Simplified Stability Analysis Method forMultiple Servo Drive Systems Connected toCommon DC Bus Line
Katsuki Miura
Dept. of Electrical, Electronics, andInformation Engineering
Nagaoka University of Technology
Nagaoka, Japank_miura@stn.nagaokaut.ac.jp

Hiroki Watanabe
Dept. of Electrical, Electronics, andInformation Engineering
Nagaoka University of Technology
Nagaoka, Japanhwatanabe@vos.nagaokaut.ac.jp

Keisuke Kusaka
Dept. of Electrical, Electronics, andInformation Engineering
Nagaoka University of Technology
Nagaoka, Japan
kusaka@vos.nagaokaut.ac.jp

Jun-ichi Itoh
Dept. of Electrical, Electronics, andInformation Engineering
Nagaoka University of Technology
Nagaoka, Japan
itoh@vos.nagaokaut.ac.jp

Takeshi Kiribuchi
OMRON Corporation
Kyoto, Japantakeshi.kiribuchi@omron.com

Hiroyuki Tokusaki
OMRON Corporation
Kyoto, Japanhiroyuki.tokusaki@omron.com

Abstract— This paper proposes a simplified stability analysis
method based on Middlebrook’s method for multiple servo
drive systems connected to the DC-bus line with the line
inductance and resistance. The impedance of the DC-bus line
causes interference among each current-control bandwidth forthe motor. This paper reveals the relationship between themultiple servo drive systems stability limit and the DC-bus
impedance with the simplified stability analysis method. The
validity of the proposed method is demonstrated by comparison
with the mathematical analysis and experiment results. As aresult, the stability limit calculated by the proposed method ismatched within an error of 2%. This paper also clarifies thecause of the errors between the mathematical analysis results
and the experimental results in terms of regarding system loss.
Keywords— DC-Bus System, Servo Drive, Impedance-based
Stability Analysis

I. INTRODUCTION
In recent years, servo-drive systems have been usedwidely in industrial applications. It is generally required toachieve high control performance, low noise, and high power
density[1-5]. In the conventional configuration of the servo
drive systems, the control panel, including servo drivers andservo motors, are connected by AC lines. However, theconventional configuration has a problem which is a radiatedswitching noise from AC lines. Thus, countermeasures areneeded to prevent interference from other pieces of equipment
such as shielding lines and the layout of power and signal lines.
However, the costs are increased due to the countermeasurefor the noises.
In order to solve this problem, the DC-bus servo drive
system, which connects the servo driver to the DC-bus line,
has been attracted [6]. This configuration reduces the radiationnoise due to the shorter length of the three-phase power line
cables and the loss due to no rectifier stage. Moreover, the DCbus servo drive systems prevent the complicated wiring andreduce the cost of the noise suppression since the number oflong power lines.
However, the DC-bus servo drive systems have theproblem of system instability due to the interference between

