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Abstract:
The experiment concentrates on various methods of measuring rotating speed such as
measurement by DC Generator Tachometer, Digital Tachometer, Stroboscope and oscilloscope
using Lissajous diagrams. These methods are combined with data analysis methods such as Least
Square Fit linear approximation, Gaussian distribution and other error analysis concepts. In the
second half of the experiment, the properties of RC filters, both low pass and High pass filters
are studied and their characteristic graph are generated.
1
Table of Content:
Topic Page No.
Introduction 2
Theoretical Principles 3
Experimental System 8
Sample Data Analysis 11
Results and Discussions 16
Conclusions 24
References 25
Nomenclature 26
Appendix 27
Data Sets 27
MatLab codes used 31
2
Introduction:
The experiments performed in this lab give an exposure to various rotational speed
measurement methods and general data analysis. Five distinct methods of rotational speed
measurement were performed on a rotating spur gear at known frequency of rotation.
The first method is that of DC generator tachometer, in this method various values of
voltages were taken by changing the rotational speed of gear at a constant interval of decrement.
This technique is important because the data generated from this method helps to develop a
strong linear relationship between voltage and frequency using method of least square. Thus by
using this method, a rotational speed can be directly related to voltage.
The second method is that of digital tachometer, in this method a set of experimental
data of rotational frequency are taken using a digital tachometer. Using a large number of data, a
Gaussian curve of frequency distribution is developed. This method is important because it helps
to build a confidence interval within which most of the measured frequency will exist. All the
outliers are eradicated in this method.
The third method is that of stroboscope method, in this method the unknown
frequency is matched with a known stroboscope frequency. This method is important because it
involves matching of frequencies from two sources. Thus, this method gives a more accurate
alternative to measure rotational speed.
The fourth method is direct measurement using oscilloscope. In this method a time
interval of direct sinusoidal input is measured. And then the respective rotatory speed is
measured by computing the frequency of the sinusoidal wave.
The fifth method is that of using Lissajou figures. This method involves adjusting the
sinusoidal signals in an oscilloscope to match the signal from the unknown rotatory motion. In
this method two sinusoidal signals (waves) are presented in a polar coordinate system (r,ө). An
ellipse would represent a superposition of two sinusoid waves with same frequency. A curve of
shape of infinity or eight aligned about x to y-axes would represent two sinusoidal waves where
frequency of one of the signal is double of the other. This method is important because it
involves matching of two sinusoidal waves, thus it is more accurate and precise method of
measuring rotating speed.
In the next set of experiment, the characteristics of high pass and low pass filter are
studied. This is important because in some electrical circuit, it is required that a prescribed value
of voltage be maintained across the circuit element. In an event if the voltage difference doesn’t
meet the requirements, then it may cause the circuit to create flaws. For example, excess voltage
across components of a refrigerator may introduce malfunction in its system. In such case, low
pass and high pass filters would control the voltage difference across the circuit element. This
would prevent malfunctioning of certain electrical circuit elements.
3
Theoretical Principles:
Data Analysis: Least Square method
The method of least squares is used to make a relationship between two sets of
variables. Suppose there are two variables namely x and y. the y can be expressed as a function
of x. for the experimental purpose, a linear relationship y= ax+b can be developed. Then method
of least squares states that for finding the leat square of the linear curve, the square needs to be
minimized,
S = ∑(y𝑖 − (ax𝑖 + b))2
𝑛
𝑘=0
By partial differentiating with respect to a and b respectively, the relation for a and b can found
as follows:
𝑎 = 𝑛 ∑ 𝑥𝑖𝑦𝑖 − ∑ 𝑥𝑖 ∑ 𝑦𝑖
𝑛 ∑ 𝑥𝑖2 − (𝑥𝑖)2
𝑏 = 𝑛 ∑ 𝑦𝑖 ∑ 𝑥𝑖
2 − ∑ 𝑥𝑖𝑦𝑖 ∑ 𝑥𝑖
𝑛 ∑ 𝑥𝑖2 − (𝑥𝑖)2
Gaussian distribution
The Gaussian distribution is an inverted dome shaped curve enclosing an area of 1
unit under x-axis. It shows the probability distribution of how likely an event should occur. A
large number of data points need to be collected to achieve a smooth Gaussian distribution. The
equation for this distribution involves terms like mean of the data, standard deviation of the data.
