電気学会論文誌●（●●●●●●●部門誌）
IEEJ Transactions on ●●●●●●●●●●●●●●●
Vol.●● No.● pp.●-●
DOI: ●.●●/ieejeiss.●●.●

Paper
Damping Control of Filter Resonance Focusing on Output Stage forMulti-Modular Matrix Converter
Hiroki Takahashi＊, Student member,

Jun-ichi Itoh＊a), Member(Manuscript received Jan. 00, 20XX, revised May 00, 20XX)
This paper discusses a damping control that is applied to multi-modular matrix converters in order to suppress the oscillation
due to the LC resonance of the input filter. The multi-modular matrix converter connects a multiple winding transformer andseveral modules. The module consists of a three-phase to single-phase matrix converter and filter capacitors. It is found that themulti-modular matrix converter has a resonance problem between the leakage inductance of the transformer and the filter
capacitors in the modules. In this paper, a damping control combined with the output current control of the multi-modular matrix
converter is proposed to suppress the resonance and to achieve better input current quality. The proposed damping control candivert the output current sensors for an ACR (auto current regulator), which is generally used in the adjustable speed drive systemof a motor. Therefore, the proposed damping control does not require additional sensors, which is an advantage over theconventional damping control applied to the input stage of the multi-modular matrix converter. From the experimental results, theproposed damping control improves the input current THD (total harmonic distortion) by 75.5%. In addition, the proposed
damping control obtains a higher damping performance than the conventional damping control.
Keywords : multi-modular matrix converter, filter resonance, damping control, high power conversion

1.

operation and to assist in the commutation similarly to generalmatrix converters. Besides, the multi-modular matrix converter
uses the filter capacitors in the matrix converter modules only
because the leakage inductance of the transformer can be treatedas the filter inductors. Nevertheless, if the LC resonance occurs
between the leakage inductance and the input filter capacitors, theresonance results in substantial distortions in the input current. A
general solution to the problem of resonance in general matrix
converters is to connect damping resistors in parallel with the
input filter inductors [11]. However, in the multi-modular matrix
converter, the damping resistors cannot be connected in parallelwith the leakage inductance of the transformer, practically.
One of other solution to suppress the resonance is to connect thedamping resistors in series with the input filter capacitors.
However, this method results in more losses in the damping
resistors due to the harmonic current caused by switching
operation in comparison with the former method [12]. Hence, thesolution using a special control to suppress the resonance isnecessary.
In the controls to suppress the filter resonance, active damping
controls for current source converters and general matrix
converters have been proposed [13]-[18]. These methods suppress
the resonance with virtual damping resistors in parallel with filter
capacitors based on the input current control of the converters. In
this paper, this active damping control based on the input current
control is defined as the conventional damping control. If theconventional damping control is applied to the multi-modular
matrix converter, the additional voltage sensors for theconventional damping control are required. Furthermore, thenumber of the voltage sensor depends on the number of modules.
This increase of the voltage sensors is not cost effective. In
addition, if the multi-modular matrix converter also employs theoutput current control with a feedback (ACR: auto current

Introduction

Recently, matrix converters have attracted a lot of attentions
among the researchers because this converter has no DC energy
buffer such as bulky electrolytic capacitors [1]-[7]. Matrix
converters promise to achieve higher efficiency, smaller size andlonger life-time compared to the conventional BTB (back to back)system which consists of a PWM (pulse width modulation)
rectifier and a PWM inverter. In addition, matrix converters
achieve bidirectional power flow, unity input power factor, andsinusoidal input and output waveforms.
In order to utilize these advantages of matrix converters forhigh power applications, such as wind turbines, pumps andblowers, the multi-modular matrix converters are investigated[8]-[10]. The multi-modular matrix converter consists of several
modules of three-phase to single-phase matrix converter and amultiple winding transformer. The multiple winding transformer isplaced between a power grid and matrix converter modules, andoutput terminals of each module are connected in series or parallelin order to provide high voltage or high power. The advantages ofthis modular structure with the multiple winding transformer are
following: 1) the circuit structure, especially placement of devices
in the modules, is simple; 2) special modulation schemes toprevent the interference between the connected modules are notrequired because the transformer isolates each module.
Consequently, high power conversion systems employing a matrix
converter topology can be designed and provided easily with theconventional modulation schemes.
The multi-modular matrix converter requires LC filters in the
input side to eliminate the harmonic current due to the switchinga) Correspondence to: Jun-ichi Itoh. E-mail: itoh@vos.nagaokaut.ac.jp＊
Nagaoka University of Technology
1603-1, Kamitomioka-cho, Nagaoka, Niigata, Japan 940-2188© 200● The Institute of Electrical Engineers of Japan.

