Download Chapter 5: Introduction to Machinery Principles

BASIC ELECTRICAL TECHNOLOGY
DET 211/3
Chapter 5: Introduction to Machinery
Principles
Rotational Motion
Angular Position, θ
The angular position θ of an object is the angle at which it
oriented, measured from some arbitrary reference point.
Unit: Radians or Degree (rad or deg)
Rotation of a rigid object P about a fixed object about a fixed axis O.
Angular velocity, ω
Angular velocity (or speed) is the rate of change in angular position with
respect to time. It is assume positive if the rotation is in a counter clockwise
direction.
d

dt
Unit: Radians per second (rads-1)
The following symbols are used in this course:
ωm
= angular velocity expressed in radians per second
fm
= angular velocity expressed in revolutions per second
nm
= angular velocity expressed in revolutions per minute
Angular velocity describes the speed of rotation and the orientation of the
instantaneous axis about which the rotation occurs. The direction of the
angular velocity vector will be along the axis of rotation; in this case
(counter-clockwise rotation).
Angular acceleration, α
Angular acceleration, α is the rate of change in angular velocity with
respect to time.

d
dt
Unit: Radians per second (rads-2)
Torque, τ (tau)
The torque on the object is defined as the product of the force applied
to the object and the smallest distance between the line action of the
force and the object’s axis of rotation.
  ( force applied )( perpendicular dis tan ce )
  Fr sin 
where θ is the angle between vector r and the vector F.
The direction of the torque is clockwise if it tends to cause a clockwise
rotation and counter clockwise if it would tend to cause a counter
clockwise rotation
In physics, a torque (τ) is a vector that measures the tendency of a force to
rotate an object about some axis. The magnitude of a torque is defined as force
times its lever arm. Just as a force is a push or a pull, a torque can be thought of
as a twist.
The Magnetic Field
Four basic principles describe how magnetic fields are used in
electrical machines (transformer, motor, and generator):
•
A current carrying produces a magnetic field in the area
around it.
•
A time changing magnetic field induces a voltage in a coil
of wire if it passes through that coil. (This is the basis of
transformer action)
•
A current carrying wire in the presence of a magnetic field
has a force induced on it. (This is the basis of motor action)
•
A moving wire in the presence of the magnetic fields has a
voltage induced in it. (This is the basis of generator action)
Faraday’s Law – Induced voltage from a time changing
magnetic field
Faraday’s law states that if a flux passes through a turn of a coil, a
voltage will induced in the turn of wire that is directly proportional to
the rate of change in the flux with respect to time.
eind
d

dt
where eind is the voltage induced in the turn of the coil and Ф is the
flux passing through the turn.
If a coil has N turns and if the same flux passes through all of them,
then the voltage induced across the whole coil is given by
eind   N
d
dt
where N = number of turns of wire in coil
The minus sign in the equation is an expression of Lenz’s Law.
Lenz’s Law
Lenz’s law states that the direction of the
voltage buildup in the coil is such that if the
coils ends were short circuited, it would be
produce current that would cause a flux
opposing the original flux change. Since the
induced voltage opposes the change that
causes it, a minus sign is included in
equation above.
If the flux shown in the figure a is increasing in strength, than
the voltage built up in the coil will oppose the increase. A
current flowing as shown in Figure b would produce a flux
opposing the increase, so the voltage on the coil must be built
up with the polarity required to drive that current through the
external circuit. Therefore, the voltage must be built up with
polarity shown in Figure b. Since the polarity of the resulting
voltage can be determine from physical considerations, the
minus sign in equation above is often left out.
If leakage is quite high or if extreme accuracy is required, a
different expression that does not make that assumption will
be needed. The magnitude of the voltage in the ith turn of
the coil always given by
eind
d ( i )

dt
If there are N turns in the coil of wire, the total voltage on
the coil is
d ( i ) d  N 
  ei  
   i 
i 1
i 1
dt
dt  i 1 
N
eind
N
Faraday’s law can be rewritten in terms of the flux linkage
as
eind 
d
dt
N
   i
i 1
(unit: weber turns)
Faraday’s law is the fundamental property of the
magnetic fields involved in transformer operation.
The effect of Lenz’s in transformer is to predict
the polarity of the voltages induced in the
transformer windings.
Production of induced force on a wire
The basic concept involved is illustrated in Figure below. The
figure shows a conductor present in a uniform magnetic field
of flux density B, pointing into the page. The conductor itself
is l meters long and contains a current of i amperes.
The force induced on the conductor is given by
F  i( lxB )
The direction of the force is given by the right hand rule. If the
index finger of the right hand points in the direction of the
vector l and the middle finger points in the direction of flux
density vector B, then the thumb points in the direction of the
resultant force on wire.
The magnitude of the force is given by the equation
F= ilBsin θ
where θ is the angle between the wire and the flux density
vector
The induction of the force in a wire by a current in the
presence of a magnetic field is the basis of motor action.
Induced voltage on a conductor moving in a magnetic field
If the wire with the proper orientation moves a through a
magnetic field, a voltage is induced in it. This idea is shown
in figure below.
The voltage induced in a wire is given by
eind  (vxB)  l
Where
v = velocity of the wire
B = magnetic flux density vector
l = length of the conductor in the magnetic field
Vector l points along the direction of the wire toward the
end making the smallest angle with respect to the
vector v x B. The voltage in the wire will be built up so
that the positive end is in the direction of the vector v x
B.
Example
Figure above shows a conductor moving with a velocity of
5.0m/s to the right in the presence of a magnetic field. The
flux density is 0.8 T into the page and the wire is 1.0m in
length. What are the magnitude and polarity of the resulting
induced voltage?
Solution Example
Direction of the quantity v x B in this example is up.
Therefore, the voltage on the conductor will be built up
positive at the top with respect to the bottom of the wire. The
direction of vector I is up. So that makes the smallest angle
with respect to the vector v X B. Since v is perpendicular to B
and since v X B is parallel to I, the magnitude of the induced
voltage reduces to:
eind  (vxB)  l
 (vB sin 90)l cos 0
 vBl
 (5.0m / s )(0.8T )(1.0m)
 4.0V
Thus, the induced voltage is 4.0V, positive at the top of the
wire
Assignment 4
Figure below shows a conductor moving with a velocity of 2.5m/s to
the right in a magnetic field. The flux density is 2.5T, out of the
page, and the wire is 2.0m in length, oriented as shown. What are
the magnitude and polarity of the resulting induced voltage?
-
-
B
- -
eind
l
v
300
++
+
+
vB