### DACs

```Digital to Analog
Converters
Alexander Gurney
Alexander Pitt
Gautam Puri
1
Digital to Analog Converters

Alexander Gurney
What is a DAC?
Applications of DACs

Alexander Pitt
Types of DACs
Binary Weighted Resistor
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Gautam Puri
Specifications
Resolution
Speed
Linearity
Settling Time
Reference Voltages
Errors
2
What is a DAC? – Alexander Gurney
What is a DAC?
A DAC converts a binary digital signal into an analog
representation of the same signal
Typically the analog signal is a voltage output, though
current output can also be used
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1
0
0
1
3
0
1
0
1
0
0
1
1
0
1
1
1
1
0
0
1
1
0
1
0
1
0
1
1
DAC
What is a DAC? – Alexander Gurney
Reference Voltage
DACs rely on an input Reference Voltage to calculate the
Output Signal
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4
What is a DAC? – Alexander Gurney
Binary to Analog Conversion
Each sample is converted from binary to analog,
between 0 and Vref for Unipolar, or Vref and –Vref for
Bipolar
Analog Output Signal
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0000 0001 0010 0011 0100 0101 0110 0111 100010011010 1011
Digital Input Signal
5
What is a DAC? – Alexander Gurney
Sampling Frequency
Sampling frequency is the number of data points sampled
per unit time
Sampling frequency must be twice the frequency of the
sampled signal to avoid aliasing, per Nyquist criteria
A higher sampling frequency decreases the sampling
period, allowing more data to be transmitted in the same
amount of time
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What is a DAC? – Alexander Gurney
Output is a Piecewise Function
This is due to finite sampling frequency
The analog value is calculated and “held” over the
sampling period
This results in an imperfect reconstruction of the original
signal
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DAC
Ideally Sampled Signal
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Output typical of a real, practical
DAC due to sample & hold
What is a DAC? – Alexander Gurney
An Example
4 Bit signal
Unipolar
Vref = 7V
8 Sample Points
Sample Frequency = 1 hertz
Duration 8 seconds
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0001 0011 0110 1100 1011 0101 0010 0111
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What is a DAC? – Alexander Gurney
Filtering
The analog signal generated by the DAC can be
smoothed using a low pass filter
This removes the high frequencies required to sustain the
sharp inclines making up the edges
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Piece-wise
Continuous Output
Digital Input
Analog
Continuous Output
0 bit
011010010101010100101
101010101011111100101
000010101010111110011
010101010101010101010
111010101011110011000
100101010101010001111
n bit DAC
nth bit
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Filter
What is a DAC? – Alexander Gurney
DACs in Audio
Digital
MP3s
CDs
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Analog
->3.5mm Audio Out
->RCA Audio Out
What is a DAC? – Alexander Gurney
DACs in Video
Digital
DVDs
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Analog
->Composite Output
->Converter Box Output
->Analog Monitor Input
DAC Types – Alex Pitt
Types of Digital to Analog Converters
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Binary Weighted
 Explanation
 Explanation
 Example
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DAC Types – Alex Pitt
Binary Weighted DAC
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Adds resistors in parallel scaled by two to divide voltage on each
branch by a power of two
Vout =
Analog
Out
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Use transistors to switch
between open and close
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Use a summing op-amp
circuit with gain
DAC Types – Alex Pitt
Binary Weighted DAC
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Circuit can be simplified by adding resistors in parallel to substitute
for Rin. *Values for A, B, C and D are either 1 or 0.
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DAC Types – Alex Pitt
Binary Weighted DAC
B0 B 1 B2 B3
General equation
 Rf 
B0 
 Bn 1 Bn  2 Bn 3
  Vin Rf 
Vout  Vin 


  n -1 
2R
4R
2 R
 R
 Rin 
MSB
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LSB
DAC Types – Alex Pitt
Binary Weighted DAC
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Works well up to ~ 8-bit conversions
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Needs large range of resistor values (2048:1 for a 12-bit
DAC) with high precision resistor values
Too much or too little current flowing through resistors
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Minimum/maximum opamp current
Noise overwhelms current through larger resistance values
DAC Types – Alex Pitt
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Each bit controls a
switch between ground
and the inverting input
of the op amp.
