Digital to Analog Conversion
Heather Humphreys
Cheng Shu Ngoo
Woongsik Ham
Ken Marek
Woongsik Ham
Topics Discussed
 What is a DAC?
 Applications
 Types of DAC circuit
 Binary weighted DAC
 R-2R Ladder DAC
 Specifications of DAC
 Resolution
 Reference Voltage
 Speed
 Settling Time
 Linearity
 DAC associated errors
Woongsik Ham
What is a DAC?
 A digital to analog converter (DAC) is a device that
converts digital numbers (binary) into an analog voltage or
current output.
Woongsik Ham
Principal components of DAC
Woongsik Ham
What is a DAC?
 Digital  Analog
 Each binary number sampled by the DAC corresponds to
Analog Output Signal
a different output level.
0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011
Digital Input Signal
Typical Output
Woongsik Ham
DACs capture and hold a number, convert it
to a physical signal, and hold that value for a
given sample interval. This is known as a
zero-order hold and results in a piecewise
constant output.
DAC
Ideally Sampled Signal
Output typical of a real, practical
DAC due to sample & hold
Woongsik Ham
Types of DAC
 Multiplying DAC*
 Reference source external to DAC package
 Nonmultiplying DAC
 Reference source inside DAC package
*Multiplying DAC is advantageous considering the external reference.
Woongsik Ham
Common Applications
 Used when a continuous analog signal is required.
 Signal from DAC can be smoothed by a Low pass filter
Piece-wise
Continuous Output
Digital Input
Analog
Continuous Output
0 bit
011010010101010100101
101010101011111100101
000010101010111110011
010101010101010101010
111010101011110011000
100101010101010001111
n bit DAC
nth bit
Filter
Woongsik Ham
Common Applications:
Function Generators
 Digital Oscilloscopes
 Digital Input
 Analog Ouput
1
 Signal Generators
 Sine wave generation
 Square wave generation
 Triangle wave generation
 Random noise generation
2
Applications – Video
Woongsik Ham
Video signals from digital sources, such as a
computer or DVD must be converted to
analog signals before being displayed on an
analog monitor. Beginning on February 18th,
2009 all television broadcasts in the United
States will be in a digital format, requiring
ATSC tuners (either internal or set-top box)
to convert the signal to analog.
Woongsik Ham
Common Applications
Motor Controllers
 Cruise Control
 Valve Control
 Motor Control
1
2
3
Woongsik Ham
Types of DAC
 Multiplying DAC*
 Reference source external to DAC package
 Nonmultiplying DAC
 Reference source inside DAC package
*Multiplying DAC is advantageous considering the external reference.
Ken Marek
Types of DAC implementations
 Binary Weighted Resistor
 R-2R Ladder
 Pulse Width Modulator (not covered)
 Oversampling DAC (used internally in HCS12)
Ken Marek
Binary Weighted Resistor
•Start with summing
op-amp circuit
•Input voltage either
high or ground
•Adjust resistor
weighting to
achieve desired Vout
Ken Marek
Binary Weighted Resistor
• Details
– Use transistors to switch
between high and ground
– Use resistors scaled by two
to divide voltage on each
branch by a power of two
– V1 is MSB, V4 LSB in this
circuit
• Assumptions:
– Ideal Op-Amp
– No Current into Op-Amp
– Virtual Ground at Inverting
Input
– Vout = -IRf
Ken Marek
Binary Weighted Resistor
 Assume binary
B5
B4
B3
B2
B1
inputs B0 (LSB)
to Bn-1 (MSB)
 Each Bi = 1 or
0 and is
multiplied by
Vref to get
input voltage
B0
Vout
B0 
B1
 Bn 1 Bn  2
  IRf   Rf Vref 

 ... n  2  n-1 
2R
2 R 2 R
 R
Ken Marek
Binary Weighted Resistor
 Example: take a 4-bit converter, Rf = aR
 B3 B2 B1 B0 
Vout  aVref  
  
2
4 8 
 1
 Input parameters:
 Input voltage Vref = -2V
 Binary input = 1011
 Coefficient a = ½
Vout
1
 1 0 1 1  11
   2         1.375V
2
1 2 4 8  8
Ken Marek
Binary Weighted Resistor
 Resolution: find minimum nonzero output
Vmin 
Rf Vref
R 2n-1
 If Rf = R/2 then resolution is
and max Vout is Vmax  Vref
Vref
2n
1 

