1. Lecture 9: D/A and A/D Converters

2. Objectives
Recognize the relationship between digital and analog values in D/A and A/D converters.
Identify two types of D/A:
the binary weighted resistor network
the R-2R ladder network.
Explain the operation of three types of A/D:
Counting
Flash
Successive approximation

3. Analog signals
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

4. A/D and D/A converters Vmax = 7.5V Vmin = proportionality 0V 1111 1110
0000
0010
0100
0110
1000
1010
1100
0001
0011
0101
0111
1001
1011
1101
0.5V
1.0V
1.5V
2.0V
2.5V
3.0V
3.5V
4.0V
4.5V
5.0V
5.5V
6.0V
6.5V
7.0V
analog to digital 4 3 2 1 t1 t2 t3 t4
0100
0110
0101
time
analog input (V)
Digital output
digital to analog 4 3 2 1
0100
1000
0110
0101 t1 t2 t3 t4
time
analog output (V)
Digital input
Vmin =
Embedded Systems Design: A Unified Hardware/Software Introduction, (c) 2000 Vahid/Givargis

5. Relationship Between Analog and Digital Values
An ideal A/D converts an analog voltage to a linearly proportional digital representation.
The A/D has two sides: analog and digital.
proportionality
Vmax = 7.5V 0V
1111
1110
0000
0010
0100
0110
1000
1010
1100
0001
0011
0101
0111
1001
1011
1101
0.5V
1.0V
1.5V
2.0V
2.5V
3.0V
3.5V
4.0V
4.5V
5.0V
5.5V
6.0V
6.5V
7.0V
Analog Digital
Vmin
: :
Vmax
Vmin =
If input voltage > Vmax the digital output is
Similarly if input < Vmin the digital output is

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

7. 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:

8. 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:

9. D/A Converter

10. 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
D/As are electronic circuits that convert digital, (usually binary) signals (for example, ) to analog electrical quantities (usually voltage) directly related to the digitally encoded input number.
Weighting 23 22 21 20
Binary Digit 1
State
+5V 0V

11. Types of D/A Converters
We will consider two types of D/A:
the binary weighted resistor network
the R-2R ladder network.

12. 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.

13. 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
= -( ) x 1000
= V

14. 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.

15. 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.
Bit Current
2 I/2
1 I/4
0 I/8
bit bit bit 0

16. The R-2R Ladder Resistor Network
The state of the bits is used to switch a voltage source

17. Analog to Digital Converter

18. A/D converter
Converts analog signals into binary words

19. 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.

20. Types of A/D Converters
There several type of A/D converters. Here, we will consider the following three types:
1. Counting type
2. Parallel or Flash
3. Successive Approximation

21. 1- Counting A/D
START
Comparator
Vin
Control Logic
clock
Counter
D/A
Digital Output
When START is received, control logic sets counter to 0, and turns on Clock sending regular pulses to the counter.
As the Clock sends regular pulses to the counter, the counter outputs a digital signal to the Digital-to-Analog converter

22. Note that the conversion time depends on the size of the input signal
As the counter counts, its output to the D/A generates a staircase ramp to the comparator.
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.
Vin
Note that the conversion time depends on the size of the input signal
V’in
Conversion time
Conv.time

23. 2- Flash Converters
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.

24. 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.

25. 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.

26. 3- Successive approximation A/D
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 ~ s, are adequate for most laboratory functions. Good tradeoff between speed and cost
Generates the digital output serially (one bit at a time).

27. Successive approximation A/D
Vref
analog input
D/A Converter
Digital Output Data
comparator
Successive Approximation Register
clock
STRT
At the beginning, all bits from the SAR are set to zero, and conversion begins by taking STRT line low.

28. Successive approximation A/D
Vref
analog input
D/A Converter
Digital Output Data
comparator
Successive Approximation Register
clock
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.

29. Successive approximation A/D
Vref
analog input
D/A Converter
Digital Output Data
comparator
Successive Approximation Register
clock
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.

30. Successive approximation A/D
Vref
analog input
D/A Converter
Digital Output Data
comparator
Successive Approximation Register
clock
STRT
If the D/A output is > Vin then the MSB is set to 0 and the next bit is set equal to 1.

31. Successive approximation A/D
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.

32. Example Calculate the maximum conversion time of
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

33. Example, continued
(b)      For a 8-bit successive approximation A/D, the conversion time is constant and equal to
It is noted that the conversion speed of successive approximation A/D is much faster than the integrating type.

34. 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.

35. 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.