Parallel Connected Multiple Drive System Using
Small Auxiliary Inverter for Numbers of PMSM
Tsuyoshi Nagano and Jun-chi Itoh
Dept. of Energy and Environmental
Nagaoka University of Technology
Nagaoka, Niigata, Japanngn244@stn.nagaokaut.ac.jp
itoh@vos.nagaokaut.ac.jp
Abstract— This paper discusses a multi-parallel drive
system for permanent magnet synchronous motors(PMSMs). This system proposes to use two different
functions of inverter, a main inverter to drive parallel
connected PMSMs and an auxiliary inverter to suppress thevibration of the motor speed. However, the power capacityof the auxiliary inverter depends on the speed response. In
this paper, the performance of the proposed system is
discussed. From the frequency characteristics, it is clarified
that the power capacity of the auxiliary inverter is lower
than 10% of the main inverter when the frequency
components included in the speed command are less than 12
rad/s. As a result, the experimental results demonstrated
that the proposed system can reduce the speed vibrationfrom 400 r/min to nearly 0 r/min with damping control in
the prototype of 1500-W PMSM drive system. The
experimental results well agree with the theoretical analysis
results.
Keywords— Permanent magnet synchronous motor,
Parallel operation, Damping control.

I.

INTRODUCTION

Recently, the adjustable speed drive techniques for thepermanent magnet synchronous motor (PMSM) haveattracted a lot of attentions among the researchers interms of the energy saving [1]-[3]. PMSM has highefficiency and high power density in comparison with theinduction motor (IM) because PMSM does not have theexcitation circuit. On the other hand, IMs can be driven
in parallel by using only one inverter. Parallel connected
multiple induction motor drive systems are applied toindustry in terms of cost reduction and simplify of thesystem, because this system only uses large power
capacity inverter to multiple induction motors [4]-[7].
The efficiency of the motor drive system is improvedby replacing IM with PMSM. However, it is difficult todrive the parallel connected PMSMs by using only one
inverter because the torque vibration occurs due to theresonance between a synchronous reactance and theinertia moment [8]. When the parallel connected PMSMs
are driven by one inverter, the current cannot be
controlled corresponding to each pole of PMSM because
the pole position is different from the pole position ofother PMSMs. Therefore, the torque vibration will occur
in this multiple motor drive system.

The damping control that uses the current feedback
has been proposed [9]. This method suppresses the torquevibration with adding the vibrational component in theeffective current to the output phase command of thevoltage command. However, this method cannot beapplied to the parallel connected PMSMs because one
inverter cannot control each current corresponding toeach pole of the PMSM.
There are some literatures which two PMSMs are
driven by one power converters [10]-[15]. These
converters can drive two PMSMs independently.
However, these converters have some problems. For
instance, the five-leg inverter [10]-[12] and the nineswitch inverter [13]-[14] is driven at low voltage
utilization ratio in comparison with a case when one
PMSM is driven by these converters because DC-link
voltage is distributed to two PMSMs. Moreover, thecurrent capacity of the switching device increases
because there is a common leg. The four-leg inverter
using the neutral point of the DC link capacitor is lessnumber of the switches than the other topologies [15].
However, this converter requires the capacitor voltage
balancing control because the capacitor neutral-point
voltage is unbalanced by the motor current. Therefore, it
is difficult to apply these converters to several motors
because the output voltage of the inverter is limited andincreasing the number of switching devices. Therefore,
for this reason, these converters are not suitable fordriving multi-parallel connected PMSMs. So, the authors
have proposed a drive system for multi-parallel
connected PMSMs by the auxiliary inverter and motor
windings [16]-[17].
In this paper, the acceleration performance of the
proposed system is evaluated from the frequency
characteristic for the speed command, the acceleration
test and the simulation results that two sets of parallel
connected PMSMs are driven. From the frequency
characteristics of the transfer function from the speedcommand to the rotational speed, it is clarified that theresonance is suppressed by the damping control applying
to the auxiliary inverter. Moreover, the minimum power
capacity of the auxiliary inverter is clarified from thefrequency characteristics of the transfer function from thespeed command to the output power of the auxiliary
inverter.
This paper is organized as follows; first, theconfiguration and the control strategy of the proposedparallel drive system are introduced. Next, thesuppression effect of the damping control from thefrequency characteristics from the speed command torotational speed in the proposed system. In addition, thestability of the proposed system with the damping control
is clarified in the Routh-Hurwitz stability criterion. Next,
the minimum power capacity of the auxiliary inverter
clarified from the frequency characteristics from the
speed command to the output power of each inverter inthe proposed system. Finally, the simulation andexperimental results demonstrate that the proposedsystem can drive multiple of parallel connected PMSMs
effectively by suppressing the torque vibration.
II.

