Dead-time Voltage Error Correction
with Parallel Disturbance Observers
for High Performance V/f Control
Tetsuma Hoshino, Jun-ichi Itoh

Takayuki Kaneko

Department of Electrical Engineering
Nagaoka University of Technology
Nagaoka, Japan

Electronics Technology Laboratory
Fuji Electric Advanced Technology Co., Ltd.
Tokyo, Japan

Abstract—This paper proposes a dead-time compensation
method with a parallel disturbance observer and a current
controller in the d-q rotational frame for V/f control. The parallel
disturbance observers consist of a fast response disturbanceobserver and a slow response disturbance observer to separate
the back electromotive force (EMF) from the estimateddisturbance voltage. As a result, the dead-time error voltage iscorrected using the proposed method. The proposed method isvalidated based on the simulation and experimental results. Thismethod can improve the current distortion to less than 1/9 that of
the conventional method.

On the other hand, dead-time compensation methods using
a disturbance observer have been previously proposed [9-10].
In these reports, the disturbance observer is used for the sensorless vector control system of a permanents magnet motor in arotational frame or a three-phase frame. However, V/f controlof the induction motor is not considered.
The dead-time compensation methods using a disturbanceobserver have some advantages as shown in next.
• Easier parameter setting than the other method.

Keywords-component; induction motor, V/f control, dead-time,
disturbance observer

I.

INTRODUCTION

Recently, inverter systems have been applied to adjustablespeed drive systems with induction motors in various fields forthe economic utilization of energy. The V/f (motor
voltage/output frequency) control method for an induction
motor is more suitable to general applications than a speed
sensor-less vector control method, because the V/f controller isvery simple, and robust for variation of the motor parameters.
All voltage source type inverters require dead-time, whichcauses output voltage errors and has a significant effect on the
low-voltage output range, such as at low speed. Therefore,dead-time compensation techniques are very important torealize a low cost motor drive system. Thus many dead-timecorrection methods are proposed [1-6]. In particular the mostpopular voltage error compensation method is feed forward
compensation of the output voltage command according to thedirection of the load current. This is very simple; however, it
does not work well for the low speed range, because ofdifficulties in detection of the load current polarity [7].
Moreover, the precise dead-time compensation method, whichbased on dead-time error voltage measured toward the load
current value in off-line, has been proposed [8]. However, thismethod is also difficult to compensate the dead-time error
voltage because the dead-time voltage error values isinfluenced by the temperature and dispersion of a switching
device.

•Possible to compensate forward drop voltage of aswitching device, simultaneously.

•

Possible to compensation in on-line at all times.

This paper proposes a dead-time compensation methodwith a parallel disturbance observer and a current controller inthe d-q rotational frame for V/f control. This work is aimed atproviding a simple and robust control system instead of speed
sensor-less vector control. The voltage error on the d- and qaxes current caused by the dead-time is suppressed by an auto
current regulator (ACR) in the d-axis and the disturbanceobservers in the q-axis. The parallel disturbance observers
consist of a fast response disturbance observer and a slow
response disturbance observer to separate the backelectromotive force (EMF) from the estimated disturbancevoltage. As a result, the dead-time error voltage is correctedusing the proposed method. The proposed method is validated
based on the simulation and experimental results. This method
can improve the current distortion to less than 1/9 that of theconventional method.
II.

PRINCIPLES OF DISTURBANCE OBSERVERS FOR
DEAD-TIME ERROR VOLTAGE CORRECTION

A. Problems of dead-time and conventional conpensation
Fig. 1 shows behavior of the voltage error during the deadtime period. Switch off time, it is so called dead-time, is added
to gate pulses of up and un in order to avoid the short circuit
between an upper arm and a lower arm. To obtain dead-time
period, the turned-on timing of the gate pulse up and un aredelayed during Td as shown in Fig. 1(b).

0197-2618/07/$25.00 © 2007 IEEE

The voltage error during the dead-time depends on the
direction of a flowing current. When the output current
direction of the leg is positive which is defined from the leg toload, the current in the leg flows through the free wheeling
diode (FWD) of the lower arm during the dead-time period.
Thus, the output voltage is decreased by the dead-time period.
On the other hands, when the output current direction isnegative, the current in the leg flows through the FWD of theupper arm. Thus, the output voltage increases. The value ofthe voltage error depends on the dead-time period and dc-link
voltage as shown in Fig. 1. Finally, the voltage error is decidedby (1).
(1)

ΔV = f sVdcTd ⋅ sign (iu )

u*

up

ON

up

iu>0
Vdc/2

iu>0

un

ON

OFF

OFF

ON

Td

Td
un

Vdc

OFF

-Vdc/2iu<0Voltage error

0Voltage error

Vdc/2

iu<0

0
-Vdc/2(a) An inverter leg.

