It serves as a model to demonstrate the use of op-amps in a variety of circuits, although the circuits will also show most of the same quantitative characteristics even if op-amps other than the JLAC are substituted. Other op-amps are available that have one or more superior or at least different characteristics. To describe the characteristics of most circuits, it is not necessary to understand the details of the op-amp internals. A simple amplifier using an op-amp, called a noninverting amplifier, is shown schematically in Figure 3.
The term noninverting means that the sign of the output voltage relative to ground is the same as the input voltage. As is typical of op-amp applications, this circuit shows a feedback loop in which the output is connected to one FIGURE 3. In the noninverting amplifier, the feedback voltage is connected to the minus input terminal and the signal input to the plus terminal.
Low resistance values lead to high power consumption. In some circuits, beyond the scope of this book, high resistance values and specialized op-amps are required. This situation is called output saturation. For the J,LAC, saturation will occur when the output is about 2 V less than the power supply voltage. As discussed by Franco , the feedback loop increases the input impedance and decreases the output impedance relative to the open-loop op-amp.
The impedance between the input terminals of the noninverting amplifier is very high. For the circuit shown in Figure 3. This means that the amplifier will not appreciably load the input device for most applications.
The circuit also shows another major advantage of op-amp amplifiers - a very low output impedance on the order of 1 0 , which results in the output voltage being relatively unaffected by connected devices. Log frequency The gain computed from Eq. It indicates that the gain is essentially constant from low frequency up to a cutoff frequency, fe. The range of frequencies between f 0 and fe is the bandwidth of this amplifier.
Above fe, the gain starts to decrease, or roll off, and this roll-off occurs at a rate of 6 dB per octave. Since an octave is a doubling of the frequency, this means that each time the frequency doubles in the range above fo the gain will decrease by 6 dB from the value computed by Eq.
Actually, the gain starts to decline at frequencies slightly below fe since fe is defined as the frequency at which the gain has declined by 3 dB. This roll off in gain at high frequencies is an inherent characteristic of op-amps. The cutoff frequency, fe, depends on the low-frequency gain of the amplifier - the higher the gain, the lower is fe.
This low-frequency gain-cutoff frequency relationship is described by a parameter called the gain-bandwidth product GBP. The noninverting circuit in Figure 3. Some more expensive op-amps have higher values of GBP.
Each amplifier has a lower gain, but the overall gain for the two stages is the same, and the bandwidth will be much higher. For the noninverting amplifier in Figure 3.
Table 4. Repeaters and hubs can also be used to extend the length. Recent advances in fiber-optic based communications have increased these distances significantly. Many standards e. In addition, most recent standards have a provision for delivering DC power to the connected device, eliminating the need for an external power source for devices requiring relatively low power. More recently, wireless connections between devices have become prevalent. The most common standards found in home networks are IEEE In addition, cellular networks are also allowing many devices to connect to a computer remotely via the Internet.
The length and speed of wireless connections depend greatly on the strength of signal between the host and receiver and generally are significantly slower than their hardwired counterpart. Wireless connections, however, allow for mobility such that a measurement system is not confined to a single location. For example, a technician may be able to travel to a remote site with a handheld device and upload the data to a central computer. Various "off-the-shelf'!
A simple, but not very efficient, approach would be to have separate sensors and displays located at each station with the values manually checked periodically to ensure that things are operating smoothly. A more elegant and useful approach, however, would to be combine all of the signals together and display them on single screen. With Ethernet connections, these components can even be spread across several states or countries.
In the above scenario, the custom software may consist of an on-screen schematic of the piping system with the current pressure and temperature displayed at each measurement station. The interface may also allow for various settings to be changed, such as the closing or opening of a valve, by simply clicking on a button located near the valve in the schematic.
Virtual instruments are recognized in contrast to measurement hardware that has a predefined function. For example, a digital multimeter uses an AID converter and a digital display to read and display a voltage level.
Other functions, such as measuring AC voltage amplitude are hard-wired into the device and new functions cannot be added as needed. A computer with an AID convertor and appropriate software can also perform the same functions, but provide additional flexibility and customizability by exploiting the capabilities of the computer to which it is attached.
