Stability Analysis of Damping Control to Suppress
Filter Resonance in Multi-modular Matrix Converter
Hiroki Takahashi

Jun-ichi Itoh

Dept. of Energy and Environmental Science
Nagaoka University of Technology
Nagaoka Niigata, Japan
thiroki@stn.nagaokaut.ac.jp

Dept. of Energy and Environmental Science
Nagaoka University of Technology
Nagaoka Niigata, Japan
itoh@vos.nagaokaut.ac.jp

Abstract— This paper discusses a stability analysis for a
damping control to suppress the LC filter resonance in a multimodular matrix converter. The damping control is combinedwith the output current control of the system to reduce thecurrent distortion which is caused by the LC resonance. The
stability is analyzed by using the Bode-diagrams of an
integrated block diagram of the multi-modular matrix
converter. In the simulation, the damping control ensures thegain margin of 4.5 dB, and the system obtains stability.
Moreover, from the experimental result, the stable operation of
the multi-modular matrix converter is confirmed and theprimary current THD of the transformer is 5.0 %. In addition,
the transient response characteristics are evaluated, which isbased on the overshoot value of the output current with respect
to the gain margin in experiments.

I.

INTRODUCTION

Medium voltage applications such as the wind turbines
have received significant focus recently. In order to achieve
high efficiency and light weight of the system, the multimodular matrix converter which is a matrix converter
topology for medium voltage applications is investigated [1][4]. The multi-modular matrix converter consists of amultiple winding transformer and several modules of a threephase to single-phase matrix converter with small filter
capacitors. Therefore, the multi-modular matrix converter
promises to achieve higher efficiency, smaller size andlonger life-time compared to the multi-modular voltage
source converter-inverter system.
However, it has been found that the multi-modular matrix
converter has a LC resonance problem which is caused by
the leakage inductances of the transformer and the capacitors
in the input filters of the modules. This filter resonance leads
to deteriorate the input current quality, and cannot be
suppressed by damping resistors practically.
In order to overcome this resonance problem in the multimodular matrix converter, a damping control combined withthe output current control, which the output current control (apart of a field oriented control) is required for an adjustablespeed drive system of a motor, has been discussed and its
validity has been confirmed [5]. The damping control
combined with the output current control is more costeffective compared with the conventional damping controls
which require voltage sensors at the input stage [6]-[8].
Previous studies regarding the conventional damping
controls for general matrix converters describe the design
equation for the control parameters [6]-[7]. However, thedesign equation cannot be expanded into the proposed
damping control. In contrast, a damping control combinedwith the output current control for a general matrix converter
topology has been discussed [9]. This control strategy
achieves the resonance suppression and also is able tocontrol the output current at the same time. However, in themulti-modular matrix converter, the stability and transient
response of not only a module but the whole system isimportant and therefore this paper discusses the design
method of the damping control parameters which is subject
to the damping effect and the transient response of the output
current.
In this paper, a stability analysis method of the multimodular matrix converter including the damping control is
described. The proposed analysis method uses Bode-diagram
based on the integrated block diagram of the whole system
which is composed of the circuit and the control block
diagram. Thus, the damping control and the output current
control can be designed easily since the effects of theresonant suppression and the output current response areyielded from the Bode-diagrams. Firstly, the system block
diagram of the multi-modular matrix converter with thedamping control and the output current control are described.
Secondly, an integrated block model of the multi-modular
matrix converter is derived based on the concept of dutycycle space vector [10]. Thirdly, the stability of the multimodular matrix converter is analyzed with the Bodediagrams. Finally, the validity of the proposed stability
analysis is confirmed in simulation and experimental results.

