Space Vector Modulation and Current Regulation of the
Four-leg Voltage-Source Inverter
Cao Yu, Shen Songhua, Wang Yong
School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics,
Beijing 100083, China
Abstract—This paper presents space vector modulation
and current regulation of the four-leg voltage-source inverter. Space vector modulation based three inverter topologies have been analyzed, and three different hysteresis current regulators have been investigated. Some simulated figures illustrate the properties of the four-leg voltage-source
converter using the 3-D space vector based the multi-level
hysteresis current regulation control.
I. INTRODUCTION
space vector based multi-level hysteresis current regulation control.
II. SPACE VECTOR MODULATION
A. Two-Leg Inverter
Fig.1 depicts a 2-Leg inverter with a LC output filter. It
is assumed that S1 and S2 as well as S3 and S4 are
switched in a complementary way. Thus, The 2-Leg inverter produces a total of four (22) switching states, depicted in Table I with the corresponding voltage vectors.
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In many commercial and industrial applications, power
is distributed through a three-phase four-wire system. To
overcome the problem of handling the neutral current
under balanced and unbalanced load conditions, two basic
topologies are applied: split dc link capacitors, and fourleg inverters [1]. With the split dc link capacitors the
three-phase inverters become three independent singlephase half-bridge inverters, and the dc link capacitor will
handle the neutral current directly. Therefore, large and
expensive dc link capacitors are needed, and the utilization of the dc link voltage is poor [1][2]. In contrast, the
three-phase four-leg inverter has more control flexibility.
Therefore, the three-phase four-leg inverters have been
investigated in this paper.
Four-leg converters can effectively provide the neutral
connection in three-phase four-wire systems. They can be
used in inverter, rectifier, and active filter applications to
handle the neutral current caused by the unbalanced
and/or nonlinear load or unbalanced source.
At first, in this paper, three-dimensional (3-D) space
vector modulation (SVM) schemes are proposed. The
proposed 3-D SVM is a superset of the traditional twodimensional (2-D) SVM, and it inherits all the merits of
the traditional 2-D SVM, also the 2-D SVM is a superset
of the one-dimensional (1-D) SVM [3][4]. So section I
presents a unified approach of the space vector modulation for voltage-source inverters. To demonstrate the proposed unified approach, three fundamental inverters topologies are analyzed: single-phase full-bridge, threephase three-wire (three-leg), and three-phase four-leg
inverters.
Second, this paper presents hysteresis current regulation techniques using three-phase inverter with neutral leg.
It has been shown that well-known techniques of hysteresis current regulation used in conventional three-phase
three leg inverters can also be applied in four-leg inverters
[5].
At last, some simulated figures illustrate the properties
of the four-leg voltage-source converter using the 3-D
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Fig.1 Two-leg inverter topology
TABLE I
TWO-LEG INVERTER SWITCHING STATES
AND VOLTAGE VECTORS
It can be seen that the output voltage space of the 2-Leg
inverter is one-dimensional, since only the voltage Vab is
applied to the load. Therefore, it is possible to represent
the switching vectors onto a line, as presented in Fig.2.
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Fig.2 Output voltage space of two-leg inverter
B. Three-Leg Inverter
Fig.3 depicts a 3-Leg inverter connected to a threephase LC filter, with a balanced load. Again, it is assumed
that S1 and S2, S3 and S4 as well as S5 and S6 are
switched in a complementary way. Thus, there are 8 eight
(23) possible switching-states, depicted in Table II with
the corresponding voltage vectors.
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Fig.3 Three-leg inverter topology
Fig.5 Four-leg inverter topology
TABLE II
THREE-LEG INVERTER SWITCHING STATES
AND VOLTAGE VECTORS
TABLE III
FOUR-LEG INVERTER SWITCHING STATES
AND VOLTAGE VECTORS
It can be verified that the output voltage space of the 3Leg inverter is two-dimensional, since it can produces
only two independent output voltages; i.e. if Vab and Vbc
are known, then Vca is implicitly defined, where a balance
load is assumed. Therefore, the output voltage space can
be constructed by projecting the leg-voltage space onto a
two-dimensional plane, using the αβ transformation matrix (1), given below:
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1
⎡
1 −
2⎢
2
⎢
Tαβ =
3⎢
3
⎢⎣0 2
1 ⎤
− ⎥
2
⎥
3⎥
−
2 ⎥⎦
(1)
In this new coordinate system ( αβ ), there are 6 nonzero vectors and two zero vectors, as shown in Fig.4.
It can be verified that the output Voltage space of the 4Leg inverter is three-dimensional, since it can produce
three independent output voltages regardless of load by
using the fourth leg to control the neutral voltage and
conduct any neutral currents. Therefore, it is possible to
represent the switching vectors in a three-dimensional
space as show in Fig.6 below, by using the αβγ transformation matrix (2), given below:
Tαβγ
Fig.4 Output voltage space of three-leg inverter in αβ -coordinates
C. Four-Leg Inverter
Fig.5 depicts a 4-Leg inverter connected to a threephase LC filter, with arbitrary loads. It is assumed that S1
and S2, S3 and S4, S5 and S6 as well as S7 and S8 are
switched in complementary way. Then, there are 16 possible switching vectors, which are presented in Table III
with the corresponding voltage vectors.
⎡
⎢
⎢
2⎢
=
3⎢
⎢
⎢
⎢
⎣
1
0
1
2
1
2
3
2
1
2
−
1 ⎤
2 ⎥
⎥
3⎥
−
2 ⎥
⎥
1 ⎥
2 ⎥⎦
−
(2)
In this new coordinate system ( αβγ ), there are 14 nonzero vectors and two zero vectors, as shown in Fig.6. Projection of all switching vectors on α − β plane creates the
well-known hexagon form Fig.7 (a), but each vector can
be obtained from two switching combination. For example V2 can be obtained from projection of switching state
vector 1100 or from projection of switching state vector
1101, as shown in Fig.7 (b).
