Control of Four-Leg Sinewave Output Inverter using
Flux Vector Modulation
Dhaval C. Patel
R. R. Sawant
M. C. Chandorkar
Department of Electrical Engineering
Department of Electrical Engineering
Department of Electrical Engineering
Indian Institute of Technology, Bombay Indian Institute of Technology, Bombay Indian Institute of Technology, Bombay
Mumbai 400076, INDIA
Mumbai 400076, INDIA
Mumbai 400076, INDIA
Email: dhavalpatel@ee.iitb.ac.in
Email: rrsawant@iitb.ac.in
Email: mukul@ee.iitb.ac.in
Abstract—The time-integral of the output voltage vector of
a three-phase inverter is often termed as the inverter flux
vector. This paper addresses the control of a three-phase fourleg sinewave output inverter having an LC filter at its output, by
controlling the flux vector in three dimensions. Flux vector control has the property that the output filter resonance is actively
damped by the output voltage control loop alone. Further, the
inverter switching action inherently regulates the output voltage
rapidly against dc bus voltage variations. Flux vector control
of sinewave output inverters finds several applications in threephase four-wire systems. This paper presents the flux modulation
method for three-phase four-leg inverters feeding unbalanced
and nonlinear loads. All the necessary steps for the digital
implementation of the flux modulator are presented. To provide
experimental validation, the modulator is implemented as part of
the control system for a stand-alone three-phase four-leg inverter
with an LC filter at its output. Control system details are also
provided. Experimental results indicate the effectiveness of the
modulator and the control system in providing balanced voltages
at the output of the LC filter even under highly unbalanced
conditions with nonlinear loads. The resonance damping and
voltage regulation properties of the modulator are also apparent
from the experimental results.
I. I NTRODUCTION
Conventional three-phase three-wire inverters are suitable
for supplying three-phase balanced loads such as induction
motors. For unbalanced three-phase loads, inverters should be
able to provide path for the neutral current. There are two
main ways for doing this with three-phase inverters.
• Inverters with split dc link capacitors [1]
• Inverters with fourth (neutral) leg [2]–[5] (see Fig. 1)
The higher dc link utilization, requirement of smaller dc link
capacitors and flexibility in control are inherent advantages of
four-leg inverters over split dc link capacitor inverters. Fourleg inverters can be used for applications such as stand-alone
sinewave output inverters for non-linear unbalanced loads,
distributed generation interfaces, microgrids, neutral current
compensators and active filters [5].
Space vector modulation methods for four-leg inverters have
been presented in [2]–[5]. Space vector modulation for four-leg
inverters is complex [2], [5]. However, it has advantages such
as low output distortion, suitability to digital implementation,
constant switching frequency and good dc bus utilization [6].
The inverter flux vector is the time-integral of the inverter
switching voltage vector. Inverter switching based on the
k,(((
Snp
Vdc
Sap
A
N
Sbp Scp
B
C
Snn
San Sbn Scn
Four leg VSC
Fig. 1.
LC filter
a Linear/Non linear
b
c Balanced/Unbalanced
Load
n
Ln
Four-leg sinewave output voltage inverter
control of the flux vector has several advantages in the control
of sinewave output inverters having LC output filters. In
contrast to voltage modulation control methods, the output
voltage control loop alone with a flux modulator is sufficient
to actively damp the output filter resonance [7]. Further,
