11 Lecture Slides ME 3222 Kinematics and Control Lab Lab 2 AD DA and Sampling Theory By Dr. Debao Zhou

2 System Review Computer: Software: Matlab (Lab 1) Hardware: Sensory 626 card (Lab 2) Controller (Labs 5 and 6) Sensors: Encoder/ accelerometer (Lab 3) Amplifier Motor External Equipment (Project) Lab 4

What will we do? What will we use? How to do? Why? –Your background EE 2111 Linear Systems and Signal Analysis Signal and system modeling concepts, system analysis in time domain, Fourier series and Fourier transform. Discrete time domain signals and systems, Z transform, applications. Just read through slide #18. 3

4 Lab 2 Task 1: Equipment, Simulink, Build the model Task 2: A/D - Set the power supply to different voltages –Check and see if you're A/D program can read the voltages correctly –Change the voltage range setting and repeat. Task 3: D/A - Send out different voltage to channel 0 from a program, check the output voltage using a multimeter to see if correct.

5 Computer: Data Acquisition and Processing Signals in data acquisition Software: Matlab Sensors e.g. voltage Amplifier Hardware: Sensory 626 card (Lab 2) A/D Analog signal Digital signal D/A Digital signal Motor What is the difference between analog/digital continues-/discrete- time signals? Analog signal

6 Data Acquisition Card - Sensory Six 24-bit counters for encoders Four 14-bit D/A outputs (analog output) Sixteen 16-bit differential A/D inputs (digital out input) 48 digital signals (digital output)

7 Data Acquisition Tasks Sampling Convert analog voltage/current to digital data (A/D) –Physical meanings of the analog data –Resolution Process of the digital data Convert digital data to analog voltage/current –Resolution –Physical meanings of the analog data

8 Expression of the Sampled data in Computer Blue: Analog data Red: sampled data

9 A/D Resolution calculation 16 bits ADC, measurement range ± 5v –The maximum number that could be represented with a 16-bit word is = The 15 th bit is reserved for sign representation, the represent numbers range is to , or 2 (16-1) -1 = = The resolution = 5v/32767 =5v/( ) = mv Saturation: if the input value is beyond the measurable range, the digital value will saturate at the max/min value. –In the input is 6v, it will give 32767; –-9v will give

10 A/D Resolution Calculation The resolution calculation is similar to that of an ADC Example: –12 bits DAC, measurement range ± 10v –The resolution = 10v/[( )-1] = 4.88 mv If the digital number is beyond the range: –Case I: saturate (similar to the ADC case) –Case II: the voltage values repeat. The above 12-bit DAC range is to +2047, if the digital value is 3000, it will give ( )/2047*10v = 4.66 v, instead of 3000/2047*10v =14.66v. If the digital value is -2500, it will give [-2500-(- 2047)]/2.47*10= mv

11 Lab 2 Task 1: Equipment, Simulink, Build the model Task 2: A/D - Set the power supply to different voltages –Check and see if you're A/D program can read the voltages correctly –Change the voltage range setting and repeat. Task 3: D/A - Send out different voltage to channel 0 from a program, check the output voltage using a multimeter to see if correct.

12 Sampling

Review: Period – Frequency Sinusoidal periodical functions Frequency, period V=cos(  t)=cos(2  ft) = cos(2  t/T)  2  f = 2  /T f = 1  /T 13

14 DT and CT Periodic Function

Displayed and Reconstructed 15

16 Sampling Theory A band-limited signal can be fully reconstructed from its samples, provided that the sampling rate (f s ) exceeds twice the maximum frequency (B) in the band-limited signal (f s >= 2B). This minimum sampling frequency (f s ) is called the Nyquist rate. This results, usually attributed to Nyquist and Shannon, is known as the Nyquist–Shannon sampling theorem, or simply the sampling theorem.

Sampling frequency, input signal frequency and output signal frequency relationship (sine wave) 17 fsfs f s /2 3f s /2 2f s Input Frequency Output Frequency Sampling Frequency f out f in f out f in fsfs

18 Lab 2 Task 4: demonstration of the sampling theorem –Set the sampling frequency to be 1000 Hz; –Change input signal frequency on the function generator to vary from 50 Hz to 1500Hz. –For each input frequency, write down the estimated frequency of the output signal from the oscilloscope

19 Sampling Theory

20 Sampling Theory Trigonometric Fourier Series

21 Sampling Theory

22 Sampling Theory

23 Sampling Theory

24 Sampling Theory

25 Sampling Theory

26 Sampling Theory (Aliasing)

27 Sampling Theory

28 Sampling - Reconstruction