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fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2021.3094583, IEEE
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Natural Speed Observer for Nonsalient AC Motors

Jiahao Chen, Member, IEEE, Jie Mei, Member, IEEE, Xin Yuan, Member, IEEE,
Yuefei Zuo, Member, IEEE, Christopher H. T. Lee, Senior Member, IEEE

Abstract—This letter addresses experimental validation of This relies on the fact that both ω̂ and θ̂d can be obtained
the reduced-order natural speed observer design for position from observer’s output error, i.e., estimated current error—
sensorless drive with nonsalient permanent magnet synchronous more specifically, ω̂ from q-axis current error, and θ̂d from
motor. The natural speed observer and the active flux estimator
are connected in a cascaded fashion, which results in a simple d-axis current error [6]. Typical example is the direct field
sensorless algorithm that needs only to tune one bandwidth oriented controlled sensorless induction motor drive [7]. There
parameter for flux estimation and one bandwidth parameter for are only a few papers in the field of sensorless permanent
speed observation. Experimental results of high speed reversal magnet (PM) motor drives that follow this design pattern. For
test, zero speed stopping test and slow speed reversal test are example, reference [8] adopts model reference adaptive system
included, where the practice of applying non-zero d-axis current
at zero speed has shown to be effective for loaded zero speed (MRAS) as a filter to the switching terms of a sliding mode
stopping test, but it causes zero-speed locked-up at slow speed observer (SMO) to get both position and speed, while a recent
reversal with acceleration rate of 50 rpm/s. Four remedies one [9] devises speed adaptive full-order observer based on
are compared to improve the slow speed reversal test and the extended emf model and elaborates its gain design with the
proposed method gives almost ramp actual speed waveform, aid of linear system tools.
non-diverging q-axis current and smooth transition in position
waveform during zero speed crossing. A new dynamic expression The observer designs that are previously applied to induc-
of the active flux is proposed and with the aid of the active flux tion motors should also be applicable to synchronous motors.
concept, the proposed sensorless algorithm is also applicable to For example, motivated by the “known regressor model” in
various types of ac motors. [10], we can replace extended emf e with a new state variable
Index Terms—permanent magnet motor drives, sensorless χ = −e − ωJ Ld i. This results in new PM motor dynamics
control, review, active flux, unified model. in which the regressor of speed consists of known signals:
N OMENCLATURE Ld pi = u − Ri + χ + (2Ld − Lq ) ωJ i
(1)
Let p denote the differentiation operator, ud , uq denote the pχ = −ωJ (u − Ri) + ω 2 (Ld − Lq )i
d-axis, q-axis voltages, id , iq denote d-axis, q-axis currents, for which one can design speed adaptive high gain observer
Ld , Lq designate the d-axis, q-axis inductances, ω the electri- [11] or speed update rule with filtered regressor [10] to get ω̂,
cal synchronous angular speed, ωr the electrical rotor angular and at the same time obtains θ̂d from the estimate of χ.
speed, R the stator resistance, KE ∈ R+ the permanent mag-
B. Position Independent Speed Acquisition
net flux linkage, Js the shaft inertia, Tem the electromagnetic
torque, TL the load The second design pattern is to first estimate speed ω̂
 npp
 torque,  the pole pair number, and
finally, J = 01 −1 0 , I = 10 01 . and then calculate rotor d-axis position with an integrator as
∗ ∗
This letter also uses bold vector notations for α-β frame θ̂d = p1 (ω̂ + ωsl ), with ωsl the commanded slip angular speed.
quantities, such as stator current i = [iα , iβ ]T , stator voltage Typical examples include the sensorless indirect field oriented
∗
u = [uα , uβ ]T , (extended) emf e = [eα , eβ ]T , and rotor active controlled, PM motor drive (with ωsl = 0) [12], [13] and
flux (linkage) ψ r = [ψαr , ψβr ]T . induction motor drive (see e.g., [10], [14]). Particularly, for
the emf based drives [12]–[14], the calculated speed signal
I. S ENSORLESS D RIVE R EVIEW: A N EW P ERSPECTIVE using calculated emf ê needs to be embedded in a low pass
filter to resolve algebraic loop problems, as is discussed in [2].

