This paper investigates the optimal values of turnon and turnoff angles, and ratio of flux linkage at turnoff angle and peak phase current positions of optimal control for accomplishing maximum output power in an 8/6 Switched Reluctance Generator (8/6 SRG). Phase current waveform is analyzed to determine optimal excitation angles (optimal turnon and turnoff angles) of the SRG for maximum output power which is applied from a nonlinear magnetization curve in terms of control variables (dc bus voltage, shaft speed, and excitation angles). The optimal excitation angles in single pulse mode of operation are proposed via the analytical model. Simulated and experimental results have verified the accuracy of the analytical model.
1. Introduction
Switched reluctance generators (SRG) has been applied to many fields because of its merits such as low manufacturing cost, low inertia, fault tolerance
[1
,
2]
high efficiency and reliability
[3]
. However, an SRG requires position sensor and produces acoustic noise and vibration
[2
,
3]
. Torque production and energy conversion process of the SRG are described
[4]
, and the SRG control systems for regulating speed and output power use a DSP controller as processor with excitation angles from a ROMbased table of switch positions. SRGs have proved for some applications, for example, starter/generator for gas turbine of aircrafts running at constant speed and connected to utility line
[3]
generating maximum output power depends on optimal excitation angles due to dc bus voltage and shaft speed are constant, wind generator connected with a fixed voltage unity line
[5]
so maximum output power depends on optimal excitation angles at each shaft speed, starter alternator in car
[6]
generating maximum output power depends on optimal excitation angles at each shaft speed and dc bus voltage due to its voltage may change during charging or discharging periods. To control generating maximum output power of SRG we must know relationship of control variables included dc bus voltage, shaft speed, and excitation angles. However there is no analytical equation for output power in terms of design parameters and control variables due to highly nonlinear characteristics of an SRG, therefore iterative simulation and experiments of an SRG have been used for finding output power profile
[7]
. Almost simulation models used to study generating maximum output power of SRG are based on lookuptable techniques
[8]
, magnetic equivalent circuit analysis
[9]
, cubicspline interpolations
[10]
, and finiteelement analysis (FEA)
[11]
which are very accurate, however these models require either numerous values of fluxlinkage and current position or information magnetic properties about an SRG. A simple model of the nonlinear magnetization characteristics of an SRG has been proposed
[12
,
13]
which is easy to build, its accuracy, and the machine geometry is unknowable. Finding the output power requires the knowledge of the current waveform that the expression of phase current based on magnetic field energy proposed
[14]
for applying to optimize method of firing based on simulations is reliable but complicated. The closed loop power control algorithm for the SRG which relies on experimental characterization at only four operating points, turnon and turnoff angles, speed, and power, is presented
[16]
. Analysis of the phase inductance and the phase current divided into five periods during the excitation period and commutation period of an SRG for creating the triggering signals of the main switches are described
[17]
. The relationship between commutation angles and output power is described
[18]
that the impact of changes in excitation angles on the output voltage or power had been examined by applying various combinations of turnon and turnoff angles. Phase current and phase flux linkage of an SRG with optimal excitation angles obtained from a nonlinear SRG model based on MATLAB/SIMULINK are analyzed to find the relationship between the conventional and freewheeling excitation patterns through SFtransform for output power maximization and optimal symmetric freewheeling excitation
[19]
.
From those described above, there is no analytical equation for output power in terms of control variables due to highly nonlinear characteristics of an SRG, and phase current is significant parameter to find optimal excitation angles of an SRG for maximum output power. In this paper the mathematical expression of a nonlinear magnetization curve used is simple that depends on the two magnetization curves in aligned and unaligned positions of the rotor, phase current waveform used to determine optimal excitation angles of an 8/6 SRG is applied from a nonlinear magnetization curve in term of control variables included dc bus voltage (
u
), shaft speed (
𝜔
), and excitation angles ((turnon (
θ_{on}
)and turnoff (
θ_{off}
) angles) and its effectiveness is validated through simulated and experimental results.
