An Overview of Space Vector PWM
Dr. P.J. Randewijk
2015-10-12
1
Introduction
Space vector Pulse Width Modulation (PWM), [1], is inherently a voltage control scheme. With
the reference voltage’s space vector known, the space vector PWM technique calculates the optimum switching pattern for the three-phase inverter to ensure that the desired space vector
voltage is obtained. These calculation are performed in the α–β or “space vector” plane. Space
vector PWM is based on the Clarke transformation, [2], that was developed by Edith Clarke1 in
1938.
2
Clarke Transformation
The Clarke transform is a mathematical transformation whereby any instantaneous three-phase
voltage or current values can be transformed to a two-phase or α–β plane. The Clark transform
was developed to determine the transient voltages following three-phase faults, as well as transients caused by open conductors. This is in contrast to sequence components, [3] developed by
Fortescue in 1918, to solve the steady state response to fault conditions.
The Clarke transform in its original form is given by,

 
 
1
1
1 −
−
α
2
2
√
√  a
 β = 2 
 0 23 − 23  b
3 1
1
1
0
c
2
2
2
(1)
or in a compact form, as
fαβ0 = Cabc→αβ0 K abc→αβ0 f abc
(2)
with f abc the matrix containing the instantaneous three-phase values of the quantities, usually
voltages or currents, that needs to be transformed to fαβ0 , the corresponding matrix with the α, β
and zero sequence quantities, K abc→αβ0 the transformation matrix and Cabc→αβ0 the transformation constant.
If we ignore the zero-sequence components, as is usually done for space vector PWM, the K abc→αβ0
transformation matrix can be replaced by K abc→αβ , with
K abc→αβ
1 Edith
"
1
1 −
√2
=
0 23
−√12
− 23
#
(3)
Clarke was the first woman to earn an M.S. in electrical engineering from MIT in 1919, the first woman to
deliver a paper at the American Institute of Electrical Engineers (AIEE predecessor to the IEEE) annual meeting in
1926, was the first female professor of electrical engineering in the USA (at the University of Texas in Austin) in 1947
and the first female Fellow of the AIEE in 1948.
page 2 of 16
An Overview of Space Vector PWM
so that fαβ0 becomes fαβ .
When the Clarke transform (1) is applied, using K abc→αβ , to the instantaneous three-phase voltages v a (t), vb (t) and vc (t), it translates into a space vector ~V with x- and y-values of Vα and Vβ
respectively as shown in Figure 1. If the space vector was plotted for a full power cycle (e.g.
20ms for 50Hz voltages), the space vector will rotate through the full 360◦ , which reminds of a
rotating MMF in a three-phase machine.
β-axis
~x
xβ
θ
α-axis
xα
Figure 1: The space vector ~x in the α − β plane.
With the α− β plane in essence being a 2D plane, this allows us to write the space vectors in a
complex form as,
(4)
~x = xα + jx β
2.1
Example:
v a (t) = V̂m cos(ωe t + ϕv )
(5)
vb (t) = V̂m cos(ωe t + ϕv − 120◦ )
◦
vb (t) = V̂m cos(ωe t + ϕv + 120 )
(6)
(7)
with V̂m the peak value of the balanced three-phase voltage and ϕv the relative phase angle
"
1
2 1 −
vα (t)
√2
=
v β (t)
3 0 23
# v (t)
−√21  a 
vb (t)
− 23
vc (t)
(8)
Using Euler’s identity
∴
e jθ = cos θ + j sin θ
1 jθ
cos θ =
e + e− jθ
2
(9)
(10)
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An Overview of Space Vector PWM
∴
vα (t) =
◦
◦
1 2