1Damping Control for Multi-modular Matrix Converter (Hiroki Takahashi et al.)regulator) for the field oriented control, the converter has several
feedback loops in both of the input and output stages. This meansthat the conventional damping control interferes with the output
current control through the converter. Hence, the design method of
the conventional damping control and the output current control
may be complicated.
On the other hand, the damping control combined with theoutput current control of a general matrix converter was suggested[19]. This method can achieve not only the resonant suppression
but also the output current control at a same time. And the input
current is fed by an open-loop control. However, the validity ofthe damping control combined with the output stage for themulti-modular matrix converter has not been clarified.
In this paper, the damping control that is combined with theoutput current control of the multi-modular matrix converter is
proposed. The proposed damping control suppresses the filter
resonance and also diverts the output current sensors for the ACR,
which is generally used in the field oriented control, instead ofusing voltage sensors on the input stage. Firstly, the mechanism of
the filter resonance is clarified in this paper. Secondly, theproposed damping control is discussed with the control block
diagrams and its features. Thirdly, the validity of the proposed
damping control for the multi-modular matrix converter isconfirmed by simulation and experimental results. Finally, thecompared results of damping effect between the proposed
damping control and the conventional damping control isevaluated, experimentally.

2.

r Module ps
3 t U1 n: 20 deg. leadwindingr Module ps
3 t U2 n: 20 deg. lag
windingr Module ps
3 t U3 nr Module ps
3 t V1 n

M

r Module ps
3 t V2 n

3r Module ps
3 t V3 nr Module ps
3 t W1 n
r Module ps
3 t W2 nr Module ps
3 t W3 nMultiplewindingtransformerPowersourceMatrix converter
modules

Load

Fig. 1. System configuration of a series multi-modular
matrix converter with nine modules.

Multi-modular Matrix Converter

2.1 System Configuration
Fig. 1 shows the system
configuration of a series multi-modular matrix converter with ninemodules as shown in [8]-[10]. The series topology outputs 7-levelhigh voltage which results in low harmonic components. The
multiple winding transformer is located between the power source
and the matrix converter modules. The transformer cancels thelow order harmonics in the primary current of the transformer andreduces the ripples in the output voltage owing to phase shift ofthe secondary windings by 20 degrees.
Fig. 2 shows a simple model of a multi-modular matrix
converter with three modules for analysis and consideration. In
Fig. 2, each module consists of a three-phase to single-phasematrix converter and filter capacitors. The bidirectional switch inthe modules uses 2 series-connected IGBT-diode. The filter
capacitor Cf is connected closely with the bidirectional switches tomitigate surge voltage to the switches. An output terminal p isconnected to load and another terminal n forms the neutral pointwith other modules.
2.2 Mechanism of Filter ResonanceFig. 3 shows the
single-phase equivalent circuit of the input filter of themulti-modular matrix converter as illustrated in Fig. 2. In Fig. 3, Llis the leakage inductance of the transformer and Cfp is theequivalent filter capacitor. Ll and Cfp, are converted to the primaryside of the transformer. Thus, the equivalent capacitance Cfp isrepresented by the following equation.

20 deg. leadwindingr Module p
s
U n
3 t

3

r Module p
s
V n
3 t

20 deg. lag
windingr Module ps
3 t W nM(a) System configuration.

p
r
s
t
Cf

n

Filter capacitors

Three-phase to single-phasematrix converter(b) Configuration of a matrix converter module.

Fig. 2. Simple model of a multi-modular matrix converter withthree modules.
iL
ic

Ll
vp

vc

Cfp

imc

2

V 
C fp  n 2  C f ............................................................. (1) V1 

Fig. 3. Single-phase equivalent circuit of the input filterin the multi-modular matrix converter.

where, n is the number of the modules and V2/V1 is the voltage
transformation ratio of the transformer. In addition, vp is thesource voltage and imc is the current controlled by the modules.
2

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Damping Control for Multi-modular Matrix Converter (Hiroki Takahashi et al.)Note that the amplitude of imc is decided by the load power. The
input power factor is assumed to unity. The exciting inductance ofthe transformer is assumed to be much larger than the leakage
inductance for the simplicity.
The secondary current of the transformer includes the harmoniccomponents caused by the power pulsation due to the single-phase
output configuration of the module. The frequencies of theseharmonics are expressed by the following equations.

Ic(s)

1
sCfp

Vc(s)

1
H1(s)
R
Imcs(s) mcs Vcs(s)Imc(s)Kenrg

Vc(s)

H2(s)Fig. 4. Block diagram of the single-phase equivalentcircuit of the input filter at a constant load.

f 2d 2  f in  2 f out ............................................................ (3)where, f2d1 and f2d2 are the frequencies of the harmonics in thesecondary current, fin is the frequency of the power source and fout
is the output frequency. However, the primary current of thetransformer iL is a sinusoidal waveform because the harmoniccomponents expressed from (2) to (3) are canceled by the multiple
winding transformer.
The filter resonance occurs because of an equivalent negative
resistor which is virtually generated at a constant power load. The
mechanism to generate the equivalent negative resistor will be
described as follows.
First of all, the filter capacitor voltage vc, the input current ofmulti-modular matrix converter imc and the output power pmc aredefined as follows.

imcs 

pmcs
1

vcs .................................................... (12)
vcs
Rmcs

imc 

pmc  vc imcs................................................... (13)vcs

For a constant load, (14) is derived because pmc becomes zero.

iimc   mcs vc   K enrg vc .................................. (14)vcs
where, Kenrg is an equivalent negative resistor gain. Equation (14)indicates the fluctuation of the input current of the multi-modular
matrix converter against the fluctuation of the filter capacitor
voltage. In other words, an equivalent negative resistor isgenerated at a constant power control, which is obvious because
the coefficient of vc is a negative value.
Fig. 4 shows the block diagram of the single-phase equivalentcircuit of the input filter in the multi-modular matrix converter at aconstant load. In order to separate the steady and differential
components, filters named H1(s) and H2(s) are introduced. H1(s)and H2(s) are a LPF (Low Pass Filter) and a HPF (High Pass
Filter) respectively, and defined by (15).

vc  vcs  vc ................................................................... (4)imc  imcs  imc ................................................................ (5)pmc  pmcs  pmc ............................................................ (6)where, suffix “s” represents steady components based on thevoltage source angular frequency in whereas suffix “” means
differential components in transient state. Then, the steadycomponents are expressed from (7) to (10).

