Design of Flux-Axis Angular Speed Estimation
using Induced Voltage in Speed Sensor-less Field
Oriented Control for Induction Motor
Jun-ichi Itoh1, Kazuya Kawai1, Koroku Nishizawa1, Hiroki Watanabe1
1

Nagaoka University of Technology, Japan

Abstract-- This paper proposes a design method of theflux-axis angular speed estimation using induced voltage in a
speed sensor-less field oriented control for induction motor(IM) drive systems. In this method, the d-axis induced voltage
is regulated to zero by a feedback control to estimate the fluxaxis angular speed. A low pass filter (LPF) is necessary in thisestimation to avoid the recursive calculation. In this paper,the design method of compensation gain kpem and the cut-off
angular frequency of LPF lpf are considered. As a result, therelation between kpem, lpf, convergence time of the d-axis
induced voltage have a match to the equation and designed
model in this paper. Furthermore, the maximum error
between the simulation and estimated results of the time
constant of the flux-axis angular speed introduced due to LPF
is 3.7% when lpf is 200 rad/s. As the experimental results,
when a torque step of 100% is applied to the induction motor,
the convergence time of the motor speed is 1.6 s as the worst
case with kpem set as 2.0 p.u. and the cut-off angular frequency
set as 1000 rad/s. However, if kpem is set as 2.0 p.u. and lpf is
set as 400 rad/s, the convergence time of the motor speed is0.43 s as the best case, which is reduced by 73.1% comparedto the worst case. As the result, kpem should decide by thetarget steady-state error, lpf should bigger than the cut-off
angular frequency of the PI controller ASR. Where, if the lpf
is high, the convergence time of the motor speed is high.
Index Terms— Induction motor, vector control, sensorless field oriented control, control design.

I. INTRODUCTION
In recent years, induction motors (IM) are used in many
fields due to their low cost and easy maintenance. As acontrol method for IM, a field oriented control (FOC) is
employed for torque control, high speed response, highefficiency, and smooth control in low-speed region. In
general, a speed sensor is necessary for the detection of theactual speed. However, the speed sensor limits cost
reduction of the motor drive systems. On the other hand, asensor-less FOC overcomes this problem by theachievement of the torque control performance close to thesensor-equipped FOC without speed sensors. The sensorless FOC is accomplished by the implementation of speed
estimation methods in the feedback control [1-19]. One of
the sensor-less FOC methods is a model reference adaptive
system (MRAS). MRAS is based on the observer and theadaptive control to calculate flux-axis angular speed. Thismethod uses both the voltage and current models [14-20].

Besides, the convergence speed of the estimation rule isslowly [20]. Furthermore, the induced voltage andsecondary flux are used in order to calculate the flux-axis
angular speed, resulting a simpler algorithm than othersensor-less FOC methods [6-13]. However, the design
method of the flux-axis angular-speed estimation using theinduced voltage, the compensation gain and the low passfilter (LPF), has not been clarified [6-9].
This paper proposes the design method of the flux-axis
angular-speed estimation using the induced voltage. The
closed transfer function of the flux-axis angular-speed
estimation with LPF is derived to evaluate the control
performance of the d-axis induced voltage regulator. The
originality of this paper is that the clarification of therelation between the compensation gain and LPF in theflux-axis angular-speed estimation, which ensures thestable operation of the speed sensor-less FOC. As a result,the estimated value of the d-axis induced voltage has amatch to the actual value. Furthermore, the maximum error
between the simulation and estimated results of the time
constant of the flux-axis angular speed introduced by LPF
is 3.7% when the cut-off angular frequency of LPF lpf is
200 rad/s. As experimental results, it is confirmed theconvergence time of the motor speed error is reduced by
73.1% in a torque step is applied.
This paper is organized as follows; in section II, thesensor-less FOC using the induced voltage to estimate
flux-axis angular-speed is introduced with the past studies.
The transfer function of the flux-axis angular-speed
estimation and the design method for kpem and lpf are
explained as the main part of this paper in the section III.
Next in the section IV, the simulation with IM model iscarried out to confirm the validation of the transfer
function of the flux-axis angular-speed estimation, whichshows the relation between the compensation gain, the cutoff frequency of LPF and the convergence time of the flux
axis angular speed. Finally, in section V, the design
method for kpem and lpf is confirmed by the experiments.
II. CONTROL METHOD AND PRINCIPLE OF FLUS-AXISANGULAR SPEED ESTIMATION
A. Configuration of Speed Sensor-less Controller
Fig. 1 shows the control block diagram of the speed
sensor-less FOC using the induced voltage. Note that r*

