Design Procedure for Output Current Control andDamping Control of Matrix Converter
Hiroki Takahashi and Jun-ichi Itoh
Dept. of Energy and Environmental Science
Nagaoka University of Technology
Nagaoka, Niigata, Japan
thiroki@stn.nagaokaut.ac.jp and itoh@vos.nagaokaut.ac.jp
Abstract— This paper presents a design procedure foran output current control and an output damping control ofa matrix converter to suppress LC filter resonance andimprove transient current response. With a conventional
design method which gives preference to stability, theoutput damping control causes a large output current
overshoot and a control bandwidth of the output current isaffected. Therefore, in order to obtain a desired transient
response keeping a stable operation, this paper describes a
modified control block diagram which a reference filter isadded in to suppress the output current overshoot and adetailed design procedure using Bode-diagrams and a
flowchart. From the experimental results, the output
damping control designed with the proposed method
suppresses the filter resonance, which results in stable
operation. In addition, the proposed design method reduces
the output current overshoot of 60% and an error between
the desired and the obtained control bandwidth incomparison with the conventional method in experiments.
Keywords— Matrix converter, Resonance, Damping
controlI.

INTRODUCTION

Recently, a matrix converter has attracted a lot of
attentions as an interface converter of a wind turbine
system and an elevator because this converter has no DC
energy buffer such as bulky electrolytic capacitors [1]-[6].
A matrix converter promises to achieve higher efficiency,
smaller size and longer life-time compared to aconventional BTB (back to back) system which consistsof a PWM rectifier and a PWM inverter.
A matrix converter requires LC filters in the grid sideto eliminate harmonic current due to switching operation.
However, a matrix converter has a problem that a filter
LC resonance is excited by a grid voltage fluctuation andan input current fluctuation. In general, the filter
resonance is suppressed by using damping resistors
connected in parallel with the filter inductors [7].
However, when a transformer is connected to the input
side of the matrix converter such as a wind turbine
system and an elevator which requires isolation between
a load and the grid, a leakage inductance of thetransformer is used as the filter inductors and thedamping resistors are not inserted. Therefore, asuppression method for the filter resonance is needed.
In past works, some papers about stability analysis of amatrix converter to prevent the filter resonance have beenpresented [8]-[11]. These papers investigate the stability
taking account into a detection of the input voltage with afilter [8], digital control [9]-[10] and matrix converter
losses [10]. In addition, an approach to the stability basedon admittance of a matrix converter has reported [11].
These investigations clarify stable limitations of a matrix
converter operation with each method and suppress thefilter resonance. However, the stability of a matrix
converter can be improved more from the point of viewof a current control of a matrix converter.
As control strategies to suppress the filter resonance,
damping controls have been proposed [12]-[14]. The
damping controls are separated into two types. The firstone is a damping control combined with the input current
control of the matrix converter [12]-[13] and another one
is combined with the output current control [14]. Then,the output damping control has an advantage when thematrix converter is applied to a wind farm and anelevator which requires a field oriented control for
accurate speed control. The advantage is that a required
feedback control is on the output stage only while thematrix converter with the input damping control needs
feedback loops on both of the input and the output sides.
The authors have already proposed the output damping
control and its parameter design method [15]. The
presented design method is based on a gain margin of thematrix converter and suppresses the filter resonance.
However, the damping control designed with theconventional method based on the gain margin generates
a large overshoot of the output current. Furthermore, adetailed design procedure to earn an intended control
bandwidth of the output current has not been presented.
This paper proposes a modified control block diagram
and a design procedure for the output current control andthe output damping control to suppress the output current
overshoot and to obtain an intended control bandwidth.
The output current overshoot is reduced by an addedreference filter for pole-zero cancellation and designedwith an approximate model of the matrix converter.
Moreover, a desired control bandwidth of the output
current and a stable operation are achieved by a proposed
design flowchart based on Bode-diagrams. The proposed
reference filter and the design procedure result in fine
output performance compared with the conventional

method. This paper is organized as follows; firstly, an
integrated block model and an approximate model of thematrix converter to design the output damping control
and the output current control including the reference
filter are described; secondly, the design flowchart for thecontrols is presented; finally, the stability and thetransient response are evaluated in simulation andexperiment.
II.