1  n 
 2 
Ken Marek
Binary Weighted Resistor
 Advantages
 Simple
 Fast
 Disadvantages
 Need large range of resistor values (2048:1 for 12-bit) with
high precision in low resistor values
 Need very small switch resistances
 Op-amp may have trouble producing low currents at the low
range of a high precision DAC
Ken Marek
R-2R Ladder
 Each bit corresponds to
a switch:
 If the bit is high, the
corresponding switch is
connected to the inverting
input of the op-amp.
 If the bit is low, the
corresponding switch is
connected to ground.
Ken Marek
R-2R Ladder
B2
B1
B0
Ken Marek
R-2R Ladder
 Circuit may be analyzed
using Thevenin’s theorem
(replace network with
equivalent voltage source
and resistance)
 Final result is:
Vout  Vref
Rf
n 1
Bi

n i
R i 0 2
Compare to binary weighted circuit:
n 1
f
i
out
ref
( n 1) i
i 0
V  V
R

R
2
B
B2
B1
B0
Rf
Ken Marek
R-2R Ladder
 Resolution
Vmin 
Rf Vref
R 2n
 If Rf = R then resolution is
Vref
and max Vout is Vmax  Vref
2n
1 

1  n 
 2 
Ken Marek
R-2R Ladder
 Advantages:
 Only 2 resistor values
 Lower precision resistors acceptable
 Disadvantages
 Slower conversion rate
Ken Marek
General comments
 Circuits as shown produce only unipolar output
 Replacing ground with –Vref will allow Vout to be positive or
negative
Cheng Shu Ngoo
DAC Specifications:





Reference Voltages
Resolution
Speed
Settling Time
Linearity
Cheng Shu Ngoo
Reference Voltage
 Determines Characteristic of DACs
 Set externally or Generated inside DAC
 Vref sets maximum DAC output voltage (if not amplified)
 Full scale output voltage:
Eo ( fs) 
Vref (2 n  1)
2
n
 Vref determines analog output voltage changes to steps taken by 1 LSB of
digital input signal (resolution)
X  k  A B
X = analog output
k = Constant
A = Vref analog
B = Binary (digital) input
Cheng Shu Ngoo
Reference Voltage
 Internal vs. External Vref?
Internal
External
•Non-Multiplier DAC
•Multiplying DAC
•Vref fixed by manufacturer
•Vary Vref
•Qualified for specified
•Consider current required
temperature range
•Consider Voltage range
•Consider dynamic effects
of inner structure
*Multiplying DAC is advantageous considering the external reference.
Cheng Shu Ngoo
Resolution
 1 LSB (digital)=1 step size for DAC output (analog)
Resolution 
Vref
2n
 Increasing the number of bits results in a finer resolution
 Most DAC - 8 to 16-bits (256 to 65,536 steps)
e.g. 5Vref DAC
1LSB=5/28 =0.0195V resolution (8-bit)
1LSB=5/23 =0.625V resolution (3-bit)
3-bit Resolution
8-bit Resolution
5
4.5
5
4
4.5
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
0.5
1
0
0.5
1 LSB
0
Cheng Shu Ngoo
Speed (Max. Sampling Frequency)
 The maximum rate at which DAC is reproducing usable analog output






from digital input register
Digital input signal that fluctuates at/ has high frequency require high
conversion speed
Speed is limited by the clock speed of the microcontroller (input clock
speed) and the settling time of the DAC
Shannon-Nyquist sampling theorem  fsampling ≥ 2fmax
Eg. To reproduce audio signal up to 20kHz, standard CD samples audio at
44.1kHz with DAC ≥40kHz
Typical computer sound cards 48kHz sampling freq
>1MHz for High Speed DACs
Settling Time
Cheng Shu Ngoo
 The interval between a command to update (change) its output value and





