Time Varying Signal Part A - Computer-Based Scientific Instrument | CEM 838, Study notes of Chemistry

Download Time Varying Signal Part A - Computer-Based Scientific Instrument | CEM 838 and more Study notes Chemistry in PDF only on Docsity! September 4, 2007 Version 2007A.1 Time Varying Signals - 1 - Part A Chemistry 838 Thomas V. Atkinson, Ph.D. Senior Academic Specialist Department of Chemistry Michigan State University East Lansing, MI 48824 Table of Contents TABLE OF CONTENTS ............................................................................................................. 1 TABLE OF TABLES.................................................................................................................... 2 TABLE OF FIGURES.................................................................................................................. 3 1. ACQUISITION AND DISPLAY OF DATA ......................................................................... 5 1.1. THE SINE FUNCTION ............................................................................................................. 5 1.1.1. Lissajou Figures – Two Sine Waves.............................................................................. 7 1.1.2. Lissajous Figures – Sine Wave versus Sawtooth ........................................................ 10 1.2. DATASETS........................................................................................................................... 10 1.2.1. Single Variable............................................................................................................ 10 1.2.2. Two Variables ............................................................................................................. 11 1.2.3. Many Variables ........................................................................................................... 12 1.3. TIME BASED ACQUISITION SCENARIOS................................................................................ 14 1.3.1. Acquisition Timing Schemes......................................................................................... 15 1.3.2. When Does Acquisition Begin ...................................................................................... 16 1.3.3. Studying Multiple Signals............................................................................................. 16 1.3.4. Analog vs. Digital......................................................................................................... 16 1.3.5. Acquisition Strategies.................................................................................................. 17 1.3.6. Multiple Acquisitions of a Time Varying Signal ......................................................... 19 1.3.7. Triggered Acquisitions ................................................................................................ 26 2. ACQUISITION HARDWARE............................................................................................. 39 2.1. ANALOG.............................................................................................................................. 39 2.2. DIGITAL .............................................................................................................................. 40 2.3. METHODS OF TIME SHARING .............................................................................................. 41 Chemistry 838 Time Varying Signals Table of Tables September 4, 2007 Version 2007A.1 - 2 - 2.3.1. Multiplexing ................................................................................................................ 41 2.3.2. Alternate Mode............................................................................................................ 42 2.3.3. Chopped Mode ............................................................................................................ 44 3. HOW TO CHOOSE .............................................................................................................. 47 4. RASTER DISPLAYS ............................................................................................................ 47 5. RANDOM ACCESS DISPLAYS ......................................................................................... 47 5.1. CRT.................................................................................................................................... 47 5.2. ANALOG OSCILLOSCOPE ..................................................................................................... 50 5.3. TIME SHARING THE BEAM................................................................................................... 51 6. THE ROLE OF THE OSCILLOSCOPE ............................................................................ 51 7. OSCILLOSCOPE (Y VERSUS TIME EXAMPLES)........................................................ 52 7.1. ASYNCHRONOUS SWEEP, WITH AND WITHOUT BLANKING................................................. 52 7.2. SYNCHRONIZED SWEEP....................................................................................................... 55 7.3. TRIGGERED SWEEP.............................................................................................................. 56 8. RASTER DEVICES (TV, MONITOR) ON THE CRT ..................................................... 58 8.1. TIMING EXAMPLES.............................................................................................................. 59 8.1.1. Black and White .......................................................................................................... 