V/f control for Switched Reluctance Motor
Takahiro Kumagai
Dept. of Science of Technology Innovation
Nagaoka University of Technology
Nagaoka, Japan
kumagai_t1125@ stn.nagaokaut.ac.jp

Jun-ichi Itoh
Masakazu Kato
Dept. of Science of Technology Innovation Nagaoka Motor Development Co., Ltd., Japan
Nagaoka University of Technology
kato@nagaoka-md.co.jp
Nagaoka, Japan
itoh@vos.nagaokaut.ac.jp

Abstract— This paper proposes a novel V/f control method
for a switched reluctance motor (SRM). The proposed method
does not require the rotor position information and thenonlinear magnetization curve model of SRM. A virtual rotor
magnetic flux is generated by controlling the zero-phase current
with a PI controller, which realizes the V/f control for SRM. In
addition, the maximum torque / current ratio control is achievedby the reactive power in the V/f control. In the proposed simple
speed control without a position sensor, the current RMS value
is reduced by 43.9% at the torque of 0.1 p.u. compared towithout high-efficiency control.
Keywords—Switched Reluctance Motor, High Efficiency, V/fcontrol, Stabilization Method

I. INTRODUCTION
Recently, high-efficiency motors have been actively
studied and developed due to increasing awareness ofenvironmental issues [1-10]. In particular, a switched
reluctance motor (SRM) has attracted attention as a variablespeed motor for industrial and home appliance applications
such as fans, pumps, and so on, thanks to its rare-earthelement-free, robustness, and cost-effectiveness [4][10].
Therefore, the adjustable speed drive without a position sensor
is expected for SRM in order to make use of robustness andcost-effectiveness.
In general, the drive method without the rotor position
sensor uses the position dependence of the inductance of SRM[11-13]. In this method, first, the instantaneous value of theinductance is detected from the voltage and the current. Next,
the detected inductance is converted to the rotor position
information based on the look-up table or the approximation
formula of the relationship between the inductance and rotor
position. However, complicated control, such as the harmonic
voltage superposition or the additional analog circuit, isrequired to acuuratecy detect the instantaneous value of theinductance [11]. In addition, the nonlinear magnetization
curve model of SRM is required for the accurate conversion
from the detected inductance into the rotor position, which
requires the complicated measurement or finite element
analysis (FEA) [12]. Furthermore, the adjustment of thecontrol parameters, such as the current amplitude, turn-on
angle, and turn-off angle, is required for the construction of
the current and speed controllers with the high torque / current
ratio, which requires pre-experiment or FEA [13][14].
This paper proposes a novel V/f control method for SRM
drive without the position sensor in order to solve the above
problems. The new contribution of the proposed method is the
combination of simple speed control and high efficient drive
without the position sensor. The V/f control essentially does
not require the rotor position information owing to its control
strategy based on the rotating coordinate system relative to theinverter output voltage. Therefore, the V/f control does nothave the problem of the inductance detection accuracy and theconversion accuracy to the rotor position. Besides, themaximum torque / current ratio control based on the reactive
power is applied into the V/f control. This result in both simple
speed control and high efficient drive without the rotor
position sensor.
Although the principle of the proposed V/f control method
for SRM drive is based on the V/f control with the stabilization
control and high efficiency control proposed for the PMSM
[15]-[18], the following features are mainly different. (A) It isnecessary to generate a virtual rotor magnetic flux
corresponding to the rotor magnet magnetic flux in PMSM.
(B) It is necessary to change the gain of stabilization control
since the rotor magnetic flux and inductance dynamicallychange according to the operating conditions. (C) It isnecessary to apply not only id=0 but also i0=iq for the highefficiency control.
In this paper, first, the proposed method for the above
features of (A), (B), and (C) is explained. Next, the usefulnessof the proposed V/f control method for SRM drive isconfirmed by the experimental results.
II. V/F CONTROL FOR SWITCHED RELUCTANCE MOTOR
Fig.1 shows the control diagram of the proposed V/fcontrol for SRM. The proposed method consists of (A) zerophase current control, (B) V/f control, and (C) high efficiency
control. The details of each control are explained in thefollowing subsections.

(A) zero-phase current ACR
i0* +
PI
-

wrm*

Pf

w

*

(B) V/f control
f/V conv.

