ME 392 University of Hartford Analysis of The Gain of An Amplifier Lab Report Follow exactly step by step what in the file , it’s really important to read

ME 392 University of Hartford Analysis of The Gain of An Amplifier Lab Report Follow exactly step by step what in the file , it’s really important to read it very carefully , please focus and follow every little details, never copy from internet , no plagiarism…. and i have attached the plots in the report file under the results section.

For this report please review the “lab report guidelines” document and use the “lab report template”. Both of these documents are on the “Course Documents” tab to the left. Because the code was provided we will be looking to see that you have a clear understanding of what was done. Remember that the lab had two parts, one related to changing the input amplitude and the other changing the input frequency. All information from the pre-report assignment must be included in the report.

The abstract should say what is contained in the report and have a brief quantitative statement of the primary results

The introduction should clearly state all objectives.

The procedure section should clearly explain what was done and have quantitative statements of all measurement parameters like sampling frequency. You must have, and cite from the text, a figure with a block diagram of the experimental setup and the caption should explain the setup. A caption like “Block diagram of system” is not a good caption.

The theory section should explain any and all calculations used to determine the results. In this case explain how the amplitudes where determined in both parts of the lab and how the gain was determined. This section should have equations with all variable defined.

The results section is the most important section. Here you need to explain the results plot that the MATLAB code produced. Please write more in this section. A plot without much explanation is not what we are looking for. One reason is that we provided the code that generates the results plots.

The conclusions section can be short but must be quantitative. State all results in a specific and quantitative way. Explain any implications from these results.

Throughout the report remember to:

1) Be quantitative and specific everywhere in the report. If, while proofreading, you find a vague or general sentence, delete it.

2) Never have yourself or any person as the subject of a sentence. You are writing about the amplifier.

3) All figures have captions that are explanations not titles and are cited from the body of the text.

4) Only use MATLAB plots have been saved using the “print” command.

