Alciatore D.G., Histand M.B. Introduction to Mechatronics and Measurement Systems

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many more video demonstrations of various music, sound, and vibration concepts,

see Internet Link 8.2.

8.2 QUANTIZING THEORY

Now we look at the process required to change a sampled analog voltage into digital

form. The process, called analog-to-digital conversion, conceptually involves two

steps: quantizing and coding. Quantizing is defined as the transformation of a con-

tinuous analog input into a set of discrete output states. Coding is the assignment of

a digital code word or number to each output state. Figure 8.3 illustrates how a con-

tinuous voltage range is divided into discrete output states, each of which is assigned

a unique code. Each output state covers a subrange of the overall voltage range. The

stair-step signal represents the states of a digital signal generated by sampling a lin-

ear ramp of an analog signal occurring over the voltage range shown.

An analog-to-digital converter is an electronic device that converts an analog

voltage to a digital code. The output of the A/D converter can be directly interfaced

to digital devices such as microcontrollers and computers. The resolution of an A/D

converter is the number of bits used to digitally approximate the analog value of the

input. The number of possible states N is equal to the number of bit combinations

that can be output from the converter:

N = 2

(8.3)

where n is the number of bits. For the example illustrated in Figure 8.3 , the 3-bit

device has 2

or 8 output states as listed in the first column. The output states are

Lin

8.2 Music,

sound, and

vibration

demonstrations

Figure 8.3 Analog-to-digital conversion.

7 111

110

5 101

4 100

3 011

2 010

1 001

0 000

output

state

output

code

0.00 –

1.25

1.25 –

2.50

2.50 –

3.75

3.75 –

5.00

5.00 –

6.25

6.25 –

7.50

7.50 –

8.75

8.75 –

10.0

discretized analog voltage ranges

8.4 Data

acquisition

experiment

8.5 LabVIEW data

acquisition and

music sampling

8.6 Sampling rate

and its effect on

sound and music

(audio only)

8.7 Music

sampling rate and

resolution effects

Video Demo

8.2 Quantizing Theory 351

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352 CHAPTER 8 Data Acquisition

usually numbered consecutively from 0 to ( N ⫺ 1). The corresponding code word for

each output state is listed in the second column. Most commercial A/D converters are

8-, 10-, or 12-bit devices that resolve 256, 1024, and 4096 output states, respectively.

The number of analog decision points that occur in the process of quantizing is

( N ⫺ 1). In Figure 8.3 , the decision points occur at 1.25 V, 2.50 V, . . . , and 8.75 V.

The analog quantization size Q, sometimes called the code width, is defined as the

full-scale range of the A/D converter divided by the number of output states:

Q = (V

max

– V

min

) ⁄ N

(8.4)

It is a measure of the analog change that can be resolved by the converter. Although

the term resolution is defined as the number of output bits from an A/D converter,

sometimes it is used to refer to the analog quantization size. For our example, the

analog quantization size is 10/8 V ⫽ 1.25 V. This means that the amplitude of the

digitized signal has an error of at most 1.25 V. Therefore, the A/D converter can only

resolve a voltage to within 1.25 V of the exact analog voltage.

Video Demo 8.7 demonstrates how both sampling rate and A/D resolution affect

the fidelity of digitized music.

8.7 Music

sampling rate and

resolution effects

Video Demo

■ CLASS DISCUSSION ITEM 8.3

Laboratory A/D Conversion

Why is a 12-bit A/D converter satisfactory for most laboratory measurements?

8.3 ANALOG-TO-DIGITAL CONVERSION

8.3.1 Introduction

To properly acquire an analog voltage signal for digital processing, the following

components must be properly selected and applied in this sequence:

1 . buffer amplifier

2 . low-pass filter

3 . sample and hold amplifier

4 . analog-to-digital converter

5 . computer

The components required for A/D conversion along with an illustration of their

respective outputs are shown in Figure 8.4 . The buffer amplifier isolates the out-

put from the input (i.e., it draws negligible current and power from the input) and

provides a signal in a range close to but not exceeding the full input voltage range

of the A/D converter. The low-pass filter is necessary to remove any undesirable

high- frequency components in the signal that could produce aliasing. The cutoff fre-

quency of the low-pass filter should be no greater than 1/2 the sampling rate. The

sample and hold amplifier (see Section 5.12) maintains a fixed input value (from

an instantaneous sample) during the short conversion time of the A/D converter.

