Analog-to-Digital and Digital-to-Analog Converters EE138 – Fall 2013

1. Analog-to-Digital and Digital-to-Analog ConvertersEE138 – Fall 2013

2. Basic Concept of Data Converters Analog signals are continuously valued signals, with infinite values in a given range. Analog signals pick up noise as they are being amplified. Analog signals are difficult to store. Digital signals are discretely valued signals, encoded in binary as 0 or 1. Digital information is different from the analog counterpart in two important respects: it is sampled and quantized. It is important to understand what information you need to retain and what information you can afford to lose. Understand of these restrictions will allow us decide how to select the sampling frequency, number of bits, and the type of filter for converting between the analog and digital realms.

3. ADC and DAC • Analog-to-digital conversion (ADC) and digital-to-analog conversion (DAC) are processes that allow microcontrollers to interact with analog signals. • From analog signals to digital signals conversion (ADC): • The sampling theorem • Signal quantization • From digital signals to analog signals conversion (DAC): • Signal reconstruction from sampled data

4. Filters for data conversion

5. The Sampling Theorem If you can exactly reconstruct the signal from the samples, then you have done a proper sampling and captured the key signal information. Definition: The sampling frequency fs, is the number of samples per second. This is to be compared with the signal cyclic frequencies Given a sinusoidal signal with f = 90 cycle/second is sampled at fs=1000 samples/second. In other words, the wave has a frequency of 0.09 of the sampling rate: f = 0.09 x fs= 0.09 x 1000 Equivalently, there are samples taken over a complete cycle of the sinusoid. These samples represent accurately the sinusoid because there is no other sinusoid that can produce the same samples

6. TheSampling Theorem Now, use f = 0.31 x fs This results into 3.1 samples per sine wave cycle. The samples are so sparse they don’t appear to follow the analog wave. But it can be proven that no other sine wave can produce the same type of samples

7. The Sampling Theorem Here, use f = 0.95 x fs This results into 1.05 samples per sine wave cycle. Clearly, this is an improper sampling of the signal because another sine wave can produce the same samples This phenomenon is called aliasing.

8. The Sampling Theorem

9. Vmax = 7.5V 4 4 1111 7.0V 1110 6.5V 1101 3 3 6.0V 1100 5.5V 1011 2 2 analog input (V) analog output (V) 5.0V 1010 4.5V 1001 1 1 4.0V 1000 3.5V 0111 time time 0110 3.0V t1 t2 t3 t4 t1 t2 t3 t4 2.5V 0101 0100 0110 0110 0101 0100 1000 0110 0101 2.0V 0100 Digital input Digital output 1.5V 0011 1.0V 0010 0.5V 0001 0V 0000 analog to digital digital to analog proportionality Analog-to-digital converters Embedded Systems Design: A Unified Hardware/Software Introduction,(c) 2000 Vahid/Givargis

10. One-Bit Comparator Vdd analog signal (Vin) Vref Vout = Vdd when Vin > Vref/2 Vout = 0 when Vin < Vref/2 R Vref/2 R

11. ADC Resolution • For an N-bit ADC, the smallest input voltage that can be resolved is 1 LSb, or: • 1/2N * (Vref+ - Vref-) • where Vref+ is the positive reference voltage and Vref- is the negative reference voltage. • Typical, we use Vref- = 0 V, and refer to Vref+ = Vref, This simplifies to: • 1/2N * Vref • Example: For Vref = 5 V, and N = 4, what is 1 LSb? • 1/24 * 4 V = 1/16 * 5 V = 0.3125 V.

12. ADC, DAC Equations • ADC: Vin = input voltage, Vref+ = reference voltage, Vref- = 0 V. • N = number of bits of precision • Vin / Vref * 2N = output_code • output_code / 2N * Vref = Vin • 1 LSB = Vref / 2N • DAC: Vout = output voltage, Vref = reference voltage. • N = number of bits of precision • Vout/ Vref * 2N = input_code • input_code/ 2N * Vref = Vout • 1 LSB =Vref / 2N

13. Sample ADC, DAC Computations 1) If Vref = 5V, and a 10-bit A/D output code is 0x12A, what is the ADC input voltage? Ans: Vin = output_code/2N * Vref = (0x12A)/210 * 5 V = 298/1024 * 5 V = 1.46 V (ADC Vin) 2) If Vref = 5V, and an 8-bit DAC input code is 0xA9, what is the DAC output voltage? Ans: Vout = input_code/2N * Vref= (0xA9)/28 * 5 V = 169/256 * 5 V = 3.3 V (DAC Vout) 3) If Vref = 4V, and an 8-bit A/D input voltage is 2.35 V, what is the ADC output code? Ans: Output code = Vin/ Vref * 2N = 2.35 V/ 4 V * 28 = .5875 * 256 = 150.4 = 150 = 0x96 (ADC output code)

14. ADC and DAC Summary ADC converts an analog signal (voltage/current) to a digital value. • DAC converts a digital value to an analog value (voltage/current). • Sample period for ADC: time between each conversion, samples are taken at a fixed rate. • Vref is analog signal varies between 0 and Vref, or between +/-Vref • Resolution is number of bits used for conversion (8 bits, 10 bits, 12 bits, 16 bits, etc). • Conversion Time is the time it takes for a analog-to-digital conversion

15. References: http://tintoretto.ucsd.edu/jorge/teaching/mae143a/lectures/8sampling.pdf www.acad.polyu.edu.hk/~mfcytang/ise204/Ninth%20Lecture.ppt‎ ecpe.ee.iastate.edu/arun/Cpre381_Sp06/lectures/adc.ppt