PS 2-49
37th IEEE Power Electronics Specialists Conference / June 18 - 22, 2006, Jeju, Korea
Six Step Modulation of Matrix Converter with
Increased Voltage Transfer Ratio
Bingsen Wang
Giri Venkataramanan
Department of Electrical and Computer Engineering
University of Wisconsin-Madison
1415 Engineering Drive
Madison, WI 53706 USA
bingsen@cae.wisc.edu; giri@engr.wisc.edu
Abstract— Conventional modulation strategies for matrix
converters based on trigonometric transformations and space
vector techniques have their voltage transfer capability limited
at about 0.866 p.u. This leads to a derating of motor capability
by 25% in the realization of variable frequency drives using
off-the-shelf line-fed motors, or requires custom designed
motors for operation at reduced voltage. In this paper, a sixstep variable frequency modulation strategy for the matrix
converter that is capable of providing more than 1 p.u.
fundamental component of output voltage is presented. Six
step operation suitable for the conventional 9-switch topology
and the reduced switch dual bridge topologies are presented,
with capability of unity displacement factor operation at the
input and output simultaneously. It is demonstrated that by
varying the switching angles, control of fundamental
component of output voltage and input current is possible, with
concomitant generation of reactive power at the input and
output terminals of the converter, with bidirectional power
flow. The paper presents analytical results, computer
simulation results along with laboratory scale experimental
results.
I.
INTRODUCTION
Potential attractive features of the matrix converter, such
as the absence of dc link energy storage, high power density,
high quality of input output waveforms are leading to
continued research efforts aimed at translating their promise
into reality [1-9]. Although the matrix converters were first
introduced as a class of frequency changers realized using
controlled switches by Gyugyi [10], significant progress
began with the introduction of high frequency synthesis
proposed by Venturini in 1980 [11-13]. Although the initial
approach was limited to a voltage transfer ratio of one half,
improved modulation strategies with voltage transfer ratio of
√3/2 was published in 1989 [14]. Further along, indirect
modulation methods based on fictitious dc link concept have
been proposed [15-19]. Except the programmed switching
pattern in [19], all these modulation strategies for the matrix
converter, based on high frequency synthesis have not been
able to break the upper bound given by √3/2 p.u. Since motor
1-4244-9717-7/06/$20.00
2006 IEEE.
output power is proportional to the square of applied voltage,
this limitation leads to reduced output power, limited to ¾ of
the motor rating, when fed from a matrix converter operating
from the line nominally rated at motor voltage. Otherwise, a
custom designed motor designed for the appropriate output
voltage level would be required. This remains one among the
many roadblocks limiting the viability of the matrix
converter. In this paper, a low switching frequency
modulation technique for frequency conversion using matrix
converters is presented. The proposed strategy is capable of
providing more than unity voltage transfer ratio. The
modulation strategy is easier developed on the basis of
operation of the dual bridge topology with a fictitious dc
link. However, the approach is applicable for the
conventional 9-switch topology as well, as will be
demonstrated through appropriate topological mapping. The
paper presents the topological mapping, followed by the
modulation strategy, performance characterization including
harmonic and reactive power flow analysis. Simulation
results are presented in the paper, and will be supplemented
with laboratory experimental results in the paper.
Fig. 1(a) illustrates the conventional matrix converter
using conventional three single-pole-triple-throw (SPTT)
switches Su, Sv, Sw. Fig. 1(b) illustrates the reduced switch
two SPTT switches Sp and Sn and three single-pole-doublethrow (SPDT) switches. The two realizations are labeled as
conventional matrix converter (CMC) and indirect matrix
converter (IMC) respectively. If the switching function of the
throws Txy, as illustrated in Fig. 1, is defined to be hxy, the
input-output transposition properties of the CMC can be
expressed as
Vo = HCMC Vi
(1)
I i = H CMCT I o
where Vi, Vo, Ii, Io are defined as [va vb vc]T, [vu vv vw]T , [ia
ib ic]T , [iu iv iw]T.
HCMC is the switching matrix with elements being switching
functions of various throws shown in Fig. 1a.
⎡ hau hbu hcu ⎤
(2)
H CMC = ⎢ hav hbv hcv ⎥
⎢
⎥
⎢⎣ haw hbw hcw ⎥⎦
- 930 -
The input-output transposition properties of the IMC can be
expressed as
Vo = H VSB HCSB Vi
(3)
I i = HCSBT H VSBT Io
II.