SYSTEM CONFIGURATION

Fig. 1 shows a system block diagram of the matrix
converter with the output damping control. For simplicity,
the input transformer is assumed as an ideal transformer
and a leakage inductance Lf, and the ideal transformer isomitted. For the same reason, a load motor is replacedwith an R-L load. In addition, an output current control
with a PI controller is used because a wind turbine
system and an elevator require a field oriented control for
accurate speed control. The output damping control
which has a damping gain Kd and a damping HPF (highpass filter) of a time constant Thpf is combined with theoutput current control feedback and suppresses the filter
resonance. However, the output damping control
generates a large current overshoot in a transient statebecause a closed-loop transfer function of the output
current control has a zero due to the damping control.
Hence, a reference filter F(s) is added for a pole-zerocancellation. In order to derive an appropriate equation ofF(s) and obtain an intended control bandwidth and astable operation, this paper deals with Bode-diagrams
derived from an integrated block model which iscomposed of a main circuit and a control diagram.
Moreover, a design procedure for the output damping
control and the output current control is proposed.
III.

DESIGN MODEL

A. Modeling of Matrix Converter based on Space Vector
In order to derive the integrated block model of amatrix converter, an output voltage and an input current
equations are required. Then, these equations of a matrix
converter based on the duty-cycle space vector have beenpresented [16]. In this paper, the integrated block modelof the matrix converter which is composed of the main
circuit and the control diagram is derived by using theduty-cycle space vector.
Fig. 2 shows the integrated block model of the matrix
converter. Variables in bold font represent their space
vectors. For example, an input voltage vector vin isdefined as following.
j 
j  
2 
vr  vs e 3  vt e 3  

3

2

v in 

2where, vr is the input R-phase voltage, vs is the input Sphase voltage and vt is the input T-phase voltage.
In order to connect the input and the output circuits ofthe matrix converter, the duty-cycle space vector md and(a) System block diagram.

(b) Output damping control.
Fig. 1. Matrix converter with the output damping control to suppress thefilter resonance and the proposed reference filter F(s) to reduce theoutput current overshoot.

Fig. 2. Integrated block model of the matrix converter which iscomposed of the main circuit and the control diagram. This block
diagram is based on the duty-cycle space vector.

mi are introduced. Then, an output voltage vector vout andan input current vector imc are yielded by (2) and (3).
33v out  m i v c  m d v c 2
2i mc 

3
3
m i i out  m d i out 
2
2





where, vc is a filter capacitor voltage vector and iout is an
output current vector. Note that a bar placed over avariable indicates complex conjugate. In contrast, md andmi are represented by follows.
md 

v in
*
v out 
3Vi



mi 

v in
*
v out 3Vi

where, vout* is an output voltage reference vector and Vi is
amplitude of the input voltage.
However, (2) and (3) are non-linear equations because
time constants of fluctuation of vc, md, mi and iout areclose to each other. Therefore, a linear approximation
method around a steady operating point is applied tothese non-linear parts in order to draw Bode-diagrams.
Equations (2) to (5) are separated into steady anddifferential components, and the differential components
are expressed as (6) to (9).



 



 v out 33
m i v cs  m is  v c   m d v cs  m ds  v c 
22

i mc 3
m i i outs  m is i out   3 m d i outs  m ds i out 
2
2





m d 

v ins*
v out 3Vim i 

v ins*
v out 
3Vi

where, suffix “s” represents the steady component basedon its fundamental frequency while “” means thedifferential component in a transient state. It should be
noted that vin is not considered because the output
current response is evaluated in this paper.
Fig. 3 shows a linearized block model of the matrix
converter regarding the differential components.
Frequency characteristics of the output current control
and the stability are obtained by using Bode-diagram.
Note that analyses should be implemented with DC modein which an input and an output angles are fixed because
a rotating frequency of the steady vector does not equal
to a rotating frequency of the differential vector.
B. Verification of the linearized model with simulation
Table 1 and Table 2 present the main circuit and thecontrol parameters of the circuit model as illustrated inFig. 1 and the linearized model as shown in Fig. 3. Thissection shows a simulation result without the output
current control and the output damping control in order toevaluate a verification of the linearized model. Note thatthe inductor and capacitor parameters are normalizedbased on 50 Hz. In addition, a carrier frequency is
increased to 100 kHz in this section only.
Fig. 4 shows an 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 passfilter) in order to observe these average waveform
without switching ripples. From Fig. 4, it is confirmedthat the linearized model waveforms correspond to thecircuit model result. Furthermore, error between thecircuit model and the linearized model waveforms is lessFig. 3. Linearized block model of the matrix converter focusing on the
differential components. Stability and a control bandwidth of the output
current are evaluated by analyzing this block model.
TABLE 1. CIRCUIT PARAMETERS OF THE MATRIX CONVERTERTABLE 2. CONTROL PARAMETERS FOR OPEN-LOOP CONTROL

