Digital Electronics: CT 304N

Unit–7
D/A and A/D Converters
Dr. Anand J. Patel
 Index
• Introduction

• D/A converters:
o the binary weighted resistor network
o the R-2R ladder network.

• A/D converters:
o Counting
o Tracking
o Flash
o Successive approximation
o Dual slope

2
 Introduction
• Analog output is typical of most transducers and
sensors. In order to use the power of digital
electronics with the real world, one must convert from
analog to digital and vice versa.

• Examples of A/D Applications

– Microphones - take your voice varying pressure waves in

the air and convert them into varying electrical signals
– Thermocouple – temperature measuring device converts
thermal energy to electric energy
– Digital Multimeters

3
 The need for Data Converters

PRE-PROCESSING DIGITAL POST-PROCESSING

ANALOG SIGNAL ANALOG
(Filtering and analog PROCESSOR (Digital to analog OUTPUT
(Speech, Images, to digital conversion) (Microprocessor) conversion and SIGNAL
Sensors, Radar, etc.) filtering)
(Actuators, antennas, etc.)

CONTROL

ANALOG A/D DIGITAL D/A ANALOG

In many applications, performance is critically

limited by the A/D and D/A performance
 A/D and D/A converters
Vmax = 7.5V 1111 4 4
7.0V 1110
6.5V 1101 3 3

analog output (V)

analog input (V)
6.0V 1100
5.5V 1011
2 2
5.0V 1010
4.5V 1001
4.0V 1 1
1000
3.5V 0111
3.0V 0110 time time
t1 t2 t3 t4 t1 t2 t3 t4
2.5V 0101
2.0V 0100 0100 0110 0110 0101 0100 1000 0110 0101
1.5V 0011 Digital output Digital input
1.0V 0010
0.5V 0001
Vmin = 0V 0000

proportionality analog to digital digital to analog

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 Relationship Between Analog and Digital Values

Vmax = 7.5V
7.0V
1111
1110
An ideal A/D converts an analog voltage to a
6.5V
6.0V
1101 linearly proportional digital representation.
1100
5.5V 1011

The A/D has two sides: analog and digital.

5.0V 1010
4.5V 1001
4.0V 1000
3.5V 0111
3.0V 0110
2.5V 0101
2.0V 0100 Analog Digital
1.5V 0011 Vmin 0..000
1.0V
0.5V
0010 : :
0001
Vmin = 0V 0000 Vmax 1..111

proportionality
• If input voltage > Vmax the digital output is 1..111.
• Similarly if input < Vmin the digital output is 0..000.
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 Relationship Between Analog and Digital Values

Vmax = 7.5V
7.0V
1111
1110
Let the A/D converter has n-bit digital output
6.5V
6.0V
1101 and let A be Analog Value and D be the
1100
5.5V 1011 equivalent Digital Number.
5.0V 1010
4.5V 1001
4.0V
3.5V
1000
0111
Then,
3.0V 0110
2.5V 0101
2.0V 0100 Analog Digital
1.5V 0011 Vmin 0..000
1.0V
0.5V
0010
A D
0001
Vmin = 0V 0000 Vmax 1..111

proportionality

A  Vmin
D (2 n  1)
Vmax  Vmin
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 Some definitions
Offset: minimum analog value Vmin

Span (or Range): is the difference between maximum and minimum analog values Vmax - Vmin

Step Size (or Resolution, Q): smallest analog change resulting from changing one bit in the digital number,
or the analog difference between two consecutive digital numbers:
Vmax  Vmin
Q
2n  1
stepsize
% Resolution :  x100%
Fullscale

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 DAC SPECIFICATIONS
• D/A converters are available with wide range of specifications
specified by manufacturer. Some of the important
specifications are Resolution, Accuracy, linearity, monotonicity,
conversion time, settling time and stability.

