Torque acting on a body is quantitatively defined
as the product of force acting on the
body and perpendicular distance of the line of action of force from the axis of
rotation.
i.e T = F×r
sinθ
Qualitatively , torque is the tendency of a force to
cause a rotational motion, or to bring about a change in rotational motion .
We will use this concept to derive an expression for torque
in a dc motor.
Visualize the below shown dc motor front-view . You
will find that each conductor experiences a force and the conductors lie near
the surface of the rotor at a common radius from its center. Hence torque is
produced at the circumference of the rotor and rotor starts rotating.
DC motor Torque equation derivation
Since all conductors experience equal force and are equidistant
from center , therefore
Total
torque = torque on one conductor × total number of conductors
Let
r=average
armature radius
L=effective
length of each conductor
Z=total
number of armature conductors
A=number
of parallel paths
Ia
=armature current
I=current
through each conductor= Ia /
A
B=average
flux density
Φ=flux
per pole
P=number
of poles
a=cross-sectional area of flux path per pole at radius
r = (2Ï€rL / P)
Force
on each conductor = BIL
Torque due to one conductor = BILr
As Z,P and A are construction features of the machine ,
therefore are constant.
∴ Ta ∝ Φ Ia
Hence, for a given dc motor, torque developed in
its armature depends on its flux per pole and armature current taken by it.
Important points
:-
(1) As back emf , Eb=(P ΦZN/60A)
, putting it in Ta , we get
EbIa is the power supplied by the source to the armature.
This gives us torque equation in terms of
power supplied by the source to the armature.
From this power, the armature has to
supply (1) iron losses in armature (2) Friction and windage losses
So, torque available at shaft Tsh
will be slightly lesser than Ta .
(2) In a dc series motor,
Φ ∝ Ia ...upto magnetic saturation
If armature reaction is
ignored and flux path reluctance is assumed constant
Therefore, Ta ∝ Ia2
(3) In
a dc shunt motor,
Φ is practically
constant if armature
reaction is ignored and flux path reluctance is assumed constant
Therefore, Ta ∝ Ia
