The American Journal of Engineering and Technology
102
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TYPE
Original Research
PAGE NO.
102-114
10.37547/tajet/Volume07Issue06-11
OPEN ACCESS
SUBMITED
19 April 2025
ACCEPTED
22 May 2025
PUBLISHED
14 June 2025
VOLUME
Vol.07 Issue 06 2025
CITATION
Emmanuel C. Obuah, Uche C. Eze, & Benjamine Akinloye. (2025). Direct-
Phase Variables Performance Analysis of Concentrated Winding
Permanent Magnet Synchronous Generator with Capacitive Assistance.
The American Journal of Engineering and Technology, 7(06), 102
–
114.
https://doi.org/10.37547/tajet/Volume07Issue06-11
COPYRIGHT
© 2025 Original content from this work may be used under the terms
of the creative commons attributes 4.0 License.
Direct-Phase Variables
Performance Analysis of
Concentrated Winding
Permanent Magnet
Synchronous Generator
with Capacitive Assistance
Emmanuel C. Obuah
Department of Electrical Engineering, Rivers State University, Port
Harcourt, Nigeria
Uche C. Eze
Department of Electrical Engineering, Rivers State University, Port
Harcourt, Nigeria
Benjamine Akinloye
Department of Electrical Engineering, Federal University of Petroleum,
Effurum, Nigeria
Abstract:
The dynamic and transient performance
analysis of a three-phase interior rotor concentrated
winding permanent magnet synchronous generator
(CW-IPMSG) with was presented. In this paper. The
study was done in direct-phase variables concentering
only the fundamental magneto-motive force (MMF).
The machine’s inductance was determined using
winding function theory (WFT). The derived
inductance was used to determine performance
characteristics of the machine’s variables such as
phase current, load current and electromagnetic
torque. The study was validated in MATLAB/Simulink
to observe the performance of the characteristics of
the generator. The study was carried out at no-load
condition, under load perturb, as well as increase and
decrease of capacitor. It was observed that the
permanent magnet synchronous generator had
slightly better output performance with capacitor
assistance.
Keywords:
Concentrated Winding; Direct-Phase
Variables.; Inductance, Permanent Magnet; Winding
Function Theory.
Introduction:
Most electric machines have distributed
winding (DW). Distributed windings were preferred over
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the concentrated winding (CW) because the distributed
winding gives a sinusoidal uniform MMF [1-3].
In recent years electric machines with concentrated
windings have become a competitive alternative to
machines with distributed windings for certain
applications. Machine with concentrated winding is
easier and cheaper to manufacture because of the short
ends winding feature. It has higher power density and
good fault tolerance capabilities especially the
fractional-slot concentrated winding machine. Again,
machine with CW has higher slot fill factor and good
efficiency compared to distributed winding permanent
magnet (PM) machine [4]. A concentrated winding is
preferred for a cost-effective application which requires
higher power density.
The benefits of concentrated windings and dual-
windings together have not been taken advantage of in
the synchronous reluctance motor. Previous researches
carried out on the synchronous reluctance motor
considered the stator of the machine as having only
distributed, or having distributed dual stator winding.
Performance characteristic of permanent magnet
synchronous motor (PMGM) with CW and DW was
carried out. The study compared the performance of the
two arrangements using certain rotor parameters such
as back EMF, resistance, efficiency, output torque etc
with two identical rotor dimensions. Parasitic
characteristics such as cogging torque, torque pulsation,
unbalanced magnetic force, mechanical vibration and
acoustic noise are always of significant concern during
the machine design. The parasitic effects could be
potentially more harmful in CW PMSM since there are
additional space harmonic contents in the stator MMF
distribution of the machine [5].
The synchronous reluctance generator with transversely
limited rotor has been studied. It was reported that the
generator is more robust, and has relatively lower core
loss, and provides spaces for embedding cage, and can
easily be skewed, and gives allowance for inserting
magnet. [6].
Reference [7] carried out comparative analysis of
synchronous reluctance machines (SynRM) with 6, 8 and
12 poles. The effect of different pole numbers on
average torque, loss, torque-ripple, d and q inductances
is investigated. It was shown that the 8-pole machine
has similar performance to the 6-pole machine but the
12-pole machine has worse performance.