· V̂m e j(ωe t+ ϕv ) + e− j(ωe t+ ϕv ) − 12 e j(ωe t+ ϕv −120 ) − 21 e− j(ωe t+ ϕv −120 ) −
2 3
1 j(ωe t+ ϕv +120◦ )
2e
◦
◦
1 2
= · V̂m e j(ωe t+ ϕv ) 1 − 12 e− j120 − 12 e j120 +
2 3
− j ( ωe t + ϕv )
1 j120◦
1 − j120◦
e
1 − 2e
− 2e
− 21 e− j(ωe t+ ϕv +120
◦)
1 2 3 j ( ωe t + ϕ v )
· V̂m
e
+ e − j ( ωe t + ϕ v )
2 3 2
= V̂m cos(ωe t + ϕv )
=
&
v β (t) =
√
◦
1 2
· V̂m 23 e j(ωe t+ ϕv −120 ) −
2 3
√
3 j(ωe t+ ϕv +120◦ )
2 e
(11)
(12)
(13)
(14)
√
3 − j(ωe t+ ϕv −120◦ )
−
2 e
√
3 − j(ωe t+ ϕv +120◦
2 e
−
√ h
i
◦
◦
◦
1 2
3 j(ωe t+ ϕv ) − j120◦
= · V̂m
e
e
− e j120 + e− j(ωe t+ ϕv ) e j120 − e− j120
2 3 √2
√
3 √ j ( ωe t + ϕ v )
1 2
− j 3e
+ j 3e− j(ωe t+ ϕv )
= · V̂m
2 3
2
1 2 3 j ( ωe t + ϕv )
e
− e − j ( ωe t + ϕv )
= · V̂m
j2 3 2
= V̂m sin(ωe t + ϕv )
(15)
(16)
(17)
(18)
(19)
Therefore, from (14) and (19), the voltage space vector
~v = V̂m cos(ωe t + ϕv ) + jV̂m sin(ωe t + ϕv )
= V̂m e
j ( ωe t + ϕv )
(20)
(21)
An typical voltage space-vector is shown in Figure 2. The angle of this space-vector, ωe t + ϕv
will vary with time. For an perfectly balanced three-phase supply voltage, as shown in Figure 3,
the space-vector voltage will rotate in a perfect circle and do one complete rotation in one period
as shown in Figure 4.
β-axis
~v = V̂m e j(ωe t+ ϕv )
vβ
ωe t + ϕv
vα
α-axis
Figure 2: The space vector ~v in the α − β plane.
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An Overview of Space Vector PWM
400
v a (t)
vb (t)
Phase Voltage [V]
300
vc (t)
200
100
0
−100
−200
−300
−400
0
5
10
15
20
Time, t [ms]
Figure 3: The three-phase voltage for one period.
We can also transform the three-phase currents to the α − β-plane
i a (t) = Îm cos(ωe t + ϕi )
(22)
◦
ib (t) = Îm cos(ωe t + ϕi − 120 )
(23)
◦
ib (t) = Îm cos(ωe t + ϕi + 120 )
(24)
∴
iα (t) = Îm cos(ωe t + ϕi )
(25)
&
i β (t) = Îm sin(ωe t + ϕi )
(26)
400
320
240
160
v β [V]
80
0
−80
−160
−240
−320
−400
−400 −320 −240 −160 −80
0
80
160
240
320
vα [V]
Figure 4: v β (t) vs vα (t) for one period.
400
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An Overview of Space Vector PWM
Similarly, from (25) and (26), for the current space vector
~i = Îm cos(ωe t + ϕi ) + j Îm sin(ωe t + ϕi )
= Îm e
j ( ωe t + ϕi )
(27)
(28)
β-axis
~i = Îm e j(ωe t+ ϕi )
iβ
ωe t + ϕi
α-axis
iα
Figure 5: The space vector~i in the α − β plane.
2.2
Calculating Power
β-axis
~v = V̂m e j(ωe t+ ϕv )
vβ
~i = Îm e j(ωe t+ ϕi )
iβ
ωe t + ϕv
ωe t + ϕi
vα
iα
α-axis
Figure 6: The space vector~i in the α − β plane.
3
s(t) = ~v ·~i∗
2
3
= V̂m e j(ωe t+ ϕv ) Îm e− j(ωe t+ ϕi )
2
3
= V̂m Îm e j( ϕv − ϕi )
2
3
3
= V̂m Îm cos ϕvi + j V̂m Îm sin ϕvi
2
2
= p(t) + jq(t)
(29)
(30)
(31)
(32)
(33)
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An Overview of Space Vector PWM
3
V̂m Îm cos ϕvi
2
= 3Vm Im cos ϕvi
(34)
p(t) =
(35)
(36)
=P
3
V̂m Îm sin ϕvi
2
= 3Vm Im sin ϕvi
(37)
q(t) =
(38)
(39)
=Q
3
s(t) = ~v ·~i∗
2
3
=
vα (t) + jv β (t) · iα (t) − ji β (t)
2
3
3
=
vα ( t )iα ( t ) + v β ( t )i β ( t ) + j v β ( t )iα ( t ) − vα ( t )i β ( t )
2
2
= p(t) + jq(t)
3
vα ( t )iα ( t ) + v β ( t )i β ( t )
2
3
v β ( t )iα ( t ) − vα ( t )i β ( t )
q(t) =
2
p(t) =
(40)
(41)
(42)
(43)
(44)
(45)
In 1984 Akagi introduced the “Instantaneous Power Theory”, [4], which later became known as
the p−q theory and proposed a change in the transformation constant,
r
2
0
0
Cabc→αβ0 = Cabc→αβ =
(46)
3
as well as in the transformation matrix,