the line impedance of the DC-bus and the current control bandfor the motor [7-13]. The over-voltage protection will trip thesystem due to oscillation in the bus current and DC-link
capacitors voltage when the DC-bus servo drive system isunstable. This interference occurs due to the negative
resistance characteristics of the servo driver, which is aconstant power load. Also, the system stability is wrong when
the servo driver’s small DC-link capacitance is used to achieve
high power density [14-16]. Therefore, the stability analysis
and damping control method for the DC-bus servo drive
systems are required. The stability analysis method for theDC-bus system is proposed by Middlebrook [17]. In
Middlebrook’s analysis method, the system is divided into
source and load sides. The system stability is analyzed by the
Nyquist plot of the product of the output impedance of thesource side and the input admittance on the load side. Ref. [6]has derived the input admittance of the servo driver by the
extra element theorem and proposed the stability analysis
method for the DC-bus servo drive system based on
Middlebrook’s method. However, the stability of the multiple
DC-bus servo drive system is not discussed in Ref. [6],although there are many situations to use many servo drivers
in a DC-bus line. In particular, the relationship between DC
line impedances and stability limit is unclear.
This paper proposes a simple stability analysis method for
multiple DC-bus servo drive systems. The new contributionof this paper is that the relationship between the systemstability and the line impedance of the DC-bus is clarified bya simple equation based on Middlebrook’s method. The
validity of the proposed analysis method is evaluated by the
analysis results using a full-order state equation andexperimental results. The stability limit calculated by the
proposed analysis agrees with that of the state equation
analysis within 2%. Furthermore, this paper discusses theinfluence of not only the DC-bus line but also the line between
the DC-bus and each servo driver because the proposed
analysis method is based on the condition that the DC-bus line
is longer than the line between the DC-bus and each servo
driver. In addition, the stability limit calculated by theproposed analysis method is evaluated by the operating
condition of the servo motors in the experiment. As a result,the stability limit in the proposed analysis agrees with that in
the experiment within 11% of the rated power. Moreover, thecause of the error is also discussed with the system losses ofthe experimental setup.
II. SYSTEM CONFIGURATION OF DC-BUS SERVO DRIVESYSTEM
A. Circuit configuration
Fig. 1 shows the system configuration of the DC-bus servo
drive system analyzed in this paper. Table 1 shows the circuit
parameters of the analyzed system. Fig. 1 shows that two
servo drivers are connected to a common DC bus with longlines. Note that this model consists of two servo drivers for
simplification of the following analysis. In Fig. 1, Lbus is theline impedance of the DC-bus, Rbus is the line resistance of theDC-bus, Lb1 and Lb2 are the line impedances between the DCbus and each servo driver, Rb1 and Rb2 are the line resistances
between the DC-bus and each driver, Cb1 and Cb2 are the DClink capacitors of the servo driver. The value of the input
capacitance of the servo driver is small in order to achieve
high power density. Note that line impedance is included in
the DC-bus line and the line between the DC-bus and each unit.
B. Controller configuration
Fig. 2 shows the control block diagram of each servo
driver with Field-Oriented Control (FOC). The PI controller isused for the current control of the d-axis and q-axis. In
addition, the zero d-axis current control and the feed-forward
compensation for canceling the interference of the permanent
magnetic(PM) motor are applied.
as

The transfer function of the PI controller PI(s) is expressed(1),

where Kp is the proportional gain, and Ti is the integration time.
Kp and Ti are designed to cancel the zero point in the closed
transfer function of the current control. The proportional gain
Kp and the integration time Ti are expressed as K p = c LmTi = Lm / RaM

vb1

Lb2 Rb2

ibus

Vdc

#1

INV #1ib1 Cb1

Lbus Rbus

iin2

INV #2

ib2

M

vb2

Cb2

#2

Common Bus
Fig .1Analyzed system configuration of DC-bus servo drive
system with two servo drivers connected to the DC-bus.
TABLE 1 Analyzed circuit parameters.

Circuit
Lbus

Wiring indutance of DC Bus

1 mH

Rbus

Wiring resistance of DC Bus

0.02 W

Cb1, Cb2

Inverter input capasitance

13 mF

Vdc

DC bus voltage

280 V

Motor (#1, #2)
Ra

Wiring resistance

1.4 W

La

Wiring inductance

3.41 mHKe

Back-EMF constant

0.051 V/(rad・s)

Pf

Pole pairsreLaiq
0

-

+
-

PI

vd*

+

iq* +
-

PI
iq

+

5

qre
dq
uvw

id

1 
PI (s) = K p 1 + sTi iin1

Lb1 Rb1

+

3

2
Vdc

vq*

vu*
vv*
vw*

re(Laid+Ke)

Fig. 2Control block diagram with the Field Oriented Control(FOC) in each controller. Zero d-axis current control and feed-forward
compensation for canceling the interference of the PM motor are
applied.

(2),where c is the cutoff angular frequency of the current
controller.
C. State equation of the DC-bus servo drive system withlong lines
The equivalent small-signal model of this systemapproximated near the operating point is expressed as aneighth-order state equation. The state equation of the analyzed
system is expressed as ib1   A11 v   A
 b1   21
 iin1   0 d  vint1   0=
dt  ib 2   A51  vb 2   0
 i   0
 in 2   vint 2   0