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 DAC SPECIFICATIONS
• Resolution: The resolution of a DAC is the reciprocal of the number of
discrete steps in the DAC. This is independent of the number of
bits. Resolution can also be shown as the number of bits converted.
• Accuracy: This is the comparison of the actual output of a DAC with the
expected output. This is expressed as a percentage of a full scale, or
maximum output voltage.
• Linearity: A linear error is a deviation from the ideal straight-line output of
a DAC.
• Monotonicity: A DAC is monotonic if it does not take any reverse steps
then it is sequenced over its entire range of input bits.
• Setting Time: This is the time it takes a DAC to settle within +/- 1/2 of a
least significant bit of its final value when a change occurs in the input
code.
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 Resolution
• Resolution:
• Resolution is defined as the number of different analog output voltage levels
that can be provided by a DAC.
• Or alternatively resolution is defined as the ratio of a change in output voltage
resulting for a change of 1 LSB at the digital input. Simply, resolution is the value
of LSB.
• Example:
• Resolution for an 8 – bit DAC for example is said to have:
• 8 – bit resolution
• A resolution of 0.392 of full-Scale (1/255)
• A resolution of 1 part in 255.
• Thus resolution can be defined in many different ways.

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 Example
Given an 4-bit A/D converter having an analog input that
ranges form 0V to 7.5V. What is the resolution of this A/D
converter?

Answer:

Vmax  Vmin 7.5  0

Q  4  0.5V
2 1
n
2 1

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D/A Converter

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 Binary representation and bit weight
In an electronic circuit, a combination of high voltage (+5V) and
low voltage (0V) is usually used to represent a binary number.
For example, a binary number 1010 is represented by

Weighting 23 22 21 20
Binary Digit 1 0 1 0
State +5V 0V +5V 0V

D/As are electronic circuits that convert digital, (usually binary)

signals (for example, 1000100) to analog electrical quantities
(usually voltage) directly related to the digitally encoded input
number.
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 Types of D/A Converters

We will consider two types of D/A:

o the binary weighted resistor network

o the R-2R ladder network.

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The binary weighted resistor network
– Comprises of a register and resistor network
– Output of each bit of the register will be low (0V) or high (5V)
– Input resistance is inversely proportional to the binary weight
of each digit.
MSB R I1 Rf

4-bit register
2R I2 If

4R I3 -
S I

LSB 8R I4 +
Vo

Vo   I f R f  ( I1  I 2  I 3  I 4 ) R f 16
 D/A Example
In the previous D/A, calculate the output voltage for an input
code word 0110 if a logic 1 is 5V and a logic 0 is 0V, and
R = Rf = 1k.

Answer:

• I1 = I4 = 0
• I2 = 5V / 2R = 5 / 2KΩ = 2.5 mA
• I3 = 5V / 4R = 5 / 4KΩ = 1.25 mA

Vo = -If Rf
= -(I4 + I3+ I2+ I1)Rf
= -(2.5+1.25) x 1000
= -3.75 V
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 The binary weighted resistor network

• Very difficult to manufacture very accurate resistors over this

range.

• Seldom used for digital numbers having more than 6 bits.

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 The R-2R Ladder Resistor Network
• Only two resistance values R and 2R are used.
• The principle of the network is based on Kirchhoff's current rule.
• Note that the network of resistors to the right of each node has an
equivalent resistance of 2R.

I I/2 R I/4 R I/8

Bit Current
2 I/2
1 I/4
2R 2R 2R 2R
0 I/8

I1=I/2 I2=I/4 I3=I/8

bit 2 bit 1 bit 0

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The R-2R Ladder Resistor Network

The state of the

bits is used to
switch a voltage
source

 I I I
Vo   R f  b2  b1  b0 
 2 4 8 20
Analog to Digital Converter

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 A/D converter

• Converts analog signals into binary words

22
 The Sample & Hold (S/H)
To measure an AC voltage at a particular instant in time, it is
necessary to sample the waveform with a ‘sample and hold’
(S/H) circuit.