T HE LITERATURE of sensorless motor drives has been

developed for decades, and there has been a trend of
unifying sensorless control for different motor types [1]–[3].
C. Speed Independent Position Acquisition
The third design pattern is exclusively targeted for syn-
When it comes to literature review of different sensorless chronous motors (i.e., PM and synchronous reluctance motors)
drives, the widely accepted classification is to divide sensorless as it directly estimates the rotor d-axis position θ̂ from which
drives into model based drives and magnetic asymmetry based the speed information ω̂ is later extracted. Typical examples
drives (see, e.g., [4], [5]). This letter, however, attempts to include high frequency signal injection based drives [15], emf
provide a new perspective in terms of how speed estimation based drives [1], [4], [16] and flux based drives [3], [17].
and position estimation are related, and to categorize the The high frequency injection method estimates θ̂d utilizing
existing sensorless drives into groups of three design patterns. the rotor saliency on the q-axis using invasive injection of
high frequency voltage. The resulting practical issues such as
A. Coupled Position and Speed Acquisition limited current control bandwidth and audible noise can be
The first design pattern is to get both speed estimate ω̂ and resolved by increasing injection frequency to PWM frequency
rotor d-axis position estimate θ̂d from one full-order observer. [15], while the theoretical development can be found in [18].

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2

The active flux can be defined in complex notations as ψ r = where the rotor flux (i.e., active flux) is determined as follows
KActive ejθd , with KActive = KE + (Ld − Lq ) id . Extended 1
emf is defined as e = [ωKActive −(Ld − Lq ) piq ] ejθd , and it KActive = (Ld −Lq )
(Ld − Lq ) id + KE (3)
becomes ωψ r if piq = 0, e.g., at steady state. rreq p +1
For salient PM motors, the extended emf e was estimated and all other symbols are defined in Nomenclature.
by Gopinath reduced-order observer [1]. For non-salient PM Model (2), (3) describes a wound field synchronous motor
motors, the emf was estimated by SMO [4], [8], [16]. with a short-circuited damper winding to allow self-starting,
The dynamics of active flux involve pure integration, which if we let KE ∝ if with if as the field winding current [22].
are also known as the critical stable voltage model in induction For PM motors and synchronous reluctance motors, ω = ωr ,
motor drive context. Stabilizing the active flux estimation is and rotor resistance rreq is ∞1 , so (3) reduces to KActive =
a popular topic, some recent development in the PM motor (Ld − Lq ) id + KE [3].
drive context, will be briefly reviewed here. As a contribution For induction motors, KE = 0 and Ld − Lq is redefined
from the control community, in [5], the stator flux estimator as the magnetizing inductance in inverse Γ circuit2 , and (3)
was corrected by the squared PM flux amplitude error, of L −L
becomes pKActive = − drreq q KActive + rreq id .
which the stability property was found to be distinct at
different speeds. While in electrical engineering community, a III. R EDUCED - ORDER NATURAL S PEED O BSERVER
disturbance observer was designed to estimate the presumed WITH ACTIVE F LUX C ONCEPT
low frequency disturbance in the presumed “sinusoidal” flux A. The Natural Speed Observer
model, using only voltage model [17]. This work is followed
The natural observer originated in [24] is applicable to dc
in [19] to further incorporate current model information, and
motor and ac induction motor. By noticing the fact that the
in [20] to cope with discretization error.
nonsalient motor model in dq frame is identical to that of a dc
With the information of θd , e, or ψ Active available, a speed
motor, we propose the 3rd-order reduced-order natural speed
estimator is then devised, e.g., Luenberger observer [15], I&I
observer that is simply a copy of (2b) and (2c) plus a load
technique [5], e-MRAS [1], [16], adaptive observer based on
torque identifier:
the dq frame voltage model [21], phase-locked-loop (PLL) [3]
and type-2 system [17]. Lq pîq = uq − Rîq − ω̂ (Lq id + KActive ) (4a)
It is worth mentioning that the θ̂d estimation process in [1], h i
[17] “weakly” depends on the estimated speed ω̂. To show or pω̂r = Js−1 npp 32 npp KActive îq − T̂L (4b)
reduce speed dependency of position estimation, it has been 1
T̂L = KP ε + KI ε + KD pε (4c)
shown with frequency analysis in [17] that an erroneous speed p
is not a big issue for active flux estimation, while in [1],
where KP , KI , KD ∈ R+ , a hat ˆ at the top of a symbol
the H∞ norm of the transfer function from estimated speed
indicates estimated value, and ε is the active power error
error to extended emf error is minimized by properly placing  
observer poles. ε = |u0q | iq − îq (5)
D. Contribution of This Letter
The contribution of this letter is twofold. First, by following when u0q = uq . The implementation of (4) and (5) relies on
the third design pattern, to propose and experimentally validate the information of rotor d-axis position θ̂d , to calculate the dq
a motor motion dynamics based speed observer with the aid of frame quantities:
natural observer concept and active flux concept. The 4th-order
active flux estimator gives θ̂d without knowledge of ω̂; and id =iα cos θ̂d + iβ sin θ̂d
by using θ̂d , a 3rd-order reduced-order natural speed observer iq = − iα sin θ̂d + iβ cos θ̂d (6)
potentially provides consistent estimate of ω̂ even during uq = − uα sin θ̂d + uβ cos θ̂d
speed transient conditions. The proposed sensorless scheme
is simple to implement with only two tuning parameters, and The observer (4) is intuitive when applied to PM motors,
iq
has potential for high performing sensorless motor drive as whereas we need to add the slip relation: ω̂ = rreq KActive + ω̂r
it removes the constant speed assumption. Second, a new before it can be applied to induction motors.
dynamic expression is proposed for active flux KActive , which The stability of the time-varying system, (4) and (5), can
is crucial for further generalizing our proposed method to be be proved via finding Lyapunov function using Krasovskii’s
applicable to various types of ac motors. stability technique [23]. However, if we treat it as a time-
II. N ONSALIENT M OTOR M ODEL IN dq F RAME invariant system by assuming KActive , id and u0q are constants,
The “3rd-order” dynamics of a nonsalient ac motor in rotor we can instead, analyze (4) through the transfer function from
field oriented synchronous d-q frame are: disturbance, TL , to output error, iq − îq

