Transcript induced voltage and torque

POWER CIRCUIT &
ELECTROMAGNETICS
EET 221
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).
Example 1 :
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 )( perpendicu lar 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
If a moment of inertia J is given, so angular velocity in t second
given by :
τ
ω  α t  xt
J
Example 2 :
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.
Example 3 :
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 
N
d
dt
   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.
Example 4 :
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 4 :
E
B
v
l
External magnetic field
The linier generator is shown in Figure above has a separation
between the rails of 1 m, and there is a 0.5 T external magnetic
field applied. If the armature is pulled at 10 m/s, how much
voltage is generated ? If 1 ohm resistor is placed across the
terminals at the end of the rails, how much force is required to
pull the armature?
Solution :
E  Blv  0.5 x1x10  5 V
The current flowing through the armature will create an
electromagnetic force opposing the motion. Thus, it is
necessery to prove an equal and opposite mechanical force.
E
5
I 

5 A
R
1
F  BIl  0.5 x5 x1  2.5 N