Let the three phase input voltages and three phase output
currents be assumed to be balanced stiff ac quantities and
expressed as
where HCSB is the switching matrix of the current source
bridge (CSB) connected with the stiff voltages. HVSB is the
switching matrix of the voltage source bridge (VSB)
connected with the stiff currents. They are given by
⎡h
HCSB = ⎢ ap
⎣ han
⎡ hup
⎢
H VSB = ⎢ hvp
⎢⎣ hwp
hcp ⎤
;
hcn ⎥⎦
hbp
hbn
hun ⎤
⎥
hvn ⎥
hwn ⎥⎦
Tau
Su
a
- vb +
Tav
b
c
Sv
iv
vv
iw
vw
-
n
-
vb
-
vc
Tcp
vu
Sw
+ ib
Twp
vpn
Twn
+ ic
Su
Tup
Tun
Sv
Tvp
vv
Sn
iu
iv
N
Tvn
vw
Tan
(
3)
= cos ( β ( t ) + 2π )
3
i
⎧1
han = ⎨
⎩0
⎧1
hbn = ⎨
⎩0
⎧1
hcn = ⎨
⎩0
ip
+ ia
o
v
mw
(
3)
π
2
= cos (α ( t ) +
3)
o
o
if (ma > mb ) ∧ ( ma > mc )
otherwise
if (mb > mc ) ∧ (mb > ma )
otherwise
(8)
⎧1 if ( mc > ma ) ∧ (mc > mb )
hcp = ⎨
⎩0 otherwise
+
va
o
Vi is source voltage amplitude;
Io is the current source amplitude;
αi(t) is the phase angle of the voltage source, given
by α i ( t ) = ωi t + α i 0 ;
⎧1
hap = ⎨
⎩0
⎧1
hbp = ⎨
⎩0
Sw
(a)
Tbp
i
The switching functions for the various throws of the CSB
and VSB are determined by (8) and (9), respectively.
Tcw
Tap
iw
i
(5)
Let the modulation functions for the CSB and VSB be
defined as
ma = cos ( β i ( t ) )
mu = cos (α o ( t ) )
;
(6)
π
2
mb = cos β i ( t ) −
m = cos α ( t ) − 2π
Tbw
Sp
vc
(
3)
= I cos ( β ( t ) + 2π )
3
The displacement angle φi on the CSB side and the
displacement angle φo on the VSB side are defined as
φi = αi 0 − β i 0
(7)
φo = αo 0 − β o 0
iu
vu
Tcv
Taw
iv = I o cos β o ( t ) − 2π
;
(
3)
= V cos (α ( t ) + 2π )
3
where β i ( t ) = ωi t + β i 0 , α o ( t ) = ωot + α o 0 .
Tbv
- vc +
vb = Vi cos α i ( t ) − 2π
mc
Tcu
va +
iu = I o cos ( β o ( t ) )
βo(t) is the phase angle of the current source, given
by β o ( t ) = ωo t + β o 0 .
Tbu
-
va = Vi cos (α i ( t ) )
where
(4)
Through terminal equivalency for CMC and IMC it can be
shown that HCMC = HCSBHVSB. Thus any modulation strategy
that provides a solution for IMC can be applied for CMC
through appropriate switching function mapping. In the
subsequent section, the modulation process is derived for the
IMC.
SWITCHING FUNCTIONS AND SYNTHESIZED
VOLTAGES/CURRENTS
if (ma < mb ) ∧ (ma < mc )
otherwise
if (mb < mc ) ∧ (mb < ma )
otherwise
if ( mc < ma ) ∧ (mc < mb )
otherwise
iw
⎧1
hup = ⎨
⎩0
⎧1
hvp = ⎨
⎩0
⎧1
hwp = ⎨
⎩0
Tbn Tcn
-
(b)
Fig. 1 Representation of (a) IMC and (b) CMC using SPDT and SPTT
switches
if ma > 0
, hun = 1 − hup
otherwise
if mb > 0
otherwise
(9)
, hvn = 1 − hvp
if mc > 0
otherwise
, hwn = 1 − hwp
The switching functions for the CSB and VSB are depicted
in Fig. 2. With this choice of switching functions, the
- 931 -
ma
mb
mc
1.5
1
0.5
vu, vv, vw, iu
resulting output voltages and input currents for unity
displacement factor operation at input and output, and the
corresponding Fourier spectra are shown in Fig. 3. It can be
observed that waveforms of output voltages and the input
currents are very similar to those of the voltage source
inverter and current source inverter except for the presence
of the ripple on the 60-degree segments of each waveform.
