Part 1: Hardware and Component Selection – Square Wave Output

Previous work in Chapter 8 has shown the limitations imposed by software overhead, and hence the rate of change of signal shape that can be displayed as it occurs or in “real time” is limited by the computer performance. The bipolar transistor 555 timer in astable mode can produce an output signal that can be made to vary from 70 kHz to about four cycles per minute. CMOS versions of the timer can generate frequencies in the megacycle range. Most 555 timer manufacturers include a standard nomograph of the relationship between the capacitance and the R1-R2 values of the resistors in the IC’s timing network. Figure 9-1 is an extended graphic that accommodates the newer versions of the CMOS-based ICs that are able to oscillate at the higher frequencies. The approximate timing network values and resulting free running output frequencies are depicted in Figure 9-1 for the astable configuration of the timer.

In order to keep the output signal from the timer chip in the low frequency range that should be suitable for the DAQFactory graphics, the resistance values should be in the megohm range (million or 10⁶ ohms) and the capacitance value in the 0.1–10 uF (micro or 1/1,000,000 F) range. The graphical data of Figure 9-1 is an approximation, and the actual resistance values chosen for use are somewhat dependent upon the capacitance value selected or available.

An electrolytic capacitor with a value of 1–10 uF should be suitable for this graphics display exercise, but for more accurate work, a higher-quality low-leakage type of capacitor may be required as detailed in the following discussion.

Electronic Components Required

Circuit Schematic

Figure 9-2 depicts the circuit configuration for the astable 555 timer.

The preceding circuit shows a single variable resistance between pins 7 and 2. The circuit will work when the variable resistor is in its mid-travel position but will produce erratic results when at the low end of its resistance value. To avoid any problems with circuit malfunctions, place a fixed value resistor in series with the variable unit to limit the lower end value of the second timing network resistance. The author used a 10 kΩ value for a 1.2 MΩ R1 + R2 network sum value.

Software

After having assembled the astable oscillator, connect the output to a differential input channel on the LabJack and configure a channel for receiving the square wave output. The graphical page component can then be created. For long-duration graphical displays, make sure the channel storage capability is large enough to support the length of the desired time display. The number of values in memory is defined by the value in the channel's “History” box. (The default entry is 3600 that can be filled quite quickly when working in experimental time frames of tens of minutes or fractions of hours.)

Page Components Required

A two-dimensional graphical screen component is created by selecting, on a new page, the 2D Graph entry from the right-click pop-up page component menu, as shown in Figure 9-3.

The default graphical screen component configuration for an x-y graph is displayed in Figure 9-4.

The graphical component is positioned and fitted to the page by pressing the Ctrl key and dragging the squares in the centers of the hatched edge lines to the desired locations on the page. Once the component has been sized, the properties menu for the screen component can be opened by right-clicking the screen component and selecting the “Properties” entry. The properties window for the graph is a multi-tabbed display with multiple entry forms for adding traces to the graph, defining the axis values used for the graphical display, adding identification titles, and selecting colors for the display. The Traces tab of the six-tab properties window, for defining the name entry, is illustrated in Figure 9-5.

A Help screen at the bottom of the properties window explains all of the data entry boxes and tabs that are found in the graphical screen component.

For the square wave being generated by the 9-volt battery-powered 555 timer oscillator, the voltage range was offset to display values from –1 to 9 volts to more clearly depict the time the waveform is at 0 V.

Part 1: Observations

With a 392 kΩ R1, a 900 kΩ R2, and a 10 kΩ series resistance limiting unit, charging a 22 uF, 25 V electrolytic capacitor from a 9 V battery, a graphical display of four high time cycles in 60 seconds was obtained with a midrange setting on R2.

Between the two extremes as depicted in Figures 9-6 and 9-7, the number of signal waveforms being generated on a unit time basis changes.

If the circuit is operated without a series resistor to limit the minimum value of R2, then waveforms such as those of Figure 9-8 may be created.