Reference [8] developed an analytical model in the d-q
reference frame to recognize the steady-state of the
self-excited reluctance generator, considering no-load
and resistive load conditions. A fast method to estimate
the minimum capacitance requirements was also
proposed, and experiments were carried out to verify
the analytical results. After that, attention was paid on
the capability of self-excitation in reluctance generator
with different residual magnetisms in the rotor.
Different levels of residual rotor magnetism are
achieved by different magnetizing DC currents. An
indicative value of phase current was defined to
determine the self-excitation, and the required
minimum residual rotor magnetism for self-excitation in
the reluctance generator connected with different
capacitances was discussed. At last, the capability of
self-excitation in reluctance generators by connecting
charged capacitors was investigated.
Reference [9] presented a performance comparison of
an interior mounted permanent magnet synchronous
generator (IPMSG) with a synchronous reluctance
generator with the same size for a wind application. It
was found that using the same geometrical dimensions,
a SynRG can convert 74 % of the power that an IPMSG
can convert, while it has 80% of the IPMSG weight.
Moreover, it is found that the efficiency for the IMPSG is
99 % at rated power compared to 98.7 % for the SynRG.
Reference [10] in their work, investigated the capability
of simple salient-pole rotor synchronous reluctance
generators at a 5 MW power level. Different salient pole
rotor profiles are considered in the finite element design
optimization of the generator. It was found that with the
simple salient pole rotor, similar torque density and
efficiency are obtained as in published equivalent
distributed flux barrier rotor reluctance synchronous
generators. Also,
Reference [11] presented a performance comparison of
a 5MW interior permanent magnet synchronous
generator (IPMSG) with a 5 MW PMa-SynRG with the
same stator, to be used for a wind energy application. It
was found that PMa-SynRG has lower rotor weight as
well as 14 % lower magnet weight with the same
maximum torque performance. For wind speeds lower
than 8.5 m/s the PMa-SynRG has less loss. Moreover,
the machine annual energy efficiency for the PMa-
SynRG is higher for average wind speeds between 5-10
m/s.
Reference [12] presented a study of permanent magnet
assisted synchronous generator for autonomous
application using the classical Park’s d
-q model. It was
observed that the generator with permanent magnet
had better output performance than the conventional
generator when compared. Also [13] carried out steady
state performance analysis of permanent magnet
synchronous generator with capacitive assistance using
d-q model, where capacitor was used to improve power
output of the generator and voltage regulation.
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The study of electric machines based on the actual
geometry of the machine is importance because it gives
the actual behaviour of the machine. Widing Function
Theory is used to study machines beheviour considering
the actual placemen of machine windings. Some studies
that adopted the theory of winding function in machine
analysis include the use of WFT on induction machines
[14], synchronous reluctance motors [15], switched
reluctant motors [16] a machine with doubly salient
structure [17], and for model of a synchronous
reluctance motor including all slot and winding
harmonics [18].
Reference [19] derived a symmetrical component for
asymmetrical multiphase windings for a motor, where
an analytical formulation is presented to relate the
harmonic content of winding functions to winding
factors. The harmonic leakage factor is accurately
formulated from the winding function, and the
suggested analysis method was validated with the star
of slots and sinusoidal functions of distribution and pitch
factors
A five-phase synchronous reluctance motor and
permanent magnet synchronous reluctance motor was
modelled and simulated in phase variables and the
results in phase variables tend to agree with that of the
Finite Element Analysis [20-21].
Reference [22] also carried out a performance analysis
of a concentrated dual-winding synchronous reluctance
machine with capacitive assistance. The machine got its
power supply directly on power line. The transient and
dynamic performance analysis of a proposed line-start,
three-phase concentrated dual-winding synchronous
reluctance motor in comparison with the conventional
concentrated winding synchronous reluctance motor
was made. The modelling of the synchronous reluctance
machine was done in direct-phase variables considering
only the fundamental magneto-motive force. The
machine inductances of both machine models were
determined using winding function theory. The derived
inductances were used to determine machine
performance characteristics such as torque, speed,
phase currents etc. The performance characteristics of
both motors were monitored using MATLAB/Simulink,
and the proposed line-start machine with capacitive
assistance was observed to have improved performance
characteristics when compared to the conventional
machine.