0
0.5
1
1.5
π
2π
0
βi(t)
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
t (s)
1
βi(t)
hbp
βi(t)
hcp
FFT(vu)
hap
0.5
βi(t)
han
0
0
10
20
hbn
βi(t)
hcn
30
40
50
nth order
(a)
βi(t)
1
βi(t)
(a)
mv
mw
π
2π
αo(t)
ia, ib, ic, va
mu
0.5
0
0.5
1
hup
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
t (s)
αo(t)
hvp
1
αo(t)
hwp
FFT(ia)
αo(t)
hun
αo(t)
hvn
0.5
αo(t)
hwn
0
0
10
20
30
40
nth order
(b)
αo(t)
(b)
Fig. 2 Waveforms of various switching functions for (a) CSB; (b) VSB for
unity power factor operation at both sets of terminals.
Fig. 3 Normalized (a) output phase-to-load-neutral voltage and spectrum;
(b) input current and spectrum for the case of φi=0 and φo=0;
- 932 -
50
III.
VOLTAGE AND CURRENT TRANSFER
CHARACTERISTICS
In addition to operation at unity displacement factor at
both input and output ports, the fundamental component of
the output voltage can be varied as the input displacement
angle varies and it is independent of the output displacement
angle. Similarly, if the output displacement angle varies, the
fundamental component of the input current varies and it is
independent of the input displacement angle. For the extreme
cases, the fundamental component of the input current is zero
if φo=π/2 and the fundamental component of the output
voltage is zero if φi=π/2. Between the two extreme cases, as
illustrated in Fig. 4a, the output voltage fundamental
component decreases as the input displacement factor
decreases while the rms value of the harmonic component
remains almost constant. Accordingly, the output voltage
distortion factor, which is defined as the ratio of the
fundamental rms value over the total rms value, decreases
with lower input displacement factor. In a dual manner, as
illustrated in Fig. 4b, the input current fundamental
component decreases as the output displacement factor
decreases while the harmonic component rms value remains
almost constant.
hap =
1
n ≠0
n ≠0
1 +∞ 1
⎛ nπ
hcp = + ∑
sin ⎜
3 n =−∞ nπ
⎝ 3
⎞ jn( βi ( t ) + 2π / 3)
⎟e
⎠
1 +∞ 1
⎛ 2nπ
han = − ∑
sin ⎜
3 n =−∞ nπ
⎝ 3
⎞ jnβi ( t )
⎟e
⎠
n ≠0
n ≠0
(10)
1 +∞ 1
⎛ 2nπ ⎞ jn( βi ( t )−2π / 3)
hbn = − ∑
sin ⎜
⎟e
3 n =−∞ nπ
⎝ 3 ⎠
n ≠0
1 +∞ 1
⎛ 2nπ ⎞ jn( βi ( t )+ 2π / 3)
hcn = − ∑
sin ⎜
⎟e
3 n =−∞ nπ
⎝ 3 ⎠
n ≠0
Similarly, the switching functions in (9) can be expressed in
a Fourier series summation as
hup =
1 1 +∞ 1
⎛ nπ ⎞ jnαo ( t )
+ ∑ sin ⎜
⎟e
2 π n =−∞ n
⎝ 2 ⎠
n ≠0
1 1 +∞ 1
⎛ nπ ⎞ jn(αo ( t )−2π / 3)
hvp = + ∑ sin ⎜
⎟e
2 π n =−∞ n
⎝ 2 ⎠
0.8
n ≠0
0.6
hwp
0.4
1 1
= +
2 π
+∞
1
⎛ nπ
2
∑ n sin ⎜⎝
n =−∞
n ≠0
1 1 +∞ 1
⎛ nπ
hun = − ∑ sin ⎜
2 π n =−∞ n
⎝ 2
0.2
n ≠0
⎞ jn(αo ( t ) + 2π / 3)
⎟e
⎠
(11)
⎞ jnαo ( t )
⎟e
⎠
1 1 +∞ 1
⎛ nπ ⎞ jn(αo ( t ) −2π / 3)
hvn = − ∑ sin ⎜
⎟e
2 π n =−∞ n
⎝ 2 ⎠
0
0
15
30
45
60
75
n ≠0
90
Input Displacement Angle φ i (degree)
hwn
(a)
1
1 1 +∞ 1
⎛ nπ ⎞ jn(αo ( t )+ 2π / 3)
= − ∑ sin ⎜
⎟e
2 π n =−∞ n
⎝ 2 ⎠
n ≠0
Substituting (10) and (11) into (3), together with (5), results
in the following relationships.