the instant it reaches its final value, within a specified percentage (± ½
LSB)
Ideal DAC output would be sequence of impulses  Instantaneous
update
Causes:
 Slew rate of output amplifier
 Amount of amplifier ringing and signal overshoot
Faster DACs have shorter settling time
Electronic switching  fast
Amplifier settling time  dominant effect
Cheng Shu Ngoo
Settling Time
tsettle
Cheng Shu Ngoo
DAC Linearity
 The difference between the desired analog output and the actual output over the full range of
expected values
 Does the DAC analog output vary linearly with digital input signal?
 Can the DAC behavior follow a constant Transfer Function relationship?
 Ideally, proportionality constant – linear slope
 Increase in input  increase in output  monotonic
Analog Output Signal
Analog Output Signal
 Integral non-linearity (INL) & Differential non-linearity (DNL)
0000
0001
0010
0011
Digital Input Signal
Linear
0100
0101
0000
0001
0010
0011
Digital Input Signal
Non-Linear
0100
0101
Heather Humphreys
Types of DAC Errors
 Gain Error
 Offset Error
 Full Scale Error
 Non-Monotonic Output Error
 Differential Nonlinearity Error
 Integral Nonlinearity Error
 Settling Time and Overshoot Error
 Resolution Error
 Sources of Errors
Heather Humphreys
Gain Error
 Slope deviation from
ideal gain
 Low Gain: Step
Amplitude Less than
Ideal
 High Gain: Step
Amplitude Higher
than Ideal
Heather Humphreys
Offset Error
 The voltage offset from
zero when all input bits are
low
*This error may
be detected when
all input bits are
low (i.e. 0).
Heather Humphreys
Full-Scale Error
 Includes gain error and
offset error
 Occurs when there is an
offset in voltage form the
ideal output and a
deviation in slope from
the ideal gain.
 Error at full scale –
contrast with offset error
at zero
Heather Humphreys
Non-Monotonic Output Error
 A form of non-linearity, due to errors in individual bits of
the input
 Refers to output that is not monotonic
Heather Humphreys
Differential Nonlinearity Error
 The largest difference between the
actual and theoretical output as a
percentage of full-scale output
voltage.
 Voltage step size differences vary as
digital input increases. Ideally each
step should be equivalent.
 In other words, DNL error is the
difference between the ideal and the
measured output responses for
successive steps.
 An ideal DAC response would have
analog output values exactly one
code (LSB) apart (DNL = 0).
Integral Nonlinearity Error
 Occurs when the output voltage is
non linear; an inability to adhere
to the ideal slope.
 INL is the deviation of an actual
transfer function from a straight
line. After nullifying offset and
gain errors, the straight line is
either a best-fit straight line or a
line drawn between the end points
of the transfer function.
 INL is often called 'relative
accuracy.'
Heather Humphreys
Heather Humphreys
Settling Time and Overshoot Error
• Settling Time: The time required
for the voltage to settle within +/the voltage associated with the VLSB.
Any change in the input time will
not be reflected immediately due
to the lag time.
• Settling time generally determines
maximum operating frequency of
the DAC
• One of the principal limiting
factors of any commercial DAC is
the settling time of the op-amp
• Overshoot: occurs when the output
voltage overshoots the desired analog
output voltage.
Resolution Errors
• Inherent errors associated with resolution
Heather Humphreys
Resolution 
Ref Voltage
2
# of bits
– More Bits => Less Error & Greater Resolution
– Less Bits => More Error & Less Resolution
– Q: How does very high resolution affect measurements?
A: LSB may be in noise range and not produce an output; it may be
difficult to find an op-amp to amplify such small current
Poor Resolution (1 Bit)
Better Resolution (3 Bit)
Heather Humphreys
Sources of Errors
 Deviation of voltage sources from nominal values
 Variations and tolerances on resistance values
 Non-ideal operational amplifiers
 Other non-ideal circuit components, temperature
dependence, etc.
Woongsik Ham
Project Applications
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Motor speed controller
Solenoid valves (pneumatics)
Digital Motor Control
Computer Printers
Sound Equipment (e.g. CD/MP3 Players, etc.)
Electronic Cruise Control
Digital Thermostat
References
 Previous student presentations and…
 http://en.wikipedia.org/wiki/Digital_to_analog
 http://www.allaboutcircuits.com/vol_4/chpt_13/index.html
 Alicatore, David G. and Michael B Histand. Introduction to Mechatronics and
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
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
Measurement Systems, 2nd ed. McGraw-Hill, 2003.
http://www.emersonprocess.com/fisher/products/fieldvue/dvc/
http://auto.howstuffworks.com/cruise-control.htm
http://www.thermionics.com/smc.htm
Maxim AN641 Glossary
http://www.electrorent.com/products/search/General_Purpose_Oscilloscop
es.html
http://www.bkprecision.com/power_supplies_supply_generators.htm
 http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/dac.html#c4