𝑥𝑚𝑒𝑎𝑛 = ∑ 𝑥𝑖
𝑛
𝜎 = [1
(𝑛 − 1)∑(𝑥𝑖 − 𝑥𝑚𝑒𝑎𝑛)2]
1/2
The Gaussian distribution is defined by the equation:
𝑓(𝑥) = 1
𝜎√2𝜋𝑒
[−(𝑥−𝑥𝑚𝑒𝑎𝑛)2
2𝜎2 ]
The following graph shows the Gaussian distribution and histogram distribution of a
large experimental data. If the bars of histogram are connected then an approximate form of
Gaussian distribution can be obtained.
4
For this experiment the data rejection will be based on 2𝜎 rule. Thus, the range of
data will be xmean + 2σ and xmean - 2σ. This range will accommodate 95.4% of the experimental
values. Thus area under such curve will be 0.954 units.
For creating the histogram, the numbers of bins/ intervals can be delected using
Sturgi’s rule as shown below:
𝑚 = 1 + 3.3𝑙𝑜𝑔10(𝑁𝑣𝑎𝑙𝑖𝑑)
Where m is the number of intervals, 𝑁𝑣𝑎𝑙𝑖𝑑 is the valid number of data sets found from 2σ rule.
Stroboscope:
When the stroboscope frequency equals the rotational speedof the gear, the line marked on the
gear will be visible. The diagrams of the possible stroboscope methods are summarized in the
following table:
case 1 2 3 4
Gear Frequency f f/2 2f f/3
Stroboscope
frequency f f f f
5
Lissajous Figures:
Lissajous figures are obtained from polar coordinate input of sinusoidal waves. An unknown
signal is fed to y-port and a known signal to x-port.
x= sin(ωt); y = sin (ωt +ᵠ)
Then various lassajious figures are obtained by varying frequencies of unknown frequency
source.
Error analysis:
The errors can be of three types:
Avoidable errors: these errors occur due to mistakes such as noting wrong data record or reading
wrong data. This error can be avoided by being more attentive during experiment.
Fixed error/ Calibration Error: These are systematic errors. These occur due to zero error in
calibration of a measuring device or backlash error.
Random errors: These are also called precision errors. These errors cannot be controlled. They
have no bias. Thus they are also called non-systematic errors.
RC filters:
6
Low pass filter: A low pass filter removes high frequency component from the signal which are
above cur off frequency. This is defined as fc = 1/(2πRC).
In the above figure a low pass filter is fed with sin (10t+100t) signal. It removes the
100t component of the frequency and only allows 10t frequency component to pass through. The
basic electrical circuit of a low pass RC filter consists of a resistor and a capacitor. The
schematic of a low pass filter circuit is shown as below.
It can be shown that: the ratio of output and input voltages can be defined as:
𝑣0
𝑣𝑖=
1
√1+𝑓
𝑓𝑐
;
and phase angle is defined as: ᵠ = tan-1(f/fc).
In decibels the ratio can be mentioned as: 𝑣0
𝑣𝑖𝑑𝐵 = 20log (
𝑣0
𝑣𝑖).
High Pass filter:
High pass filter works on a similar principle as that of the low pass filter except, the
position of resistor and capacitor are interchanged in the circuit. It removes low frequencies from
the signal which are below cut off frequency (fc).
0 2 4 6 8-1
-0.5
0
0.5
1sin(10t+100t)
0 1.57 3.14 4.71 6.28-1
-0.5
0
0.5
1sin(10t)
data1 data2
7
In the above figure a high pass filter is fed with sin (10t+100t) signal. It removes the
10t component of the frequency and only allows 100t frequency component to pass through. The
basic electrical circuit of a low pass RC filter consists of a resistor and a capacitor. The
schematic of a low pass filter circuit is shown as below.
The equations governing the working of a high pass filter are shown below:
𝑣0
𝑣𝑖=
𝑓𝑓𝑐
√1 +𝑓𝑓𝑐
The phase angle is defined as: : ᵠ = π/2 - tan-1(f/fc).
In decibels the ratio can be mentioned as:
𝑣0
𝑣𝑖𝑑𝐵 = 20log (
𝑣0
𝑣𝑖).
Error Analysis:
Error may arise due to charge jump or leakage in the capacitor.