(b) Relationship between reference
pulse and voltage error

Figure 1. Relations between reference pulse and voltage error.

where, fs: switching frequency, Vdc: dc-link voltage, Td:
dead-time period, iu: output current of the leg, sign(x):signfunction. If x>0 then sign(x) =1, if x<0 then sign(x) =ҟ1, if x=0
then sign(x) =0.
It should be noted that the magnitude of the voltage error
does not depends on the amplitude of the output voltage andthe output current. Therefore, when the output voltage is small
such as a low speed operation, the affect of the dead-time isstrongly appeared because the ratio of the voltage error to the
output voltage becomes large.

Figure 2. Equivalent circuit of induction motor.

B. Dead-time correction method using disturbance observers
Fig. 2 shows the equivalent circuit, in which a secondary
leakage inductance is converted to a primary side, of an
induction motor on d-q rotating frame. This paper proposes a
dead-time compensation method with a disturbance observer.
The voltage error of the dead-time is estimated by the output
current and the motor parameters. The relation between the
motor voltage v1 and current i1 is obtained by (2).
ª R1 + pLσªv1d º « ω L« » « 1 σ«v1 q » = « − R
2
«0» «
« » «
«¬ 0 »¼ «
0«¬−ω1 LσR1 + pLσ

p

ω1

0

R2
+p
Lm

− R2

ω1 − ω m

Lm: magnetizing inductance, ΔV: disturbance voltage,
Tf, Ts: faster and slower time-constant of each disturbance observer,
KACR: gain of the auto current regulator.

−ω1
p

º
» ª i1d º»« »i− ω1 + ω m »» « 1q »«φ 2 d »»« »R2+ p » «¬φ 2 q »¼
»¼
Lm

(2)

where, R1 is the primary resistance, R2 is the secondary
resistance, p is differential operator p = d / dt , Lm is themagnetizing inductance, Lσ is the equivalent leakage
inductance, ω1 is the primary angular frequency, ωm is thesecondary angular frequency, i1d is d-axis components of the
primary current on the d-q frame, i1q is q-axis components ofthe primary current on d-q frame, v1d is d-axis components ofthe primary voltage, v1q is q-axis components of the primaryvoltage, φ2d is d-axis components of secondary flux, and φ2q is
q-axis components of secondary flux.
From equation (2), when the d-axis corresponds to thevector for the secondary flux φ2 and the q-axis component φ2q
is equal to zero, the q-axis voltage v1q is calculated using (3).

Figure 3. Block diagram of an induction-motor drive system using dead-timeerror correction using disturbance observers.

v1q = ( R1 + R 2 + pLσ )i1q − ω1 Lσ i1d + ω m φ 2 d .

(3)In the equation (3), the 1st term of the right side change
according to the electrical time constant Te = Lσ / (R1+R2)
because the q-axis current respond at the electrical time
constant. In contrast, the 3rd term of the right side changes
slower than the 1st term, because the secondary angularfrequency responds at the mechanical time constant. Then, the
2nd term means the cross-term.
The voltage error can be considered as the voltage
disturbance for the q-axis current. In case of the low speed
operation, ω1 and ωm in (3) become the small to other terms.
Thus, if those small terms can be neglected, then the voltage
error can be estimated using a voltage disturbance observer asshown in (4).

1
(R1 C + R2C + sLσC )I 1q
1 + sT

where, suffix C means controller parameter, T is time
constant of the observer, s is Laplace operator, and I1q is
Laplace transfer of i1q.
Fig. 3 shows a block diagram for the proposedcompensation method using V/f control. The feedback filter
using the Tf (fast response disturbance observer) estimates theback EMF in addition to the disturbance voltage. To estimate
only the voltage error from dead-time in the middle- or highspeed range, the feedback filter of the Ts (slow response
disturbance observer) is used to cancel the back EMF. On theother hand, the ACR on the d-axis corrects the dead-time
voltage error on d-axis and maintains the rated excitation
current of the motor.
III. ANALYSES OF THE PROPOSED SYSTEM
One of advantages of a V/f control method is that a V/fcontrol method is not sensitive for motor parameters in contrast
to a speed sensor-less vector control. However the proposed
voltage disturbance observers need some motor parameters.
We are afraid that the proposed observer spoils the advantageof the conventional V/f method. Therefore, the parameter
sensitivity of the proposed method is discussed based ontransfer functions of the proposed system shown in Fig. 3, in
this section.
A. Transfer functions of the proposed system
Before analysis of the proposed system, two low-passfilters written in connected in parallel, as shown in Fig. 3, isassembled as

