Modeling Design for a
Matrix Converter with a Generator as Input
Junnosuke Haruna
Jun-ichi Itoh
Nagaoka University of Technology
Kamitomioka-cho Nagaoka City Niigata, Japan
itoh@vos.nagaokaut.ac.jp
Abstract - This paper proposes a model and design based on
the behavior of a matrix converter and a generator. Presently,
applications for grid connection have been considered widely
such as micro-gas turbines, wind power systems, engine
generators and so on. However, a very large impedance power
supply such as the generator has not been sufficiently
examined. Especially the PM generator which is used invarious applications has large synchronous reactance. To
perform a simple characteristic analysis, an ideal mathematicalmodel excluding switching parts is introduced. An input filter
characteristic is discussed by using an ideal mathematicalmodel. Thus, the cause of the unstable behavior is clarified. Inorder to verity about the stability operation, a proposed
stability control on the rotating frame is applied to thegenerator. The proposed method can control the generator
current and the output voltage regardless of the generator
speed. In addition, a control of optimal input power factor is
proposed to obtain maximum modulation index of the matrix
converter. Moreover, the fluctuation in the output voltage isable to compensate by the proposed voltage compensation
method. This paper confirms the validity of the proposed
strategy by simulation and experiment. The large oscillationon the generator terminal voltage and current are suppressedby the proposed stability control. Moreover, the proposed
control achieves stability at motor load 1.5kW and 3.7% of the
input current THD.

I.

INTRODUCTION

Recently, matrix converters, that can directly convert an
AC power supply voltage into an AC voltage of variable
amplitude and frequency without large energy storage, suchas electrolytic capacitor, have been actively studied [1-9].
Matrix converters have advantages of size reduction, lightweight, long-life and high efficiency power supplies,
compared with the use of a PWM rectifier and inverter
system.
However, matrix converters are mainly considered for
motor drive system applications, such as elevators and air
conditioners. Lately, applications for grid connection havebeen considered, such as micro-gas turbines, wind powersystems, engine generators and so on. The behavior of
matrix converters under the large line impedance of a power
grid has been investigated [4] [8]. However, very large
impedance power supplies, such as generators have not been
sufficiently examined.
For example, it has not been sufficiently discussed thatthe terminal voltage of the generator and input current are
unstable due to the resonance between synchronous
reactance and the input capacitor. The stability control
method of the input current considering the high line

impedance has been already proposed [4]. However, that
effect has not been reported in case of the generator as input.
For a stable operation, it is important to obtain a systemmodeling before control it. However, it is difficult to
discuss the behavior of the matrix converter because thebehavior of the input current interference directly to the
output control. In order to analyze the operation of thematrix converter easily, a simple model such as
mathematical model is required.
This paper discusses a model for a matrix converter with ahigh impedance power supply such as a generator. In thispaper, the ideal mathematical model for the matrix converter
is introduced to discuss the stabilization of the input filter byusing synchronous reactance of the generator. In addition,
the design of a stability control is sufficiently provided
based on the ideal mathematical model. Since, the control
method of the generator using the matrix converter has notbeen discussed, yet. Therefore, the output voltage of thematrix converter is depended on an input power factor
because a voltage transfer ratio is limited by the input power
factor. Therefore, the input power factor control has to beconsidered from the point of view of the maximum voltage
transfer ratio. Note that, the output voltage of the matrix
converter depends on the input terminal voltage. The output
voltage should keep to constant for the stable operation.
Therefore, this paper also proposes the output voltage
compensation method using frequency information of theinduced e.m.f. instead of the input terminal voltage of thematrix converter.
This paper provides the control characteristics of thematrix converter based on simulation and experimental
results as follows; (1) the fundamental operation with R-L
load, (2) the relation among the input current phase, theterminal voltage and the output voltage, (3) the operation of
the induction motor load, (4) the acceleration anddeceleration characteristics of the induction motor, and (5)the acceleration and deceleration of the generator with an
output voltage compensation method. The results of thesimulation and the experiment confirmed the validity of the
proposed system.
II. MATHEMATICAL MODEL OF MATRIX CONVERTER
A. Mathematical model without switching operation
Fig. 1 shows a circuit diagram of the matrix converter.
The simulation of the matrix converter including motor or
generator consists of control part to obtain the output