Multiple windingtransformer
20 deg. leadwinding
iin
vin

iu

Matrix converter modules

3

r
s
t

U-phase modulerstV-phase modulepvuSecondaryvoltageangle
calculation

2

LPF

tan-1

3f
/
2 ab

iv

iw
p

Damping control
vw

v2Wr
v2Wt

i2d* = 1i2q* = 0 2

*

dq
/
3f

i2r
i2s*
i2t*
3

sThpf
1+sThpf 2

Idout
3f Iqout
/
2
dq

Cf

2

q2U
q2V

Ro

vv
n

n

q2W

Lo

p

r
s
t

20 deg. lag
winding

vuv

n

qout

12
Switchingsignalcalculation

vu*
vv*
vw*

dq
/
3f

2

Kd

PI control
1+sTi
Kp
sTi

2

2
2

Idout*
2 Iqout*

Output current control(ACR)

W-phase module

Figure 1. System diagram of a multi-modular matrix converter employing three modules with the damping control. Note that the configuration of U,
V-phase modules is the same as W-phase. The output voltage control block includes the damping control and an ACR.

II. MULTI-MODULAR MATRIX CONVERTER EMPLOYINGDAMPING CONTROL COMBINED WITH OUTPUT STAGE
Fig. 1 shows the system diagram of a three modules
multi-modular matrix converter This paper only considers athree-module model due to the simplicity, which in practicala nine modules multi-modular matrix converter is required
for a medium voltage application [1]-[4]. For the same
reason, a constant power load composed by an R-L load andan ACR (auto current regulator) for the output current is used,
not a motor controlled by a field oriented control. A multiple
winding transformer is located between the power source
and matrix converter modules. The transformer cancels thelow order harmonics in the primary current of thetransformer and reduces the ripples in the output voltage dueto phase shift of the secondary windings by 20 degrees. Eachmatrix converter module consists of a three-phase to singlephase matrix converter which controls output phase voltage
and secondary current of the transformer, and the small filter
capacitors.
As mentioned previously, the multi-modular matrix
converter has the LC resonance problem due to leakage
inductances of the transformer and the capacitors in the input
filter of the modules. In order to suppress the distortion, adamping control combined with the output current control,
which has a damping gain Kd and a damping HPF (high passfilter) with a time constant Thpf, has been proposed [5]. The
damping control is applied to the feedback path of the ACR
as illustrated in Fig. 1.
The filter resonance is separated into the series resonancedue to harmonics in the power source voltage and the parallel
resonance triggered by a dynamic characteristic of the multimodular matrix converter. A solution for the series resonanceis to redesign the filter parameters, which changes theresonant frequency to avoid the frequency of the voltage
source harmonics. However, it is difficult to suppress theparallel resonance with the redesign only. The parallel
resonance is excited by the output current control with highnatural frequency [11]. Therefore, the parallel resonance canbe referred as the destabilization of the output current control.
In order to stabilize the multi-modular matrix converter, thispaper reveals Bode-diagrams derived from an integrated
block diagram which is composed of the circuit and thecontrol diagram, and the damping control is designed basedon the desired gain margin. However, as shown in Fig. 1, thedamping control interferes with the response of the output
current control. Thus, it is necessary to evaluate the transient
response of the output current with respect to the designed
stability.
III.

ANALYSIS MODEL

A. Derivation of the Analysis Model
In order to derive the analysis model of the multimodular matrix converter, the output voltage equation andthe input current equation are required. The output voltage
and the input current equations of a general three-phase to
three-phase matrix converter based on the duty-cycle space
vector have been presented [10]. However, these equations
cannot be applied to the multi-modular matrix converter
directly because the multi-modular matrix converter employs
three-phase to single-phase matrix converter module.
Therefore, an equivalent three-phase to three-phase matrix
converter model which provides the same output voltage andthe input current as the multi-modular matrix converter isconsidered.
Fig. 2 shows the equivalent model based on a three-phaseto three-phase matrix converter corresponding to the multimodular matrix converter. This equivalent model has a three-phase LC filter, a three-phase transformer and a three-phaseto three-phase matrix converter. The filter inductor Lfp
corresponds to the leakage inductance of the multiple
winding transformer in Fig. 1. In contrast, the filter capacitor
is converted to the primary side of the three-phasetransformer and the equivalent capacitance Cfp is presentedby (1).
2

