As a general rule, conversion of electrical power into mechanical power takes place in the rotating part of an electric motor. In D.C motors, the electric power is conducted directly to the armature or the rotating part with the help of brushes and commentator. Hence, in this sense a D.C motor can be called or referred to as a conduction motor. But in the case of A.C motors, the rotor receive power (electric) by induction in exactly the same way as the secondary of a 2-winding transformer receives its power from the primary and not by conduction. That is why such motors are branded as induction motors. In some cases an induction motor is treated as a rotating transformer, since its primary winding is at a standstill but the secondary rotates freely.
The multi-phase induction motors is widely used in terms of applications of A.C. motors, the main advantages and disadvantages attributed to it are as follows:
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It has an extremely strong and harsh physical structure.
It is relatively less expensive and very reliable.
It has high efficiency
It does not require major maintenance work
It starts from rest and does not require a separate starting motor
Its efficiency is affected with variations in speed
Its speed decreases with increase in load
Its starting torque is lower as compared to D.C. shunt motor
There are two major parts in an Induction motor:
The stator of an induction motor is in theory the same as that of a synchronous motor or generator. It is made up of a number of slots which hold the windings. Different stators carry different windings such as single phase or three phases depending on the power that it is fed with. The number of poles upon which it is wound depends on the speed, more the number of poles, lesser the speed and lesser the number of poles, more the speed.
a. Squirrel-cage rotor, also known as induction motors:
Fig. 1.1 Squirrel-cage induction motor
Almost 90% of induction motors are of this type because this arrangement has the strongest and simplest physical structure. It essentially consists of a cylindrical laminated core with slots that are parallel carrying conductors made up or copper and aluminum or alloys. These bars or rods are welded to end rings that are short circuited at either end, thus deriving its name as squirrel cage.
Though the slots are anticipated to be kept in parallel, it is in most occasions deliberately given a slight skew, which is useful in the following ways:
It helps the motor to run quietly
It reduces the locking tendency of the rotor, which is the affinity of the rotor teeth to remain under the stator teeth due to direct magnetic attraction between the two. In small rotors another method of construction is used which consists of placing the entire rotor core in a moulds and casting all the bars and end rings in one piece, this process commonly uses aluminum.
b. Phase-wound rotor, also known as slip-ring motors or wound motors.
This type of rotor is provided with 3-phase distributed windings consisting of coils. The rotor is wound for as many poles as the number of stator poles and is always wound 3-phase even when the stator is wound two-phase. The point of this type of rotor is that the three winding terminals are brought out and connected to three insulated slip rings, this make the accumulation of external resistance possible and thus assists the starting of the motor by increasing the starting torque of the motor and for changing its variable characteristics like speed and torque.
1.3 Single Phase Induction Motor
Fig. 1.2 Single phase induction motor
When we look at the physical makeup of the single phase induction motor it is quiet similar to the multi-phases induction motors with the variation that its stator it provided with a single phase winding and a switch is used in some types of motors in order to cut out a winding generally used only for starting purposes. This type of motor has dispersed stator winding and a squirrel-cage rotor. When it is given an input from a single-phase supply, its stator winding produces a field which is alternating. It is not a synchronously revolving flux as in the case of a two or three phase stator winding getting input from two or three phase power supply.
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We say that a single phase motor is not self-starting because the alternating flux acting on the squirrel cage rotor which is at a standstill cannot make it rotate.
However, as seen in the lab, if the rotor of such a machine is given an initial start by hand, then a torque arises and the motor accelerates to its final speed.
This behavior of the motor can be explained by different theories like the ones called two-field or double-field revolving theory and the cross field theory among others.
Chapter 2: Equivalent Circuit of Single Phase Induction Motor (Without Core Loss)
Fig.2.1 Equivalent Circuit of Single Phase Induction Motor
A single-phase motor may be treated as a compilation of two motors, having the same stator winding but with the respective rotors revolving in opposite directions. The figure shown above is the equivalent circuit of such a motor. The single phase motor has been imagined to be made up of one stator winding and two imaginary rotors.
The stator impedance is:
The impedance of each rotor is: () ; where , and represent half the actual rotor values in stator terms, that is, stands for half the standstill reactance of the rotor, as referred to stator.
Each rotor has been assigned half the magnetizing reactance or represents half the actual reactance.
The impedance of 'forward running' rotor is : (taking into account all the halved values)
And it runs with a slip of ' ' .
The impedance of the 'backward running' rotor is: (taking into account all the halved values)
It runs with a slip of (2-s). Under standstill conditions, , but in running condition, is about 95% of the applied voltage.