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The converter should have a resolution and analog quantization size appropriate to

the system and signal. The computer must be properly interfaced to the A/D con-

verter system to store and process the data. The computer must also have sufficient

memory and permanent storage media to hold all of the data.

The A/D system components described above can be found packaged in a vari-

ety of commercial products called data acquisition (DAC or DAQ) cards or modules.

As shown in Figure 8.5 , DAC products are available in a variety of form factors

including PC and instrument panel plug-in cards, laptop computer PCMCIA cards,

and stand-alone external units with standard interfaces (e.g., USB). Internet Link 8.3

provides links to vendors and sources of information for various products available

on the market.

Data acquisition cards and modules usually support various high-level lan-

guage interfaces (e.g., C ⫹ ⫹ , Visual Basic, FORTRAN) that give easy access to the

product’s features. Easy-to-use software applications are also available to provide

graphical interfaces to program a DAC module to acquire, process, and output sig-

nals. National Instruments’ LabVIEW software is an example (see Internet Link 8.4

and Section 8.5 for more information). With a commercial DAC system, interfac-

ing signals between a mechatronic system or laboratory experiment and a desktop

computer is a simple matter of calling a function from a custom-written program or

clicking and dragging icons in a graphical interface. Lab Exercise 12 explores the

basics of how to use the LabVIEW software to sample and store an external voltage

signal, including sampled music. Video Demo 8.5 demonstrates all of the steps in the

process, and Section 8.6 explores the details.

In addition to A/D converters, commercial DAC systems usually provide

other input and output functionality including binary (TTL-compatible) I/O, D/A

buffer

amplifier

sample

& hold

low-pass filter

0011 0010 0100

0011 0010 1011

0011 0011 1100

0011 0101 0111

: : :

A/D

converter

computer/

memory

sensor

Figure 8.4 Components used in A/D conversion.

Lin

8.3 Data

acquisition online

resources and

vendors

Lin

8.4 National

Instruments’

LabVIEW software

Figure 8.5 Various form factors of data acquisition products. ( Courtesy of National

Instruments, Austin, TX )

8.3 Analog-to-Digital Conversion 353

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Lab Exercise

Lab 12 Data

acquisition

354 CHAPTER 8 Data Acquisition

converters, counters and timers, and signal conditioning circuitry. Important features

when selecting a DAC system include the A/D and D/A resolutions (the number of

bits used) and the maximum sampling rate supported. These features are key to accu-

racy and reliability in computer monitoring and control applications. An example

data acquisition and control card that can be inserted into a PC motherboard slot is

shown in Figure 8.6 . Its internal architecture is illustrated in Figure 8.7 . The card’s

functions include two 12-bit D/A converters, one 12-bit A/D converter, 24 lines of

digital I/O, and three 16-bit counter-timers.

The process of analog-to-digital conversion requires a small but finite interval

of time that must be taken into consideration when assessing the accuracy of the

results. The conversion time, also called settling time, depends on the design of the

converter, the method used for conversion, and the speed of the components used in

the electronic design. Because analog signals change continuously, the uncertainty

about when in the sample time window the conversion occurs causes corresponding

uncertainty in the digital value. This is of particular concern if there is no sample and

hold amplifier on the A/D input. The term aperture time refers to the duration of the

time window and is associated with any error in the digital output due to changes in

the input during this time. The relationship between the aperture time and the uncer-

tainty in the input amplitude is shown in Figure 8.8 . During the aperture time Δ T

the input signal changes by Δ V, where

ΔV

dVt()

-------------

ΔT

≈

(8.5)

Sampling at or above the Nyquist frequency will yield the correct frequency

components in a signal. However, to also obtain accurate amplitude resolution, we

must have an A/D converter with a sufficiently small aperture time. It is often in the

nanosecond range for 10- and 12-bit resolution.

8.5 LabVIEW data

acquisition and

music sampling

Video Demo

Figure 8.6 Example data acquisition and control card. (Courtesy of National

Instruments, Austin, TX)

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Figure 8.7 Example data acquisition and control card architecture. (Courtesy of

National Instruments, Austin, TX)

ΔT

ΔV

V(t)

Figure 8.8 A/D conversion aperture time.