A12
0
A32
A42
A52
0
0
0

0
A23
A33
A43
0
0
0
0

0
0
A34
0
0
0
0
0

A15
0
0
0
A55
A65
0
0

A16
0
0
0
A56
0
A76
A86

0
0
0
0
0
A67
A77
A87

0   ib1   B1 
0   vb1   0 
0   iin1   0   
0   vint1   0 +v
0   ib 2   B5  dc  
0   vb 2   0 
A78   iin 2   0   
0   vint 2   0 

(3),where each element of the state equation are in the appendix.
There are four state variables per unit. The state variables are
as follows;

ib1, ib2: Current flowing through the wiring inductance
between the DC bus and the inverter.
vb1, vb2: Inverter input capacitor voltage.
iin1, iin2: Motor q-axis current converted to DC side of
the inverter.
vint1, vint2: Integral value of the PI controller.
The state equation of the current flowing into unit #1 (ib1)
includes another unit state variable (ib2, vb2). In addition,
each element, such as A11, A12, A15, A16, and B1, includes thebus impedance (Lbus, Rbus) and the line impedance of otherunits (Lb2, Rb2). Therefore, interference among each unit
connected to the DC-bus occurs in this system.
The analysis method by the state equation of the wholesystem is difficult when many units are connected to the DCbus line due to the complexity of deriving the state equation.
In this paper, the interference among each unit is analyzed
by assuming that the line from DC-bus to the servo driver issufficiently longer than the DC-bus line. The state equation of
the current flowing into unit #1 from the DC-bus when Lbus <<Lb1, Lb2, and Rbus << Rb1, Rb2 is approximated as
ib1
R
1
1
 − b1 ib1 −vb1 +
vdc
dt
Lb1
Lb1
Lb1

(4).

Equation (4) does not include other unit state variables andparameters. Therefore, the interference among each unit
connected to DC-bus does not occur under the condition thatthe line between the DC-bus and the servo driver issufficiently longer than the DC-bus line. In this case, it ispossible to analyze the stability of the whole system by the
Middlebrook method for each unit.
III. PROPOSED STABILITY ANALYSIS METHOD
In this section, the stability analysis method with the casethat the impedance of the DC-bus is larger than the line
impedance between the DC-bus and the unit is proposed. This
proposed analysis method is based on the impedance method
proposed by Middlebrook [17].
A. The impedance modelFig. 3 shows the equivalent impedance model when theline impedance of the DC-bus is larger than the line
impedance between the DC-bus and each unit. In this model,
the servo driver is expressed as the input admittance Yin1(s),Yin2(s). The equivalent impedance model is expressed as amodel in that the LC filter and the input admittance of theservo driver are cascaded.
as

The output impedance of the LC filter Zo(s) is expressed
Z o ( s) =sLbus + Rbus
s 2 LbusCbus + sCbus Rbus + 1

Vs

The input admittance of each servo driver Yin(s) is derived
by the extra element theorem [6]. The input admittance of aconstant power load is expressed as

Rbus

A

Yin1(s)+Yin2(s)=Yin_all(s)Cb1+Cb2

A
Fig. 3The equivalent impedance model when the line impedanceof the DC-bus is larger than the line impedance between the DC-bus
and each unit.

iq* +
-


1 
K p 1 +sTi Controller

1
Ra + sLm

iq

Motor

T(s)
Fig. 4The simplified system block diagram. This model ignores
the interference term and d-axis current control loop due to feedforward compensation and zero d-axis current control.

steady-state value of the servo driver input voltage. The load
impedance of the motor is converted to the input impedanceby the conversion ratio.
The input impedance with ideal feedback ZN(s) isexpressed asV
V 2
Vb,0 2Z N ( s) = − b,0 = − b,0 = −
I b,0
Pout
( I q,0 Ra + re Ke ) I q,0(8),

where Ib,0 is the steady-state value of the DC-link current, andPout is the motor output. Note that ZN(s) is a negative
impedance depending on the motor output power.
Fig. 4 shows the simplified system block diagram. The
system block diagram in fig. 4 ignores the d-axis current
control loop and the interference term of the PM motor due tothe feed-forward compensation and the zero d-axis current
control. The open-loop transfer function of the current control
loop T(s) is expressed as
1 
1T (s) = PI (s)G(s) = K p 1 +

sT
R
+
sLm
i  a


(5),

where Cbus is the sum of the input capacitance of the servo
drivers.