H1 s   H 2 s   1 ............................................................ (15)Fig. 4 has a positive feedback loop because of the negative resistor
gain -Kenrg. As a result, this system becomes unstable with LC
resonance. This is also obvious from the transfer function of Fig. 4
presented from the following equation.
1
Ll C fp
Vc s V p s 

1  1
1

s2 
H 1 s   K enrg H 2 s  s C fp  Rmcs
L
C
l
fp

....................................................................................... (16)H1(s) passes the steady component in the filter capacitor voltage
based on the angular frequency in. On the other hand, in atransient state disturbed from the output stage, the filter capacitor
voltage fluctuates with the resonant frequency because thecapacitor is charged from the power source through the leakage
inductance. Then, H2(s) passes the resonant component whose
frequency is generally over 10 times as large as the fundamental
frequency. Thus, H2(s) can be assumed as 1 in this resonant
frequency region and (17) is obtained. However, the full analysis
about this issue is a future work whereas this paper discusses thefilters on the separated frequency bandwidth for simplicity.

vcs  2Vc cosin t .......................................................... (7)imcs  2 I mc cosin t ........................................................ (8)pmcs  vcs imcs  Vc I mc 1 cos 2in t  ................................. (9)vcsV c .......................................................... (10)imcs I mc

where, Vc and Imc are the RMS values of the filter capacitor
voltage and the input current of the multi-modular matrix
converter, and Rmcs is an equivalent resistor defined as an
impedance model of the multi-modular matrix converter. Note thatthe output power of the single-phase model oscillates with thefrequency by twice of in though the output power of the threephase model is constant.
From (4) to (6), imc is represented by (11).
imc 

IL(s)
Imc(s)

f 2d1  f in  2 f out ............................................................ (2)Rmcs 

1
sLl

Vp(s)

pmc
...................................................................... (11)vc

Equation (11) is applied the linear approximation method aroundthe operating point in steady state because (11) is a nonlinear
equation. The input current imc is separated into steady anddifferential components and they are expressed as (12) and (13),respectively.

Vc s V p s 

3

1
Ll C fps2 K enrgC fp

s

1
Ll C fp...................................... (17)IEEJ Trans. ●●, Vol.●●, No.●, ●●●Damping Control for Multi-modular Matrix Converter (Hiroki Takahashi et al.)The configuration of U and V-phase modules is the same as W-phase module.
U-phase module*MV-phase module

3

3f – 1f
Matrixconverter

v2cr v2ct
2

Damping
12
control i2r**i2s2
dq i2t* Switching/
signals
2 3f 3 calculation
[3]

2

Tan-1

3f
sThpf
/
Kd
1+sT
hpf 2
dq 2
3f
/
ab

i2d* = 1i2q* = 0

1
s

2

q2

q

ASR
Iqout*
Idout*=0

iU
iW

3f
/
dq

vU*
dq
vV*
/
3f
*

vW

Idout
Iqout
ACR
Vdout*
Vqout*

W-phase module

Fig. 5. Control block diagram of the multi-modular matrix converter employing the conventional active damping control.
This equation indicates the damping factor of the system becomes
negative and this characteristic appears strictly when the fast
feedback control is applied to the output stage of themulti-modular matrix converter [20]. A typical feedback control inan adjustable speed drive system of a motor is the field oriented
control including an ACR as a minor loop. The field oriented
control is necessary for industry. However, for the multi-modular
matrix converter, the fast output current control in the fieldoriented control behaves as the equivalent negative resistor.
Therefore, the multi-modular matrix converter must cancel thenegative resistor gain in Fig. 4 with its control strategy when the
output current control is applied.

3.

matrix converter is proposed. The proposed method diverts thecurrent sensors for the ACR and does not require the voltage
sensors at the filter capacitors. Thus, the required sensors for the
proposed damping control are independent from the number ofmodules. Furthermore, the design of controllers for the ACR andthe damping control are simpler than the conventional method
because the feedback loop is applied to the output stage only.
Fig. 6 shows the system block diagram of the multi-modular
matrix converter with the proposed damping control. The output
voltage control block includes the proposed damping control tosuppress the input filter resonance, and a field oriented control
composed of the ACR and the ASR (auto speed regulator). On theother hand, the secondary current of the transformer is fed by an
open-loop control. The secondary voltage phase angles of thetransformer of U-phase and V-phase modules, q2U and q2V, arecalculated from (18) to (19) using the secondary voltage phaseangle of W-phase module q2W.