V 
C fp  n 2  C f
 V1 

Lfp

1:a

Cfp
Power
sourceLC filter

Transformer

Figure 2. Equivalent three-phase to three-phase matrix converter
model corresponding to the multi-modular matrix converter in Fig. 1.

where, n is the number of the modules and V2/V1 is thevoltage transformation ratio of the multiple windingtransformer in Fig. 1. The three-phase transformer in Fig. 2
is adopted to obtain the same output voltage as the multimodular matrix converter. Then, the output voltage
amplitude of the multi-modular matrix converter Vout_mmmc is
expressed by (2).

vL

vin

1 iL
sLfp

ic

mn  V2  Vin2  V1 where, m is the modulation index and Vin is the amplitude of
the source voltage. In contrast, the output voltage amplitudeof the equivalent model Vout_emc is presented by (3).
Vout _ emc 

3
maVin
2



where, a is a turn ratio of the three-phase transformer in Fig.
2. In order that the equivalent model yields the same output
voltage as the multi-modular matrix converter, a is defined
as following.

imc

Fig. 3 shows the integrated block model of the equivalentmatrix converter circuit and the control diagram. Variables inthe bold font represent their space vectors. For example, theinput voltage vector vin is defined as following.
v in 

2 
vr  vs e
3  vt e

2
j 
3








where, vr is the input R-phase voltage, vs is the input S-phasevoltage and vt is the input T-phase voltage.
In order to connect the input and output circuits of theequivalent model, the duty-cycle space vector md and mi are
introduced and the output voltage vector vout is yielded by (6).

vc

a
vc2

iout*

PI

Non-linear equations
vout vout cal.
1
imc cal. iout
Eq. (7)
Eq. (6)
Ro+sLovout*

md cal.
Eq. (8)dq/
ab

mi cal.
Eq. (9)

qout
Kd

md
mi
ab /dqsThpf
1+sThpf

Figure 3. Integrated block model of the equivalent matrix converter
which is composed of the circuit and the control diagram. This block
diagram is based on the duty-cycle space vector.



Therefore, the analysis block model is obtained by using theequivalent model in Fig. 2 and the concept of the duty-cycle
space vector.

2
j 
3

Control 1
Vinv out n  V2  a
3  V1 

1
sCfp

a

Circuit

Vout _ mmmc Three-phaseto three-phasematrix converter33m i v c2  m d v c2
2
2



where, vc2 is the filter capacitor voltage converted to thesecondary side of the transformer. Note that a bar placed
over a variable indicates complex conjugate. On the other
hand, the input current of the equivalent matrix converter imc
is expressed by (7).
i mc 

3
3
m i i out  m d i out
2
2



where, iout is the output current vector. Hence, the input andoutput circuits of the equivalent model are connected by (6)and (7). In contrast, when the input power factor reference ofthe multi-modular matrix converter is unity and the filter
inductor voltage in steady state is much smaller than the
input voltage, md and mi are represented by follows.

md 

v in
*
v out
3Vi



mi 

v in
*
v out
3Vi



where, vout* is the output voltage reference. Therefore, theintegrated block model is derived by using (6) to (9) because
vout* is also the output of the control block diagram.
However, Fig. 3 has a problem, which the transient
responses of vout and imc indicate non-linear characteristics
because the time constants of fluctuation of vc2, md, mi andiout are close to each other. Therefore, a linear approximation
method around the operating point at a steady state is applied
to these non-linear parts. The output voltage vector vout isseparated into the steady and differential components, where
the differential component is expressed as (10).
v out 



 



3
3
m i v c2s  m is v c2  m d v c2s  m ds v c2 22

where, suffix “s” represents steady component based on its
fundamental frequency whereas “ ” means differential
component in a transient state. In a similar way, thedifferential component of imc is expressed as (11).
i mc 