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 Types of A/D Converters

There several type of A/D converters. Here, we

will consider the following types:
1. Counting type
2. Tracking type
3. Parallel or Flash
4. Successive Approximation
5. Dual slope

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 ADC Specifications
• Like DAC, ADCs are also having many important specifications.
Some of them are Resolution, Quantization error, Conversion
time, Analog error, Linearity error, DNL error, INL error & Input
voltage range.

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 ADC Specifications
• Resolution:
• The resolution refers to the finest minimum change in the signal
which is accepted for conversion, and it is decided with respect to
number of bits. It is given as 1/2n, where n is the number of bits in
the digital output word. As it is clear, that the resolution can be
improved by increasing the number of bits or the number of bits
representing the given analog input voltage.
• Resolution can also be defined as the ratio of change in the value of
input voltage Vi, needed to change the digital output by 1 LSB.
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 ADC Specifications
• Quantization error:
• If the binary output bit combination is such that for all the
values of input voltage Vi between any two voltage levels,
there is a unavoidable uncertainty about the exact value of Vi
when the output is a particular binary combination. This
uncertainty is termed as quantization error. Its value is ± (1/2)
LSB.

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 ADC Specifications
• Analog Error:
• An error occurring due to the variations in DC switching point of the comparator, resistors, reference
voltage source, ripples and noises introduced by the circuit components is termed as Analog error.
• Linearity Error:
• It is defined as the measure of variation in voltage step size. It indicates the difference between the
transitions for a minimum step of input voltage change. This is normally specified as fraction of LSB.
• DNL (Differential Non-Linearity) Error:
• The analog input levels that trigger any two successive output codes should differ by 1 LSB. Any
deviation from this 1 LSB value is called as DNL error.
• INL (Integral Non-Linearity Error:
• The deviation of characteristics of an ADC due to missing codes causes INL error. The maximum
deviation of the code from its ideal value after nulling the offset and gain errors is called as Integral
Non-Linearity Error.
• Input Voltage Range:
• It is the range of voltage that an A/D converter can accept as its input without causing any overflow
in its digital output.

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 Counting ADC
Comparator START

Vin Control Logic

clock

Counter
D/A

Digital Output

o When START is received, control logic sets counter to 0, and turns on Clock
sending regular pulses to the counter.
o As the Clock sends regular pulses to the counter, the counter outputs a digital
signal to the Digital-to-Analog converter
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• As the counter counts, its output to the D/A generates a staircase ramp to
the comparator.

Vin

Note that the conversion

V’in time depends on the size
of the input signal

Conversion time Conv.time

• As the ramp voltage increases to the comparator, it rises closer and closer to
Vin. When the ramp voltage exceeds Vin , the comparator output shifts which
signals the control logic to turn off the clock. With the clock off, the counter
reading is proportional to Vin.
• With a counting type A/D, if the signal is varying rapidly, the counter must
count up and reset before each cycle can begin, making it difficult to follow
the signal.
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Tracking type ADC

31
Tracking type ADC
• Tracking uses an up-down counter and is faster that the counting type ADC
because the counter is not reset after each sample.
• It tracks analog input hence the name tracking.
• In order for this to work, the output reference voltage should be lower than the
analog input.
• When the comparator output is high, the counter is in counting up mode of
binary numbers.
• As a result, this increases stair step reference voltage out until the ramp reaches
the input voltage amount.
• When reference voltage equals the input voltage the comparators output is
switched to low mode and starts counting down.
• If the analog input is decreasing the counter will continue to back down to track
input. If the analog input is increasing the counter will go down one count and
resume counting up to follow the curve or until the comparison occurs.
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 Flash Converters ADC

• If very high speed conversions are needed,

e.g. video conversions, the most commonly
used converter is a Flash Converter.

• While such converters are extremely fast,

they are also very costly compared to other
types.

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Flash Converters

The resistor network

is a precision voltage
divider, dividing Vref
into equal voltage
increments to one
input of the
comparator.
The other comparator
input is the input
voltage.
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 Flash Converters
• The encoder logic implements a truth table to convert the ladder of
inputs to the binary number output.