Control Strategy to Suppress Filter ResonanceFig. 5 shows the control block diagram of the multi-modular
matrix converter employing the conventional active damping
control. The basic concept of this technique is to insert a virtual
harmonic damping resistor in parallel with the filter capacitor Cf
owing to the input current control in Fig. 5. As a result, the virtual
harmonic damping resistor cancels the negative resistor gain withno influence on the fundamental component. However, the active
damping control requires high speed response and high accurate
voltage sensors in order to detect high frequency harmoniccomponents. In addition, the required voltage sensors are
increased with the number of modules. For example, themulti-modular matrix converter with nine modules as shown in
Fig. 1 needs 18 voltage sensors at least for the active damping
control. Therefore, the damping control which is independent fromthe number of modules is preferred.
In addition, the multi-modular matrix converter has two
feedback loops in both of the input and output stages because of
the conventional active damping control and the ACR for the
output current included in a field oriented control. This means thatthe design method of the conventional damping control and theoutput current control may be complicated. Hence, the damping
control which is applied to the output stage is preferred.
In order to overcome these issues, the damping control
combined with the output current control of the multi-modular

q2U  q2W  2 9 [rad] ................................................... (18)q2V  q2W   9 [rad] ..................................................... (19)Note that this voltage detection is required to one module only andthis detection is independent from the number of modules.
Next, this paragraph explains the output voltage control block.
First of all, a HPF which has a time constant Thpf separates the
harmonic components caused by the filter resonance from theoutput current, Idout and Iqout. This is based on the principle that theresonant distortion appears in the output current since the matrix
converters has no energy buffer. The cut-off frequency of the HPF
is set lower than the resonant frequency because the fundamental
frequency component in the output current is converted to a
constant value by a synchronous reference frame. Besides, theharmonic components are converted to ripples. The extracted
ripples are multiplied by the damping gain Kd, and subtracted fromthe output current. The above shows the theory of the damping
control in order to suppress the resonant distortion. On the other
hand, the output current is controlled by the ACR for the deviation
obtained from the output current commands Idout* and Iqout*, and
4

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Damping Control for Multi-modular Matrix Converter (Hiroki Takahashi et al.)The configuration of U and V-phasemodules is the same as W-phase module.

U-phase moduleM
V-phase module31s3f – 1f
Matrixconverter

Tan-1

q2W

*

3f
/
2 ab

Secondaryvoltageangle
calculation(18)-(19)

v2Wr
v2Wt

Cf

iU

12

iW

2

q2U
q2V

Damping control

i2d* = 1i2q* = 0 2

i2r*
i2s*
dq i2t*Switching/signals3
3f
calculation [3]W-phase module

vU*
vV*
vW*

3f Idout/dq Iqout

dq
/
3f

sThpf
1+sThpf

KdsThpf
1+sThpf

Kd

ASR
Iqout*

q

Idout*=0

ACR

Vdout*
Vqout*

Fig. 6. Control block diagram of the multi-modular matrix converter with the proposed damping control.
feedback values from the damping control. As a result, thedamping control path in Fig. 6 becomes positive feedback
eventually.
Fig. 7 shows the block diagram of the single-phase equivalentcircuit of the input filter with the conventional and the proposed
damping controls. Kio is defined as the following equation.

Vp(s)

IL(s)

1
sLl

Ic(s)

Imc(s)

1sCfp

Imcs(s)1
Rmcs

Imc(s)Kenrg

1 V2.............................................................. (20)Kio 
Rmcs V1

Imcd(s)where,  is the modulation index of the multi-modular matrix
converter at steady state and Kio is a coefficient to connect the
input stage and the output stage of the multi-modular matrix
converter. In addition, in Fig. 7 (a), Vc2(s) is the filter capacitor
voltage converted to the secondary side of the transformer,
Imcd(s) and Imcd2(s) are the compensated primary and secondary
current of the transformer owing to the damping control. In
contrast, in Fig. 7 (b), Iout(s) is the output current, Ioutd(s) is theoutput of the damping control, Voutd(s) and Voutd_m(s) are thecompensated output voltages of the system and the module owing
to the damping control. As discussed in section 2.2, the filter
resonance is excited by the fast feedback control for the output
current. In other words, the behavior of the input current of themodule affected by the output current control generates theequivalent negative resistor gain. In the conventional damping
control as illustrated in Fig. 7 (a), the damping control pathcontaining the damping HPF and the damping gain Kd is added tothe block diagram. This is because the damping compensation
current Imcd(s) is obtained by the detection of the filter capacitor
voltage and is added to the input current command i2d* and i2q* in
Fig. 5. On the other hand, if the proposed damping control isintroduced as shown in Fig. 6, the output voltage command isseparated to the component generating the equivalent negative
resistor gain and the damping compensation voltage Voutd(s).
This is equivalent to adding the damping path to the block
diagram such as the conventional method. Thus, the damping pathin Fig. 7 (b) contains the parameters on the output stage and Kio.
The transfer function of Fig. 7 (b) focusing on the frequency range
which the damping HPF and H2(s) are assumed as 1 is presented
as following.

