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ME 310 Chapter 7 - Digital Methods Sampling, Digital Devices & Data Acquisition ME 310 - Instrumentation Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Digital Sampling and Conversion Mechanical Engineering University of Kentucky 1 ME 310 Chapter 7 - Digital Methods Introduction to the Digital World • What is a digital signal and what differentiates it from other signals? – All electronic signals (and thus circuits) are either analog or digital. Both are voltage (or current) signals. (See Chap. 1 & 2 slides for examples.) • An analog signal is continuous; i.e., infinitely divisible into smaller and smaller parts. • A digital signal is quantized; i.e., the information is divided into discrete quantities of a finite size. • Up to c. 1950, prior to the advent of the transistor (and thereafter the computer) all signals were essentially analog. Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Analog versus Digital Numbers • Since analog numbers are continuous, there are an infinite number of values between any two readings. – Between 12.7 and 12.8 there exists 12.71, 12.75, 12.762572, etc. • In a digital system, the values are limited to the smallest digital increment. – If 0.1 is the smallest increment, then there are no values between 12.7 and 12.8. – Thus, any digital system can be represented by integer values alone by proper scaling; 12.7 and 12.8 become 127 and 128 (×10), respectively. Since the system is represented by integers alone, no real values exist between 127 and 128. Mechanical Engineering University of Kentucky 2 ME 310 Chapter 7 - Digital Methods Binary Representation • Digital information is transmitted using binary; each number is made up of a pre-determined quantity of bits and 8 bits make a byte. • Every bit has a value of 0 or 1; either off or on. • Examples of a 4-bit number are 0000, 1111, and 1101. Logic 1 (5 V), high Logic 0 (0 V), low 1 1 0 1 Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Binary Ranges • The total number of possible values in a digital system is given by 2N where N is the number of bits in each number. N 1 4 8 16 32 Total 2 16 256 65,536 4x109 Min 0 0 0 0 0 Max 1 15 255 65,533 Big! • Alternatively, one can use the range -total/2 to total/2-1. Mechanical Engineering University of Kentucky 3 ME 310 Chapter 7 - Digital Methods Counting in Binary • What is the decimal (base 10) equivalent of 1101? – (1 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 8 + 4 + 0 + 1 = 13 • 0 = 0 • 8 = 1000 1-bit • 1 = 1 • 9 = 1001 2-bit • 2 = 10 • 10 = 1010 • 3 = 11 • 11 = 1011 3-bit 4-bit • 4 = 100 • 12 = 1100 • 5 = 101 • 13 = 1101 • 6 = 110 • 14 = 1110 • 7 = 111 • 15 = 1111 Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Digital Measurements • Why use digital signals? – Most measurement inputs are analog in nature - why bother converting to digital? • Computers are digital - signal measurement can be easily automated for recording, analysis and process control. • A single system can handle different types of input; e.g., temperature, pressure, force, etc. • Digital signals are noise-resistant; increased accuracy. • Digital readouts are typically easier to read and less prone to error than analog gages. Mechanical Engineering University of Kentucky 4 ME 310 Chapter 7 - Digital Methods Digital Conversion • The analog signal of the measuring input must be converted to a digital signal for the digital system (computer) to read it. – Analog-to-digital (A/D) converter. • Likewise, digital output can be converted into an equivalent analog signal. – Digital-to-analog (D/A) converter. A/D V V D/A Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Digital Conversion • To convert an analog signal from a device, one needs Correct cabling A/D converter (internal or external) Input + + Signal Computer Software + Mechanical Engineering University of Kentucky 5 ME 310 Chapter 7 - Digital Methods Resolution • The resolution, Q, of a A/D converter is determined by the input range (for example, -10 V to +10 V) and the number of bits. Q = ΔVFS 2N • Thus, a -10 to +10 V range has a voltage resolution of 78 mV, 4.8 mV, or 0.3 mV for a 8-bit (28=256), 12-bit € (212=4096), or 16-bit (216=65,536) converter, respectively. (These are typical bit depths of standard A/D converters.) Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Quantization Error • The finite resolution of an A/D converter results in a fixed error in the measurement since the actual analog signal will almost always fall somewhere between two bit values. • This is called quantization error and can be estimated by uq = εV 2 • The error can only be reduced by using more bits. € Mechanical Engineering University of Kentucky 6 ME 310 Chapter 7 - Digital Methods Problem 7.14 • A 12-bit A/D has full scale range of E=5 V and an accuracy of 0.03% full scale. Estimate (i) quantization error; (ii) total error; (iii) relative uncertainty. The quantization error is determined by FSR and depth: Q = 5V /212 = 1.2 mV eQ = ±Q /2 = ±0.6 mV The maximum possible error is the sum of all errors: € emax = eQ + eaccuracy = 0.6 + 5(0.0003) = 2.1 mV The probable error is given by the relative uncertainty: € 2 2 uE = ± eQ + eaccuracy = ± 0.6 + 5(0.0003) = ±1.6 mV € Mechanical Engineering University of Kentucky € ME 310 Chapter 7 - Digital Methods Saturation • The upper and lower bounds of A/D converters limit the range of input values that can be converted. • If the signal falls outside this range, it is saturated. saturation V V t t • If a signal becomes saturated, it must be appropriately conditioned before conversion from analog to digital. Mechanical Engineering University of Kentucky 7 ME 310 Chapter 7 - Digital Methods Sampling • The sampling rate is the rate at which the analog signal is converted to a digital record. • Each converter has a hardware determined maximum sampling rate. • Typical values are in the 1 kHz (1000 samples/second) to 100 MHz (100,000,000 samples/second) range. • Any analog signal which varies with a frequency greater (or even equal) than the sampling cannot be accurately measured. The signal is undersampled and aliasing may result. Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Aliasing Example (cf. 7.1, 7.2) • Aliasing occurs not only in acquiring data in an experiment, but can also occur in creating data in a simulation • Say we wish to simulate a sine wave; in MATLAB, type »x=[0:6:100] »y=sin(x) »plot(x,y,'bs-') • We get something that looks like a sine wave, right? Mechanical Engineering University of Kentucky 8 ME 310 Chapter 7 - Digital Methods Aliasing Example (cont.) • Since we are already familiar with the behavior of a sine wave, we can immediately tell that this is wrong • Try plotting using finer step sizes, increasing the resolution »x=[0:5:100] »y=sin(x) »plot(x,y,’ro-') »x=[0:1:100] »y=sin(x) »plot(x,y,’gd-') Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Aliasing (cont.) • In each of these cases, we have undersampled the signal • The frequency is determined by the period, f=1/T=1/2π • As a first guess, determine step size by dividing the period by 10, or ~0.6 »x=[0:0.1:100] »y=sin(x) »plot(x,y,'bs-') • Here we used a finer resolution to get more detail Mechanical Engineering University of Kentucky 9 ME 310 Chapter 7 - Digital Methods Sampling Rate & Nyquist Frequency • The sampling rate is determined by the time interval between samples, δt fs = 1 δt • The sampling theorem states that to accurately represent the measured signal, the sampling rate should be at least of twice that€ the highest frequency in the signal f s = 2 f N > 2 f signal • fN is known as the Nyquist frequency € Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Determining Alias Frequencies • As mentioned, if fs is too low, aliasing occurs - misinterpretation of these frequencies as “real” signals or frequencies are called alias frequencies • Using a Fourier series representation of the signal, one can show that the frequencies f and f+m/δt are indistinguishable, where m is an integer • The phase of the alias signal will vary from in-phase to out-of-phase as one goes from one alias frequency to the next (see the folding diagram, Fig. 7.3) Mechanical Engineering University of Kentucky 10 ME 310 Chapter 7 - Digital Methods Fold-Back Diagram • The observed alias frequency and the phase relation can be determined from the fold-back diagram Out- of-p hase 3fN ase In-ph 2fN Out- of-p hase 0 fN • Frequency content that is at frequencies above fN will appear as alias frequencies less than fN; Ii.e., superimposed (folded back) and appear as lower frequencies Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Problem 7.4 • Find the alias frequency that results from sampling the signal f1 at sampling frequency fs 3fN • f1=60 Hz, fs=90 Hz 2fN – Nyquist frequency is fN=fs/2=45 Hz 0 fN – The ratio f/fN=1.33 – From Figure 7.3, a ratio of 1.33 corresponds to a “measured” frequency of 0.67fN – Thus, fmeasured=0.67fN=30 Hz and is out-of-phase with the original signal Mechanical Engineering University of Kentucky 11 ME 310 Chapter 7 - Digital Methods Further Aliasing Issues • Aliasing also crops up when examining frequencies in phase space (spectra) • Cutting of a periodic signal either with too few periods or truncating it out-of-phase will produce (or “leak”) multiple