is the rotor angular speed, ˆ r is the calculated rotor
angular speed, s is the slip angular speed, id* and iq* arethe currents command on dq-axis, id and iq are the currents
on dq-axis, ed and eq are the dq-axis induced voltages, vu*,
vv*, vw* and iu, iu, iw are the of output voltages commandand the motor currents, and  is the flux angle of thecontroller.
As shown in Fig. 1, this control system does not require
the detection values from the speed or position sensors.
The flux-axis angular speed is estimated from the dq-axis
voltages and the dq-axis currents. In addition, ̂ is an
instantaneous value which implies that if the motor current
or motor voltage changed rapidly, a large error between thereal value and calculated value might occur, leading to theinstability of the system. In the flux-axis angular speed
estimation in this paper, LPF is required to prevent therecursive computation.
A decoupling control is used in this system to remove
the dq-axis voltage interference factors by feed-forward
control. The d-axis voltage command vd is calculated fromthe output of the d-axis current PI output vd*, the electricalangular speed ˆ r , the stator inductance Ls, and the q-axis
current iq,(1)ˆ  L1  M )iq
vd  vd *  

R2,

s 

M
 L1id )
L

ˆ 

ˆ 

-

r +
*

-

PI(ASR)

Iq

*

+
-

Cut-off angularfrequency speed

PI(ACR)

(7)

2d

eq

2d

 k pem ed(8)Note that the definition of the unit of kpem is (rad/s)/V.
The estimated induced voltage ed and eq are given by (9)and (10),where  is defined as  = 1 – M 2 / (L1L2).
The secondary magnetic flux is calculated from themutual inductance M, the rotor resistance R2, the rotor
inductance L2 and id,
MR2L22 d  id
×(3)
L2 L2 s  R2
where s is the differential operator.
The slip angular speed is calculated from iq, id, L2 andd)id  ˆ L iqdtdeˆq  vq  ( R1  L )iq  ˆ L id
dt

＾r +

(10)

where L is leakage inductance.

V d*

V u*

Vq

+

dq /3 F

*

Vv*
Vw*

PWM
Inverter+
Decoupling control
based on (1), (2)
Iu

Id

s

(9)

eˆd  vd  ( R1  L+PI(ACR)eq

Fig. 2 shows the induced voltage vector in dq-frame.
Note that Lσ is the leakage inductance, R1 is the primary
resistance, and s is the differential operator. As shown in
Fig. 2(b), the secondary d-axis flux 2d aligns with the daxis when the induced voltage vector aligns to q-axis. In
other words, if the secondary d-axis flux 2d is leading or
lagging to the d-axis as shown in Fig. 2(a) and (c), theinduced voltage vector includes the d-axis component. In
these case, the flux axis angular speed cannot be calculatedby (7). Instead, the flux-axis angular speed is estimated byusing the dq-axis induced voltage and the compensation
gain kpem as:(2)

Id * +

(4)

L
id 2
R2

B. Principle for Flux Axis Angular Velocity Estimation
MethodThe d-axis induced voltage ed of the induction motor iscontrolled to zero. Hence, the flux-axis angular speed iscalculated from the q-axis induced voltage eq and thesecondary d-axis flux 2d as:

The q-axis voltage command vq is calculated from theoutput of q-axis current PI vq*, the secondary magnetic flux
2d, the d-axis current id, the leakage coefficient , Ls andre,vq  vq* +ˆ ( 2 d

iq

Slip frequency
calculation
based on (4)LPF
Cut-off angularfrequency lpf

3F / dq

Iq
Secondary magnetic
flux calculation
based on (3)Inverter angularfrequency calculation
based on (8)Iw

ed Induced voltage
calculation
eq based on (9), (10)

 dt

Iv



Fig. 1. Block diagram of flux-axis angular speed estimation.