V2nKd
V1 Imcd2(s)Vcs(s)

Vc(s)

Vc(s)

H1(s)
H2(s)

sThpf
V2
1+sThpf Vc2(s) V1

Damping control path(a) With the conventional damping control.
Vp(s)

IL(s)

1
sLl

Ic(s)

Imc(s)

1sCfp

Imcs(s)1
Rmcs

Imc(s)Kenrg

Imcd(s)nKioVoutd_m(s)Vcs(s)

Vc(s)

Vc(s)

H1(s)
H2(s)

3
1+sTi
sThpf
Kp
Kd
Kio
n
sTi
1+sThpf
Voutd(s) Ioutd(s)Iout(s)Damping control path(b) With the proposed damping control.

Fig. 7. Block diagram of the single-phase equivalentcircuit of the input filter in the multi-modular matrix
converter with the damping controls.

Vc s V p s 

1
Ll C fps2 1
1 1 K  K  K enrg s 
C fp
C fp  Ll Ti ........... (21)2

K  3Kio K d K p ............................................................. (22)Therefore, the proposed damping control maintains the positive
5

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Damping Control for Multi-modular Matrix Converter (Hiroki Takahashi et al.)200 Input phase voltage [V]0-200 Primary current of the transformer [A]200-20
Output line voltage [V]
1000
0
-1000
20 Output current [A]0

Table 1. Simulation conditions.
(a) Common conditions.

Input line voltage

200 Vrms

ASR natural frequency 63.7 Hz

Input frequency

50 Hz

ACR natural frequency 650 Hz

Rated power

3 kW

Carrier frequency

10 kHzLeakage inductance

9.42 %

Input filter C

8.55 %

Damping HPF
cut off frequency

30 Hz(b) Conditions depending on the number of modules.

3-module model

9-module model

Voltage transformation ratio
V2/V1

1

1/3

Damping gain

0.6 p.u.

0.55 p.u.

-200.241800 rpm (1 p.u.)

Rated line voltage

180 Vrms

Rated power

1.5 kW

Rated current

6.1 Arms (1 p.u.)d-axis inductance

11.5 mH

q-axis inductance

23.0 mH

Stator resistance

0.783 W

Inertia moment

0.00255 kgm2

0.27

0.28

0.29

0.3[s]

200 Input phase voltage [V]0-200 Primary current of the transformer [A]200-20
Output line voltage [V]
1000
0
-1000
20 Output current [A]0-200.24

0.25

0.26

0.27

0.28

0.29

0.3[s](b) With the proposed damping control which results in the primary
current and the output current THDs under 1%.

damping factor and suppresses the filter resonance.

4.

0.26

(a) Without damping controls which results in the primary current and theoutput current THDs by 62.9% and 15.4%, respectively.

Table 2. IPM motor parameters.
Rated speed

0.25

Fig. 8. Input and output waveforms of the three-module
model of the multi-modular matrix converter as drawn in
Fig. 2 at steady state in simulations.

Simulation Results

Table 1 presents simulation conditions for the multi-modular
matrix converter with three modules as drawn in Fig. 2 and ninemodules as shown in Fig. 1. The simulations are tested with a fieldoriented control which is composed of the ASR and the ACR tocontrol rotation speed of the IPM (interior permanent magnet)
motor. The negative resistor appears because the ACR keeps
constant output power in the motor load. In addition, note that thefilter inductors are added in the primary side instead of the large
leakage inductance of the transformer. On the other hand, Table 2
shows the IPM motor parameters as the load in the simulation.
4.1
Three-module modelFig. 8 shows the input andoutput waveforms of the three-module model of the multi-modular
matrix converter at steady state in simulations. Note that theseresults are obtained when the IPM motor operates at rated speed
and rated torque output. Fig. 8 (a) shows the result without
damping controls and (b) shows the result with the proposed
damping control. It can be confirmed that the primary current of
the transformer contains large distortion caused by the filter
resonance from Fig. 8 (a), and the primary current THD (total
harmonic distortion) is 62.9%. Moreover, the output line voltage
and the output current have large distortion too because matrix
converter modules have no energy buffers. The output current
THD is 15.4% from Fig. 8 (a). On the other hand, in Fig. 8 (b), theresonant distortion in the output current is mitigated by the
proposed damping control. As a result, the primary current of thetransformer contains less distortion compared to Fig. 8 (a). Boththe primary current of the transformer and the output current
THDs are under 1% in Fig. 8 (b). In addition, the output line
voltage obtains 5- level waveform.
Fig. 9 shows the transient waveforms regarding the IPM motor.

Fig. 9 (a) shows the result without damping controls and (b)shows the result with the proposed damping control. In Fig. 9 (a),the dq-axis currents and the output torque oscillate during
acceleration period and at rated speed and rated torque output due
to the filter resonance. On the other hand, in Fig. 9 (b), the current
and torque oscillations are suppressed by the proposed damping
control. However, the overshoot of the d-axis current by 1.2 p.u.
occurs when the rated torque is provided although the d-axis
current command is set to zero. This problem will be overcome by
the optimal design of the controllers in future works. Therefore, it
is confirmed from the simulation results that the proposed
damping control suppresses the filter resonance and achieves
stable operation with the IPM motor load.
4.2
Nine-module modelFig. 10 shows the input andoutput waveforms of the multi-modular matrix converter with ninemodules. This simulation result clarifies that the proposed
damping control is applicable to the practical multi-modular
matrix converter. In common with the simulation of thethree-module model, these results are obtained when the IPM
motor operates at rated speed and rated torque output. Fig. 10 (a)shows the result without damping controls and (b) shows theresult with the proposed damping control. From Fig. 10 (a), theprimary current THD of the transformer and the output current
THD are yielded by 72.9% and 18.6% respectively because of thefilter resonance. However, from Fig. 10 (b), both of these currents
contain less resonant distortion than the result of (a) owing to the
proposed damping control. The primary current THD of thetransformer is 1.77% and the output current THD is under 1%. In
addition, the output line voltage waveform has multi-level like a
6