Fig. 1.Configuration of the proposed system. The
proposed system uses two different power rating
inverters, the large power capacity inverter to drive themotors and the small power capacity inverter tosuppress the vibration.

PROPOSED SYSTEM AND CONTROL STRATEGY

Fig. 1 shows the configuration of the proposed system.
In PMSMs, the auxiliary windings which are used in thedamping control (with the auxiliary inverter) are placedin the slots together with the main windings. The
proposed system uses two different power rating inverters.
The first one is the large power capacity inverter for themain windings to control the speed of the parallel
connected PMSMs. The second one is a small power
capacity inverter for the auxiliary windings to suppress
the torque vibration. In term of the effectiveness of the
proposed system, it is very important that the power
capacity of the auxiliary inverter is enough small.
Fig. 2 shows the control block diagram of the proposedsystem. In the proposed system, the V/f control is applied
to the main inverter and the field-oriented control and thedamping control are applied to the auxiliary inverter.
Each of the auxiliary inverter controls the current inauxiliary windings of the PMSM in order to suppress thetorque and speed vibrations. Since these vibrations arecaused by the phase difference between the rotational
coordinates of the inverter and of the PMSM, it can besuppressed by the damping control in the auxiliary
inverter.
Fig. 3 shows the relationship between the d-q rotating
frame and the γ-δ rotating frame for a V/f control. In thecontrol of the auxiliary invereter, the direction of the flux
vector with permanent magnet is defined as d-axis assame as the conventional FOC. In the V/f control, theoutput voltage vector is defined as δ-axis, the axis whichlags by π/2 rad from δ-axis can be defined as γ-axis. The
lag of the load angle ϕ occurs between the d-q rotating
frame and γ-δ rotating frame as shonwn in Fig. 3.
Therefore, the load angle ϕ, rotational speed ωre and the
speed command ω* can be expressed as
p ϕ = ω re − ω * ............................................................(1)where p is differential operator.
When the vibration of the speed is caused by theresonance between the inertia moment and thesynchronous reactance, the load angle ϕ also is vibrating
as shown (1). Then, the changes of the load angle pϕ is

1sFig. 2. Control block diagram of the proposed system.
The V/f control is applied to the main inverter. The
auxiliary inverters uses the field-oriented control for thedamping control.

Fig. 3. Relationship between the d-q rotating frame and theγ-δ rotating frame. The γ-δ rotating frame lags by the load
angle ϕ from the d-q rotating frame.

the difference between the rotational speed and the speedcommand. In order to compensate the changes of the load
angle pϕ, the q-axis current command iq* is calculated
from the damping controller as shown in Fig. 2. As aresult, the vibration of the speed and torque caused by theresonance are suppressed by compensating the changes ofthe load angle pϕ with the current controlled by FOC andthe damping control.
III.