The total torque is also found to be:
Chapter 3: Analysis of the Equivalent Circuit
To proceed with the analysis of the equivalent circuit, the values of the main parameters as assumed to be as follows:
Resistance of the stator main winding
Reactance of the stator main winding
Magnetizing reactance of the stator main windings
Rotor resistance at standstill
Rotor reactance of standstill
Voltage v=110 volts
Fig. 3.1 Equivalent Circuit with respect to
Upon substituting the values that have been taken into consideration,
Hence we get in Eqn(3.1),
a== 3.56/2= 1.78
b== 2.56/2= 1.28
c== 53.5/2= 26.75
Substituting the values in eqn. 2,
1.6146 + j (1.2166)
Fig.3.2 Equivalent Circuit with respect to
Upon substituting the values that have been taken into consideration,
Hence we get in eqn. 3
= 3.56/2(2-0.04)= 0.908
= 2.56/2= 1.28
= 53.5/2= 26.75
Substituting the values in eqn. 4,
= 0.8260 + j( 1.248)
Fig. 3.3 Equivalent Circuit with respect to
Upon substituting the values that have been taken into consideration,
We know; and is 110volts
Merging Step-D and Step-A
Fig. 3.4 Equivalent Circuit with respect to and
Substituting the values,
CHAPTER 4: Application of MATLAB Software
4.1 Introduction to MATLAB
MATLAB which is the short form for "MATrix LABoratory", a numerical computing language. Developed by The MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, and Fortran.
Fig 4.1 Screen shot of the MATLAB desktop
4.1.1 Application of the Windows on the MATLAB software desktopÂ
*Command History View a log of or search for the statements you entered in the Command Window, copy them, execute them, and more.
*Command Window Run MATLAB language statements.
* Current Directory Browser View files, perform file operations such as open, find files
and file content, and manage and tune your files.
*Editor Create, edit, debug, and analyze M-files (files containing MATLAB language statements).
*Figures Create, modify, view, and print figures generated with MATLAB.
*File and Directory Comparisons View line-by-line differences between two files.
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*Help Browser View and search the documentation and demos for all your MathWorks products.
*Profiler Improve the performance of your M-files.
*Start Button Run tools and access documentation for all your MathWorks products.
*Variable Editor View array contents in a table format and edit the values.
*Web Browser View HTML and related files produced by MATLAB.
*Workspace Browser View and make changes to the contents of the workspace.
4.2 Steps for Implementation of the formulated problem in MATLAB
4.2.1 Step A: Initialization of Variables
As in most computing languages, the know values of the constants are initialized at the start, in MATLAB, the format for initialization of the values is
(Alphanumeric name of variable) = (Value the user wants to assign);
As per the above syntax, the known constant values from Chapter 3 (pg 8) are initialized as:
v=230.0; //The voltage have assumed for the application in the equation
poles=4; //The number of poles in the Induction machine
f=50; //The frequency (in Hz)
r1=1.86; //The resistance of the stator main winding
r2=3.56; //The rotor resistance at standstill
x1=2.56; //The reactance of the stator main winding
x2=2.56; //The rotor reactance at standstill
xm=53.5; //The magnetizing reactance of the stator main windings
j=sqrt(-1);//Initialization of imaginary part
4.2.2 Step B: Initialization of Arrays
In MATLAB, the format for initialization of an array is give by:
(Alphanumeric name of array) = (Starting point value) : (Interval between the array) : (Ending point value)
As per the above syntax, the value of Slip is initialized as:
Columns 1 through 16
2.0000 1.9900 1.9800 1.9700 1.9600 1.9500 1.9400 1.9300 1.9200 1.9100 1.9000 1.8900 1.8800 1.8700 1.8600 1.8500
Columns 17 through 32
1.8400 1.8300 1.8200 1.8100 1.8000 1.7900 1.7800 1.7700 1.7600 1.7500 1.7400 1.7300 1.7200 1.7100 1.7000 1.6900
Columns 33 through 48
1.6800 1.6700 1.6600 1.6500 1.6400 1.6300 1.6200 1.6100 1.6000 1.5900 1.5800 1.5700 1.5600 1.5500 1.5400 1.5300
Columns 49 through 64
1.5200 1.5100 1.5000 1.4900 1.4800 1.4700 1.4600 1.4500 1.4400 1.4300 1.4200 1.4100 1.4000 1.3900 1.3800 1.3700
Columns 65 through 80
1.3600 1.3500 1.3400 1.3300 1.3200 1.3100 1.3000 1.2900 1.2800 1.2700 1.2600 1.2500 1.2400 1.2300 1.2200 1.2100
Columns 81 through 96
1.2000 1.1900 1.1800 1.1700 1.1600 1.1500 1.1400 1.1300 1.1200 1.1100 1.1000 1.0900 1.0800 1.0700 1.0600 1.0500
Columns 97 through 112