It would also be possible to display multiple signals on a single screen or perform mathematical operations on the signals. Figure 4. Digitizers are available with speeds in the GHz range and typically have 8-bit precision. As the digitization occurs at speeds greater than that at which the signal can be processed and displayed, the digital output is immediately stored into memory.
The stored signal is then sent to a microprocessor and a display unit, such as an LCD screen. For example, the scope can also be used to measure the signal frequency, amplitude, pulse width, and rise time, or set to trigger on a particular type of event, just to name a few of the functions commonly available. In addition, digital scopes are capable of acquiring and displaying multiple signals simultaneously on the same screen.
The first is the dB Le. The bandwidth takes into account the frequency response of all elements of the oscilloscope prior to digitization e. More details about the limitations associated with bandwidth and 4. A general purpose data logger might be temporarily installed for a building air conditioning system in order to obtain diagnostic performance data over a period of time such as temperature, airflow, and fan operation.
Some buildings are permanently instrumented using data loggers to obtain acceleration and other data if and when an earthquake occurs. The flight data recorder in a commercial aircraft is a specialized data logger and provides important information after accidents. Recent automobiles often include a data logger that records a small amount of speed and other data just before and after an accident may be part of the airbag system.
To take a data sample, for example, the following instructions must be executed: 1. Instruct the multiplexer to select a channel.
Instruct the AID converter to make a conversion. Retrieve the result and store it in memory. In most applications, other instructions are also required, such as setting amplifier gain or causing a simultaneous sample-and-hold system to take data. The software required depends on the application. Using selections from various menus, the operator can configure the program for the particular application.
These programs can be configured to take data from transducers at the times requested, display the data on the screen, and use the data to perform required control functions. There are a number of very sophisticated software packages now available for personal computer-based data-acquisition systems. The programs are configured for a particular application using menus or icons.
They may allow for the incorporation of C program modules. These software packages are the best choice for the majority of experimental situations. Instrumentation Reference Book, 3rd ed. IEEE Std IEEE Board-level systems set the trend in data acquisition, Computer Design, Apr.
Convert the decimal number to 8-bit simple binary. Convert the decimal number 1 to bit simple binary. Convert the decimal number to bit simple binary. Find the bit 2's-complement binary equivalent of the decimal number The number is an 8-bit 2's-complement number. What is its decimal value? How many bits are required for a digital device to represent the decimal number 27, in simple binary?
How many bits for 2's-complement binary? How many bits are required for a digital device to represent the decimal number 12, in simple binary? How many bits are required to represent the number in 2's-complement binary? Find the output in decimal if the input is a b c d 4. Find the output in decimal if the input is 4. Find the output in decimal if the input is o v. Find the output in decimal if the input is a b c d 6. The output of the AID converter is in 2's-complement format.
Find the output of the AID converter if the input to the amplifier is a 1. Find the output of the AID converter if the input to the amplifier is a 0. Estimate the quantization error as a percent of reading for an input of 1. Estimate the quantization error as a percent of reading for an input of 2. If the input is 8. Estimate the quantization error as a percentage of reading for an input An amplifier is connected to the input and has selectable gains of 10, , and Select the best value for the gain to minimize the quantizing error.
What will be the quantizing error as a percentage of the reading when the transducer voltage is 3. Could you attenuate the signal before amplification to reduce the quantizing error? The connected transducer has a maximum output of 7. The connected transducer has a maximum output of 10 mV. Estimate the analog voltage output if the input is simple binary and has the decimal value of Problems 4.
Simulate the successive-approximations process to determine the simple binary output. Specify whether these errors are of bias or precision type. List the questions that you want to discuss with an application engineer working for a supplier. D i screte Sa m p l i n g a n d CHAPTER Ana lys i s of Ti me-Va ryi n g Sig nals Unlike analog recording systems, which can record signals continuously in time, digital data-acquisition systems record signals at discrete times and record no information about the signal in between these times.
In this chapter we introduce restrictions that must be placed on the signal and the discrete sampling rate. For example, a reading sample may be taken every 0. The experimenter is then left with the problem of deducing the actual measurand behavior from selected samples. Figure 5. The important characteristic of the sampling system here is its sampling rate normally expressed in hertz.
Figures 5. To infer the form of the original signal, the sample data points have been connected with straight-line segments. In examining the data in Figure 5. However, we know that the sampled signal is, in fact, a sine wave. The amplitude of the sampled data is also 1 02 5.
