1

An automatic procedure for induction motor

parameter estimation at standstill
Luca Peretti∗ , Associate Member, IET, and Mauro Zigliotto∗∗ , Member, IET
∗ ABB Corporate Research Center, Västerås, Sweden (e-mail: luca.peretti@se.abb.com).
∗∗ University of Padova, Department of Technique and Management of Industrial Systems, Vicenza, Italy (email:
mauro.zigliotto@unipd.it)

Abstract—The paper presents a self-commissioning procedure drawbacks are related to the simplifications introduced in the
for the automatic parameter estimation of three-phase induction estimation equations and, again, the supposed linearity of the
motor drives. The procedure consists of a step-by-step approach system.
with different test signals to obtain the parameter values, while
maintaining the motor at standstill. The actual implementation Recently, and using a similar approach, [5] presented a
is capable of mapping both inverter and motor parameters non- very simple and handy standstill test. The paper claims that
linearities, providing accurate data for the tuning of common two standstill impedance measurements, using two sinusoidal
current regulators and for advanced sensorless drives as well. voltage signal injections, are mathematically sufficient to de-
Theoretical and experimental results are provided, proving the termine the IM parameters of the well-known inverse-Γ four-
effectiveness of the procedure.
elements equivalent circuit. The solution is to be considered as
Index Terms—AC converter machines, Induction motors, Pa- the simplest one in the scenario, although the least accurate. A
rameter estimation, Non-linear models, Digital control. slightly more precise procedure, based on two sinusoidal signal
injection, is reported in [6]. The paper includes a detailed
I. I NTRODUCTION discussion about the frequency dependence of some of the
motor parameters, but the problem of inductance saturation is
Parameter estimation of AC machines is a research topic
not solved yet.
in the electric drive area for almost three decades. From the
More recently, a research branch focused on the realization
early scalar controllers to the more recent vector field-oriented
of reliable sensorless drives, in which AC motors are elec-
control (FOC), a precise knowledge of motor parameters has
tronically speed-driven without the need of a position/speed
always been a key issue.
sensor, which is substituted by a mathematical speed observer
Induction motor (IM) parameter estimation represents one
that heavily relies on the motor model. An accurate knowledge
of the most widely studied topics in the electric drive literature,
of model parameters is mandatory for a stable sensorless drive
mainly due to the widespread use of IMs in industry. Early
with acceptable performances. All in all, this implies that
papers, as for example [1], were devoted to parameter sensi-
parameter non-linearities, like magnetic saturation, should be
tivity analysis. Then, many papers discussed how to provide
included.
constant motor parameters to FOC algorithms for IMs. Among
Two different approaches exist to track parameter non-
the presented solutions, [2] and [3] are quite representative. In
linearities. The first approach is the on-line tracking of the
[2], one of the first standstill procedures for IM parameter
most critical parameters of the IM like stator and rotor resis-
estimation was presented. The procedure exploits DC and
tances, in order to update on-the-fly the tuning of sensorless
transient measurements on currents and references voltages
algorithms. Interesting solutions have been presented in [7],
from the controllers to estimate the stator resistance, the
[8] and [9].
leakage inductance and the rotor time constant of the machine
The second approach makes use of off-line procedures,
under test. As mentioned by the Author, the procedure is still
which use test signals to compute a detailed map of IM
limited to linear models. Magnetic saturation is not considered,
parameters non-linearities, prior or during the final commis-
even for the calculation of the rotor time constant, which is
sioning phase. [10] proposed a standstill solution which takes
obtained through specific transient tests that do not saturate
into account the magnetising inductance variation. The flux
the machine. A more comprehensive solution for parameter
linkage-current relationship is estimated with a third-order
estimation of FOC drives is presented in [3], which claims a
polynomial by means of a sinusoidal current injection and a
complete drive tuning within 60 s. However, the motor rotates
recursive least square algorithm. However, the procedure needs
during the tuning process. Moreover, it requires a speed sensor
a prior knowledge of other IM parameters. In [11], a fully-
and a phase voltage measurement, which is not common in
comprehensive series of laboratory tests has been performed
industrial drives. The same solution is recalled in [4], where
on an IM, with the aim of obtaining a complete description
the parameter estimation is performed at standstill using a
of the motor parameters. As a matter of fact, an excellent
PWM inverter and injecting DC and sinusoidal currents. Major
self-commissioning scheme should obtain results as close as
L. Peretti was formerly with the University of Padova, Department of possible as the ones shown in that work. Procedures like that
Technique and Management of Industrial Systems, Vicenza, Italy. are not suitable for self-commissioning, since they are not
 2

