Page 109 - Servo Motors and Industrial Control Theory -

6.3 Mathematical Model 103

For normal loading conditions, it can be fairly accurately assumed that the air-gap
flux is constant. The magnitude of the rotor e.m.f. is then proportional to the frac-
tional slip s . If the rotor runs in exact synchronism with the air-gap field, the in-
ℓ
duced e.m.f. is zero and the rotor develops zero torque. In practice, even on no load,
a small motor torque is required to overcome windage and friction, and the motor
runs at a speed which is slightly less than the synchronous speed. When the load is
applied to the motor shaft, the speed drops further below synchronous speed and
larger rotor currents are induced. When a rotor current flows, an additional compo-
nent of stator current is induced. As in a transformer, this load component of stator
current neutralizes the rotor m.m.f. and leaves the resultant air-gap flux practically
unaltered, as previously assumed. With the above assumption, the torque developed
by the motor can be written as

T : K · · I · cos( )= φ 2 φ 2 (6.3)
m
where
K is a constant
I is the r.m.s. current in the rotor conductor
2
φ is the rotor phase lag
2
I cos φ 2 is the in-phase current
2
φ is the air-gap flux.
The rotor current and the stator current are related to each other with a constant
factor. Therefore, like for DC motors the output torque can be assumed to be pro-
portional to the stator current as
T : K ·I= (6.4)
m t
Where K is the constant of proportionality and I is the stator current. As will be
t
discussed later, Eq. (6.4) is only valid for small range of motor operating point.
In order to obtain the voltage equation of an AC induction motor the complete
characteristics of stator and rotor must be known. To simplify the analysis, the
equivalent circuit diagram of an AC motor for one of the supply phase must be
studied. This is necessary because to obtain a mathematical model, a linear relation
between the applied voltage and the stator current must be found. Because of the
interaction between the rotor and stator the circuit diagram is fairly complex. The
circuit diagram for one phase is shown in Fig. 6.1.
In Fig. 6.1, R is the stator resistance, X = L ω is the stator reactance (ohms),
1
1
1
R is the rotor resistance (ohms), X = L ω is the rotor reactance (ohms), R is the
2
2
m
2
magnetizing resistance (ohms), X = L ω is the magnetizing reactance (ohms), s ℓ is
m
m
the slip ratio, and the resistance R /s ℓ resistance provide the equivalent back e.m.f.
2
of DC motors.
The circuit diagram of Fig. 6.1 can be reduced to a single resistance and reac-
tance as shown in Fig. 6.2.