This behavior a constant value of the output occurs if the wave is sampled at any rate that is an integer fraction of the base frequency fm e. The data in Figure 5. The frequency, 1 Hz, is the difference between the sampled-data frequency, 10 Hz, and the sampling rate, 11 Hz.
These incorrect frequencies that appear in the output data are known as aliases. It turns out that for any sampling rate greater than twice fm ' the lowest apparent frequency will be the same as the actual 1 04 Chapter 5 Discrete Sampling and Ana lysis of Time-Va rying Signals 1.
The theorem also specifies methods that can be used to reconstruct the original signal. The amplitude in Figure 5. The sampling-rate theorem has a well-established theoretical basis.
There is some evidence that the concept dates back to the nineteenth-century mathematician Augustin Cauchy Marks, The theorem is often known by the names of the latter two scientists. A comprehensive but advanced discussion of the subject is given by Marks This process will be discussed in some detail later in the chapter.
Even if the signal is correctly sampled i. For example, Figure 5. The sampled data are shown as the small squares. However, these data are not only consistent with a Hz sine wave but in this case, the data are also consistent with Actually, there are an infinite number of higher frequencies that are consistent with the data. The higher frequencies can be eliminated from consideration since it is known that they don't exist. In some cases, the requirements of the sampling-rate theorem may not have been met, and it is desired to estimate the lowest alias frequency.
The lowest is usually the most obvious in the sampled data. A simple method to estimate alias frequencies involves the folding diagram as shown in Figure 5. The use of this diagram is demonstrated in Example 5.
Example 5. The lowest alias frequency is the difference between frequency. In part b , the sampling frequency is less than the signal frequency. The folding diagram is the simplest method to determine the lowest alias frequency. In part c , the requirement of the sampling-rate theorem has been met, and the alias frequency is in fact the signal frequency. To know that the frequency is correct, we must insure that the sampling rate is at least twice the actual frequency, usually by using a filter to remove any frequency higher than half the sampling rate.
The process of determining these component frequencies is called spectral analysis. There are two times in an experimental program when it may be necessary to perform spectral analysis on a waveform. The first time is in the planning stage and the second is in the final analysis of the measured data.
In planning experiments in which the data vary with time, it is necessary to know, at least approximately, the frequency characteristics of the measurand in order to specify the required frequency response of the transducers and other instruments and to determine the sampling rate required. In many time-varying experiments, the frequency spectrum of a signal is one of the primary results.
To examine the methods of spectral analysis, we first look at a relatively simple waveform, a simple Hz sawtooth wave as shown in Figure 5. At first, one might think that this wave contains only a single frequency, Hz.
However, it is much more complicated, containing all frequencies that are an odd-integer mUltiple of , such as , , and Hz. The lowest frequency, to, in the periodic wave shown in Figure 5. Of course, Eq. If J t is even, it can be represented entirely with a series of cosine terms, which is known as a Fourier cosine series.
If fit is odd, it can be represented entirely with a series of sine terms, which is known as a Fourier sine series. Many functions are neither even nor odd and require both sine and cosine terms. If Eqs. These have frequencies of , , , and Hz, respectively. As can be seen, the sum of the first and third harmonics does a fairly good job of representing the sawtooth wave. The main problem is apparent as a rounding near the peak- a problem that would be reduced if the higher harmonics e.
If, for example, the experimenter considers the first-plus-third harmonics to be a satisfactory approximation to the sawtooth wave, the sensing instrument need only have an upper frequency limit of Hz. Solution: The fundamental frequency for this wave is 10 Hz and the angular frequency, w is Also, by examination, we can conclude that it is an odd function and that the cosine terms will be zero and only the sine terms will be required. Using Eq. One problem associated with Fourier-series analysis is that it appears to only be useful for periodic signals.
In fact, this is not the case and there is no requirement that f t be periodic to determine the Fourier coefficients for data sampled over a finite time. We could force a general function of time to be periodic simply by duplicating the function in time as shown in Figure 5. If we directly apply Eqs. However, if the resulting Fourier series were used to compute values off t outside the time interval O-T, it would result in values that would not necessarily and probably would not resemble the original signal.