Damping control
K1
HPF

0

v0*
vg*

3f/gd0

vd*

+

vu*
vv*
vw*

Δvd*
w 1* 1 q 1*+s-(C) High
efficiency
controlig
id
3f/gd0i0
Fig. 1.

V/f control method based on gd axis for SRM.

iu
iv
iw

A. Zero-phase current control
This paper adopts the mathematical model in the FieldOriented Control (FOC, i.e. vector control) for SRM [19]-[21].
The FOC of SRM uses the dq-axis Cartesian coordinate
system with the d-axis defined as the direction of the virtual
rotor flux vector generated by the zero-phase current and theq-axis defined as the direction 90 degrees behind the d-axis.
In the zero-phase current control, the virtual rotor flux is
generated by controlling the zero-phase current with a PI
controller, which realizes the V/f control for SRM describedin Section B.
B. V/f control
Fig. 2 shows the control block diagram of each servo
driver with FOC. PI controller is used for the current controlof the d-axis and q-axis. In addition, zero d-axis current
control and the feed-forward compensation for canceling theinterference of the PM motors are applied.
The proposed control principle is based on the V/f control
with the stabilization control [15]-[18], which has been
proposed for PMSM. The V/f control uses the gd-axis
Cartesian coordinate system with the g-axis defined as the
direction of the inverter output voltage and the d-axis definedas the direction 90 degrees behind the g-axis. In the V/f control
for PMSM, the torque oscillation occurs due to the resonancebetween the moment of inertia and the armature inductance.
Then, the torque oscillation is suppressed by the damping
control, which feeds back the active current id to the electric
angular frequency command w* [15]-[18]. In SRM, the same
damping control as that for the PMSM is achieved bygenerating the virtual rotor flux by the zero-phase current
control.
Next, the stability and the responsivity of the proposed
control are explained. The voltage equation and the torqueequation in the gd-axis Cartesian coordinate system areexpressed as in (1) and (2), respectively,
Lac
L
Lac


− ac sin(6q1 − 2 )cos(2 )  Ldc + 2 cos(6q1 − 2 )2
2
  ig   vg 
ig    
Lac
Lac
Lac
 
  
v
=
R
i
+
−
sin(6
q
−
2

)
L
−
cos(6
q
−
2

)
−
sin(2

)

 p  id  
d
d
1
dc
1
 
 
2
2
2
  i   v0  i0    0 
Lac
Laccos(2 )−sin(2 )
Ldc




2
2
Lac


− Ldc − Lac cos(6q1 − 2 )sin(2 )  − Lac sin(6q1 − 2 )2 ig  Lac
+2w1  Ldc − Lac cos(6q1 − 2 )Lac sin(6q1 − 2 )cos(2 )  id 2 i 

 0
0
0
0




Lac
Lac
 Lac

sin(6
q
−
2

)
cos(6
q
−
2

)
−
sin(2

)
1
1
 2

2
2

 ig 
 Lac
 
Lac
Lac
+2 p (d ) cos(6q1 − 2 ) −sin(6q1 − 2 ) −cos(2 )  id 22
 2
 i 

 0
LL− ac cos(2 )0 − ac sin(2 )



2
2

The state equation is derived by first ignoring theasynchronous term of (1) appearing as the sixth harmonic for
simplification and then performing a linear approximationnear the steady-state in the same way as Ref. [15]-[18], andexpressed as in (3), ig   A11 A12 i   A
 d   21 A 22
 i   A
A 32
p  0  =  31
 w   A 41 A 42
 d   A
A 52
   51
 x   A 61 A 62

A13
A 23
A 33
A 43
A 53
A 63

A14
A 24
A 34
A 44
A 54
A 64

A15
A 25
A 35
A 45
A 55
A 65

A16   ig   B11

A 26   id   B21

 B31


A 36 i0 +
A 46   w   B41
A 56   d   B51
  
A 66   x   B61

B12
B22
B32
B42
B52
B62

B13
B23
B33
B43
B53
B63

B14 

B24   vg 
B34   vd B44   v0 
B54   w * 
B64 

(3),
The specific formulas for each element are shown in the
appendix.
The following conditions are assumed in order to briefly
discuss the stability and response.
(ⅰ) No-load condition: φ = 0, ig = 0, id = 0(ⅱ) High-speed rotation condition: w0Ldc, w0Lac >> R, w0Ldc >> K1id(ⅲ) The mechanical time constant is sufficiently larger thanthe electrical time constant: p(∆ig) = 0, p(∆id) = 0, p(∆i0) = 0(ⅳ) The effect of HPF on stability is small: p(∆x) = 0
The natural angular frequency wn and the damping factor
 are calculated from the characteristic equation of the second-order system, which is derived by approximating the derived
sixth-order state equation expressed as in (1) as a second-ordersystem from the assumption of from (ⅰ) to (ⅳ), and expressed
as in (4) and (5),