59 8.1.2. Black and White (Multiple Frames Example) ............................................................. 60 8.1.3. Gray Scale ................................................................................................................... 61 8.1.4. Gray Scale (Multiple Frames Example)...................................................................... 62 8.1.5. Interlaced .................................................................................................................... 63 8.2. RASTER IMAGES.................................................................................................................. 64 8.2.1. Black and White .......................................................................................................... 64 8.2.2. Gray Scale ................................................................................................................... 65 8.2.3. Interlaced .................................................................................................................... 66 9. CRT MODES SUMMARY ................................................................................................... 67 10. GRAPHICAL REPRESENTATIONS............................................................................... 67 10.1. FULL BITMAP ..................................................................................................................... 67 10.2. GRAYSCALE....................................................................................................................... 69 10.3. COLOR ............................................................................................................................... 70 Table of Tables TABLE 1 - TWO VARIABLE MEASUREMENTS................................................................................................................11 TABLE 2 - TRIG, POSITIVE SLOPE PARAMETERS ...........................................................................................................27 TABLE 3 - ACQUISITION PARAMETERS .........................................................................................................................28 TABLE 4 - ACQUISITION PARAMETERS .........................................................................................................................29 TABLE 5 - ACQUISITION PARAMETERS .........................................................................................................................32 TABLE 6 - ACQUISITION PARAMETERS .........................................................................................................................33 TABLE 7 - SIGNAL PARAMETERS ..................................................................................................................................42 TABLE 8 - OSCOPE PARAMETERS .................................................................................................................................43 TABLE 9 - OSCOPE PARAMETERS .................................................................................................................................44 Chemistry 838 Time Varying Signals Acquisition and Display of Data 1. Acquisition and Display of Data This section examines issues impacting the acquisition and display of data. This generalized discussion has applicability to analog oscilloscopes, digital oscilloscopes, and computerized data acquisition. 1.1. The Sine Function The mathematical sine function is a periodic function of two variables that has great importance in analog and digital electronics, and in experimental science and will be used in many sections of this document. In many sections of this document the sine function will be used as an example signal. The concepts developed using the sine function as an example can be easily extrapolated to more complicated signals. Indeed, Fourier says that all periodic functions is the sum of sine functions. The following is a brief review of the sine function. The sine function is a function of one dependent and one independent variable. In one manner of description, the dependent variable (y) is the y-component of a vector of length A, A = 1 in Figure 1, that is rotating around the point at the tail of the vector as illustrated in the left side of Figure 1. The independent variable is the angle, θ, of rotation. The right side of Figure 1 is a plot of three cycles of y versus θ. If the vector is rotating at a constant velocity ω, then θ = ωt. In this latter case the x-axis of the plot could be expressed in time as well. September 4, 2007 Version 2007A.1 - 5 - 1.0 -1.0 0 Vp Vpp 0 0.5 1.0 1.5 2.0 3.02.5 Sine.cdr 29-Aug-2007 π/2 (90 )° 0 3 /2 (270 )π ° π (180°) π/4 (45 )°3 /4 (120 )π ° 5 /4 (225 )π ° 7 /4 (315 )π ° y (Periods) 1800 360 540 720 900 1080 (Degrees) π 2π0 3π 4π 5π 6π Radians Figure 1 - Sine Function The sine function is described in mathematical notation as follows. )sin()( θ φ+= Aty )sin()( If the vector is rotating at a constant velocity ω, then θ = ωt. ω φ+= tAty )2sin()( π φ+= ftAty Chemistry 838 Time Varying Signals Acquisition and Display of Data ))(2sin() 22sin()( φ φ πππ tt p A p t t p Aty +=+= fp /1= fω π2= September 4, 2007 Version 2007A.1 - 6 - p t ft φφ π πφ 2 2 == Where: y is the amplitude of the signal at time t A is the amplitude of the sine wave ω is the frequency of the sine wave in radians per unit time f is the frequency of the sine wave in number of cycles per unit time p is the period of the sine wave in units of time φ is the phase angle (or simply the phase) tφ is the time of the offset of the beginning of the cycle of one sine wave to the beginning of a second sine wave of the same frequency. Chemistry 