# Digital to Analog Converters Alexander Gurney Alexander Pitt Gautam Puri 1.

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Digital to Analog Converters Alexander Gurney Alexander Pitt Gautam Puri 1

Digital to Analog Converters  Alexander GurneyWhat is a DAC? Applications of DACs  Alexander PittTypes of DACs Binary Weighted Resistor R-2R Ladder  Gautam PuriSpecifications Resolution Speed Linearity Settling Time Reference Voltages Errors 2

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 01010101 00110011 01110111 10011001 10011001 10101010 10111011 DAC What is a DAC? – Alexander Gurney 3

Reference Voltage  DACs rely on an input Reference Voltage to calculate the Output Signal What is a DAC? – Alexander Gurney 4

Binary to Analog Conversion 1011 1001101001111000011001010100 0011001000010000 Digital Input Signal Analog Output Signal  Each sample is converted from binary to analog, between 0 and Vref for Unipolar, or Vref and –Vref for Bipolar What is a DAC? – Alexander Gurney 5

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 What is a DAC? – Alexander Gurney 6

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 Ideally Sampled Signal Output typical of a real, practical DAC due to sample & hold DAC What is a DAC? – Alexander Gurney 7

An Example  4 Bit signal  Unipolar  Vref = 7V  8 Sample Points  Sample Frequency = 1 hertz  Duration 8 seconds 00010011011011001011010100100111 What is a DAC? – Alexander Gurney 8

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 0 bit n th bit n bit DAC 011010010101010100101 101010101011111100101 000010101010111110011 010101010101010101010 111010101011110011000 100101010101010001111 Digital Input Filter Piece-wise Continuous Output Analog Continuous Output What is a DAC? – Alexander Gurney 9

DACs in Audio DigitalAnalog MP3s ->3.5mm Audio Out HD Radio ->Signal received by speaker CDs ->RCA Audio Out What is a DAC? – Alexander Gurney 10

DACs in Video DigitalAnalog DVDs ->Composite Output OTA Broadcast ->Converter Box Output Youtube ->Analog Monitor Input What is a DAC? – Alexander Gurney 11

Types of Digital to Analog Converters  Binary Weighted  Explanation  Advantages and disadvantages  R-2R Ladder  Explanation  Example  Advantages and disadvantages DAC Types – Alex Pitt 12

Binary Weighted DAC  Use transistors to switch between open and close  Use a summing op-amp circuit with gain  Adds resistors in parallel scaled by two to divide voltage on each branch by a power of two DAC Types – Alex Pitt 13 V out = Analog Out

Binary Weighted DAC  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. DAC Types – Alex Pitt 14

Binary Weighted DAC MSB LSB General equation B0B0 B1B1 B2B2 B3B3 DAC Types – Alex Pitt 15

Binary Weighted DAC  Advantages  Works well up to ~ 8-bit conversions  Disadvantages  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  Minimum/maximum opamp current  Noise overwhelms current through larger resistance values DAC Types – Alex Pitt 16

R-2R Ladder DAC  Requires only two resistance values (R and 2R) V ref 4 bit converter  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. DAC Types – Alex Pitt 17 RFRF

R-2R Ladder Example  Convert 0001 to analog V0V0 V1V1 V2V2 V3V3 V1V1 V ref V0V0 DAC Types – Alex Pitt 18 RFRF

R-2R Ladder Example 19  Convert 0001 to analog 2R R V0V0 V ref DAC Types – Alex Pitt RFRF RFRF

R-2R Ladder By adding resistance in series and in parallel we can derive an equation for the R-2R ladder. DAC Types – Alex Pitt 20

R-2R Ladder By knowing how current flows through the ladder we can come up with a general equation for R-2R DACs. MSB LSB DAC Types – Alex Pitt 21

R-2R Ladder RfRf  4-Bit Equation  Substituting  General Equation DAC Types – Alex Pitt 22

R-2R Ladder DAC  Advantages  Only two resistor values  Can use lower precision resistors DAC Types – Alex Pitt 23

Specifications of DAC Lets discuss some terms you’ll hear when dealing with DACs  Reference Voltage  Resolution  Speed  Linearity  Settling Time  Some types of Errors Specifications - Gautam Puri 24

Reference Voltage V ref  The reference voltage determines the range of output voltages from the DAC  For a ‘Non-Multiplying DAC’, V ref is a constant value set internally by the manufacturer  For a ‘Multiplying DAC’, V ref is set externally and can be varied during operation  V ref also affects DAC resolution (which will be discussed later). Specifications - Gautam Puri 25

Full scale voltage  Full scale voltage is the output voltage when all the bits of the digital input signal are 1s.  It is slightly less than reference voltage V ref  V fs = V ref - V LSB Specifications - Gautam Puri 26

 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  Example: A DAC with 10 bits has a resolution of  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 Resolution Specifications - Gautam Puri 27

Speed (Sampling frequency)  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  Sampling frequency is limited by the input clock speed (depends on microcontroller) and the settling time of the DAC Specifications - Gautam Puri 28

Settling Time  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 V LSB.  A faster DAC will have a smaller settling time t settle Specifications - Gautam Puri 29

Linearity  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 LinearNon-linear Specifications - Gautam Puri 30

Types of DAC Errors  Non-monotonic output error  Non-linear output error ― Differential ― Integral  Gain error  Offset error  Full scale error  Resolution error  Settling time and overshoot error Specifications - Gautam Puri 31

Non-monotonic Output Error  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. Specifications - Gautam Puri 32

Differential non-linear output error  For a change in the LSB of input, the output of an ideal DAC is V LSB  However in a non-linear DAC the output may not be exactly the LSB but rather a fraction (higher or lower) of it Specifications - Gautam Puri 33

Differential non-linear output error  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. Specifications - Gautam Puri 34

Integral non-linear output error  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 adhere to a straight line.  The maximum deviation from this line is called INL. Specifications - Gautam Puri 35

Integral non-linear output error  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. Specifications - Gautam Puri 36

More important - DNL or INL ?  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 37

Gain Error  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 external potentiometer Specifications - Gautam Puri 38

Offset Error  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 Specifications - Gautam Puri 39

Full Scale Error  It is a combination of gain and offset error  It is measured at the full scale input Specifications - Gautam Puri 40

Resolution Error  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)Higher resolution (3 bits) Specifications - Gautam Puri 41

Settling Time and Overshoot Error  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. Specifications - Gautam Puri 42

References Previous semester lecture slides http://www.hitequest.com/Hardware/a_dac.htm http://www.national.com/appinfo/adc/files/ms101157.pdf http://www.hitequest.com/Hardware/a_dac.htm http://www.national.com/appinfo/adc/files/ms101157.pdf http://www.noise.physx.u- szeged.hu/DigitalMeasurements/ADConversion/ADSpecs. pdf http://www.noise.physx.u- szeged.hu/DigitalMeasurements/ADConversion/ADSpecs. pdf Scherz, Paul. Practical Electronics for Inventors. 2nd Edition, McGraw Hill. 2007. http://masteringelectronicsdesign.com/an-adc-and-dac- differential-non-linearity-dnl/ http://masteringelectronicsdesign.com/an-adc-and-dac- differential-non-linearity-dnl/ http://masteringelectronicsdesign.com/an-adc-and-dac- integral-non-linearity-inl/ http://masteringelectronicsdesign.com/an-adc-and-dac- integral-non-linearity-inl/ 43

Questions ? Alexander Gurney What is a DAC? Alexander Pitt Types of DACs Guatam Puri Specifications 44

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