Most study on CW machine considered the machine as
motor. There is paucity of literature where the machine
was used as generator. This study shall look at the
analysis of the synchronous reluctance generator with
assistance from permanent magnet for excitation.
To accomplish the goal of the study, the following
objectives were addressed.
i.
To develop the clock diagram of the CW-IPMSG
machine based on the arrangement
ii.
To present mathematical models for the
inductance of the CW-IPMSG in direct-phase
variable using Winding Function Theory
iii.
To use the calculated inductance to obtain the
phase voltage, phase voltage, load current and
electromagnetic torque of the CW-IPMSG at no-
load condition using MATLAB/Simulink
iv.
To study of the performance of the CW-IPMSG
on sudden addition and removal of load using
MATLAB/Simulink
v.
To study of the effect of variable capacitance on
the performance of the CW-IPMSG on using
MATLAB/Simulink.
2. MATERIALS AND METHODS
There are several methods used to model or analysis of
electric machine in literature. In this section, a list of
materials and method adopted is presented.
2.1. Materials
The materials used for the study include typical
machine parameters winding presented in Table 1, and
MATLAB/Simulink.
2.2. Method
Winding Function Theory was used to accomplish the
study's aim. Winding Function involves calculating
machine inductances based on actual geometry or
placement of the machine coil.
2.1.1. Modeling of the CW-IPMSG
In modeling the CW-IPMSG, it is assumed that
i.
the magnetic flux across the air gap is
perpendicular. only radial flux is considered
ii.
the magnets are seen to be placed adjacent to
each other; and the flux density between the adjacent
magnets is assumed to be equal to zero. By this, there is
no flux linkage between the adjacent magnets.
iii.
two coils are series connected and behave like a
single phase.
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With these assumptions, the magnets can be modeled
as coils which are wound in opposite direction with
number of turns. So, because of the way they are
stacked, the maximum turn is 2. The task here is to use
Winding Function Theory to calculate the machine
inductance and used the calculated inductance to
obtained other relevant quantities like power output,
voltage etc. Figure 1 shows the coil clock diagram of the
machine based on the winding pattern. The machine is
a 4 pole 12 slot machine with double-layer winding.
Figure 2 shows an ideal 4 pole machine based on the
assumption made
Figure 1 Clock diagram for the machine with double layer
Figure 2 A basic machine
The winding function method is used for calculation of
the machine inductances along with the actual machine
geometry. Winding function corresponding to the stator
windings are defined as a function of stator angle
according to their winding layouts in [23]. The procedure
used in [24] was adopted to obtain the expression for
turn function in (1).
⦑𝑛⟮𝜑
𝑠
⟯⦒ = ⦋
1
2𝜋
∫ 𝑛(𝜑
𝑠
)
𝜋
0
⦌ =
𝑁
𝑐
4
(1)
The turn function shows the number of turns as a
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function of the stator angle.
According to
[24]
,
the inductances will be calculated for
the case in which the two coils, A and B, are treated as
distinct coils, and ones which are connected in series.
Equation (2) gives the expression for the first series.