the inverter switching inherently regulates the output voltage
against dc bus voltage variations. The method also lends itself
to easy digital implementation on a processor or an FPGA.
Two dimensional flux vector modulation of three-leg inverters was presented in [7], [8]. Grid connected applications of
three-leg sinewave output inverters using flux vector modulation were discussed in [7]. A flux vector modulator for a fuel
cell inverter was presented in [9]. An application for active
filter was discussed in [10].
An undesirable feature of flux modulators is the variable
inverter switching frequency that results from the tracking of
the flux reference vector using inverter switching within a
hysteresis band. The switching frequency characteristics of the
flux modulator for a three-leg inverter were discussed in [7].
A solution to the problem of variable switching frequency was
presented in [11], [12], which resulted in constant switching
frequency.
The concept of two dimensional inverter flux vector control
for a three-phase three-wire system can be extended to three
dimensional inverter flux vector control for a three-phase fourwire system. Two dimensional control is restricted to situations
in which the reference flux vector is confined to the q−d plane.
With three dimensional control the reference flux vector can
be anywhere in the q−d−0 space. This can be used effectively
to control a four-leg inverter.
TABLE I
S WITCHING POSITION WITH CORRESPONDING PHASE VOLTAGES AND
V0 = Vdc
V14
TRANSFORMED VOLTAGES
Switch
States
0000
0100
0110
Van
Vbn
Vcn
Vq
0
Vdc
Vdc
0
0
Vdc
0
0
0
0
2
V
3 dc
1
V
3 dc
− 31 Vdc
− 32 Vdc
− 31 Vdc
1
V
3 dc
0010
0
Vdc
0
0011
0001
0
0
Vdc
0
Vdc
Vdc
0101
0111
1111
1011
1001
Vdc
Vdc
0
-Vdc
-Vdc
0
Vdc
0
0
-Vdc
Vdc
Vdc
0
0
0
1101
0
-Vdc
0
1100
1110
0
0
-Vdc
0
-Vdc
-Vdc
1010
1000
-Vdc
-Vdc
0
-Vdc
-Vdc
-Vdc
Vd
0
0
Vdc
1
√
3
1
√
V
3 dc
0
− √1 Vdc
3
0
1
V
3 dc
2
V
3 dc
3
Vdc
0
− 13 Vdc
− 32 Vdc
1
V
3 dc
2
V
3 dc
1
V
3 dc
1
− 3 Vdc
− √1 Vdc
− 31 Vdc
0
− 23 Vdc
− 31 Vdc
0
0
0
0
− 32 Vdc
− 31 Vdc
3
1
√
V
3 dc
1
√
V
3 dc
V0 = 2Vdc/3
−Vdc/ 3
V10
V4
1
V
3 dc
2
V
3 dc
1
V
3 dc
2
V
3 dc
− √1 Vdc
3
0
0
0
− √1 Vdc
V12
Vd = Vdc/ 3
V6
Vd = 0
V0
V8
V0 = Vdc/3
V2
0 d
q
V0
V15
V0 = 0
V13
V0 = −Vdc/3
V7
− 32 Vdc
-Vdc
V11
V5
V9
This paper presents a three dimensional flux vector modulator for four-leg sinewave output inverters used to feed
nonlinear and unbalanced loads. The implementation steps
are discussed in detail. Closed loop voltage control of an
experimental four-leg inverter with an LC filter is implemented
using synchronous reference frame PI controllers for the q−
and d− axis flux vector components. The experimental results
indicate that the modulator and control system is very effective
in providing balanced regulated output voltages even with
highly unbalanced nonlinear loads. Filter resonance damping
and good dynamic response are also apparent.
II. T HREE D IMENSIONAL S PACE V ECTORS FOR F OUR -L EG
I NVERTER
In four-leg inverters the load neutral wire is connected to the
fourth leg as shown in the Fig. 1. This provides the flexibility
to control the neutral voltage and hence produces balanced
voltages across the load. The maximum voltage across each
phase is Vdc . This is an advantage in terms of dc link voltage
utilization in comparison with a split dc link capacitor inverter.
Although the fourth leg introduces complexity, it gives more
flexibility to control the voltage using advanced pulse width
modulation techniques.
In four-leg inverters with three-phase unbalanced loads,
electrical variables in a − b − c coordinates can be transformed
to q − d − 0 coordinates as follows.