executed at standstill and they need a varying mechanical load. electromechanical speed, which is equal to pωm where p is the
They also require a quite high computational effort. However, pole pair and ωm is the mechanical speed. Further expansions
results clearly show the dependence and the non-linearities of (1) require some additional considerations, reported below.
of every IM parameter with respect to IM state variables
(for example, the magnetising inductance as function of the A. Magnetically saturated machines
magnetising current).
The constitutive equation of a generic non-linear inductor
This work presents a complete off-line self-commissioning
is the following:
procedure for IM parameter estimation at standstill. The dλ(i(t))
algorithm is not limited to linear models and it improves u(t) = (2)
dt
the results obtainable with the procedures of [12], [13] and
[14], by innovating and merging them all with a step-by- where u, i are the voltage and current in the bipole, and λ(i(t))
step approach. The setup is composed by a conventional indicates the magnetic flux linkage as function of the current.
induction motor fed by a three-phase PWM inverter. Several The instantaneous link between the flux linkage and the current
different test signals are used in sequence, with the aim of can be expressed by the relation λ(i(t)) = La (i(t))i(t), in
extracting and mapping parameter non-linearities, as proven which the apparent inductance La is function of the current
in the experimental section. The procedure is performed at only. By utilising the chain differentiation rule, it can be
standstill, and without any change in motor connections, so expanded as:
that it fits for motors already placed on site, too. When the dλ(i(t)) d (La (i(t))i(t))
machine is supplied from an inverter, there will appear high u(t) = = =
frequency components of the current, superimposed to the  dt dt 
dLa (i(t)) di(t)
sinusoidal signals. These higher harmonics are of few interest = La (i(t)) + i(t) =
di(t) dt (3)
here, since the goal is to get a model valid for the first
harmonic of the supply. Actually, this makes sense since any di(t)
= [La (i(t)) + Ld (i(t))] =
vector controller (and sensorless scheme as well) operates to dt
control fundamental quantities only [15]. Of course, a model di(t)
= L(i) .
valid also at high frequency could improve the accuracy in fast dt
dynamic responses (e.g. step response) but the trade-off is with Equation (3) introduces the differential inductance Ld . The
a unitary, simple and easy to handle lumped-parameter model. final expression is formally similar to the linear case, with the
The limited improvement in the accuracy that derives from obvious distinction of an explicit dependence on the current
the modelling of high-frequency effects on parameters would of the instantaneous inductance L
hardly justify the rise of complexity of the model topology. dLa (i)
The paper is organised as follows: Sect. II recalls the L(i) = La (i) + i(t) . (4)
di
basics of the induction motor, with particular focus on the
A graphical interpretation of the apparent and instantaneous
magnetically saturated machines and their equivalent circuit
inductances is reported in Fig. 1. The instantaneous inductance
representation at standstill. Sect. III details the theory and the
L represents the actual inductance around the operating point
sequence of the step-by-step procedure for each parameter
(idc , λdc , ) and it expresses the derivative dλ/di of the flux
of the machine. Sect. IV shows some experimental results
linkage with respect to the current in that point.
obtained from the implementation of the procedure in the
laboratory, by using a fast-control prototyping test bench.
Considerations and comments on the obtained results are given
as well, before the conclusive remarks of Sect. V.

II. I NDUCTION MOTOR BASICS

The general IM space vector voltage balance equations for
a symmetric machine can be expressed in the stationary αβ
reference as follows [16]:

 dλs
 u s = R s is +

dt (1) Fig. 1. Graphical interpretation of L and La .
 0 = Rsr isr + dλsr − jωme λsr


dt
In (1), the space vectors are relative to the stator phase B. Inverse-Γ equivalent circuit at standstill
voltages (usa , usb , usc ) and currents (isa , isb , isc ), the rotor In case of a linear system, the IM mathematical model (1)
currents referred to the stator (isra , isrb , isrc ), the stator flux yields the well-known inverse-Γ electrical equivalent circuit
linkages (λsa , λsb , λsc ) and the rotor flux linkages referred to [16], which features a transient inductance Lt in the stator
the stator, (λsra , λsrb , λsrc ). Rs is the stator resistance, Rsr branch and the magnetising inductance Lϕ in one of the two
is the rotor resistance referred to the stator, and ωme is the derived branches (Fig. 2).
 3

isb = −isa , isc = 0

(7)
iϕb = −iϕa , iϕc = 0

and, of course, usb = −usa , usc = 0. If this is the case,

equations (6) collapse to the following:

Fig. 2. Inverse-Γ equivalent circuit for linear systems. 
 u disa diϕa
 sa = Rs isa + Lt (isa ) + Lϕa (iϕa )
dt dt (8)
 di ϕa
0 = Rsr isra + Lϕ (iϕa )


In case of saturation of the motor magnetic paths, establish- dt
ing the flux density will require an additional component of
the magnetising current iϕ , which will be a nonlinear function The per-phase equivalent circuit, including saturation ef-
of the stator flux. The combined effect of the magnetomotive fects, which will be the reference for the next Sections, is
force (mmf) required by the saturated stator teeth and yoke reported in Fig. 4. It is worth to recall that due to the setting
may be incorporated in the inverse-Γ circuit by adding an ωme = 0, the representation is valid at standstill only.
appropriate nonlinear inductive element across the stator flux
linkage ([17]). To maintain the same circuit topology of Fig. 2,
the saturation effect has been split into the inductances Lt (is )
and Lϕ (iϕ ), which then become functions of the respective
current space vectors, according to [16].

Fig. 4. Per-phase inverse-Γ equivalent circuit at standstill, including satura-

tion.

According to Sect. II-A, the inductances of the circuit are all

of instantaneous type and each of them depends on the current
that flows in its own circuit branch; in case of absence of
Fig. 3. Inverse-Γ vector equivalent circuit including stator iron saturation, at
standstill. saturation, their differential component is null, and they reduce
to the apparent (and constant) inductance type. Once more, it
At standstill (ωme = 0), and consistently with the general is worth to highlight that the instantaneous inductances are
discussion of Sect. II-A, which applies also to vector quanti- suitable for writing voltage balance equations, as in (8). The
ties, the extension to the non-linear case can be obtained by conventional nomenclature still holds, referring to Lϕ and Lt
expressing the flux linkages derivatives in (1) as functions of as magnetising and transient (or total leakage) inductances
the instantaneous inductances: respectively. The equation set (6) is the base for the proposed
 procedure of parameters estimation. As mentioned, the aim
 u = R i + L (i ) dis + L (i ) diϕ
 is to make a set of measurements with different test signals,
 s s s t s ϕ ϕ
dt dt (5) and at different current levels, wanting to get the link between
 0 = Rsr isr + Lϕ (iϕ ) ϕ

 di the electrical parameters of Fig.3 and the different operating
dt conditions. Those parameters are essential in most of the
sensorless algorithms reported in literature, as for example thus
where the dependences of the inverse-Γ circuit inductances related to the model reference adaptive systems (MRAS) class.
to the current space vectors are made explicit. The resulting In the following, the focus will be brought to the derivation
equivalent circuit is reported in Fig.3. The voltage balance of the four-parameter set under the hypothesis (7), but the
equation for the phase a is obtained by taking the real results can be readily extended to the case of a current vector
components of (5): positioned anywhere in the stationary reference frame, by the
 use of (6) and the vector version of (4).
di di
 usa = Rs isa + Lt (is ) sa + Lϕ (iϕ ) ϕa


dt dt (6) III. T HEORY OF OPERATIONS

 0 = Rsr isra + Lϕ (iϕ )
 di ϕa
dt The estimation procedure refers to the model of Fig. 4
and it consists of four consecutive steps, each exploiting the
Clearly, further simplifications to (6) are possible only if results of the preceding one. The parameters are estimated
either the system is linear, so that inductances are independent by imposing a known voltage and measuring the current in
to the currents, or in the particular case in which one phase is a single phase (let it be phase a). The constraint on the
disconnected, which realises the conditions voltages which makes valid the representation at standstill
 4