ud = Rid + Lq (pid − ωiq ) + pKActive (2a) (iq − îq ) (p) p npp

= 3 L J (Lq id + KActive ) (7)
uq = Riq + Lq (piq + ωid ) + ωKActive (2b) TL (p) (p + ωob ) q s
Js
pωr = 32 npp KActive iq − TL (2c) 1 The unified model in [2], however, needs to put rreq = 0 for PM motors.
npp | {z } 2 To be specific, following the inverse-Γ circuit notations in [23], Lq , Lσ
Tem is the total leakage inductance, and Ld , Lµ + Lσ is the stator inductance.

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3

where we have tuned the PID coefficients to place the three

observer poles at −ωob ∈ R<0
 npp 0 R
3ωob = Lq Js (Lq id + KActive ) KD uq + Lq

n
(Lq id +KActive ) 32 npp KActive +KP u0q
2


3ωob = Lqpp
Js
n

 3 0
ωob = LqppJs (Lq id + KActive ) KI uq
(8)
However, with u0q = uq , u0q becomes a time-varying function
of speed ωr , and this has made (8) not feasible for variable (a) (b)
speed operation. To make tuning rule (8) time-invariant with Fig. 1. Block diagram of (a) the proposed position sensorless control system,
and (b) the active flux estimator (9).
regard to speed changes, we propose to use q-axis current error
as ε, by putting u0q = 1.
used for Park transformation and the resulting dq frame signals
B. The Active Flux Estimator for Acquiring θ̂d id , iq , uq are the inputs to the speed observer (4) that outputs
The adopted flux estimator in the PM motor context is ω̂r for speed control.
   The tuning process of (9) and (4) is rather simple. First,
ki ψ
pψ 1 = u−Ri+wv + kp + KActive 2 − ψ 2 (9) tune the flux estimator (9) as a second-order system via pole
p |ψ 2 | placement using ωest and ζest = 1, and then tune the speed
where stator flux estimate is ψ 1 = [ψα1 , ψβ1 ]T , the rotor active observer (4) as a third-order system via pole placement using
flux estimate is ψ 2 = ψ 1 −Lq i, and the disturbance is denoted ωob .
by wv ∈ R2 . In this estimator, only the amplitude of flux
estimate is corrected by the active flux amplitude parameter
IV. E XPERIMENTAL R ESULTS
KActive . Then, the θ̂d is computed as arctan2 (ψβ2 , ψα2 ).
If we designate two auxiliary rotor active flux estimates as The test motor is a 750 W, 3 Arms, 2.4 Nm, 750 rpm3
voltage model (ψ VM ) and current model (ψ CM ) to satisfy (at 50 Hz), npp = 4 pole pair, surface mounted PM motor,
ψ CM = KActive ψ 2 |ψ 2 |
−1 and its parameters are: R = 1.9 Ω, Ld = Lq = 5.0 H, KE =
(10) 0.10 Wb and Js = 7.5 kg · cm2 . The test motor is driven
p (ψ VM + Lq i) = u − Ri + wv
by a voltage source SiC inverter that receives gate signals
we can express the flux estimate from (9) in terms of ψ VM from the digital signal processor (DSP), TMS320-F28377D.
and ψ CM as follows (see also [19]) The carrier signal frequency of the space vector pulse width
kp p + ki p2 modulation is 10 kHz, and the current sampling period and
ψ2 = ψ CM + 2 ψ (11) code execution period are 1 × 10−4 s. The DSP has two cores,
p2 + kp p + ki p + kp p + ki VM
one is for implementing control algorithm, and the other is
and we can also derive the transfer function from the distur- used to generate 6 channels of digital-to-analog (DAC) signals
bance wv to flux mismatch ψ 2 − ψ CM as that will be captured by a scope. A dc power supply is used as
p the dc bus. An encoder with 2500 ppr is used for verification
(ψ 2 − ψ CM ) = 2 wv (12)
p + kp p + ki purpose. Current loop bandwidth is tuned to be 200 Hz, the
From (11) and (12), kp , ki can be tuned by placing the poles bandwidth of the closed loop speed control is 40 Hz, and speed