Counting frequency and phase difference:
Frequency can be counted by equation f = 1/period (Hz)
Phase Difference can be calculated as ᵠ = (N/M)*3600; where N = phase difference in cm and M
is the period in cm.
0 2 4 6 8-1
-0.5
0
0.5
1sin(10t+100t)
0 1.57 3.14 4.71 6.28-1
-0.5
0
0.5
1sin(100t)
8
Methodology and experimental procedures:
Least square method application:
For checking linear relationship between voltage and frequency, a least square
methodology is used. First connect magnetic pickup to electronic frequency counter. The counter
would produce number of counts per revolution equal to the number of teeth on gear. For the
purpose of this experiment a gear of 60 teeth is used. Now connect the DC generator to digital
voltmeter. Set motor speed to 1500 rpm on the electronic frequency counter with motor speed
control knob.
Now turn on DC amplifier knob to 15 volts and lock the knob on test rig. Now change
the motor speed from 1500 rpm to 0 rpm at a decrement of 300 rpm. Record voltage and
frequency counter reading.
Random Data Analysis: rotating speed by digital Tachometer
This experiment consists of collecting a large amount of data points using a digital
Tachometer. Set the motor speed to 1000 ± 10 rpm. Read the electronic frequency counter.
Repeat this procedure to obtain 60 measurements. A large number of data points will yield better
Gaussian distribution of the data.
Additional rotational speed measurement methods:
Stroboscope
Stroboscope is a light illuminating source, which flashes light at a given frequency.
Set the motor speed to 1000 rpm. Record the electronic frequency counter reading. Set
stroboscope frequency to approximately that of motor frequency.
Now adjust the stroboscope frequency such that timing mark on the gear appears
stationary. Sketch the picture of the gear this time. Again set stroboscope frequency that is
double of the motor frequency, then record the gear picture. Repeat this procedure for a
stroboscope frequency that is 3 times and then half times that of true shaft speed. Record the
observations.
fs = f motor fs = fmotor/2
9
Lissajou figures
Lissajou figures are the graphs obtained in the polar coordinates of two sinusoidal
signals. In order to obtain these figures, first connect output of the electronic frequency counter
to the vertical amplifier of the oscilloscope. Set the motor speed to 1500 rpm. Adjust the vertical
amplifier gain until a 4 or 5 cm peak to peak sine wave appears on the screen. Temporarily
disconnect the input to vertical amplifier of the oscilloscope.
Now connect the output of the audio oscillator to the horizontal amplifier of the
oscilloscope. Adjust oscillator frequency to equal approximately the frequency of the electronic
frequency output, turn the trigger control to x-y position. Adjust the resulting image so that a line
would appear on x-y plane. Reconnect the output of the electronic frequency counter to the
vertical amplifier of the oscilloscope. The resulting Lissajou figure should remain stationary.
Sketch the Lissajou figure; record both oscillator frequency and motor speed from the
electronic frequency counter. Increase the frequency until a double loop figure appears. Record
this frequency and its sketch.
Now, similarly decrease the oscillator frequency until a double loop figure appears.
Sketch the Lissajou figure and record this frequency. Again leave the oscillator frequency
unchanged and set motor speed until a single loop appears. Sketch the loop and record the
frequency.
The above diagram shows all the four cases performaed in this part of the experiemnt
Two sinusoidal signals fx = fy
2fx = fy fx = 2fy
10
Signal to noise ratio improvement by RC Filters:
Procedure for setting up low pass filter:
First set oscillator frequency to 1000Hz. Connect he output to the electronic counter
and to the input terminals A and B of the low pass RC filter and also to CH 1of the oscilloscope
as shown in the circuit diagram below. Adjust vertical gain to produce 4 cm peak to peak valley.
Connect the output of the low pass filter to CH2 vertical input of the oscilloscope.
Now record peak to peak valley, vertical peak to valley and gain setting. Then record
phase of the output to input signal with respect to the input signal name it N. the period of the
signal should be recorded and recognized as M.
Repeat the above procedure for the oscillator frequencies (in Hz): 200,500, 1000,
2000, 4000, 8000, 20,000, 40,000.
Procedure for setting up High pass filter:
This procedure is same as that for above except, connect the wires to high pass filter
instead of low pass filter. First set oscillator frequency to 1000Hz. Connect he output to the
electronic counter and to the input terminals A and B of the High pass RC filter and also to CH
1of the oscilloscope as shown in the circuit diagram below. Adjust vertical gain to produce 4 cm
peak to peak valley. Connect the output of the High pass filter to CH2 vertical input of the
oscilloscope.