3
m i i outs  m is i out   3 m d i outs  m ds i out 2
2

Fig. 4 shows the linearized block model of the equivalentmatrix converter regarding the differential components. The
products of space vectors which become non-linear
equations as shown (6) and (7) are linearized by using sumof products of the steady vector and the differential vector.
Thus, the stability and the transient characteristics of themulti-modular matrix converter are obtained by using Bodediagram derived from Fig. 4. Note that the analysis should be
implemented with DC mode in which the input and output
angles are fixed because the rotating frequency of the steadyvector does not equal to the rotating frequency of thedifferential vector.
B. Verification of the linearized model with simulation
Table 1 and Table 2 present the simulation conditions of
the circuit model as illustrated in Fig. 1 and the linearized
model as shown in Fig. 4. This section shows the result
without the ACR and the damping control in order toevaluate the verification of the linearized model. The resultswith the ACR and the damping control are described in next
chapter. Note that the inductor and capacitor parameters arenormalized based on 50 Hz.
Fig. 5 shows the indicial response of the circuit modeland the linearized model with an open-loop control in asimulation. It should be noticed that all waveforms areapplied with a 1 kHz cut-off frequency LPF (low pass filter)in order to observe these average waveform withoutswitching ripples. From Fig. 5, it is confirmed that thelinearized model waveforms correspond to the circuit model
result. Furthermore, the error between the circuit model and

1

vin

iL

ic

vL sLfp
imc
vins

Control1Viniout*

vout*
dq
/
ab

PI

vc

a

vc2
Linearized equations
vout vout cal.
1
imc cal.
Eq. (11)Eq. (10)
Ro+sLoa
Circuit

1
sCfp

iout

md cal.

md

Eq. (8)mi cal.

mi

Eq. (9)qout

ab /dqsThpf
1+sThpf

Kd

Figure 4. Linearized block model of the equivalent matrix converter
regarding the differential components. The stability of the multimodular matrix converter is obtained by analyzing this block model.
TABLE I.

CIRCUIT PARAMETERS OF THE MULTI-MODULAR
MATRIX CONVERTER.

Input line voltage

200 Vrms

Rated output voltage

300 Vrms

Rated power

3 kW

Carrier frequency

100 kHz

Input filter L (Lfp)

6.65 mH(15.7%)

Load resistance (Ro)25.2 W(84.0%)

Input filter C (Cf)

6.7 mF(8.42%)

Load inductance (Lo)

5.5 mH(5.76%)

Input voltage angle

105 deg.

Output voltage angle

5 deg.

TABLE II.

CONTROL PARAMETERS FOR OPEN-LOOP CONTROL.

Open-loop
control

0.51
0.5
-0.94
-0.98

d-axis voltage command (steady state)

0.5 p.u.

d-axis voltage command (step input)

0.01 p.u.

q-axis voltage command

0 p.u.

d-axis voltage command [p.u.]

Input R-phase filter inductor current [A]Green: Original circuit model (Fig. 1)
Red: Linearized model (Fig. 4)
Output U-V line voltage [V]

178
174

Output U-phase current [A]

4.95
4.85
0

2

7

12

17

[ms]

Figure 5. Indicial response of the circuit model and the linearized modelwith an open-loop control in a simulation. It is confirmed that thelinearized model result corresponds to the circuit model result.

the linearized model waveforms is lesser than 1% in steadystate. The input S-phase and T-phase waveforms, and theoutput V-phase and W-phase waveforms are in the samemanner. Thus, the linearized model is valid for modeling of
the multi-modular matrix converter.