√1
2
√1
2
1
−
√2
3
2

0
Kabc
→αβ0 =  1

√1
2

−√12 
− 23
so that we now can define the modified Clarke transform as
 
  r  √1
√1
√1
0
a
2
2
2
1
1  b
α = 2 
−
1 −

√2
√2
3
3
β
c
− 3
2
and the modified inverse Clarke transform as
 1
  r
√
a
 12
2
√
b =
3 2
c
√1
2
or in a compact form, as
1
− 12
− 12
(47)
(48)
2

 
0

3  
2√  α
β
− 3
√
(49)
2
f abc = Cαβ0→abc Kαβ0→abc fαβ0
(50)
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An Overview of Space Vector PWM
which allows us to define
C = Cabc→αβ0 = Cabc→αβ = Cαβ0→abc = Cαβ→abc
If we again ignore the zero sequence components, (48) reduces to
#  a
r "
1
1
−
1
−
2
α
√2
√2
b
=
3
3
β
3
−
2
2
c
(51)
(52)
which again allows us to write the space vector in complex plane in stead of the α− β plane. Also
the inverse modified Clarke transform, with the zero sequence components ignored, can now be
written as

  r 
1
a
√
3  α
1
b = 2 
(53)
− 2
2√  β
3
3
1
c
−2 − 2
The instantaneous power, or p−q theory allow us to define the instantaneous active- and reactive power, p and q respectively, for a three-phase system (with the zero sequence components
ignored) as
vα v β iα
p
=
(54)
q
−v β vα i β
or in the more elegant complex form as
s = ~v ·~i∗
(55)
(56)
= (vα + jv β ) · (iα − ji β )
= (vα iα + v β i β ) + j (v β iα − vα i β )
{z
}
|
{z
}
|
p
(57)
q
= p + jq
(58)
with ~v the space vector voltage and ~i∗ the complex conjugate of the space vector current as
shown. The beauty of the p−q theory is that not only has it a familiar form in the complex
domain, but also for a balance sinusoidal three-phase system,
p=P
(59)
q=Q
(60)
and
which implies that the average three-phase active and reactive power, i.e. P and Q respectively,
can be obtained instantaneously! For non-balanced and non-sinusoidal three-phase system, the
definition of p and q differs a bit from the tradition definition for P and Q. For more information,
the reader is referred to [5] where the p−q theory is discussed in extreme detail.
page 8 of 16
An Overview of Space Vector PWM
3
Applying the Clarke Transform to a Three-Phase Inverter
A three-phase inverter has only eight possible switching combinations. This assumes that the
top- and bottom- (power electronic) “switches” of a phase arm are switched complementary
“out of phase” i.e. they are never both “OFF” or both “ON” at the same time, or STOP =SBOTTOM .
This then results in the following eight different switching states for the three-phase inverter, as
shown in Figure 7. When applied to a three-phase load, the eight different switching states of
the three-phase inverter, translate into eight different switching configurations of the connected
three-phase load as shown in Figure 8.
+
Vd
+
1
A
B
C
−
3
A
C
4
A
6
A
8
A
B
C
Vd
B
C
−
+
+
5
A
B
C
Vd
−
−
+
+
Vd
A
+
B
−
Vd
2
−
+
Vd
Vd
7
A
B
C
−
Vd
B
B
C
C
−
Figure 7: Three-phase inverter switching states.
The switching states are almost similar to binary’s “Grey Codes”. To change from one state
to the other, only one phase-arm changes state, or put differently only one switching instance
accompanies a state change to an adjacent state. This can more clearly be seen in Table 1.
Table 1: Inverter switching states in tabular format.
Switching
State
0
1
2
3
4
5
6
7
Phase
A B C
0 0 0
1 0 0
1 1 0
0 1 0
0 1 1
0 0 1
1 0 1