The switch is
connected to ground if
the corresponding bit
is zero.
Vref
RF
4 bit converter
Requires only two resistance values (R and 2R)
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DAC Types – Alex Pitt

Convert 0001 to analog
V3
Vref
V2
V1
V0
Req 
V1
RF
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1
R
1/ 2 R  1/ 2 R
V0
V0 
R
1
V1  V1
RR
2
V1 
R
1
V2  V2
RR
2
V2 
R
1
V3  V3
RR
2
DAC Types – Alex Pitt

Convert 0001 to analog
R
RF
Vref
V0
2R
1
V0  Vref
8
Vout  
19
R
RF
2R
V0  
1
R 
Vref  f 
16
 R
DAC Types – Alex Pitt
By adding resistance in series and in parallel we can derive an
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DAC Types – Alex Pitt
MSB
LSB
By knowing how current flows through the ladder we can come
up with a general equation for R-2R DACs.
 B3 B2 B1 B0 
Vout   I sum Rf   I  
   Rf
4 8 16 
 2
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DAC Types – Alex Pitt

4-Bit Equation
 Substituting
Vout
I 
Vref
R
Vref  B3 B2 B1 B0 
  I sum Rf  
   Rf
 
R  2
4 8 16 

Rf
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General Equation
R f n1 Bi
Vout  Vref

R i  0 2n  i
DAC Types – Alex Pitt
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Only two resistor values
Can use lower precision resistors
Specifications - Gautam Puri
Specifications of DAC
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Lets discuss some terms you’ll hear when dealing with DACs
Reference Voltage
Resolution
Speed
Linearity
Settling Time
Some types of Errors
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Specifications - Gautam Puri
Reference Voltage Vref
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The reference voltage determines the range of output voltages
from the DAC
For a ‘Non-Multiplying DAC’, Vref is a constant value set
internally by the manufacturer
For a ‘Multiplying DAC’, Vref is set externally and can be varied
during operation
Vref also affects DAC resolution (which will be discussed later).
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Specifications - Gautam Puri
Full scale voltage
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Full scale voltage is the output voltage when all the bits of the
digital input signal are 1s.
Vref (2 N  1)
Vfs 
2 N voltage Vref
It is slightly less than reference
Vfs = Vref - VLSB
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Specifications - Gautam Puri
Resolution
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Resolution of a DAC is the change in output voltage for a change in
the least significant bit (LSB) of the digital input
Resolution is specified in “bits”.
Most DACs have a resolution of 8 to 16 bits
V
Resolution  ref
 VLSB
N
2
Example: A DAC with 10 bits has a resolution of
V
1
Resolution  ref

Vref
10
2
1024
Higher resolution (more bits) = smoother output
A DAC with 8 bits has 256 steps whereas one with 16 bits has
65536 steps for the given voltage range and can thus offer smoother
output
Specifications - Gautam Puri
Speed (Sampling frequency)
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Sampling frequency is the rate at which the DAC accepts
digital input and produces voltage output
In order to avoid aliasing, the Nyquist criterion requires that
fsampling  2 f max
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Sampling frequency is limited by the input clock speed
(depends on microcontroller) and the settling time of the
DAC
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Specifications - Gautam Puri
Settling Time
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It takes the DAC a finite amount of time to produce the exact
analog voltage corresponding to the digital input
The settling time is the time interval from when the DAC
commands the update of its output to when the voltage
actually reaches ± ½ VLSB.