voltage command and switching models for the PWM
operation. The simulation of these applications require avery long time due to short time step for the PWM operation.
Therefore, a mathematical model is introduced in order toachieve the simulation at a shorter time.
Fig. 2 shows an ideal mathematical model of the matrix
converter. The behavior of the input side of the matrix
converter is expressed by current sources. Likewise, thebehavior of the output side is expressed by voltage sources.
An active power of the input side always equals to the
output side in the mathematical model, where the converter
loss can be neglected. Then the relationship between input
current source and output voltage is given by (1)Vin I in cos φ = Vout I out cos φl(1)where Vin is the input voltage, Vout is the output voltage, Iin is
the input current, Iout is the output current, cosφ is the input
power factor and cosφl is the output power factor.
However, it is complicated to calculate the instantaneous
value of the input power factor and load power factor.
Hence, this paper considers the relation between the output
and input side on a rest frame known as α-β frame. The
output voltage is obtained by multiplying input current
command by the input voltage. Likewise, the input current
is obtained by multiplying the output voltage command by
the load current. Therefore the output voltage and input
current are expressed in (2) and (3) considering is active
power.
⎧voα = (viα iicmdα + viβ iicmdβ )vocmdα(2)⎨⎩voβ = (viα iicmdα + viβ iicmdβ )vocmdβ⎧iiα = (vocmdα ioα + vocmdβ ioβ )iicmdα⎨⎩iiβ = (vocmdα ioα + vocmdβ ioβ )iicmdβ

Input side

Output side

vr

ir

vu

iu

vs

is

vv

iv

vt

it

vw

iw

Fig. 2. Ideal mathematical model of the Matrix converter.

(3)

where viα, viβ are the input voltage, iiα, iiβ are the input
current, iicmdα, iicmdβ are the input current command
normalized by the output current, voα, voβ are the output
voltage, vocmdα, vocmdβ are the output voltage command
normalized by the input voltage and ioα, ioβ are the output
current.
Fig. 3 shows a simulation results using ideal mathematicalmodel with the synchronous reactance and filter capacitor.
In case of the generator as input, the synchronous reactanceis added to the input part of the mathematical model. In
addition, the input filter capacitor is also connected. As aresult, the resonance between synchronous reactance andfilter capacitor occurs. The input terminal voltage and input
current have large oscillation due to the filter resonance.
Likewise, the output voltage also has distortion. That is, thesynchronous reactance is the cause of unstable.
B. Behavior of the synchronous reactance in a generator
Fig. 4 shows a single phase equivalent circuit of the input
filter with synchronous reactance. The output active power
must be kept constant even if the input terminal voltage has
fluctuation. The relationship among the terminal voltage,
the load current and the output power are obtained as (4)Pos + ΔPo = vcs ios + Δvc ios + vcs Δio + Δvc Δio

Fig. 1. Configuration diagram of the matrix converter.

(4)

Fig. 3. Simulation results using ideal mathematical model.

Fig. 4. Single phase equivalent circuit of the input filter with synchronous
reactance in the generator.

Fig. 5. Block diagram of the input filter.

where Pos, ΔPo are output power, vcs, Δvc are terminal
voltage, ios, Δio are load current and the subscript “s” issteady term, the symbol “Δ” means ripple components.
The product of the ripple components Δvc and Δio can beneglected due to small value. Therefore, the steady terms
and ripple components of the load current are given by (5)and (6)

ios =

Pos1=
vcs
vcs Ro(5)

