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**Digital to Analog Conversion**

Heather Humphreys Cheng Shu Ngoo Woongsik Ham Ken Marek

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**Topics Discussed Woongsik Ham 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

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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. Explain picture in detail. 1 min

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**Principal components of DAC**

Woongsik Ham Principal components of DAC Explain picture in detail. 1 min

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**What is a DAC? Digital Analog**

Woongsik Ham What is a DAC? Digital Analog Each binary number sampled by the DAC corresponds to a different output level. Digital Input Signal Analog Output Signal Explain picture in detail. 1 min

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**Ideally Sampled Signal**

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. Output typical of a real, practical DAC due to sample & hold Ideally Sampled Signal DAC Explain graphs in detail min

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**Types of DAC Multiplying DAC* Nonmultiplying DAC**

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.

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**Common Applications Used when a continuous analog signal is required.**

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 Analog Continuous Output Digital Input n bit DAC 0 bit Filter nth bit

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**Common Applications: Function Generators**

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

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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.

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**Common Applications Motor Controllers**

Woongsik Ham Common Applications Motor Controllers Cruise Control Valve Control Motor Control 1 2 3

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**Types of DAC Multiplying DAC* Nonmultiplying DAC**

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.

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**Types of DAC implementations**

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

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**Binary Weighted Resistor**

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

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**Binary Weighted Resistor**

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

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**Binary Weighted Resistor**

Ken Marek 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 B5 B4 B3 B2 B1 B0

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**Binary Weighted Resistor**

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

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**Binary Weighted Resistor**

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

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**Binary Weighted Resistor**

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

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**R-2R Ladder Each bit corresponds to a switch:**

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.

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Ken Marek R-2R Ladder B2 B1 B0

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

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**R-2R Ladder Resolution If Rf = R then resolution is and max Vout is**

Ken Marek R-2R Ladder Resolution If Rf = R then resolution is and max Vout is

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**R-2R Ladder Advantages: Disadvantages Only 2 resistor values**

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

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**General comments Circuits as shown produce only unipolar output**

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

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**DAC Specifications: Reference Voltages Resolution Speed Settling Time**

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

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**Reference Voltage Determines Characteristic of DACs**

Cheng Shu Ngoo 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) To a large extent, the characteristics of a DAC are defined by its reference voltage, whether generated within the DAC or applied externally. First, the reference voltage (VREF) sets the DAC's maximum output voltage if the output signal is not amplified by an additional output stage. VREF also defines the voltage step by which the output changes in response to a 1-LSB transition at the input. One step equals VREF/2N, where N is the DAC resolution. When connecting an external reference, you should consider not only the current required and the voltage range of the DAC's reference input, but also any dynamic effects produced by the DAC's inner structure. With variation of the applied digital value, the reference input resistance can also change. Thus, the reference selected must be capable of following each load step within the required time, or you must add a capacitor or an op-amp buffer. X = analog output k = Constant A = Vref analog B = Binary (digital) input 27

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**Reference Voltage Internal vs. External Vref? Internal External**

Cheng Shu Ngoo Internal vs. External Vref? Internal External Non-Multiplier DAC Vref fixed by manufacturer Qualified for specified temperature range Multiplying DAC Vary Vref Consider current required Consider Voltage range Consider dynamic effects of inner structure Multiplying mode - The variable voltage is multiplied with the adjusted digital input value and transferred to the output, producing the effect of an accurate digital potentiometer. For this operating mode you should consider the DAC's bandwidth and voltage range, as well as dynamic characteristics of the reference input; such as voltage feedthrough from the reference input to the output at a digital value of zero. *Multiplying DAC is advantageous considering the external reference. 28

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**Resolution 1 LSB (digital)=1 step size for DAC output (analog)**

Cheng Shu Ngoo 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) 1 LSB Smallest output voltage change for change in 1 LSB digital input Diff between successive values

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**Speed (Max. Sampling Frequency)**

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 Human hearing range - ~20Hz – ~20kHz

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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

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Settling Time Cheng Shu Ngoo tsettle

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**DAC Linearity Cheng Shu Ngoo**

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) Linear Non-Linear

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**Types of DAC Errors Gain Error Offset Error Full Scale Error**

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

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**Gain Error Slope deviation from ideal gain**

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

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**Offset Error The voltage offset from zero when all input bits are low**

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).

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**Full-Scale Error Includes gain error and offset error**

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

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**Non-Monotonic Output Error**

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

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**Differential Nonlinearity Error**

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).

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**Integral Nonlinearity Error**

Heather Humphreys 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.'

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**Settling Time and Overshoot Error**

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.

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**Resolution Errors Inherent errors associated with resolution**

Heather Humphreys 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)

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**Sources of Errors Deviation of voltage sources from nominal values**

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.

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**Project Applications Motor speed controller**

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

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References Previous student presentations and… Alicatore, David G. and Michael B Histand. Introduction to Mechatronics and Measurement Systems, 2nd ed. McGraw-Hill, 2003. Maxim AN641 Glossary es.html

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