G ( s) =()

s Ts − T f
1
1−=
1 + sT f 1 + sTs 1 + s Ts + T f + s 2Ts T f()

1 − G (s)·§ RC 1 + sTeC
1 + G ( s )¨¨− 1¸¸¹© R 1 + sTe
Vq1=*§ R 1 + sTeC
·
Vq
− 1¸¸
1 + G( s)¨¨ C
R
sT
1
+
e
©
¹
Vq

=

Parameters

ValuesPrimary resistance R1

2.78Ω

Secondary resistance R2

2.44Ω

Leakage inductance Lσ

11.0mHParameters
Fast response disturbanceobserver time constant Tf
Slow response disturbanceobserver time constant Ts

Values1msec

500msec(a)Disturbance to output voltage transmission characteristics(b) Output voltage command to output voltage transmission characteristics(5).

From q-axis control part in Fig.3, the transfer functions to
the output voltage V1q from the voltage error ΔV or from thecommand voltage V1q* in the Laplace plane are leaded as (6),
(7).

ΔV

Table 1. Analysis conditions.

(4)

Magnitude [dB]

ΔVˆ =

(6)

Figure 4. Frequency response of parallel connected disturbanceobserver system.

condition, where RC = R and LσC = Lσ. In addition, the dashedlines indicate various conditions, where either RC or LσC isequal to R or Lσ multiplied or divided by ten. The otherparameters used to calculate the Bode diagrams are given in
Table. 1. In Fig. 4(a), lower magnitude indicates higher
disturbance rejection performance. In Figure 4(b), 0 [dB] ofmagnitude means that controller outputs the same voltage tocommand voltage v1q*.
Then, parameter error sensitivities of each RC and LσC areconsidered.

(7)

where R=R1+R2, RC=R1C+R2C, G(s) is the transfer function
shown in (5)B. Analyses of parameters mismatches using Bode diagrams
Fig. 4(a),(b) shows Bode diagrams for (6) and (7),respectively. The solid lines in Fig. 4 indicate an ideal

1) Ideal conditon ( RC=R, LσC=Lσ )Equation (8) is the transfer function from the voltage error
ΔV to the output voltage V1q based on (6) in case of the ideal
condition in which the controller parameter agree with the
motor parameter. It should be noted that the frequency
response is same to a notch filters one at this ideal condition,.
The center frequency of the notch is obtained by (9).

Imaginary(a)Placement of six roots.
(b) Trackings of the most vibratile root No.5.
Figure 5. Placement and tracking of roots. O indicates no parameter variations, Δ indicates +20%, and ∇ indicates –20%.
The solid line, dash-dot line and dashed lines represents the KACR current regulator gain, and the Tf and Ts observer time-constant variations, respectively.

Vq
ΔV
f0 =

=

( )

•

1 + s 2T f + s 2T f Ts()

1 + s T f + Ts + s 2T f Ts

Ҟ

(8)

1
2π Ts T f

(9)

As a result, the disturbance rejection gain in Fig. 4(a) is
0 [dB] in a very low frequency region. It means that error
voltage in the very low frequency region such as a low speed
region can not be corrected. Therefore slow response
disturbance observer time constant Ts have to be decided
according to operation region of the system. Depending on theapplications, the slow response disturbance observer may notuse in the very low speed region.
2) Affection of the parameter RC
In a middle and low frequency region, the disturbancevoltage is suppressed by the proposed method, very well. Thusa speed control range of the system should be designed in thisregion. On the other hand, the performance of the disturbancevoltage suppression is varied by the parameter RC. If RC issmaller than the R, then the performance will be worse. In
contrast, if the RC is bigger than the R, the performance will bebetter. Therefore, the control parameter RC should be set to alittle larger than the actual motor parameter R.
3) Affection of the parameter LσC
The affection of the parameter LσC only appears in the highfrequency region. However, the behavior is similar to theaffection of the parameter RC. That is, smaller LσC improves
the performance of the disturbance voltage suppression. In
addition, bigger LσC degrades the performance. Therefore, thecontrol parameter LσC should be set to a little smaller than theactual motor parameter Lσ. It should be note that the affectionof the parameter Lσc is not too bigger than R.
Finally, we summarize the considerations about theparameter mismatch.
•

The parameter mismatch of R affects considerably inthe middle frequency region.