Part 2: Hardware and Component Selection – Triangular and “Sawtooth” Outputs

In addition to the creation of a square wave signal with a varying duty cycle, the astable 555 timer can be used to generate a “sawtooth” and asymmetrical triangular wave. Basic electronics teaches that when a capacitor is charged or discharged through a fixed value resistor, an increasing or decreasing exponential voltage value is seen across the terminals of the capacitor. When a capacitor is charged with a constant current source, a linear voltage increase is seen across the capacitor terminals. The linear voltage change forms a triangular waveform that can be used to generate a voltage ramp having several applications in chemical analysis and other experimental work.

By assembling the circuit depicted in Figure 9-10, two additional waveforms can be generated.

Part 2: Observations

When the capacitor is charged through a series resistor, the familiar exponential voltage change is observed and recorded as seen in Figure 9-11, between test points 1 and 3.

When the voltage between TP1 and TP2 is measured, a triangular waveform is recorded as illustrated in Figure 9-12.

The charging of the capacitor with a constant current generates the linear voltage ramp across the capacitor plates as seen on the left-hand side of the waveform. The steep right-hand portion of the signal is caused by the rapid discharge of the capacitor when the discharge pin of the timer is connected to ground through the emitter of an internal NPN switching transistor in the 555 timer IC. Although the discharge trace appears to be a straight line, it is in reality the initial portion of the inverse exponential curve of the positive charging curvature seen in Figure 9-11. Figures 9-12 and 9-13 illustrate the effects of adding a fixed series resistance between the capacitor and the discharge pin of the timer IC.

Simple logic would suggest that to obtain a linear triangular waveform, charging and discharging a capacitor through constant current sources and sinks would achieve the desired result, but a simpler solution can be found by using “function generators” that can produce signals of various shapes and frequencies.

Part 3: Hardware and Component Selection – Dual-Slope Triangular Waveform

To obtain a symmetrical triangular waveform, it is easier to use a “function generator.” Like the 741 and 555 ICs, a very successful function generator is the Exar XR-2209 that has been available in an eight-pin dual in-line package (DIP) since 1975. The integrated circuit is built around a circuit known as a voltage-controlled oscillator (VCO). The XR-2209 VCO can simultaneously generate both a square and triangular voltage waveform signal from a single eight-pin DIP. The chip can be powered from a single- or dual-voltage supply as required by the application at hand. (See “Discussion.”) The function generator chip requires some care when single-ended or dual-voltage supplies are used for power as detailed in the manufacturer’s data sheets. The recommended circuit schematic for using the 2209 function generator with a dual-voltage power supply, in the author’s case +/–9-volt batteries, is depicted in Figure 9-14.

To create the positive and negative voltage ramps, the circuit is powered by a pair of 9 V batteries configured as a bipolar +/–9 V supply with a common ground as depicted in Figure 9-15.

If the circuit, when properly assembled on a breadboard, fails to operate as expected, consult the manufacturer’s data sheets and the “Discussion” section.

Part 3: Observations

The XR-2209 can produce symmetrical triangular waveforms as depicted in Figure 9-16 and with an appropriate “pull-up” resistor the square waveform of Figure 9-17. (See discussion on design limitations for “pull-up” resistor selection.)

X vs. Time Recordings

The operating sequence for the 555 timer has been outlined in Chapter 8, and from that summation together with the information in Figures 9-6 and 9-7, we can see that the waveform generated in the astable mode changes frequencies as the R2 resistance value in the timing network varies.

Thus, the total signal period is

T = t₁ + t₂ = 0.693(R1 + 2R2)C

and the frequency of operation is

f = 1/T = 1.44/(R1 + 2R2)C

The capacitor charges through both resistors while discharging through only one. When the R2 resistor in the network is a variable resistance, then the time that the output is low is proportional to the value of the varying resistance. The varying analog resistance in the 555 timing network could thus be digitized by measuring the width of the recording during which the output is low.

There are limitations as to the relative width of both the high and low times that can be generated with the circuit shown in Figure 9-2. Special circuitry is required to keep the oscillator frequency relatively constant, while the widths of the high and low times (the duty cycle) of the oscillator are varied.

Expanding or contracting the time scale of the graphical display can vary the resolution of the waveform displayed.