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B. α-β-γ Three-Level Hysteresis Current Regulator
To obtain better results from hysteresis regulator represented above, we follow the same techniques as in conventional three-phase three-leg inverters. Keeping in mind
that in four leg inverters after the projection in α-β plan
each vector has two possibilities, for example V2 (1100 or
1101), as shown in Fig.7 (b), we need another regulator
(γ-regulator) to select the proper vector as shown in
Fig.9. Therefore the three level hysteresis regulator is
used for γ (< 0, = 0, or >1). Example V2, is the projection of (1100 or 1101). The γ regulator can obtain the
proper vector selection.
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Fig.6 Output voltage space of four-leg inverter in αβγ -coordinates
(a)
(b)
Fig.7
(a) Switching projection of inverter vectors in αβ plane
(b) Example of the projection of vector V2
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Fig.9 α-β-γ Three-level hysteresis PWM current regulator
III. CURRENT REGULATION SCHEMES
α-β-γ Multi Level Hysteresis Current Regulator
C.
A. Four independent Two-Level Hysteresis Regulator
The block scheme of four independent two level hysteresis regulators is shown in Fig.8. Each regulator determines the state of one inverter leg keeping the error of the
related leg within the hysteresis band [2][6][7]. In four
independent two-level hysteresis regulators there is no
information between the individual hysteresis regulators,
and there is no strategy for zero selection. Keeping in
mind that the projection of the all switching vectors in αβ plane produces the well-known hexagon-form, the
improvement of four-leg hysteresis regulator can be
achieved by using the same techniques as in three-leg
inverter to regulate the α-β components and another
regulator for the neutral leg.
We can see that there is still no complete definition of
the output voltage in case of α=0 and β=0. Therefore,
for further improvement we use multi-level hysteresis
regulator as shown in Fig.10. In thus scheme the voltage
in α-axis is divided into four different levels and β-axis
into three levels.
From α-β regulators, the digital output signals (dα,
d β ) selection the state of inverters switches using the
switching table, which defines the inverter outputs as
shown in Table IV. But this table is not sufficient to define the four-leg inverter states, because each vector has
two possibilities. By using the third regulator and Fig.6 it
is possible to define the switching table for four-leg inverter as shown in Table V.
Fig.8 Four independent two-level hysteresis PWM current regulator
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Fig.10 α-β-γ Multi-level hysteresis PWM current regulator
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TABLE IV
SELECTION TABLE FOR α-β MULTI-LEVEL
CURRENT REOULATOR
(a)
TABLE V
SELECTION TABLE FOR γ MULTI-LEVEL
CURRENT REGULATOR
(b)
Fig.11α-β-γ Multi Level hysteresis current regulator waveforms
with unbalance load
(a) Output voltages Van, Vbn, Vcn (b) Output current ia, ib, ic
IV. SIMULATION RESULTS
In order to prove the approach of the 3-D space vector
based multi-level hysteresis PWM current regulator for
the four-leg voltage-source is correct, the circuit model
was built in the soft of Saber. Simulation results are
shown in Fig.11. The inverter and the load data are given
below:
Inverter data:
DC link voltages =370V;
LC filter: La=Lb=Lc=0.4mH, Ln=0.1mH,
Ca=Cb=Cc=40μF;
f=50Hz;
Hα=0.5/3, Hβ=0.5/2, Hγ=0.5.
Load (unbalanced) data:
Ra=5Ω, Rb=50000Ω, Lc=16mH, Rc=3Ω.
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[1] R.Zhang, V.H.Prasad, D.Boroyevich, F.C.Lee, “Three-dimensional
space vector modulation for four-leg voltage-source converters”,
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2002.
[2] Sun Chi, Bi Zengjun, Wei Guanghui, “Modeling and simulation of a
three-phase four-leg inverter based on a novel decoupled control
technique”, in Proceedings of CSEE, 2004, vol.24, no.1, pp. 124 –
130.
[3] H.Pinheiro, F.Botteron, C.Rech, L.Schuch, R.F.Camargo, H.L.Hey,
H.A.Grundling, J.R.Pinheiro, “Space vector modulation for voltage-source inverters: a unified approach”, in IEEE IECON Proc.,
2002, vol.1, pp. 23 – 29.
[4] M.J.Ryan, R.D.Lorenz, R.W. De Doncker, “Modeling of sinewave
inverters: a geometric approach”, in IEEE IECON Proc., 1998,
vol.1, pp. 396 - 401.
[5] S.M.Ali, M.P.Kazmierkowski, “PWM voltage and current control of
four-leg VSI”, in IEEE ISIE Proc., 1998, vol.1, pp. 196 – 201.
[6] Ruan Xinbo, Yan Yangguang, “The control strategy for three-phase
inverter with four bridge legs”, Transactions of China electro technical society, vol.15, no.1, pp. 61 – 64, 2000.
[7] Yang Hong, Ruan Xinbo, Yan Yangguang, “Maximum error current
regulated three-phase inverter with four bridge legs”, Power electronics, vol.23, no.1, pp. 32 – 33, 43, 2003.
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In this paper, space vector modulation based three inverter topologies have been analyzed, and three different
hysteresis current regulators have been investigated. The
simulation model is built in the soft of Saber for the fourleg voltage-source converter using the 3-D space vector
based multi-level hysteresis current regulation control.
Simulation results are presented to validate the proposed
approach.
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REFERENCES
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V. CONCLUSION
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