838 Time Varying Signals Acquisition and Display of Data 1.1.1. Lissajou Figures – Two Sine Waves -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 S in e 2 Sine 1 1 11 0 20 0 40 0 60 0 80 0 10 00 -6-4-20246 0 20 0 40 0 60 0 80 0 10 00 -6-4-20246 Sine1 Ti m e 0 200 400 600 800 1000 -6 -4 -2 0 2 4 6 0 200 400 600 800 1000 -6 -4 -2 0 2 4 6 S in e 2 Time 2 1 3 2 3 4 4 5 5 6 7 8 9 10 6 7 8 9 10 11 2,7 3,8 1,6,11 4,9 5,10 Lissajou1.cdr 3-Sep-2007 Figure 2 - Lissajou Figure (2:4) September 4, 2007 Version 2007A.1 - 7 - Chemistry 838 Time Varying Signals Acquisition and Display of Data e.) 5:3 1.1.2. Lissajous Figures – Sine Wave versus Sawtooth Figure 6 – Projection of a Sine and Sawtooth Figure 6 illustrates how the trajectory of the dot on an analog oscilloscope can be determined given the signals applied to the horizontal and the vertical channels. If one were to carry out this procedure for longer periods of time, one sees that if the two signals are periodic, the trajectory is reproducible and the dot retraces the same closed path. 1.2. Datasets The choice of a scenario used to acquire data depends on a large number of factors. For instance, is the desired function periodic or aperiodic? Is the sample stable, i.e. can the experiment be reproduced? Will a single acquisition of data be sufficient, or will multiple acquisitions be needed, say, to average out the noise or to study the changes in the sample as a function of time? 1.2.1. Single Variable In a few situations, the results of an experiment is a set of measurements that yield a set of single variables. The measurements can be made at any time, i.e., there are no constraints as to when. Furthermore, there are no other variables are considered. An example is flipping coins in a probability experiment. The only information needed whether each flip (yi) results in “heads” or “tails” Equation 1 nnnn yyyyyyyy ,,,, 1,23,43,21 −−−L September 4, 2007 Version 2007A.1 - 10 - Chemistry 838 Time Varying Signals Acquisition and Display of Data 1.2.2. Two Variables Many experiments produce datasets of two variables such as the typical spectrum seen in Figure 7. Classically, the measurements of the individual x,y pairs are made at anytime, merely set x and measure y. An example is when the spectrum of a stable sample is acquired with a spectrometer with manually selected wavelength. Another example is the classical titration where the independent variable is the volume of titrant added up to that point in the titration. The dependent variable is the state of the indicator, e.g. the pH of the solution, or the color of the indicator at that point. September 4, 2007 Version 2007A.1 - 11 - ),(),,(),,(),,( 112211 nnnn yxyxyxyx −−L 3.5 4.0 4.5 Equation 2 0 100 200 300 400 500 600 0.5 1.0 1.5 2.0 3.0 2.5 D ep en d ar ia bl e Independent Variable GeneralMeasurement.opj - TwoLines - 29-Aug-2007 en t V Figure 7 - Typical Experimental Measurement Table 1 lists a number of two variable experiments of importance to the modern chemist. Table 1 - Two Variable Measurements Measurement Independent Variable Dependent Variable Absorption Spectrometry Wavelength Absorption of incident light Emission Spectrometry Wavelength Emitted light intensity EPR, NMR Magnetic field Absorption of incident electromagnetic radiation Voltametry Voltage across the Electrochemical Cell Current through cell Kinetics Time Concentration of a given species Titration Volume of titrant State of the indicator Single Crystal Crystallography Orientation of detector in 3 space X-ray intensity Mass spectrometry Magnetic field Ion current Chemistry 838 Time Varying Signals Acquisition and Display of Data An important variant of this type experiment occurs when the independent variable is time. An example of this is the kinetic experiment where the concentration of a chemical species is to be measured as a function of time. These experiments produce datasets of the form shown below. September 4, 2007 Version 2007A.1 - 12 - ),(),,(),,(),,( 112211 nnnn ytytytyt −−L (),(( tytx ),,(),,,(),,,(),,,( ,2,11,21,112,22,121,21,11 nnnnnn yyxyyxyyxyyx −−−L Time In te ns ity Equation 3 Another example is the typical modern spectrometer where the output wavelength of the monochromator can be scanned over a region of wavelengths, usually linearly with time. As the wavelength is scanned, the amplitude of the output signal, e.g. absorbance, transmittance, or light intensity, is automatically recorded. This precludes the need for the operator to manually set the wave length, make a reading, change the wave length, etc. ))(),(()),(),(()),(),(()), tytxtytxtytx L 112211 nnnn −− Equation 4 where x(ti) is the wavelength at time ti, y(ti) is the intensity of the signal at time ti. The actual dataset acquired is typically of the form of Equation 3. The conversion of the data set to the form of Equation 4 is achieved with the parameters of the wave length sweep, i.e. the initial wavelength and the rate of change of wavelength. 