ΔPo − Δvc ics(6)vcs
where Ro is the load resistance equivalent to the constant
power load.
In (6), ΔPo equals zero when the output power iscontrolled to constant. Then, the input current ripple isobtained as (7)iΔio = − os Δvc = − K 0 Δvc(7)vcs
where K0 is a feedback gain defined by ios / vsc.
Fig. 5 shows a block diagram of the input filter with theoutput power is controlled constantly. H1(s) and H2(s)represent as a filter to divide the steady term and the ripple
component respectively as in fig. 5. For example, each filter
has characteristics as low pass filter and high pass filter. In
this case, a transfer function of the input filter is given by (8)1
Lx C fVc ( s )(8).
=
Vg ( s )
⎞
⎛
1 1
1
2
⎜
s +
H1 (s ) − H 2 (s )K 0 ⎟⎟ s +C f ⎜⎝ Ro
Lx C f⎠
It seems that K0, which is a feed back gain of the ripple
component, behaves a negative susceptance. The positive
feed back gain K0 becomes larger when the output current
control is high response because the high control gainstrongly suppresses the input voltage disturbance. As aresult, the system becomes more unstable.
In a matrix converter with power grid as input, the input
filter resonance is suppressed by a damping resistor
connected to the input filter reactor in parallel. However,for the generator as input, the damping resistor can not be
inserted to the synchronous reactance in parallel practically.
Therefore, the control for suppressing resonance is required
to the input side.
Δio =III. CONTROL STRATEGY
A. Input Current Stability Control
To suppress the oscillation of the input side, this paper
applies a damping control [4] [10] for the generator. In thedamping control, the distortion component of the terminal
voltage is removed by using a band pass filter (BPF). After
that, the distortion component is subtracted to the input
current command. In this case, a band pass frequency
depends on the input frequency. Therefore, the band passfrequency has to be adjusted by the output frequency of thegenerator. On the basis of these problems, this paper
achieves the stability control method using an adaptive BPF
on a rotating frame known a d-q frame.
Fig. 6 shows a block diagram of the stability control on
the d-q frame. The fundamental frequency component on
the d-q frame becomes a constant value, i.e. DC signal. In
addition, harmonics components appear as a ripple.
Therefore, harmonics components can be completely
removed by using a low pass filter (LPF) which has a longtime constant. The cut-off frequency of the LPF is set to a

Fig. 6. Block diagram of the proposed input current stability control.

Fig. 7. Block diagram of the transfer function of the input current stability
control.
jωLx Iin

Vg

Vg

Vc

jωLx Iin
Vc

Iin

Iin

(a) Lag power factor (θpfc < 0) (b) Lead power factor (θpfc > θvg)
Fig. 8. Relationship among the input current, generator terminal voltage andelectromotive force of the generator.

quick response, faster than the speed response of thegenerator. This method is acceptable for generator input
since the adaptive BPF can be constructed by LPF.
Fig. 7 shows a block diagram of the input filter
considering only having ripple factor with the proposed
stability control. The transfer function from generator
voltage to capacitor voltage is described by (9)1LΔVc ( s )
xC f
=
(9).
ΔVg ( s ) s 2 + 1 (K − K )s + 1
d0
Cf
Lx C fThen the damping factor ζ is calculated by (10)1 Lx
ζ =(K d − K 0 )
2 Cf

(10).

The damping factor ζ can be increased by the feedback
gain Kd. Then the feedback gain Kd is given by (11)Cf
(11).
Kd = K0 + 2
ζLx
As a result, the stability of the system increases.

P = V g iq

200

Generator terminal voltage vc

150
vc,vout[V]B. Input Power Factor Control
Fig.8 shows two phasor vector diagrams when the input
power factor is controlled at a lag and lead phase,respectively. It should be noted that the input power factor
is decided by the θpfc as shown in Fig. 6. The input terminal
voltage Vc that is smaller than the induced e.m.f. Vg isobtained in the case of lag power factor as shown in Fig.8(a). On the other hand, the input terminal voltage becomeslarger than induced e.m.f. in the case of lead power factor as
shown in Fig.8 (b).
In these cases, an apparent power S and an active power P
are obtained by (12) and (13)S = vc (id + jiq )(12)Output voltage vout

100
50

-30

-20

-10

0

10

20

30

40

50

pfc[deg]

Fig. 9. Input power factor between generator terminal voltage and output
voltage.

(13).

From (12) and (13), the input power factor cosθvg is
calculated as shown in (14)iq
P Vg
cos θ vg = =(14).
S vc id2 + iq2