2 Pr

wn =
=

JLdc

2
P

(4),

J
K1
Ldc

(5),

i0

(6),

where

r =

Lac
2

(1),

where K1 is the damping gain, r is the virtual rotor flux, andi0 is the zero-phase current. As shown in Eq.(1), the damping
factor  is the function of the feedback gain K1. In other words,
the proposed V/f control is stabilized by setting K1
appropriately. Comparing the natural angular frequency andthe damping factor in IPMSM [15]-[18], the rotor magnet
magnetic flux corresponds to the virtual rotor magnetic flux.
In addition, the q-axis inductance corresponds to the dc
components of the self-inductance Ldc.

(2),

Table 1 shows the motor parameters. The test motor is an
18S/12P type SRM of 750W, 5000r/min (=1 p.u.), and2.39Nm (=1 p.u.).

T = 2 PLac i0 iq = 2 PLac i0 (ig sin(2 ) + id cos(2 ))where R is the winding resistance, Ldc and Lac are the dc andac components of the self-inductance, p is the differential
operator, f is the deviation angle between gd-axis Cartesian
coordinate and dq-axis Cartesian coordinate, P is the numberof pole pairs, and J is the moment of inertia.

Fig.2 shows the loot locus of the sixth-order state equation
when damping gain K1 is increased at no-load and base speed.
Note that from No. 1 to No.6 are the characteristic roots of thesixth-order state equation. In this paper, the damping gain K1
is set so that the real parts of No. 1 and 2 are equal to the realparts of No. 3 and 4 in order to suppress the overshoot of theTABLE 1 Motor parameters of the test motor

Imaginary part

2000
1000

K1=0.043 p.u.

No.5

-3000
-4000
-120

Gain [dB]Gain [dB]

K1=0.043 p.u.
No.4
-100

-80

-60
-40
Real part

-20

-60
-60

0

20

300
300
250
250
200
200
150
150
100
100
50
50
0
0

Fig. 2.
Loot locus of sixth-order state equation when damping gain
K1 is increased at no-load and base speed.

Fig.3 shows the frequency characteristics of the command
speed to the motor speed under no-load and base speed. Notethat the figure shows the three conditions of zero-phasecurrent of 0.33 p.u., 0.66 p.u., and 1 p.u. In addition, thedamping gain K1 is set so that the real parts of No. 1 and 2 areequal to the real parts of No. 3 and 4 in each condition. As
shown in Fig.3, the calculation results by (3) and thesimulation results are almost the same, i.e., the derived sixthorder equation of state-model is valid.
Fig.4 shows the relationship between zero-phase current
and natural angular frequency. Note that the calculation resultsof the natural angular frequency are obtained by using (4). As
shown in Fig.4, the calculation results and the simulation
results are almost the same, i.e., the characteristic derived by
the quadratic system approximation is valid. In this paper, thecutoff frequency of the HPF for stabilization control is set to
be 1/20 of the natural frequency wn [15]-[18]. In other words,
the HPF cutoff frequency wc is expressed as in (7).

wn 1 2 Pr=
20 20 JLdc

10000
10000

(7),

Fig.5 shows the relationship between zero-phase current
and optimized damping gain K1. As shown in Fig. 5, therelationship between zero-phase current and optimized
damping gain K1 is inverse proportional. In this proposed
method, the damping gain K1 is applied by varying the gain
with respect to the value of the zero-phase current, as shownin Fig.5.
C. High efficiency control
Fig.6 shows the control diagram of the high efficiency
control for SRM. The torque / current ratio is improved byadjusting the inverter output voltage in the high efficiency

