Rotational Motion, Newton's Law and Power Relationship
Almost all electric machines rotate about an axis, called the shaft of the machines. It is important to have a basic understanding of rotational motion.
Angular position, - is the angle at which it is oriented,
measured from some arbitrary reference point. Its measurement units are in radians (rad) or in degrees. It is similar to the linear concept of distance along a line.
Conventional notation: + ve value for anticlockwise rotation; - ve value for clockwise rotation.
Angular Velocity, - Defined as the velocity at which the measured point is moving.
Similar to the concept of standard velocity where:
where: r - distance traverse by the body; t - time taken to travel the distance r
For a rotating body, angular velocity is formulated as:
where: - Angular position/ angular distance traversed by the rotating body;
t - time taken for the rotating body to traverse the specified distance, .
Angular acceleration, - is defined as the rate of change in angular velocity with respect to time.
Its formulation is as shown:
1. In linear motion, a force applied to an object causes its velocity to change. In the absence of a net force on the object, its velocity is constant. The greater the force applied to the object, the more rapidly its velocity changes.
2. Similarly in the concept of rotation, when an object is rotating, its angular velocity is constant unless a torque is present on it. Greater the torque, more rapid the angular velocity changes.
3. Torque is known as a rotational force applied to a rotating body giving angular acceleration, a.k.a. 'twisting force'.
4. Definition of Torque: (Nm)
Work, W - is defined as the application of Force through a distance. Therefore, work may be defined as:
Assuming that the direction of F is collinear (in the same direction) with the direction of motion and constant in magnitude, hence,
Applying the same concept for rotating bodies,
Assuming that is constant,
Power, P - is defined as rate of doing work. Hence,
Applying this for rotating bodies,
This equation can describe the mechanical power on the shaft of a motor or generator.
Newton's Law of Rotation
Newton's law for objects moving in a straight line gives a relationship between the force applied to the object and the acceleration experience
by the object as the result of force applied to it. In general,
F - Force applied
m - mass of object
a - resultant acceleration of object
Applying these concept for rotating bodies,
J - moment of inertia
- angular acceleration
The Magnetic Field
Magnetic fields are the fundamental mechanism by which energy is converted from one form to another in motors, generators and transformers.
First, we are going to look at the basic principle - A current-carrying wire produces a magnetic field in the area around it.
Production of a Magnetic Field
Ampere's Law - the basic law governing the production of a magnetic field by a current:
where H is the magnetic field intensity produced by the current Inet and dl is a differential element of length along the path of integration.
H is measured in Ampere-turns per meter.
Consider a current currying conductor is wrapped around a ferromagnetic core;
Applying Ampere's law, the total amount of magnetic field induced will be proportional to the amount of current flowing through the conductor wound with N
turns around the ferromagnetic material as shown. Since the core is made of ferromagnetic material, it is assume that a majority of the magnetic field will be
confined to the core.
The path of integration in Ampere's law is the mean path length of the core, lc. The current passing within the path of integration Inet
is then Ni, since the coil of wires cuts the path of integration N times while carrying the current i. Hence Ampere's Law becomes,
In this sense, H (Ampere turns per metre) is known as the effort required to induce a magnetic field.
The strength of the magnetic field flux produced in the core also depends on the material of the core. Thus,
B = magnetic flux density (webers per square meter, Tesla (T))
= magnetic permeability of material (Henrys per meter)
H = magnetic field intensity (ampere-turns per meter)
The constant may be further expanded to include relative permeability which can be defined as below:
where: o - permeability of free space (a.k.a. air).
Hence the permeability value is a combination of the relative permeability and the permeability of free space. The value of relative permeability is dependent
upon the type of material used. The higher the amount permeability, the higher the amount of flux induced in the core.
Relative permeability is a convenient way to compare the magnetizability of materials.
Also, because the permeability of iron is so much higher than that of air, the majority of the flux in an iron core remains inside the core instead of travelling
through the surrounding air, which has lower permeability. The small leakage flux that does leave the iron core is important in determining the flux linkages
between coils and the self-inductances of coils in transformers and motors.
In a core such as in the figure,
Now, to measure the total flux flowing in the ferromagnetic core, consideration has to be made in terms of its cross sectional area (CSA). Therefore,
Where: A - cross sectional area throughout the core
Assuming that the flux density in the ferromagnetic core is constant throughout hence constant A, the equation simplifies to be:
Taking into account past derivation of B,