1.2.3. Many Variables The next level of complexity is the case where there are two dependent variables (y1,i and y2,i) and one independent variable (xi). Figure 8 is an example of this type experiment where the intensity of light emitted from a sample is simultaneously measured at two wave lengths providing the monitoring of two different species in the sample. In this case, time is the independent variable. The intensity of the signal at wavelength λ1 and the intensity of the signal at wavelength λ2 the dependent variables. Equation 5 Figure 8 - Time course of Two Variables Chemistry 838 Time Varying Signals Acquisition and Display of Data September 4, 2007 Version 2007A.1 - 15 - StandardWindow.cdr 20-JUL-1997 ymin tmin tmax (x )min ( )xmax ymax Δy Δt (Δx) Figure 10 - Acquisition Window An experiment can be thought of a series of measurements of one or more dependent variables with time as the independent variable. An acquisition window, i. e. Figure 10, describes how the data is acquired for a given dependent variable. In essence, the measurement process is the discovery of the set of points on the grid of the acquisition window that are the closest to the signal or parameter being measured. Of course, what actually happens is that the grid point nearest the physical parameter for tmin is determined and then that for the next time increment, etc. sequentially in time across the window. The typical goal is to optimize the window so that the signal being acquired fills the window giving the maximum resolution possible. The window is defined by the choices of the parameters tmin, tmax, Δt, ymin, ymax, and Δy. The choices are constrained by the needs of the experiment and the abilities of the acquisition system. 1.3.1. Acquisition Timing Schemes Figure 11 shows a constant Δt, which is the most common strategy. Figure 11 contains an example of an acquisition using equal acquisition intervals. Figure 13 and Figure 12 suggest other, nonlinear strategies that may be desirable. The ultimate goal is to gather the best information possible about the signal of interest. More data points are usually desired for the portions of a signal that are changing more rapidly, hence, the non-linear cases could be considered in such cases. Chemistry 838 Time Varying Signals Acquisition and Display of Data September 4, 2007 Version 2007A.1 - 16 - -2 0 2 4 6 8 10 12 0 5 10 15 20 25 30 Time A m pl itu de -2 0 2 4 6 8 10 12 0 10 20 30 40 50 60 70 80 Time A m pl itu de Signal Timebase Signal Timebase Figure 11 - Equal Acquisition Intervals Figure 12 - Varied Acquisition Intervals -2 -1 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 Time A m pl itu de -2 0 2 4 6 8 10 0 20 40 60 80 100 120 140 Time A m pl itu de Signal 1 Signal Timebase Timebase Signal 2 Figure 14 - Multiple Signals Figure 13 Exponential Acquisition Intervals 1.3.2. When Does Acquisition Begin Another important consideration is the specification of tmin, i.e. when does the acquisition of information begin. Typically, the acquisition of a signal is to begin at a particular time. The identification of when that time, i.e. the trigger event, has occurred causes the acquisition to begin. 1.3.3. Studying Multiple Signals Figure 14 illustrates a frequent need to acquire more than one signal at a time. A common approach is to use a multiplexed ADC which results in the timing shown in Figure 68. 1.3.4. Analog vs. Digital The implied quantized nature of the measurements in this discussion is slanted toward the use of Analog to Digital Converters to make the measurements. However, the use of analog Chemistry 838 Time Varying Signals Acquisition and Display of Data oscilloscopes, analog recorders, and manual recording to acquire a set of data is similar. In those cases the Δt and Δy are the horizontal and vertical resolutions of the analog device. As with all measurements, the best results occur for these devices when the signal being measured fills the oscilloscope display, the width of the recorder, etc. That is, the best results are when the signal of interest fills the acquisition window. 1.3.5. Acquisition Strategies High Duty Cycle Signal 5 10 15 20 25 30 A m pl itu de -10 -5 0 0 100 200 300 400 500 600 700 800 900 1000 Time Figure 15 - High Duty Cycle Signal September 4, 2007 Version 2007A.1 - 17 - Chemistry 838 Time Varying Signals Acquisition and Display of Data September 4, 2007 Version 2007A.1 - 20 - acquisition window results in the starting points (t = 0, 600, 1200 time units) for the windows to not fall on the same relative points of the cycles of the signal. Chemistry 838 Time Varying Signals Acquisition and Display of Data 1.3.6.1.1. Synchronized 1 0 500 1000 1500 2000 2500 3000 -4 -2 0 2 4 0 500 1000 1500 2000 2500 3000 -4 -2 0 2 4 A m pl itu de Time (Absolute) Figure 18 - Original Signal 0 100 200 300 400 500 -4 -2 0 2 4 0 100 200 300 400 500 -4 -2 0 2 4 A m pl itu de Time (Absolute) Figure 19 - Acquisition A September 4, 2007 Version 2007A.1 - 21 - 0 500 1000 1500 2000 2500 3000 -6 -4 -2 0 2 4 6 0 500 1000 1500 2000 2500 3000 -6 -4 -2 0 2 4 6 Window BWindow A Window C A m pl itu de Time 500 600 700 800 900 1000 500 600 700 800 900 1000 -4 -2 0 2 4 Figure 20 - Acquisition Windows A m pl itu de Time (Absolute) 0 100 200 300 400 500 -4 -2 0 2 4 Figure 21 - Acquisition B 0 100 200 300 400 500 -4 -2 0 2 4 Am pl itu de Time (Relative to Acquisition Window) 1000 1100 1200 1300 1400 1500 -4 -2 0 2 4 1000 1100 1200 1300 1400 1500 -4 -2 0 2 4 A m pl itu de Time(Absolute) Figure 23 - Acquisition C Figure 22 - Overlay A, B, C Chemistry 838 Time Varying Signals Acquisition and Display of Data 1.3.6.1.2. Synchronized 2 0 500 1000 1500 2000 2500 3000 -4 -2 0 2 4 0 500 1000 1500 2000 2500 3000 -4 -2 0 2 4 A m pl itu de Time (Absolute) Figure 24 – Original Signal 0 100 200 300 400 500 -4 -2 0 2 4 0 100 200 300 400 500 -4 -2 0 2 4 A m pl itu de Time (Absolute) Figure 25 – Acquisition A September 4, 2007 Version 2007A.1 - 22 - 0 500 1000 1500 2000 2500 3000 -6 -4 -2 0 2 4 6 0 500 1000 1500 2000 2500 3000 -6 -4 -2 0 2 4 6 Am pl itu de Time (Absolute) 1000 1100 1200 1300 1400 1500 Window BWindow A Window C 1000 1100 1200 1300 1400 1500 -4 -2 0 2 4 Figure 26 – Acquisition Windows A m pl itu de Time (Absolute) 0 100 200 300 400 500 -4 -2 0 2 4 Figure 27 – Acquisition B 0 100 200 300 400 500 -4 -2 0 2 4 A m pl itu de Time (Relative to