1 1 1
page 9 of 16
An Overview of Space Vector PWM
+
+
v an
Za
Vd
−
Zb
−
−
vbn
n
Zc
−
1
Vd
Za
−
−
−
v an
n
Zc
+
+
−
3
Vd
Za
−
−
−
v an
n
Zb
+
+
Za
Zb
Zc
+
+
Zb
vcn
−
−
4
v an
Zc
n
vcn
−
−
6
vbn
n
Vd
Za
−
n
+
v =0
7
Zc
+
v an
−
vbn
+
vcn
+
Vd
−
2
+
Za
−
n
Vd
Za
5
−
+
vbn
−
+
−
vbn
−
+
Zb
−
vcn
Zb
n
+
vcn
+
Zc
Vd
Zc
+
+
v an
−
vbn
Zb
+
−
vcn
+
+
Vd
Za
+
+
+
+
Zc
−
Zb
−
0
v =0
+
Figure 8: The effect of the three-phase inverter switching states on a balanced three-phase load.
When the Clarke transform is applied to switching state 1, the resultant space vector,
# 2 
r "
1
1
3
1
−
−
√2
√2
− 1  Vd
~V1 = 2
3
3
3
3
− 2
2
− 13
r 2 1
=
V
3 0 d
r
2
· V 0◦ V
=
3 d
(61)
(62)
(63)
where Vd is equal to the DC bus voltage of the inverter.
Similarly for switching state 2,
# 1 
r "
1
1
−√2  31 
√2
~V2 = 2 1 −
Vd
3
3
3
− 23
2
− 23
r
2
=
· V 60◦ V
3 d
(64)
(65)
When the Clarke transformation is performed on the eight inverter switching states, it translates
into six voltage vectors. e.g.
r
2
~
~
~
~
~
~
| V1 | = | V2 | = | V3 | = | V4 | = | V5 | = | V6 | =
·V
(66)
3 d
but with different angles, as well as 2 vectors ~V0 and ~V7 of zero length. These space vectors
are shown graphically in Figure 9. It can be seen that the adjacent switching state transform to
adjacent space vectors in the transformed two-phase α–β plane.
page 10 of 16
An Overview of Space Vector PWM
β-axis
Vβ > |Vα | tan 60◦ 0
~V3 [010]
~V2 [110]
II
~V4 [011]
III
~V0 [000]
I
Vβ > 0
~V1 [100]
α-axis
~V7 [111]
VI
IV
V
~V5 [001]
Vβ < −|Vα | tan 60◦ 0
~V6 [101]
Vα > 0
Figure 9: Three-phase inverter switching space vectors.
4
Calculation of the optimum PWM switching pattern
From the previous section it can be seen that, for the inverter to produce a specific reference
voltage (or after using the Clarke transform a specific reference vector ~Vre f ), the inverter have to
“make up” this reference vector using only the six available switching vectors and the two zero
state vectors. From Figure 10 it can be seen that if the reference vector is situated in, say sector I,
switching vectors ~V1 , ~V2 as well as the two zero switching vectors ~V0 and ~V7 will produce the
minimum associated current errors.
To generalise: It can be seen that once the sector, in which the reference voltage are situated,
has been determined, the minimum amount of switching states are obtained when only the
two states adjacent to this sector, as well as the two zero state, are used.
The PWM switching pattern can be optimised even further by minimising the amount of switching transitions to occur during a switching period. This is done by rearranging the switching
sequences in a particular way, depending on in which sector the reference voltage is situated.
If the reference vector was found to be once again in say sector I, the optimum switching sequence
will be as follows: . . . 012721012721. . . This will ensure that a state change will only occur to
an adjacent state to ensure that a state change accompanies only one switching transition. The
switching sequences for the different sectors are shown in Table 2.
5
Determining the sector in which the reference vector is situated
With the reference voltages Vre f α and Vre f β known, the sector in which the reference vector is
