Electrical and Mechanical
angles are frequently used in the study of Electrical Machine theory and
therefore it is very important to have a crystal clear concept of these two
terms and the relationship between them. In this post we will also discuss
about the Synchronous Speed and their relationship with Electrical and
Mechanical angle.
As we know that emf is
induced in a conductor is given by Faraday’s Law of Electromagnetic Induction
which can be expressed as below.
The magnitude of emf
generated in a conductor of length L moving at a speed of v perpendicular to
the Magnetic Field B is given by
E = BLv
For better understanding of
Faraday’s Law and generation of emf in a Conductor, read How does Motional EMF Produced?
Thus from the above
expression of emf, we observe that the waveform of emf depends on the wave shape
of Magnetic Field. If the waveform of Magnetic Field is square then the
generated or induced emf in the conductor will also be square. Likewise if the
waveform of Magnetic Field is Sinusoidal then waveform of E will also be
sinusoidal.
Consider a two pole machine
as shown in figure below. Suppose a conductor is rotating at a constant angular
velocity. When the conductor is at a, emf induced in the conductor is zero as
the magnetic field at a is zero.
Similarly, when conductor is
at b, the emf induced in maximum as magnetic field is maximum at b. Let us
consider this magnetic field at b to be positive i.e. magnetic field entering
the south pole of stator to be positive then the for the whole rotation of the conductor,
the waveform of emf induced will be as shown in figure (a) below.
Mind that the waveform of
magnetic field as well as the induced emf are same as discussed earlier in the
post. Thus one complete revolution of conductor results in one complete cycle
of induced emf. In other words, we can say that 360° mechanical rotation of
conductor results in 360° change in the induced emf for two pole machine.
Now, assume that the same
conductor in now moving at a constant angular speed but in four pole machine as
shown in figure above. Here when the conductor is at a, the induced emf is zero
as magnetic field at a is zero. Similarly for different positions of the
conductor, the magnitude of induced emf with sign is tabulated below for the
sake of better understanding.
Sr.
No.

Position
of Conductor

Value
of Magnetic Field

Value
of Induced emf

1)

b

+Maximum

+Maximum

2)

c

Zero

Zero

3)

d

Maximum

Maximum

4)

e

Zero

Zero

5)

f

+Maximum

+Maximum

6)

g

Zero

Zero

7)

h

Maximum

Maximum

The above values of the
induced emf in the conductor when plotted with the rotation results in the
waveform shown in figure (b) above. Thus we observe that, two cycles of emf is
generated in one complete revolution of conductor. Therefore, though the
mechanical rotation of conductor is only 360° but the electrical rotation is
two times i.e. 720°. This point must be noted and carefully understood.
Thus based on the above two
cases, we can say that P/2 cycle of emf is generated in one revolution for a P
pole Electrical Machine and hence we can write the relationship between the
Electrical and Mechanical angles as
Ɵ_{elect} = (P/2) Ɵ_{mech} …………………..(1)
Hope you understood how the
relationship between the Electrical and Mechanical angles derived and what their
bases are.If you still have doubt, feel free to write in comment box.
Now we will understand the
how the synchronous speed is related with the relationship between the Electrical
and Mechanical angles?
From the relationship
between the Electrical and Mechanical angles,
Ɵ_{elect} = (P/2) Ɵ_{mech}
Differentiating both sides
with respect to time, we get
d(Ɵ_{elect}) /
dt = (P/2)d(Ɵ_{mech}) / dt
⇒
ω_{elect}
= (P/2)ω_{mech}
_{}
But we know that, ω = 2πf
⇒
2πf_{elect}
= (P/2) 2πf_{mech}
_{}
⇒
f_{elect}
= (P/2) f_{mech}
_{}
But mechanical frequency
simply means the number of revolution of conductor per second. If we take
number of revolution per second to be n then,
f = Pn/2
where f = frequency of
generated emf.
Again, if we take number of
revolution of conductor per minute to be N then number of revolution per second
n = N/120, therefore
Frequency of emf f = PN/120
This frequency f is known as
synchronous frequency. That’s all. So simple?? Please write your views and
comments. Thank you!
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