EVALUATION OF SUPPRESSION PERFORMANCE

The damping control has to suppress the speed andtorque vibration caused by the resonance between theinertia moment and the synchronous reactance. In thischapter, the suppression effect of the damping control isclarified from the frequency characteristics from the
speed command to rotational speed in the proposedsystem.
Fig. 4 shows the simplified system in order to confirm
the operation of the damping control. The proposedsystem uses the PMSM placed the auxiliary windings in
the slots together with the main (conventional) windings,
and so the mutual magnetic interference occurs between
the main and the auxiliary windings. Due to the above
reason, the control for the auxiliary inverter becomescomplicated. Therefore, the proposed system is validated
using a model where two PMSMs are connected in series
via single shaft. Then, the rear end of the main PMSM isconnected to the load machine. It means that themagnetic coupling was neglected in the simulation andthe experiment.
The voltage equation of the motor that is connected tothe main inverter is represented as the voltage equation in
the γ-δ rotating frame because the main inverter is driven
by the V/f control. Note that the γ-δ rotating frame lags
by the load angle ϕ from the d-q rotating frame as shownin Fig. 3. On the other hand, the voltage equation of the
motor that is connected to the auxiliary inverter isrepresented as the voltage equation in the d-q reference
frame because the auxiliary inverter control is FOC.
Therefore, the voltage equation of the motor that isconnected to the main inverter in the estimated rotating γδ frame is given by⎡vMγ ⎤ ⎡ RM + pLdM
⎢ ⎥=⎢*⎣vMδ ⎦ ⎣⎢ ω LdM− ω * LqM ⎤ ⎡iMγ ⎤⎡ sin ϕ ⎤⎥ ⎢ ⎥ + ωreψ m M ⎢⎥ .......... (2)RM + pLqM ⎦⎥ ⎣iMδ ⎦⎣cos ϕ ⎦On the other hand, the voltage equation of the motor thatis connected to the auxiliary inverter in d-q reference
frame can be given by⎡v Ad ⎤ ⎡ R A + pLdA
⎢ ⎥=⎢⎣⎢v Aq ⎦⎥ ⎣⎢ ω re LdA−ω re LqA ⎤ ⎡i Ad ⎤ ⎡ 0 ⎤................... (3)⎥⎢ ⎥ +R A + pLqA ⎦⎥ ⎣⎢i Aq ⎦⎥ ⎢⎣ω reψ m A ⎥⎦Moreover, torque and speed equations can be given by
T = TM − TA =p ω re =()

3
3
PfMψ mM iMγ sinϕ + iMδ cosϕ − PfAψ mAiAq ........... (4)
2
2

(

1
PfM TM − PfAT A
J) .................................................. (5)where p is differential operator, R is armature resistance,
L is synchronous reactance, Pf is pairs of the pole, Ψm ismagnet flux linkage, ϕ is the load angle, and J is theinertia moment of motors, Suffix ‘A’ represents theparameter of the motor that is connected to the auxiliary
inverter, ‘M’ represents the parameter of the motor that isconnected to the main inverter.
Equations (2) and (3), (4) are non-linear so that these
R
⎡
− M
⎢
LM
⎢
⎡ pΔi Mγ ⎤ ⎢
ω
−
0
⎥ ⎢
⎢
⎢ pΔi Mδ ⎥ ⎢⎢⎢ pΔi Ad ⎥0⎥=⎢⎢
⎢ pΔi Aq ⎥ ⎢⎢ pΔω ⎥ ⎢
0
re ⎥
⎢
⎢
⎢
⎣⎢ pΔϕ ⎦⎥ ⎢ 3 PfM 2ψ mM
sin ϕ 0
⎢2
J
⎢
0
⎣

ω0
−

RM

0
0

0

0

R
−
LA

0

0

0

LM

3 PfM ψ mM
cos ϕ 0
2
J
0

0

−

−

R
LA
2
3 PfA ψ mA
2

0

ψ mM

sin ϕ 0

LM

ψ mM
LM

cos ϕ 0
0

−

2

0

−

0

J

0
0
−1

Figs.4 Analysis and verification model of the PMSM in
addition the auxiliary windings for damping control.

∫

1
Ti

Fig. 5. State variable diagrams of the auxiliary inverter
control. If the current control response is faster than thedamping control response, the loop gain in the current
control is 1. Therefore, the q-axis voltage command
ΔvAq* can be given by (10).

equations are linearized about the steady state. The state
equation after the linearization can be given by (6).
Equation (6) shows the 6th order of the state equationthat is complicated to evaluate the stability. In order tosimplify this equation, it is assumed that the mechanicaltime constant is larger than electrical time constant, then(6) can be approximated as the 2nd order state equation, as(7).
2
2⎡ 3 PfM 2ψ mM ψ mM RM
3 PfM ψ mM ⎤ Δω
⎡ pΔωre ⎤ ⎢−⎥ ⎡ re ⎤2
ω0 LdM LqM 2 JLqM ⎥ ⎢J⎢ pΔϕ ⎥ = ⎢ 2⎥⎣⎦ ⎢⎣ Δϕ ⎦
⎥
−
1
0
⎣
⎦
2
⎡ 3 Pf 2ψ mM RM⎤ ⎡ΔvMδ ⎤ .........(7)3 Pf ψ mA−0⎢⎥ ⎢ Δv ⎥+ ⎢ 2 Jω0 2 Ld Lq
2 JRA⎥ ⎢ Aq ⎥⎢⎣00
1 ⎥⎦ ⎢⎣ Δω1 ⎥⎦where suffix ‘0’ is the steady state value.
Fig. 5 shows the state variable diagrams of the auxiliary
inverter control that is applying the damping control. In
Fig. 5, the q-axis voltage command ΔvAq*, it is assummedthat the loop gain in the current control is 1 (assumingthat the current control response is faster than thedamping control response), can be given by
Δ v Aq = − K d (Δ ω * − Δ ω re ) ................................... (8)p Δ x 1 = 0 .............................................................. (9)where Kd is the damping gain.
Moreover, in Fig. 3, the δ-axis voltage command ΔvMδ*and the γ-axis voltage command ΔvMγ* of the mainω 0ψ mM