situated can easily be determined. If Vre f β > 0 then it can either be sector I, II or III. The next step
would be to determine in which of these three sectors it actually is. If Vre f β > Vre f α tan 60◦ , the
reference vector in situated in sector II. With “tan 60◦ ” being a constant, this expression is easy
and fast to calculate. If Vre f α > 0 the reference vector is in sector I, else it is situated in sector III.
page 11 of 16
An Overview of Space Vector PWM
β-axis
~V3
~V2
II
III
I
die dt 0
~V4
IV
die dt 2
~Vre f
die dt 1
~V1
α-axis
VI
V
~V5
~V6
Figure 10: Three-phase inverter switching space vectors.
Table 2: Switching sequences for the different sectors.
Sector
Sector I
Sector II
Sector III
Sector IV
Sector V
Sector VI
Switching Sequence
. . . 0127210. . .
. . . 0327320. . .
. . . 0347430. . .
. . . 0547450. . .
. . . 0567650. . .
. . . 0567650. . .
If however Vre f β < 0 the calculations are similar to determine if the reference vector is in either
sectors IV, V or VI. A flow diagram to determine in which sector the reference vector is situated,
is shown in Figure 11.
6
Calculating of the duty cycles
The following assumptions have to be made in order to calculate the duty cycles for the different
switching vectors,
• the vector Vre f is assumed to be constant during one switching cycle Ts and
• the duty cycle of the switching states bordering to the sector in which Vre f was found to be,
Vx and Vx+1 , will remain constant during the entire switching period.
This then results in the following equation to be solved:
Vx · Tx + Vx+1 · Tx+1 + V0 · T0 = Vre f · Ts
• with Tx and Tx+1 the “ON” times for Vx and Vx+1 respectively,
• the zero state actually comprising of two zero states, V0 = V7 = 0 and finally
• the requirement that Tx + Tx+1 6 Ts
(67)
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An Overview of Space Vector PWM
Begin
Vβ >0
yes
no
Vβ <−|Vα | tan 60◦
yes
Vβ >|Vα | tan 60◦
no
Vα >0
no
yes
yes
Vα >0
no
Sector IV
yes
no
Sector VI
Sector V
Sector III
Sector I
Sector II
End
Figure 11: Three-phase inverter switching space vectors.
The voltage reference vector can also be expressed in terms of the duty cycles of the switching
vectors.
Vx · d x + Vx+1 · d x+1 + V0 · d0 = Vr e f
(68)
with
dx =
Tx
Tx+1
,d x+1 =
Ts
Ts
(69)
and
d0 =
T0
Ts
(70)
The solving of (68) for the duty cycles d x and d x+1 involves some elementary trigonometry and
varies from sector to sector.
6.1
Sector I
For sector I the duty cycles can easily be calculated using Figure 12 as a guideline.
With Vre f α and Vre f β known, d1 V1 and d2 V2 can be calculated as follows:
d1 V1 = Vre f α −
d2 V2 =
Vre f β
Vre f β
(71)
tan 60◦
(72)
sin 60◦
so that from (66), it follows that
d1 =
Vre f β
Vre f α −
tan 60◦
q
2
3 · Vd
(73)
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An Overview of Space Vector PWM
β-axis
~V2 [110]
Vre f β
~Vre f
d2 V2
~V1 [100]
Vre f α
d1 V1
α-axis
Figure 12: Calculation of the duty cycle for sector I.
and
d2 =
Vre f β
q
sin 60◦ 23 · Vd
(74)
If d1 + d2 > 1 , then the values of d1 and d2 will have to be scaled. The new values, d10 and d20 will
then be as follows:
d10 =
d1
d1 + d2
(75)
d20 =
d2
d1 + d2
(76)
and
If however d1 + d2 < 1 , the “slack” will have to be taken up by incorporating the zero state