Acquisition Window) 2000 2100 2200 2300 2400 2500 -4 -2 0 2 4 2000 2100 2200 2300 2400 2500 -4 -2 0 2 4 A m pl itu de Time(Absolute) Figure 28 – Overlay A, B, C Figure 29 – Acquisition C Chemistry 838 Time Varying Signals Acquisition and Display of Data 1.3.6.1.5. Synchronized 5 September 4, 2007 Version 2007A.1 - 25 - 0 500 1000 1500 2000 2500 3000 -4 -2 0 2 4 0 500 1000 1500 2000 2500 3000 -4 -2 0 2 4 A m pl itu de Time (Absolute) 0 100 200 300 400 500 600 -4 -2 0 2 4 0 100 200 300 400 500 600 -4 -2 0 2 4 A m pl itu de Time (Absolute) Figure 42 - Original Signal Figure 43 - Acquisition A 0 500 1000 1500 2000 2500 3000 -6 -4 -2 0 2 4 6 0 500 1000 1500 2000 2500 3000 -6 -4 -2 0 2 4 6 A m pl itu de Time (Absolute) 600 700 800 900 1000 1100 1200 Window BWindow A Window C 600 700 800 900 1000 1100 1200 -4 -2 0 2 4 Figure 44 - Acquisition Windows A m pl itu de Time (Absolute) 0 100 200 300 400 500 600 -4 -2 0 2 4 Figure 45 - Acquisition B 0 100 200 300 400 500 600 -4 -2 0 2 4 A m pl itu de Time (Relative to Acquisition Window) 1200 1300 1400 1500 1600 1700 1800 -4 -2 0 2 4 1200 1300 1400 1500 1600 1700 1800 -4 -2 0 2 4 A m pl itu de Time(Absolute) Figure 46 - Overlay A, B, C Figure 47 - Acquisition C Chemistry 838 Time Varying Signals Acquisition and Display of Data 1.3.7. Triggered Acquisitions Vtrigger source Trigger Source Trigger_0.cdr Time TP TP TP TN TN TN T - Trigger Event (Trigger Slope = Positive)P T - Trigger Event (Trigger Slope = Negative)N Figure 48 - Simple Trigger (Level/Slope) 1. Trigger sources: a. Internal (Channel 1, Channel 2, …) Signal of interest b. External c. Line September 4, 2007 Version 2007A.1 - 26 - F F F Trigger_1.cdr A A AS S SD D D D - Post Trigger Delay (if any) is in process F - House keeping: process data, prepare for next acquisition sequence. T - Trigger Event occurs and the acquisition sequence begins S - Sampling occurs, acquisition of the data is in progress. A - Trigger is armed and the acquisition is ready to begin A T T T Time Figure 49 - Time Course of a Triggered Acquisition Chemistry 838 Time Varying Signals Acquisition and Display of Data 1.3.7.1. Triggered Acquisition – Positive Slope 0 500 1000 1500 2000 September 4, 2007 Version 2007A.1 - 27 - 0 500 1000 1500 2000 -4 -2 0 2 4 -4 -2 0 2 4 S ig na l 54321 Time 0 1 Tr ig O n 0 1 Tr ig Ar m ed 0 1 Tr ig ge re d 0 1 Oscope_Triggered_2.opj - Signals2 - 26-Aug-2007 A cq S ig Figure 50 - Signals Table 2 - Trig, Positive Slope Parameters Trigger Slope 1 Trigger Level 2 Initial Delay 0 Post Trigger Delay 0 Flyback 75 Number of Samples 300 0 50 100 150 200 250 300 350 -4 -2 0 2 4 0 50 100 150 200 250 300 350 -4 -2 0 2 4 Oscope_Triggered_2.opj - Sample2 - 26-Aug-2007 Positive SLope Si gn al A m pl itu de Time (Relative to the Acquisition Window) Figure 51 - Triggered, Positive Slope - Resultant Chemistry 838 Time Varying Signals Acquisition and Display of Data 1.3.7.4. Triggered Acquisition – Positive Slope September 4, 2007 Version 2007A.1 - 30 - 0 1000 2000 3000 4000 -6 -4 -2 0 2 4 6 8 0 1000 2000 3000 4000 -6 -4 -2 0 2 4 6 8 Oscope_Triggered_3.opj - SignalAll - 27-Aug-2007 S ig na l A m pl itu de Time Chemistry 838 Time Varying Signals Acquisition and Display of Data 0 1000 2000 3000 4000 -4 0 4 8 -4 0 4 8 Oscope_Triggered_3.opj - Signal2 - 27-Aug-2007 Time Si gn al 0 1 Tr ig O n 0 1 Tr ig Ar m ed 0 1 Tr ig ge re d 0 1 0 1000 2000 3000 4000 Ac qS ig Figure 56 - Signals Parameter Value Trigger Slope 1 Trigger Level 2.5 Initial Delay 0 Post Trigger Delay 0 Flyback Time 75 Number of Samples 150 0 50 100 150 -6 -4 -2 0 2 4 6 8 0 50 100 150 -6 -4 -2 0 2 4 6 8 Oscope_Triggered_3.opj - Sample2 - 27-Aug-2007 Si gn al A m pl itu de Time (Relative to Acquisition Window) Figure 57 - Resultant Display September 4, 2007 Version 2007A.1 - 31 - Chemistry 838 Time Varying Signals Acquisition and Display of Data 0 1000 2000 3000 4000 -4 0 4 8 -4 0 4 8 Time S ig na l 0 1 Tr ig O n 0 1 Tr ig A rm ed 0 1 Tr ig ge re d 0 1 0 1000 2000 3000 4000 Oscope_Triggered_3.opj - Signal3 - 27-Aug-2007 A cq S ig Figure 58 - Signals Table 5 - Acquisition Parameters Parameter Value Trigger Slope 1 Trigger Level 0 Initial Delay 0 Post Trigger Delay 0 Flyback Time 75 Number of Samples 150 0 20 40 60 80 100 120 140 160 -3 -2 -1 0 1 2 3 4 0 20 40 60 80 100 120 140 160 -3 -2 -1 0 1 2 3 4 Oscope_Triggered_3.opj - Sample3 - 27-Aug-2007 Si gn al A m pl itu de Time (Relative to Acquisition Window) Figure 59 - Resultant Display September 4, 2007 Version 2007A.1 - 32 - Chemistry 838 Time Varying Signals Acquisition and Display of Data 0 1000 2000 3000 4000 -8 -4 0 4 8 Time S ig na l 0 1 Tr ig O n 0 1 Tr ig A rm ed 0 1 Tr ig ge re d 0 1 0 1000 2000 3000 4000 Oscope_Triggered_6.cdr - Signal2 29-Aug-2007 A cq S ig Parameter Value Trigger Slope 1 Trigger Level 8 Initial Delay 0 Post Trigger Delay 0 Flyback Time 75 Number of Samples 300 1500 1550 1600 1650 1700 1750 1800 1850 4 5 6 7 8 9 10 1500 1550 1600 1650 1700 1750 1800 1850 4 5 6 7 8 9 10 Oscope_Triggered_6.cdr - Sample2 29-Aug-2007 S ig na l A m pl itu de Time September 4, 2007 Version 2007A.1 - 35 - Chemistry 838 Time Varying Signals Acquisition and Display of Data 0 1000 2000 3000 4000 -8 -4 0 4 8 Time S ig na l 0 1 Tr ig O n 0 1 Tr ig A rm ed 0 1 Tr ig ge re d 0 1 0 1000 2000 3000 4000 Oscope_Triggered_6.cdr - Signal3 29-Aug-2007 A cq S ig Parameter Value Trigger Slope -1 Trigger Level 2 Initial Delay 0 Post Trigger Delay 0 Flyback Time 75 Number of Samples 300 2000 2050 2100 2150 2200 2250 2300 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 2000 2050 2100 2150 2200 2250 2300 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 Oscope_Triggered_6.cdr - Sample3 29-Aug-2007 Si gn