Detection
Value

III. BLOCK DIAGRAM OF FLUX AXIS ANGULAR SPEED
ESTIMATION
Fig. 3 shows the analysis model of the flux-axis angularspeed estimation with LPF. The closed-loop transfer
function of the d-axis induced voltage is expressed by (11).
k pem

ωˆ
ed *

1  k pem Lσ iq


1 s

1

ˆ sL I q

＾
eq

R1I q

Vq

(11)

＾

lpf (1  k pem Lσ iq )Besides, the relationship between ed* and eˆd as follows,edk pem Lσ iq
1  k pem Lσ iqeˆd
1ed * 1  slpf (1  k pem Lσ iq )According to (12), the steady-state error ederror of the daxis induced voltage is calculated by (13).
k pem Lσ iq
ederror  1 (13)
1  k pem Lσ iq

2d lagging behind d-axis.

(a)

(12)

F 2d

＾

ˆ sL I q
R1I q

eq

Vq

＾

The compensation gain kpem is calculated from (13) as,
1  ederror(14)
k pem Lσ iq (ederror  1)

F 2d

However, according to Fig.3, the d-axis induced voltage
should be calculated with the disturbance parts as,eˆd 

11  k pem Lσ iq1 s1
lpf (1  k pem Lσ iq )

eq


 (vd  R1id )  (15)
 Lσ iq2d


According to (12) and (15), the time constant Tmodel andcut-off angular frequency model of the designed model
shown in Fig. 3 is calculated by (16).
11(16)
Tmodel model lpf (1  k pem Lσ iq )where the cut-off angular frequency of LPF lpf iscalculated by (17).
model(17)
lpf 
(1  k pem Lσ iq )Fig. 4 shows the design flowchart for the flux axis
angular speed estimation block. Firstly, the steady-state
error and the cut-off angular frequency of speed controller
is decided. Note that, êd should be close to zero. However,
it is difficult to obtain the zero of êd because the steadystate error remains by P control as shown in Fig. 3. Besides,
the cut-off angular frequency of the speed controller
should be calculated from the motor parameter. Next, thekpem is calculated by (14). Then, the cut-off angularfrequency of the designed d-axis induced voltage modelmodel should be decided. The motor will be uncontrollable
when model is lower than ASR because the designed modelinterferes the motor speed control. On the other hand, if themodel is high, the convergence time of the d-axis induced
voltage is long. As the result, the convergence time of
motor speed is long when a torque is given. Then, model(b)

2d matching to d-axis.

＾ˆ sL I q
R1I q

eq

Vq

＾

F2 d
ed

(c) 2d leading to d-axis.
Fig. 2. Induced voltage vector in dq-frame.

eq2d
ed * +



eˆd

+

k pem
+
+

+

1
1 s

1



lpf

iq Lσ

vd  R1id

Fig. 3. Block diagram of flux axis angular speed estimation.

should be decreased. As the result, lpf is calculated by (17).