2. Analysis of SRG Operation
SRG is a machine which excitation energy is supplied in every stroke. During the conducting period
θ_{on}
–
θ_{off}
(
Fig. 1
), the excitation energy is converted to electrical energy after the aligned angle. No energy is supplied to the load during the conducting period. During the defluxing period
θ_{off}
–
θ_{ext}
(
Fig. 1
), the store field energy is released as output energy through the freewheeling diodes (
Fig. 2
). The electrical energy produced during the defluxing period exceeds the excitation energy. The phase voltage has just
θ_{on}
and
θ_{off}
switching angles as shown in
Fig. 1
.
Phase of voltage and current, and flux linkage
Generator circuit for a phase
Figs. 1(a)

(c)
show idealized current waveforms with single pulse control that the peak current occurs at
θ_{off}
–
θ_{peak}
.
Fig. 1(a)
shows the case that the current increase after turning off the switches at
θ_{off}
, when the back emf in the coil is larger than the dc bus voltage (
e
>
u
).
Fig 1(b)
shows the case that the constant current after turning off the switches at
θ_{off}
until
θ_{peak}
, when the back emf and the dc bus voltage balance (
e
=
u
).
Fig. 1(c)
shows the case that the current decrease after turning off the switches at
θ_{off}
, when the back emf in the coil is smaller than the dc bus voltage (
e
<
u
) . The control method which regulates
u
with speed in order to maximize the output power by keeping the condition of
u
=
e
is proposed
[1
,
15]
.
In this paper, the control scheme of an 8/6 SRG for maximum output power which controls the current waveform like the case of
e
=
u
is proposed. The analytical model and its accuracy are validated through simulated and experimental results.
The voltage equation for a phase of the switched reluctance machine by neglecting the mutual inductance between the phases is given as:
Where
u
is the dc bus voltage,
i
is the phase current,
R
is the phase resistance,
L
is the phase inductance,
e
is the back emf,
θ
is the rotor position, and
𝜔
is the shaft speed. The back emf is defined as
Phase resistance variation is small and ohmic drop on phase resistance is usually negligible compared to dc bus voltage, so we don’t consider phase resistance variation in our analysis, flux linkage equals to:
Where extinct angle(
θ_{ext}
) = 2
θ_{off}
–
θ_{on}
, The circuit diagram of a generator with one phase leg has been proposed
[1]
as shown in
Fig. 2
. The integral of the currents in
Fig. 2
can be defined as:
The net generated current (
I_{o}
) =
I_{out}

I_{in}
, and phase output power (
P_{out}
) =
u
·
I_{o}
.
3. Phase Current Formulation
In earlier contribution of magnetization curve in
Fig. 3
proposed
[12
,
13]
; at unaligned position (
θ_{u}
) is linear to phase current, and at aligned position (
θ_{a}
) approximated by two curves composes of a straight line from coordinate origin
O
to point
S
and a curve from point
S
to point
M
. An analytical model proposed
[12
,
13]
can describe magnetization characteristics of machines with sufficient accuracy. This analytical model described by flux linkage depends on phase current and rotor position as shown in (5).
Flux linkage curves in θ_{u} and θ_{a}
When
Where
L_{u}
is inductance of the coil for the unaligned position,
L_{a}
is inductance of the coil for the aligned position,
N_{r}
is pole number of rotor,
θ
is rotor angle in radian. Note that
Ψ_{S}
,
i_{S}
and
Ψ_{M}
,
i_{M}
are the values of the flux and current taken at points
S
and
M
respectively.
Table 1
shows parameters of the candidate SRG used in this paper which has eight poles on the stator and six poles on the rotor as shown in
Figs. 4(a)
, and
(b)
shows idealized inductance versus rotor position.
Parameters of the candidate SRG
Parameters of the candidate SRG
The candidate SRG used in paper: (a) Structure of the 8/6 SRG; (b) Idealized inductance versus rotor position
Fig. 5
shows the magnetization curves of the candidate SRG at different rotor positions, both from measurement and the analytical model proposed
[12
,
13]
. Parameters of the analytical model used to calculate in (5) are
L_{u} = 40μH, L_{a} = 490μH, i_{s} = 25A, i_{M} = 45A, Ψ_{s} = 0.0125Wb
, and
Ψ_{M} = 0.017Wb
.