spikes in the spectrum plot (see Fig. 7.4) • This can be solved by taking many periods (at least 10) and starting and stopping the data set at the same point in the period • One can also use anti-aliasing filters or windows to help alleviate these problems Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Calibration • Since the analog signal is measured in volts and we really want to look at the signal in some real units (K, Pa, N, lb, etc.), we need to calibrate the signal during the conversion process. • This requires using some known inputs on the device – another instrument is usually required if a calibration signal is not available. Calibration curve V A/D V quantity Mechanical Engineering University of Kentucky 12 ME 310 Chapter 7 - Digital Methods In Class Demo • DAQ System Computer Force External transducer A/D converter Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Digital Circuits Mechanical Engineering University of Kentucky 13 ME 310 Chapter 7 - Digital Methods Digital Circuits • Digital circuits can be used for a number of applications, including control and signal conditioning. • A digital circuit uses high and low voltages to indicate if the control is on or off. on 5V off off 0V • Typical values are 0 to 5 V. For example, if 0 < V < 2.5 V, the control is off. If 2.5 V < V < 5 V, the control is on. Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Mechanical Digital Circuits • Digital circuits don’t have to be electronic (though almost all are) they can be mechanical as well (see Babbage’s difference engine to the right, c. 1833, it could keep 32 decimal places) • The last mechanical computer was Harvard-IBM Mark I, completed in 1944. It kept 23 decimal places, had a memory of 72 counters, and could do a multiply in six seconds. Mechanical Engineering University of Kentucky 14 ME 310 Chapter 7 - Digital Methods Electronic Digital Circuits • Electronic digital circuits Vacuum tube include – Vacuum tubes – Relays – Magnetic cores or amplifiers – Superconducting junctions – Transistors Superconducting Josephson junction Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Transistors • A transistor is a simple circuit that can produce high or low output from high or low input. • The exact relationship between the input and output (or multiple inputs and outputs) depends on the transistor design. • Transistors are the most basic element of CPU design - they determine the overall processing power. Mechanical Engineering University of Kentucky 15 ME 310 Chapter 7 - Digital Methods The First Transistor • First transistor developed in 1947 by Bardeen, Brattain, and Shockley at Bell Labs (later awarded Nobel prize for discovery) • Associates later developed IC chip and formed Intel • Less than a decade later, all computers used transistors (instead of vacuum tubes or relays, for example) for logic, and later for memory Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Transistor Operation Arrow indicates direction of current Mechanical Engineering University of Kentucky 16 ME 310 Chapter 7 - Digital Methods Moore’s Law • 8086 (c. 1974) computed in 8- bits and had 6,000 transistors • Pentium (1993) computes in 32- bits and has 3,000,000 transistors • P4 (2001) computers in 32-bits and has 42,000,000 transistors TI’s First Transistor Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods IC Chips • Transistors have been combined with other electronic elements such as resistors and diodes to make more complex digital circuits - these have been combined into single devices called integrated circuit or IC chips. • Each IC chip consists of several logic units along with input power (+5 V) and ground. Mechanical Engineering University of Kentucky 17 ME 310 Chapter 7 - Digital Methods Logic Units • Logic units are devices which produce high or low output for given input conditions. • The basic logic units are the inverter (which flips the signal) and the AND, OR, NAND, and NOR gates. • These can all be combined in a single IC chip to perform a one function or serve as a single platform to perform a single type of operation for different parts of a circuit. • The input-output relation of a logic unit or gate can be determined from a logic or truth table. Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods The Inverter • Inverts signal; low (0) ⇒ high (1) or high (1) ⇒ low (0). • It has one input and one output. A Q A Q 0 1 1 0 Mechanical Engineering University of Kentucky 18 ME 310 Chapter 7 - Digital Methods The AND Gate • Both inputs must be high to yield a high output. A B Q 0 0 0 A Q 1 0 0 B 0 1 0 1 1 1 • AND gates can have more than 2 inputs as well. Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods The NAND Gate • The opposite of the AND gate. A B Q 0 0 1 A Q 1 0 1 B 0 1 1 1 1 0 • A single NAND gate contains 4 transistors, 3 resistors, a diode, and is powered by a separate input voltage (+5 V). Mechanical Engineering University of Kentucky 19 ME 310 Chapter 7 - Digital Methods OR and XOR Gates • OR gate is high if either input is high. A B Q 0 0 0 A Q 1 0 1 B 0 1 1 1 1 1 • Exclusive OR (XOR) is high only if either output is high, but not both. Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods IC Families • Various logic units can be combined so that a IC can perform a single function. • Families of IC chips have been created that have special characteristics; the most common of these is the TTL family (transistor-transistor logic). • Some sample TTL chips include – 7400 quad 2-input NAND gate – 7408 quad 2-input AND gate – 7430 8-input NAND gate – 7474 dual D-edge-triggered flip-flop Mechanical Engineering University of Kentucky 20 ME 310 Chapter 7 - Digital Methods 7408 TTL • Quad 2-input AND Gate TTL Voltage Power 4 2-input AND gates TTL Ground Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods 7430 TTL • 8-input NAND gate; VCC max is 5.25 V, min is 4.75 V Mechanical Engineering University of Kentucky 21 ME 310 Chapter 7 - Digital Methods 7474 TTL • The flip-flop; used to store 1 bit of information for later use; very important for CPUs. A B Q Q’ A Q 1 1 1 0 1 1 0 1 0 1 1 0 Q’ 1 0 0 1 B – If A and B are opposite of one another, then Q follows A and Q' is the inverse of Q. – If both A and B are switched to 1 simultaneously, then the circuit remembers what was previously presented on A and B. Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods 7474 D-Type Flip-Flop Mechanical Engineering University of Kentucky 22 ME 310 Chapter 7 - Digital Methods Sample IC Circuits • Counter • Frequency meter • Timer – Most commonly used is the 555. • Multiplexer Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Frequency Meter • Counts how many times a signal pulses per second (Hz). 1/2 Hz square wave 5 MHz crystal- Cascade of seven controlled divide by 10 oscillator IC chips Event counter Input signal Mechanical Engineering University of Kentucky 23 ME 310 Chapter 7 - Digital Methods TTL Frequency Divider A 7474 flip-flop and a 7400 NAND Gate Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods CMOS Logic Families • The complementary metal oxide semiconductor family (CMOS) has equivalents to most of the TTL chips. • CMOS chips are much lower in power requirements (drawing about 1 mA) and operate with a wide range of supply voltages (typically 3 to 18 volts). • The CMOS model number will have a C in the middle of it, e.g., the 74C04 is the CMOS equivalent to the TTL 7404. A big drawback is extreme sensitivity to static electricity - they must be carefully protected from static discharges. Mechanical Engineering University of Kentucky 24 ME 310 Chapter 7 - Digital Methods Final Comments • Test • Expect to see – Normal and student-t tables – Propagation of error – Simple filters (passive, active) – Low-pass, high-pass, band-pass filters – DAQ systems (issues related to A/D, D/A conversion) Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Additional Digital Examples Mechanical Engineering University of Kentucky 25 ME 310 Chapter 7 - Digital Methods Analog versus Digital • Any signal is either a digital or analog signal – the difference is obvious in their appearance. Analog (continuous) Digital (discrete) Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Analog versus Digital Examples • Most any type of measurement can be digitized – let’s look at time and temperature, for example. – While the change in the height of the mercury is gradual, the readings occur in discrete units - we can guess at a more accurate reading but are limited to the tick- marks for accurate measurements. Mercury thermometer Mechanical Engineering University of Kentucky 26 ME 310 Chapter 7 - Digital Methods Analog versus Digital Examples • Thermocouple – The thermocouple provides a continuous (analog) signal into the meter, which is digitized and displayed on a LCD panel. 12.8 Analog signal Digitized signal Digital Readout Mechanical Engineering University of Kentucky ME 310 Chapter 7 - Digital Methods Analog versus Digital Examples • Timekeeping is another example. – Time can be measured and displayed in either analog or digital fashion. – Time displays usually (but not always!) indicate if the system is keeping track of time digitally or analog, but beware - an analog display may sweep seconds using discretized ticks!. Mechanical Engineering University of Kentucky 27