Next, the relation among the input terminal voltage,
induced e.m.f. of the generator and the output voltage isconsidered.
A maximum voltage transfer ratio of the matrix converter
is limited to 0.866 of the input terminal voltage. In addition,
the voltage transfer ratio depends on the input power factor
as shown (1). As a result, the maximum voltage Vout isconstrained as (15)
Vout = λVin cos θ (0 ≤ λ ≤ 1)
(15)
where λ is a modulation index and cosθ is the input power
factor.
From (14) and (15), the output voltage Vout is given by (16)iq
(16).
Vout = λvc cos θ vg = λVg
2
id + iq2
In (16), id is reactive current. In order to maximize
generator efficiency, the input current is controlled as id = 0.
Therefore, the maximum output voltage Voutmax is obtainedby (17)
iq
= λVg
(17).
Vout max = λVg
iq2
Fig.9 shows a simulation result among the phase angleθpfc, the generator terminal voltage and the output voltage.
The maximum value of the output voltage is obtained at θpfc
= 26 deg. In contrast, the output voltage Vout becomes
smaller than the output voltage command in the case of lag
power factor (θpfc < 0 deg) and lead power factor (θpfc > 20deg). In addition, the generator terminal voltage increases
in the case of lead power factor (θpfc > 0 deg). As a result,the operation of the vector diagram in Fig. 8 is confirmed.
Therefore, when a generator is used as the power supply, theinput current phase should be controlled that the phase of
the input current is agreed with the induced e.m.f. in order toobtain the maximum voltage transfer ratio and highgenerator efficiency.

Fig. 10. System control block diagram using the proposed output voltage
compensation.

C. Output Voltage Compensation Control
The terminal voltage of the matrix converter is changedby the generator speed and current because the induced
e.m.f of the generator depends on the generator speed andthe voltage drop is caused by synchronous reactance.
Therefore, in order to obtain the constant output voltage, theoutput voltage amplitude compensation is required.
The output voltage can not compensate by only the
amplitude of the terminal voltage because the output voltageof the matrix converter depends on the input voltage andinput power factor as described in (15). Generally, the input
power factor of the matrix converter is always controlled tounity. Then, the output voltage is compensated according to
the amplitude of the terminal voltage of the matrix converter.
In case for the generator as input, the input power factor ischanged by the load condition of the generator.
This paper proposes the output voltage compensation
method using the frequency of the induced e.m.f. From (17),when a reactive input current id is controlled to zero, theoutput voltage of the matrix converter is proportional to the
induced e.m.f. Therefore, compensation value is inversed
proportional to the induced e.m.f. The induced e.m.f. can bedetected from the generator speed. It should be noted thatthe generator speed is detected using a pole position sensor
or estimated by the generator current and the input voltage.
Therefore a compensated output voltage command vout** isobtained by (18)
f
**
*
vout
= rate vout
(18)
f in

where fin is frequency of the generator terminal voltage, frate
is rated frequency of the generator and vout* is the output
voltage command.
The compensated output voltage command increases by
the decrease of the generator speed.
Fig. 10 shows a system block diagram using the input
stability control and proposed output voltage compensation
method. Firstly, the pole position of the generator is
detected by the pole position sensor. Secondly, the induced
e.m.f. phase and generator frequency are calculated bydigital Phase Locked Loop (PLL). The induced e.m.f. phaseis used for not only the output voltage compensation but
also the input power factor control.
The maximum value of the output voltage is limited to
0.866 of the induce e.m.f. even if the proposedcompensation method is applied. When the output voltage
command is over the 0.866, the harmonic distortion occurs
on the output voltage.

Fig. 13 shows the experimental result of a generator –motor system. A general purpose induction motor of 1.5kW
is used with V/f control. As mentioned in the experimental
result in Fig. 11 (b), clean sinusoidal input current
waveforms are obtained and total harmonic distortion (THD)
for the input current is 3.7%. Moreover, the output current
THD is 1.7%. The proposed stability control is also
Terminal voltage (phase) [250V/div]0
Input current [10A/div]0

Output voltage (line-to-line) [250V/div]
00

IV. EXPERIMENTAL RESULTS
Fig. 11 shows the experimental results of a generator
supply with a R-L load. The experimental parameters ofTable 1 and 2 were used. Fig. 11 (a) shows waveforms of
the input and output voltage and current with theconventional control method. Fig. 11 (b) shows waveforms
using the proposed stability control method.
In the case of Fig. 11 (a), as mentioned in the simulation
result in Fig. 3, large oscillation occurs in the input terminal
voltage and input current. In addition, output waveforms
have a distortion due to the resonance of the input side.
As for Fig. 11 (b), the oscillation of the input side isdrastically suppressed in comparison with Fig. 11 (a).
Moreover, the distortion of the output side is eliminated.
These results lead to the conclusion that the proposed
stability control is extremely effective for a generator power
supply.
Fig. 12 shows the experimental results that relation
among the phase angle θpfc, the generator terminal voltage
and the output voltage. As mentioned in the simulation
result in Fig. 9, the output voltage Vout becomes smaller thanthe output voltage command in the case of lag power factor(θpfc = 0 deg). The maximum value of the output voltage is
obtained at θpfc = 26 deg. On the other hand, the output
voltage decreases at θpfc > 26 deg. However, the generator
terminal voltage still rises. Therefore, when a generator isused as the power supply, the input current phase should becontrolled at the same phase as the induced e.m.f. in order toobtain the maximum voltage transfer ratio.

