Environmental Engineering Reference
In-Depth Information
in each phase at standstill is V r . Since the induced voltage is proportional to the rate
of change of flux (2.10), the rotor voltage at a particular slip s will be
rotor voltage ¼ s V r
The torque developed by the induction machine will depend on the current
flowing in each rotor phase (2.7). The rotor phase current for a given slip will be
determined by the rotor phase voltage s V r applied across the rotor impedance. This
consists of resistance R r and inductive reactance L r . The reactance depends on the
rotor current frequency. If the standstill rotor reactance is X r ¼w s L r , then its value
at slip s will be sX r . The rotor current is therefore given by
s V r
R r þ j sX r
I r ¼
Note that the slip-dependent rotor voltage may be replaced by the standstill rotor
voltage as follows:
V r
R r = s þ j X r
I r ¼
The standstill rotor voltage can be referred to the stator circuit, given the effective
rotor/stator turns ratio N :
V r
The rotor current referred to the stator is
V 0 r ¼
I 0 r ¼ N I r
It is instructive to develop an equivalent circuit representing each phase of the
induction machine. This is analogous to the equivalent circuit for a transformer
shown in Figure 2.3, with machine rotor in place of transformer secondary and
machine stator in place of transformer primary. It is convenient to refer all quan-
tities to the stator. The current in a rotor phase will be 1/ N times the current in a
stator phase, to achieve the m.m.f. balance required for transformer action (see
Section 2.2.5). Rotor impedance referred to the stator must therefore be scaled by
1/ N 2 to give the same relative voltage drop and power loss in the stator. Thus
R r
N 2
R r ¼
X r
N 2
X r ¼
The rotor circuit referred to the stator is shown to the right of aa 0 in Figure 3.12.
The stator resistance R s and stator leakage reactance X s are also shown in
Figure 3.12. The equivalent circuit is completed by the magnetising reactance X m .
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