Now record peak to peak valley, vertical peak to valley and gain setting. Then record
phase of the output to input signal with respect to the input signal name it N. the period of the
signal should be recorded and recognized as M.
Repeat the above procedure for the oscillator frequencies (in Hz): 200,500, 1000,
2000, 4000, 8000, 20,000, 40,000.
Circuit diagram for Low pass and High pass
filter.
11
Sample Data Analysis:
Method of Least Square fit:
Sr. no. RPM(x-axis) Frequency (Hz) Voltage(y-axis) x2 y2 xiyi
1 1500 1500 14.8 2250000 219.04 22200
2 1200 1210 12.2 1464100 148.84 14762
3 900 905 9 819025 81 8145
4 600 615 6.1 378225 37.21 3751.5
5 300 305 3 93025 9 915
6 0 0 0 0 0 0
Sum 4500 4535 45.1 5004375 495.09 49773.5
From the equation discussed in the theory,
𝑎 = 𝑛 ∑ 𝑥𝑖𝑦𝑖 − ∑ 𝑥𝑖 ∑ 𝑦𝑖
𝑛 ∑ 𝑥𝑖2 − (𝑥𝑖)2
𝑎 = 6(49773.5) − (4535)(45.1)
6(5004375) − (4535)2
𝑎 = 0.009948
𝑏 = 𝑛 ∑ 𝑦𝑖 ∑ 𝑥𝑖
2 − ∑ 𝑥𝑖𝑦𝑖 ∑ 𝑥𝑖
𝑛 ∑ 𝑥𝑖2 − (𝑥𝑖)2
𝑏 = 6(45.1)(5004375) − (49773.5)(4535)
6(5004375) − (4535)2
𝑏 = 0.0027
Thus equation of the line/ linear relationship becomes (of form y = ax+b):
Thus a linear relationship between frequency and voltage is verified.
V = aω + b
V = 0.009948ω + 0.0027
12
Gaussian Distribution:
The sample data collected is as follows. Each student took 12 readings.
Student 1 Student 2 Student 3 Student 4 Student 5
975 977 951 973 970
958 978 954 975 975
968 977 956 977 974
976 978 955 978 963
973 979 953 960 974
977 978 956 961 949
978 979 959 957 976
980 977 957 954 974
983 979 959 963 976
981 976 958 958 977
975 975 963 959 977
980 978 962 963 971
The mean of the data is calculated as follows:
𝑥𝑚𝑒𝑎𝑛 = ∑ 𝑥𝑖
𝑛
𝑥𝑚𝑒𝑎𝑛 = 58152
60= 969.2 𝑟𝑝𝑚
𝜎 = [1
(𝑛 − 1)∑(𝑥𝑖 − 𝑥𝑚𝑒𝑎𝑛)2]
1/2
𝜎 = [1
(60 − 1)∑(𝑥𝑖 − 𝑥𝑚𝑒𝑎𝑛)2]
1/2
𝜎 = 9.7037
For this experiment the data rejection will be based on 2𝜎 rule. Thus, the range of data will be
xmean + 2σ and xmean - 2σ.
xmax = xmean + 2σ = 969.2 + 2(9.7037) = 949.79.
xmin = xmean - 2σ = 969.2 - 2(9.7037) = 988.6.
Thus, range of data is from 949.79 to 988.6.
13
By the rejection rule, we rule out the reading 949 from the set because it is an outlier and doesn’t
fit well into the given range.
Thus number of valid data points now becomes Nvalid = 59.
For creating the histogram, the numbers of bins/ intervals can be delected using
Sturgi’s rule as shown below:
𝑚 = 1 + 3.3𝑙𝑜𝑔10(𝑁𝑣𝑎𝑙𝑖𝑑)
𝑚 = 1 + 3.3𝑙𝑜𝑔10(59)
𝑚 = 6.843 ≅ 7
The histogram will have 7 bins. This histogram will be presented in the results
section.
The Gaussian distribution is defined by the equation:
𝑓(𝑥) = 1
𝜎√2𝜋𝑒
[−(𝑥−𝑥𝑚𝑒𝑎𝑛)2
2𝜎2 ]
𝑓(𝑥) = 1
9.703√2𝜋𝑒
[−(𝑥−969.2)2
188.29]
The above equation will generate the required Gaussian distribution curve.