The analyst must be careful to select a large enough value of T so that all wanted effects can be represented by the resulting Fourier series. An alternative method of finding the spectral content of signals is that of the Fourier transform, discussed next. The Fourier transform is a generalization of Fourier series.
The Fourier transform can be applied to any practical function, does not require that the function be periodic, and for discrete data can be evaluated quickly using a modern computer technique called the Fast Fourier Transform. In presenting the Fourier transform, it is common to start with Fourier series, but in a different form than Eq.
This form is called the complex exponential form. These relationships can be used to transform Eq. L cn ejnwot n - oo 5. In Section 5. If a longer value of T is selected, the lowest frequency will be reduced. This concept can be extended to make T approach infinity and the lowest frequency approach zero. In this case, frequency becomes a continuous function. It is this approach that leads to the concept of the Fourier transform. The Fourier transform of a function. Such a signal is not well suited to analysis by the continuous Fourier transform.
The Fs are complex coefficients of a series of sinusoids with frequencies of 0, Af, 2Af, 3Af,. The amplitude of F for a given frequency represents the relative contribution of that frequency to the original signal. Only the coefficients for the sinusoids with frequencies between 0 and N 12 - 1 Af are used in the analysis of signal.
The coefficients of the remaining frequencies provide redundant information and have a special meaning, as discussed by Bracewell A sophisticated algorithm called the Fast Fourier Transform FFf has been developed to compute discrete Fourier transforms much more rapidly.
This algorithm requires a time proportional to N log2 N to complete the computations, much less than the time for direct integration. The only restriction is that the value of N be a power of 2: for example, , , , and so on. Programs to perform fast Fourier transforms are widely available and are included in major spreadsheet programs. The fast Fourier transform algorithm is also built into devices called spectral analyzers, which can discretize an analog signal and use the FFf to determine the frequencies.
It is useful to examine some of the characteristics of the discrete Fourier transform. If we discretize one second of the signal into samples and perform an FFf we used a spreadsheet program as demonstrated in Section A. As expected, the magnitudes of F at f 10 and f 15 are dominant. However, there are some adjacent frequencies showing appreciable magnitudes.
This is a consequence of the definition of the discrete Fourier transform and the FFf algorithm. To get the correct amplitude of the input sine wave, I F I should be multiplied by 21N. In Figure 5. Similarly for Figure 5.
For Figure 5. The actual maximum frequency is Hz. For the FFTs shown in Figures 5. In general, the experimenter will not know the spectral composition of the signal and will not be able to select a sampling time T such that there will be an integral number of cycles of any frequency in the signal.
This complicates the process of Fourier decomposition. To demonstrate this point, we will modify Eq. Although 15 complete cycles of the Hz components are sampled, The results of the DFT are shown in Figure 5.
The first thing we notice is that the Fourier coefficient for It should be recognized that without a priori knowledge, the user would not be able to deduce whether the signal had separate lO-Hz and Hz components or just a single component at 1O. An unexpected result is the fact that the entire spectrum outside of 10, 11, and Hz has also been altered, yielding significant coefficients at frequencies not present in the original signal.
This effect is called leakage and is caused by the fact that there are a non-integral number of cycles of the 1O. The actual cause is that the sampled value of a particular frequency component at the start of the sampling interval is different from the value at the end.
A common method to work around this problem is the use of a windowing function to attenuate the signal at the beginning and the end of the sampling interval. A windowing function is a waveform that is applied to the sampled data. The sampled data is multiplied by this window function producing a new set of data with smoother edges.
The Hann function is superimposed on top of the data with the sinusoidal shape apparent. The central portion of the signal is unaffected while the amplitude at the edges is gradually reduced to create a smoother transition.
Compared to Fig 5. This should not come as a surprise as the windowing function clearly suppresses the average amplitude of the original signal. This tradeoff between frequency resolution and amplitude is inherent for all window types.
Windows that present good resolution in frequency but poor determination of amplitude are often referred to as being of high resolution with low dynamic range. Each type of window has its own unique characteristics, with the proper choice depending on the application and preferences of the user.
See Engelberg , Lyons , Oppenheim et. It is common, however, to plot one or both axes on a logarithmic scale. In the case of amplitude, it is common to plot the spectral power density, which is typically represented in units of decibels dB. The majority of signals encountered in practice will have record lengths much longer than the examples presented here.