StartTable I. Induction motor parameters.
Quantity
Poles paris
Rated power
Rated voltage
Rated current
Primary resistance
Secondary resistancePrimary leakage inductance
Secondary leakage inductanceleakage inductance
Mutual inductance
Moment of inertia
Rated speed

Symbol
Pf
P
Vn
An
R1
R2
L1
L2
L
M
J

ωn

Value
2
3.7 kW
188 V
18 A
0.414 Ω0.423 Ω34.54 mH
34.54 mH
1.24 mH
34.3 mH
0.0163 kg・N
1500 rpm

Decision of ederror and ASR
Calculation of kpem to achieve
steady-state deviation ederror (14)Decision of modelNomodel > ASR

Increasing modelYes
Calculation of lpf (17)FinishIV. SIMULATION RESULTS
Table I lists the induction motor parameters. In order toconfirm the validity of the proposed sensor-less vector
FOC, a 3.7 kW prototype of the general-purpose induction
motor was tested. Note that, the cut-off frequencies of thecurrent and speed regulations are 600 Hz and 30 Hz (188.4
rad/s), respectively, whereas both the sampling frequency
and switching frequency are 20 kHz.
Fig. 5 shows the simulation result with the speed
sensor-less FOC using the proposed flux-axis angularspeed estimation. As shown in Fig. 5, the speed command
starts to increase after 0.8 second with no load, then, therated load torque step is applied after 1.5 second. At all
times, the actual speed follows the command value. As theresult, when a torque step is applied to the motor, a speed
error occurs. Note that the estimated value agrees with theactual value of the d-axis induced voltage.
Fig. 6 shows the simulation result with the speed
sensor-less FOC using the proposed design model of fluxaxis angular speed estimation. As shown in Fig. 6(a),100% of the motor torque step is applied after 1.5 second.
As the result, when a torque step is applied to the motor,
the q-axis current ripple and motor speed ripple occurs.
When the compensation gain kpem is set as 8.0 p.u. and lpf
is set as 200 rad/s, the convergence time of the motor speed
is the shortest. Meanwhile, when kpem is set as 1.0 p.u. andlpf is set as 1000 rad/s, the convergence time of the motor
speed is the longest. As shown in Fig. 6(b), the same
condition is applied as Fig. 6(a). As the result, when the
LPF cut-off angular frequency lpf is set as 107 rad/s, themotor is uncontrollable. Meanwhile, when lpf is set as1000 rad/s, the convergence time of the motor speed is thelongest. The relation between kpem, lpf, convergence timeof the d-axis induced voltage follow the designed model,(14) and (17).
Fig. 7 shows the value and the error rate of the steadystate error of the d-axis voltage against the compensation
gain kpem. The compensation gain kpem are varied from 1.0
p.u. to 15.0 p.u.. That is calculated form (15). As the result,the standard deviation of the d-axis voltage from thesimulation almost follows the estimated value. In
particular, the maximum error between the simulation andestimated results of the time constant of the flux-axis

Fig. 4 Design flowchart for compensation gain kpem and cut-off angularfrequency of LPF lpf
q-axis current 1.0 p.u.=22.1A
d-axis current 1.0 p.u.=8.1A
d-axis current
Current 1[p.u.]

q-axis current0
1.0 p.u.=1500 rpm
1.0
actual speed
reference speed
estimated speed

Speed[p.u.] 0.5
0.0

1.0 p.u. = 188V
0.00

d-axisInducedVoltage-0.01[p.u.]

actual voltage
estimated voltage

0.8

1.0

1.2

1.4

1.6

1.8

Fig. 5. Speed control simulation result with the sensor-less FOC.

angular speed is 2.3% at the compensation gain kpem of 3.0
p.u. due to the parameter mismatch of the vd, eq, iq, as
shown in Fig.3. These results confirm the validation of the
proposed design method for the flux-axis angular speed
estimation. Moreover, the effects of kpem on the d-axis
induced voltage regulation can be observed from Fig. 7 asfollows; if the compensation gain kpem is low, the standard
deviation of the d-axis voltage is high.
Fig. 8 shows the relation between the time constant andcut-off frequency obtained from simulation and estimatedresults. As shown in Fig. 8, the d-axis induced voltage
command ed* is set as a step change from 0 p.u. to 0.02 p.u.
in the closed-loop. Note that the time constant of the flux
axis angular speed obtained from the simulation almost
follows the estimated value. In particular, the maximum
error between the simulation and estimated results of thetime constant of the flux-axis angular speed is 3.7% at the
LPF cut-off angular frequency lpf of 200 rad/s. These
results confirm the validation of the proposed design
method for the flux-axis angular speed estimation.