5) Be sure that the results match the objectives

6) Write in past tense. You have completed the experiment and you are writing about what you did. Theory: Digital Data Acquisition
2.1
Continuous and Discrete Signals
Experimental data are typically collected using sensors that produce signals related to states of the measured
system, such as temperature, pressure, velocity, etc. These quantities in general change in time, sometimes
rapidly. For example the sounds we hear are in fact changes in air pressure — the ear of a typical healthy young
adult can perceive acoustic pressure oscillations up to about 20,000 cycles per second (Hz). Often, signals are
also created by the experimentalist in order to cause a certain effect in a physical system. A signal sent to a
loudspeaker causes acoustic pressure waves to occur in a particular way, while a signal coming from a
microphone carries information about the particular acoustic pressure fluctuations at the microphone location.
Any quantity that varies can be categorized as either continuous or discrete. A discrete variable can take
on only a finite number of possible values within any given range, while a continuous variable can have any of
an infinite number of values. For example, the weight of passengers in an elevator is a continuous variable:
there are an infinite number of possible weights in between 0 and the maximum weight at which the cable
snaps. The number of passengers, on the other hand, is discrete: there can be 0, 1, 2, and so on, but not 11/2 or
0.876 people. A signal describing a continuous variable is often called an analog signal.
Time itself is a continuous variable. Each minute can be divided into seconds, each second into milliseconds,
each millisecond into microseconds, and so on, so that there are an infinite number of particular times in
between any two given times. The next 5 seconds is certainly a finite amount of time, but time passes
continuously through an infinite number of values on its way from 0 to 5 sec. As a practical matter, however,
one can only make a finite number of measurements in any given period of time. The process of recording a
certain number of values of some quantity is called sampling, and it has the effect of changing time from a
continuous to a discrete quantity. Instead of a continuously varying function, the measured signal now becomes
a set containing a finite number of discrete measured values. Other continuous variables can also be converted
to discrete variables by sampling, for instance, a finite span of distance, which encompasses an infinite number
of possible locations, can be sampled at a finite set of locations.
Additionally, the value of a measured quantity is typically discretized by the measurement process. For
example, although weight is strictly speaking a continuous variable, if you measure your own weight using a
bathroom scale, you typically round off the result to the nearest pound or half-pound, because the scale is not
precise enough to let you discern differences of fractions of an ounce. Let’s say you record your weight for an
entire year and find that it fluctuates between 150 and 160 pounds. If you sample your weight once per day,
then at the end you have 365 discrete times at which your weight was measured. If you round to the nearest
half-pound, then the record of your weight will also contain only 21 discrete levels of weight: 150, 150.5, 151,
151.5, and so on up to 160. Time has been discretized by the process of sampling, and weight has been
discretized by the process of rounding off.
When a computer or other digital device records the result of any measurement, there is a rounding-off
process that is unrelated to the precision of the sensor. Because the information is stored in a digital fashion,
there are only a finite number of possible different values that can be represented, which number depends on
the number of bits devoted to the task. The actual value indicated by the sensor must be rounded to the nearest
value that can be represented digitally. This process is called digitization. Thus, in order for a computer to
record an analog signal that varies continuously in time, the signal is simultaneously sampled at discrete times
and digitized to discrete values. Each of these processes is described in detail below.
2.2
Sampling Signals
The data acquisition cards (DAQs) in the computers in the laboratory measure voltage levels repeatedly at even
intervals of time. We will refer to this interval of time as the “sample interval” or si, or equivalently as the “time
step” ∆t. The reciprocal of the sample interval is the sampling frequency or sampling rate and typically has units
of cycles per second, called Hertz (Hz).
1
or
(1)
Figure 1 shows an example of an analog voltage (in this case, a simple sine wave), and a digital signal that
results from sampling the continuous signal to create a discrete signal. The sampled signal only exists at a set
of specific, discrete times. Note that the sampling rate is fast enough that there is a large number of sample
intervals for each period of oscillation of the sine wave.
The sampling rate can be adjusted by the user, up to a maximum limit determined by the particular DAQ
card being used (and, in many cases, the number of signals being measured at any given time). The simple rule
for how fast is fast enough to sample is that you need to sample at least twice as fast as the fastest changes in
the signal. Figure 2 shows how sampling fast enough gives a discrete signal that is an adequate representation
of the original analog signal, while sampling too slowly gives a poor representation, which may be
misinterpreted as a much slower fluctuation. This phenomenon is called aliasing, and the need to sampling
twice as fast as the fastest changes in the signal is called the Nyquist criterion; both will be discussed in more
depth during Lab 3.
2.3
Digitizing Signals
The discrete values that a signal has at discrete moments in time must be converted into binary numbers for
representation in the computer. Unlike decimal numbers, which are written using 10 different numerals (0
through 9), binary numbers are represented by only two numerals, 0 and 1, representing the two possible
states of a bit of electronic memory. In the decimal number 305, the 5 in the “ones” place simply means five, the
0 in the “tens” place means 0 times ten, and the 3 in the “hundreds” place means 3 times one hundred, for a
total of three hundred plus five. In the binary number 101, there is a 1 in the “ones” place, a 0 in the “twos”
place, and a 1 in the “fours” place, for a total of four plus one, or five. The number five is represented