Outputcurrent
control(ACR)

STABILITY ANALYSIS USING SIMULATION

Fig. 6 shows the indicial responses of the linearized
model with the designed output current control in asimulation. This simulation adds a small step of 0.01 p.u. tothe output current reference at 2 ms. Fig. 6 verifies that thefilter resonance is excited immediately after the step input of
the output current reference, and the filter inductor current
and the filter capacitor voltage contain the resonant
oscillation. Moreover, the output current diverges from its
command due to the resonance. Then, the oscillation
frequency of the output current is 352 Hz. Therefore, in orderto evince the mechanism of this destabilization, the analysisof the open-loop transfer function of the output current
control is needed.
Fig. 7 shows a Bode-diagram of the open-loop transfer
function of the output current control in the linearized blockmodel without the damping control. Fig. 7 indicates theinstability due to the LC filter. Fig. 7(a) shows the gain
characteristics and (b) shows the phase characteristics. From
Fig. 7, the gain curve without the LC filter decreases
uniformly, which the ratio is -20 dB/dec. In addition, thephase curve is constant at -90 deg.. However, the gain curve
taking the LC filter into account has a resonance point at 535
Hz. Then, the designed gain-crossover frequency is shiftedfrom 650 Hz to 874 Hz and the phase is lagged drastically by
the LC filter characteristic. As a result, the phase margin andthe gain margin of the linearized model are obtained by -234
deg. and -4.56 dB, respectively. In this way, the multimodular matrix converter can be unstable due to the LC
resonance.
B. Parameter Design of Damping Control
Table 4 shows the damping control parameters. Thissection mentions the design method of the damping control.
Fig. 8 shows a Bode-diagram of the open-loop transfer
function of the output current control in the linearized model
as shown in Fig. 4. The transfer function of the damping
control Hdamp(s) is presented as following.
H damp s  

1  sThpf 1  K d 
1  sThpf

1  sT2
1  sT1



PARAMETERS FOR OUTPUT CURRENT CONTROL.
d-axis current command (steady state)0.75 p.u.

d-axis current command (step input)

0.01 p.u.

q-axis current command

0 p.u.

ACR natural frequency

650 HzInput R-phase filter inductor current [A]
-1.5
-2.5
-30
-40
-50
7

Input R-phase filter capacitor voltage [V]Output U-phase current [A]60.85

Output d-axis current [p.u.]
Green: Command
Red: Response

0.75
Output q-axis current [p.u.]
0.5
Green: Command
Red: Response
0
-0.5
2
4
6
8
0

10

[ms]

Figure 6. Indicial response of the linearized model with the output
current control only in a simulation. This result shows that the filter
resonance is excited and the system becomes unstable.


Gain [dB]

A. Mechanism of Destabilization
Table 3 shows the parameters of the output current
control of the linearized model. This section explains themechanism of the destabilization due to the output current
control and the LC filter. Thus, the damping control is notapplied to the linearized model. In addition, the output
current control is designed to yield the natural frequency of
650 Hz without taking the LC filter into account.

With the LC filter

Gain margin: -4.56 dB
Without the LC filter





Frequency [Hz]







(a) Gain characteristic.
Phase [deg.]IV.

TABLE III.

Without the LC filter







Phase margin: -234 deg.
With the LC filterFrequency [Hz]





(b) Phase characteristic.
Figure 7. Bode-diagram of the open-loop transfer function of the output
current control in the linearized block model without the damping
control. The LC filter resonance makes the system unstable.

g a  g m   20 log10 1  K d   0

TABLE IV.
Damping
control

Fig. 9 shows the indicial response of the linearized modelwith the output current control and the designed damping
control in a simulation. The resonant oscillation in the output
current is suppressed by the damping control. As a result, allwaveforms converge toward their steady points. Therefore,
the proposed damping control achieves the resonant
suppression and the output current control at the same time.
However, the overshoot of the output current response is
obtained by 261% and the cause of this overshoot is thedamping control.
C. Transient Response of Output Current Control
Fig. 10 shows a Bode-diagram of the closed-loop transfer
function of the output current control in the linearized blockmodel in order to evaluate the overshoot of the output current
in a transient response. When the damping control is not
introduced, the gain curve is less than 3 dB under 250 Hz
and has a resonance point at 352 Hz. On the other hand, thedamping control increases the gain curve in the lower
frequency bandwidth and generates larger overshoot of theoutput current than the designed value. In addition, the gainin the lower frequency bandwidth increases with respect tothe designed gain margin. Thus, a tradeoff relationshipDamping HPF time constant (Thpf)

7.0 msgaWith the output
damping control


gm







Frequency [Hz]

fcp

k

fcp

k

(a) Gain characteristic.
Without the output damping control

Phase [deg.]

