DAC counts from 0 to 15 (since it is a 4-bit DAC, it can represent 2⁴ = 16 levels). The comparator monitors the output of the DAC and determines when it crosses the threshold voltage.

It is observed that when the DAC count reaches 8, the comparator erroneously indicates threshold crossing. Upon analysis, it is found that one of the input bits of the DAC is stuck at '1'. We need to determine which bit is stuck at '1'.

Step-by-Step Solution:

1. Understanding the DAC Operation:

A 4-bit unipolar DAC converts a digital value (binary) into an analog voltage. The output voltage is given by:

V_out = (Digital Value / Maximum Digital Value) × Reference Voltage

Since the DAC is 4-bit, the maximum digital value is 15 (binary 1111). The reference voltage is 10V. Therefore:

V_out = (Digital Value / 15) × 10

For each binary value, the corresponding analog voltage is calculated as follows:

2. Comparator Threshold Analysis:

The comparator is set to a threshold voltage of 7.5V. It should ideally indicate threshold crossing when the DAC output voltage exceeds 7.5V. From the table above, this happens when the binary value exceeds 1011 (decimal 11), as for binary 1100 (decimal 12), V_out becomes 8V.

However, the problem states that the comparator erroneously indicates threshold crossing at binary 1000 (decimal 8). This suggests that the DAC is producing an incorrect output voltage due to one of its input bits being stuck at '1'.

3. Identifying the Stuck-at-'1' Bit:

To determine which bit is stuck at '1', we analyze the effect of each bit being stuck at '1' on the output voltage:

• B₃ (MSB): If the most significant bit (B₃) is stuck at '1', the output voltage would be significantly higher than expected for lower binary values, as B₃ contributes the largest weight (8 × Reference Voltage / 15). For binary 1000 (decimal 8), the output voltage would be correct, so B₃ is not the stuck bit.
• B₂: If B₂ is stuck at '1', it adds an additional weight (4 × Reference Voltage / 15) to the output voltage. For binary 1000 (decimal 8), the output voltage would incorrectly include this additional weight, resulting in an erroneous threshold crossing at the comparator. This matches the problem description, so B₂ is the stuck bit.
• B₁: If B₁ is stuck at '1', it adds a smaller weight (2 × Reference Voltage / 15). For binary 1000 (decimal 8), the output voltage would not exceed the threshold, so B₁ is not the stuck bit.
• B₀ (LSB): If the least significant bit (B₀) is stuck at '1', it adds the smallest weight (Reference Voltage / 15). This would not cause the erroneous threshold crossing at binary 1000 (decimal 8), so B₀ is not the stuck bit.

Correct Answer: The stuck-at-'1' bit is B₂.

Important Information

To further understand the analysis, let’s evaluate the other options:

• B₀ (LSB): The least significant bit contributes the smallest weight to the output voltage. If it is stuck at '1', the error in output voltage would be minimal and insufficient to cause the threshold crossing at binary 1000 (decimal 8).
• B₁: The second least significant bit contributes a slightly larger weight than B₀. If it is stuck at '1', the output voltage would still not exceed the threshold at binary 1000 (decimal 8).
• B₃ (MSB): The most significant bit contributes the largest weight to the output voltage. If it is stuck at '1', the output voltage would be significantly higher than expected for lower binary values, but it would not match the observed erroneous behavior at binary 1000 (decimal 8).

Conclusion:

By analyzing the effect of each bit being stuck at '1', we identify that B₂ is the stuck bit, as its incorrect contribution to the output voltage causes the comparator to erroneously indicate threshold crossing at binary 1000 (decimal 8). This analysis highlights the importance of understanding DAC operation and bit weighting in digital systems.

Concept:

For a ladder-type D/A Converter:

Output Voltage (V0) = Resolution × Decimal Equivalent of binary input.

Where Resolution is given by:

\(Resolution=\frac{{{V}_{ref}}}{{{2}^{n}}}\)

Application:

Given n = 5 and the Digital input = 11010

∵ The Resolution will be:

\(R=\frac{{{V}_{ret}}}{{{2}^{n}}}=\frac{10}{{{2}^{5}}}=~0.3125\)

Since the decimal Equivalent of 11010 = 26

So, V0 = 26 × 0.3125

V0 = 8.125 V

Note: If the full-scale voltage is given, then:

Resolution \(=\frac{{{V}_{fs}}}{{{2}^{n}}-1}\)

For n-bit conversion, the conversion time for different ADC are:

Counter type ADC: (2n – 1) Tclk

Successive approx. time ADC: n Tclk

Flash type ADC: Tclk

Dual slope ADC: (2n+1 – 1) Tclk

The successive approximation A/D converter has shorter conversion time compared to the counter ramp A/D converter.

