Transcript Digital to Analog Converters (DAC)

Digital to Analog Converters
(DAC)
1
©Paul Godin
Created March 2008
Updated March 2010
Digital and Analog
◊
Digital systems are discrete, meaning they have a finite
numerical value. Sometimes referred to as “fixed” or
“stepped” values.
◊
Analog values are continuous, meaning they have a value
that can vary continuously. The values can be to a great
degree of precision and may contain more information such
as frequency, phase, etc…
◊
Analog values make up real-world values that can be
measured.
◊
This presentation describes methods for converting digital
values to analog values.
DAC 1.2
Digital to Analog
◊ Digital electronics offers advantages over analog
in processing, data manipulation, storage and
analysis of values.
◊ Often these digital circuits must interface with the
real world:
◊ as inputs to analyze, process and manipulate
◊ as outputs to control the physical environment
◊ It is important to establish a means of converting
between digital systems and the real world.
DAC 1.3
Transducers
◊ Transducers are devices that convert physical
quantities into electrical quantities. There are
many possible physical measurements requiring
many types of transducers:
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Light
Pressure
Speed
Flow
Angle
Temperature
Rotation
Vibration
Sound, …
DAC 1.4
Actuators
◊ Actuators are electrically controlled devices that
control the physical environment. There are many
types of actuators available. These include:
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motors
solenoids (electromagnetic non-rotational motion)
relays
pumps
valves
lifts
heaters
acoustic devices, …
DAC 1.5
Analog versus Digital
Distorted Analog signal
Original Analog signal
000000100000010000101000
101000011010010011001110
101000100000101000101000
011010010011001110101000
100000001000010100000010
000101000101000011010010
011001110101000100001010
000110100100110011101010
001010111011011010001001
Binary signal
DAC 1.6
Analog to Digital
Original Analog signal
A to D Conversion
The voltage is converted to a
binary value at regular intervals.
Animated
000100110111101010001
000111000000100000010
011100101001001011101
011110010101010010101
010101001001010101001
000101001010101111010
000001001011101011101
000000010101110101010
000000000001001111010
000000000000111111010
000000000001010101010
000000000001011011101
000000000001101101100
000000001100010111010
000000100011111010110
000001001010101000100
000001010111101111000
000011001101010100101
000110111000010100101
…
Binary signal
DAC 1.7
Digital to Analog
000100110111101010001
000111000000100000010
011100101001001011101
011110010101010010101
010101001001010101001
000101001010101111010
000001001011101011101
000000010101110101010
000000000001001111010
000000000000111111010
000000000001010101010
000000000001011011101
000000000001101101100
000000001100010111010
000000100011111010110
000001001010101000100
000001010111101111000
000011001101010100101
000110111000010100101
…
Digital signal
Animated
D to A Conversion
Analog signal
The binary value is converted to
a voltage at regular intervals.
DAC 1.8
Digital to Analog Converters
◊ Digital to Analog Converters take a digital value
and convert it to voltage or current over time.
◊ Converting discrete values to analog values has
some challenges. Invariably, the analog value will
retain some of the discrete steps from the digital
value.
◊ Note that our discussions will focus on the output
voltages but these are also applicable to output
current.
DAC 1.9
Scale
◊ The range of the available digital values, based on
the number of bits in the binary number,
represents the full scale.
◊ lowest binary value represents the lowest voltage
◊ highest binary value represents the highest voltage
◊ The maximum and minimum voltage must be
known to determine the scale.
◊ The Full Scale Output is the maximum value that
the DAC can produce for the design.
DAC 1.10
Min binary = 0000
Max binary = 1111
Scale Example
MSB
LSB
D
C
B
A
VOUT
DAC
Min VOUT = 0V
Max desired VOUT = 7.5V
There are 16 values from 0000 to
1111, but the first step (0000)
equals 0. Therefore, 15 steps for
an output range of 7.5 Volts. The
scale will be 0.5 Volts per binary
increment.