Two major issues are worth consideration when performing

the stator resistance estimation, the precision and the meaning
of the results. The former is related to the precision by which
u∗ is known. A voltage sensor is usually avoided and the phase
voltage is approximated by its reference (9), provided that an
adequate compensation of inverter non-linearities is performed
[9]. Further details will be provided in the Sect. IV.
The estimation meaning depends on whether the voltage is
measured on motor or inverter terminals. In the first case, the
estimated resistance value coincides exactly with the phase
resistance Rs . In the present case, which makes use of the
voltage references, the Usa,dc /Isa,dc ratio encompasses the
whole chain of resistive elements, including cables and IGBTs
(a) on-state resistances. This approach fits for sensorless control
algorithms, since they also use the voltage references as input
to their motor models.

B. Transient inductance estimation

The parameter Lt is estimated by means of the injection
of a high frequency sinusoidal test signal. At steady state, the
total impedance of the circuit in Fig. 4 is
jΩh Lϕ Rsr
Z(jΩh ) = Rs + jΩh Lt + (10)
jΩh Lϕ + Rsr

(b) where Ωh = 2πfh , with fh frequency of the injected signal.

It is recognised that the last term of (10) converges to Rsr
Fig. 5. The selected topology for induction motor voltage supply. (a) One
phase open. (b) Suitable voltage SVM reference. as the frequency increases. Consequently, for sufficiently high
values of Ωh the following approximation applies:

(usb = −usa , usc = 0) is easily obtained by the topology Z (jΩh ) ≈ Rs + Rsr + jΩh Lt (11)
reported in Fig. 5a. In principle, the real part of the impedance could be used
The voltage generator of Fig. 5a is computed to get a net for the estimation of Rs + Rsr . Since Rs is already known,
phase voltage (referred to the motor neutral point 0) usa = the value Rsr could be evaluated. However, this way is not
ua0 = u∗ . When the IM is fed by a conventional three-phase advisable, due to skin and proximity effects that increase
space-vector-modulated (SVM) inverter, the candidate voltage the resistances at high frequency, giving fake results for the
vector reference is the following: estimation [18]. On the contrary, the imaginary part gives a
2 ∗  straightforward estimate of Lt .
u∗s = u − u∗ ej2π/3 (9)
3 As the frequency of the test signal increases, the error
It is easy to see that (9) implements the topology of Fig. 5(a). between the measured = [Z (jΩh )] /Ωh and Lt becomes negli-
The voltage vector u∗s is obtained by applying u∗ to the phase gible. For a IM with a rated frequency of 50 Hz, a test angular
a, −u∗ to the phase b and a null voltage to the phase c. The frequency Ωh ≥ 2π300 rad/s is appropriate.
current flows from phase a to b, while c remains virtually Unfortunately, the validity of the aforementioned method
open. In the stationary frame, the reference space vector is is restricted to the linear case only. Even if the transient
√ inductance is mainly associated to a leakage flux linkage,
then u∗ − ju∗ / 3, as shown in Fig. 5(b).
As a relevant add-on feature, the motor will remain con- nevertheless it may suffer of some saturation effects. It means
nected to the plant with unmodified electrical and mechanical that the linearity is lost and the impedance (10) makes no
connections, and this fits for a specific requirement of many sense, since the current is distorted.
industrial applications. Here, a more general approach is proposed. It consists in ap-
plying a small sinusoidal voltage usa,ac (t) = Usa,ac cos(Ωh t)
superimposed to a predefined set of increasing DC voltage
A. Stator resistance estimation levels, while measuring the phase currents at each stage. The
The first parameter to be estimated is the stator resistance DC component is selected so to span the whole current range
Rs . A DC current is imposed in the circuit of Fig. 4 by on the basis of the previous knowledge of Rs , and it sets the
setting u∗ =Usa,dc in (9). The inductances in Fig. 4 behave working point (as for example the point P of Fig. 1). The low
as short-circuits and no current flows into the Rsr branch. amplitude nature of the AC signal permits the exploitation
Consequently, the ratio between the phase voltage and current of the small-signal theory, with supposed linearity around
returns the expected estimation of the Rs value. an operating point. This approach is indicated as DC+AC
 5

method, and it represents a marked difference with respect In the first step, when the inverse-Γ circuit of Fig. 4
to the procedure described in [12] where only an AC method is fed by a DC current, the inductances are short-circuits
at various current levels is exploited. and the magnetic energy is stored without being transformed
AC voltages and currents are used to compute the instan- into mechanical energy. The flux linkages are obtained by
taneous inductance Lt . The voltage amplitude and frequency integrating the equation (8) using the definition (3):
are imposed, and thus they are known a priori. The real and Z t
imaginary components of the fundamental AC current phasor (usa − Rs isa ) dt = usa (0) − Rs isa (0)+
(13)
I˙sa,ac = Isα + jIsβ are provided by the Goertzel algorithm, 0
+λta (t) − λta (0) + λϕa (t) − λϕa (0)
synchronised to the phasor U̇sa,ac of the applied voltage. The
Goertzel algorithm is a novel alternative to [12]. It is a single- At t = 0 the circuit is in a steady state DC condition, and
harmonic discrete Fourier transformation that is particularly therefore usa (0) − Rs isa (0) = 0. Afterwards, a null voltage
suitable for the on-line detection of one or few harmonics usa (t) = 0 is applied to the motor (step b)) and both the
(see [19] and Appendix B for details). By construction, it is current and the motor flux linkages converge to zero, so that
U̇sa,ac = Usα + j0 = Usa,ac + j0 and from (11) it is: Z t
= [Z] 1