Graphical displays require a large amount of computer processing resources and, as noted in the previous exercises dealing with time and timing, have a limited ability to respond to a rapidly changing signal. Rapidly changing signals are best digitized with hardware for storage in memory and then, after collection, converted into a graphical format for display.

For slower signal changes, DAQFactory’s ability to store graphical data in its channels and then be able to display it as a strip chart recording can be very useful in revealing hidden information. If the triangular waveform of Figure 9-12 is recorded for 8 to 10 minutes and then the time scale of the graphical display is reconfigured to display an 8-minute window of the data (i.e., an 8-minute window would be 8 minutes × 60 = 480 for the time base), then a host of variations become evident, as displayed in Figures 9-24 and 9-25.

In Figures 9-24 and 9-25, extending the time scale over which the repetitive voltage cycles are displayed has brought out visually the influences of several ubiquitous experimental sources of error.

Most individuals are familiar with the propagation of water waves in a body of still water. Water waves from two sources caused by stones thrown into a pond appear to our eyes to pass through one another without interference. However, if an object is floating on the surface of the water at the same point where the waves pass through one another, a violent pitching of the object is seen. The violent pitching is caused by the superposition principle that sums the amplitudes of the two water displacement waves passing through one another. The distortions visible at 7:14:30 and 7:19:00 in Figure 9-24 could be caused by a second voltage variation wave with an amplitude of ½ volt and a frequency of one cycle in 4 1/2 minutes blending with or interfering with the main signal.

An additional source of pattern distortion caused by a more complex electronics problem involving timed repetitive digital sampling of cyclic analog waveforms is known as aliasing and is discussed in some electronics textbooks.¹ (Aliases are a RPi-Python programming code utility concept.)

The recorded triangular waveforms are reasonably reproducible with respect to their frequency of occurrence as the author’s breadboarded circuit can be seen to be producing 19 cycles in 5 minutes. The reproducibility of the voltage levels however can be seen to be both drifting and oscillating. The lower values for the voltage vary from 3.0 to 3.4, while the upper values vary from 5.7 to 5.0. Although the upper and lower voltage values are varying, the display has a distinct pattern that suggests the system is both drifting and oscillating due to the factors discussed previously.

Finger heat applied to the left- and right-hand transistors in the constant current source produces the expansion and compression in cycle time illustrated in Figures 9-25 and 9-26.

The erratic amplitude seen in addition to the altered cycle time is also a result of thermal effects.

The materials from which electronic components are fabricated also contribute to the noise seen in electronic circuits. Wire-wound resistors are the least noisy, metal films are intermediate, and carbon-based components exhibit the greatest contribution to resistor circuit noise.

Electrolytic capacitors are inexpensive and available in higher values, but virtually all high-value electrolytic units have sizable leakage currents. Leakage currents can cause problems in systems that require cyclic or repetitive reproducibility. Traditional low-leakage capacitors are generally not available in high capacitance values, but when the limited higher-value units are located, they are usually very expensive and large in physical size as depicted in the photo of Figure 9-27.

Creation of a symmetrical triangular waveform can be done with op-amps and capacitors, but a circuit known as a voltage-controlled oscillator has been designed to simultaneously produce both square and symmetrical triangular waveforms. The Exar XR-2209 IC is a module that with an external capacitor and resistor can be powered by dual or single, 4- to +/–13-volt supplies to produce the required signal. Figures 9-16 and 9-17 are typical outputs from the IC. The triangular wave in the author’s breadboard setup can be seen to systematically vary in the peak voltages achieved. The breadboard setup also proved to be very sensitive to the value of the “pull-up” resistor used to develop the square waveform. The component sensitivity is probably due to operating the circuitry in an area near to the extremes of the circuit design.

X-Y Recordings

x-y recordings are often used when the signal to be recorded is cyclic in nature. Because of the cyclic nature of the signal, it is desirable to clear old traces from the x-y screen as new ones will be overlaid on the older data traces. By specifying the number of data points to plot, using the [n] channel value notation, any fraction or multiple of signal cycles can be displayed.