A FS
K
n
(2  1)
D
C
B
A
VOUT
0
0
0
0
0.0
0
0
0
1
0.5
0
0
1
0
1.0
0
0
1
1
1.5
0
1
0
0
2.0
0
1
0
1
2.5
0
1
1
0
3.0
0
1
1
1
3.5
1
0
0
0
4.0
1
0
0
1
4.5
1
0
1
0
5.0
1
0
1
1
5.5
1
1
0
0
6.0
1
1
0
1
6.5
1
1
1
0
7.0
1
1
1
1
7.5
AFS = Analog Full Scale Voltage
DAC 1.11
Scale Calculation
◊ A 4-bit DAC has an output of 1.0 Volts with a
binary value of 1010.
◊ What is the output voltage if the binary value is 1100?
◊ What is the Full Scale Output voltage?
◊ An 8-bit DAC has an output of 12.0 Volts with a
binary value of 00111100.
◊ What is the K value?
◊ What is the Full Scale Output voltage?
DAC 1.12
Scale Example
◊ Analyzing the voltage output from the example it
becomes evident that the output voltage, although
analog, still follows a pattern of discrete values.
DAC 1.13
Input Weights
◊
Binary numbers have positional weighting. The value
adjacent to the LSB has a weight of 2 times the LSB, the
position adjacent to this has a weight of 4 times the LSB and
the MSB has a weight of 8 times the LSB.
10112
23
◊
◊
22
21
20
In our example, each LSB change represented a 0.5V
change to the output. By following the weights of the inputs
starting from the LSB, the next value represents 1.0V (2x
the LSB), the next value represents 2.0V (4x the LSB) and
the MSB represents 4.0V (8x the LSB).
With this in mind, a binary 1011 for our example should
represent: 4.0V + 1.0V + 0.5V = 5.5 Volts
DAC 1.14
Resolution
◊ The resolution represents the smallest change, or
step, in the analog output. The greater the
resolution, the smaller the steps.
◊ To increase resolution increase the number of bits
in the binary value.
◊ In our example, a 4-bit number represented a 0.5
volt change per step. By increasing the number
to 5 bits, each change would represent
approximately 0.25 volt change per step,
increasing the resolution.
DAC 1.15
Improved Resolution
◊ By increasing the binary number size
by one bit the voltage between steps
decreases.
4-bit resolution
5-bit resolution
DAC 1.16
Percentage Resolution
◊ Resolution can be expressed as the percentage of
the Full Scale Output.
% Re s olution
◊ 4-bit example:
◊ 5-bit example:
S tep_ S ize
1
1

 n
VFS
# S teps (2  1)
% Re s olution
% Re s olution
0.5V
1

 6.6 7%
7.5V 1 5
0.2 5V
1

 3.2 2%
7.7 5V 3 1
DAC 1.17
Bipolar DAC
◊ The examples shown so far represented positive
output voltage values. Analog values can also be
negative.
◊ To represent a negative value a signed 2’s
compliment is used.
DAC 1.18
Signed Magnitude
◊ Binary systems utilize only 1’s and 0’s. The
negative symbol cannot be used.
◊ In a signed magnitude value, the bit in the
leftmost position of a binary number is used to
indicate if the value is positive or negative. This is
the sign bit. The value following the sign bit is
the magnitude.
01001101 = positive value, 10011012
11001101 = negative value, 10011012
The leftmost bit is the sign bit.
DAC 1.19
Signed Magnitude
◊ The signed magnitude is difficult for digital devices
to utilize. Digital systems are designed to add
values together.
◊ Another system is used instead: 2’s compliment.
Sign Bit:
0 is positive
1 is negative
DAC 1.20
1’s Compliment
◊ The 1’s compliment simply means taking the
compliment of each binary bit in a binary number.
10110 : Original Number
01001 : One’s Compliment
DAC 1.21
2’s Compliment
◊ The 2’s compliment simply means taking the
compliment of each binary bit in a binary number
and adding a 1 to the LSB. This is considered the
equivalent of a negative number for purposes of
addition.