14
RC filters:
Low Pass Filter
For the given RC Filter, the properties of the circuit components are defined below:
R = 1000 Ω
C = 0.066 µF
Thus, the cut off frequency can be calculated as :
𝑓𝑐 = 1
2𝜋𝑅𝐶=
1
2𝜋 (1000)(0.066 ∗ 10−6)= 2411.4 𝐻𝑧
The cutoff frequency is found to be fc = 2411.4Hz.
For the first set of reading:
f = 200 Hz,
fc = 2411.4 Hz
From above values:
𝑣0
𝑣𝑖=
1
√1+𝑓
𝑓𝑐
;
𝑣0
𝑣𝑖=
1
√1+200
2411.4
= 0.9609;
Phase angle is defined as:
ᵠ = -tan-1(f/fc) = -tan-1(200/2411.4) = -4.740
In decibels the ratio can be mentioned as:
𝑣0
𝑣𝑖𝑑𝐵 = 20 log (
𝑣0
𝑣𝑖) = 20 log(0.9609) = 0.346𝑑𝐵.
Similarly calculations for other frequency cases were performed using excel.
For experimental values, ᵠ = N/M*(180/π) in degrees = -0.2/4*(180/π) = -180.
15
High Pass Filter
For the given RC Filter, the properties of the circuit components are defined below:
R = 1000 Ω
C = 0.066 µF
Thus, the cut off frequency can be calculated as :
𝑓𝑐 = 1
2𝜋𝑅𝐶=
1
2𝜋 (1000)(0.066 ∗ 10−6)= 2411.4 𝐻𝑧
The cutoff frequency is found to be fc = 2411.4Hz.
For the first set of reading:
f = 200 Hz,
fc = 2411.4 Hz
From above values:
𝑣0
𝑣𝑖=
𝑓
𝑓𝑐
√1+𝑓
𝑓𝑐
;
𝑣0
𝑣𝑖=
200
2411.4
√1+200
2411.4
= 0.0797.
Phase angle is defined as:
ᵠ =900 - tan-1(f/fc) =900 - tan-1(200/2411.4) = 85.260
In decibels the ratio can be mentioned as:
𝑣0
𝑣𝑖𝑑𝐵 = 20 log (
𝑣0
𝑣𝑖) = 20 log(0.0797) = −21.97𝑑𝐵.
Similarly calculations for other frequency cases were performed using excel.
For experimental values, ᵠ = 900 - N/M*(180/π) in degrees = -2/7.8*(180/π) = 85.250.
16
Results and Discussions:
Throughout the experiment, several methods of measuring rotational speed and their
data analysis were performed. The respective results and key characteristics of each methods are
discussed under this section.
Method of least square and DC Tachometer
This method was used in the experiment to generate a linear relationship between
voltage and that of frequency. This method helped to generate a strong linear relationship
between voltage and that of frequency. Thus, it could be noted that voltage can be used to
measure the rotational speed of a rotating object. For the purpose of this experiment, the linear
relation was found to be:
V = aω + b
V = 0.009948ω + 0.0027
The corresponding graph depicting this relationship is shown below:
The slope of the graph was found to be 0.009948 and the y intercept of the graph is 0.0027. The
very low value of y-intercept shows that the line is symmetric about origin. This fact further
endorses the theoretical assumption that voltage varies linearly with the rotational speed. Due to
strong relation the percent error involved in this case is very small.
y = 0.0099x - 0.0027R² = 0.9997
0
2
4
6
8
10
12
14
16
0 500 1000 1500 2000
Vo
ltag
e (
V)
Frequency (Hz)
Method of least square-Voltage Frequency graph
17
Gaussian distribution and Digital Tachometer:
This method involved collection a large number of data points. Thus, the confidence
interval in this case would help to get an accurate value of the frequency. A total of 60 data were
recorded out of which one of the data was an outlier. Then a bar graph with the help of these
frequencies was generated. The bar graph is shown below:
The Gaussian Distribution Curve is shown as below; the graph was obtained using a
MATLab code generated for the given set of data:
The mean value of the experimental data was found to be 969.2 Hz. The actual value
of the frequency for the experiment was 1000 Hz. Thus the Gaussian curve for this experiment
05
1015202530
req
ue
ncy
of
occ
ura
nce
Range of the frequencies
The histogram for Gaussian Distribution of experiemntal data
950 955 960 965 970 975 980 985 9900.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
frequency
frequency o
f occura
nce
gaussian Distribution for the experiemntal data
18
was found to be offset about 30 units to the left side. This could be possibly due to some
experimental errors.