A faster DAC will have a smaller settling time
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tsettle
Specifications - Gautam Puri
Linearity
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If the change in analog output voltage per unit change in digital
input remains constant over the entire range of operation, the
DAC is said to be linear
Ideally the DAC should have a proportionality constant which
results in a linear slope
Non-linearity is considered an error, and will be further
discussed in the errors section
0000
Analog Output Signal
Non-linear
Analog Output Signal
Linear
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0001
0010
0011
Digital Input Signal
0100
0101
0000
0001
0010
0011
Digital Input Signal
0100
0101
Specifications - Gautam Puri
Types of DAC Errors
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Non-monotonic output error
Non-linear output error
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―
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Differential
Integral
Gain error
Offset error
Full scale error
Resolution error
Settling time and overshoot error
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Specifications - Gautam Puri
Non-monotonic Output Error
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A monotonic function has a slope whose sign does not change
Non-monotonic error results when the analog output changes direction
for a step or a few steps of digital input
In a closed loop control system this may cause the DAC to toggle
continuously between 2 input codes and the system will be unstable.
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Specifications - Gautam Puri
Differential non-linear output error
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For a change in the LSB of input, the output of an ideal DAC is
VLSB
However in a non-linear DAC the output may not be exactly
the LSB but rather a fraction (higher or lower) of it
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Specifications - Gautam Puri
Differential non-linear output error
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Basically “differential” non-linearity expresses the error in step
size as a fraction of LSB
The DNL is the maximum of these deviations over the entire
transfer function
One must choose a DAC with DNL less than 1 LSB. A DNL >
1 LSB will lead to non-monotonic behavior. This means that for
certain steps in digital input, the output voltage will change in
the opposite direction. This may cause a closed loop control
system to become unstable as the system may end up
oscillating back and forth between two points.
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Specifications - Gautam Puri
Integral non-linear output error
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The integral non-linearity error is the difference between the
ideal and actual output. It can also be defined as the difference
between ideal and a best fit line
INL occurs when the output is non-linear and thus unable to
The maximum deviation from this line is called INL.
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Specifications - Gautam Puri
Integral non-linear output error
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INL is expressed as fraction of LSB.
INL cannot be calibrated out as the non-linearity is
unpredictable and one does not know where the
maximum deviation from the ideal line will occur.
One must choose an ADC with an INL (maximum
deviation) within the accuracy required.
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Specifications - Gautam Puri
More important - DNL or INL ?
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The DNL and INL are both important non-linear errors to be
aware of.
In the case of an application such as an imaging one, where
slight differences in color densities are important, the
“differential” non-linearity error is more important.
In an application where the parameters vary more widely, such
as speed of a vehicle, the “integral” non-linearity error may be
of greater importance
Specifications - Gautam Puri
Gain Error
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The difference between the output voltage (or current) with
full scale input code and theideal voltage (or current) that
should exist with a full scale input code
2 Types of Gain Error
1. Low Gain: Step
Amplitude Less than
Ideal
2. High Gain: Step
Amplitude Greater
than Ideal
Gain Error can be adjusted to zero by using
an
38 external potentiometer
Specifications - Gautam Puri
Offset Error
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It is the difference in ideal and actual output voltage at a digital input of
zero
All output values will differ from the ideal values by that same amount,
hence the output is “offset” from the input
Offset can be ‘positive’ or ‘negative’
It can be fixed by adding/subtracting the difference to the digital input
before passing through the DAC
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Specifications - Gautam Puri
Full Scale Error
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It is a combination of gain and offset error
It is measured at the full scale input
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Specifications - Gautam Puri
Resolution Error
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If the resolution is not high enough, the DAC cannot
accurately output the required waveform
Lower resolution results in higher resolution error
Low resolution (1 bit)
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Higher resolution (3 bits)
Specifications - Gautam Puri
Settling Time and Overshoot Error
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If settling time is too high, the DAC will not produce the
ideal output waveform fast enough and there will be a
delay or lag.
This will also lower the maximum operating frequency of
the DAC.
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References
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Previous semester lecture slides
http://www.hitequest.com/Hardware/a_dac.htm
pdf
Scherz, Paul. Practical Electronics for Inventors. 2nd Edition,
McGraw Hill. 2007.