Electric Power Components and Systems, 34:115–118, 2006
Copyright © Taylor & Francis Inc.
ISSN: 1532-5008 print/1532-5016 online
DOI: 10.1080/15325000691001449
Letter to the Editor: A Formula for the External
Rotor Resistance of Induction Motor
EMANUEL GLUSKIN
Electrical Engineering Department
Holon Academic Institute of Technology
Holon, Israel
and
Electrical Engineering Department
Ben-Gurion University
Beer-Sheva, Israel
A useful formula for calculation of the external resistance, added (per phase) to the
rotor of an induction motor, is derived. The criterion is to decrease the starting current
of the motor in a prescribed proportion.
Keywords power systems, induction motor, external rotor resistance, starting current
1.
Introduction
One of the well known possibilities of decreasing the starting current of an induction
motor with wound rotor is to use added (external) resistors connected in series with the
rotor’s windings. Such an added resistance r2added (the same in each phase) also may
be subject to requirements associated with the needed starting torque, and the finally
established motor’s speed. However, it is reasonable to develop a suitable formula for
the resistance, based only on the requirement of the starting current limitation, similar
as it is done for a DC motor. The formula thus obtained gives relatively high value of
r2added , and this value should be used, first of all, for choosing the rheostat. In any case,
the limitation on the current is important per se.
We require the starting current, Istart determined in the standstill conditions, be
decreased in a prescribed ratio:
Istart → Istart /b,
with b given, and we have to find the function
r2added
= f (b).
r2
Manuscript received in final form on 7 March 2005.
Address correspondence to Emanuel Gluskin, Holon Academic Institute of Technology, Electrical Engineering Department, 52 Golumb Street, P.O. Box 305, Holon 58102, Israel. E-mail:
gluskin@ee.bgu.ac.il or gluskin@hait.ac.il
115
116
E. Gluskin
This function depends on the angle/phase of the equivalent impedance of the motor
(determined in the blocked-rotor test, using the -equivalent scheme, or ignore the parallel
elements in the T -equivalent scheme [1, 2]), written, in the standard notations, as
zeq = r1 + r2 + j (x1 + x2 ) = req + j xeq .
2.
Derivation of the Formula
At the start, we have for slip s, that s = 1, and in the per phase calculation, we have,
using the standstill parameters,
Istart = V
2 + x2
req
eq
.
(1)
Here V is the nominal motor per phase voltage that may result in an unacceptably large
Istart if req is not properly increased.
Resolving Eq. (1) for req = r1 + r2 , we obtain
r2
=
2
V
2
− xeq
− r1 .
Istart
(2)
In this equation we simultaneously perform the replacements
r2 → r2 + r2added
and
Istart → Istart /b,
obtaining
r2added
=
b2
2
V
Istart
2
− xeq
− (r1 + r2 ).
(3)
Now using Eq. (1) in the form
V
Istart
2
2
2
= req
+ xeq
,
we have from Eq. (3) that
r2added
=
2 + x2 ) − x2 − r ,
b2 (req
eq
eq
eq
=
r2added
2 + (b2 − 1)x 2 − r .
b2 req
eq
eq
or
(4)
Letter to the Editor
117
Introducing
tan(ϕeq ) =
xeq
,
req
we rewrite Eq. (4) as
r2added
2
2
2
=
b + (b − 1) tan (ϕeq ) − 1 req .
(5)
We can somewhat simplify this formulae by using the typical assumption [1, 2] that
req = 2r2 ,
and substituting this in Eq. (5), we obtain that
r
r2added
2 + (b2 − 1) tan2 (ϕ ) − 1 ,
= 2added
=
2
b
eq
r2
r2
(6)
which is the final form of the formulae for r2added suggested.
For b 1, we have from Eq. (6)
r2added
2b
,
=
r2
cos(ϕeq )
and for b = 1 + ε, ε 1, and ε tan2 (ϕeq ) 1,
r2added
2ε
= 2ε[1 + tan2 (ϕeq )] =
.
2
r2
cos (ϕeq )2
Since the typical tan(ϕeq ) is about 2 or 3, in the latter case,
r2added
ε,
r2
for example, 14ε. This shows high sensitivity of r2added to the requirement for decrease
in Istart , for a small decrease.
Example
Given
zrotor = (1/2)zeq = 0.5 + j 1.5 (or, r2 = 0.5 ; tan(ϕeq ) = 3), and it is required to decrease the starting current by 3,
that is, b = 3. We have, according to Eq. (6),
√
r2added
= 2[ 9 + 8 · 9 − 1] = 16,
r2
(7)
which shows how (physically) large the rheostat has to be. We also find r2added = 16r2 =
8 .
118
3.
E. Gluskin
Conclusion and Final Remarks
Useful Eqs. (5) and (6), relevant to starting induction motors having wounded rotor, are
suggested. The per unit value for r2added defined by Eq. (6) certainly does not contradict
any possible requirement for r2added relevant to the state of developed speed, because with
the development of the speed, r2added can be reduced. As for the possible requirement for
r2added , associated with the needed starting torque—this required value and that given by
Eq. (6) have to be considered against each other in order√to find the optimum/compromise.
√
External eddy-currents impedances for which z ∼ ω (Re(z) ∼ ω) should also be
mentioned here. The value thus added to zeq is automatically decreased with the decrease
in the rotor’s electrical frequency. This is an excellent compact and reliable tool for soft
starting induction motors.
References
1. B. S. Guru and H. R. Hiziroglu, Electric Machinery and Transformers, New York: Oxford
University Press, 2001.
2. G. McPherson and R. D. Larmore, An Introduction to Electrical Machines and Transformers,
New York: John Wiley & Sons, 1990.