The gain and the phase curves with the damping control in
Fig. 8 are the result when the damping control is designed inorder to obtain the gain margin of 5 dB. Note that thedesigned Kd and Thpf are shown in Table 4. From Fig. 8, thedesigned damping control decreases the gain at fcp and yields
the actual gain margin of 4.5 dB. As a result, the systemachieves a stable operation. The error between the desired
and the actual gain margins is caused by the break point
approximation of a Bode-diagram of (14).

0.67 p.u.

Without the output damping control





Damping gain (Kd)where, 0 < Kd < 1, ga > 0, gm > 0. In contrast, in order toensure gm, a phase lag at fcp should be zero even if thedamping control is applied. Thus, a time constant of the HPFof the damping control Thpf is designed by using (14), which
is based on a break point approximation of a Bode-diagram.

5
Thpf 
2 1  K d  f cp

PARAMETERS FOR DAMPING CONTROL.

Gain [dB]

From (12), the damping control is equivalent to a phase lagcompensator because T2 is smaller than T1. Hence, in order toensure the desired gain margin gm, the gain at the phasecrossover frequency should be decreased by the designed
damping control. Note that the damping control parameters
are designed based on a gain margin not a phase margin
because the design based on a phase margin cannot stabilize
the system for some conditions. First, the phase-crossover
frequency fcp is obtained from the phase curve without thedamping control. Next, the gain ga at fcp is measured from thegain curve without the damping control and the damping
gain Kd is calculated by the following equation.

With the output damping control









Frequency [Hz](b) Phase characteristic.
Figure 8. Bode-diagram of the open-loop transfer function of the output
current control in the linearized block model in Fig. 4. fcp is the phasecrossover frequency, and gm is the desired gain margin. The damping
control makes the system stable with the gain margin of 4.5 dB.
Input R-phase filter inductor current [A]
-1.5
-2.5
-30
-40
-50
7

Input R-phase filter capacitor voltage [V]Output U-phase current [A]60.85

Output d-axis current [p.u.]
Green: Command
Red: Response

0.75
Output q-axis current [p.u.]
0.5
Green: Command
Red: Response
0
-0.5
2
4
6
8
0

10

[ms]

Figure 9. Indicial response of the linearized model with the output
current control and the damping control in a simulation. This resultshows that the damping control stabilize the system. However, thedamping control generates the overshoot of the d-axis current of 261%.



gm = 10 dB
gm = 7.5 dB
gm = 5 dB

Gain [dB]Output current 10 A/div

Output line voltage 500 V/div
10 ms/div

gm = 2.5 dB

0



0

Without the output damping control





Frequency [Hz]k0(a) Gain characteristic.
(a) Without damping controls which results in the primary current THD
of 47.4% and the output current THD of 14.4%.


Phase [deg.]Primary currentof transformer 10 A/div

0





Primary phase voltageof transformer 250 V/div

gm = 2.5 dB

gm = 5 dB
gm = 7.5 dB

Primary currentof transformer 10 A/div

Output current 10 A/div

Output line voltage 500 V/div
10 ms/div0

gm = 10 dB




Primary phase voltageof transformer 250 V/div

Without the output damping control





Frequency [Hz]

0

k
0

(b) Phase characteristic.
Figure 10. Bode-diagram of the closed-loop transfer function of theoutput current control in the linearized block model as drawn in Fig. 4.
gm is the desired gain margin. The damping control increases the gain in
the lower frequency bandwidth and generates larger overshoot of theoutput current than the designed value.

between the stability and the transient response isdemonstrated.
V.