⎤
⎡ 1
⎥
⎢
⎥
L
ω 0ψ mM⎥ ⎡ Δi Mγ ⎤ ⎢ M
sin ϕ 0⎥⎢⎥ ⎢⎢ 0LM⎥ ⎢ Δi Mδ ⎥⎥ ⎢ Δi ⎥ ⎢0⎥ ⎢ Ad ⎥ + ⎢ 0⎥ ⎢ Δi Aq ⎥ ⎢⎥⎢⎥ ⎢0⎥ ⎢Δω re ⎥ ⎢ 0⎥ ⎢ Δϕ ⎥ ⎢2⎦ ⎢3 PfM ψ mM⎥⎣
i Mγ 0 cos ϕ 0 − i Mδ 0 sin ϕ 0 ⎥
⎢ 0
2
J
⎢⎣ 0
⎥
0
⎦
−

LM

(

cos ϕ 0

)

0
1
LM
0

0

0

0

0

1
LA

0

0

0
0

0
0

0
1
LA
0
0

⎤
i Mδ 0 ⎥⎥⎥− i Mγ 0 ⎥ ⎡ Δv Mγ ⎤⎥⎢⎥ ⎢Δv Mδ ⎥0 ⎥ ⎢ Δv Ad ⎥⎥⎢⎥⎥ ⎢ Δv Aq ⎥0 ⎥ ⎢ Δω * ⎥
⎥⎣
⎦
0 ⎥⎥
1 ⎥⎦

......................................................................................................................................................................................... (6)Table 1. Analysis and experimental conditions

inverter are given byΔv Mγ = 0 ............................................................... (10)Δ v M δ = ψ mM Δ ω ................................................. (11)-1

Rated speed [min ]Rated current [A]Number of pole pairs
Armature resistance [Ω]d-axis inductance [mH]q-axis inductance [mH]Electro-motive force constant [Vs/rad]Assuming as δ0 = 0, iMγ0 = 0, iMδ0 = 0, ωL is larger thanR, substituting (8), (10) and (11) into (7) yields the state
equation ofpx = Ax + Bu ................................................. (12)where x = [Δω re

Δδ ] , u = Δ ω *⎡ 3 PfM 2ψ mM
Kd
⎢−
A=⎢ 2
J
⎢⎣
−1

3 PfM ψ mM
2 JLqM02

2⎤⎡ 3 PfM 2ψ mM⎤⎥, B=⎢
Kd ⎥
J
⎥
⎢2
⎥
1
⎥⎦
⎣⎢
⎦⎥

3 Pf ψ mM
J
2

3 Pf ψ mM
3 Pf ψ mM
Kd s +J
2
2 JLqM2

2

2

Fig. 6. Frequency characteristics of the speed command to
the rotational speed in the proposed system. Table I showsthe evaluated condition. By applying the damping control, themagnitude at the resonance frequency decreases to 5 dB.

×

Therefore, the suppression effect of the damping control
is clarified from the frequency characteristics from the
speed command to rotational speed in the proposedsystem. The proposed damping control can suppress theresonance between the moment inertia and synchronous
reactance as show in Fig. 6. In the experiment, when theconstant speed command adds sinusoidal wave as the
speed command, the amplitude in the frequency of thesinusoidal wave is drawn by observing the speed
response. The gain between the speed commands androtational speed in the experimental results has beenplotted in Fig. 6 as dot marks. The experimental results
well agree with the analysis results.