0 − 21 − 12
Xq
Xa
√ 
√




3
 Xd 



− 23 
(1)
  Xb 

 = 2/3  1
2




 1
1
1
X0
Xc
2
2
2
There are sixteen switch combinations possible in fourleg inverters. The switching vectors are represented by states
[Sn ,Sa ,Sb ,Sc ] of the inverter legs. Each leg is denoted by 1
and 0, when upper switch and lower switch of the leg is closed
k,(((
V3
Vq = −Vdc/3
Vq = 0
Vq = −2Vdc/3
Fig. 2.
V0 = −2Vdc/3
Vq = Vdc/3
V1
Vq = 2Vdc/3
V0 = −Vdc
Switching vectors in q − d − 0 coordinate space
respectively. The switch positions determine the phase to
neutral voltages, which are transformed to q−d−0 coordinates
using (1). Table I shows the phase to neutral voltages and the
transformed q − d − 0 voltages for each inverter switching
state. Fig. 2 shows the entries of Table I as vectors in the
three dimensional q − d − 0 space.
III. F LUX M ODULATION FOR F OUR -L EG I NVERTER
A. Principle of the Flux Modulator
The inverter flux vector is defined as
Z t
~ dτ + Ψ(0)
~
~
V
Ψ(t) =
(2)
0
~ is the inverter output voltage vector (Fig. 2.)
In this, V
In three dimensional flux vector modulation, the vector
~ is made to track a reference vector Ψ
~ ∗ by choosing an
Ψ
appropriate sequence of inverter output voltage vectors. An
inverter voltage vector is selected on the basis of the error
~ ∗ and Ψ,
~ so that Ψ
~ moves towards Ψ
~ ∗ . Fig. 3 shows
between Ψ
~
~∗
the actual flux vector Ψ tracking the reference flux vector Ψ
in the three dimensional q − d − 0 space. The flux vector error
is sampled at regular intervals ∆T , and the the inverter output
voltage vector is chosen so as to keep the vector error within
a tolerance band. This is detailed in the next section.
B. Implementation of the Flux Modulator
The flux modulator is implemented in discrete-time on a
digital signal processor (DSP). The sampling time step for
0 V5
Reference Flux Vector Ψ
d
T3
T2
Ψ4
Ψ1
V1 V4
V6
III
T4
q
T2
T3
Ψ*
qd
II
III
T4
II
I
q
I
IV
q
IV
T1
Actual Flux Vector Ψ
VI
T1
V
T6
T6
Graphical representation of reference flux tracking
Fig. 4.
T5
VI
V
T5
Fig. 3.
Flux Reference Trajectory
Ψ3
V7
V14
d
d
∗
Sector identification for balanced and unbalanced flux trajectories
TABLE II
S ECTOR WITH CORRESPONDING LIMITS OF TANGENT SLOPE
Limits of
π/3
≤
2π/3
≤
π
≤
4π/3
≤
5π/3
≤
0
≤
tangent slope
T <
2π/3
T <
π
T <
4π/3
T <
5π/3
T <
2π
T <
π/3
TABLE III
E RROR BITS GENERATION
Sectors
I
II
III
IV
V
VI
Comparison of Vectors
Ψ∗x − Ψx ≥ h
Ψ∗x − Ψx ≤ h
−h < Ψ∗x − Ψx < h
the discrete-time implementation is ∆T . This is the time step
at which the error between the reference and the actual flux
~ ∗ − Ψ,
~ is sampled for corrective action. In order to
vector, Ψ
realize flux modulator for a four-leg inverter it is necessary to
~ ∗ , the
1) identify the sector on the q − d plane in which Ψ
qd
~ ∗,
q − d plane projection of the reference flux vector Ψ
is located, as shown in Table II and Fig. 4
2) generate the error bits for the q−, d− and 0−axis