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Damping Control for Multi-modular Matrix Converter (Hiroki Takahashi et al.)Motor speed [p.u.]
1 Speed command
Actual speed0Output torque [p.u.]2
10d-axis current [p.u.]
Current command Idout* = 0
10Actual current Idout-1
q-axis current [p.u.]2
Current command Iqout*
1
Actual current Iqout
0
0.15
0.2
0.25
0.1
0.3[s]

sinusoidal waveform. Therefore, it is confirmed that the proposed
damping control is applicable to the multi-modular matrix
converter with nine modules in common with the three-module
model.

5.

Experimental Results

5.1 Verification of the Proposed Damping Control
Table 3 presents experimental conditions for the multi-modular
matrix converter with three modules drawn in Fig. 2 because of
the experimental limitation and the experiment with nine modules
is a future work. Note that the experiments are tested with aconstant power load composed of a R-L load and the ACR, not an
field oriented control. The negative resistor appears even if theR-L load is used instead of the motor because the ACR keeps theoutput power constant. In addition, the filter inductors are addedin the primary side instead of the large leakage inductance of thetransformer in common with the simulation condition. The
commutation sequence of the semiconductor switches isdetermined by the direction of the output current, that is, it is also
called the current type commutation.
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 proposed damping control. In Fig. 11 (a), thefilter resonance is excited, and the primary current of thetransformer and the output current have resonant distortions. The
primary current of the transformer and the output current THDs
are obtained as follows, 18.6% and 6.56%. The current THDs
measured from experiments are lower than the simulation results
because loss of the transformer and the modules behave as
damping resistors. In contrast, the filter resonance is suppressedby the proposed damping control in Fig. 11 (b). Consequently, theprimary current of the transformer and the output current obtainlower distortion compared to Fig. 11 (a). The primary current of
the transformer and the output current THDs are 4.55% and 2.89%,
respectively. Thus, the proposed damping control mitigates theresonance distortions by 75.5%. Furthermore, the output line
voltage obtains 5-level waveform without the voltage fluctuation
due to the filter resonance.
Fig. 12 shows the harmonic components analysis of the primary
current of the transformer. Fig. 12 (a) shows the result without the
proposed damping control and (b) shows the result with theproposed control. In Fig. 12 (a), the primary current of thetransformer includes the 340-Hz and 440-Hz components byapproximately 10% caused by the filter resonance. In contrast, inFig. 12 (b), the resonant components are mitigated by the
proposed damping control. In addition, the harmonic components
within a range from 100 Hz to 1 kHz are suppressed by the
proposed damping control since the proposed control has low
cut-off HPF (30 Hz). This upper limit of suppressible harmonics is
decided by the control bandwidth due to the control frequency.
According to [18], the control delay due to the sampling and thePWM generation leads to the phase delay of the damping
compensation command. The phase delay becomes 90 deg. when
the resonant frequency is 1/8 of the control frequency in [18]. At
this frequency, the damping control behaves as an inductance.
This equivalent inductance cannot suppress the current distortion
including the resonance. From this consideration, the proposed
damping control suppresses the current distortion within 1.25 kHz
in this experiment because the control frequency is set to 10 kHz(a) Without damping controls which results in steady perturbation in the
output torque and dq-axis currents due to the filter resonance.

Motor speed [p.u.]
1 Speed command
Actual speed0Output torque [p.u.]2
10d-axis current [p.u.]
Current command Idout* = 010-1
q-axis current [p.u.]2
Current command Iqout*
1
0
0.15
0.2
0.1

Actual current Idout

Actual current Iqout
0.3[s]
0.25(b) With the proposed damping control which results in suppression of thesteady fluctuation.

Fig. 9. Transient waveforms regarding the IPM motor.
200 Input phase voltage [V]0-200 Primary current of the transformer [A]200-20
Output line voltage [V]
1000
0
-1000
20 Output current [A]
0
-20
0.26
0.27
0.28
0.24
0.25

0.29

0.3[s](a) Without damping controls which results in the primary current THD
and the output current THD by 72.9% and 18.6%, respectively.

200 Input phase voltage [V]0-200 Primary current of the transformer [A]200-20
Output line voltage [V]
1000
0
-1000
20 Output current [A]
0
-20
0.26
0.27
0.28
0.24
0.25

0.29

0.3[s](b) With the proposed damping control which results in the primary
current THD and the output current THD by 1.77% and under 1%,
respectively.