EXPERIMENTAL RESULTS

This chapter evaluates the damping effect and thetransient response of the multi-modular matrix converter
prototype with the designed damping control as shown in
Fig.1. The prototype parameters are almost consistent withthe parameters shown from Table 1 to Table 4. However, theinput voltage is 50 Hz while the output voltage is 30 Hz. In
addition, the output d-axis current reference is separated into
the steady component of 0.75 p.u. and the step component of
0.05 p.u.. Furthermore, the carrier frequency is set to 10 kHz.
The modulation of the matrix converter module employs themethod as introduced in [12].
Fig. 11 shows the input and output waveforms of themulti-modular matrix converter obtained by the experiments.
Fig. 11 (a) shows the result without damping controls and (b)shows the result with the damping control which is designed
from the gain margin of 5 dB. In (a), the filter resonance is
excited, and the primary current of the transformer and theoutput current have resonant distortions. It is noted that theexperimental result in (a) shows the stability limit although
the simulation result in Fig. 6 becomes unstable. This isbecause the loss of the multiple winding transformer and themulti-modular matrix converter behaves as a damping0(b) With the proposed damping control which results in the primary
current THD of 5.0% and the output current THD of 2.7%.
Figure 11. Input and output waveforms of the multi-modular matrix
converter as illustrated in Fig. 1 in experiments.

resistor. The primary current THD (total harmonic distortion)of the transformer is 47.4% and the output current THD is
14.4% from Fig. 11 (a). In contrast, the filter resonance issuppressed and the system is stabilized by the damping
control in Fig. 11 (b). As a result, the primary current THDof the transformer is 5.0% and the output current THD is
2.7%. Thus, the damping control mitigates the resonancedistortions in the primary current by 89%. Furthermore, theoutput line voltage shows a 5-level waveform without thevoltage fluctuation due to the filter resonance.
Fig. 12 shows the dq-axis output current in a transient
response with the damping control which is designed fromthe gain margin of 5 dB. Note that this experiment fixes theoutput phase to 5 deg. in common with the simulation. The
output current converges toward the reference because thedamping control stabilizes the system. However, the d-axis
current has an overshoot of 256% due to the damping control.
Fig. 13 shows the output current overshoot characteristics
with respect to the designed gain margin. The experimental
and simulation results of the output d-axis current overshootincrease together with the designed gain margin. This isbecause the damping control increases the gain in the lower

bandwidth as shown in Fig. 10 though the gain amplitude of
the unit step is inversely proportional to the frequency.
Therefore, the trade-off relationship between the stability andthe transient response regarding the damping control isconfirmed. The maximum error between the experimental
and the simulation results is 11.2% and this error is due tothe control and detection delays. In the future, the fullconsideration about this error will be reported.

ACKNOWLEDGMENT
A part of this study was supported by Industrial
Technology Grant Program in 2009 from New Energy andIndustrial Technology Development Organization (NEDO)
of Japan.
REFERENCES
[1]

[2]

[3]

[4]

[5]

q-axis current command
0.133 p.u./div (Red)CONCLUSION

This paper discusses the stability analysis for a damping
control combined with the output current control to suppressa LC filter resonance in a multi-modular matrix converter.
Past works related to the design method of the conventional
damping control applied to the input current control cannot
be expanded into the proposed damping control. The analysis
method uses Bode-diagrams based on the integrated block
diagram of the whole system which is composed of thecircuit and the control block diagram. Thus, the damping
control can be designed easily since the effects of theresonant suppression and the output current response areyielded from the Bode-diagrams. From the simulation
analyses, the designed damping control ensures the gain
margin of 4.5 dB against a requirement specification of 5 dB,
and a stable operation is achieved. From the experimental
result, the multi-modular matrix converter with the designed
damping parameters operates in stable condition, and theprimary current of the transformer THD is suppressed by
89%. Therefore, the validity of the proposed stability
analysis method is confirmed. In addition, the transient
response characteristics of the output current control which isaffected by the damping control are evaluated. As a result,
the tradeoff relationship between the stability and thetransient performance is illustrated clearly. In the future, theoptimal design of the damping control will be considered.