Important points:

• Counter type ADC and successive approximate ADC uses DAC
• Counter type ADC uses linear search and successive approximation type ADC uses binary search
• Ring counter is used in successive approximation type ADC
• Flash type ADC is fastest ADC
• Flash type ADC requires no counter
• For an n-bit ADC, flash type ADC requires (2n – 1) comparators
• Dual slope ADC is most accurate

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

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:

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

Calculation:

Number of bits (n) = 8

Resolution \( = \frac{{1}}{{{2^8} - 1}} = \frac{{1}}{{255}}\)

No of comparators required for n bit flash type ADC is (2ⁿ - 1)

Given that, n = 12

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:

For a ladder-type D/A Converter:

Output Voltage (V0) = Resolution × Decimal Equivalent of binary input.

Where Resolution is given by:

\(Resolution=\frac{{{V}_{fs}}}{{{2}^{n}}-1}\)

Where, V_fs = Full scale voltage or maximum voltage

Application:

Given n = 6 and the Digital input = 101001

∵ The Resolution will be:

\(R=\frac{{{V}_{ret}}}{{{2}^{n}-1}}=\frac{10}{{{2}^{6}-1}}=~0.1587\)

Since the decimal Equivalent of 101001 = 41

So, V0 = 41 × 0.1587

V0 = 6.5067 V

V₀ ≈ 6.5 V

Note:  If the reference voltage is given, then:

\(Resolution=\frac{{{V}_{ref}}}{{{2}^{n}}}\)

Comparator:

• A comparator is a circuit which compares a signal voltage applied at one input of an Op-Amp with a known reference voltage at the other input.
• It produces either a high or a low output voltage, depending on which input is higher.
• Since, a comparator output has two voltage levels, either high or low. (1 or 0) So it acts as an Analog to Digital convertor.
• it is not linearly proportional to the input voltage.

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The common Analog to digital converters are:

Ramp-type:

• The principle of Ramp-type DVM is based on the measurement of the time it takes for a linear ramp voltage to rise from 0 V to the level of the input voltage (or) to decrease from the level of the input voltage to zero.
• This type of Analogue to Digital Converter is very slow (but cheap and simple).
• It is ideal for data that changes fairly slowly such as vehicle or aircraft control systems.

• Audio signals are slow enough to be converted.

Dual-slope converter:

• In the dual-slope technique, an integrator is used to integrate an accurate voltage reference for a fixed period of time. The same integrator is then used to integrate with the reverse slope, the input voltage, and the time required to return to the starting voltage is measured.
• The automatic zero correction function is performed before each conversion so that changes in the offset voltages & current will be compensated.

Successive Approximation:

• The basic principle is that binary regression, in which analog input is compared with DAC reference voltage which is repeatedly divided in half.
• A successive approximation A/D converter consists of a comparator, a successive approximation register (SAR), output latches, and a D/A converter.
• It is capable of high speed and is reliable.

For n-bit conversion, the conversion time for different ADC are:

Counter type ADC: (2n – 1) Tclk

Successive approx. time ADC: n Tclk

Flash type ADC: Tclk

(Flash Type ADC is also known as Parallel comparator type)

Dual slope ADC: (2n+1 – 1) Tclk

The fastest type of Analog to Digital converter is the Flash type / Parallel comparator type.

Important points:

• Counter type ADC and successive approximate ADC uses DAC
• Counter type ADC uses linear search and successive approximation type ADC uses binary search
• Ring counter is used in successive approximation type ADC
• Flash type ADC is the fastest ADC
• Flash type ADC requires no counter
• For an n-bit ADC, flash type ADC requires (2n – 1) comparators
• Dual slope ADC is the most accurate.

The correct answer is option 3):(15)

Concept:

Flash Type ADC:

1) It is the fastest ADC among all the ADC types.

2) An n-bit flash type ADC requires: 2ⁿ -1 comparators, 2n resistors, and one 2ⁿ × n priority encoder.

Analysis: Number of bits(n) = 4

Number of comparators required = 2⁴ -1 = 15