10110 : Original Number
01001 : One’s Compliment
00001 : Add a 1
01010 : 2’s Compliment
DAC 1.22
2’s Compliment
Example using 2’s compliment:
Represent the following operation in binary:
1100 is 12
12
-3
9
0011 is 3
1100 is 1’s compliment
1101 is 2’s compliment
1100 (12)
+1101 (-3)
11001 (9)
Note the extra bit is always disregarded
DAC 1.23
2’s Compliment Example
13
-10
3
1101 is 13
1010 is “10”
0101 is 1’s compliment
0110 is 2’s compliment
1101 (13)
+0110 (-10)
10011 (3)
DAC 1.24
Signed 2’s Compliment Example
13
-10
3
01101 is 13 (signed 2’s compliment)
01010 is “10”
10101 is 1’s compliment
10110 is 2’s compliment
01101 (+13)
+ 10110 (-10)
100011 (+3)
Sign bit
DAC 1.25
Signed 2’s Compliment Example
10
-13
-3
Sign bit
01010 is 10 (signed 2’s compliment)
11101 is “-13”
10010 is 1’s compliment
10011 is 2’s compliment
01010 (+10)
+ 10011 (-13)
11101 (-3, in 2’s compliment)
11101 is “-3” in 2’s compliment
10010 is 1’s compliment
10011 is -3
DAC 1.26
Exercise
1. Determine the decimal values of the following
signed binary numbers:
a) 01001
b) 10001
c) 10101
2. Determine the decimal values of the following
signed, 2’s compliment binary numbers:
a) 01001
b) 10001
c) 10101
DAC 1.27
Exercise (continued)
3. What is the range of signed decimal values in 1
byte of data, and what is the most negative value
in a signed byte?
4. What is the most negative value in a signed, 2’s
compliment byte?
DAC 1.28
Answers
1. a: +9, b: -1, c: -5
These values are not 2’s compliment, so what follows the
sign is magnitude
2. a: +9, b: -15, c: -11
3. 0111 1111 (most positive, +127) to 1000 000 (most
negative, -128)
4. -128 (1000 000). This is an unusual exception as the 2’s
compliment equals the magnitude, but is valid.
DAC 1.29
Bipolar DAC
◊ If structured accordingly, DACs can produce
negative and positive output voltages.
Value
Signed 2’s Compl.
DAC input
VOUT
Most +
0111 1111
1111 1111
+VMAX
Zero
0000 0000
1000 0000
0V
Most -
1000 0000
0000 0000
-VMAX
◊ Some DACs will accept a signed, 2’s compliment
value, but this would be specified.
DAC 1.30
Review Questions
◊ An 8-bit DAC has an input of 0100 1001 and
produces an output of 2.19 Volts.
1.
2.
3.
4.
What
What
What
What
is the voltage resolution?
is the FSO?
are the input weights of the bits?
is the percent resolution?
1001001 = 73 > 2.19/73=0.03V/bit
1111 1111 = FSO =>255*0.03=7.65V
1st=0.03v, 2nd=0.06, 3=0.12, 4=0.24, 5=0.48, 6=0.96, 7=1.92, 8=3.84
1/(28-1)=1/255=0.392%
DAC 1.31
Review Questions 2
◊ An 6-bit DAC has an input of 010 111 and
produces an output of 172.5 mV.
1.
2.
3.
4.
What
What
What
What
is the voltage resolution?
is the FSO?
are the input weights of the bits?
is the percent resolution?
010111 = 23 => 172.5m/23= 7.5mV/bit
111 111 = FSO =>*0.03=472.5mV
1st=7.5v, 2nd=15, 3=30, 4=60, 5=120, 6=240mV
(verify 120+30+15+7.5=172.5mV)
1/(26-1)=1/63=1.59%
DAC 1.32
End of Part 1
©Paul R. Godin
prgodin°@ gmail.com
DAC 1.33