Fig.4.4
Fig.
frequeny.
frequency

Error rate > 0.5%

Error rate > 0.5%

Line: Calculation results
Line: calculationfrom Eq.(4)from Eq.(6)Dot: Simulation results
Dot: Simulation
results

00

0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
11
Zero
phase
current
i[p.u.]0
Zero phase current i0 [p.u.]
Relationshipbetween
betweenzero-phasezero phasecurrent
currentand
andnatural
naturalangularangular
Relationshipgaindanminggain
Optimized
Optimized danming[p.u.]
K11[p.u.]Kmotor rotation speed [15]-[18]. As shown in Fig. 2, thedamping gain K1 is the optimum value at 0.043 p.u. in thiscondition. In addition, all roots are located in the left half ofthe s-plane, i.e., the system is stable.

wc =

1000
1000

=0.33p.u.
i0i0=0.33
p.u.
=0.66p.u.
ii00=0.66
p.u.

-40
-40

No.1
No.2

-2000

100
100

i0=1p.u.
Line: Calculation from Eq.(3)-80 Line:
i0=1 p.u.
Calculation from Eq.(3)Dot: Simulation results-100 Dot: Simulation results-100
Angular frequency [rad/s]
Angular frequency [rad/s]
Fig. 3Frequency characteristics of speed command to motor speed
Fig.
3. no-load
Frequency
characteristics
of speed command to motor speed
underand base
speed (0.6p.u.).
under no-load and base speed (0.6p.u.).

0
-1000

00
10
-2010
-20

-80

No.3

No.6

600Hz
600Hz

2020

4000
3000

46Hz
15Hz 30Hz 46Hz
15Hz

4040

750W
5000r/min
2.39Nm
62V
18S/12P
0.102W
0.615mH
1.17mH
25.4A (i0, iq)

Naturalfrequency
angularfrequency
Naturalangularwwn [rad/s]n [rad/s]

PowerMax. speed (1p.u.)
Max. torque (1p.u.)
Inverter DC voltage
Poles
ResistanceAC inductance
DC inductance
Rated current (1 p.u.)

0.2
0.2

0.15
0.15
0.1
0.1

0.05
0.05

00

00

10
2020
10
Zero
Zerophase
phasecurrent
currenti0 i[A]
[A]

30 30

0

Fig.
RelationshipFig.5.5
Relationshipbetween
betweenzero
zerophase
phasecurrent
currentand
andoptimized
optimized
damping
dampinggain
gainKK1 1
ii00 =icontrol=iqq control
igig

×× ++

idid

×× ++

××

**
Q
Qdq
dq
+
+
2L
2Ldc
dc
--

PP

ww1*1*
vdv*d*

××

Fig.6.5
Fig.

i0i*0

√
√
LPF
LPF

*

Δvd*d*
Δv
Limit
Limit
[0,Vdcdc]]
[0,V

Q
Qgdgd
id =0 control
controlHighefficiency
efficiencycontrol
controlmethod
methodHighcontrol. The following conditions achieve the maximum
torque/current ratio in SRM [19]-[21].

i0 = iq(8),

id = 0(9),Therefore, it is necessary to apply not only id=0 but also i0=iq
for the high efficiency control.

First, the i0=iq control is explained. In the id=0 control
condition described below, iq is equal to the AC current
amplitude Iac. Therefore, i0 should be controlled by Iac. In otherwords, the command value expressed by the following
equation should be applied to the command value of zerophase current control.

SRM(test)

torquemeter

servo motor(load)

i0* = iq = I ac = ig 2 + id 2
(10),where ig and id are g-axis and d-axis current, respectively.
Next, the id=0 control is explained. The id=0 control isachieved indirectly in the gd-axis coordinate by focusing on
the reactive power of SRM. In the dq-axis coordinate, thereactive power Qdq input into SRM is expressed as in (11).



Qdq = vq id − vd iq = 2w Ldc (id 2 + iq 2 ) + Lac i0id

2 (11),

Fig. 7.

Stabilization control
onoff
Torque(1Nm/div)0

where vd and vq are d-axis and q-axis current respectively. In
addition, the reactive power Qdq under the condition of id=0 is
expressed as in (12),Qdq = 2w Ldciq 2 = 2w Ldc I ac 2 = 2w Ldc (ig 2 + id 2 )(12),On the other hand, in the gd-axis coordinate, the reactive
power Qgd is expressed as in (13),Qgd = vd ig

Experimental equipment.