The histogram was relatively skewed to the right side of the distribution. However,
the range of values was acceptable as they were close to the actual value of frequency. The error
developed can be calculated as follows:
Percent error = 𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑟𝑖𝑒𝑚𝑒𝑛𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒
𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒× 100 =
1000−969.2
1000× 100 = 3.08%
Percentage error = 3.08%
A major concern may be that the values were recorded incorrectly i.e. there might
have been miscommunication between data observer and that of data recorder this error could
have been avoided. There might have been unbiased error. This error could arise as a result of
calibration error in the digital measurement device. There could be a possibility of zero error in
the device.
Other Methods of measuring rotational speed:
Stroboscope:
Under this method, the frequency of stroboscope was matched with that of the gear rotational
speed frequency. The values of stroboscope frequency and that of gear rotation frequency are
summarized in the following table along with the sketch of the visuals observed during the
experiment.
case 1 2 3 4
Gear Frequency 1000 1000 1000 1000
Stroboscope
frequency 995.9 499.8 1992.7 2999
Diagrams
It was found that only one line was visible on gear when the stroboscope frequency
was equal or double that of gear frequency. In order to distinguish between this, half the
frequency of the stroboscope and if one line appears then previous stroboscope frequency equals
motor frequency. If one line appears then previous stroboscope frequency would equal 2 times
the motor frequency.
19
For other cases, of two lines appear then stroboscope frequency is double that of
motor frequency. If three lines appear, then stroboscope frequency equals three times the motor
frequency.
This case involves matching the stroboscope frequency with that of motor frequency.
Thus, this method of measuring the frequency or rotational speed is more accurate as it can be
seen in the diagram. The error involved in this case was very small. This could arise from
unbiased error source.
Percent error = 𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑟𝑖𝑒𝑚𝑒𝑛𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒
𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒× 100 =
1000−995.9
1000× 100 = 0.41%
Percentage error = 0.41%
It can be observed that percent error in this case is relatively very small.
Lissajous Figures:
Lissajous figures involved comparison of two sinusoidal waves, where one of the signal had
unknown frequency and other had known adjustable frequency. The known frequency was
adjusted to coincide with the unknown frequency. Using this fact, Lissajous figures were formed
using polar coordinates. There were a total of four cases studied; these are summarized below
with the help of a diagram.
Two sinusoidal signals
fx = fy
fx = fy = 1500 rpm
2fx = fy
fx = 1500rpm, fy = 3000rpm
fx = 2fy
fx = 1500rpm, fy = 750rpm
Case 1
Case 2 Case 3
20
Signal to Noise Ratio Improvement by RC Filters:
Low pass Filter:
The low pass filter removes the contribution of high frequency (above cut off
frequency (fc)) from the sinusoidal signal. For the purpose of this experiment, the cutoff
frequency was calculated to be 2411.4 Hz. The results of the required values of v0/vi dB are
summarized in the table attached at the end of the report. The characteristic graphs for the
experiment are shown as below:
The log(f/fc) vs (v0/vi)dB plot is as shown below. It can be concluded that the
experimental values are in good agreement with that of the theoretical values. This satisfies the
equation derived for the low pass filter.
Error Analysis
The variation or the error in the graph increases as the frequency of the oscillator
increases. Thus may be due to the fact that as the values of oscillator frequency went high, the
precision and accuracy in setting the exact value of frequency was lost for example, the actual
frequency taken during the experiment for the last run was 39,500Hz instead of 40.000 Hz. This
was because the frequency of the oscillator fluctuated very often when the frequency was
increased.
-30
-25
-20
-15
-10
-5
0
-1.5 -1 -0.5 0 0.5 1 1.5
(v0
/vi)
dB
log(f/fc)
Log(f/fc) vs (v0/vi)dB Graph for Low pass RC Filter
Experiemntal
values
Theoretical
values
21
The log(f/fc) vs phase angle (∅) plot is as shown below. Again, it can be concluded
that the experimental values are in good agreement with that of the theoretical values. This
satisfies the equation derived for the low pass filter. A similar perception of error analysis can be
implied for this graph.