of the second-order systems via the relations: kp = 2ωest ζest loop execution period is 5 × 10−4 s.
2
and ki = ωest , where ζest is the damping ratio and ωest is the
undamped natural frequency. In addition, the relation among A. Voltage Error Issues
ψ VM , ψ CM , and ψ 2 is clarified in Fig. 1b. The active flux estimate ψ Active is prone to be erogenous
Please note the above analysis does not make any conclusion due to voltage error at low speeds, so it is important to remove
regarding the convergence to actual flux. In order to prove the the drift in dc bus voltage measurement as well as compensate
stability of the estimated flux error ψ r − ψ 2 , a fourth order inverter nonlinearity.
dynamical system with ψ r − ψ CM and ψ CM − ψ 2 as states When measuring the inverter’s voltage and current charac-
must be analyzed using a similar approach elaborated in [25]. teristics at standstill, it should be emphasized that instead of
applying α-axis current, the dc current should be applied at β-
C. Brief Summary axis such that the phase U current (i.e., iα ) is null and phase V
The implementation of the proposed sensorless algorithm and phase W have the same current amplitude with opposite
is summarized by the block diagram in Fig. 1a. In Fig. 1a, signs. Otherwise, by regulating iα , the current amplitude of
the voltage u is substituted with the voltage command u∗ , phase U is two times as large as the amplitude of phase V or
and the current i is obtained by applying amplitude invariant W , so the distorted phase voltage that is a nonlinear function
Clarke transformation to measured phase currents. Then, u of phase current cannot be acquired. Finally, multiplying a
and i are the inputs to the flux estimator (9), as shown in scaling factor of sin( −2π 3 ) to β-axis quantities gives phase
Fig. 1b, which outputs ψ 2 to provide its angle information quantities.
as θ̂d = arctan2 (ψβ2 , ψα2 ). Rotor position estimate θ̂d is 3 In this letter, rpm means mechanical revolution per minute.

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(a) (b) (c) (d)

|ω̂ |
Fig. 2. Sensorless high speed reversal test using (a) ωob = 170; (b) ωob = 340; (c) ωob = 510; (d) SMO from [4] with k = 400, τc−1 = 20r +10 and PLL
with proportional gain of 1000 and integral gain of 1 × 105 (all in SI units).

B. Load Motor Manipulation id = 2 A is applied at t = 2.9 s

The shaft of the test motor is coupled with another servo
motor as load. The load motor is controlled with some speed
id = 2 A enlarges speed ripples
command and its speed loop output limit is set to 2.1 A. (main component is of 6.67 Hz)
In Fig. 2, the speed command for the load motor is 0 rpm,
so the q-axis current has the same sign of the speed at steady
state. The overall load torque that includes friction is estimated
to be about 1.5 Nm, which equals 62.5% of the motor rated
torque.
In Fig. 3 and Fig. 5, in order to test the sensorless drive for
various operating conditions, the speed command of the load
motor is set to −300 rpm. This means that when the speed
is positive, the test motor is motoring; while the test motor
is regenerating when the speed is negative. In addition, it is
observed that the steady state value of iq is higher for positive
speed operation and is lower in negative speed operation in
Fig. 6c, because the load motor “helps” to fight against the Fig. 3. Sensorless zero speed stopping test under step speed command. The
friction as long as rotor shaft is spinning at a negative speed. flux estimator tuning is kp = 20, ki = 4 (all in SI units).