While equation (3) gives the expression for the second
series
𝑁
𝐴
(𝜑
𝑠
) =
3𝑁
𝑐
4
(2)
𝑁
𝐵
(𝜑
𝑠
) =
−𝑁
𝑐
4
(3)
The magnetizing-inductance of phase A is calculated by
integrating the turn function with respect to the stator
angular position from the reference period zero degree
as the machine rotates to 360 degrees for complete
revolution. as found in (4)
𝐿
𝐴
=
𝜇
𝑜
𝑟
𝑠
𝑙
𝑠
𝑔
𝑒𝑓𝑓
∫
𝑁
𝐴
2
(𝜑
𝑠
)𝑛
𝐴
2𝜋
0
𝑑𝜑
𝑠
(4)
Solving gives (6)
𝐿
𝐴
=
1
8
𝜋𝑁
2
𝜇
𝑜
𝑟
𝑠
𝑙
𝑠
𝑔
𝑒𝑓𝑓
∗ 𝑔
𝑜
(5)
Similarly, the magnetizing-inductance of B is calculated
as
𝐿
𝐵
=
𝜇
𝑜
𝑟
𝑠
𝑙
𝑠
𝑐
∫
𝑁
𝐵
2
(𝜑
𝑠
)𝑛
𝐵
𝑑𝜑
𝑠
2𝜋
0
(6)
Again, Solving gives (8)
𝐿
𝐵
=
3
8
𝜋𝑁
2
𝜇
𝑜
𝑟
𝑠
𝑙
𝑠
𝑔
𝑒𝑓𝑓
∗ 𝑔
𝑜
(7)
where
0
is the relative permeability of free space
𝑟
𝑠
is the stator radius
𝑙
𝑠
is the stator length
𝑁
𝑐
is the number of turns of coil
𝑔
𝑒𝑓𝑓
is the effective air-gap
𝑔
𝑜
is the amplitude of the first order harmonic
From (5) and (7), the magnetizing-inductance is
proportional to the square of the number of turns per
tooth, stator length and stator radius, and inversely
proportional to the effective air gap length. It is also
clear that total inductance does not depend on the
number of stator slot. Therefore, air-gap inductance
does not depend on the number of slots. Furthermore,
the inductance does not depend on the number of
poles. So, air-gap inductance is not a function of a
combination of number of poles per slot.
To achieve a sinusoidal field, each of the stator windings
of machine is shifted in space relative to the others by
2π/3
. In the stator reference frame, the self-inductances
of the stator for phase B and C describing the electrical
circuit of a three-phase synchronous machine using
conventional notations are given as (8) through (11), if
leakage inductance is accounted for
[25]
.
𝐿
𝑎𝑎
= 𝐿
𝑙𝑠
+ 𝐿
𝐴
− 𝐿
𝐵
𝑐𝑜𝑠 2 𝜃
𝑟
(8)
𝐿
𝑏𝑏
= 𝐿
𝑙𝑠
+
1
2
𝐿
𝐴
− 𝐿
𝐵
𝑐𝑜𝑠 2 (𝜃
𝑟
−
𝜋
3
)
(9)
𝐿
𝑐𝑐
= 𝐿
𝑙𝑠
+
1
2
𝐿
𝐴
− 𝐿
𝐵
𝑐𝑜𝑠 2 (𝜃
𝑟
+
𝜋
3
)
(11)
The mutual inductances are in (12) through (17)
𝐿
𝑎𝑏
= −
1
2
𝐿
𝐴
− 𝐿
𝐵
𝑐𝑜𝑠 2 (𝜃
𝑟
−
𝜋
3
)
(12)
𝐿
𝑎𝑐
= 𝐿
𝑙𝑠
+ 𝐿
𝐴
− 𝐿
𝐵
𝑐𝑜𝑠 2 (𝜃
𝑟
−
2𝜋
3
)
(13)
𝐿
𝑏𝑎
= −
1
2
𝐿
𝐵
− 𝐿
𝐴
𝑐𝑜𝑠 2 (𝜃
𝑟
+
𝜋
3
)
(14)
𝐿
𝑏𝑐
= −
1
2
𝐿
𝐵
− 𝐿
𝐴
𝑐𝑜𝑠 2 (𝜃
𝑟
+
𝜋
3
)
(15)
𝐿
𝑐𝑎
= −
1
2
𝐿
𝐴
− 𝐿
𝐵
𝑐𝑜𝑠 2 (𝜃
𝑟
+ 𝜋)
(16)
𝐿
𝑐𝑏
=
1
2
𝐿
𝐴
− 𝐿
𝐵
𝑐𝑜𝑠 2 (𝜃
𝑟
−
2𝜋
3
)
(17)
The stator total inductances can be given in matrix form
in equation (18) for phase A, B and C
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=
cc
cb
ca
bc
bb
ba
ac
ab
aa
S
L
L
L
L
L
L
L
L
L
L
(18)
Also, considering stator phase displacement angle, the
phase A, B and C flux linkage is expressed as in equations
(19) (20) (21)
𝜆
𝑎𝑠
= 𝐿
𝑎𝑎
𝑖
𝑎
+ 𝐿
𝑎𝑏
𝑖
𝑏
+ 𝐿
𝑎𝑐
𝑖
𝑐
+ 𝜆
𝑚
(𝑠𝑖𝑛 𝜃
𝑟
)
(19)
𝜆
𝑏𝑠
= 𝐿
𝑏𝑏
𝑖
𝑎
+ 𝐿
𝑏𝑎
𝑖
𝑏
+ 𝐿
𝑏𝑐
𝑖
𝑐
+ 𝜆
𝑚
𝑠𝑖𝑛 (𝜃
𝑟
−
2
3
𝜋)
(20)
𝜆
𝑐𝑠
= 𝐿
𝑐𝑐
𝑖
𝑎
+ 𝐿
𝑐𝑏
𝑖
𝑏
+ 𝐿
𝑐𝑐
𝑖
𝑐
+ 𝜆
𝑚
𝑠𝑖𝑛 (𝜃
𝑟
+
2
3
𝜋)
(21)
where
m
is the permanent magnet constant flux.