component errors as shown in Table III
3) select the inverter voltage vector that reduces the errors
in the q−, d− and 0−axis components as shown in
Tables IV and V.
~ ∗ identifies one
1) Sector identification: The location of Ψ
qd
of six sectors (I. . .VI) on the q − d plane. This is shown
in Table II and Fig. 4. The sector is identified by limits
~ ∗ . These
to the slope of the tangent to the trajectory of Ψ
qd
limits are given in Table II. It is important to note that,
~ ∗ may or
depending on the application, the trajectory of Ψ
qd
may not be a circle. In applications with balanced loads, the
trajectory would typically be a circle. However, if the inverter
has to produce balanced output voltages when unbalanced and
~ ∗ will not be
nonlinear loads are present, the trajectory of Ψ
qd
a circle. Both situations are shown in Fig. 4. In Fig. 4, the
tangents are denoted as T1. . .T6 and the sectors as I. . .VI.
2) Error bits generation: The errors in the q−, d− and
0−axis flux vector components are Ψ∗q − Ψq , Ψ∗d − Ψd and
Ψ∗0 − Ψ0 . These errors are used to determine three bits Sq ,
Sd and S0 as shown in Table III. In this table, the subscript x
stands for one of q, d and 0. The error tolerance band is h.
3) Inverter voltage vector selection: The sector information
and error bits determined above are used to select an appropriate inverter voltage vector for output during the current time
~∗−Ψ
~ during the
step. The selected vector reduces the error Ψ
time step.
k,(((
Error Bit
Sx = 1
Sx = 0
Sx = Sx−
Next Action
Increase Ψx
Decrease Ψx
No Change
There are eight possible inverter voltage vectors which can
be selected for any given sector. These are given in Table IV.
Further, there are eight possible combinations of the three error
bits Sq , Sd and S0 . Each possible vector can correct for a
specific combination of error bits based on the following rules.
• Select an inverter voltage vector that can reduce the flux
component errors in all three axes simultaneously.
• If no voltage vector can correct all three errors, select a
vector which can correct any two errors without affecting
the third error.
• If no voltage vector can correct two errors without
affecting the third error, use 0-axis vectors. Here 0-axis
vectors are V0 , V1 , V14 and V15 , shown in Fig. 2.
• From among the 0-axis voltage vectors, use a vector
which can reduce the 0-axis error.
These rules are tabulated as shown in Table V.
The flux modulator described above is implemented on a
digital signal processor. In the processor, the inverter flux
components are updated at constant time intervals of ∆T using
Euler explicit integration as given below.
Ψq,k = Ψq,k−1 + Vq,k−1 × ∆T
Ψd,k = Ψd,k−1 + Vd,k−1 × ∆T
(3)
Ψ0,k = Ψ0,k−1 + V0,k−1 × ∆T
TABLE IV
P OSSIBLE SWITCHING VECTORS FOR EACH SECTOR
Sector
I
II
III
IV
V
VI
V0
V0
V0
V0
V0
V0
V1
V1
V1
V1
V1
V1
V4
V4
V2
V2
V8
V8