ω φ
ω φ
⎡ 3
⎤ (12)
⎛e
⎞ ⎤ ⎡ 3I
⎛ eφ
e
e φ ⎞
i =⎢
∑ ⎜ 6k + 1 − 6k − 1 ⎟⎥ ⎢ 2π ∑ (−1) ⎜ 6k + 1 − 6k − 1 ⎟ e ω ⎥
π
Total Input Current
Fundamental Input Current
Distortion Power Factor
Harmonics of Input Current
1.2
j (6 k +1)( i t − i )
+∞
j (6 k −1)( i t − i )
+∞
⎣
k =−∞
⎝
⎠⎦ ⎣
j (6 k +1)ωo t
⎡ 1 +∞
e j (6k −1)ωot
k ⎛e
−
vu = ⎢ ∑ ( −1) ⎜
+
π
6
k
1
6k − 1
k
=−∞
⎝
⎣
0.8
0.6
j
k
o
a
Input Current
1 +∞ 1
⎛ nπ ⎞ jnβi ( t )
+ ∑
sin ⎜
⎟e
3 n =−∞ nπ
⎝ 3 ⎠
1 +∞ 1
⎛ nπ ⎞ jn( βi ( t ) −2π / 3)
hbp = + ∑
sin ⎜
⎟e
3 n =−∞ nπ
⎝ 3 ⎠
Total Output Voltage
Fundamental Ouput Voltage
Distortion Power Factor
Harmonics of Output Voltage
1.2
Output Voltage
The results shown in Fig. 4 are obtained through
numerical evaluations. These results can be well explained if
we conduct Fourier analysis on the input currents and output
voltages. The switching functions in (8) can be expressed in
a Fourier series summation as
k =−∞
⎞ ⎤ ⎡ 3 3Vi
⎟⎥ ⎢
⎠ ⎦ ⎣ 2π
o
⎝
−j
o
j 6k
⎠
ot
⎦
⎛ e − jφi
e jφi ⎞ j 6 k (ωi t −φi ) ⎤
−
⎥
∑⎜
⎟e
6k − 1 ⎠
k =−∞ ⎝ 6k + 1
⎦
+∞
The fundamental component of the input current and output
voltage can be determined to be:
0.4
I a1 =
0.2
0
0
15
30
45
60
75
90
Output Displacement Angle φ o (degree)
(b)
Fig. 4 Variation of the input current and output voltage versus output
displacement angle and input displacement angle.
6 3I o
π2
cos(φo );
Vu1 =
6 3Vi
π2
cos φi
(13)
From (13), it can be observed that the maximum voltage
transfer ratio occurs when the input displacement power
factor is unity, which is 6 3 / π 2 =1.05. The maximum
current transfer ratio occurs when the output displacement
factor is unity. With the analytical expression in (13), we
can study the real and reactive power flow across the power
converter under different operating conditions, which is
discussed in the following section.
- 933 -
IV.
POWER FLOW
To calculate the real power, only the fundamental
component of the input currents and output voltages need to
be considered since the active power transfer takes place
with voltage and current of the same frequency. The input
power and output power can be found with the expressions of
the input and output quantities derived in (13) together with
the input voltage sources and output current sources in (5).
⎞
3 ⎛ 6 3I o
9 3
Pin = Vi ⎜
cos φo ⎟ cos(φi ) = 2 Vi I o cos φi cos φo
2 ⎝ π2
π
⎠
Pout =
(14)
reactive power (with inductive load) if φo is positive.
Together with other quadrants, the characteristics of the
input and output reactive power are illustrated in Fig. 6.