Using Kickoff’s voltage law, the generator voltage
equation is given as (22).
(
)
c
S
S
S
S
V
I
L
dt
d
R
I
E
+
+
=
(22)
Rearranging (23) gives (24)
( )
( )
c
r
S
r
r
S
r
S
S
S
V
L
d
dL
R
I
E
dt
dI
−
+
−
=
−
1
*
*
(23)
where
𝑉
𝑐
is the capacitor voltage
𝐼
𝑆
is the stator three phase current given in matrix form
as:
𝐼
𝑆
= ⦋𝐼
𝑎
𝐼
𝑏
𝐼
𝑐
⦌
(24)
𝑅
𝑆
is the stator three-phase resistance, given in matrix
form as (24).
𝑅
𝑆
= 𝑑𝑖𝑎𝑔 [
𝑅
𝑎
0
0
0
𝑅
𝑏
0
0
0
𝑅
𝑐
]
(25)
where
𝑅
𝑎
= 𝑅
𝑏
= 𝑅
𝑐
𝑬 is given as = [𝑬
𝒂
; 𝑬
𝒃
; 𝑬
𝒄
; ]
(26)
where
wt
m
E
a
cos
*
=
(27)
−
=
3
2
cos
*
wt
m
E
b
(28)
−
=
3
4
cos
*
wt
m
E
c
(29)
For the sake of making the capacitor voltage as a system
state variable for simulation, the integral equation for
the capacitor voltage is given as (30).
dt
I
C
V
s
ca
=
1
(30)
where
𝐶
is the capacitance of the capacitor.
The load equation is given in as
[26]
as:
1
cos
1
2
−
=
r
aL
aL
R
L
(31)
where
𝑅
𝑎𝐿
is the per phase resistive load
Consequently, the output power is given as (32).
𝑃
𝑜𝑢𝑡
= 3𝐸𝐼
𝑎
𝑐𝑜𝑠 𝛷
(32)
where
𝛷
is the load power factor.
The electromagnetic torque can be derived in (33) using
co-energy method.
(33)
3. Results and Discussion
The performances of the CW-IPMSG at no-load
condition, load perturbs and capacitance variation are
( )
m
r
r
S
S
e
d
L
d
I
T
+
=
2
2
1
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shown in Figures 3 to 13. The results were obtained
using equations (5), (7) through (21), (23) through (25),
(27) through (33) with the relevant parameters shown in
Table 1. Excitation capacitor of 90 µF was used while the
permanent magnet flux used was 0.8 at 0.8 power
factor.
3.1. Dynamic Performance of CW-IPMG at No-Load
Condition
Figure 3 through 7 shows the phase voltage, stator
phase current, capacitor phase current per phase load
current and electromagnetic torque of the CW-IPMSG
at no-load condition. The CW-IPMSG exhibited the
quality of producing voltage of about 230 V. This
closely match the value given in the machine
nameplate. The corresponding phase current is shown
in Figure 4. The phase current for the CW-IPMSG is
about 7 A while the capacitor current is about 7.5 A as
illustrated in Figure 5. In Figure 6, the load current is
about 0.0015 A. The electromagnetic torque is
illustrated in Figure 7. At initial start, the value of the
torque rose to about 5.6 Nm. The generator continues
to run, it started gaining stability. That was from 0.2
seconds through 0.4 seconds when it stopped running.