1.0400 1.0300 1.0200 1.0100 1.0000 0.9900 0.9800 0.9700 0.9600 0.9500 0.9400 0.9300 0.9200 0.9100 0.9000 0.8900
Columns 113 through 128
0.8800 0.8700 0.8600 0.8500 0.8400 0.8300 0.8200 0.8100 0.8000 0.7900 0.7800 0.7700 0.7600 0.7500 0.7400 0.7300
Columns 129 through 144
0.7200 0.7100 0.7000 0.6900 0.6800 0.6700 0.6600 0.6500 0.6400 0.6300 0.6200 0.6100 0.6000 0.5900 0.5800 0.5700
Columns 145 through 160
0.5600 0.5500 0.5400 0.5300 0.5200 0.5100 0.5000 0.4900 0.4800 0.4700 0.4600 0.4500 0.4400 0.4300 0.4200 0.4100
Columns 161 through 176
0.4000 0.3900 0.3800 0.3700 0.3600 0.3500 0.3400 0.3300 0.3200 0.3100 0.3000 0.2900 0.2800 0.2700 0.2600 0.2500
Columns 177 through 192
0.2400 0.2300 0.2200 0.2100 0.2000 0.1900 0.1800 0.1700 0.1600 0.1500 0.1400 0.1300 0.1200 0.1100 0.1000 0.0900
Columns 193 through 200
0.0800 0.0700 0.0600 0.0500 0.0400 0.0300 0.0200 0.0100
4.2.3 Step C: Initialization of Equations
In MATLAB, it is possible to initialize equations to alphanumeric variables for simplicity:
4.2.4 Step D: Initialization of Loops
Similar to other computing languages, it is possible to initialize loops to avoid repetition of the same syntax again and again, thus saving time and memory space,
Here we make use of the 'for' loop to achieve this target. The syntax of the same is same is given by:
for (Name assigned to the counter) = (Starting point value) : (Ending point value)
body of the loop
As per the above syntax, the 'for' loop is initialized as:
for count=1:length(s)-1 // name of the counter :count
// starting point value :1
// ending point value :length of the loop s - 1
//start of the body in the loop
rf(count)=l(count); // real part of z1f
xf(count)=(m(count))/(n(count)); // imaginary part of z1f
rb(count)=l1(count); // real part of z1b
xb(count)=(m1(count))/(n1(count)); //imaginary part of z1b
u(count)=v/((r1+(j*x1))+(z1f(count))+(z1b(count))); // Current
// end of the body in the loop
end //end of the loop
4.2.5 Step E: Initialization of Plots
Syntax for the plotting of graphs
First we have to define both the variables to 0 so that they will have the same array size.
Let us define the first variable as : x
Let us define the second variable as: y
x(length(y))=0; //To make the array length the same for both the variables.
plot (x , y); //Command to plot the graph.
4.3 Source Code of Program created as pas per the formulated problem in MATLAB
% specify the equivalent circuit parameters
%sub parameters for simplified calculation
rf(count)=l(count);% real part of z1f
xf(count)=(m(count))/(n(count));%imaginary part of z1f
rb(count)=l1(count);% real part of z1b
xb(count)=(m1(count))/(n1(count));%imaginary part of z1b
pgb(length(s))=0; // Vary as per the graph required
4.4 Graphical representation of results in MATLAB:
4.4.1 Torque Vs. Slip Characteristic
Fig 4.2 Torque vs. Slip Characteristics
a. This graph above helps us calculate electromagnetic torque at different values of slip
b. It facilitates the theory that the output power of the drives is lower than the output power at load.
c. It also assists us to determine at what slip the machine will run.
d. The above graph is an important tool used by researchers to determine the breaking region and the breaking current of the machine.
4.4.2 Torque Vs. Current Characteristic
Fig 4.3 Torque vs. Current Characteristics
With help from the above graph the loading of the machine can be decided since it is know that the product of torque and angular velocity gives the power of the machine
4.4.3 Pgf Vs. Slip Characteristic
Fig 4.4 Pgf vs. Slip Characteristics
4.4.4 Pgb Vs. Slip Characteristic
Fig 4.5 Pgb vs. Slip Characteristics
Based on the Fig 4.4 and Fig 4.5, the designer can predict the relative strength of the component field developed in the loading.
CHAPTER 4: Conclusion
A single phase induction motor is an asynchronous A.C motor , the rotor does not receive electric power by conduction but by induction in exactly the same way as the secondary of a 2-winding transformer receives its power from the primary. Its primary winding is stationary but the secondary rotates freely. Induction motors are among the strongest and simplest in terms of physical structure in motors
This project is proceeding forward in analysis of its equivalent circuit. In this report we have derived the first phase of the analysis and have used hypothetical values to equate the formula
The analysis of the circuit will continue into the next phase and will eventually be synchronized with MATLAB in the final phase of the project to obtain a graphical output and design, enabling us to vary the different parameters.