Fig. 4. 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.

than 1% in a steady state. Thus, the linearized model isvalid for modeling of the matrix converter. In
consequence, the output current control including the
proposed reference filter and the output damping control
are designed with Bode-diagrams derived from thelinearized model in the next chapter.

IV.

PROPOSED DESIGN PROCEDURE OF CONTROLPARAMETERS

A. Approximate Model to Design Proposed Reference
Filter and Control Parameters
In order to design the output current control with theproposed reference filter and the output damping control,
Bode-diagrams of the linearized block model in Fig. 3
should be clarified. However, the linearized block model
is a fifth order transfer function, which is caused by the
input LC filter, the load inductor, the PI controller andthe output damping control, and the transfer function isvery complicated. Therefore, an approximate model
based on a third order transfer function is used to design
these controls.
Fig. 5 shows the approximate block model between
iout_LPF* and iout in Fig. 3 without the output damping
control. The approximate model is based on the gain
curve of the linearized block model in Fig. 3 in order toobtain the proposed reference filter configuration easily.
It should be noted that the output damping control is notapplied when a forward transfer function in theapproximate model is calculated. The approximate model
is a second order transfer function which has a damping
factor  and a natural angular frequency n. The transfer
function in Fig. 5 approximates a resonance point
characteristic of the linearized block model. Therefore, and n are expressed by (10) and (11).
 

1
1 1


1
2
2 2
Mp



n   p 1  1 M p 2

 


1
4

Fig. 5. Approximate block model between iout_LPF* and iout without
the output damping control to simplify the design procedure. The
transfer function of this block diagram becomes the standard form of thesecond order system.
TABLE 3. PARAMETERS OF THE PI CONTROLLER

Fig. 6. Gain characteristic of the closed-loop transfer function between
iout_LPF* and iout without the output damping control. The gain curveof the approximate model is similar to the linearized model.

Fig. 7. Approximate block model between iout* and iout taking theoutput damping control into account. The proposed reference filter F(s)is derived from this block model.

where, Mp is a peak gain of the resonance point and p is
a resonance angular frequency which are measured withthe gain characteristic of the linearized block model ofthe matrix converter in Fig. 3.
Table 3 shows parameters of the PI controller. These
parameters should be designed with a flowchart aspresented in the next section. However, in order to clarify
the validity of the approximate block model in Fig. 5, thePI parameters are set to the linearized model in Fig. 3.
Fig. 6 shows a gain characteristic of the closed-looptransfer function between iout_LPF* and iout without theoutput damping control. From Fig. 6, the resonance peak
gain and the resonance frequency of the approximate
model designed with (10) and (11) corresponds to thelinearized block model of the matrix converter. In
addition, the gain characteristic of the approximate model
is similar to the linearized model. In a frequency region
higher than 1 kHz, gain error between the linearized
model and the approximate model can be neglected
because the region is not available for the current control
due to detection delay and PWM delay of the real system
if the carrier frequency is set to 10 kHz as shown in Table
1. Thus, the approximate model in Fig. 6 is valid for themodeling of the linearized block model of the matrix
converter without the output damping control.
Fig. 7 shows the approximate block model between
iout* and iout taking the output damping control into
account. First, a transfer function of the proposed
reference filter F(s) to suppress the output current
overshoot is derived. A closed-loop transfer function
between iout_LPF* and iout is presented by (12).
n 2
iout