J. Kang, E. Yamamoto, M. Ikeda, E. Watanabe: "Medium-Voltage
Matrix Converter Design Using Cascaded Single-Phase Power Cell
Modules", IEEE Trans. Ind. Electron., Vol. 58, No. 11, pp. 50075013 (2011)
E. Yamamoto, H. Hara, T. Uchino, M. Kawaji, T. J. Kume, J. K.
Kang, H. P. Krug: "Development of MCs and its Applications in
Industry", IEEE Industrial Electronics Magazine, Vol. 5, No. 1, pp. 412 (2011)
J. Wang, B. Wu, D. Xu, N. R. Zargari: "Multimodular Matrix
Converters With Sinusoidal Input and Output Waveforms", IEEE
Trans. Ind. Electron., Vol. 59, No. 1, pp. 17-26 (2012)
J. Wang, B. Wu, D. Xu, N. R. Zargari: "Indirect Space-Vector-Based
Modulation Techniques for High-Power Multimodular Matrix
Converters", IEEE Trans. Ind. Electron., Vol. 60, No. 8, pp. 30603071 (2013)
H. Takahashi, J. Itoh: “Damping Control Combined to Output Stage
for a Multi-Modular Matrix Converter”, Proc. 28th IEEE APEC,(2013)

d-axis current
0.133 p.u./div

0.75

q-axis current
0.133 p.u./div (Green)0

2 ms/div

Figure 12. Output dq-axis current response with the damping control
designed to obtain the gain margin of 5 dB. The output current
converges toward the reference. However, the d-axis current overshootof 256% is generated by the damping control.
d-axis current overshoot [%]VI.

d-axis current command
0.133 p.u./div




Experimental point
Approximate curve (Experiment)

Simulation point
Approximate curve (Simulation)





.


.
Desired gain margin [dB]



.

Figure 13. Output current overshoot characteristic with respect to thedesigned gain margin. This result shows the trade-off relation between
the stability and the transient response regarding the damping control.
[6]

T. Nunokawa, T. Takeshita: "Resonance Suppression Control in
Complex Frame for Three-Phase to Three-Phase Matrix Converters",
Proc. EPE2007, (2007)[7] J. Haruna, J. Itoh: "Modeling Design for a Matrix Converter with a
Generator as Input", Proc. 11th IEEE Workshop on Control andModeling for Power Electronics, COM352 (2008)[8] M. Rivera, C. Rojas, J. Rodriguez, P. W. Wheeler, B. Wu, J.
Espinoza: "Predictive Current Control With Input Filter Resonance
Mitigation for a Direct Matrix Converter", IEEE Trans. PowerElectron., Vol. 26, No. 10, pp. 2794-2803 (2011)[9] J. Haruna, J. Itoh: "Control Strategy for a Matrix Converter with a
Generator and a Motor", Proc. 26th IEEE APEC, pp. 1782-1789(2011)[10] D. Casadei, G. Serra, A. Tani, L. Zarri: "Matrix Converter
Modulation Strategies: Anew General Approach Based on SpaceVector Representation of the Switch State", IEEE Trans. Ind.
Electron., Vol. 49, No. 2, pp. 370-381 (2002)[11] Q.Guan, P. Yang, X. Wang, X.Zhang: "Stability Analysis for Matrix
Converter with Constant Power Loads and LC Input Filter", Proc. 7th
IPEMC 2012, pp. 900-904 (2012)[12] J. Itoh, I. Sato, A. Odaka, H. Ohguchi, H. Kodachi, N. Eguchi: "ANovel Approach to Practical Matrix Converter Motor Drive System
With Reverse Blocking IGBT", IEEE Trans. Power Electron., Vol.
20, No. 6, pp. 1356-1363 (2005)