Usα + j0
 lim (usa − Rs isa ) dt = −λta (0) − λϕa (0) = −λsa (0)
t→+∞
Lt = = = 0
(14)
Ωh Ωh Isα + jIsβ
1 Usα Isβ (12) At t = 0 the currents in the two inductances of Fig. 4 are the
= − 2
Ωh Isα + Isβ2 same (isa (0) = iϕa (0) = Isa,dc ) and they are linked to the
flux linkages by the apparent inductances, as outlined in Sect.
A simulation may help in understanding the error that II-A. Merging this information with (14) gives:
would be introduced by neglecting the saturation effect. Let
us consider a simple R-L circuit where R = 1 Ω and La (Isa,dc ) = Lϕa (Isa,dc ) + Lta (Isa,dc ) =
Z t
L is a (known) non-linear function of the current (Fig. 6). 1 (15)
=− lim (usa − Rs isa ) dt
As mentioned, the proposed DC+AC method imposes a DC Isa,dc t→∞ 0
voltage to set different operating points, and a superimposed The expression (15) suggests the way to compute the mag-
AC voltage of 2 V at 300 Hz to detect the instantaneous netising apparent inductance Lϕa , which will be used in
inductance, by means of expression (12). Results are shown in conjunction with (4) to calculate the instantaneous inductance
Fig. 6. The proposed estimation is able to track precisely the Lϕ . There is an obvious advantage in considering (15), since
it is not affected by inductance saturation, allowing different
0.06
Real Lt profile DC current values Isa,dc to be set in order to map λϕa , and
DC+AC method Lϕa immediately after, as function of the current.
0.05 AC method only As mentioned earlier, the computation requires the knowl-
edge of both the resistance Rs and the transient inductance
Inductance [H]

0.04 Lt , which at this point are available from the previous steps.
An effective procedure consists in prefiguring the total flux
linkage λsa (i) = λϕa (i) + λta (i) as a third-order polynomial
0.03
function of the current Isa,dc ([10], [20]) so that

0.02 λsa (i) p3 i3 + p2 i2 + p1 i + p0

La (i) = = =
i i (16)
0.01 2
= p3 i + p2 i + p1 + p0 i −1
−10 −5 0 5 10
Current [A]
The coefficients (p3 , p2 , p1 , p0 ) can be obtained by a poly-
Fig. 6. Effect of Lt saturation in the accuracy of the estimation methods.
nomial fitting algorithm, which elaborates the measurements
of λsa obtained from (14) by imposing different Isa,dc current
original (known) outline of the saturated inductance, while the
levels. Substituting the last of (16) in (4) returns
conventional AC method, which applies only AC voltages of
increasing amplitudes, clearly fails even for medium current dLa (i)
L(i) = La (i) + i = 3p3 i2 + 2p2 i + p1 = Lϕ (i) + Lt (i)
levels. di
(17)
that will be used to get Lϕ from the value of Lt obtained in
C. Magnetising inductance estimation Sect. III-B.
The Lϕ estimation at standstill is based on the model In principle, the main drawback of expression (15) relies
discussed in Sect. II-B. The proposed method consists of two in the digital implementation of a pure integrator. This is a
steps: a) impose a DC current level Isa,dc , to fix a steady state rather annoying and well-known problem, due to the DC offset
working point; b) set a null voltage vector and integrate the that always affects the integrator input, causing output drift.
resistive component of the current, until the system reaches Effective countermeasures, taylored on the singularity of the
the null state. proposed method, are discussed in Sect. IV.
 6

D. Rotor resistance estimation known [13] that the voltage distortion behaviour at low current
The rotor resistance referred to the stator Rsr (Fig. 4) can levels diverges from the theoretical profile, being a constant
be obtained by exciting the system with a sinusoidal, low- only for sufficiently high current magnitudes.
frequency and low-amplitude voltage usa = Usa cos(Ωl t). The non linear behaviour of the inverter can be mitigated
Similarly to Sect. III-B and for an effective mathematical by prefiguring an input-output mapping. To this aim, different
elaboration, a α-β reference frame fixed to the phasor U̇sa DC current levels Isa,dc are first applied to the motor, and the
is chosen, so that the sinusoidal phasor I˙sa = Isα + jIsβ can corresponding voltage references u∗sa are recorded accordingly.
be extracted fairly easily by the Goertzel algorithm, which Fig. 7 shows the static u∗sa = f (Isa,dc ) curve measured in
represents a novel and distinctive feature of the proposed laboratory.
method over the existing ones, as [14].
15
The voltage phasor U̇sra = Usrα + jUsrβ across Rsr is
given by:

U̇sra = U̇sa − (Rs + jΩl Lt )I˙sa = u*sa

(18) 10 Rs isa
= Usα − (Rs + jΩl Lt )(Isα + jIsβ )

u*sa [V]
The resistance value is the ratio between the voltage and
current amplitudes: 5
|U̇sra |
Rsr = (19)
|I˙sra |
Due to the resistive nature of the branch, the current phasor 0
0 2 4 6 8 10
I˙sra is the component of I˙sa in phase with U̇sra : Isa,dc [A]