⎧⎪ψ⎛(iMγ 0 cos ϕ0 + iMδ 0 sin ϕ 0 ) − ψ mM sin ϕ ⎞⎟⎫⎪
mM
− s⎜ K d +
⎨
0 ⎬
⎜
⎟⎪
L
ω0
LqM ω0⎪⎩ qM⎝⎠⎭.............................................................................. (13)The damping factor ζ and the natural angularfrequency ωn are given byζ=

3 PfM Kd
2 2ωn =3 PfM ψ mM
2
JL qM

LqM
J

........................................... (14)................................................ (15)Without the damping control (ζ = 0), the magnitude at
the resonance frequency rises. By contrast, by applying
the damping control (ζ = 0.3), the magnitude at theresonance frequency decreases to 5 dB. The highsuppression effect is obtained by increasing the damping
gain as shown (14) because the damping factor increases.
⎡
R
−
⎢
L
⎢
⎡ pΔiMainγ ⎤ ⎢
ω
−
0
⎢
⎥ ⎢
⎢ pΔiMainδ ⎥ ⎢
⎢ pΔi
⎥ ⎢
0
Auxd
⎢
⎥ ⎢
⎢ pΔi Auxq ⎥ ⎢0⎢⎥=⎢⎢ pΔωre ⎥ ⎢⎢ pΔδ ⎥ ⎢ 3 P 2ψ
f
m
⎢
⎥ ⎢
sin δ
⎢ pΔxacr _ d ⎥ ⎢ 2 J
⎢ pΔx
⎥ ⎢
0
acr _ q ⎦
⎣
⎢
0
⎢
⎢⎣
0

ω0

0

0

−

R
L

0

0

−

R Ki
−
L
L

0

−

0
0

−

0

2

3 Pf ψ m
cos δ
2 J
0

0

0
0

− Ki
0

0

R Ki
−
L
L
2
3 Pf ψ m

−

2

1800
8.2
4
3
3
1.55
1.98
11.5
15.2
23
33.2
0.368 0.338

2

2

s2 +

PMA
750

0.0051 0.0026
Inertia moment [kgm ]Rotational speed in stationary state [rad/s]
900Fig. 6 shows the frequency characteristic of thetransfer function from the speed command to rotational
speed of the V/f control from the main motor. Table 1
shows the analysis and experimental conditions. In Fig.
6, the magnitude becomes lower at the resonancefrequency ωn by applying the damping control incomparison to the proposed system without the damping
control. It means that the suppression effect becomes
higher.
The transfer function is derived from the 2nd state
equation of (12). The speed command Δω* to therotational speed Δωre transfer function is given by
Δωre
=
Δω *

PMM
1500

Rated power [W]*IV.

STABILITY OF PROPOSED SYSTEM

In the previous section, it is confirmed that the PMSM
can suppress the resonance by applying the damping
control in the proposed system. In this chapter, thestability of the proposed system with the damping control

ψm
L

ψm
L

sin δ 0

−

ω0ψ m

0

sin δ 0

0

0

1
LTi

0

0

ω0ψ m

cos δ 0L

0−cos δ 0L

K d Ki
L
2

J
0

−1

3 Pf ψ miγ cos δ − iδ sin δ
2 J
0

0
− Ki

0
− K d Ki

0
0

0

(

)

0
0
0
0

⎤
0 ⎥
⎥
⎡
⎤ ⎡ iδ 0 ⎤0 ⎥ ⎢ ΔiMainγ ⎥ ⎢ψ⎥⎥ Δi
⎢ m − iγ 0 ⎥⎥ ⎢ Mainδ ⎥ ⎢ L
0 ⎥ ⎢ Δi Auxd ⎥ ⎢ 0 ⎥⎥⎥⎥⎢⎢ K K ⎥
1 ⎥ ⎢ ΔiAuxq ⎥ + ⎢ d i ⎥ Δω *⎥ ⎢ Δω ⎥ ⎢ L ⎥re ⎥LTi ⎥ ⎢⎢ 0 ⎥⎥ ⎢ Δδ ⎥ ⎢ 1 ⎥
⎥
0 ⎥⎢
⎥
⎢
⎥ ⎢Δxacr _ d ⎥ ⎢ 0 ⎥⎥0 ⎥ ⎢ Δx⎣ acr _ q ⎦ ⎢⎣ K d K i ⎥⎦
0 ⎥
⎥
0 ⎥⎦

........................................................................................................................................................................................ (16)is clarified in the Routh-Hurwitz stability criterion.
The state equation of the proposed system which isapplied with the damping control is given by (16).
There are two integrators in the auxiliary inverter
control because the field-oriented control has d-axis andq-axis current control as shown in Fig. 3. Thus, the statevariables becomes from six to eight state variables when
the auxiliary inverter is applied with the damping control.
The Routh-Hurwitz stability criterion is based onordering the coefficients of the characteristic polynomialwhich have the real coefficients of (17).
a0 s n + a1 s n−1 + " + an−1 z + an = 0 ............................. (17)The proposed system is stable when all roots thecharacteristic polynomial of (17) is negative. In Hurwitz
stability criterion, the necessary and sufficient conditions
for all roots of the characteristic polynomial of (17) are as
follows;
1) There are All coefficients (a0, a1, a2,…., an).
2) All coefficients have same sign.
3) All Hurwitz determinants (Δ1, Δ2,.., Δn-1) are positive.

