Digital to Analog Conversion

1. Digital to Analog Conversion Heather Humphreys Cheng ShuNgoo Woongsik Ham Ken Marek

2. 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

3. 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.

4. Woongsik Ham Principal components of DAC

5. Analog Output Signal Digital Input Signal Woongsik Ham What is a DAC? • Digital  Analog • Each binary number sampled by the DAC corresponds to a different output level.

6. Ideally Sampled Signal Output typical of a real, practical DAC due to sample & hold Woongsik Ham Typical Output 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

7. 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.

8. Common Applications Woongsik Ham Piece-wise Continuous Output Analog Continuous Output Digital Input n bit DAC 0 bit 011010010101010100101 101010101011111100101 000010101010111110011 010101010101010101010 111010101011110011000 100101010101010001111 Filter nth bit Used when a continuous analog signal is required. Signal from DAC can be smoothed by a Low pass filter

9. Common Applications:Function Generators Woongsik Ham 1 2 • Digital Oscilloscopes • Digital Input • Analog Ouput • Signal Generators • Sine wave generation • Square wave generation • Triangle wave generation • Random noise generation

10. Woongsik Ham Applications – Video 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.

11. Common ApplicationsMotor Controllers Woongsik Ham 1 2 3 Cruise Control Valve Control Motor Control

12. 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.

13. Ken Marek Types of DAC implementations • Binary Weighted Resistor • R-2R Ladder • Pulse Width Modulator (not covered) • Oversampling DAC (used internally in HCS12)

14. Binary Weighted Resistor Ken Marek • Start with summing op-amp circuit • Input voltage either high or ground • Adjust resistor weighting to achieve desired Vout

15. 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

16. Binary Weighted Resistor Assume binary inputs B0 (LSB) to Bn-1 (MSB) Each Bi = 1 or 0 and is multiplied by Vref to get input voltage Ken Marek B5 B4 B3 B2 B1 B0

17. Example: take a 4-bit converter, Rf = aR Input parameters: Input voltage Vref = -2V Binary input = 1011 Coefficient a = ½ Ken Marek Binary Weighted Resistor

18. Ken Marek Binary Weighted Resistor • Resolution: find minimum nonzero output • If Rf = R/2 then resolution isand max Vout is

19. 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

20. 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

21. R-2R Ladder B2 B1 B0 Ken Marek

22. Circuit may be analyzed using Thevenin’s theorem (replace network with equivalent voltage source and resistance) Final result is: B2 B1 B0 Ken Marek R-2R Ladder Rf Compare to binary weighted circuit:

23. Ken Marek R-2R Ladder • Resolution • If Rf = R then resolution isand max Vout is

24. R-2R Ladder Advantages: Only 2 resistor values Lower precision resistors acceptable Disadvantages Slower conversion rate Ken Marek

25. General comments Circuits as shown produce only unipolar output Replacing ground with –Vref will allow Vout to be positive or negative Ken Marek

26. Cheng Shu Ngoo DAC Specifications: • Reference Voltages • Resolution • Speed • Settling Time • Linearity

27. 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: Vref determines analog output voltage changes to steps taken by 1 LSB of digital input signal (resolution) Cheng Shu Ngoo X = analog output k = Constant A = Vref analog B = Binary (digital) input

28. Reference Voltage Internal vs. External Vref? Cheng Shu Ngoo *Multiplying DAC is advantageous considering the external reference.

29. Resolution 1 LSB (digital)=1 step size for DAC output (analog) 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) Cheng Shu Ngoo 1 LSB

30. 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 Cheng Shu Ngoo

31. Settling Time 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

32. Settling Time Cheng Shu Ngoo tsettle

33. 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 Integral non-linearity (INL) & Differential non-linearity (DNL) Cheng Shu Ngoo Linear Non-Linear

34. 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

35. Heather Humphreys Gain Error • Slope deviation from ideal gain • Low Gain: Step Amplitude Less than Ideal • High Gain: Step Amplitude Higher than Ideal

36. 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).

37. 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

38. 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

39. 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).

40. Heather Humphreys 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.'

41. 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.

42. Heather Humphreys Resolution Errors • Inherent errors associated with resolution • 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 Better Resolution (3 Bit) Poor Resolution (1 Bit)

43. 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.

44. Woongsik Ham Project Applications • Motor speed controller • Solenoid valves (pneumatics) • Digital Motor Control • Computer Printers • Sound Equipment (e.g. CD/MP3 Players, etc.) • Electronic Cruise Control • Digital Thermostat

45. 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 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_Oscilloscopes.html • http://www.bkprecision.com/power_supplies_supply_generators.htm • http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/dac.html#c4