Consider a microphone measurement sampled at 40 kHz over a s period of time. The record length in this case would 5. In this case, the apparent frequency resolution, tlf, of the FFT would be 0. As already seen, spectral leakage is likely to limit the resolution to much higher values and it is unlikely that this level of resolution would practically be needed. Rather, it is more common to use methods such as Bartlett's or Welch's methods. In these methods, the sampled signal is divided into equal length segments with a window function applied to each segment.
For example, if one were to take the example above, it could be divided into approximately segments with values in each segment. The FFT of each segment would have a frequency resolution The major advantage to this form of analysis is that any uncertainty in the Fourier coefficients is reduced through the averaging of multiple FFT coefficients.
This typically yields a much smoother curve than a single FFT. See Lyons and Oppenheim, et. However, the signal frequently contains significant energy at frequencies higher than fe. If the signal is to be recorded with an analog device, such as an analog tape recorder, these higher frequencies are usually of no concern. They will either be recorded accurately or attenuated by the recording device.
If, however, the signal is to be recorded only at discrete values of time, the potential exists for the generation of false, alias signals in the recording. The sampling-rate theorem does not state that to avoid aliasing, the sampling rate must be twice the maximum frequency of interest but that the sampling rate must be greater than twice the maximum frequency in the signal, here denoted by fm ' As an example, consider a signal that has Fourier sine components of 90, , , and Hz.
If we are only interested in frequencies below Hz, we might set the sampling rate at Hz. In our sampled output, however, we will see frequencies of Hz and 40 Hz, which are aliases caused by the and Hz components of the signal. Section 5. In the foregoing example, in which fm is Hz, we would select a samplIng rate, fs, greater than Hz. Real sets of sampled data are finite. However, the series converges and discrete samples in the vicinity of the time t contribute more than terms not in the vicinity.
As an example, consider a function sin 2'IT0. It is sampled at a rate of 1 sample per second and samples are collected. Note that fs exceeds 2fm' so this requirement of the sampling-rate theorem is satisfied. A portion of the sampled data is shown in Figure 5. The sampled data have been connected with straight-line segments and do not appear to closely resemble the original sine wave.
The original sine wave has been recovered from the data samples with a high degree of accuracy. Reconstructions with small sample sizes or at the ends of the data samples will, in general, not be as good as this example.
Although Eq. In most cases, the use of very high sampling rates can eliminate the need to use reconstruction methods to recover the original signal. As we noted in Section 5. Assume that we have set the sampling rate such that the minimum alias frequency la has a value just equal to Ie.
The highest frequency in the signal that could cause aliasing is 1m. All frequencies above 'm have zero amplitude. Digital filtering techniques USIng software can be used to eliminate these alias frequencies. Solution: a Using Eq. It is used to determine the probability that a random variable has a value less than or equal to a specified value.
Example 6. Solu tion: a Using Eq. Either by substituting 15 into the equation or by reading from the graph, we find that the probability that the lifetime is less than 15 h is 0. Binomial Distribution The binomial distribution is a distribution which describes discrete random variables that can have only two possible outcomes: "success" and "failure.
The following conditions need to be satisfied for the binomial distribution to be applicable to a certain experiment: L Each trial in the experiment can have only the two possible outcomes of success 2. The probability of success remains constant throughout the experiment.
The experiment consists of n independent trials. Determine the probability that in a batch of 20 computers, 5 will require repair during the warranty period. Success will be defined as not needing repair within the warranty period. Other assumptions underlying the application of this distribution are that all trials are independent and that the probabilities of success and failure are the same for all computers. Using Eqs. If we buy four of these bulbs, what are the probabilities of finding that four, three, two, one, and none of the bulbs are defective?
Again, we can use the binomial distribution. The probability of having four, three, two, one, and zero defective light bulbs can be calculated by using Eq. Solution: We use Eq. Poisson Distribution The Poisson distribution is used to estimate the number of random occurrences of an event in a specified interval of time or space if the average number of occurrences is already known.
For example, if it is known that, on average, 10 customers visit a bank per five-minute period during the lunch hour, the Poisson distribution can be used to predict the probability that 8 customers will visit during a particular five-minute period. The Poisson distribution can also be used for spatial variations. Worked examples are provided for theoretical topics and sources of uncertainty are presented for measurement systems.
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