Figure 1: An analog signal (left) changes continually with time. The dots in the plot on the right represent
discrete points sampled from the original signal at a regularly spaced sample interval.
2
Figure 2: The same analog signal discretized by sampling at two different sample rates: (top) fast enough,
(bottom), too slow, causing aliasing. From page 407 of LabVIEW 8 Student Edition, R. H. Bishop
with a single numeral 5 in ordinary decimal notation, but by three numerals, 101, in binary notation. The chart
in Table 1 indicates how various numbers are represented in each form. Note that decimal numbers require
one additional numeral each time you reach the next higher power of 10, whereas binary numbers require one
more numeral each time you reach the next higher power of 2.
The data acquisition cards (DAQs) in the computers in the lab have 12 bit digitizers: they use 12 bits of
memory to represent each measured value. This means that there are 212 or 4096 possible states, which, if you
wish, could be labeled 0 to 4095 in decimal notation, or 0 to 111 111 111 111 in binary. When the card receives
a certain input voltage, it must round this voltage to one of 4096 voltages representing the entire range of
possible voltages. In most digitizers, this voltage range can be adjusted by the user, usually by selecting from
one of several available options.
Let’s say we have a DAQ set to measuring −10 to +10 volts, or a 20 volt range. This means that the lowest
voltage −10 volts would correspond to “voltage number 0,” or a binary representation 000 000 000 000, and
the largest voltage value +10 volts would be “voltage number 4096,” or 111 111 111 111. In between −10 and
+10 volts there are at infinite number of possible voltages for the incoming analog signal, but only 4094
remaining voltages that the digitizer can represent. Thus, extremely small changes in input voltage will not be
able to be discerned by the digitizer, because their difference is not enough to switch from one of the 4096
possible voltages to the next. The smallest change in voltage that the DAQ card can resolve is referred to as the
code width (Vcw), and is the voltage change that alters the last digit of the binary number (the “ones” place),
referred to as the least significant bit or LSB.
voltage range
Vcw =
Nbits
(2)
2
For example, a 12 bit DAQ with a voltage range set to ±10 volt (20 volts wide) is (20 V)/2 12, or 4.9 mV. Thus,
it is possible for the incoming analog voltage to vary by amounts less than 4.9 mV without changing the digital
representation recorded by the computer. Any given digitizer has a fixed number of bits, so the only way to
decrease the code width is to decrease the overall voltage range. The ratio of the full range of the DAQ to the
smallest discernible voltage change is called the dynamic range, and is determined by the number of bits. It is
often reported in decibels (recall that each 20 dB represents one order of magnitude).
voltage range
dynamic range =
Nbits
=2
(3)
Vcw
dynamic range in dB = 20log10(2Nbits)
= 20log10(2) × Nbits
= 6.02 × Nbits
Table 1: Decimal numbers and binary representations
Number
Decimal
0
1
zero
Binary
0
1
one
2
two
3
10
three
3
4
11
100
5
6
7
8
101
110
111
1 000
9
10
1 001
1 010
11
12
14
15
16
1 011
1 100
1 110
1 111
10 000
31
32
11 111
100 000
33
64
100 001
1 000 000
99
100
1 000 011
1 100 100
one twenty eight
128
10 000 000
two fifty six
256
100 000 000
four
five
six
seven
eight
nine
ten
eleven
twelve
fourteen
fifteen
sixteen
thirty one
thirty two
thirty three
sixty four
ninety nine
one hundred
512
1 000 000
000
one thousand
1000
one thousand twenty four
1024
1 111 101
000
10 000 000
000
1 000 000 000
000
five twelve
4096
four thousand ninety six
For a 12 bit digitizer, the largest voltage fluctuation that can be measured is 4096 times greater than the
smallest voltage difference that can be discerned, so the dynamic range is 4096, or about 72 dB. In order to take
full advantage of the available dynamic range, and thus have the best possible digital representation of the input
signal, the voltage range should be set to be only a little larger than the maximum expected fluctuation in input
signal. For instance, if you expect an incoming signal to be sinusoidal with roughly 1 V amplitude, set the
digitizer range to ±1.5 V rather than ±10 V: instead of 4.9 mV, the code width will be reduced Vcw to 0.73 mV,
allowing you to capture the input signal more accurately.
Figure 3 compares the digital representation of a signal measured using a DAQ with 3 bits to a DAQ with 16
bits, both with a voltage range from 0 to 10 V. The code width for the 3-bit DAQ is only (10 V)/23 or 1.25 V,
while the 16-bit DAQ has a code width so small, (10 V)/2 16 or 0.15 mV), that it cannot be resolved in the image.
4
Figure 3: The bit resolution is shown for a sinusoid digitized at 3 bit and 16 bit resolution. From page 405 of
LabVIEW 8 Student Edition, R. H. Bishop
As mentioned above, the overall range of a digitizer can often be adjusted by the user, usually by choosing
among several fixed ranges. For instance, a user may be able to select ranges of , −100 mV to +100 mV, −1 V to
+1 V, or −10 V to +10 V, among others. The range must be larger than any value the digitized signal will have,
or else the signal will be clipped. This means that any value of the signal that exceeds the maximum value of
the range will simply appear to be that maximum value, while any value below the minimum will simply appear
to be that minimum. For instance, if the range of the digitizer is −1 V to +1 V, any value of the incoming signal
that is greater than +1 V will erroneously appear to be +1 V, and any value below −1 V will appear to be −1 V.
Figure 4 shows an example of a signal whose value sometimes exceeds this range being clipped as it is digitized.
Ideally, the incoming signal should fill most of the range of the digitizer, but never exceed it. Because the
levels expected from the incoming signal are not always know precisely, a user should err on the side of
selecting a range that is larger rather than smaller. However, if the range selected is too large compared to the
level of the incoming signal, the digitizer will not be making effective use of all the dynamic range available. For
instance, if a signal has amplitude of 50 mV, but the digitizer range selected is −10 V to +10 V, then only the
middle 0.5% of the range will be used. If a 12 bit digitizer is being used, there are 4096 discrete levels to
represent voltages between −10 V and +10 V but for this particular signal, the digitizer only makes use of the
roughly 20 discrete levels that fall in between −50 mV and +50 mV. Figure 5 illustrates this concept. If the same
