TELKOMNIKA ISSN: 1693-6930
A Novel Technique for Fault-Tolerant Control of Single-Phase IM M. Jannati 787
In Figure 3,
1 2
2 2
1 2
2 2
4 3
4 3
j Z
jZ Z
Z Z
j Z
jZ Z
Z Z
Z Z
Z Z
Z
b la
lm b
lm f
la lm
f lm
11 In 11, Z
3
and Z
4
are the parameters in terms of inductances L
ds
andor L
qs
andor M
ds
andor M
qs
. As can be seen using 10, the equivalent circuit of single-phase IM Figure 2 changed into a balanced circuit Figure 3. Equation 10 can be re-written as follows:
m a
m a
m a
m a
I I
N N
j I
jI I
N N
jI
1 1
12 And,
m a
m a
V Z
V jZ
V V
jZ V
Z jV
3 4
1 3
4 1
13 Equation 12 and 13 are transformation matrices for transformation of variables from
unbalanced set eg., Figure 2 to the balanced set eg., Figure 3. Based on 12 and 13, following transformation matrices for stator voltage and current variables can be derived:
Transformation matrix for stator voltage variables:
s qs
s ds
e e
e e
s qs
s ds
e vs
e qs
e ds
Z Z
Z Z
T
cos
sin sin
cos
3 4
3 4
14 Transformation matrix for stator current variables:
s qs
s ds
e e
qs ds
e e
qs ds
s qs
s ds
e is
e qs
e ds
i i
M M
M M
i i
T i
i
cos sin
sin cos
15 To obtain 14 and 15, following substituting have been considered in 14 and 15
superscript “e’’ indicates that the variables are in the rotating reference frame. Moreover, “ θ
e
’’ is the angle between the stationary reference frame and the rotor flux oriented reference frame.
s qs
m s
ds a
e qs
e ds
s qs
m s
ds a
e qs
e ds
ds qs
a m
e e
i I
i I
i I
i jI
v V
v V
v V
v jV
M M
N N
j
, ,
, ,
, ,
, ,
cos 1
, sin
1 1
1 1
16 It is expected by using 14 and 15 the unbalanced equations of single-phase IM
become similar to the balanced equations.
4.2. Equations of RFOC for Single-Phase IM Based on Proposed Method
In this section vector control equations of single-phase IM based RFOC is presented. Using 15 and after simplifying, the equations of single-phase IM are obtained as follows:
ISSN: 1693-6930
TELKOMNIKA Vol. 13, No. 3, September 2015 : 783 – 793
788 Rotor voltage equations:
s qr
s dr
e s
e s
r r
r r
r r
r r
e s
s qs
s ds
e is
e is
qs ds
r qs
r ds
e s
e s
i i
T T
dt d
L R
L L
dt d
L R
T i
i T
T dt
d M
M M
dt d
M T
T
1 1
17 After simplifying Equation 17 can be written as:
e qr
e dr
r r
r e
r r
e r
r r
e qs
e ds
qs qs
e r
qs e
r qs
i i
dt d
L R
L L
dt d
L R
i i
dt d
M M
M dt
d M
18 Electromagnetic torque equation:
s qs
s ds
e is
e is
ds qs
T e
s T
e s
s qr
s dr
s qs
s ds
ds qs
s qr
s dr
s qr
s ds
ds s
dr s
qs qs
e
i i
T T
M M
T T
i i
Pole i
i M
M i
i Pole
i i
M i
i M
Pole
1 1
2 2
2
19 After simplifying Equation 19 can be written as:
2
e qr
e ds
e dr
e qs
qs e
i i
i i
M Pole
20 In summery based on Equation 17-20, equations of RFOC for single-Phase IM are
obtained as following equations. In the process of obtaining these equations the assumption λ
dr e
= ǀλ
r
ǀ and λ
qr e
=0 is considered in RFOC method, the rotor flux vector is aligned with d-axis; λ
dr e
= ǀλ
r
ǀ and λ
qr e
=0:
dt d
T i
M
r e
ds qs
r
1
21
e qs
r qs
r e
i L
M Pole
2
22
r r
e qs
qs r
e
T i
M
23
TELKOMNIKA ISSN: 1693-6930
A Novel Technique for Fault-Tolerant Control of Single-Phase IM M. Jannati 789
In Equation 21, T
r
is rotor time constant T
r
=L
r
R
r
. As can be seen from equations 21-23 the structure of RFOC equations of single-phase IM using proposed transformation
matrix for stator current variables, become like balanced equations. Consequently, Figure 4 is proposed for IRFOC of single-phase IM. As mentioned
before, by substituting L
ds
=M
ds
=R
ds
=0, this control system can be used for faulty single-phase IM. In This Figure, 2 to 2 transformation matrix for stator currents is as follows:
s qs
s ds
bs as
i i
i i
1 1
24
Figure 4. Block diagram of proposed IRFOC for both healthy and faulty single-phase IM
5. Simulation Results