C. Natural Speed Observer based Sensorless Drive The 6.67 Hz Component

1) Sensorless High Speed Reversal: Fix ωest = 25 rad/s

and ζest = 1. Three sets of ωob values are tested. According
to Fig. 2a, 2b, 2c, the 0%–0.97% rising time for the motor to
step from −1500 rpm to 1400 rpm is about 0.12 s regardless
of ωob values, implying that the speed observer bandwidth Fig. 4. Fourier analysis of the actual speed waveform from Fig. 3. Spectrum
is not the bottleneck for reducing rising time. There exists of id = 0 A corresponds to t ∈ (1.1, 2.9) s in Fig. 3, and spectrum of
undesired oscillation in ω̂r when ωob is too large, which leads id = 2 A corresponds to time domain data of t ∈ (7, 9) s in Fig. 3.
to oscillated i∗q and iq profiles in Fig. 2c. The above results
validate the effectiveness of the one parameter speed observer In Fig. 3, the speed ripples are found to have vanished
tuning rule (8). at zero speed. After applying nonzero id , the speed ripples
2) Extreme Low Speed Tuning Guidelines: For extreme low become more apparent at 100 rpm, and are found to have the
speed operation, we recommend to select a large damping ratio same frequency of the mechanical speed of 6.67 Hz (i.e., 100
ζest such that the transfer function in (12) behaves as a first- rpm). This is further supported with Fourier analysis as shown
order one, e.g., no overshoot. According to our studies, we in Fig. 4. When id = 0 A, the average value of the actual speed
adopt ζest = 5 and ωest = 2 rad/s. is 39.8 dB (i.e., −97.2 rpm), and the amplitude of the 6.67
3) Sensorless Zero Speed Stopping Test: In order to sta- Hz speed harmonic is 5.9 dB; while id = 2 A, the average
bilize zero speed operation when the motor is loaded, we value of the actual speed is 39.6 dB (i.e., 95.5 rpm), and the
propose to excite the motor with d-axis current, i.e., id = 2 A. amplitude of the 6.67 Hz speed harmonic is increased to 23
Note this nonzero id is not an effort to “mitigate” the influence dB. How to reduce the speed ripples is beyond the scope of
of the inverter nonlinearity, as the motor is loaded and iq is this letter, but the thing to note here is that the speed estimate
large. The reason is that, in a detuned controller frame, the ω̂ even tracks the speed ripples in ω at 100 rpm steady state.
nonzero id helps to pull the rotor d-axis back to align with 4) Sensorless Slow Speed Reversal: The point of a slow
the controller d-axis. In Fig. 3, at zero speed operation, one speed reversal test is to keep decreasing the acceleration
can observe that iq and θ̂d are slowly deviating even though rate of the ramp speed command until the sensorless drive
the load motor outputs constant load torque. In this case, the shows some undesired behaviors. In our case, the acceleration
nonzero id helps to “lock” the shaft at zero speed if θd 6= θ̂d . is ±50 rpm/s in Fig. 5. At a glance, one might consider

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loss of speed control 3) Additional Flux Limiter Correction: The estimator dy-
namics (9) are modified with an additional flux limiter [17]

pψ 01 =u − Ri − klimiter ψ 02 − ψ 02limited


zero speed locked-up
!
ψ 02
 
ki 0
+ kp + KActive 0 − ψ 2 (14a)
p ψ2
ψ 02 =ψ 01 − Lq i (14b)

ψ 02 , if ψ 02 ≤ κKActive
0
ψ 2limited = ψ 02 (14c)
detuned field orientation κKActive , others
| 02 |
ψ