Table 2 Calculation of Hurwitz determinants
Results

Hurwitz determinants
Δ1

8458

>0

Δ2

2.138x1011
9.616x1021

>0

Δ4
Δ5

5.479x1035
9.657x1051

>0
>0

Δ6

5.691x1070>0Δ7

3.486x1091

>0

Δ3

>0

Table 3 Coefficient of the characteristic polynomial
Coefficients

Coefficients

a0

1

a4

7.404x1013

a1

8458

a5

2.765x1016

a2

3.339x107

a6

9.413x1018

a3

6.866x1010

a7

9.888x1020

a8

9.143x1022

Δ 1 = a1

Δ2 =

a1

a3

a0

a2

a1

a3

a 5 " a 2 n −3

a0
0

a2
a1

a4 " a2n−4
a3 " a 2 n−5

Δ n −1 = 0

a0

a 2 " a 2 n −6

0
#

0
#

a1 " a2 n −7
# %
#

0

0

0

"

an −1

Table 2 shows calculation of Hurwitz determinants andTable 3 shows the coefficient of the characteristic
polynomial in the state equation of the proposed system
which is applied with the damping control of (16). The
analysis condition is shown in Table 1. All Hurwitz
determinants are positive as shown Table 2. Moreover,there are All coefficient which have same sign. Therefore,
because of the necessary and sufficient conditions, theproposed system is stable.
V.

MINIMUM POWER CAPACITY OF AUXILIARY
INVERTERIn the previous section, it is confirmed that the PMSM
can be stabilized by applying the damping control in the
proposed system. However, if the damping gain Kd
increases to obtain the larger suppression effect, theoutput power of the auxiliary inverter becomes larger.
Moreover, the power capacity of the auxiliary inverter
depends on the speed command response. Thus, in thissection, the minimum power capacity of the auxiliary
inverter is clarified from frequency characteristics.
Fig. 7 shows the frequency characteristic of thetransfer function from speed command to output powerof the main inverter, the auxiliary inverter, and the sumof the inverters in the proposed system. In the proposedsystem, the main torque cannot be controlled directly due

Fig. 7. Frequency characteristics of the transfer function
from speed command to output power of the main inverter,
the auxiliary inverter, and the sum of the inverters in the
proposed system.