1
T ( s)
1
1+
Z N ( s ) 1 + T ( s ) Z D ( s) 1 + T ( s)

Lbus

(9),

where G(s) is the transfer function of the motor.
The input admittance of the servo driver is expressed bysubstituting (7)-(9) for (6) as (10).
I q ,0 ( Ra I q ,0 + re K e ) T ( s )( Ra I q,0 + re K e ) 1+
2
Vbus ,0
1 + T ( s ) Vb ,0 2 ( Ra + sLm ) 1 + T ( s )2

Yin ( s ) = −

(6),

(10)

where ZN(s) is the input impedance with ideal feedback, Zd(s)is the input impedance without feedback, and T(s) is the openloop transfer function of the current control loop. The input
impedance without feedback ZD(s) is expressed as

The servo driver input admittance consists of the negative
term depending on the motor output and the positive term
depending on the square of the output voltage. Thus, the input
admittance of the servo motor changes with the operation
points, such as torque (q-axis current) and motor speed.

Yin (s) =

2



Vb,0
Z D ( s) = 2 ( Ra + sLm ) =  ( Ra + sLm ) (7), Ra I q ,0 + re Ke 
1where  is the conversion ratio, Iq,0 is the steady-state value ofq-axis current, re is the motor angular velocity, and Vb,0 is theB. The proposed stability analysis method
The stability of the model shown in Fig. 3 is analyzed with
the Nyquist plot of TMLG(), which is the product of the output
impedance of the LC filter Zo(s) and the combined input
admittance of the servo driver Yin_all(s). TMLG() is expressedasIm

TMLG ( ) = Z o ( )  Yin _ all ( )=  ReZ o ( ) ReYin _ all ( ) − Im Z o ( ) Im Yin _ all ( )+ j  Re Z o ( ) Im Yin _ all ( ) + Im Z o ( ) Re Yin _ all ( )Stable(-1,0)(11).

Re

Figure 5 shows the Nyquist plots. The Nyquist stability
criterion determines the system stability from the value of thereal part of TMLG() at the angular frequency o which theimaginary part is zero. The stability condition is expressed as Re Z o (o ) Re Yin _ all (o ) − Im Z o (o ) Im Yin _ all (o )  −1

UnstableFig. 5The Nyquist plot. The system stability is determined by thereal part value when the imaginary part is zero.

1
Lbus ( Cb1 + Cb 2 )(13).

Fig. 6 shows the relationship between o and res on thebode diagram and the Nyquist plot of TMLG(). In Fig. 6 (a),res and o are close on the bode diagram because the phaseof TMLG(s) is changed sharply near the resonant angularfrequency of the LC filter. Fig. 6 (b) shows that Re{TMLG(o)}is almost agreed with Re{TMLG(res)}. Thus, the systemstability is determined by the real part value of TMLG(res).
Note that the error between Re{TMLG(res)} and Re{TMLG(o)}is dependent on the square of the ratio Rbus to Lbus.
The stability condition equation is expressed bysubstituting the resonance angular frequency of LC filter resfor Re{TMLG(res)} as
ReYin _ all (res )  −Rbus ( Cb1 + Cb 2 )
Lbus

(14).

The proposed analysis method expresses the stable conditionby the equation, which consists of the real part of the
combined input admittance of the servo drivers, the line
inductance, the resistance of the DC-bus, and the input
capacitor of the servo driver.
Fig. 7 shows the relationship between the output power ofthe servo motor and the real part of the input admittance of theservo driver. The stable condition by (14) is also shown in Fig.
7. The stability limit is expressed clearly by (14). The real partof TMLG(res) is decreased linearly with increasing the motor
output power. Besides, the stable output range when the motor
rotates is expanded because the real part of TMLG(res) is
increased.
IV. ANALYSIS RESULTS
In this section, the validity of the proposed analysis
method is evaluated by comparison with the stable limit output
calculated by the polar arrangement of (3). The analysis
parameters are based on Table 1.

o(8759rad/s)res(8771rad/s)

Frequency [rad/s](a) The bode diagram1

Imaginary partres =

Phase [deg.]

