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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 R-2R Ladder 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 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 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 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 6 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 DAC Ideally Sampled Signal 7 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 0001 0011 0110 1100 1011 0101 0010 0111 8 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 Piece-wise Continuous Output Digital Input Analog Continuous Output 0 bit 011010010101010100101 101010101011111100101 000010101010111110011 010101010101010101010 111010101011110011000 100101010101010001111 n bit DAC nth bit 9 Filter What is a DAC? – Alexander Gurney DACs in Audio Digital MP3s HD Radio CDs 10 Analog ->3.5mm Audio Out ->Signal received by speaker ->RCA Audio Out What is a DAC? – Alexander Gurney DACs in Video Digital DVDs OTA Broadcast Youtube 11 Analog ->Composite Output ->Converter Box Output ->Analog Monitor Input DAC Types – Alex Pitt Types of Digital to Analog Converters Binary Weighted Explanation Advantages and disadvantages R-2R Ladder Explanation Example Advantages and disadvantages 12 DAC Types – Alex Pitt Binary Weighted DAC Adds resistors in parallel scaled by two to divide voltage on each branch by a power of two Vout = Analog Out Use transistors to switch between open and close 13 Use a summing op-amp circuit with gain DAC Types – Alex Pitt 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. 14 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 15 LSB DAC Types – Alex Pitt 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 16 Minimum/maximum opamp current Noise overwhelms current through larger resistance values DAC Types – Alex Pitt R-2R Ladder DAC 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) 17 DAC Types – Alex Pitt R-2R Ladder Example Convert 0001 to analog V3 Vref V2 V1 V0 Req V1 RF 18 1 R 1/ 2 R 1/ 2 R V0 V0 R 1 V1 V1 RR 2 V1 R 1 V2 V2 RR 2 V2 R 1 V3 V3 RR 2 DAC Types – Alex Pitt R-2R Ladder Example 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 R-2R Ladder By adding resistance in series and in parallel we can derive an equation for the R-2R ladder. 20 DAC Types – Alex Pitt R-2R Ladder 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 21 DAC Types – Alex Pitt R-2R Ladder 4-Bit Equation Substituting Vout I Vref R Vref B3 B2 B1 B0 I sum Rf Rf R 2 4 8 16 Rf 22 General Equation R f n1 Bi Vout Vref R i 0 2n i DAC Types – Alex Pitt R-2R Ladder DAC Advantages 23 Only two resistor values Can use lower precision resistors Specifications - Gautam Puri 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 24 Specifications - Gautam Puri Reference Voltage Vref 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). 25 Specifications - Gautam Puri Full scale voltage 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 26 Specifications - Gautam Puri Resolution 27 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) 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 Sampling frequency is limited by the input clock speed (depends on microcontroller) and the settling time of the DAC 28 Specifications - Gautam Puri 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 VLSB. A faster DAC will have a smaller settling time 29 tsettle Specifications - Gautam Puri 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 0000 Analog Output Signal Non-linear Analog Output Signal Linear 30 0001 0010 0011 Digital Input Signal 0100 0101 0000 0001 0010 0011 Digital Input Signal 0100 0101 Specifications - Gautam Puri 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 31 Specifications - Gautam Puri 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. 32 Specifications - Gautam Puri Differential non-linear output error 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 33 Specifications - Gautam Puri 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. 34 Specifications - Gautam Puri 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. 35 Specifications - Gautam Puri 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. 36 Specifications - Gautam Puri More important - DNL or INL ? 37 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 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 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 39 Specifications - Gautam Puri Full Scale Error It is a combination of gain and offset error It is measured at the full scale input 40 Specifications - Gautam Puri 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) 41 Higher resolution (3 bits) Specifications - Gautam Puri 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. 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.noise.physx.uszeged.hu/DigitalMeasurements/ADConversion/ADSpecs. pdf Scherz, Paul. Practical Electronics for Inventors. 2nd Edition, McGraw Hill. 2007. http://masteringelectronicsdesign.com/an-adc-and-dacdifferential-non-linearity-dnl/ http://masteringelectronicsdesign.com/an-adc-and-dacintegral-non-linearity-inl/ 43 Questions ? Alexander Gurney What is a DAC? Alexander Pitt Types of DACs Guatam Puri Specifications 44