•

The parameter mismatch of Lσ affects in the highfrequency region.

The gain of the controller from reference voltage V1q*
to output voltage V1q is strongly affected by theparameter mismatch of RC. The affection of the LσC issmaller than affection of the RC.

Therefore RC must be determined with consideration for theerror in R. However, the proposed system is not as sensitive as
the speed sensor-less vector control system.
C. Stability analyses using roots trackings
The stability of the proposed system is analyzed using roots
placement and tracking in the complex plane. The conditions
for analyses use parameters of a general purpose 750 W
induction motor under constant-speed in addition to thosegiven in Table 1.
Fig. 5(a) shows the roots placement under the condition that
Tf = 1 ms, Ts = 100 ms, KACR = 0.5 and ω1 = ωm = 0.65 PU.
This system has six roots because it is a sixth dimensional
system. The placement of roots No.1-4 in Fig. 3 indicatessufficient stability, because those roots are dispersed on the real
axis in an area that is negatively distant from the imaginary
axis. However, the remaining roots, Nos. 5 and 6, are locatednearest to the imaginary axis. Therefore, the stability of the
proposed system can be discussed based on root No. 5.
Fig. 5(b) shows the root-No.5 trackings of variations in
KACR, Tf, and Ts. Each of the parameters is varied ±20% fromeach of these values. To stabilize the system, the roots should
be located in the area negatively distant from the imaginary
axis, in addition to minimizing the imaginary part. Thus, Tf
and Ts are set to slower values, and KACR is set to a higher value
in order to realize stabilization of the system. It should be
noted that a slow Tf avoids the error voltage correction, and aslow Ts deteriorates the acceleration performance due to theback EMF. In contrast, KACR should be set as high as possible
for stabilization with the consideration about a delay time of
the feedback loop. As a result, the parameters should bedesigned according to the next terms.
•

Tf is set as faster as possible for controller.

•

Ts is set to faster than the mechanical time-constant.

•

After that, KACR is adjusted for the stability of thesystem.

i1d*
䋭V/f controller
f-econversion

*

v1d*

KACR

2
䜯
v1q** 3

i1d
v1q*

vu*
vv*

VSI

IM

vw* Inverter
iu iw

Disturbanceobserverin Fig. 3.

i1d
i1q

3 䜮2

θ

1
s

Figure 7. Experimental system.
Table 2. Experimental conditions.
(a) D-axis ACR gain set to KACR = 0.5(b) D-axis ACR gain set to 2KACR = 1.0

Figure 6. Damping result with parameter tuning.

Fig. 6 shows the d- and q-axis current, and the q-axis
compensating voltage for verification of the stabilization with
KACR. Each parameter has the same values as those used for the
analyses except KACR. Note that the load condition is changed
from no load to rated load at 2.0 sec, and from rated load to noload at 4.0 sec, as shown in Fig. 6. As shown in Fig. 6(a), thevibration in the current occurs because damping is not enough.
In contrast, the vibration is converged by adjusting ACR gain
KACR in Fig. 6(b). Therefore, the valid of the stabilization
method is confirmed by the simulation.
IV.

EXPERIMENTAL RESULT

Fig. 7 shows a control block diagram of an experimental
system using the proposed V/f control. The experimental
system is composed of a general induction motor and an
inverter. The output voltage of the inverter is controlled on dand q rotating frame. The voltage command v1q* is obtainedon q-axis. In additions d-axis has the current regulator tomaintain the exciting current and the stabilization.
Then, the proposed method is applied to the experimental
system shown in Fig.7 so that the performance of the
disturbance voltage rejection is inspected. Table 2 gives theexperimental conditions of the controller and the motor. Notethat slow response disturbance observer does not work in the
low speed region according to the result in the section III-B.