Various characteristics graphs of Low pass RC Filter are plotted using MATLab code
-100
-80
-60
-40
-20
0
-1.5 -1 -0.5 0 0.5 1 1.5
ph
ase
dif
fere
nce
(∅
)
log(f/fc)
Log(f/fc) vs phase difference (∅)Graph for Low pass RC Filter
Experiementl values
Theoretical values
0 1 2 3 4 5 6 7-1
0
1Voltage -time graph for low pass filter where f<<fc
0 1 2 3 4 5 6 7-1
0
1Voltage -time graph for low pass filter where f=fc
0 1 2 3 4 5 6 7-1
0
1Voltage -time graph for low pass filter where f>>fc
ein
eo output signal
ein
eo output signal
ein
eo output signal
22
High pass Filter:
The high pass filter removes the contribution of low frequency (below cut off
frequency (fc)) from the sinusoidal signal. For the purpose of this experiment, the cutoff
frequency was calculated to be 2411.4 Hz. The results of the required values of v0/vi dB are
summarized in the table attached at the end of the report. The characteristic graphs for the
experiment are shown as below:
The log(f/fc) vs (v0/vi)dB plot is as shown below. It can be concluded that the
experimental values are in good agreement with that of the theoretical values. This satisfies the
equation derived for the low pass filter.
Error Analysis
The variation or the error in the graph increases as the frequency of the oscillator
increases. Thus may be due to the fact that as the values of oscillator frequency went low, the
precision and accuracy in setting the exact value of frequency was lost for example, the actual
frequency taken during the experiment. This was because the frequency of the oscillator
fluctuated very often when the frequency was decreased in this case.
The log(f/fc) vs phase angle (∅) plot is as shown below. Again, it can be concluded
that the experimental values are in good agreement with that of the theoretical values. This
-25
-20
-15
-10
-5
0
-1.5 -1 -0.5 0 0.5 1 1.5
(v0
/vi)
dB
log(f/fc)
Log(f/fc) vs (v0/vi)dB Graph for High pass RC Filter
Experiementalvalues
Theoreticalvalues
23
satisfies the equation derived for the low pass filter. A similar perception of error analysis can be
implied for this graph.
Various characteristics graphs of High pass RC Filter are plotted using MATLab code
-20
0
20
40
60
80
100
-1.5 -1 -0.5 0 0.5 1 1.5
ph
ase
dif
fere
nce
(∅
)
log(f/fc)
Log(f/fc) vs phase difference (∅)Graph for High pass RC Filter
Experimentalvalues
Theoreticalvalues
0 1 2 3 4 5 6 7-1
0
1Voltage -time graph for High pass RC filter where f<<fc
ein
eo output signal
0 1 2 3 4 5 6 7-1
0
1Voltage -time graph for High pass RC filter where f=fc
ein
eo output signal
0 1 2 3 4 5 6 7-1
0
1Voltage -time graph for High pass RC filter where f>>fc
ein
eo output signal
24
Conclusions:
With the help of experimental data it can be concluded that all the methods of rotating
speed measured accurate values of the rotational speed of the motors with insignificant error in
each case. The data analysis of individual methods were also significant in deriving the
relationship among the data points. The characteristic graphs for the Low pass and High pass
filter were found to be in good agreement with the theoretical predictions.
25
References:
J.p. Holman, ‘Experimental Methods for Engineers’, seventh Edition, Mc-graw Hill production,
2001, pg 48-220.
Lab Manual ME 343, Mechanical Engineering laboratory 1, ME Department, NJIT.