The effects of non-reproducible signals that are seen in Figures 9-22 and 9-23 arise from the same causes that are evident in the variations of the recorded x vs. time signals of Figures 9-24, 9-25, and 9-26.

Experimental

The code presented in Figure 9-29 plots two straight lines and two sinusoidal traces with different frequencies that are graphically displayed in Figure 9-30.

Observations

Examination of the microprocessor plotter demonstration code and the displayed frequencies of the sinusoids validates the expected 20:1 frequency ratio between the sine wave and cosine. The constant values plot as the expected straight lines.

Discussion

Inclusion of the plotter in the Arduino’s IDE has made a very powerful visualization technique available to the experimental investigator. The plotter is both very easy to use and useful. Plots generated by an experimental process being controlled by the microprocessor can be recorded for archiving with the print screen function available on host computers. Experimental plot archiving has been used in experimental work involving the measurements of temperature, motion, and vibration and in light and optics investigations.

Although the plotter is a very useful function, it is at the time of manuscript preparation limited in several aspects of operation. The colors of the traces are fixed by the operating system of the IDE and can be difficult to see at times. The scales are auto-adjusting and unlike the DAQFactory plotter cannot be independently set to different values.

Occasionally on initial start-up, the plotter will produce spurious images such as depicted on the left in Figure 9-31 or improperly auto-scale the y axis.

As is seen in the preceding figure, the plotter settles into reproducibility reasonably quickly but on occasion may plot erroneously until the 500-point window refreshes itself and the auto-scale functions also settle into a reproducible plot mode.

Introduction

As noted, graphical plotting of experimental data can take two forms. If the data is generated at a high rate, it is best saved by streaming into memory for storage and analyzed graphically at a later time. Experimental data generated at a slower rate can often be displayed as it is created in a “live” or “real-time” display. Python and the RPi use a graphical plotting library called matplotlib for display of both live and stored data.

An example of a Python matplotlib code that plots out the values of the voltage at the wiper arm of a 10 kΩ potentiometer biased between the 3.3-volt RPi power source on the GPIO array and its ground is provided in Listing 9-1 at the end of the chapter. The code has been modified from the strip chart recorder program that can be found as “animation example code: strip_chart_demo.py” in the matplotlib documentation. The documentation contains a full development tutorial for the use of this type of animated graphical display.

Although the RPi does not have an extensive selection of commercial graphical display software applications available, the matplotlib can provide a substantial basis from which the required application can be developed. The relatively short program used to monitor the varying potentiometer voltage in this exercise is equipped with several advanced utilities for in-depth examination of the recorded graphical presentation. A section of the matplotlib documentation entitled “Interactive Navigation” describes the actions of the seven buttons seen in the bottom-left corner of the plotting display as seen in Figures 9-32 and 9-34. The left button restores the focus of the display when any of the display manipulation or storage buttons has been used. Buttons allow sections of the recorded trace to be saved as seen in Figure 9-33 and enlarged as seen in Figures 9-34, 9-35, and 9-36. In addition to the button-activated utilities, the library example also displays the coordinates of the mouse cursor so that exact points can be identified by placing the cursor pointer at a point of interest in the tracing and reading the x and y coordinates of the point in terms of the display time and the measured data value from the numerical values displayed in the lower right-hand corner of the plotter frame.

The matplotlib program is also very easy to alter the scale of either plotting axis, but because of the time scale inconsistencies seen in previous exercises, the plotter time base displayed needs to be calibrated as described in the following experimental section.

Experimental

To demonstrate the plotting facility available with the RPi, an example can be created from the gpiozero library and an MCP3008 ADC IC reading the voltage from the wiper of a biased potentiometer. The wiper voltage is digitized by an MCP3008, 10-bit ADC configured as described in Chapter 6, Figure 6-17. To facilitate programming with the ADC, the gpiozero library has been used to provide the plotting data through accessing the “pot.value” attribute of the object instantiated in the line “pot = MCP3008(0)”. The creation of the pot object with the gpiozero library enables the programmer to access the wiper voltage value connected to the first channel on the ADC chip. The value is automatically normalized to a dimensionless floating-point value between 0 and 1 by setting the code variable to be plotted equal to the pot.value attribute.

