0

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;