al A m pl itu de Time September 4, 2007 Version 2007A.1 - 36 - Chemistry 838 Time Varying Signals Acquisition and Display of Data 0 1000 2000 3000 4000 -8 -4 0 4 8 Time S ig na l 0 1 Tr ig O n 0 1 Tr ig Ar m ed 0 1 Tr ig ge re d 0 1 0 1000 2000 3000 4000 Oscope_Triggered_6.cdr - Signal4 29-Aug-2007 A cq S ig Parameter Value Trigger Slope -1 Trigger Level -6 Initial Delay 0 Post Trigger Delay 0 Flyback Time 75 Number of Samples 300 3000 3050 3100 3150 3200 3250 3300 3350 -8 -7 -6 -5 -4 -3 -2 3000 3050 3100 3150 3200 3250 3300 3350 -8 -7 -6 -5 -4 -3 -2 Oscope_Triggered_6.cdr - Sample4 29-Aug-2007 S ig na l A m pl itu de Time September 4, 2007 Version 2007A.1 - 37 - Chemistry 838 Time Varying Signals Acquisition Hardware 2.2. Digital (A - Single Channel Data Logger) Analog Pretreatmente1 Controller Buffer Time Base AcqDisplay_2.cdr 31-Aug-2007 Analog Pretreatmente1 Controller Buffer Time Base (B - Single Channel Digital Oscilloscope) Controller Horizontal Control Vertical Control Display Time Base (C - Two Channel, Individual ADC) Analog Pretreatmente1 Analog Pretreatmente1 Controller Buffer Buffer Controller Horizontal Control Vertical Control Display Figure 66 - Digital Acquisition Systems September 4, 2007 Version 2007A.1 - 40 - Analog Pretreatment Analog Pretreatmente2 Analog Pretreatmente3 Analog Pretreatmente3 e1 AcqDisplay_3.cdr 31-Aug-2007 Controller Buffer Computer Controller Horizontal Control Vertical Control Display Figure 67 - Digital Oscilloscope with Computer Connection Chemistry 838 Time Varying Signals Acquisition Hardware 2.3. Methods of Time Sharing This section examines ways of time sharing ADCs and displays among multiple signals. 2.3.1. Multiplexing In this case, one point of one signal is processed, and then a point of the second signal, etc. until all signals have been processed for that time point. Then the next time point is processed for the first signal, and then the next time point for the next signal, etc. This method is the limiting case of the chopped mode. MultiplexADC.cdr 20-JUL-1997 tk tk+1 tk+2 September 4, 2007 Version 2007A.1 - 41 - yk zk yk+1 yk+2 zk+1 zk+2 Δtacq ΔtData y(t) z(t) Figure 68 - Multiplexed ADC Chemistry 838 Time Varying Signals Acquisition Hardware 2.3.2. Alternate Mode Sine 1 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 Time Vo lta ge Sine 2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 Time Vo lta ge Horizontal Voltage -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 Time Fr ac tio n of s w ee p Z-Axis -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 1 = O n, 0 = O ff Table 7 - Signal Parameters Parameter Sine 1 Sine 2 Amp 1 1 phase 0 0.3 Period 0.8 0.4 Freq 1.25 2.5 Offset 2 -2 September 4, 2007 Version 2007A.1 - 42 - Chemistry 838 Time Varying Signals Acquisition Hardware Vertical Switch Position -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 Time C ha nn el Vertical Beam Deflection -4 -3 -2 -1 0 1 2 3 4 0 0.5 1 1.5 2 2.5 3 Time Ve rt ic al P os iti on OScope Presentation -4 -3 -2 -1 0 1 2 3 4 0 0.5 1 Fraction of the sweep Ve rti ca l P os iti on Table 10 Oscope parameters Parameter Value Sweep Time 0.6 Sweep Amplitude 1 Flyback Time 0.2 Sweep+Flyback 0.8 Beam Switch Time 0.02 September 4, 2007 Version 2007A.1 - 45 - Chemistry 838 Time Varying Signals Acquisition Hardware Vertical Switch Position -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 Time C ha nn el Vertical Beam Deflection -4 -3 -2 -1 0 1 2 3 4 0 0.5 1 1.5 2 2.5 3 Time Ve rt ic al P os iti on OScope Presentation -4 -3 -2 -1 0 1 2 3 4 0 0.5 1 Fraction of the sweep Ve rti ca l P os iti on Table 11 Oscope parameters Parameter Value Sweep Time 0.6 Sweep Amplitude 1 Flyback Time 0.2 Sweep+Flyback 0.8 Beam Switch Time 0.01 September 4, 2007 Version 2007A.1 - 46 - Chemistry 838 Time Varying Signals How to choose September 4, 2007 Version 2007A.1 - 47 - 3. How to choose If the periods of the signals are much less than the beam switching time, use the alternate mode. If the periods of the signals are much greater than the beam switching time, use the chopped mode. The Tektronix 2254 have a fixed chop frequency of 10Khz or a beam switching time of 100 microseconds. 4. Raster Displays CRT LCD Plasma 5. Random Access Displays 5.1. CRT Many of the figures in this section are from Section 3-4 and following in "Making the Right Connection" Chemistry 838 Time Varying Signals Random Access Displays found in the section on raster devices. Figure 75 shows how the intensity of the dot will decay after the beam is turned off. As you will see shortly, the beam in these devices is never static but is always rapidly moving across the screen of the CRT. Thus, the length of time a given point (pixel) on the screen remains lit after the beam moves on depends on the kinetics of the phosphor used. As you will see, the utility of these CRT devices depends on the image being retraced and the human eye integrating the trace into a “stable” image. The choice of the speed of the phosphor used will determine the operating characteristics of a given CRT and how it can be used. September 4, 2007 Version 2007A.1 - 50 - Electron Beam Intensity (Flux) of the Light Emitted by the Phosphor (at a given wavelength) Oscope Intensity _1.cdr Time Intensity of the Light Emitted by the Phosphor (at a given wavelength) Oscope_2.cdr Slow Phosphor Beam On Beam Off Fast Phosphor Figure 75 - Phosphor Persistence Figure 74 - Dot Intensity vs. Beam Flux 5.2. Analog Oscilloscope Figure 76 - Oscilloscope Schematic Chemistry 838 Time Varying Signals The Role of the Oscilloscope 5.3. Time Sharing the Beam Figure 77 - Multi-Trace Oscilloscope Schematic 6. The Role of the Oscilloscope