*
iout_LPF

Thpf

1  sT 

hpf

 2
 2

 1
s3  
 n 1  K d  s  n 2n  s 2  n 
 Thpf
 Thpf ThpfIt is obvious that the output damping control generates alarge current overshoot because the closed-loop transfer
function between iout_LPF* and iout has a zerodepending on Thpf. In order to suppress the output current
overshoot, the zero in (12) should be cancelled. Hence,F(s) is defined as (13).
F s  1
1  sThpf

Hence, the proposed reference filter is designed easily
when the Thpf is decided in the proposed design procedure.
B. Design Flowchart
In order to design the output current control and theoutput damping control, this section proposes a design
flowchart.
Fig. 8 shows a proposed design flowchart for the
output current control and the output damping control toensure a desired control bandwidth and the stability.
Input parameters into the flowchart are determined by
specifications and outputs of the flowchart are theproportional gain Kp, the integral time Ti of the PI
controller, Kd and Thpf of the output damping control.
1) Step 1: Calculation of Kp and Ti of PI ControllerFirst, Kp and Ti should be calculated in order to draw a
Bode-diagram as shown in Fig. 6. Then, Kp and Ti aredesigned by (14) and (15), supposing that a plant of thecontroller is an R-L model.

K p  c _ designLo 



Lo

Ro



Ti 

where, c_design is a desired cut-off angular frequency ofthe output current control. It should be noted that thedesign concept of (14) and (15) is based on the first ordertransfer function of the output current control loop
neglecting the input LC filter and the output damping
control. However, the actual cut-off angular frequency ischanged by the resonance characteristic and the output
damping control.
2) Step 2: Simulation to Measure Mp and p
In order to calculate parameters of the approximate
model for a simple design process, a Bode-diagram
between iout_LPF* and iout of the linearized matrix
converter block model such as Fig. 6 is drawn by asimulator. In this paper, Piece-wise Linear Electrical
Circuit Simulation (PLECS) is used as a simulator.
3) Step 3: Calculation of  and n
In order to obtain the approximate model of the matrix
converter,  and n are calculated by (10) and (11).
4) Step 4: Decision of Kd and Thpf
By using (12) and (13), a Bode-diagram between iout*and iout of the approximate block model in Fig. 7 isillustrated. Then, Kd and Thpf is decided with the Bodediagram to earn an intended control bandwidth of theoutput current.
Fig. 9 shows gain characteristics between iout* andiout of the approximate block model when the Kd and fhpf
are changed respectively. Note that fhpf is a cut-off
frequency of a HPF of the output damping control. Fig. 9(a) shows a gain characteristic when Kd is changed andfhpf is a constant of 50 Hz, and (b) shows a result when fhpf

Fig. 8. Proposed design flowchart for the output current control and theoutput damping control.

(a) Kd is changed (fhpf is a constant of 50 Hz).

(b) fhpf is changed (Kd is a constant of 0.5 p.u.).
Fig. 9. Gain characteristics between iout* and iout of the approximate
block model when the Kd and fhpf are changed respectively.

is changed and Kd is a constant of 0.5 p.u.. From Fig. 9(a), the cut-off frequency of the output current control
increases as Kd is increased from 0.3 p.u. to 0.6 p.u..
However, the cut-off frequency is decreased in the region
where Kd is over 0.6 p.u.. This is affected thecharacteristic around the resonance point. On the other
hand, it is confirmed that the cut-off frequency increases
as the fhpf is increased from Fig. 9 (b). Therefore, anintended control bandwidth of the output current isyielded by adjusting the damping parameters Kd and fhpf.
5) Simulation to Confirm Cut-off Frequency of
Output Current Control
A Bode-diagram between iout* and iout of thelinearized block model in Fig. 3 with the decided
damping parameters is drawn and the control bandwidth

of the output current c is evaluated. If the obtained
control bandwidth is not acceptable, the damping
parameters should be redesigned by going back to Step 4.
6) Stability Analysis
Finally, the stability of the system is analyzed with aNyquist-diagram. If the obtained stability does not satisfy
the specification of stability, the PI parameters and thedamping parameters which influence the frequency
characteristic of the open-loop transfer function of theoutput current control should be adjusted again. For
example, c_design is reduced or Thpf is increased from theoriginal value in order to increase the stability and Kd isredesigned to earn the intended control bandwidth.
V.

TABLE 4. DESIGNED PARAMETERS OF THE OUTPUT DAMPING CONTROL.