III. EXPERIMENTAL RESULTS
Fig. 7 shows the experimental system. The load side servo
outputs the arbitrary constant torque. In addition, theinstantaneous torque is measured by a high-response torquemeter (UTMII-5Nm, 1kHz bandwidth, UNIPULSE). The V/fratio is set so that the modulation index is 1 at the base speed(0.6 p.u.).
Fig. 8 shows the effect of stabilization control. The speed
is 0.6 p.u. and the torque is 0.5 p.u. In the section with thestabilization control, there is no oscillation in the U-phasecurrent and torque. On the other hand, the oscillation occurs
and eventually leads to over current after the invalidation of
the stabilization control. These results indicate theeffectiveness of the stabilization control.
Fig. 9 shows the speed-torque curve. The proposed control
achieves the wide range of the drive without the position
sensor. Note that the stepping out occurs in the low-speed andlarge-torque due to the effect of the output voltage error
caused by the dead time. In addition, the stepping out also
occurs in the high-speed and large-torque. This is because theVibration

Over current

U-phase current(10A/div)0

Fig. 8.

Effect on the stabilization control.

Not operable

1.2

Torque [p.u.](1p.u.=2.39Nm)(13),

Therefore, the vd should be adjusted by the P controller inorder to match the reactive power Qgd calculated by (4) to thereactive power Qdq calculated by (3). As shown in Fig.2, thevalue calculated by (2) is applied as the command value forzero-phase current control for the i0=iq control. On the other
hand, the deviation between (3) and (2) is controlled to be zeroby adjusting the vd with the P controller for the id=0 control.
Note that the LPF in the second stage of the P controller shownin Fig.2 is to achieve the id=0 control at a steady-state by
reducing the gain for high frequencies [15]-[18]. In addition,a limit is provided in order to prevent overcompensation.

100ms/div

Not operable

1
0.8
0.6
0.4
0.2
0
0

Fig. 9.

0.2
0.4
0.6
0.8
1
Rotation speed [p.u.] (1p.u.=5000r/min)

Speed and torque curves

test SRM is designed for the single-pulse operation at highspeed.
Fig. 10 shows the effect on the stabilization control withthe optimized damping gain. The speed was set to 0.6 p.u. In
addition, the torque was set to two conditions of (a) 0.25 p.u.
and (b) 1 p.u. The damping gain optimized under the conditionof (a) torque 0.25 p.u. is K1 = 0.059 p.u. On the other hand,
The damping gain optimized under the condition of (b) torque1 p.u. is K1 = 0.018 p.u. First, the experiments are conductedunder inappropriate conditions, and the optimized damping
gain K1 is replaced under both conditions. Next, the optimized
appropriate damping gain K1 is applied under both conditions.
As shown in Fig. 11 (a) and Fig. 11 (b), continuous vibration
occurs in the torque and current when an inappropriate
damping gain K1 is applied at each operating point. On theother hand, continuous vibration does not occur in the torqueand current when an appropriate damping gain K1 is applied
at each operating point. Therefore, the effect on thestabilization control with the optimized damping gain is
confirmed.
Fig.11 shows the effect of the high efficiency control at a
speed of 0.2 p.u. The current of the the proposed method (V/fcontrol with i0=iq and id=0) is compared with that FOC with
i0=iq and id=0 and that of V/f control with only i0=iq. As a result,the current RMS value is reduced by 43.9% compared withthe only i0=iq control at the torque of 0.1 p.u. On the other hand,the current RMS value is increased by 14.8% compared with
FOC with the position sensor. However, the increase of thecurrent RMS value becomes small at large torque. Therefore,
the proposed drive without the position sensor brings out thesame performance as the drive with the position sensor.

100ms/div
Torque(1Nm/div)0

IV. CONCLUSIONSThis paper proposed a novel V/f control method for SRM.
In the control method, a virtual rotor magnetic flux was
generated by controlling the zero-phase current with a PI
controller, which realized the V/f control for SRM. In addition,
the high efficiency control is applied to the V/f control. This
result in the achievement of both simple speed control andhigh efficient drive without the position sensor.

0.25 p.u.(0.60Nm)
With
K1=0.018 p.u.