Thus, the oscilloscope is actually two independent voltage measuring devices. Just measure the deflection of the beam in the horizontal direction and this is linearly related to the voltage being applied to the horizontal deflection plates. Likewise the vertical deflection is a measure of the voltage being applied to the vertical deflection plates. If the oscilloscope is carefully constructed and calibrated and the measurement of the deflection is carefully done, one should be able to determine the corresponding voltage to about +- 1%. One might ask why spend several thousand dollars to buy an instrument that cannot measure voltages with any better resolution than a $5 VOM. The first answer to this question is that the device can be used to simultaneously compare two signals. The second reason is anchored in the September 4, 2007 Version 2007A.1 - 51 - Chemistry 838 Time Varying Signals Oscilloscope (y versus Time Examples) fact that the beam can be driven from any point within the operational area of the screen to any other point in nanoseconds or even less. Thus, the true role of the oscilloscope is in observing very high frequency signals and/or comparing two time varying signals. 7. Oscilloscope (y versus Time Examples) The most common application of the oscilloscope is to observe a signal as a function of time. This is achieved by putting the signal being studied on the vertical axis of the scope and putting a standard saw tooth signal on the horizontal axis. Since the amplitude of the saw tooth is linear in time, the horizontal component of the beam position will move across the screen of the CRT linearly in time, i.e. sweep (usually left to right) across the screen linearly in time. The horizontal position of the beam will, hence, be a measure of time. The following sections will examine details of this approach and the exact nature of the function applied to the horizontal axis. 7.1. Asynchronous Sweep, With and Without Blanking September 4, 2007 Version 2007A.1 - 52 - Min Off Vhorizontal z-Axis (Beam) Max On S F Sweep_1.cdr Time S FS F S - Sweep is on and the beam is moving from left to right F - Flyback - Sweep is returning rapidly to the right of the screen. Figure 78 - Simple Sweep Chemistry 838 Time Varying Signals Oscilloscope (y versus Time Examples) 7.2. Synchronized Sweep Figure 81 - Synchronized Sweep Figure 81 illustrates the results of carefully adjusting the sweep time (time to sweep across the screen) to be an integral multiple of the period of the signal being studied. As you can see, the resultant image is much simplified. Since the vertical axis can be calibrated in measured deflection per volt applied to the vertical deflection input, quantitative measurements of the amplitude of the signal being studied can be made. However, since the horizontal sweep rate must be variable to perform the synchronization, quantitative measurement in the horizontal direction is problematic. In addition, parameters of real signals tend to drift leading to a continual need for adjusting the sweep rate to stay in sync. September 4, 2007 Version 2007A.1 - 55 - Chemistry 838 Time Varying Signals Oscilloscope (y versus Time Examples) 7.3. Triggered Sweep September 4, 2007 Version 2007A.1 - 56 - Vtrigger source Trigger Source Trigger_0.cdr Time TP TP TP TN TN TN T - Trigger Event (Trigger Slope = Positive)P T - Trigger Event (Trigger Slope = Negative)N Figure 82 - Simple Trigger (Level/Slope) Min Off z-Axis (Beam) On Vhorizontal Max S F Trigger_1.cdr Time A AS FA S F S - Sweep is T - Trigger E F - Flyback - A - Sweep is A on and the beam is moving from left to right vent occurs and the sweep begins Sweep is returning rapidly to the right of the screen. armed and ready to begin T T T Figure 83 - Time Course of a Triggered Scope Chemistry 838 Time Varying Signals Oscilloscope (y versus Time Examples) Figure 84 - Triggered Sweep September 4, 2007 Version 2007A.1 - 57 - Chemistry 838 Time Varying Signals Raster Devices (TV, Monitor) on the CRT 8.1.2. Black and White (Multiple Frames Example) -15 -5 5 15 0 200 400 600 800 1000 V er tic al -15 -5 5 15 0 200 400 600 800 1000 H or iz on ta l -2 3 8 0 200 400 600 800 1000 Time B ea m September 4, 2007 Version 2007A.1 - 60 - Chemistry 838 Time Varying Signals Raster Devices (TV, Monitor) on the CRT 8.1.3. Gray Scale -15 -5 5 15 0 100 200 300 400 500 600 V er tic al -15 -5 5 15 0 100 200 300 400 500 600 H or iz on ta l -2 3 8 0 100 200 300 400 500 600 Time B ea m September 4, 2007 Version 2007A.1 - 61 - Chemistry 838 Time Varying Signals Raster Devices (TV, Monitor) on the CRT 8.1.4. Gray Scale (Multiple Frames Example) -15 -5 5 15 0 200 400 600 800 1000 1200 V er tic al -15 -5 5 15 0 200 400 600 800 1000 1200 H or iz on ta l -2 3 8 0 200 400 600 800 1000 1200 Time B ea m September 4, 2007 Version 2007A.1 - 62 - Chemistry 838 Time Varying Signals Raster Devices (TV, Monitor) on the CRT 8.2.2. Gray Scale 1 2 3 4 5 6 7 8 Pixel 1 2 3 4 5 6 7 8 H or iz on ta l L in e Raster (8 x 8) Display Gray Scale Horizontal flyback Vertical flyback Raster8x8gray.cdr 20-JUL-1997 T V Atkinson - Department fo Chemistry - Michigan State University September 4, 2007 Version 2007A.1 - 65 - Chemistry 838 Time Varying Signals Raster Devices (TV, Monitor) on the CRT 8.2.3. Interlaced 1 2 3 4 5 6 7 8 Pixel 1 2 3 4 5 6 7 8 H or iz on ta l L in e Raster (8 x 8) Display Interlaced Gray Scale Horizontal flyback Vertical flyback Raster8x8grayinterlaced.cdr 20-JUL-1997 T V Atkinson - Department fo Chemistry - Michigan State University The traditional television does not transmit a complete frame (raster) of pixel intensity information at one time. The information is