Possible Vectors
V5
V12
V13
V5
V6
V7
V3
V6
V7
V3
V10
V11
V9
V10
V11
V9
V12
V13
V14
V14
V14
V14
V14
V14
V15
V15
V15
V15
V15
V15
TABLE V
F LUX MODULATOR SWITCHING TABLE
II
III
Sq Sd S0
111
110
101
100
011
010
001
000
111
110
101
100
011
010
001
000
111
110
101
100
011
010
001
000
Vector
V12
V13
V14
V1
V4
V5
V14
V1
V14
V1
V14
V1
V4
V5
V6
V7
V14
V1
V14
V1
V6
V7
V2
V3
Sector
IV
V
VI
Sq Sd S0
111
110
101
100
011
010
001
000
111
110
101
100
011
010
001
000
111
110
101
100
011
010
001
000
Vector
V14
V1
V10
V11
V14
V1
V2
V3
V8
V9
V10
V11
V14
V1
V14
V1
V12
V13
V8
V9
V14
V1
V14
V1
0.2
0.15
0.1
0.05
Ψ0
Sector
I
0
−0.05
−0.1
−0.15
−0.2
0.4
0.3
0.2
0.1
Ψd
−0.1
−0.2
−0.3
−0.4
Fig. 5.
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Ψq
Inverter flux vector tracking for unbalanced reference
the q− and d−axis synchronous frame PI controllers shown
in Fig. 6 are computed accordingly. The control of the 0-axis
voltage is given below.
The model of an LC filter in q − d − 0 coordinate is
~qd0
V
Here, the subscript k refers to the sample number. The
previous voltage vector components are calculated by the
digital signal processor on the basis of the information of the
previous switching state of the inverter and the dc bus voltage
~ ∗ , is usually provided
feedback. The flux vector reference, Ψ
by the sinewave output voltage control loop of the control
system.
An example of the flux modulator working is shown in
Fig. 5. For this example, the modulator is programmed on a 32bit floating point DSP. The a−, b− and c−phase components
of the reference flux were made intentionally unbalanced.
~ in three dimensional
Fig. 5 shows the inverter flux vector Ψ
space, for a dc bus voltage Vdc = 320V . The angular
velocity of the reference flux vector is 2π × 50rs−1 and the
reference flux magnitudes are Ψa = 0.4Vs, Ψb = 0.6Vs and
Ψc = 0.05Vs. The error tolerance band h = 0.01Vs. The
sampling interval ∆T is 20µs. As seen in Fig. 5, the actual
flux vector tracks the reference closely. The trajectory in the
three dimensional q − d − 0 space is an ellipse that is inclined
to the q − d plane.
IV. F OUR -L EG S INEWAVE O UTPUT I NVERTER C ONTROL
Fig. 6 shows the closed loop control system for a standalone four-leg sinewave output inverter with an LC filter. The
inverter and filter are required to supply regulated and balanced
sinusoidal voltages to unbalanced and nonlinear loads. As
shown in Fig. 6, the reference voltage vector components are
e
e
denoted by Eqref
, Edref
and E0ref . The superscript e denotes
quantities in the synchronously rotating q e − de reference
frame. These are derived through transformation of phase
reference voltages from the a − b − c frame.
The q − d plane component vector of the reference voltage
~ ∗ . This vector component is controlled by the twovector is E
qd
dimensional flux control method detailed in [7]. The gains of
k,(((
0
Lqd0
d~
~ qd0
iqd0 + E
= Lqd0 dt