With
K1=0.059 p.u.

U-phase current(10A/div)0(a) Torque: 0.25 p.u.
100ms/div
Torque(1Nm/div)0With
K1=0.059 p.u.

With
K1=0.018 p.u.

1.0 p.u.
(2.39Nm)

REFERENCES[1]

U-phase current(10A/div)0(b) Torque: 1.0 p.u.
Fig. 10. Effect on the stabilization control with the optimized
damping gain.
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582-593 (2022)[9] T. Hirayama, F. Hayashibara, S. Kawabata, "Current Control andMover Design Method for Thrust Characteristics Improvement ofLinear Synchronous Motor with Half-Wave-Rectified Self-Excitation",in IEEJ Journal of Industry Applications, Vol. 10, No. 6, pp. 643-651(2021)[10] N. Tashiro, K. Nakamura, "A Novel Control Method for In-wheel SR
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IEEE Trans. on IE, Vol.58, No.12, pp.5352–5360 (2011)[12] G.G. Lopez, P.C. Kjaer, and T.J.E. Miller: “A new sensorless method
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20.0

V/f control
Reduced (only i0=iq)
by 43.9%

16.0

12.0

FOC with position sensor(i0=iq, id=0)
Increased by 14.8%
1p.u=2.39Nm

8.0
4.0
0

Fig. 11.

Proposed V/f control(i0=iq, id=0)

0.1

0.2

0.4
0.6
Torque [p.u.]

0.8

1

Effect on the high efficiency control (0.2p.u. speed).

[15] J. Itoh, N. Nomura, and H. Ohsawa, “A comparison between V/fcontrol and position-sensorless vector control for the permanent
magnet synchronous motor”, PCC2002, pp.1310-1315 (2002)[16] J. Itoh, T. Toi, M. Kato: "Maximum Torque per Ampere Control Using
Hill Climbing Method Without Motor Parameters Based on V/fControl", 18th European Conference on Power Electronics andApplications (EPE'16), Vol. , No. DS3d-Topic 4-0283, (2016)[17] J. Itoh, T. Toi, K. Nishizawa: "Stabilization Method Using Equivalent
Resistance Gain Based on V/f Control for IPMSM with Long Electrical
Time Constant", The 2018 International Power Electronics Conference,Vol. , No. 23E1-5, pp. 2229-2236 (2018)[18] J. Itoh, T. Toi, K. Nishizawa: "Stabilization Method for IPMSM withLong Electrical Time Constant Using Equivalent Resistance Gain
Based on V/f Control", IEEJ Transactions on Industry Applications,Vol. 8, No. 4, pp. 592-599 (2019)[19] N. Nakao and K. Akatsu:“Vector control specialized for switched
reluctance motor drives”, International Conference on Electrical
Machines (ICEM), pp.943-949 (2014)[20] N. Nakao and K. Akatsu:“Vector control for switched reluctance motor
drives using an improved current controller”, IEEE Energy Conversion
Congress and Exposition (ECCE), pp.1379-1386 (2014)[21] K.Tokui, T. Kumagai, J. Itoh: "Torque Ripple Suppression Method
based on FOC for SRM without FEM Analysis", European Conference
on Power Electronics and Applications (EPE) (2021)

APPENDIX
Each element of the state equation (3) are expressed asfollows. Note that the subscript "0" indicates the value of theoperating point in each variable.
 L L 2L 2
− R  Ldc 2 − ac sin 2 (20 )  − 2w0  − dc ac sin(20 ) cos(20 ) 
2
2




A11 =
L L 2
Ldc 3 − dc ac
2

(A-1)

 L 2



L L 2
L L 2
− R  − ac sin(20 ) cos(20 )  − 2w0  − Ldc 3 + dc ac sin 2 (20 )  − 2 K1 ( Ldc 3 − dc ac )id 0
2
2
2




A12 =
L L 2
Ldc3 − dc ac
2

(A-2)

Lac3 
1  2
 L L

− R  − dc ac cos(20 )  − 2w0
 Ldc Lac − sin(20 )
2 
2
2 


A13 =
2
L L
Ldc3 − dc ac
2

(A-3)

2



 L L 2

Lac 3 1  2
 L L−2  − dc ac sin(20 ) cos(20 )  ig 0 +  − dc ac cos 2 (20 )  id 0 + Ldc Lac − sin(20 )i00 
2
2
2 
2 