I˙sra = I˙sa cos (ϑsr ) (20) Fig. 7. Static usa = f (isa ) for Rs estimation.
where ϑsr is the angle between I˙sa and U̇sra . The value of
The curve accounts for both the voltage distortion effects
cos (ϑsr ) can be obtained by the reverse expression of the
at low current levels and the constant voltage drop at higher
scalar product between I˙sa and U̇sra , as follows:
current levels. The curve of Fig. 7 can be profitably used for
Usrα Isα + Usrβ Isβ an estimation of Rs , provided that the derivative is computed
cos (ϑsr ) = (21)
|U̇sra ||I˙sa | on the higher-current part only. In this work, the last five u∗sa
Substituting expression (21) in (19) gives: values were used with the corresponding Isa,dc samples, and
2 2
then processed through a simple least-square method to obtain
Usrα + Usrβ the estimation of Rs . The final result was Rs = 0, 814 Ω,
Rsr = (22)
Usrα Isα + Usrβ Isβ confirmed by an off-line DC test with a high precision multi-
which is the resolutive equation for the estimation of Rsr . meter (Agilent 34401A). Of course, the resistance varies with
Different frequencies are used to map the Rsr value in the temperature and, in case of sensorless control applications,
the range of interest, which is usually from zero to the on-line tracking would be highly recommended.
IM nominal slip frequency (5% ÷ 10% of the nominal IM Fig. 7 also reports the ideal Rs Isa,dc curve, that is, the
speed). Frequencies higher than the nominal slip speed are not portion which is directly related to the motor and cables linear
meaningful because they do not fall under normal the workin resistive elements. The difference between the ideal and the
conditions of a vector-controlled IM drive. measured curve represents the non-linear elements, and it is
used to compensate the voltage reference values generated by
IV. E XPERIMENTAL RESULTS the control algorithm (via a look-up table or a polynomial
approximation), to obtain a close match between the reference
A series of experimental results were performed on an and the real phase voltage. In absence of voltage sensors,
IM whose rated values and nameplate data are reported in the outlined procedure has revealed good usability and it was
Appendix A. Experiments were carried out with a space-vector essential to perform the subsequent steps of the estimation
modulation of 10 kHz, IGBT dead times of 4 µs and a DC procedure.
bus of 100 V, although results are not critically affected by an
increased value of the DC-bus voltage. B. Transient inductance estimation
The second step of the procedure, as discussed in Section
A. Stator resistance estimation III-B, estimates the transient inductance Lt . A superimposed
As illustrated in Sect. III-A, a DC test is performed to AC voltage of 2 V at 300 Hz was used to investigate the
estimate the stator resistance. It is known that the dead- asymptotic effect of (11). Voltage and current fundamental
time negatively affects the phase voltage generation of the harmonics were acquired and processed to obtain the imagi-
PWM, introducing an ideally-constant voltage drop whose sign nary part of the impedance Z (jΩ). Results are shown in Fig.
depends on the sign of the phase current [21]. It is also well 8.
 7

0.022 experimental value represents the unavoidable offset which

0.02
must be subtracted from the integrator input when calculating
the stator flux linkage using (13). The current is then forced
0.018
again to zero by imposing a zero voltage, and calculating the
0.016 integral (13) over the whole transient. Different transients are
repeated for different DC current levels, in order to get the
Lt [H]

0.014
profile of the integral in the whole IM current range. The curve
0.012 was then compared with the reference one obtained with the
0.01 DC-power supply.
Two important aspects must be considered while performing
0.008
the automatic detection of the magnetising inductance as
0.006 function of the stator current. Such aspects have not been
2 4 6 8 10 12
isa [A] previously considered in [13] and are meant to improve the
overall accuracy of the procedure.
Fig. 8. Lt estimation by DC+AC method. The first is related to the presence of the voltage drop ut
caused by the IGBTs and the threshold voltage ud of the
free-wheeling diodes within the inverter. During a step-down
The Lt profile shows a certain degree of saturation, which
transient of the current isa from positive values to zero, the
could be explained considering the saturation of the stator
topology outlined in Fig. 5a becomes that depicted in Fig. 9.
slot corners in the motor for higher currents, affecting the
value of the total leakage flux linkage in the air gap. These
measurements confirm that Lt is a delicate parameter in an IM
model, since its value is very small and difficult to calculate.
On the other hand, its impact on sensorless motor models is
very low, as documented in [22]. Authors also found that
Lt values obtained with the proposed procedure are very
repeatable by changing the amplitude and/or the frequency
of the superimposed AC voltage.

C. Magnetising inductance estimation

The experimental measurement of the magnetising induc-
tance was found to be the most challenging part of the Fig. 9. Step-down current transient scheme.
procedure. First, a reference curve was obtained by off-
line measurements, in order to compare the results of the The phase-to-phase voltage usab is obtained by writing
procedure. The data were obtained by measuring the current of twice (one for each phase a, b) the equation (8), then by taking
the phase a and the differential voltage between the phases a the difference on both sides with the assumption (7). It is:
and b, at the inverter output. Different transients were imposed λt (isa ) λϕ (iϕa )
by forcing different DC currents in the motor with a DC-power usab = 2Rs isa + 2 +2 (23)
dt dt
supply. The step-down transient of both current and voltage
were recorded and post-processed numerically, calculating the Kirchoff’s second law, applied to the mesh of Fig. 9, returns:
stator resistance before the transient and the integral (13) until ud + usab + ut = 0 (24)
the current reached the zero state.
Then, the automatic detection of the magnetising inductance Substituting (23) in (24), integrating side by side and applying
was implemented by feeding the motor with the inverter exactly the same considerations carried out in Sect. III-C, one
directly. As mentioned in Sect. III-C, a drawback of the obtains the phase flux linkage λsa (0), computed at time t = 0
proposed approach relies in the digital implementation of a and current isa = Isa,dc :
pure integrator, which suffers of drift problems. Such issues Z t
ud + ut