Magnetization curves of the candidate SRG
The significant variable to control an SRG for maximum output power is phase current. The phase current equation in this paper obtains from substituting (2) into (5). Since the magnetization curve in
Fig. 3
approximated by two curves composes of a straight line from coordinate origin
O
to point
S
therefore phase current in this period (
i
≤
i_{s}
) can be obtained from (6) and a curve from point
S
to point
M
therefore phase current in this period (
i
>
i_{s}
) can be obtained from (7).
Where
4. Optimal Excitation Angles
The case of
P_{out}
=
f
(
θ_{on}, θ_{off}
) when dc bus voltage (
u
) and shaft speed (
𝜔
) are constant, therefore output power will be a constant function of just switching angles proposed
[7]
. In
Fig. 6
shows the relationship of phase current and flux linkage waveforms, when conducting period is equals to defluxing period. Flux linkage at
θ_{off}
and
θ_{peak}
positions are defined as
Ψ_{f}
and
Ψ_{k}
respectively, and their equations are given as:
Idealized of phase current and flux linkage.
To simplify the analysis,
x
is defined as a ratio of
Ψ_{k}
and
Ψ_{f}
that it can be expressed as:
Then (10) can be rewritten as:
Phase current depends on dc bus voltage, shaft speed, and excitation angles that peak phase current (
i_{peak}
) at
θ_{peak}
position can occur in during generation period.
Then
θ_{peak}
can be obtainedas follow:
An SRG operates in single pulse mode, the phase current waveforms of the SRG limited the peak value to be equal can occur in three cases as shown in
Fig. 1
, and when the SRG is controlled with the constant of dc bus voltage and shaft speed therefore the peak phase current can be known by; to simplify the analysis,
y
is defined as a ratio of the dc bus voltage and the shaft speed and
y_{opt}
is defined as a ratio of the dc bus voltage and the shaft speed at peak phase current, in (12) implies that the peak phase current can exist either at turnoff position or after, the value of
in (12) never exceeds 1. Those mean a ratio of the dc bus voltage and the shaft speed at peak phase current
can be expressed as:
To find the optimal turnon angle (
θ_{on.opt}
) the analytical model is simulated for three cases of
y
;
y
= 0.042,
y
= 0.044, and
y
= 0.046 that all cases are fixed
u
at 27V when the excitation angles were adjusted to limit the peak value of the phase current to 45A, and the output power (4phase) can be known by
P_{out}
= 4 × (
u
·
I_{o}
). There are many combinations of turnon and turnoff angles as shown in
Fig. 7
. Apparently the maximum output power exists at turnon angle at 15° for all cases as shown in
Fig. 7(a)
. This simulation result has been also confirmed
[7]
. Also in this paper the value of optimal turnon angle (
θ_{on,opt}
) is 15°.
Fig. 7(b)
shows output power versus turnoff angle.
Output power at different excitation angles: (a) Output power versus turnon angle (b) Output power versus turnoff angle.
Fig. 8(a)
shows the phase current waveforms with single pulse control at
y
= 0.038,
y
= 0.042, and
y
= 0.048 with
u
= 27V for all cases when excitation angles were adjusted to limit the peak value of the phase current to 45A,
Fig. 8(b)
shows their flux linkage waveforms for three cases based on (5) with
L_{u}
= 40μH,
L_{a}
= 490μH,
i_{s}
= 25
A
,
i_{M}
= 45A,
Ψ_{s}
+ 0.0125
Wb
, and
Ψ_{M}
= 0.017Wb, and
Fig. 8(c)
shows their energy conversion loops. For peak phase current at 45A, consequently the
y_{opt}
based on (13) is 0.042. Simulation results; the case of
y
= 0.048 or
y
>
y_{opt}
back emf in the coil is smaller than dc bus voltage (
e
<
u
) and the current decreases after
θ_{off}
, the case of
y
= 0.042 or
y
=
y_{opt}
back emf in the coil and dc bus voltage balance (
e
=
u
) and the current stays constant from
θ_{off}
until
θ_{peak}
that the SRG generates maximum output power confirmed
[1
,
15]
, and the case of
y
= 0.038 or
y
<
y_{opt}
back emf in the coil is larger than dc bus voltage (
e
>
u
) and the current increases after
θ_{off}
.