transmitted in two parts, each part containing information in alternative rows. The image is recreated by two “interleaved” sweeps across the screen. In the figure above one half of the “interleave process” is represented by the heavy solid line. The other half is represented by the lighter solid line. September 4, 2007 Version 2007A.1 - 66 - Chemistry 838 Time Varying Signals CRT Modes Summary September 4, 2007 Version 2007A.1 - 67 - 9. CRT Modes Summary Type Horizontal Drive Vertical Drive Beam Drive X-Y plot remote signal source remote signal source on Time base Oscope (Simplest) local sweep generator (free running) remote signal source on Time base Oscope (Simple) local sweep generator (free running) remote signal source Blanked on flyback Time base Oscope (Typical) local sweep generator (Triggered) remote signal source Blanked on flyback and when armed Raster (TV, Monitor) local sweep generator local sweep generator remote source (Beam Intensity contains the visual information for a given point (pixel) in the image being displayed.) The longer the persistence, the lower the refresh rate needed to keep an image visible. The longer the persistence, the slower the motion (i.e. the changes from one frame to the next) can be. 10. Graphical representations The representation and visualization of 2D and 3D objects, i. e. graphics, is a major concern in the computer industry. Often graphical representation will provide a more appropriate and efficient method of storing and presenting information. This is a very complicated subject1. Methods for representing 2D and 3D objects are required. 3D objects usually must be represented, i.e. projected, in 2D. Recreating the color of an object in the presentation is a major topic. In addition, there is a wide variation in the nature, e. g. resolution, color characteristics, speed, etc. of the devices and mechanisms used to present an image that can be viewed by the human. Some aspects of this large area of endeavor will be covered in this section. 10.1. Full bitmap The vast majority of modern graphical output devices, e.g. CRT, liquid crystal, and other displays, ink jet, laser, and other printers, are raster devices. In such cases, the image is represented by a collection of picture elements (pixels) arranged in a two-dimensional array (raster) as shown in Table 12. In such representations, graphical elements, e.g. lines, circles... are a set of pixels that are turned "on", against a background of pixels that are "off". In this particular example (32 x 32), a line between 3,5 and 13,15 is thus a set of 11 "darkened" pixels. 1 “Computer Graphics: Principles and Practicies,” 2nd edition, James D. Foley, Andries van Dam, Steven K. Feiner, John F. Hughes, Addison-Wesley, 1993. Chemistry 838 Time Varying Signals Graphical representations Figure 90 - Abraham Lincoln Detail (120 pixels per inch magnified) 10.3. Color Figure 91 shows the response of the human eye to different colors of light. The three responses are for three different sets of cones that respond to light differently. The three sets of cones are sensed separately by the brain and determine the “color” perceived for each point of the image. This phenomenon is the basis of the tri-stimulus theory of color perception. For the human eye, all colors can be achieved by placing varying intensities of the three basic colors in close proximity of each other (See Figure 92). The human eye can resolve about 256 different levels of gray or each of the three basic colors. Thus an 8 bit grayscale is usually sufficient. A 24 bit tri- color representation usually is called “true color.” September 4, 2007 Version 2007A.1 - 70 - Chemistry 838 Time Varying Signals Graphical representations Human Eye's Response to Light 0 10 20 30 40 50 60 70 80 90 100 350 400 450 500 550 600 650 700 750 Wavelength Se ns iti vi ty Gree Red Blu Figure 91 - The Response of the Human Eye to Light Red Green Red Green Blue Blue Resolution Element for Color CRT Display InlineDelta rgb.cdr 21-Mar-1997 Figure 92 - Resolution Element for Color CRT September 4, 2007 Version 2007A.1 - 71 - Chemistry 838 Time Varying Signals Graphical representations 72 - September 4, 2007 - Version 2007A.1 10.3.1. REVISION HISTORY Revision History for Time Varying Signals Version Date Authors Description 1997 21-Jul-1997 T V Atkinson This document is the transcription of my lecture notes as distilled over 3 decades of CEM 838. This is the first edition of the material in this form. Contained the Oscilloscope material. 2000 8-Nov-2000 T V Atkinson Added data acquisition systems 2001 31-Oct-2001 T V Atkinson Added instrument systems section and more acquisition systems. 2002 4-Nov-2002 T V Atkinson Expanded DAC and ADC section. 2003 9-Oct-2003 T V Atkinson Added Data Analysis. Expanded switch section. 2004 12-Oct-2004 T V Atkinson Added ideal/real switch section. 2004.1 4-Nov-2004 T V Atkinson Reformatted and added commentary to the computer interfacing hardware section. Major reorganization and changes to the Acquisition and Instrument Systems Sections. Moved the data analysis section to a separate document. 2004.2 9-Nov-2004 T V Atkinson Added the communication section. 2005.1 3-Nov-2005 T V Atkinson Added ADC system with local buffer, Trigger, S/H, multiplexer. Fixed several typos. 2006.1 31-Aug-2006 T V Atkinson Expanded the introductory oscilloscope sections. Shared Oscope sections with class. 2006.2 4-Oct-2006 T V Atkinson Minor expansion of successive approximation ADC. Corrected some typos. 2006.3 5-Oct-2006 T V Atkinson Corrected more typos. Rearranged acquisition windows. 2007.1 8-Aug-2007 T V Atkinson Modified to include digital oscilloscopes.