Figure 3: Per phase voltage of the CW-IPMSG on no-load condition
Figure 4: Per phase current of the CW-IPMSG at no-load condition
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Figure 5: Per phase capacitor current of the CW-IPMSG at no-load condition
Figure 6: Per phase load current of the CW-IPMSG at no-load condition
Figure 7: Electromagnetic torque of the CW-IPMSG at no-load condition
3.2. Performance of the CW-IPMG on Sudden
Addition and Removal of Load
Performance of the of the CW-IPMSG on sudden
addition and removal of load was also studied in Figure
8 through Figure 10. The generator operated with load
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from zero seconds to 0.7 second when load was
suddenly removed and the added again at 2 seconds. As
seen in Figure 8, there was voltage flicker observed as
the load was added. There was also some level of
oscillation on the electromagnetic torque, from 0.5
second to 1 second when the load was added and
removed in Figure 9. Figure 10 shows the same effect for
the load current. The load current was 0.4 A when the
generator operated on load. When load was removed,
the load current is about zero
Figure 8: Phase voltage of the CW-IPMSG on sudden removal and addition of load condition
Figure 9: Electromagnetic torque of the CW-IPMSG on sudden removal and addition of load condition
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Figure 10: Load current of the CW-IPMSG on sudden removal and addition of load condition
3.3. Performance of the CW-IPMG on Varriable
Capacitance
Since the generator is equipped with a balanced three
phase capacitor, which can be varied, the analysis
extended to the effect of change in excitation capacitor.
The capacitance is made to change from 32 µF to 52 µF
and then to 92 µF. The corresponding time of change is
from 0 to 0.5 second and then 1.0 second. The change
in the capacitance and the corresponding voltage build-
up, load current and the electromagnetic torque is
illustrated in Figure 11 through Figure 13. The behavior
of the generator with respect to the performance
parameters indicates that increase in capacitor
capacitance yields more voltage and electromagnetic
torque and this can also be applicable to other
parameters or quantities. The simulation was done at
resistive-
inductive load of 500 Ω-300 mH and permanent magnet flux of 0.8
Figure 11: Voltage of the CW-IPMSG on variable capacitance
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Figure 12: Load current of the CW-IPMSG on variable capacitance
Figure 13: Electromagnetic torque of the CW-IPMSG on variable capacitance
Table 1: Studied Machine Parameters
S/N
Parameters
Value
1
Rated voltage
220 V
2
Rated power
110 kW
3
Rotor speed
2400 rpm
4
Number of turns per coil
115
5
Amplitude of the 1st order
harmonics
5.8
6
Stator Stack length
800 mm
7
Frequency
50Hz
8
Resistance
0.03 Ω
9
Effective airgap
1.2 mm
10
Stator radius
82.4 mm
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CONCLUSION
Dynamic and transient performance analysis of
concentrated winding permanent magnet synchronous
generator with capacitor assistance has been studied.
The coil winding diagram of the studied permanent
magnet synchronous generator with concentrated
winding was developed. The inductance of the
machines was also calculated based on Winding
Function Theory. It was seen that inductance is
proportional to the square of the number of turns per
tooth, stator length and stator radius, and inversely
proportional to the effective air gap length. It is also
clear that total inductance does not depend on the
number of stator slot. The calculated inductance was
used to obtain the voltage, electromagnetic torque etc
of the generator at no-load condition, sudden addition
and removal of load, and capacitor variation using
MATLAB/Simulink tool. At no-load condition, it was
observed that CW-IPMSG maintain the desired
performance where the output voltage is 230 V and
electromagnetic torque is 5 Nm. When load was
suddenly added to the generator, there were
oscillations at the time when the load was added, and
another oscillation was observed when load was
removed. Increase in value of capacitor also increase
the output performance of the generator at particular
load and permanent magnet flux. This study was
performed for a 4 pole 12 slot concentrated winding of
permanent magnet synchronous generator.
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