The variation of the input reactive power and output
reactive power with the input and output displacement angle
is plotted against φi and φo in Fig. 8. Again, both the input
and output reactive power are normalized to 9 3
π
φo
2 Vi I o , is
plotted against φi and φo in Fig. 5. Obviously, the power
throughput reaches its maximum when φi = φo = 0, i.e. unity
displacement power factor at both input and output ports.
In contrast to the active power, the reactive power across
the converter is not conserved. Thus, both the input and
output ports can supply reactive power or absorb reactive
power simultaneously. If the reactive power associated with
fundamental components alone is considered, the reactive
power on the input side and output side can be found as
follows.
⎞
3 ⎛ 6 3I o
9 3
Qin = Vi ⎜
cos φo ⎟ sin(φi ) = 2 Vi I o sin φi cos φo
2 ⎝ π2
π
⎠
Qout =
Vi I o . In
both plots, the absolute value of the reactive power is shown.
It is easy identify the direction of the reactive power flow
with reference to Fig. 6.
⎞
3 ⎛ 6 3Vi
9 3
cos φi ⎟ I o cos(φo ) = 2 Vi I o cos φi cos φo
⎜
2
π
2⎝ π
⎠
The active power throughput, normalized to 9 3
π2
Input supplies reactive power
Output supplies reactive power
Input absorbs reactive power
Output supplies reactive power
φi
Input supplies reactive power
Output absorbs reactive power
Input absorbs reactive power
Output absorbs reactive power
(15)
⎞
3 ⎛ 6 3Vi
9 3
cos φi ⎟ I o sin(φo ) = 2 Vi I o cos φi sin φo
⎜
2
2⎝ π
π
⎠
Since the phase angle φi and φo are defined as the lagging
phase angle of the input an output current, the input side
absorbs reactive power or the converter looks inductive to
the voltage source if φi is positive. The output side supplies
Fig. 6 Characteristic of input reactive power and output reactive power in
different quadrants on the operation φi-φo plane.
Fig. 5 Variation of the real power throughput with the output and input
displacement angles.
Fig. 7 Variation of the input reactive power with the output and input
displacement angles.
- 934 -
100
vu (V)
50
0
-50
-100
0
0.01
0.02
0
0.01
0.02
0.03
0.04
0.05
0.03
0.04
0.05
10
i u (A)
5
0
-5
-10
t (s)
(b)
Fig. 8 Variation of the output reactive power with the output and input
displacement angles.
V.
Fig. 9 Simulation results of the output voltage and output current for RL
load
SIMULATION AND EXPERIMENTAL RESULTS
Although the modulation process has been discussed with
the sinusoidal input voltage and sinusoidal currents, the
modulation process still applies if the output sinusoidal
currents are replaced by the RL load. Waveforms from
computer simulation of output voltages and output currents
of a matrix converter fed from a 60 Hz, 70V, three phase
source feeding a 70 V, 120 Hz to a three phase R-L load
(R=8 Ω, L=10 mH) using Simulink® are shown in Fig. 9.
The similar operating condition is used in the experiment
except the voltage amplitude has lower value. The
experimental results are shown in Fig. 10.
VI.
CONCLUSIONS
In this paper, a new modulation/control strategy for the
matrix converter featuring up to unity voltage transfer ratio is
proposed. The voltage/current transfer ratio is studied using
Fourier analysis. Furthermore, the real and reactive power
flow across the converter under the phase control mode has
been presented based on the Fourier analysis results. The
simulation and experimental results for an RL load case
reveals the practical feasibility of this approach.
Such a low frequency modulation approach, combined
with proper filter design and/or multi-pulse techniques,
practical designs may be realized for application of the
matrix converter at medium voltage high power applications
using devices such as IGCTs. The combination of the
proposed phase modulation together with conventional pulse
width modulation provides independent specification of
power factor at both the input and output ports of the matrix
converter. Furthermore, the proposed approach makes it
possible to develop, extend and apply appropriate modulation
strategies beyond the limits of conventional scalar and vector
PWM approaches in the transition region between linear
operation and six-step operation.
Fig. 10 Experimental results of the output voltage and output current for
RL load.
ACKNOWLEDGMENT
The authors would like to acknowledge support from the
Wisconsin Electric Machine and Power Electronics
Consortium (WEMPEC) at the University of WisconsinMadison. The work made use of ERC shared facilities
supported by the National Science Foundation (NSF) under
AWARD EEC-9731677.
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