V. EXPERIMENTAL RESULTS
The effectiveness of the proposed design method of theflux-axis angular-speed estimation using the induced
voltage is verified. In the experiment, a three-phaseinduction motor (MVK8115A-R, Fuji Electric Co., Ltd)
with the parameters shown in Table 1 is used as the test
motor. The motor is driven by the proposed design method.
The load torque is varied by controlling of the load motor.
The cut-off frequencies of the current and speed
regulations are 600 Hz and 30 Hz, respectively, whereas
both the sampling frequency and the switching frequency
are 20 kHz, which are the same conditions as simulation.
Fig. 10 shows the experimental waveforms of the phasecurrent and the line-to-line voltage with the conventional
vector control and the proposed sensor-less FOC,
respectively.
Fig. 11 shows the experimental waveforms of the dqaxis currents, the detected and estimated motor speeds. In
any case, the detected speed follows the speed commandof 1500 rpm. In the conventional vector control, the motor
speed does not contain any ripple. In contrast, as shown in
Fig. 11 (b) and (d), when the compensation gain kpem is set
as 2.0 p.u., a current ripple occurs when the motor is 0 rpm.
That is caused by the d-axis induced voltage calculation in(9) which is using the q-axis current. In addition, when the
motor does not have any load, the detection relative errorof the q-axis current is big. Furthermore, if kpem is high, thecalculation relative error of the flux-axis angular speed isbig. This is because the d-axis induced voltage calculation
uses the q-axis current. When the compensation gain kpem
is set as 2.0 p.u. and lpf is set as 1000 rad/s as shown in
Fig. 11(d), a speed ripple occurs at beginning of the motor
acceleration because the time constant of the d-axis
voltage is low, which follows the (17).
Fig. 12 shows the experimental waveforms of the dqaxis current, the detected speed when the torque step isapplied. In any case, the speed error occurs when thetorque step is applied to the induction motor. In theconventional vector control, the maximum speed error
speed is 3%, and the convergence time is 100 ms. In
contrast, in the proposed speed sensor-less FOC, the speed
ripple of 8% occurs and the convergence time is longer
than that of the conventional vector control. In particular,when the compensation gain kpem is set as 2.0 p.u. and lpf
is set as 200 rad/s, the convergence time is 430 ms, whichq-axiscurrent[p.u.]k pem =1. 0 p.u.,  lpf = 1000 rad/sk pem =8. 0 p.u . , lpf = 1000 rad/sk pem =8. 0 p.u . ,  lpf = 200 rad/s
kpem=1.0 p.u

2
10

1. 0 p.u.= 1500 rpm
1.05
Motor
speed
[p.u.] 1.00

kpem=1.0 p.u

0.95
1. 0 p.u . = 188V
kpem=8.0 p.u

d-axisInduced 0.00
Voltage[p.u.]-0.05

kpem=1.0 p.u
1.5

1.6

1.7

1.8

(a). With different compensation gain kpem and lpf.
6
4
2

Current[p.u.]

q-axis current 1.0 p.u.=22.1A

lpf=107 rad/s Uncontrollablelpf=1000 rad/slpf=200 rad/s

0-2
1.0 p.u.=1500 rpm
1.0

Motor
0.5
Speed
[p.u.]
0.0lpf=200 rad/slpf=1000 rad/slpf=107 rad/s Uncontrollable
1.0 p.u. = 188V

0.05
d-axisInduced 0.00
Voltage[p.u.] -0.05

lpf=200 rad/slpf=1000 rad/slpf=107 rad/s Uncontrollable

-0.10
0.8

1.0

1.2

1.4

1.6

(b) With different LPF cut-off angular frequency lpf.
Fig. 6. Speed control simulation result with proposed sensor-less FOC
with different design parameters.
d-axis induced voltage 1.0 p.u. =188V

0.00

d-axis induced voltage[p.u.]