• The cost of this type of converter stems from the circuit complexity
since the number of comparators and resistors required increases
rapidly. The 3-bit example required 7 converters, 6-bits would
require 63, while an 8-bits converter would need 255 comparators
and equivalent precision resistors.
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 Successive approximation ADC

• This is the most common A/D used in the

laboratory environment.
• It is reasonably priced for large bit values, i.e. 10,
12.
• Its conversion times, typically ~ 10-20 s, are
adequate for most laboratory functions. Good
tradeoff between speed and cost
• Generates the digital output serially (one bit at a
time).
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 Successive approximation ADC

analog D/A Converter Vref

input

Digital
Output
Data
comparator
Successive clock
Approximation
Register
STRT

At the beginning, all bits from the SAR are set to zero, and
conversion begins by taking STRT line low.
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 Successive approximation ADC

analog D/A Converter Vref

input

Digital
Output
Data
comparator
Successive clock
Approximation
Register
STRT

First the logic in the SAR sets the MSB bit equal to 1
(+5 V). Remember that a 1 in bit 7 will be half of full
scale.
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 Successive approximation ADC

analog D/A Converter Vref

input

Digital
Output
Data
comparator
Successive clock
Approximation
Register
STRT

The output of the SAR feeds the D/A converter producing an

output compared to the analog input voltage. If the D/A output is
< Vin then the MSB is left at 1 and the next bit is then tested.
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 Successive approximation ADC

analog D/A Converter Vref

input

Digital
Output
Data
comparator
Successive clock
Approximation
Register
STRT

If the D/A output is > Vin then the MSB is set to 0 and
the next bit is set equal to 1.

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 Successive approximation ADC

• Successive bits are set and tested by comparing the

D/A output to the input Vin in an 8 step process (for
an 8-bit converter) that results in a 8-bit binary
output that represents the input voltage.

• Note that the successive approximation process

takes a fixed time - 8 clock cycles for the 8-bit
example.

41
 Example
Calculate the maximum conversion time of
(a)a 8-bit counting A/D and
(b) a successive approximation A/D,
if the clock rate is 2MHz.

Solution:
(a) For a 8-bit counting A/D, the maximum number of count is
nc = 28 = 256
Therefore, the maximum conversion time is

nc 256
Tc    128  10 6
s  128s
f 2  10 6
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 Example, continued

(b) For a 8-bit successive approximation A/D, the

conversion time is constant and equal to

n 8
Tc    4  10 6
s  4s
f 2  106

It is noted that the conversion speed of successive

approximation A/D is much faster than the counting type.

43
Dual slope ADC

44
Dual slope ADC
• Dual-Slope ADC operate on the principle of integrating the
unknown input and then comparing the integration times with
a reference cycle. The basic way is to use two slopes (dual) as
in this diagram:

45
Dual slope ADC
• This circuit operates by switching in the unknown input signal and then
integrating for full scale number counts.
• During this cycle the reference is switched in and if the reference is of
opposite polarity, the ramp will be driven back towards ground.
• The time that is takes for the ramp to again reach the comparator threshold
of ground will be directly proportional to the unknown input signal.
• Since the circuit uses the same time constant for the integrator, the
component tolerances will be the same for both the integration and
differentiation cycles.
• Therefore the errors will cancel except for the offset voltage that will be
additive during both the cycles.
• The main benefits of this type is the increased range, the increased accuracy
and resolution, and the increased speed. 46
 Accuracy of A/D Conversion
There are two ways to improve accuracy of A/D conversion:

• increasing the resolution (by increasing the number of bits e.g. 10-
bit, 12-bit, etc.) which improves the accuracy in measuring the
amplitude of the analog signal.

• increasing the sampling rate which increases the maximum

frequency that can be measured.

47
 Aliasing
• Occurs when the input signal is changing much faster
than the sample rate.

For example, a 2 kHz sine wave being sampled at 1.5

kHz would be reconstructed as a 500 Hz (the aliased
signal) sine wave.

Nyquist Rule:
• Use a sampling frequency at least twice as high as
the maximum frequency in the signal to avoid
aliasing.
48
The End

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