Phase current and flux linkage waveforms, and energy conversion loops in single pulse mode of operation
To find the optimal value of
x
(
x_{opt}
) that the
θ_{peak}
needs to be resolved using the relation of
θ_{peak}
with
i_{peak}
based on (12). Phase current in case of
e
=
u
in
Fig. 8(a)
, the
x_{opt}
can be known by calculating in (10) that is 0.266.
The optimal turnoff angle (
θ_{off.opt}
) can be known by substituting
x_{opt}
and
θ_{peak}
into (11).
To validate the value of
x_{opt}
,
Fig. 9
shows three phase current waveforms based on (6) and (7) that limit the peak value of the phase current to 35A, 40A, and 45A. The values of
θ_{on.opt}
,
x_{opt}
, and
u
for three cases are fixed at 15°, 0.266 and 27V respectively.
θ_{peak}
and
θ_{off.opt}
can be known by calculating in (12) and (11) respectively. Simulation results, waveforms of the phase current for all three cases are similar to phase current waveform in case of
e
=
u
.
Phase current waveforms with optimal excitation angles
5. Experimental Results
The proposed model is verified via comparison with laboratory measurements, the schematic layout of the experimental system is shown in
Fig. 10(a)
, and the testbed in laboratory is established shown in
Fig. 10(b)
. The 3phase induction motor drives the 8/6 SRG which is excited through a 4phase asymmetrical converter. This converter uses the same dc source for excitation through the IGBTs and demagnetization through the diodes. The converted energy supplies to a sufficiently stiff dc source to prevent the dc bus level running away during generation period. The rotary encoder provides rotor position as pulse train (3,600count/rev) to TMS320F2812 DSP controller. The values of excitation angles which drive the gate of converter are processed by the controller. The
y
ratio can be controlled by adjusting dc bus voltage and shaft speed. The output power has been measured on a testbed of variations of ratio of dc bus voltage and shaft speed, and excitation angles.
Experimental setup, converter, load and SRG
For experiment to validate the value of
x_{opt}
;
Fig. 11(a)
shows the measured waveforms of phase of voltage and current and average output of voltage and current when the SRG is controlled to limit
i_{peak}
to 45A with the optimal excitation angles. The
u
and
𝜔
used for experiment are 27V and 642 rad/s respectively that the values of
θ_{on,opt}
and
x_{opt}
obtained from the simulation results of analytical model are 15° and 0.266 respectively. For
θ_{off.opt}
is 6.34° based on (11),
Fig. 12(a)
shows the measured waveforms of phase of voltage and current and average output of voltage and current when the SRG is controlled to limit
i_{peak}
to 45A with
u
and
𝜔
used for experiment are 27V and 717 rad/s respectively that the values of
θ_{on}
and
x
are 15° and 0.253 respectively. For
θ_{off}
is 6.75° based on (11),
Fig. 13(a)
shows the measured waveforms of phase of voltage and current and average output of voltage and current when the SRG is controlled to limit
i_{peak}
to 45A with
u
and 𝜔 used for experiment are 27V and 558 rad/s respectively that the values of
θ_{on}
and
x
are 15° and 1 respectively. For
θ_{off}
is 4.40° based on (11), and
Figs. 11(b)
,
12(b)
, and
13(b)
show phase current waveform both from measurement and analytical model.
Optimal excitation angle for x = x_{opt} (I_{L} = 39.86A, U_{L} = 27.6V, P_{out} = 1100.14W)
Excitation angle for x < x_{opt} (I_{L} = 31.97A, U_{L} = 27.52V, P_{out} = 879.81W)
Excitation angle for x > x_{opt} (I_{L} = 20.6A, U_{L} = 27.68V, P_{out} = 570.21W)
Experimental results; the SRG controlled with optimal excitation angles in
Fig. 11
generates maximum output power that the optimal excitation angles are
θ_{on}
= 15° and
θ_{off}
= 6.34°.
For experiment to validate the value of
x_{opt}
,
Fig. 14
shows the measured phase current waveforms for all three cases of
i_{peak}
; 35A, 40A, and 45A.