Output current [10A/div]

10[ms/div](a) Conventional control.
Terminal voltage (phase) [250V/div]0
Input current [10A/div]0

Output voltage (line-to-line) [250V/div]00

Output current [10A/div]

10[ms/div](b) Proposed input current stability control.
Fig. 11. Experimental waveforms with RL load.

Fig. 12. Experimental waveforms with RL load.
Terminal voltage (phase) [250V/div]0
Input current [10A/div]

Table 1. Generator parameters.
0

Output voltage (line-to-line) [250V/div]

Table 2. Simulation and experimental conditions.

00

Output current [10A/div]

10[ms/div]

Fig. 13. Experimental waveforms with an induction motor load.

applicable to the motor load condition.
Fig. 14 shows accelerating and decelerating results of theinduction motor. The induction motor speed is changed
from 150 r/min to 1500 r/min at 0.3 sec. After that, theinduction motor decelerated from 1500 r/min to 150 r/min at
0.3 sec. In this system, the generator terminal voltage may
increase instantly by the acceleration and deceleration. In
Fig. 14, the input current rises by the acceleration anddeceleration. However, the generator terminal voltage does
not change instantly by the acceleration and deceleration.
Thus, the input current distortion does not increase as well.
Moreover, the output voltage and current can be controlled
to stable condition. These results lead to the conclusion thatthe proposed stability control is effective for accelerationand deceleration of the induction motor.
Fig. 15 shows accelerating and decelerating result of thegenerator with an R-L load. The generator speed decreased
from 1800 r/min to 1350 r/min at 0.5sec, after that,
increased to 1800 r/min at 0.5sec. The modulation index of
the matrix converter is 0.4. Fig. 15 (a) and (b) show
waveforms of the input/output voltage and input/output
current where it without or with output voltage
compensation, respectively. If without compensation, theterminal and output voltage decreased while the generator
speed decreased. As a result, the output side becomes
unstable when the generator speed decreases greatly. On theother hand, with compensation method, the output voltage
can be kept constant even if the terminal voltage decreases.
In addition, both the output voltage and current and the
input current are in stable. The proposed output voltage
compensation method does not affect the input side.

0
Terminal voltage (phase) [250V/div]0
Input current [5A/div]0
Output voltage (line-to-line) [250V/div]0
Output current [10A/div]
10[ms/div]

Fig. 14. Accelerating and decelerating result of the induction motor.
0
Terminal voltage (phase) [250V/div]0
Input current [5A/div]0010[ms/div](a) without output voltage compensation.
0
Terminal voltage (phase) [250V/div]0

V. CONCLUSION
This paper proposes a model and analysis based on the
behavior of the matrix converter with a generator as input.
The feature of the proposed methods is to clarify the cause
of instability conduction by using an ideal mathematicalmodel.
In order to solve the instability, this paper proposed the
input current stability control on the rotating frame. In
addition, the relation among the input current phase, theterminal voltage, output voltage and the induced e.m.f. of
the generator was analyzed, and the optimal power factor
control is proposed.
Moreover, the output voltage
compensation using generator speed was proposed.
The experimental and simulation results leaded to theconclusion that the matrix converter was valid for thegenerator power supply using the proposed stability control
and the optimal power factor control. The proposed control
method could achieved stability at motor load 1.5kW and
the input current THD of 3.7%.
This study was supported by Industrial Technology Grant
Program in 2005 from New Energy and Industrial
Technology Development Organization (NEDO) of Japan.
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Output voltage (line-to-line) [100V/div]
Output current [10A/div]

Input current [5A/div]00

Output voltage (line-to-line) [100V/div]
Output current [10A/div]

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using proposed output voltage compensation.
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