Moreover, the effects of LPF on the d-axis induced voltage
regulation can be observed from Fig. 8 as follows; if the
cut-off angular frequency of LPF is high, the time constantbecomes high and the convergence time becomes long. In
addition, if the compensation gain kpem is high, the time
constant also becomes high and the convergence time ofthe d-axis induced voltage also becomes long.
Fig. 9 shows the relation between the convergence timeof the motor speed, the cut-off angular frequency of LPF
lpf and the compensation gain kpem. If the cut-off angularfrequency of the LPF is high, the convergence time of the
motor speed is long. In addition, if the compensation gain
kpem is high, the convergence time also becomes long. Notethat these characteristics the result shown in Fig. 8.

Simulation result-0.01

Calculated results-0.02

Maximum error rate 2.3%

-0.03
0.0

10.0
5.0
compensation gain kpem [p.u.]

15.0

Fig. 7. Relation between compensation gain kpem and standard
deviation from simulation and estimated results

is 62.3% of that when the compensation gain kpem is set as
2.0 p.u. and lpf is set as 1000 rad/s. As the result, if thetime constant of the d-axis induced voltage is low as shownVI. CONCLUSION

Time constant of model TModel [s]

6

4

2
0
200

400
600
800
Cutoff angular frequency lpf [rad/s]

1000

Fig. 8. Relation between time constant and cutoff frequency obtained
from simulation and estimated results.
Convergence time of motor speed [s]in (19), the convergence time of the d-axis induced voltage
is long. In other words, the motor speed convergence time
becomes long.
Fig. 13 shows the relation among the experimental
results of the motor speed convergence time, thecompensational gain kpem and the LPF cut-off angularfrequency lpf. When a torque step of 100% is applied, theconvergence time of the motor speed is 1.6 s as the worst
case with the compensation gain kpem set as 2.0 p.u. and thecut-off angular frequency set as 400 rad/s. However, if thecompensation gain kpem is set as 2.0 p.u. and the cut-off
angular frequency is set as 400 rad/s, the convergence timeof the motor speed is 0.43 s as the best case, which is
reduced by 73.1% compared to the worst case. The effectsof LPF in the flux-axis angular speed  calculation can beobserved from Fig. 13 as follows; if the cut-off angularfrequency of the LPF is high, the motor speed convergencetime is long. In addition, if the compensation gain kpem ishigh, the motor speed convergence time is long, which
follows (19) and the simulation result shown in Fig. 9. The
cause of the large convergence time error between
simulation result in Fig. 9 and experiment result in Fig.13
is that the real motor parameters is not as same as thedesign parameters in PI current controller and PI speed
controller.

1
kpem=10.8

kpem=2
0.6

0.71s

0.4
0.2
0

0

200

400
600
800
1000
Cutoff angular frequency lpf [rad/s]

1200

Fig. 9. Relation between time constant and cutoff frequency obtained
from simulation and estimated results.
U-phase current [10A/div]

W-phase current [10A/div]

This paper proposed the design method of the flux-axis
0▶
angular speed estimation using the induced voltage for IM
drive systems. LPF was required in the flux-axis angularLine to line voltage Vuv [250 V/div]
speed estimation to stabilize the d-axis induced voltage
0▶
regulation. The effects of this LPF on the d-axis induced
voltage regulation was evaluated. In particular, theLine to line voltage Vuw[250 V/div]
maximum error between the simulation and estimatedresults of the time constant of the flux-axis angular speed0▶
was 3.7% when the cut-off angular frequency of LPF lpf
4 ms
as 200 rad/s.
As the experimental results, LPF highly affected the flux(a) With conventional vector control.
axis angler speed . Besides, when the torque step of 100%
U-phase current Iu [10A/div]
W-phase current Iw [10A/div]
was applied, the convergence time of the motor speed was0▶
1.6 sec. as the worst case with the compensation gain kpem
set as 1.0 p.u. and the cut-off angular frequency set as 1000
Line to line voltage Vuv [250 V/div]
rad/s. However, if the compensation gain kpem was set as
2.0 p.u. and the cut-off angular frequency lpf was set as
0▶
400 rad/s, the convergence time of the motor speed was
0.43 sec. as the best case, which was reduced by 73.1 %Line to line voltage Vuw [250 V/div]
compared to the worst case.
0▶
As the conclusion, the cut-off angular frequency of LPF
lpf highly affected the flux-axis angular-speed. The
4 ms
relation among the flux-axis angular-speed, lpf and the
(b)
With
the
speed
sensor-less
FOC(k
=
1.0
p.u.,