COMPARISON RESULTS BETWEEN THE PROPOSEDAND CONVENTIONAL METHODS IN SIMULATION

Table 4 presents the designed parameters of the output
damping control with the proposed design flowchart in
Fig. 8 and the conventional method. It should be notedthat the conventional method is based on the desired gain
margin [15]. In the conventional method, the PI
parameters are designed with (14) and (15) in common
with the proposed procedure. In addition, the damping
parameters are calculated by the following equations.

Fig. 10. Gain characteristics of the closed-loop transfer function between
iout* and iout with the proposed and the conventional methods. The
proposed method mitigates the error between the desired and the yielded
cut-off frequencies by 1/4 compared to the conventional method.

ga  gm   20log10 1 Kd   0 
Thpf 

5 
2 1  K d  f cp

where, fcp is a phase-crossover frequency of a gain curveof the output current control without the output damping
control, ga is a gain at fcp without the damping control andgm is a desired gain margin which is set to the same as thedesigned result with the proposed method for comparison.
Fig. 10 shows gain characteristics between iout* andiout with the proposed and the conventional methods.
The gain curve of the approximate model meets -3 dB at
650 Hz as shown in Table 3 by the proposed design
flowchart. In addition, the proposed method mitigateserror between the desired and the obtained cut-off
frequencies by 1/4 in comparison with the conventional
method. This is because the conventional method gives
preference to the stability design over adjusting thecontrol bandwidth. Moreover, the gain curve with theconventional method increases in the region around 100
Hz because of the zero generated by the output damping
control and causes a large output current overshoot.
Fig. 11 shows the Nyquist-diagram of the linearizedblock model of the matrix converter between iout_LPF*
and iout with the proposed and the conventional
procedures. As a result of the proposed procedure, theNyquist-diagram of the proposed method has a gain
margin of 3.85 dB and the system is stable, which isequivalent to suppression of the filter resonance. It should
be noticed that the experimental setup has more stability
because the loss of the matrix converter behaves as a

Fig. 11. Nyquist-diagram of the linearized block model of the matrix
converter between iout_LPF* and iout with the proposed and theconventional design procedures. The proposed design procedure yields again margin of 3.85 dB and the system is stable.

damping resistor [10]. On the other hand, theconventional method also stabilizes the system with thegain margin of 4.30 dB. Then, error of the gain margin
between the intended value of 3.85 dB and the actual iscaused by the break point approximation of a Bodediagram for deriving (17) [15]. Therefore, it is confirmedthat the proposed procedure yields the stability and thedesired control bandwidth of the output current.
Fig. 12 shows the output d-axis current waveform insimulations with the linearized block model in Fig. 3.
From Fig. 12 (a), the system without the damping control
becomes unstable right after the step input because thesystem gain margin is -3.20 dB as shown in Fig. 11.
However, the output damping control designed with theproposed and the conventional methods stabilizes thesystem. Then, the proposed method reduces the d-axis
current overshoot by 67% because of the proposed
reference filter F(s) in comparison with the conventional
method. Hence, it is confirmed that the proposed method
achieves less overshoot of the output current than theconventional method in the simulations.
VI.