signal is digitized with the digitizer range set to −100 mV to +100 mV the signal will occupy about half of the
range, so will make use of over 2000 discrete levels rather than a mere 20. Thus, the digitizer range should
always be set to be larger than the expected signal level but not by any more than is necessary to account for
uncertainty and fluctuations in the input signal amplitude.
5
Figure 4: Clipping of a digitized signal when the signal level exceeds the digitizer range (in this example the
range is −1 V to +1 V)
Figure 5: Digitization of a signal that fills only a small part of the digitizer range. For a 12 bit digitizer, there are
1012 = 4096 different levels within the selected range of −10 V to +10 V (shown at left), but this means there
are only about 20 different levels used to represent a small signal with amplitude around 50 mV (as shown in
the zoomed in version on the right). A better choice would be to select a smaller range for the digitizer so that
there are more discrete levels in between −50 mV and +50 mV.
2.4
Noise and Averaging
In the context of experimental data, noise refers to random fluctuations, typically across a wide variety of
frequencies simultaneously. In practice, any measurement signal will be contaminated by some amount of
noise, that is, random fluctuations will added to the desired signal by undesired effects such as vibration of a
6
test fixture, electromagnetic interference, etc. Often, the word “signal” is used to refer specifically to the desired
components of the measured signal, and all undesired portions are referred to as “noise,” so that the quality of
a measured signal can be characterized by its signal-to-noise ratio (SNR), which is the ratio of the strength of
the desired part of the signal to the undesired parts. The SNR is often expressed in decibels, for example, an
SNR of 40 dB means that the desired component of a signal is 100 times stronger than the random noise
fluctuations.
If a measurement has particularly low amounts of external noise, a digitizer with a small dynamic range
(low number of bits) is used, or the measured voltage signal is too small compared to the available voltage
range, then the errors from rounding off introduced by the process of digitization may be the most significant
source of noise (this is called bit noise). A signal distorted by bit noise will resemble the stair-step signal shown
in Fig. 3 for the 3-bit digitization, or the one shown in Fig. 5, while a signal distorted by other, external types of
noise will have unevenness that appears more random when viewed close-up.
If the exact same measurement is repeated more than once, one would expect the signal itself to be the
same, but the random noise to be different every time. By averaging the results of multiple measurements, noise
can be reduced while leaving the desired signal itself unaffected. The desired signal is coherent from one
measurement to the next, that is, the average of N identical signals gives the same signal. Random noise,
however, is incoherent— averaging N different noise patterns gives a level of noise roughly
times as big
each of the individual noise levels. Thus, averaging four measurements of the same signal will give a result with
half as much noise as an individual measurement, while average 100 measurements will reduce noise by a
factor of 10 (on a decibel scale, these reductions are 6 dB and 20 dB respectively).
One interesting thing to note is that if there are no sources of random noise, the bit noise will be exactly the
same with every measurement, and so cannot be averaged away. However, the inevitable presence of some
other sources of noise means that the measured voltage will not be digitized exactly the same way each time.
In other words, the presence of other sources of noise makes the bit noise incoherent, so that it too can be
averaged away. This allows averaging to give results with higher accuracy than the bit resolution would
normally seem to allow.
2.5
Triggering
In order for averaging multiple measurements to work, the signals must be “lined up” properly so that they are
coherent, that is, the same behavior must occur at the same time in each measurement. Slowly varying signals
can be aligned manually, for example, if you measure barometric pressure once an hour and wish to average
the results from several different days, it isn’t too difficult to make sure one 7 AM measurement gets averaged
with the other 7 AM measurements, the 8 AM measurements all get averaged with each other, and so on.
However, manually lining up the data becomes impossible for rapidly varying signals and measurements with
high sample rates. To get multiple measurements to line up properly in practice, it is usually necessary to have
each measurement be triggered by some event that also triggers (or is triggered by) the behavior of the system
being studied. For example, if the same electrical pulse is used as a trigger to turn on a loudspeaker and begin
recording with a microphone 100 cm away, then each time the experiment is run, the sound will arrive at the
microphone at the same time, about 35 msec after the measurement begins (the speed of sound is roughly 350
m/sec). If multiple measurements are averaged, the sound from the loudspeaker, always beginning at 35 msec,
will average coherently and remain. Sounds from other sources in the room, as well as other sources of
electrical noise in the measurement system, will not always occur at the same time, so they will average
incoherently and averaging greater numbers of measurements will reduce the influence of noise.
Triggering can be accomplished either “in hardware” or “in software.” In the “hardware” case, some
triggering event such as an electrical pulse is used as a signal to begin both the experiment and the data
acquisition. This signal can be generated by an external device or created on the computer and sent to the
system via an output port on the data acquisition card. In the “software” case, data acquisition is begun earlier,
at some arbitrary (untriggered) time, and then after data acquisition is complete, the recorded data is examined
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to identify some triggering condition, and the relevant portion of the data is extracted for use. For instance, in
the above example of a loudspeaker and microphone, the data would be examined to identify the sudden rise
in sound pressure level when the sound from the speaker arrives at the microphone. The portion of the data
immediately following that rise should be s…
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