to the V/f control. However, by calculating the torquetransfer function from the speed one, the frequency
characteristic of each output power is derived. If theelectrical and mechanical losses are neglected, the output
power of the auxiliary inverter depends on themechanical output of the motor. In other words, thetransfer function of the output power for the auxiliary
inverter is derived from the mechanical output transfer
function of the motor. However, the mechanical output isnonlinear because the mechanical output is the product of
two variables, the rotational speed and the torque. Due tothis reason, the output power transfer function of theauxiliary inverter is derived after the mechanical output islinearized. The speed command to the output power of
the auxiliary inverter transfer function is given bys 2ω 0 K dψ mA
ΔPA..........(18)
=
2
2
2
Δω *
PfM ψ mM
PfM ψ mM2s +Kd s +JJLqMIn Fig. 7, the output power of the auxiliary inverterrises near the resonance frequency. On the other hand,when the frequency components into the speed command
are lower and lower, the output power of the auxiliary
inverter is suppressed more. In other words, the auxiliary
inverter does not contribute steady state i.e. constant
speed. Nevertheless, although the output power of themain inverter is constant because the main inverter drives
the motor. Note that the damping factor is designed by
the equations which are derived by the authors.
Figs. 8 show the experimental results that each output
power is measured at each operating point in Fig. 5 when
0.5 p.u. of the constant speed command adds sinusoidal
wave as the speed command. Assuming that the inverter
output voltage is equal to the voltage command, each
output power is calculated from the voltage command andthe measured current. The waveforms in Fig. 8(a), (b) and(c) mention the operation at the point (a), (b) and (c) in
Fig. 7, respectively. Comparing with each result, theamplitude ratio between the output power of the auxiliary
inverter and the main inverter is changed corresponding to
the frequency components included in the speed command.
When the frequency component included in the speedcommand is lower than 32 rad/s, the output power of theauxiliary inverter can become lower than 10% of the main
inverter as shown Fig. 7. This boundary condition isdecided by inertia moment J, q-axis inductance LqM,
magnet flux linkage Ψm and the damping gain Kd as
shown (14). When the damping gain Kd increases, theoutput power of the auxiliary inverter increase from 10%of the main inverter at 32 rad/s because the damping
factor increases. Therefore, the results show that thepower capacity of the auxiliary inverter can be designed tonearly several percent of the main inverter when the
proposed system is applied to the application which
requires a slow speed response such as fan applications. In
this session, in order to confirm the validity of proposedsystem in terms of acceleration which means the rate of
the speed command, the relationship between the output
power of the auxiliary inverter and the overshoot of therotational speed for speed command is demonstrated in
the experiment.
The output power of the auxiliary inverter was
discussed by frequency characteristics. However, theacceleration or deceleration time is set as the speedcommand in practically. In this session, the output powerof the auxiliary inverter is discussed by acceleration
which means the rate of the speed command.
The time response of the rotational speed is derived
from the inverse Laplace transform of the ramp responseof (13). In this section, in order to simplify (13), it isassumed as ϕ0 = 0, iMγ0 = 0 and iMδ0 = 0 in (13). Moreover,
the overshoot of the rotational speed can be derived fromthis time response, as given by
α sin⎛⎜ωn t peak 1 − ζ 2 ⎞⎟⎝⎠Δω re _ Overshoot =............... (19)2
ωn 1 − ζ exp(ζωn t peak )t peak =)(2
2
1 log 2ζ − 1 + 2ζ ζ − 1................. (20)2
ω ζ 2 −1nwhere α [rad/s2] is acceleration.
The overshoot of the rotational speed can be calculated
by using (19).
Similarly, the time response of the output power of theauxiliary inverter during the acceleration is also derived
from the inverse Laplace transform of the ramp response

Figs. 8. Experimental results that each output power ismeasured at each operating point ((a) is at 1 Hz, (b) is at 3
Hz, (c) is 6 Hz) in Fig. 7 when 0.5 p.u. of the constant
speed command adds sinusoidal wave as the speedcommand.
ζ =3 Pf K d
2
2
TRn =

LM
J

Jω n

αn =

Tn
Tn

J

Fig. 9. Relationship between the output power of theauxiliary inverter and the overshoot of the rotational speed
the against speed command. Table 1 shows also theevaluated condition. The higher suppression effect isobtained in exchange for the increase of the output powerof the auxiliary inverter by increasing the damping gain.

of (18). In this section, in order to simplify (18), it isassumed as δ0 = 0, iMγ0 = 0, iMδ0 = 0 in (18). Moreover, themaximum output power of the auxiliary inverter can bederived from this time response, as given by
ΔPaPeak =(Pf ψ mA K d ω 0α sin ω n 1 − ζ 2 t peak

) .. (21)ω n 1 − ζ exp(ζω n t peak )The maximum output power of the auxiliary inverter
can be calculated by (21).
Fig. 9 shows the relationship among the damping gain
Kd, the maximum output power of the auxiliary inverter
during the acceleration and the overshoot of the rotational
speed when the PMSM is accelerated during one, half,quarter of the rated acceleration αn on the condition asshown in Table 1. The damping gain is standardized by
the damping factor ζ as given by (14). The output powerof the auxiliary inverter rises with the increase of thedamping factor. On the other hand, the overshoot of therotational speed decreases with the increase of thedamping factor, as shown in Fig. 9. The maximum output
power of the auxiliary inverter (i.e. the power capacity of
the auxiliary inverter) depends on the acceleration factor
α in the speed command as shown in Fig. 9. The
experimental results well agree with the calculation.
2

Fig. 10. Acceleration and deceleration test without/with damping control in motor-generator set. (a) After the acceleration,
the 400 r/min - speed vibration is maintained. (b) The speed vibration is reduced from 400 r/min to nearly 0 r/min in comparedwith (a)VI.