However, it is difficult to solve the complicated equation of
the imaginary part in TMLG() for o. Therefore, this paper
proposes the simple stability analysis method using theresonant angular frequency of the LC filter res instead of o.
The resonant angular frequency of the LC filter res isexpressed as

Gain [dB](12).

res (8771rad/s)
0.5

o(8759rad/s)

0

-0.5
-1.5

-1

-0.5

0

0.5

Real part(b) The Nyquist plot.
Fig. 6
The relationship between o and res. The real part value of
TMLG(o) and TMLG(res) are almost agreed.

A. Stability analysis when two motors are driven.
Fig. 8 shows the pole map of the state equation in the DCbus servo drive system by (3) when unit #1 is driven at 200 W,
3000 r/min, and unit #2 is driven at 1500 r/min. The poles ofthe system move to the right half-plane (RHP) by increasing
the q-axis current of unit #2. When the q-axis current of unit
#2 is larger than 1.92 A, the poles are located in the RHP. It
means the system is unstable.
Fig. 9 shows the stability limit calculated by the proposedand full-order state equation analysis methods when two servo
drivers are connected to the DC-bus. Note that unit #1 isdriven at 3000 r/min, and unit #2 side motor is driven at 1500
r/min. The stability limit of single-unit changes depending on

V. EXPERIMENTAL RESULTS
Table 1 shows the experimental conditions. Note that theline resistance of the DC-bus Rbus includes the DC and AC
resistance at the resonant frequency of the LC filter. The
inductance and AC resistance of the line inductor is measured
with an LCR meter (HIOKI 3522-50). Also, DC resistance ismeasured by the voltage drop calculation method. The loadside motor controls each servo motor rotation speed. The
stable limit is searched by changing the q-axis current
command of the tested servo system.
Fig. 11 shows the bus current waveform and input
capacitor voltage waveform of each unit when the q-axis
current command value on unit #2 is increased. The oscillation
occurs in the bus current and capacitor voltage when thesystem is unstable. Note that unit #1, which does not change
the operating point, becomes unstable due to interference
among each unit.
Fig. 12 shows the stability limit by the proposed analysis
method and the experimental result with two units. The motorof unit #1 rotates 3000 r/min, and that of unit #2 rotates 1500
r/min. From Fig. 10, the result of the analysis andexperimental shows that the stable limit of one unit is reduced
by increasing the output of another unit. However, the stability
limit of the experiment is increased by 42 W on average, 11%of the rated output power, compared with the stability limit by
the proposed method.
Fig. 13 shows the simulation circuit model with theinverter loss. The simulation, including the inverter and motor
losses, is performed in order to discuss the cause of the error
between the experimental result and the analysis result. The
inverter loss is simulated by inserting the variable resistor
based on efficiency characteristics into the DC side of the

input
Real
input
the ofof partReal part
servo driveradmittance of theaddmitance(jresin)}
Re{YinRe{Y(j[mS]res)}[mS]-

3000 r/min

2(Cb1+Cb2)Rbus
Lbus
Stable

1
0

-1

0

100

200

300

400

-2

1000 r/min-3

2000 r/min Unstable

Iq2 = 3 A

Iq2
q1 = 0 A
Imaginary Part

Fig. 10 shows the error ratio of the stable limit between the
proposed analysis method and the full-order state equation
when the ratio of the DC-bus line inductance to the line
inductance between the DC-bus and each unit is changed.
Also, the value of Rb1 is changed so that the ratio of Lb1/Rb1 isthe same. In addition, the line impedance of unit #2 (Lb2, Rb2)
are same as unit #1 as the line length between the DC-bus andeach unit is equal. These conditions are that the stable limit
output of unit #2 is compared when unit #1 output is 200 W.
In Fig. 10, the error ratio of the calculated stable limit between
the proposed analysis method and the full-order state equation
is increased with an increased ratio Lb1/Lbus. When the ratio
Lb1/Lbus is 1/10, the proposed analysis method determines thesystem’s stability within 1% compared to the state equation
method.

3

OutputPowerpowerP[W]
Outputout[W]
Fig. 7
The relationship between motor output and the real part ofthe servo input admittance. The stable condition is also shown. The
real part value of TMLG() at the resonant angular frequency res isdecreased linearity with increasing output power.