EXPERIMENTAL RESULTS

This chapter evaluates the stability and the transient
response of a matrix converter prototype as shown inFig.1. The prototype parameters are almost consistent
with the parameters shown from Table 1 to Table 4.
However, the input voltage is 50 Hz while the output
voltage is 30 Hz. In addition, an output d-axis current
reference is separated into a steady component of 0.4 p.u.
and a step component of 0.05 p.u.. A modulation strategyof the prototype employs the method as introduced in [3].
Fig. 13 shows the input and output waveforms of thematrix converter obtained by experiments. Fig. 13 (a)shows a result without the output damping control and (b)shows a result with the output damping control designedwith the proposed method. In Fig. 13 (a), the filter
resonance is excited, and the input current and the output
current have resonant distortions. It is noted that theexperimental result in Fig. 13 (a) shows a stability limit
although the simulation result in Fig. 12 becomes
unstable. This is because loss of a matrix converter
behaves as a damping resistor as mentioned in theprevious chapter. The input current THD (total harmonic
distortion) is 52.9% and the output current THD is 15.7%
from Fig. 13 (a). In contrast, the filter resonance issuppressed and the system is stabilized by the output
damping control designed with the proposed method. As
a result, the input and output current THDs are reduced to10.3% and 4.43%, respectively. A main cause ofremaining distortion in the input current is generated bycommutation failure because the input voltage detection
for the commutation is implemented at a power supply
not the filter capacitor voltage in the experiment. The
input current will be improved by applying a hybrid
commutation method [17]. Thus, the designed output
damping control mitigates the resonance distortions in the
input current by 81%.
Fig. 14 shows the dq-axis output current in a transient
response with the damping control designed with theconventional and the proposed methods. It should be
noted that the experimental results after this paragraphare obtained in the condition of a fixed output current
angle in common with simulations. The conventional andthe proposed methods stabilize the matrix converter andthe output current converges toward the reference.
However, the damping control designed with theconventional method generates a d-axis current overshootof 188% due to the zero in the closed-loop transfer
function. On the other hand, the proposed method
reduces the d-axis current overshoot to 76% because of
the pole-zero cancellation owing to the proposed
reference filter F(s). Hence, the proposed method reduces
the output current overshoot in transient response by 60%.
Fig. 15 shows gain curve characteristics of the output
current control designed with the proposed and theconventional methods in simulations and experiments. It
should be noted that the intended control bandwidth is
650 Hz and the sinusoidal waveform as the d-axis current
reference is used in the experiment. The proposed method
mitigates the error between the desired and the obtained
cut-off frequencies in comparison with the conventional
method since the cut-off frequency of the conventional(a) Without any damping controls. (unstable)(b) Damping control designed with the conventional method. (stable)(d-axis current overshoot: 159%)(c) Damping control designed with the proposed method. (stable)(d-axis current overshoot: 52.6%)
Fig. 12. Output d-axis current waveform in simulation. The d-axis
current overshoot is suppressed by 67 % owing to the proposed method.

(a) Without the output damping control which results in the input
current THD of 52.9%.

(b) With the output damping control designed with the proposed method
which results in the input current THD of 10.3%.
Fig. 13. Input and output waveforms of the matrix converter as
illustrated in Fig. 1 in experiments. The output damping control
designed with the proposed method suppresses the filter resonance andreduces the input current THD by 81%.

method exceeds 1 kHz because of the damping control.
In addition, the proposed method reduces gain in lower
frequency band around 100 Hz compared to theconventional method because of the proposed reference
filter F(s). As a result, the proposed method reduces theoutput current overshoot as shown in Fig. 14. It should be
noted that the error between the experimental and thesimulation results in Fig. 15 is caused by digital control
and a full consideration about the difference will bereported in the future. Therefore, it is confirmed that the
proposed design method reduces the output current
overshoot and the error of the control bandwidth of theoutput current.
VII. CONCLUSION
This paper presents a design procedure for the output
current control and the output damping control of amatrix converter to suppress the LC filter resonance andimprove a transient response. With the conventional
design method, the output damping control causes a large
output current overshoot and a control bandwidth of theoutput current is affected. In the proposed design
procedure, a reference filter to suppress an output current
overshoot and a design flow chart to obtain a desired
control bandwidth of the output current and the stability
are proposed. From experimental results, the output
damping control designed with the proposed method
suppresses the filter resonance, which results in a stable
operation. In addition, the proposed design method
reduces the output current overshoot of 60% and error
between the desired and the obtained control bandwidth
in comparison with the conventional method which isbased on a gain margin. Therefore, the validity of the
proposed design procedure is confirmed.

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

d-axis current
0.1 p.u./div

q-axis current command
0.1 pu/div (Red)q-axis current 0.1 pu/div(Green)0.4

01 ms/div(a) With the conventional method. The d-axis current overshoot is
188%.
d-axis current command
0.1 p.u./div

d-axis current
0.1 p.u./div

q-axis current command
0.1 pu/div (Red)q-axis current 0.1 pu/div(Green)0.4

01 ms/div(b) With the proposed method. The d-axis current overshoot is 76%.
Fig. 14. Output dq-axis current response with the damping control
designed with the conventional and the proposed methods. The proposed
method reduces the current overshoot in transient response by 60%.

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