Table 2
shows the optimal control variables for all three cases that the values of
θ_{on,opt}
,
x_{opt}
and
u
for three cases are fixed at 15°, 0.266 and 27V respectively and
θ_{off.opt}
can be known by calculating in (11). Experimental results, waveforms of the phase current for all three cases are similar to phase current waveform in case of
e
=
u
.
Phase current waveforms with x_{opt} = 0.266 for all three cases of i_{peak}; 35A, 40A, and 45A
Parameters for three cases of phase current
Parameters for three cases of phase current
For experiment to validate the optimal excitation angles for three cases when
u
is fixed;
Figs. 15
,
17
, and
19
show output power versus different excitation angles when excitation angles were adjusted to limit
i_{peak}
= 45A for all three cases with the values of
u
and 𝜔 shown in
Table 3
. The values of optimal excitation angles based on the proposed method are shown in
Table 3
that the SRG generates maximum output power and the waveforms of phase current for case 1 to case 3 are shown in
Figs. 16
,
18
, and
20
respectively.
Output power at different excitation angles; case 1
Optimal excitation angles (θ_{on}=15°θ_{off}=6.34°) (I_{L} = 39.86A, U_{L} = 27.6V, P_{out} = 1100.14W)
Output power at different excitation angles; case 2
Optimal excitation angles (θ_{on}=15°θ_{off}=6.13°) (I_{L} = 35.97A, U_{L} = 27.6V, P_{out} = 992.77W)
Output power at different excitation angles; case 3
Optimal excitation angles (θ_{on}=15°, θ_{off}=5.91°) (I_{L} = 31.78A, U_{L} = 27.6V, P_{out} = 877.13W)
Parameters for three cases
Parameters for three cases
For experiment to validate the optimal excitation angles with different
u
and 𝜔, all dashed lines obtained from measurements in
Fig. 21
show output power versus different excitation angles when excitation angles were adjusted to limit peak phase current for three cases; 30A, 45A, and 60A, with
u
and 𝜔 as shown in
Table 4
. Experimental results; maximum output power apparently exists at turnon angle at 15° for all cases as shown in
Fig. 21(a)
, three points; (a), (b), and (c), of noted by asterisk in
Fig. 21(b)
are the
θ_{off.opt}
based on (11) as shown in
table 4
and
Figs. 22(a)

(c)
show phase of voltage and current, and average output of voltage and current when the SRG controlled with the optimal excitation angles generates maximum output power.
Output power at different excitation angles
Optimal control variables
Optimal control variables
Optimal excitation angles at (a) i_{peak} = 30A, (b)i_{peak} = 45A, and (c)i_{peak} = 60A
6. Conclusion
Optimal excitation angles of an 8/6 SRG analyzed from phase current waveform are presented in this paper. The phase current equation is significant factor for determining the optimal excitation angles that depends on optimal values of excitation angles (
θ_{on,opt}
and
θ_{off,opt}
), a ratio of flux linkage at turnoff angle and peak phase current positions (
x_{opt}
). The analytical model is applied from a nonlinear magnetization curve in terms of dc bus voltage(
u
), shaft speed (
𝜔
), and excitation angles (
θ_{on}
and
θ_{off}
). The experimental results have confirmed the accuracy of the analytical model for determining the optimal excitation angles of the SRG for maximum output power. All cases for the SRG controlled with the optimal excitation angles in single pulse mode of operation generate maximum output power and phase current waveforms are similar to the case of
e
=
u
. Simulation and experimental results; the value of
θ_{on,opt}
is 15°, the value of
θ_{off,opt}
can be known by substituting
x_{opt}
= 0.266 and
θ_{peak}
into (11).
BIO
Pairote Thongprasri He received M. Eng. Degree in electrical engineering from KMITL. Now he is studying in D. Eng. Program (electrical engineering) at KMITL. His research interests are switched reluctance machine and power electronics.
Supat Kittiratsatcha He received the M.S. and Ph.D. degrees in electric power engineering from Rensselaer Polytechnic Institute, Troy, NY. He is an Associate Professor with the Department of Electrical Engineering at KMITL. His research interests include switched reluctance machine design and solid state lighting.
Miller T. J. E.