Stable limit
Iq2 = 1.92 A

Real Part
Fig. 8The pole map of the DC-bus servo drive system when unit
#1 is driven at 200 W, 3000 r/min, and #2 is driven at 1500 r/min.

400
Unit #2 Stable limit Ouptput
Power [W]B. The influence of the line between the DC-bus and eachunit
The proposed analysis method is based on the condition
that the DC-bus line is sufficiently longer than the line
between the DC-bus and each unit. This section discusses theinfluence of the line impedance between the DC-bus and each
unit on the accuracy of the proposed analysis method.

300
200

Unit#1 : 3000 r/min
Line: Stability limit by
Unit#2 : 1500 r/min
the full-order state
equation
Dot: Stable limit byproposed analysis
method100
0

Lbus : 1 mH
Rbus : 0.02 W
0

100

200

300

400

Unit #1 Stable limit Ouptput Power [W]

Fig. 9The relationship of the stable limit power at two motors
connected DC bus. The stable limit of single-unit changes dependingon the output power of another unit.

Unit#1 : 3000 r/min, 200W
Unit#2 : 1500 r/min
Lbus : 1mH, Rbus = 20mW
Lb1=Lb2, Rb1=Rb2
Lb1/Lb2 = const.

10

Error ratio of unit #2 sidestable limit output [%]the output power of another unit. The stability limit calculatedby the proposed analysis agrees with the state equation
analysis within 2%. Therefore, the proposed simple analysis
method determines the stability of the DC-bus servo drive
system with high accuracy.

1

0

0.01

0.1

1

Lb1/Lbus [-]Fig. 10The stable limit power error relationship between the
proposed analysis method and analysis by state equation when the line
inductance between the DC-bus and each unit is changed. The
resistance value is also changed so that the ratio.

servo driver. The efficiency of the servo motor is measured bya power meter (HIOKI PW6001). The simulation model also
includes deadtime and discretized control systems.
Fig. 14 shows the stability limit when a motor is driven as
calculated by the proposed analysis method, the simulation
result, and the experimental result. The error between thesimulation result and the experimental result is reduced byapproximately 50%. Therefore, the losses of the inverter andmotor also influence the stability of the DC-bus servo drive
system.

Table 2 Experimental condition.

Circuit

Lbus

Wiring indutance of DC Bus

10 mH

Rbus

Wiring resistance of DC Bus

1.8 W

Cb1, Cb2

Inverter input capasitance

6.8 mFVdc

DC bus voltage

280 V

fsw

Switching frequency

20 kHzDead time

2 ms

Td

Drive Side Motor (Unit #1, #2)
R88M-1M40030T (Omron)

APPENDIX
Each element of the state equation (3) are expressed asfollows.

A11 =


−1
Rbus + Rb1 
Lbus Rbus
1 −

2
Lbus
Lbus + Lb1  ( Rbus + Rb1 )( Lbus + Lb1 ) 1−
( Lbus + Lb1 )( Lbus + Lb 2 )

(15)
−1
1
A12 =
2
Lbus
Lbus + Lb1
1−( Lbus + Lb1 )( Lbus + Lb 2 )

A15 =

(16)

Load Side Motor (Unit #1, #2)
SGMJV-08ADA21 (YASKAWA Electric Corporation)#1 Capacitor Voltage
vb1 100V/div#1 Capacitor Voltage
vb1 100V/div#2 Capacitor Voltage
vb2 100V/div#2 Capacitor Voltage
vb2 100V/div

Bus Current ibus
1A/div 40ms/div

Bus Current ibus
1A/div

300
200

Proposed Analysis
method

100
0

0

100

200

300

400

Fig. 12The comparison of stability limit between the proposed
analysis method and experiment when two servo drivers are driven.