The configuration of the RPi with the gpiozero library to access the MCP3008 ADC also allows the plotting program to be modified to accept any sensor or transducer that is able to supply a voltage value of 3.3 V or less. Figures 35 and 36 are two traces that have been made from the output of a 555 IC timer that has been wired to the first or 0 channel of the digital converter. The configuration of the 555 IC is illustrated in the right-hand drawings and circuitry of Chapter 8, Figure 8-8. For this experiment, R1 and R2 were 4.7 kΩ, and C1 was a 100 μF electrolytic capacitor in the 555 timer RC network. The output circuitry also included an LED and current limiting resistor to aid in circuitry assembly, verification of electrical operation, and validation of recorder display by observing a continuous LED flashing at a rate of 59 flashes in 60 s. The final two expanded scale figures, Figures 9-37 and 9-38, were made with the “save a trace” button of the options row at the bottom left of the plotter display.

In order to aid in the development of the adaptation of the published matplotlib strip chart recorder code that uses an internal random number generator to create the y values for the plotter example output, the author inserted a number of diagnostic print statements in the code being modified. The print statements stream out the values of certain variables at points in the executing code to the Python console to aid in the development of different methods for adapting the code to follow data from different sources. Commenting out the diagnostic print lines will clear the console display. The streamed-out variable data is seen in the console displays as the left-hand screens in Figures 9-32 to 9-34. When no longer required, the print lines can be commented out.

Several factors must be taken into account when using graphical data displays on the RPi. As has been noted in previous exercises and previously, the time base of the system is not constant, and hence the time scale at the bottom of the plotter display is of limited reliability and must be semi-quantitatively calibrated for semi-quantitative use (see “Discussion”).

Observations

Figure 9-32 displays the voltage value trace from 80 to 120 minutes into the experiment in which the author has manually turned the potentiometer shaft at the times recorded on the un-calibrated relative time axis of the display. The trace is relatively quick to respond, but rapid twisting of the shaft can overrun the display’s ability to keep up with the changing data value.

Figure 9-33 illustrates the actions invoked when the “save a figure” icon at the extreme right of the row of options is clicked. The graphical figure is saved as a png image in the documents file of the RPi.

In the screen capture of Figure 9-34, the cursor of the mouse had been placed on the vertical response line just past 17 minutes, and the exact coordinates of the point were then printed in the bottom right-hand corner of the display.

Figure 9-34 illustrates the scale expansion option that expands the area enclosed by a mouse-drawn box to a full-screen display. The expanded image can then be saved as noted previously, or the “return to previous view” icon can be used to restore or resume the normal plotting action.

In the following two figures, the output from a 555 timer configured as detailed was recorded at expanded scales with the “save a figure” option. Figures 9-35 and 9-36 illustrate the ability of the software to save the plotted data in shorter time scales from external voltage-generating sources.

Discussion

Graphical data recording with the strip chart recorder program from the Python matplot library is a very adaptable and flexible system that can be used to display data directly from sensors attached to the GPIO array or from the Python serial port.

In Figure 37, the output from a 555 timer configured with R1 = 5.83 kΩ, R2 = 4.7 kΩ, and a 420 μF C1 timing capacitor created ten LED flashes in 33 sec. The timer was powered by the 3.3 V supply of the GPIO array and calibrated for a 4-minute display window as detailed in the “Experimental” section.

Figure 9-38 illustrates the scale expansion capability available with the display option buttons of the data plotting program.

A significant number of sensors have been coded into the gpiozero library that could be used to provide data for the matplotlib plotting program.

One of the advantages of graphical data displays becomes obvious when the variation in the time width of the rectangular pulses is presented in the visual format of Figure 9-38.

Table 9-1 demonstrates a limitation of the time base used for the RPi graphical data displays. A progressive incremental halving of the Dt value increased the time measurement, but the return to the 0.005 value produced a 2-second difference from the original measured value. The differential validates the earlier caution noted in the manuscript with respect to the RPi operating system priorities that can interfere with the timekeeping of the input and output operations of the computer.