Aperture Time

EXAMPLE 8.2

Consider a sinusoidal signal A sin( ␻ t ) as an input to an A/D converter. The time rate of

change of the signal A ␻ cos( ␻ t ) has a maximum value of A ␻ . Using Equation 8.5 , the maxi-

mum change in the input voltage during an aperture time Δ T

ΔVAωΔT

To eliminate uncertainty in the value of the digital output, we need to ensure that Δ V is less

than the analog quantization size:

ΔV

------ -

(continued )

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356 CHAPTER 8 Data Acquisition

8.3.2 Analog-to-Digital Converters

A/D converters are designed based on a number of different principles: successive

approximation, flash or parallel encoding, single-slope and dual-slope integration,

switched capacitor, and delta sigma. We consider the first two because they occur

most often in commercial designs. The successive approximation A/D converter

is very widely used because it is relatively fast and cheap. As shown in Figure 8.9 ,

it uses a D/A converter in a feedback loop. D/A converters are described in the

next section. When the start signal is applied, the sample and hold (S&H) amplifier

latches the analog input. Then the control unit begins an iterative process, where the

digital value is approximated, converted to an analog value with the D/A converter,

and compared to the analog input with the comparator. When the D/A output equals

the analog input, the end signal is set by the control unit, and the correct digital out-

put is available at the output.

where 2 A is the total voltage range and N is the number of output states. At the limit of this

constraint,

ΔV

------ -

Therefore, the required aperture time, if a sample and hold amplifier is not being used, is

ΔT

ΔV

Aω

--------

Nω

--------

Consider, for example, converting a signal using 10-bit resolution, which provides

(1024) output states. If the signal were speech (from a microphone) with a bandwidth of

10 kHz,

Δ T

would have to be less than

ΔT

Nω

--------

1024 2π 10,000⋅()

---------------------------------------------

3.2 10

8–

× 32 nsec== = =

This is a very short required aperture time compared to the required minimum sample period

given by 1/[2(10 kHz)] ⫽ 50 ␮ sec. Even for this low-resolution converter, the required aper-

ture time (32 nsec) is much smaller than the required sample period (50,000 nsec).

(continued )

analog

input

D/A

digital

output

S&H

comparator

successive

approximation

latch

control unit

start

signal

end

signal

Figure 8.9 Successive approximation A/D converter.

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If n is the resolution of the A/D converter, it takes n steps to complete the con-

version. More specifically, the input is compared to combinations of binary fractions

(1/2, 1/4, 1/8, . . . , 1/2

) of the full-scale (FS) value of the A/D converter. The control

unit first turns on the most significant bit (MSB) of the register, leaving all lesser bits

at 0, and the comparator tests the DAC output against the analog input. If the analog

input exceeds the DAC output, the MSB is left on (high); otherwise, it is reset to 0.

This procedure is then applied to the next lesser significant bit and the comparison

is made again. After n comparisons have occurred, the converter is down to the least

significant bit (LSB). The output of the DAC then represents the best digital approxi-

mation to the analog input. When the process terminates, the control unit sets the end

signal signifying the end of the conversion.

As an example, a 4-bit successive approximation procedure is illustrated graphi-

cally in Figure 8.10 . The MSB is 1/2 FS, which in this case is greater than the signal;

therefore, the bit is turned off. The second bit is 1/4 FS and is less than the signal,

so it is left on. The third bit gives 1/4 ⫹ 1/8 of FS, which is still less than the analog

signal, so the third bit is left on. The fourth provides 1/4 ⫹ 1/8 ⫹ 1/16 of FS and is

greater than the signal, so the fourth bit is turned off and the conversion is complete.

The digital result is 0110. Higher resolution would produce a more accurate value.

An n -bit successive approximation A/D converter has a conversion time of n Δ T,

where Δ T is the cycle time for the D/A converter and control unit. Typical conversion

times for 8-, 10-, and 12-bit successive approximation A/D converters range from

1 to 100 ␮ s.

The fastest type of A/D converter is known as a flash converter. As Figure 8.11

illustrates, it consists of a bank of input comparators acting in parallel to identify

the signal level. The output of the latches is in a coded form easily converted to the

required binary output with combinational logic. The flash converter illustrated in

Figure 8.11 is a 2-bit converter having a resolution of four output states. Table 8.1

input

signal

011

1/2 FS

bit

3 (MSB) 0 (LSB)

1/4 FS

(1/4 + 1/8) FS

Figure 8.10 4-bit successive approximation A/D conversion.