Fig. 10. Input and output waveforms of the nine-module
model of the multi-modular matrix converter as drawn in
Fig. 1 at steady state in simulations.
7

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Damping Control for Multi-modular Matrix Converter (Hiroki Takahashi et al.)Table 3. Experimental conditions.
Input line voltage

200 Vrms

Input frequency
Rated power

Output frequency

30 Hz

50 Hz

Load resistance

83.7 %

3 kW

Load inductance

1.88 %

Trans. turn ratio

1

ACR natural frequency

650 HzLeakage inductance

9.42 %

Input filter C

8.55 %

Damping gain

0.6 p.u.

10 kHzCommutation time

2.5 msDamping HPF
cut off frequency

30 Hz

fundamental
340Hz

440Hz

10%
1%

Load current command 0.95 p.u.

Carrier frequency

Primary phase voltageof transformer 250 V/div

100%

Primary currentof transformer 10 A/div

0

0

500

1k
1.5k
Frequency [Hz]2k(a) Without damping controls.

0

100%

fundamental

0

10%
0
Output current 10 A/div

340Hz

440Hz

1%

Output line voltage 500 V/div
10 ms/div(a) Without damping controls which results in the primary current THD
and the output current THD by 18.6% and 6.56%, respectively.

Primary phase voltageof transformer 250 V/div

Primary currentof transformer 10 A/div

0

0

500

1k
1.5k
Frequency [Hz]2k(b) With the proposed damping control.

0Fig. 12. Spectrum of the primary current of the transformer
with experiments.
proposed damping control suppresses the filter resonance andachieves stable operation.
5.2
Comparison with the Conventional Active
Damping Method Applied to Multi-modular Matrix Converter
This section discusses the comparison between the conventional
active damping control as drawn in Fig. 5 and the proposed
damping control as illustrated in Fig. 6. This comparison results
clarify the performance of the filter resonant suppression of eachmethod. The evaluation subjects are the primary current THD
characteristics against the output power and the leakage
inductance of the transformer.
Fig. 14 shows the THD characteristics obtained from theprimary current of the transformer which is subject to the output
power. Note that the output line voltage is kept to 240 V rms
constantly. The damping gains of the conventional and the
proposed damping controls are set to 1.0 p.u. and 0.6 p.u.
respectively, and the cut-off frequency of the HPF in the damping
controls is 30 Hz in common. Without any damping controls, theprimary current THD of the transformer is increased toapproximately 20% in the range over 60% of load due to theresonance. However, the conventional and proposed damping
controls mitigate the primary current THD of the transformer.
Then, the conventional control cannot reduce the primary current
THD of the transformer and the primary current THD isapproximately 10% in the range over 90% of load. On the other
hand, the proposed damping control suppresses the THD under00
Output current 10 A/div

Output line voltage 500 V/div
10 ms/div(b) With the proposed damping control which results in the primary
current THD and the output current THD by 4.55% and 2.89%.

Fig. 11. Input and output waveforms of the multi-modular
matrix converter with three modules in experiments.
same as the carrier frequency. However, the full consideration
about this issue has to be reported in the future.
Fig. 13 shows the input and output waveforms of a module
obtained from experiments. Fig. 13 (a) shows the waveforms
without any damping controls and (b) shows the results with theproposed damping control. In Fig. 13 (a), the filter resonancegenerates distortion in the filter capacitor voltage and thesecondary current of the transformer. However, in Fig. 13 (b), theresonant distortion in the filter capacitor voltage and the secondary
current waveforms are suppressed by the proposed damping
control. The secondary current THDs of transformer without andwith the proposed damping control are 27.5% and 5.19%,
respectively. Note that the secondary current THD calculationomits the power pulsation components expressed from (2) to (3).
Furthermore, the filter capacitor voltage THD is reduced from
10.7% to 2.19%. From these results, it is confirmed that the
8

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Damping Control for Multi-modular Matrix Converter (Hiroki Takahashi et al.)Filter capacitor
phase voltage 250 V/div

Secondary currentof transformer 10 A/div

25
No damping controls

Primary current THDof transformer [%]

0

0

0

20
15

Conventional
damping control

1050Output current 10 A/div0

Output phase voltage 250 V/div
10 ms/div(a) With no damping controls which results in the capacitor voltage THD
and the secondary current THD by 10.7% and 27.5%, respectively.

Filter capacitor
phase voltage 250 V/div

Proposed damping control
0

20

40
60
Output power [%]

80

100

Fig. 14. THD characteristics obtained from the primary
current of the transformer that is subject to the output
power.

Secondary currentof transformer 10 A/div

25

0

No damping controls
20

Primary current THDof transformer [%]000
Output current 10 A/div

Output phase voltage 250 V/div
10 ms/div

15
10

Proposed
damping control

50(b) With the proposed damping control which results in the capacitor
voltage THD and the secondary current THD by 2.19% and 5.19%.

Conventional
damping control

0

5
10
Leakage inductance of transformer %XL [%]15

Fig. 15. THD characteristics of the primary current of thetransformer that is subject to the leakage inductance of thetransformer.