MOTOR DRIVE PERFORMANCE EVALUATIONIn this chapter, in order to confirm the effectiveness ofthe damping control, the experiments are conducted witha motor –generator set.
Fig. 10 shows the experimental results that illustrate
the motor speed vibration when the proposed system isapplied (a) without the damping control and (b) with thedamping control ( ζ = 0.3 ) in an acceleration anddeceleration test. In this experiment, it is difficult tomeasure the torque response directly. Therefore, thespeed vibration is evaluated instead of the torqueresponse. In Fig. 10(a), the proposed system isimplemented without the damping control. The speed
vibration occurs during the acceleration. After theacceleration, a 400 r/min of speed vibration and the 10
Ap-p of current vibration in q-axis of the main inverter are
maintained. On the other hand, Fig. 10(b) demonstrates
the experimental results, where the proposed system isimplemented with the damping control ( ζ = 0.3 ). The
effectiveness of the auxiliary inverter from the results
confirms that the speed vibration is reduced from 400
r/min to nearly 0 r/min in compared with the acceleration
test of Fig. 10(a). The 10 Ap-p of the current vibration in
the q-axis of the main inverter is suppressed as well.
Then, the output power of the auxiliary inverter issuppressed to 13% of the main inverter when thedamping factor is designed at 0.3. In the same way as theacceleration test of Fig. 10(a), after the deceleration, a
500 r/min of speed vibration and the 15 Ap-p of current
vibration in q-axis of the main inverter are maintained asshown in the deceleration test of Fig. 10(a). In contrast, it
is confirmed that, as shown the deceleration test of Fig.
10(b), the speed vibration is reduced from 500 r/min tonearly 0 r/min and the current vibration is reduced from15 Ap-p to 0 Ap-p in compared to Fig. 10(a) in the same

way as acceleration test. Nevertheless, it is confirmedthat the q-axis current of the auxiliary inverter flows only
during acceleration and deceleration. Moreover, themaximum q-axis current of the auxiliary inverter is 20%of the q-axis current of the main inverter. Therefore, it isconfirmed that the auxiliary inverter can suppress the
speed vibration via auxiliary windings with a small q-axis
current of the auxiliary inverter even if in the accelerationand deceleration test.
VII.

EVALUATION OF PARALLEL DRIVE SYSTEM

In previous chapter, the effectiveness of the damping
control in the proposed system is discussed. This chapter
discusses the effectiveness of the proposed system when
two PMSM are driven.
Fig. 9 shows the simulation models that are used toverify the operation of two sets of parallel connected
PMSM. Since it is difficult to construct the prototype ofthe drive system using two PMSMs, the simulation isused to evaluate the validity of the proposed controller as
first step.
Fig. 10 shows the simulation results when two sets ofparallel connected PMSMs are driven by the proposedsystem with the damping control as shown in Fig. 8.
Smooth acceleration progresses are confirmed in the two
set of parallel connected PMSMs. Besides, when the
rated motor speed, load step applies to PMSM1 at 0.38s,
and later also applies to PMSM2 at 0.42s, the operationof the two auxiliary inverters can be observed from theoutput power. The maximum output power of theauxiliary inverter is approximately 0.25 p.u. of the rated
power of the main inverter. Although the power capacityof main inverter increases with the increase of parallel
units, the power capacity of auxiliary inverter does notchange according to the numbers of parallel units.

VIII.

CONCLUSION

This paper discusses the acceleration performance ofthe proposed parallel connected multiple PMSM drive
system from the frequency characteristics and theacceleration test. From the frequency characteristics ofthe speed response, it was clarified that the resonance issuppressed by applying the damping control. In addition,
the output power of the auxiliary inverter is suppressed to
10% of the main motor in the speed response of lower
than 32 rad/s in the prototype of 1500-W PMSM drive
system. The results show that the power capacity of theauxiliary inverter is designed to nearly several percent of
the main inverter when the proposed system is applied tothe application which requires a slow speed responsesuch as fan applications.
In the future work, the driving several units of the
motor-generator in parallel with the proposed system
drives will be evaluated in the experimental system.

Fig. 9. Simulation model for parallel connected dual
motor drive. In order to neglect the magnetic coupling,
two parallel drive of the proposed system is validated
using Fig. 8.

ACKNOWLEDGMENT
A part of this study was supported by Industrial
Technology Grant Program in 2011 from New Energy andIndustrial Technology Development Organization(NEDO) of Japan.
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