have been addressed in some papers like [23] and [24], but lim + Rs isa dt = λsa (0) (25)
t→+∞ 0 2
unfortunately proposed solutions do not work in a standstill
condition, where a single motor phase is equivalent to a simple and then
RL circuit.
In order to reduce the drift effect, the procedure adopts La (Isa,dc ) = Lϕa (Isa,dc ) + Lta (Isa,dc ) =
the following approach. A first transient from zero to a DC Z t  (26)
1 ud + vt
current level is performed, by imposing a suitable phase- = lim + Rs isa dt
to-phase DC voltage (u∗ = Usa,dc in Fig. 5). When the Isa,dc t→∞ 0 2
DC condition is reached, the average back electromotive The value ud + ut can be either inferred from the data sheet
force Esa,dc = Usa,dc − Rs Isa,dc is calculated. In principle, or found by means of a dedicated off-line measurement, as
Esa,dc should be equal to zero but, in practice, the non-zero specified hereafter. By imposing a voltage step-down transient,
 8

which forces the current to move from the steady-state rated of only a constant term was enough to obtain very reliable
value to zero, and by measuring the usab voltage at the inverter results.
output by a differential probe, a map of ud + ut as function of According to (25), the experimental measurements returned
the current can be obtained. The result, for the present case, is the phase flux linkage λsa as function of the initial steady-state
shown in Fig. 10. The profile was approximated by piece-wise current value, λsa = λsa (Isa,dc ). The profile of the stator flux
continuous lines and implemented into equation (26), for each closely matches the reference one obtained by means of the
current sample. DC power supply. Only low-current results deviate from the
reference profile, because it is difficult to integrate the flux
4 linkage at low current levels. In any case, they can be easily
replaced by a linear approximation of the curve, since the flux
3.5
is within the linear region of the magnetic B − H curve. To
3 have an objective verification of the result, a finite element
2.5
analysis (FEA) based on the IM geometry and windings data
ut+ud [V]

has been carried out, too, as suggested in [25] and [26]. The
2 FEA simulation was performed by imposing growing levels of
1.5 current density to both phases a and b, while setting ic = 0,
to reproduce the conditions of the proposed procedure. The
1
measured and simulated flux linkages are reported in Fig.
0.5 12a, while Fig. 12b shows a flux map during computer FEA
simulations.
0
0 2 4 6 8 10 12 14
isa [A]
0.9

Fig. 10. Profile of the ut + ud curve as function of isa . 0.8

0.7
The second aspect is related to the accuracy of the current 0.6
sensor within the inverter. Even if the measurement offset can
λsa [Vs]

0.5
be removed by simple automatic procedures, low-accuracy
sensors could lead to potential wrong results in the flux 0.4
FEA simulation
linkage estimation. To this purpose, the measurements of the 0.3 Measured λsa
inverter built-in Hall current sensor were compared, before
0.2
the activation of the estimation procedure, to the current mea-
surements obtained with a high-accuracy ammeter (QinetiQ 0.1

PPA2530). Fig. 11 shows the percentage error of the Hall 0

0 5 10 15
sensor, normalised to the rated motor current IN (see Table isa [A]
I). (a)

0.8
Error [% of IN]

0.6

0.4

0.2
(b)
0 Fig. 12. (a) λsa , measurement and simulation by finite element analysis, (b)
0 2 4 6 8 10 12 14 IM flux map resulting from FEA simulations.
isa (Hall sensor) [A]

Fig. 11. Percentage error of the current sensor measurement. The error between the measured and FEA-simulated flux
linkage remains within 7% of the rated one, and the small
The discrepancy increases at higher current levels, po- mismatch can be largely attributed to the use of a 2-D (instead
tentially affecting the correct estimation of the magnetising of 3-D) finite element analysis tool. The result is therefore
inductance. As a rough-and-ready countermeasure, a constant quite accurate if one consider that, for example, the analysis
bias of +0,4% was added to the Hall sensor measurement for carried out in [27] about the influence of parameter variations
the calculations of the automatic procedures. The simple use in sensorless rotor flux oriented IM, shows that a incorrect
 9

setting of magnetising inductance by 20% produces a speed LUT for inverter non-linearities compensation could give some
detuning of only 0.2% and a torque error of 8%. benefits.
For the sake of completeness, the estimation procedure has Unlike [14], here the problem is overcome by using the
been applied to a second motor (fed by the same inverter), same DC-bias technique presented in Sect. III-B for the
whose nameplate data are reported in Table II. The accuracy estimation of Lt . In this case, a single bias is used to shift
of the results matches the expectations. In particular, it is to the reference voltage (and the current as well) so that the
stress out the importance of the correction suggested in Fig. superimposed sinusoidal voltage are sufficiently far from the
10, and the related discussion. The voltage drop across the near-zero region, where the voltage compensation shows its
switching devices cannot be neglected, because it turns out weakness. Fig. 15 reports the Rsr outline as function of
to play a crucial role in the integration (25), as shown in the frequency. Meaningful frequencies are from zero to the
Fig. 13, where the flux linkages (with DC power supply and nominal slip speed ΩslN , which from the data of Table I
PWM inverter, respectively, as voltage sources) are compared. corresponds to 2,5 Hz.
The inaccuracy suggests that some countermeasures have to be
taken anyway. Of course, the finest the estimation of voltage 1
drops in any working condition, the better the accuracy of flux
linkage estimation.
0.8
1.4
DC tests
Inverter tests
1.2
0.6

Rsr [Ω]
Stator flux linkage λsα [Vs]

0.8

0.6
0.4
0.4

0.2

0
0 1 2 3 4 5 6
0.2
Current isα [A]

Fig. 13. λsα estimation with DC pwer supply and PWM inverter as voltage 0
sources. 0 0.5 1 1.5 2 2.5
Slip frequency [Hz]