■ CLASS DISCUSSION ITEM 8.4

Selecting an A/D Converter

What are some reasons why a designer might select a 10-bit A/D converter instead

of a 12-bit or higher resolution converter?

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358 CHAPTER 8 Data Acquisition

lists the comparator output codes and corresponding binary outputs for each of the

states, assuming an input voltage range of 0 to 4 V. The voltage range is set by the

min

and V

max

supply voltages shown in Figure 8.11 (0 V and 4 V in this example).

The code converter is a simple combinational logic circuit. For the 2-bit converter,

the relationships between the code bits G

and the binary bits B

( Question 8.11 ) are

+⋅=

(8.6)

(8.7)

Adding more resolution is a simple matter of adding more resistors, comparators,

and latches. The combinational logic code converter would also be different. Unlike

with the successive approximation converter, adding resolution does not increase the

time required for a conversion.

Several analog signals can be digitized by a single A/D converter if the analog

signals are multiplexed at the input to the A/D converter. An analog multiplexer

simply switches among several analog inputs using transistors or relays and control

signals. This can significantly reduce the cost of a system’s design. In addition to

cost, other parameters important in selecting an A/D converter are the input voltage

range, output resolution, and conversion time.

–

code

converter

max

min

trigger

comparators

latches

Figure 8.11 A/D flash converter.

Table 8.1 2-bit flash converter output

State Code (G

) Binary (B

) Voltage Range

0 000 00 0–1

1 001 01 1–2

2 011 10 2–3

3 111 11 3–4

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8.4 DIGITAL-TO-ANALOG CONVERSION

Often we need to reverse the process of A/D conversion by changing a digital value

to an analog voltage. This is called digital-to-analog (D/A) conversion. A D/A con-

verter allows a computer or other digital device to interface with external analog

circuits and devices.

The simplest type of D/A converter is a resistor ladder network connected to an

inverting summer op amp circuit as shown in Figure 8.12 . This particular converter is a

4-bit R -2 R resistor ladder network, which requires only two precision resistance values

( R and 2 R ). The digital input to the DAC is a 4-bit binary number represented by bits

, b

, and b

, where b

is the least significant bit and b

is the most significant bit.

Each bit in the circuit controls a switch between ground and the inverting input of the

op amp. To understand how the analog output voltage V

out

is related to the input binary

number, we can analyze the four different input combinations 0001, 0010, 0100, and

1000 and apply the principle of superposition for an arbitrary 4-bit binary number.

If the binary number is 0001, the b

switch is connected to the op amp, and the

other bit switches are grounded. The resulting circuit is as shown in Figure 8.13 .

Because the noninverting input of the op amp is grounded, the inverting input is also

at ground. The equivalent resistance between node V

and ground is R, which is the

parallel combination of two 2 R values. Therefore, V

is the result of voltage division

of V

across two series resistors of equal value R:

-- -

(8.8)

–

2R 2R 2R

MSB LSB

out

RRR

Figure 8.12 4-bit resistor ladder D/A converter.

Figure 8.13 4-bit resistor ladder D/A with digital input 0001.

–

out

8.4 Digital-to-Analog Conversion 359

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360 CHAPTER 8 Data Acquisition

Similarly, we can show that

andV

-- -

= V

-- -

(8.9)

Therefore,

-- -

(8.10)

is the input to the inverting amplifier circuit, which has a gain of

------ -

–

-- -

–=

( 8.11)

Therefore, the analog output voltage corresponding to the binary input 0001 is

out

----- -

–=

(8.12)

Similarly, we can show ( Questions 8.16 through 8.18 ) that, for the input 0010,

out

-- -

–=

(8.13)

and for the input 0100,

out

-- -

–=

(8.14)

and for the input 1000,

out

-- -

–=

(8.15)

The output for any combination of bits comprising the input binary number can now

be found using the principle of superposition:

out

+++=

(8.16)

If V

is 10 V, the output ranges from 0 V to ( ⫺ 15/16)10 V for the 4-bit binary

input, which has 16 values ranging from 0000 (0) to 1111 (15). A negative ref-

erence voltage V

can be used to produce a positive output voltage range. Either

case yields a unipolar output, which is either positive or negative but not both.

A bipolar output, which ranges over negative and positive values, can be produced

by replacing all ground references in the circuit with a reference voltage whose sign

is opposite to V

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