where κ = 1.15 [17], and klimiter is active and set to 2π ×

100 rad/s only if the absolute value of the speed command is
less than 1.5 s−1 (i.e., 90 rpm in our case). Note (14) is a
combination of (9) and (13).
4) Experimental Results: More experimental results of slow
Fig. 5. Sensorless slow speed reversal test with ramping speed command and speed reversal are presented in Fig. 6. In Fig. 6, both the loss
id = 2 A. The flux estimator tuning is kp = 20, ki = 4 (all in SI units). of speed control and zero-speed lock-up phenomena are gone.
Even though the speed waveform in Fig. 6a trends to diverge, it
Fig. 5 shows a successful slow speed reversal test. However, is pulled back to speed command before it develops oscillation.
there exist problems of “zero speed locked-up”, “detuned field However, in Fig. 6a and 6c, a new phenomenon of zero-speed
orientation”, and “loss of speed control” as annotated in Fig. 5. oscillation occurs during slow zero speed crossing (instead of
Zero speed locked-up occurs when the actual motor speed ω speed waveform being a straight horizontal line at zero speed
becomes zero regardless of the speed command during speed as in Fig. 5). Besides, the 6.67 Hz speed ripples are mitigated
reversal near zero speed. Detuned field orientation essentially in Fig. 6, because id is regulated to 0.
means θ̂d from the active flux estimator deviates from θd . In a In conclusion, the experimental results using (9) with kp =
sensorless drive, a clear indicator for loss of field orientation 200 rad/s and ki = 0 are preferred, as there is no speed
is that iq deviates from its desired value, as shown in Fig. 5, oscillation at zero speed, and the waveforms of θd and θ̂d
at t = 4 s when the motor escapes from zero speed locked-up, resemble that of sensored control the most.
and at t = 9 s when the motor is locked up at zero speed.
B. Comparative Studies to EMF based Drive
A severer consequence of loss of field orientation is loss of
speed control at t = 4 s when the actual speed reaches over A popular4 sliding mode observer (SMO) from [4] is
−100 rpm when the speed command is about −30 rpm. During adopted for the comparison between the emf-based drive and
loss of speed control, the actual speed diverges from speed flux-based drive. Corresponding successful experimental slow
command, but as θ̂d converges, speed control is attained again. speed reversal results are shown in Fig. 6d, and high speed
V. D ISCUSSION reversal results are shown in Fig. 2d. The fast switching nature
of SMO causes the motor speed ripples in Fig. 6d and Fig. 2d,
A. Improving Slow Speed Reversal Performance and we have been using a time-varying SM gain k for low
1) Integrator with Nonlinear Amplitude Limiter: In [26], an speed operation to mitigate the chattering issues.
integrator with a flux amplitude limiter is proposed for flux According to our studies, emf-based drive using SMO can
estimation achieve comparable performance as ours. However, it is much

ψ
 more complicated to tune. Particularly, implementing SMO
pψ 1 = u − Ri + kp LCM 2 − ψ 2 (13a) needs to tune an SM gain, a speed dependent low pass filter
|ψ 2 |
( 0
pole, and a coefficient to mitigate chattering of switching
|ψ 2 | , if ψ 2 ≤ KActive function (e.g., saturation function in [4] or sigmoid function in
LCM = (13b)
KActive , others [8], [16]), in order to get rotor position, while (9) needs to tune
only one parameter, i.e., ωest for high speed, or only kp for
where the limiter correction is only active when |ψ 2 | is larger extreme low speed. Specifically, during low speed operation,
than KActive , and in other words, when |ψ 2 | is smaller than the tuning is as simple as increasing kp until it stabilizes the
KActive , (13a) becomes an unstable pure integrator. pure integrator, and moreover, choosing a large kp does not
2) Integrator with Linear Amplitude Correction: The inte- bring penalty to system performance, i.e., no chattering issues.
grator with limiter (13) is suitable for induction motor, but it
does not make sense for PM motor whose rotor flux amplitude VI. C ONCLUSION
is always equal to KE , so when |ψ 2 | is smaller than KActive , After categorizing existing sensorless drives into groups of
it should also be corrected. In fact, (13) becomes (9) if we three design patterns, this letter proposes a simple load torque-
put LCM = KActive and ki = 0. Consequently, the transfer adaptive natural speed observer using the active flux concept,
function in (12) reduces to a first-order one, which is consistent
with our tuning guideline to select a large damping ratio ζest . 4 For example, it is later adopted in [27] to reconstruct extended emf.

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Transactions on Power Electronics
6

(a) (b) (c) (d)

Fig. 6. Sensorless slow speed reversal test (a) using (13) with kp = 200; (b) using (9) with kp = 200, ki = 0; (c) using (14) with ωest = 2 and ζest = 5
(all in SI units); (d) using SMO, with k = 10KE |ω ∗ | + 5 V and τc−1 = 5 rad/s, where k and τc are defined in [4].

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