The availability of the flux linkage enables the computation Fig. 15. Rsr estimation.
of the magnetising inductance, which can be obtained by using
the equations (16) and (17) and the values of Lt (Fig. 8). The It is evident that the Rsr parameter suffers of the skin effect
behaviour of both the sum of the two inductances and the even in the range of interest. It is worth to note that skin effect
magnetising inductance alone are reported in Fig. 14. represents only a part of the Rsr increment during normal
operation, the other effect being the resistance increase as
0.14
Lφ function of the temperature. Therefore, the proposed procedure
0.12 Lφ + Lt is useful for an initial knowledge of the Rsr profile, but
it is advisable to exploit an on-line Rsr tracking procedure
0.1 to preserve the validity of the model during normal drive
Lφ, Lφ + Lt [H]

operation.
0.08

0.06
V. C ONCLUSIONS
0.04
In this paper, an automatic procedure for IM parameters
0.02 estimation was presented. The procedure is performed at
complete standstill, which is particularly appreciated when
0
0 2 4 6
isa [A]
8 10 12 14 the motor is already connected to the load. The estimation
fully identifies the parameters of the inverse-Γ model, and
Fig. 14. Computation of Lϕ from λsa and Lt . it includes the non-linearities of both the motor and the
voltage inverter. The estimated parameters are ready-to-use for
conventional vector-controlled drives, thus representing a step
towards the complete self-commissioning. The procedure has
D. Rotor resistance referred to the stator estimation been implemented in laboratory and tested on IM prototypes.
The last part of the procedure estimates the Rsr parameter. The results were validated by comparison with an accurate
Since the maximum frequency of interest for the injected sinu- finite element analysis. Further research activity will include
soidal signals spans over few Hertz, it would be very easy to extensive tests on motors of different sizes and power ratings.
force hazardous values of current even with very low voltages. The focus will be also on the on-line tracking of the temper-
This poses a potential problem, since the estimation of low- ature and frequency-dependant parameters, as stator and rotor
amplitude sinusoidal voltages is rather difficult, although the resistances, and of iron losses influence as well.
 10

ACKNOWLEDGEMENTS [22] S. Bolognani, L. Peretti, and M. Zigliotto, “Parameter sensitivity analysis

of an improved open-loop speed estimate for induction motor drives,”
Authors would like to thank SAEL s.r.l., Torri di Quartesolo IEEE Trans. Power Electron., vol. 23(4), pp. 2127–2135, 2008.
(VI), Italy, for its financial support. Special thanks to prof. [23] J. Hu and B. Wu, “New integration algorithms for estimating motor flux
S. Bolognani, N. Bianchi and dr. A. Rais for their wise and over a wide speed range,” IEEE Trans. Power Electron., vol. 13(5), pp.
969–977, 1998.
unconditioned support to our research activity and to Dr. L. [24] R. Bojoi, P. Guglielmi, and G. Pellegrino, “Sensorless direct field
Alberti, for his great help in FEA analysis. oriented control of three-phase induction motor drives for low cost ap-
plications,” Conference Record of the 41st Industry Application Society
R EFERENCES (IAS) Annual Meeting, vol. 2, pp. 866–872, 2006.
[25] L.Alberti, N.Bianchi, and S. Bolognani, “A very rapid prediction of im
[1] K. B. Nordin, D. W. Novotny, and D. S. Zinger, “The influence of motor performance combining analytical and finite-element analysis,” IEEE
parameter deviations in feedforward field orientation drive systems,” Trans. Ind. Appl., vol. 44(5), pp. 1505–1512, 2008.
IEEE Trans. Ind. Appl., vol. 21(4), pp. 1009–1015, 1985. [26] L.Alberti, N.Bianch, and S. Bolognani, “Variable-speed induction ma-
[2] H. Schierling, “Self-commissioning - a novel feature of modern inverter- chine performance computed using finite-element,” IEEE Trans. Ind.
fed induction motor drives,” Proceedings of the IEEE Conference on Appl., vol. 47(2), pp. 789–797, 2011.
Power Electronics and Variable Speed Drives, pp. 287–290, 1988. [27] E.Levi and M.Wang, “Impact of parameter variations on speed es-
[3] A. M. Khambadkone and J. Holtz, “Vector-controlled induction motor timation in sensorless rotor flux oriented induction machines,” IEE
drive with a self-commissioning scheme,” IEEE Trans. Ind. Electron., Conference on Power Electronics and Variable Speed Drives, vol. 456,
vol. 38(5), pp. 322–327, 1991. pp. 305–310, 1998.
[4] J.-K. Seok, S.-I. Moon, and S.-K. Sul, “Induction machine parameter
identification using pwm inverter at standstill,” IEEE Trans. Ind. Elec-
tron., vol. 12(2), pp. 127–132, 1997. A PPENDIX A
[5] A. Wolfram, “Induction motor parameter estimation at standstill by
means of advanced signal processing methods,” Proceedings of the In- IM NAMEPLATE DATA
ternational Exhibition and Conference for Power Electronics Intelligent
Motion Power Quality (PCIM), pp. 105–110, 2006.
[6] Y.-S. Kwon, J.-H. Lee, S.-H. Moon, B.-K. Kwon, C.-H. Choi, and J.-K. Table I
Seok, “Standstill parameter identification of vector-controlled induction IM MOTOR PARAMETERS
motors using the frequency characteristics of rotor bars,” IEEE Trans.
Ind. Appl., vol. 45(5), pp. 1610–1618, 2009.
Nominal power P 3,7 kW
[7] K. Akatsu and A. Kawamura, “Sensorless very low-speed and zero-
speed estimations with online rotor resistance estimation of induction
motor without signal injection,” IEEE Trans. Ind. Appl., vol. 36(3), pp. Nominal speed ΩmN 1500 rpm
764–771, 2000.
[8] J. Maes and J. A. Melkebeek, “Speed-sensorless direct torque control Nominal current IN 11,8 Aef f
of induction motors using an adaptive flux observer,” IEEE Trans. Ind.
Appl., vol. 36(3), pp. 778–785, 2000. Nominal voltage VN 280 Vef f
[9] J. Holtz and J. Quan, “Sensorless vector control of induction motors
at very low speed using a nonlinear inverter model and parameter Nominal slip speed ΩslN 150 rpm
identification,” IEEE Trans. Ind. Appl., vol. 38(4), pp. 1087–1095, 2002.
[10] M. Bertoluzzo, G. S. Buja, and R. Menis, “Self-commissioning of rfo cos(ϕ) 0,77
im drives: one-test identification of the magnetization characteristic of
the motor,” IEEE Trans. Ind. Appl., vol. 37(6), pp. 1801–1806, 2001.
[11] A. B. Proca and A. Keyhani, “Identification of variable frequency
induction motor models from operating data,” IEEE Trans. Energy
Convers., vol. 17(1), pp. 24–31, 2002. Table II
[12] R. J. Kerman, J. D. Thunes, T. M. Rowan, and D. W. Schlegel, S ECOND IM MOTOR PARAMETERS
“A frequency-based determination of transient inductance and rotor
resistance for field commissioning purposes,” IEEE Trans. Ind. Appl., Nominal power P 2,2 kW
vol. 32(3), pp. 577–584, 1996.
[13] M. Ruff, A. Bünte, and H. Grotstollen, “A new self-commissioning
Nominal speed ΩmN 1500 rpm
scheme for an asynchronous motor drive system,” Conference Record
of the IEEE Industry Applications Society (IAS) Annual Meeting, vol. 1,
pp. 616–623, 1994. Nominal current IN 5,1 Aef f
[14] J. Godbersen, “A stand-still method for estimating the rotor resistance of
induction motors,” Conference Record of the IEEE Industry Applications Nominal slip speed ΩslN 110 rpm
Society (IAS) Annual Meeting, pp. 900–905, 1999.
[15] A.Lamine and E.Levi, “Dynamic induction machine modelling consid-
ering the stray load losses,” Universities Power Engineering Conference,
vol. 2, pp. 582–586, 2004.
[16] P. Vas, Vector Control of AC Machines. New York, USA: Oxford A PPENDIX B
University Press, 1990. G OERTZEL ALGORITHM DETAILS
[17] G. R. Slemon, Electric Machines and Drives. New York, USA: Addison
Wesley, Inc., 1992. The derivation of the Goertzel algorithm for single-tone
[18] T. J. White and J. C. Hinton, “Compensation for the skin effect in
vector-controlled induction motor drive systems,” Seventh International
detection starts from the normalised definition of the discrete
Conference on Electrical Machines and Drives, pp. 301–305, 1995. Fourier transform (DFT):
[19] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing,
N −1
Third Edition. Upper Saddle River, NJ, USA: Prentice-Hall, Inc., 2009. X
[20] M. Ruff and H. Grotstollen, “Identification of the saturated mutual X(k) = x(r)WNkr 0≤k ≤N −1 (27)
inductance of an asynchronous motor at standstill by recursive least r=0
squares algorithm,” Conference Record of the 5th European Conference
on Power Electronics and Applications, vol. 5, pp. 103–108, 1993. where N is the number of samples, WN = e−j2π/N , x(r) and
[21] A. R. Munoz and T. A. Lipo, “On-line dead-time compensation tech-
nique for open-loop PWM-VSI drives,” IEEE Trans. Power Electron., X(k) are the sampled signal (normalised sample time T = 1)
vol. 14(4), pp. 683–689, 1999. and its frequency-domain transformation (sample frequency
 11

