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Digital to Analog Converters Alexander Gurney Alexander Pitt Gautam Puri 1

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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

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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

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

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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

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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

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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

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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

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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

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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

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

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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

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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

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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

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

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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

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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

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

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

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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

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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

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

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

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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

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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

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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

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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 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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

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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

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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

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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

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

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