2001
“Electronic Control of Switched Reluctance Machines”
Newnes
Oxford, UK
Xin K.
,
Zhan Q.
,
Luo J.
2006
“A New Simple Sensor less Control Method for Switched Reluctance Motor Drives”
Journal of ElecEng& Tech
1
(1)
52 
57
. Ferreira C. A
,
Jones S. R.
,
Heglund W. S.
,
Jones W. D.
1995
“Detailed Design of a 30kW Switched Reluctance Starter/Generators System for a Gas Turbine Engine Application”
IEEE Trans on IndAppl
31
(3)
553 
561
Torrey D. A.
2002
“Switched Reluctance Generators and Their Control”
IEEE Trans on IndElec
49
(1)
3 
14
Cardenas R.
,
Ray W. F.
,
Asher G. M.
1995
“Swithced Reluctance Generators for Wind Energy Applications”
Proc IEEE
559 
564
Fahimi B.
,
Emadi A.
,
Sepe R. B.
2004
“A switched Reluctance MachineBase Startor/Alternator for More Electic Cars”
IEEE Trans on EnergConv
19
(1)
116 
124
Asadi P.
,
Ehsani M.
,
Fahimi B.
2006
“Design and Control Characterization of Switched Reluctance Generator for Maximum Output Power”
Proc IEEE
1639 
1644
Soares F.
,
Branco P. J. C.
2001
“Simulation of a 6/4 Switched Reluctance Motor Based on Matlab/ Simulink Environment”
IEEE Trans on Aero and Elec Sys
37
(3)
989 
1009
DOI : 10.1109/7.953252
Kokernak J. M.
,
Torrey D. A.
2000
“Magnetic Circuit Model for the Mutually Coupled SwitchedReluctance Machine”
IEEE Trans on Mag
36
(2)
500 
507
DOI : 10.1109/20.825824
Pulle D. W. J.
1991
“New Data Base for Switched Reluctance Drive Simulation”
Proc. IEE on ElecPowAppl
138
(6)
331 
337
Xu Y.
,
Torrey D. A.
2002
“Study of the Mutually Coupled Switched Reluctance Machine Using the Finite ElementCircuit Coupled Method”
Proc. IEE on ElecPowAppl
149
(2)
81 
86
Roux C.
,
Morcos M. M.
2002
“On the Use of a Simplified Model for Switched Reluctance Motors”
IEEE Trans on EnerConv
17
(3)
400 
405
Cai Y.
,
Yang Q.
,
Su L.
,
Wen Y.
,
You Y.
2010
“Nonlinear Modeling for Switched Reluctance Motor by Measuring Flux Linkage Curves”
ProcIEEE on Com Eng and Tech
V647 
V651
Cao X.
,
Deng Z.
,
Yao T.
,
Cai J.
,
Zhuang Z.
2010
“Analysis and Application of Phase Current in Switched Reluctance Generators”
IEEE Trans. on Appl Sup
20
(3)
1063 
1067
DOI : 10.1109/TASC.2010.2043081
Sawata T.
,
Kjaer P. C.
,
Cossar C.
,
Miller T. J. E.
1998
“A Control Strategy for the Switched Reluctance Generator”
ProcICEM on Elec Mach and Sys
2131 
2136
Sozer Y.
,
Torrey D. A.
2004
“Closed Loop Control of Excitation Parameters for High Speed Switched Reluctance Generators”
IEEE Trans on PowElec
19
(2)
355 
362
Chen H.
,
Gu J. J.
2010
“Implementation of the ThreePhase Switched Reluctance Machine System for Motors and Generators”
IEEE/ASME Trans on Mech
15
(3)
421 
432
DOI : 10.1109/TMECH.2009.2027901
Ziapour M.
,
fjei E. A
,
Yousefi M.
2013
“Optimam Commutation Angles for Voltage Regulation of a High Speed Switched Reluctance Generator”
Proc PEDST
271 
276
Nasirian V.
,
Kaboli S.
,
Davoudi A.
2013
“Output power Maximization and Optimal Symmetric Freewheeling Excitation for Switched Reluctance Generators”
IEEE Trans on IndAppl
49
(3)
1031 
1042