(18)

A21 = 1 / Cb1(19)

A23 = −1/ Cb1

(20)

K I 
1 
A32 =  12 − p b1,0 
Lm 
Vdc 

Unit #1 : 3000 r/min
Unit #2 : 1500 r/min

Stable limit
Power [W]
UnitUnit#1 #1
Stability
limitOuptput
output power[W]−1
Rbus  Lbus (1 + Rb 2 / Rbus ) 1 −Lbus 2
Lbus + Lb1 ( Lbus + Lb 2 ) 1−
( Lbus + Lb1 )( Lbus + Lb 2 )−1
Lbus
Lbus 2
Lbus + Lb1 )( Lbus + Lb 2 )(1−( Lbus + Lb1 )( Lbus + Lb 2 )Experimental
result400

Lbus Rbus

(17)
A16 =

40ms/div(a)Stable condition.
(b)Unstable condition.
Fig. 11
The waveform of the bus current ibus and the input capacitor
voltage on each unit vb1, vb2 in an unstable condition. The oscillation
occurs when the system is unstable.

#2#2Stability
Unit
Ouptput
Stable limitlimit
Unit[W]
power[W]output Power

VI. CONCLUSIONSThis paper proposed the simple stability analysis method of
the DC-bus servo drive system. The proposed methodclarified the influence of the line impedance of the DC-bus on
the stability of the DC-bus servo drive system. The stability
limit calculated by the proposed method agreed with the resultof the full-order state equation analysis method to within 2%.
The influence of the line impedance between the DC-bus andeach unit for the accuracy of the proposed analysis method is
also discussed. In addition, the stability limit by the proposed
method agreed with 11% of the rated power by experimental
result. The cause of error between the analysis result and theexperimental result is discussed with the system losses. In
future work, the active damping method in order to expand thestable operation area and the parameter design method will be
provided.

(21)

Ibus

Vdc

INV #1
Cb1
Rcb1

Inverter
Loss

#1
M

Fig. 13The simulation circuit model. The model includes the ESR of
the input capacitor and the variable resistance as the inverter loss.

A33 = −Ra + K p
Lm

A34 = 1 / Lm

(22)
(23)

poweroutputpower
limitoutput
Stability
Stable limit
[W]
[W]

500

Experiment
34 W
Sim. with losses
Prop. Analysis Method
24 W
26 W
19 W
21 W

400
300
200

35 W

100
0
0

1000

2000

3000

4000

Motor
speed
[r/min]M otorrotatespeed[rpm]
Fig. 14
The comparison of stability limit by the proposed method, the simulation with losses, and the experiment according to motor speeds. The
error between the simulation result and the experimental result is reduced than the error between the analysis result and the experimental byapproximately 50%.

A42 = −

A43 = −A51 =K p I b1,0
TV
i dc

Kp

(24)
(25)

Ti

−1
Rbus  Lbus (1 + Rb1 / Rbus ) 
1 −

2
Lbus
Lbus + Lb 2 
( Lbus + Lb1 ) 1−
( Lbus + Lb1 )( Lbus + Lb 2 )

(26)
A52 =

A55 =

−1
Lbus
2
Lbus
( Lbus + Lb1 )( Lbus + Lb 2 )1−( Lbus + Lb1 )( Lbus + Lb 2 )

(27)


−1
Rbus + Rb 2 
Lbus Rbus
1 −
Lbus 2
Lbus + Lb 2  ( Rbus + Rb 2 )( Lbus + Lb 2 ) 1−
( Lbus + Lb1 )( Lbus + Lb 2 )

(28)
A56 =

−1
1
Lbus 2
Lbus + Lb 2
1−
( Lbus + Lb1 )( Lbus + Lb 2 )(29)

A65 = 1 / Cb 2

(30)

A67 = −1 / Cb 2

(31)

A76 =

1  2 K p Ib 2,0 
2 −

Lm 
Vdc 

A77 = −Ra + K p
Lm

A78 = 1 / Lm
A86 = −

A43 = −

B1 =

B1 =

K p I b 2,0
TV
i dc

Kp
Ti

(32)
(33)
(34)
(35)
(36)


1
1
Lbus 
1 −
Lbus 2
Lbus + Lb1  Lbus + Lb 2 
1−( Lbus + Lb1 )( Lbus + Lb 2 )

(37)


1
1
Lbus 
1 −
Lbus 2
Lbus + Lb 2  Lbus + Lb1 1−( Lbus + Lb1 )( Lbus + Lb 2 )(38),
where 1 and 2 are the conversion ratio from the bus voltage
to the output voltage of each unit, Ib1,0 and Ib2,0 are the steadystate value of each DC-link current.
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