vectors, so that d1 + d2 + d0 = 1. With d0 , d1 and d2 calculated, the duty cycles for phase-arms A,
B and C can easily be calculated.
D A = d1 + d2 +
D B = d2 +
DC =
d0
2
d0
2
d0
2
(77)
(78)
(79)
This can graphically be depicted as shown in Figure 13.
6.2
Sector II
With the reference vector located in Sector II, the duty cycles for switching vectors ~V2 and ~V3 can
be calculated as:
1 Vre f β
·
+ Vre f α
2 sin 60◦
q
(80)
d2 =
2
·
V
d
3
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An Overview of Space Vector PWM
SA 0
1
1
1
SB 0
0
1
1
SC 0
0
0
1
T1
T2
To
2
DA
DB
DC
To
2
Figure 13: Calculation of the duty cycle for sector I.
and
d3 =
1 Vre f β
− Vre f α
·
2 sin 60◦
q
2
3 · Vd
(81)
For sector II, the switching sequence are: . . . 0327230. . .
Therefore:
d0
2
DB = d3 + d2 +
D A = d2 +
DC =
6.3
d0
2
d0
2
(82)
(83)
(84)
Sector III
With the reference vector located in Sector III, the duty cycles for switching vectors ~V3 and ~V4
are:
Vre f β
q
d3 =
(85)
2
◦
sin 60
3 · Vd
and
d4 =
Vre f β
−Vre f α −
tan 60◦
q
2
3 · Vd
(86)
For sector III, the switching sequence are: . . . 0347430. . .
Therefore:
D B = d3 + d4 +
DC = d 4 +
DA =
d0
2
d0
2
d0
2
(87)
(88)
(89)
page 15 of 16
An Overview of Space Vector PWM
6.4
Sector IV
With the reference vector located in Sector IV, the duty cycles for switching vectors ~V4 and ~V5
are:
Vre f β
−Vre f α +
tan 60◦
q
(90)
d4 =
2
·
V
d
3
and
d5 = −
Vre f β
q
sin 60◦ 23 · Vd
(91)
For sector IV, the switching sequence are: . . . 0547450. . .
Therefore:
DC = d 5 + d 4 +
D B = d4 +
DA =
6.5
d0
2
d0
2
d0
2
(92)
(93)
(94)
Sector V
With the reference vector located in Sector V, the duty cycles for switching vectors ~V5 and ~V6 can
be calculated as:
1 Vre f β
− ·
− Vre f α
2 sin 60◦
q
d5 =
(95)
2
·
V
d
3
and
d6 =
−
1 Vre f β
·
+ Vre f α
2 sin 60◦
q
2
3 · Vd
(96)
For sector V, the switching sequence are: . . . 0567650. . .
Therefore:
DC = d 5 + d 6 +
D A = d6 +
DB =
d0
2
d0
2
d0
2
(97)
(98)
(99)
page 16 of 16
An Overview of Space Vector PWM
6.6
Sector VI
With the reference vector located in Sector VI, the duty cycles for switching vectors ~V6 and ~V1
are:
d6 = −
and
d1 =
Vre f β
q
2
◦
sin 60
3 · Vd
Vre f β
−Vre f α +
tan 60◦
q
2
3 · Vd
(100)
(101)
For sector IV, the switching sequence are: . . . 0167610. . .
Therefore:
D A = d1 + d6 +
DC = d 6 +
DB =
d0
2
d0
2
d0
2
(102)
(103)
(104)
References
[1] H. van der Broeck, H.-C. Skudelny, and G. Stanke, “Analysis and realization of a pulsewidth
modulator based on voltage space vectors,” IEEE Transactions on Industry Applications, vol. 24,
no. 1, pp. 142–150, 1988.
[2] E. Clarke, “Problems Solved by Modified Symmetrical Components, Part I,” General Electric
Review, vol. 11, no. 41, pp. 488–494, 1938.
[3] C. L. Fortescue, “Method of Symmetrical Co-Ordinates Applied to the Solution of Polyphase
Networks,” Transactions of the American Institute of Electrical Engineers, vol. XXXVII, no. 2, pp.
1027–1140, Jul. 1918.
[4] H. Akagi, Y. Kanazawa, and A. Nabae, “Instantaneous Reactive Power Compensators Comprising Switching Devices without Energy Storage Components,” IEEE Transactions on Industry Applications, vol. IA-20, no. 3, pp. 625–630, May 1984.
[5] H. Akagi, E. H. Watanabe, and M. Aredes, Instantaneous Power Theory and Applications to Power
Conditioning. IEEE Press, 2007.