Software usage:
MATLab 2014
26
Nomenclature:
Symbol Meaning
V voltage
ω Frequency in RPM
a Slope in the linear relation graph
b y-intercept in the linear relation graph
f Frequency in Hz
∑ Summation of a large sum
xmean Average value in the data set
N Number of data points
Nvalid Number of valid data points
σ Standard deviation for the data set
fc Cut off frequency of low or high pass RC Filter
V0 Output voltage in RC Filter circuit
Vi Input voltage in RC Filter circuit
∅ Phase angle
27
Appendix:
Data Set:
Method of least squares:
Sr. no. RPM(x-axis) Frequency (Hz) Voltage(y-axis) x2 y2 xiyi
1 1500 1500 14.8 2250000 219.04 22200
2 1200 1210 12.2 1464100 148.84 14762
3 900 905 9 819025 81 8145
4 600 615 6.1 378225 37.21 3751.5
5 300 305 3 93025 9 915
6 0 0 0 0 0 0
Sum 4500 4535 45.1 5004375 495.09 49773.5
Gaussian Distribution:
60 readings of rotational speed (frequency)
Student 1 Student 2 Student 3 Student 4 Student 5
975 977 951 973 970
958 978 954 975 975
968 977 956 977 974
976 978 955 978 963
973 979 953 960 974
977 978 956 961 949
978 979 959 957 976
980 977 957 954 974
983 979 959 963 976
981 976 958 958 977
975 975 963 959 977
980 978 962 963 971
Data for calculating mean and standard deviation:
Values in Ascending
order (x-xi) (x-xi)^2
949 -20.2 408.04
951 -18.2 331.24
953 -16.2 262.44
954 -15.2 231.04
954 -15.2 231.04
28
955 -14.2 201.64
956 -13.2 174.24
956 -13.2 174.24
957 -12.2 148.84
957 -12.2 148.84
958 -11.2 125.44
958 -11.2 125.44
958 -11.2 125.44
959 -10.2 104.04
959 -10.2 104.04
959 -10.2 104.04
960 -9.2 84.64
961 -8.2 67.24
962 -7.2 51.84
963 -6.2 38.44
963 -6.2 38.44
963 -6.2 38.44
963 -6.2 38.44
968 -1.2 1.44
970 0.8 0.64
971 1.8 3.24
973 3.8 14.44
973 3.8 14.44
974 4.8 23.04
974 4.8 23.04
974 4.8 23.04
975 5.8 33.64
975 5.8 33.64
975 5.8 33.64
975 5.8 33.64
975 5.8 33.64
976 6.8 46.24
976 6.8 46.24
976 6.8 46.24
976 6.8 46.24
977 7.8 60.84
977 7.8 60.84
977 7.8 60.84
977 7.8 60.84
977 7.8 60.84
977 7.8 60.84
977 7.8 60.84
29
978 8.8 77.44
978 8.8 77.44
978 8.8 77.44
978 8.8 77.44
978 8.8 77.44
978 8.8 77.44
979 9.8 96.04
979 9.8 96.04
979 9.8 96.04
980 10.8 116.64
980 10.8 116.64
981 11.8 139.24
983 13.8 190.44
58152 -2.7E-12 5555.6
Data for Histogram intervals:
Bar ranges interval Frequency of occurrences
951-955.57 5
955.57-960.14 10
960.14-964.71 6
964.71-969.28 1
969.28-973.85 4
973.85-978.42 25
978.42-982.99 7
30
Matlab codes used:
For Low pass filter
t = 0:pi/180:2*pi; f1 = sin(t); f2 = 0.99*sin(t-0.07);
subplot(3,1,1); hold on plot(t,f1); plot(t,f2,'r'); legend('ein','eo output signal') title('Voltage -time graph for low pass filter where f<<fc'); hold off
f3 = sin(t); f4 = 0.707*sin(t-pi/4);
subplot(3,1,2); hold on; plot(t,f3); plot(t,f4,'r'); legend('ein', 'eo output signal') title('Voltage -time graph for low pass filter where f=fc'); hold off;
f5 = sin(t); f6 = 0.01*sin(t-pi/2);
subplot(3,1,3); hold on; plot(t,f5); plot(t,f6,'r'); legend('ein','eo output signal') title('Voltage -time graph for low pass filter where f>>fc'); hold off;
31
For high pass filter
t = 0:pi/180:2*pi; f1 = sin(t); f2 = 0.01*sin(pi/2-t);
subplot(3,1,1); hold on plot(t,f1); plot(t,f2,'r'); legend('ein','eo output signal') title('Voltage -time graph for High pass RC filter where f<<fc'); hold off
f3 = sin(t); f4 = 0.707*sin(pi/4-t);
subplot(3,1,2); hold on; plot(t,f3); plot(t,f4,'r'); legend('ein','eo output signal') title('Voltage -time graph for High pass RC filter where f=fc'); hold off;
f5 = sin(t); f6 = 0.99*sin(t-0.07);
subplot(3,1,3); hold on; plot(t,f5); plot(t,f6,'r'); legend('ein','eo output signal') title('Voltage -time graph for High pass RC filter where f>>fc'); hold off;
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