Lf
= 0
0
0
Lf
0

0

0
Lf + 3Ln
(4)
(5)
Vqd0 , iqd0 and Eqd0 are inverter voltage, inverter output current
and filter terminal voltage respectively. The state equations of
the inverter and filter for the 0-axis components are
0 −ωf2 0
Ė0
E0
=
Ψe0
1
0
Ψ̇e0
2
1 ωf 0 − Cf
Ψv0
+
(6)
i0
0
0
Equation (6) remains unchanged in synchronous reference
1
frame. The frequency ωf 0 = √
. The flux comCf (Lf +3Ln )
ponent Ψe0 is associated with the 0-axis voltage component
across the filter capacitor, and Ψv0 is associated with the 0-axis
voltage component at the inverter terminals.
With a PI regulator, the close loop state equations for the
0-axis component are


 

0
−ωf2 0
ωf2 0
Ė0
E0
 Ψ̇e0  =  1
  Ψe0 
0
0
Ψv0
−ki0 kp0 ωf2 0 −kp0 ωf2 0
Ψ̇v0



1
0
0
− Cf
E0ref
  Ė0ref  (7)
0
0
+ 0
1
ki0 kp0 kp0 Cf
i0
The characteristic polynomial for this system is
Fs = s3 + kp0 ωf2 0 s2 + ωf2 0 (1 + ki0 ) s
(8)
This can be used to determine the PI regulator gains for a
specified dynamic response.
e
Eq ref
+
e
ψq ref
PI
ω
ψq ref
controller
e
Ed ref
+
e
Switching Pulses
ψd ref
PI
controller
E0 ref
+
jωt
e
ψd ref
FLUX
MODULATOR
Vabc
ψ0 ref
PI
controller
+
Lf
FOUR LEG
INVERTER
E0
e
Eq
−jωt
e
e
Ed
Ed
Fig. 6.
i abc
Cf
n
ω
Eq
Eabc
Ln
abc
to
qd0
Linear/Nonlinear
Balanced/Unbalanced
Load
Eabc
Stand-alone four-leg sinewave output inverter system
V. E XPERIMENTAL R ESULTS
To provide experimental validation, the flux modulator and
voltage control system described above was implemented to
control a stand-alone four-leg sinewave output inverter. The
power circuit was built with four IGBT legs. The IGBT
assembly was rated for 35A rms current and 1200V dc
bus with 850µF /1200V dc link capacitors. The LC filter
components values were Lf = 3mH, Ln = 3mH and
Cf = 200µF . The entire control system including the flux
modulator was implemented on a platform with a Texas
Instruments 32-bit floating point DSP TMS320VC33 with a
13.3ns instruction cycle. The sampling time for the control
system was ∆T = 20µs.
The synchronous reference frame PI controller gains for
~ qd were set at Kpq = 0.0023, Kiq = 0.175,
controlling E
Kpd = −0.000346, and Kid = 0.538. The gains for the 0-axis
controller were Kp0 = 0.1 and Ki0 = −0.08. The suffixes p
and i stand for proportional and integral gains respectively.
The suffixes q, d and 0 stand for the q − d − 0 coordinates.
Fig. 7 shows phase and line voltages at the inverter terminals. These were obtained with only the flux modulator,
for a 50 Hz output frequency, with balanced flux references.
The q− and d− axis flux references each had a magnitude of
0.4V s, and the 0-axis flux reference was 0V s. The value of
the tolerance band h = 0.01V s. The inverter dc bus voltage
was 320V .
Experiments were performed to test different load conditions, such as balanced/unbalanced and linear/nonlinear threephase loads. Here the waveforms for two different load conditions are shown.
Fig. 8 shows waveforms for a three-phase diode bridge
rectifier load on the inverter. The dc side of the rectifier has
a filter capacitor and a resistive load. The upper three traces
show phase voltages and the corresponding phase currents.
The lowest trace shows the inverter dc link voltage. Initially
the rectifier is not connected to the inverter. It is switched
on to the inverter at a certain time. Fig. 8 shows the noload, transient, and loaded steady state performance of the
k,(((
Fig. 7.
Fig. 8.
Experimental phase and line voltage at inverter terminals
Experimental waveforms with three-phase rectifier load
Here the flux modulator is operated without the output voltage
control loop. The reference flux magnitudes are set as Ψa =
0.4, Ψb = 0.4 and Ψc = 0.4. The upper trace shows the output
phase voltage across the LC filter capacitor. The lower trace
shows the inverter dc link voltage. In the experiment, the dc
link voltage is reduced from 330V to 250V . It is apparent that
there is no change in the output voltage amplitude even after
the dc link voltage is reduced.
VI. C ONCLUSION
Fig. 9. Experimental waveforms with unbalanced linear load and three-phase
rectifier
A flux vector modulation method has been proposed for the
control of a sinewave output four-leg inverter. Digital processor implementation of the flux modulator for a four-leg inverter
is simple. This paper has described the implementation details
of the modulator. It has also described the implementation
of a voltage control system to regulate the output sinewave
voltages feeding unbalanced and nonlinear loads using the flux
modulator. The paper has presented experimental validation
of the modulator and control system. The results show that
the flux modulator proposed here works satisfactorily under
balanced and unbalanced, linear and nonlinear load conditions
on the inverter.
R EFERENCES
Fig. 10.
Experimental result of voltage regulation of flux Modulator
system. The high quality sinewave voltage output under no
load shows the effectiveness of the active damping of the LC
filter resonance.
Fig. 9 shows waveforms with unbalanced linear load and
balanced nonlinear loads connected to the inverter. The upper
three traces show phase voltages and the corresponding phase
currents. The next trace shows the neutral current supplied by
the inverter to the unbalanced load. The lowest trace shows
the inverter dc link voltage. As in the previous experiment,
the inverter operates on no load initially. The load is switched
on to the inverter subsequently.
Voltage regulation of the flux modulator is shown in Fig. 10.
k,(((
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