Parameters

ValuesParameters

ValuesRated power

750W

Rated current

3.6A

Poles

4

Rated exciting current

2.0A

Rated voltage

200V

Primary resistance R1

2.78Ω

Rated frequency

50Hz

Secondary resistance R2

2.44Ω

Rated speed

1420r/minLeakage inductance Lσ

11.0mH

Switching frequency

20kHz

Dead-time period

3μsec

D-axis ACR gain

2.0

Fast response disturbanceobserver time constant Tf
Slow response disturbanceobserver time constant Ts

1msec

10msecA. Comparison of THD between the conventional method
and the proposed method
Fig. 8 shows the u-phase, d-axis and q-axis current
waveforms under conditions where the output frequency is 1
Hz with no load. It should be noted that the conventional
compensation method shown in Fig. 8(a) uses the voltage error
feed-forward based on the current direction. The rated
excitation current is maintained by the disturbance observers
used in the proposed method as shown in Fig. 8(b). The total
harmonic distortion (THD) of the current shown in Fig. 8(b) is
0.98%, with a 7.93 point reduction from the result as shown in
Fig. 8(a).
Fig. 8(c) shows each harmonic components of u-phasecurrent. The proposed method corrects the 2nd, 5th, and 7th
harmonic component because of the decreasing of current
stagnation at zero-crossing point.
B. Comparison of steady state torque between theconventional method and the proposed method
Fig. 9 shows the steady state torque curves. The output
frequency is set to the rated slip of 2.67 Hz, and then decreases
in speed. The starting torque (at 0 r/min) with the proposed
method is 119%, which is six times the torque with theconventional method. Using the proposed method, the current
stagnation at zero-crossing point of the conventional method isdecreased, so that the current waveform is improved. It should
be note that the torque curve of the proposed method exceeding
the ideal line is caused by the over compensation for thevoltage drop which depend on the primary resistance R1 .

Torque [%](a) Without proposed method

Ide
a

l to
rqu
e

(Ba
sed

on
r

ate
d

slip
)

Figure 9. Improvement of steady state torque curvewith proposed compensation method.

(b) With proposed method

Figure 10. Characteristics against step-shape load torque.
.
Acceleration

Brakingmm : 0.2pu/div.

(c) Harmonic component in iu waveform

Figure 8. Current waveforms at 1Hz under no-load condition.
(Output frequency 1Hz, no load, 750W induction motor)

C. Dynamic state charactraistics
Fig. 10 shows the characteristic responses to the step-shapeload torque. The output frequency was set to 5 Hz, and thestep load was applied at 1 sec. The proposed method works
without stall against step-shape load, which means that the
proposed method keeps stable in the dynamic state.
Fig. 11 shows the acceleration and braking characteristic
between zero and the rated speed. The acceleration andbraking time is set to 0.5 sec under the no load condition. The
proposed method can generate high starting torque andacceleration torque though the middle and high speed region,

iq, id : 0.2pu/div.
id
iq
iu

iu : 2A/div.

(200ms/div.)
Figure 11. Characteristics against acceleration and braking.

because the slow response disturbance observer cancels backEMF from the output of the fast response disturbance observer.
In addition to acceleration, the proposed method can generate
braking torque at all speed range.

iu THD [%]D. Verification of the affection of parameter mismatch
Fig. 12 shows the affection of the THD by the variations ofparameter mismatches of RC or LσC.
When the RC was smaller than the R, the THD is increased.
Likewise, when the RC was bigger than the R, the THD isdecreased. On the other hands, when the parameter LσC was
different to the actual Lσ, the THD is increased. Those results
agree with the conclusion in the section III-B. Thus, thevalidity of the analyses was confirmed.
V.

CONCLUSIONS
Figure 12. THD variations about parameter mismatching.

A parallel disturbance observer, used for dead-time error
voltage correction, was proposed and verified. A summary of
the outcomes indicated in this study is as follows:•

The proposed method can suppress the disturbancevoltage except back-EMF.

•

The THD of the motor current is improved by less than
1/9 that for the conventional method.

•

Using the proposed method, the starting torque isimproved to 119%, which is six times that for theconventional method.

•

The parameter mismatch does not causes significantly
performance degradation.

•

The proposed method is robust against the dynamic
loads e.g. step load torque and acceleration.

ACKNOWLEDGMENT
This study was supported by Industrial Technology Grant
Program in 2005 from New Energy and Industrial Technology
Development Organization (NEDO) of Japan.

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