F = 1/N ), respectively. The Goertzel algorithm is obtained multiplication and two additions in each sample period. In this
by manipulating (27), considering the following identity: way, the algorithm can be effectively used in real time without
affecting the execution time of control routines. Moreover, the
WN−kN = e(j2π/N )N k = ej2πk = 1 (28)
value of N can be increased as needed with no effect on the
Using (28) into the first of (27), it follows: computational time requirements, obtaining a very high DFT
N −1 N −1
selectivity.
−k(N −r)
X X
X(k) = WN−kN x(r)WNkr = x(r)WN (29)
r=0 r=0

Based on (29), the following sequence can be introduced:

N −1
−k(n−r)
X
y(n) = x(r)WN (30)
r=0

The sequence (30) is equal to X(k) when n = N . It represents

a convolution between x(n) and WN−kn or, equivalently, the
output of a linear system whose impulse response is equal to
WN−kn . The Z-transform of WN−kn is:
1
1 − WNk z −1 (31)


H(z) =
1 − 2 cos(2πk/N )z −1 + z −2 | {z }
FIR filter
| {z }
IIR filter
Expression (31) represents the transfer function between the
sequence y(n) and the input x(n), split into an IIR filter and
a FIR filter. As regards the former, let s(n) be the output of
the IIR filter and S(z) its Z-domain value. It follows:
X(z) = 1 − 2 cos (2πk/N ) z −1 + z −2 S(z)


(32)
Back into the time domain, it is:
s(n) = x(n) + 2 cos (2πk/N ) s(n − 1) − s(n − 2) (33)
With the input x(n) being sampled during real-time control,
the sequence s(n) is computed online. As regards the FIR
filter, its output y(n) is computed only for n = N , provided
that s(N ) and s(N − 1) have been previously saved:
X(k) = y(n) n=N
= s(N ) − WNk s(N − 1) (34)
Since WNk is a complex number, the result of (34) is complex,
too. The real and imaginary parts are:

< [X(k)] = s(N ) − cos (2πk/N ) s(N − 1)

(35)
= [X(k)] = sin (2πk/N ) s(N − 1)

Since x(n) is a real signal, (33) shows that the computa-

tion of s(n) requires two additions and one multiplication,
since 2 cos(2πk/N ) is stored as a coefficient. Thus, s(N )
is obtained with N multiplications and 2N additions. The
calculation of < [X(k)] and = [X(k)] requires two multipli-
cations and one addition. Altogether, the Goertzel algorithm
then requires N + 2 multiplications and 2N + 1 additions.
This is a net saving, if compared to the DFT in (27) which
needs 2N multiplications and 2N additions to compute a real
and imaginary parts of X(k) from a real signal x(k) [19].
However, the real advantage of the Goertzel algorithm for
single harmonic analysis resides in its recursive definition (33).
The N + 2 multiplications and 2N + 1 additions are spread
over the whole time window, and s(n) is updated with one