=200
rad/s).
pem
lpf
compensation gain kpem was as follows; if the lpf was high,Fig. 10. Experimental waveforms of line-to-line voltages and phasethe time constant of flux-axis angular-speed was high andcurrents.
the convergence time of motor speed was long. Moreover,
if the compensation gain kpem was high, the time constantdesign value of kpem and the cut-off angular frequency ofof flux-axis angular-speed was high and the convergence
LPF lpf should be clarified by calculation.
time of motor speed was long, which followed the modeland equation shown in this paper. Note that if theconvergence time of the motor speed was long, the rippleof the motor speed might occur. In the future work, the0▶

d-axis current

100%

d-axis current
100%

0▶

q-axis current

q-axis current

100%

100%
Maximum error rate 3%0▶

0▶

100 ms1500 rpm
0▶

Detection speed

Detection speed

1500 rpm

1.0 s

40 ms
0▶

(a)With conventional vector control.

(a) With conventional vector control.
d-axis current
0▶

q-axis current

100%

100%

0▶

100%

0▶

Detection speed

q-axis current

Maximum error rate 8%

Detection speed
1500 rpm

520 msCalculation speed

1500 rpm

0▶
1500 rpm

200 ms0▶
1.0 s(b)With speed sensor-less FOC (kpem = 2.0 p.u., lpf=200 rad/s).

0▶

(b)With speed sensor-less FOC (kpem = 1.0 p.u., lpf=200 rad/s).

d-axis current
0▶

q-axis current

q-axis current

100%

100%
0▶

100%

0▶

Detection speed

Maximum error rate 8%

Detection speed

1500 rpm

Calculation speed

430 ms0▶

1500 rpm

200 ms1500 rpm
0▶
1.0 s(c) Convergence time of motor speed increase by kpem with speed
sensor-less FOC (kpem = 1.0 p.u., lpf=200 rad/s)..

0▶

(c) Convergence time of motor speed increase by kpem with speed
sensor-less FOC (kpem = 2.0 p.u., lpf=200 rad/s).

d-axis current
0▶

q-axis current
100%

100%

0▶

100%

0▶

Detection speed

q-axis current

1500 rpm

690 ms

Detection speed
Calculation speed0▶

1500 rpm

1500 rpm0▶

Maximum error rate 8%

200 ms1.0 s(d) Convergence time of motor speed increase by lpf with speed
sensor-less FOC (kpem = 2.0 p.u., lpf=1000 rad/s).
Fig. 11. Experimental waveforms of dq-axis current, detection speed
and calculation speed.

0▶

(d) Convergence time of motor speed increase by lpfwith speed
sensor-less FOC (kpem = 2.0 p.u., lpf=1000 rad/s).
Fig. 12. Experimental waveforms of speed when a torque was given.

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Convergence time of motor speed [s]REFERENCES

1.6
kpem=1 Figs. 10(c), 11(c)1.2

1.17 skpem=2
0.8

Figs. 10(b), 11(b)0.4
0Figs. 10(a), 11(a)
0

200

400

600

800

1000

1200

Cut-off angular velocity [rad/s]

Fig. 13. Relation between time constant and cutoff frequency
obtained from simulation and estimated results.

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