Fig. 13. Input and output waveforms of a module withexperiment.
5% in the range more than 60% load. In the light load region
especially under 40% load, the proposed damping control cannot
reduce the primary current THD under 5%. The main reason ofincreasing THD is caused by the small amplitude of thefundamental component of the primary current although thedistortion components are almost constant and unrelated to theload. This distortion is caused by the commutation failure, is notcaused by the filter resonance. Thus, the proposed damping
control suppresses the resonant distortion in larger range of theoutput power than the conventional damping control.
Fig. 15 shows the THD characteristics of the primary current of
the transformer which is subject to the leakage inductance of thetransformer. Without any damping controls, the primary current
THD of the transformer increases with the leakage inductance. In
particular, the data in the region over 10% of the leakage
inductance of the transformer could not be taken because theprimary current of the transformer diverges due to thedestabilization. In contrast, the conventional and the proposed
damping controls suppress the primary current THD. However,
although the conventional damping control cannot suppress theprimary current THD in the region over 9.42% of the leakage
inductance, the proposed damping control mitigates the primary
current THD under 5% in the condition of the leakage inductancefrom 4.71% to 14.1%. Hence, the proposed damping control
mitigates the resonant distortion in larger range of the leakage
inductance of the transformer than the conventional damping
control.

Next, the reason why the proposed damping control has higher
resonant suppression ability than the conventional damping
control is discussed. The conventional damping control in the
input stage has a saturation problem. Basically, the amplitude of
the input current is dominated by the output power in the matrix
converter. Therefore, the secondary current of the transformer
command i2d* on the synchronous reference frame in Fig. 5 is set
to 1 p.u. constantly in order to obtain the largest voltage transfer
ratio of the system. For the conventional method, the modules
operate in over modulation region because the input current
commands of the modules including the damping compensation,i2r*, i2s* and i2t*, exceed 1 p.u.. Consequently, the damping effect of
the conventional method is reduced due to the over modulation.
Note that the voltage transfer ratio is decreased when the input
current command i2d* is set under 1 p.u. in order to keep thedamping effect in the conventional method. The reduction of thevoltage transfer ratio is not preferred for adjustable speed drive
systems. In contrast, the damping effect of the proposed control is
not decreased because the matrix converter modules can control
the amplitude of the output voltage within the limitation of theoutput voltage. From these results, the proposed damping control
has higher performance than the conventional method regarding
the suppression of the filter resonance.

6.

Conclusion

This paper discussed the damping control to suppress the LC
resonance caused by the input filter of the multi-modular matrix

9

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Damping Control for Multi-modular Matrix Converter (Hiroki Takahashi et al.)converter. Past works related to the damping control for current
source converters and general matrix converters have not clarified
the validity for the multi-modular matrix converter. Especially, theconventional damping control applied to the input current controlof the multi-modular matrix converter has two problems in termsof the number of sensors and design of the control parameters.
In this paper, a damping control suitable for the multi-modular
matrix converter was proposed. The proposed damping control is
combined with the output current control of the multi-modular
matrix converter. Therefore, the proposed damping control
suppresses the filter resonance and also diverts the output current
sensors for the ACR instead of using high speed response voltage
sensors on the input stage.
From simulation and experimental results, the multi-modular
matrix converter without damping controls has large resonant
distortion in the primary current of the transformer and the output
current. However, the proposed damping control mitigates the
distortion by 75.5% at steady state in experiments using thethree-module model. The simulation results indicate the proposed
damping control is applicable to the practical multi-modular
matrix converter with nine modules. In addition, the comparison
results of the damping effect between the proposed damping
control and the conventional damping control are evaluated. From
the THD characteristics of the primary current of the transformer
which is subject to the output power, the proposed damping
control suppresses the THD under 5% over 60% load region
although the conventional damping control cannot reduce the
THD in the region more than 90% of load. Moreover, theproposed damping control mitigates the resonant distortion under
5% in conditions of the leakage inductance from 4.71% to 14.1%
while the conventional control cannot suppress the filter resonancein the region over 9.42% of the leakage inductance. Thus, thevalidity of the proposed damping control is confirmed bysimulations and experiments. In the future, the optimal design
method of the proposed damping control and the experiments withnine modules are implemented.

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)
(18)

(19)
(20)

Hiroki Takahashi

(2)
(3)

(4)

(5)

(6)

(7)

(8)

(9)

(Student member) received his B.S. and M.S.
degrees in electrical, electronics and informationengineering from Nagaoka University ofTechnology, Nagaoka, Japan in 2011 and 2013,
respectively. He is currently working towards to a
Ph.D. degree in electrical and electronicengineering. His research interests include matrix
converters and high power conversion systems.

Jun-ichi Itoh

(Member) received his M.S. and PhD degrees inelectrical and electronic systems engineering from
Nagaoka University of Technology, Niigata, Japan
in 1996 and 2000, respectively. From 1996 to 2004,
he was with Fuji Electric Corporate Research andDevelopment Ltd., Tokyo, Japan. Since 2004, He
has been with Nagaoka University of Technology
as an associate professor. He received the IEEJ
Academic Promotion Award (IEEJ TechnicalDevelopment Award) in 2007 and the Isao Takahashi Power Electronics
Award in 2010. His research interests include matrix converters, DC/DC
converters, power factor correction techniques and motor drives. He is amember of the Institute of Electrical Engineers of Japan.

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IEEJ Trans. ●●, Vol.●●, No.●, ●●●