A14 =
L L 2
Ldc 3 − dc ac
2

(A-4)

 L L 2 
−4w0 − dc ac i00 
2


A35 =
L L 2
Ldc 3 − dc ac
2(A-17)A41 =2P 2 Lac
i00 sin(20 )J(A-18)A42 =2P2 Laci00 cos(20 )J(A-19)A43 =

2P2 Lac(ig 0 sin(20 ) + id 0 cos(20 ))J(A-20)A45 =

2P2 Lac( 2ig 0i00 cos(20 ) − 2id 0i00 sin(20 ) )J(A-21)

A52 = −K1

(A-22)

A54 = −1

(A-23)

A56 = K

(A-24)

1
(A-25)A 62 =A 66 = − L 2 L−4w0  dc ac cos(20 )i00 
2


A15 =
L L 2
Ldc 3 − dc ac
2

(A-5)

A16 = 2K1id 0

(A-6)

 L 2



L L 2
− R  − ac sin(20 ) cos(20 )  − 2w0  Ldc 3 − dc ac cos 2 (20 ) 
2
2




A 21 =
L L 2
Ldc 3 − dc ac
2

(A-7)

B12 =

(A-8)

B13 =



L L 2
L 2
L L 2− R  Ldc 2 − ac cos 2 (20 )  − 2w0  dc ac sin(20 ) cos(20 )  + 2 K1 ( Ldc 3 − dc ac )ig 0
2
2
2




A 22 =
L L 2
Ldc 3 − dc ac2
Lac3 
1  2
L L

− R  dc ac sin(20 )  − 2w0
 Ldc Lac − cos(20 )
2 
2 
 2

A 23 =2
L L
Ldc 3 − dc ac
2

2



L L 2

Lac3 
1  2
 L L−2  dc ac sin 2 (20 )  ig 0 +  dc ac sin(20 ) cos(20 )  id 0 + Ldc Lac − cos(20 )i00 
2 
2 


 2

 2



A 24 =2
L L
Ldc 3 − dc ac
2(A-10)
 L 2 L−4w0 − dc ac sin(20 )i00 2
A 25 =
L L 2
Ldc 3 − dc ac
2(A-26)(A-27)

Aotherwise = 0

B11 =

Lac 2
sin 2 (20 )2
LL 2
Ldc 3 − dc ac2

Ldc 2 −

−

−

(A-9)

1


Lac 2
sin(20 ) cos(20 )2
L L 2
Ldc 3 − dc ac
2

Ldc Lac2

cos(20 )L L 2
Ldc − dc ac
2(A-28)(A-29)(A-30)

3

B21 =

B22 =

−

Lac 2
sin(20 ) cos(20 )2
L L 2
Ldc 3 − dc ac2
Lac 2
cos 2 (20 )2
LL 2
Ldc 3 − dc ac2

Ldc 2 −

Ldc Lac
2

B23 =

Ldc −
3

(A-11)−B31 =sin(20 )Ldc Lac 2
2

Ldc Lac2

cos(20 )L L 2
Ldc − dc ac
2(A-31)(A-32)(A-33)(A-34)

3(A-12)A 26 = −2K1ig 0

 L 2L L L

− R  − dc ac cos(20 )  − 2w0  dc ac sin(20 ) 
2
2




A31 =
L L 2
Ldc3 − dc ac2

 L 2L
L L

− R  dc ac sin(20 )  − 2w0  dc ac cos(20 ) 
2
 2



A32 =
2
L L
Ldc 3 − dc ac
2
A 33 =− RLdcLdc Lac 222

Ldc Lac(A-13)

2

B32 =Ldc 3 −

B33 =(A-14)(A-15)(A-16)

Ldc Lac 22

Ldc 2
Ldc Lac 22(A-35)(A-36)

Ldc 3 −

B14 = 2id 0(A-37)

B24 = −2ig 0(A-38)

B54 = 1

(A-39)

Botherwise = 0(A-40)

Ldc 3 −2
 L 2L  L L−2  dc ac sin(20 )  ig 0 +  dc ac cos(20 )  id 0 
2
2



 


A 34 =
L L 2
Ldc 3 − dc ac2

sin(20 )