This article has been accepted for publication in a future issue of this journal, but has not been

fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2016.2544242, IEEE
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A DSP-Based Resolver-To-Digital Converter for

High Performance Electrical Drive Applications
Massimo Caruso, Antonino O. Di Tommaso, Fabio Genduso, Rosario Miceli, Member, IEEE, and Giuseppe Ricco
Galluzzo

Abstract—This paper presents a low cost, simple and highly I. I NTRODUCTION

accurate resolver-to-digital converter (RDC) for electrical drive
applications based on an integrated software approach, thus ANY SYNCHRONOUS electric motors require very
allowing a significant reduction of hardware components count
with significant improvements in terms of reliability, reduction
M accurate position sensing for sinusoidal control. The
purpose of such a control is to enable an efficient and smooth
of fault rate and susceptibility to electromagnetic interference.
Particular attention has been addressed to cost which is 25% operation enhancing comfort while limiting vibrations. In
off over conventional RDC. Simulations and experimental tests some cases related to mechanical constraints, we have to
confirm the high quality of the proposed system. deal with through-shaft design [1]–[3]. One can quote, for
Index Terms—Electrical Drives, Speed, Position, Resolver, example, power drives for electric or hybrid electric vehicles
Tracking Loop. as well as for electric power steering motor. More generally,
these sensors need to keep a simple and robust design and
a restricted number of parts as they are subject to high
N OMENCLATURE
vibration levels, a wide temperature range and speeds of
vsin ,vcos resolver stator output voltages several thousands rpm.
k resolver stator-to-rotor transformation ratio Resolvers have been highly investigated [4], [5], especially
ve excitation voltage signal amplitude from the signal conditioning point of view [6]–[9]. In [10] a
ωr angular frequency of the excitation voltage low-cost method to convert the amplitudes of the sine/cosine
t time variable signals into an angle without using look-up tables has been
t0 initial time instant proposed. This method, by using the alternating pseudo-linear
ts time instant corresponding to the signal sampling segments of the signals, improves precision. In [11] Authors
θ resolver shaft absolute angular position improve software demodulation by taking samples at positive
ϕ estimated resolver angular position peak values of excitation signal. The position is determined
T sampling period through the inverse tangent function in a similar way as in
KP proportional gain of the PI controller [12]. The paper [13] focuses on a technique to get the angular
KI integral gain of the PI controller position with the help of a neural network.
FP I transfer function (TF) of the PI controller
The problems of self calibration and high accuracy in
Fol open-loop transfer function (TF)
resolver to digital converters (RDC) are the object of the paper
Fe input/output position error transfer function (TF)
[14]. It considers a combination of classical resolver and a
Θ input position signal in the z-domain
second order decoupled double synchronous reference frame-
Φ output position signal in the z-domain
based phase-locked loop (DSRF-PLL). Despite the apparent
Φx maximum value of the output position signal
initial mathematical difficulty, using a standard digital sig-
ζ damping factor of the system
nal processor (DSP) with a 12-bit analog-to-digital converter
ωn characteristic frequency of the system
(ADC), up to 14 bits resolutions can be achieved with a
ω0 PMSM rated speed
small computation cost (about 13% of the 100 overall Million
ω rotor speed of the resolver
Instructions Per Second) [15]. In [16] a modified angle-
a acceleration constant
tracking observer suitable for DSP is proposed. The angle
v speed constant
estimation algorithm is based on the sign and absolute values
fref reference frequency of the input position signal
of sine and cosine of the rotor position.
θref sinusoidal time-varying reference position signal
In [17] the Authors concentrate their investigations on the
θx amplitude of the sinusoidal reference position
classical demodulation problem focusing on the integration
signal
technique starting from the zero crossing detection of the
Manuscript received May 21, 2015; revised September 14, 2015 and signals. The result is the obtainment of actual envelopes of
December 23, 2015; accepted February 10, 2016. the signal waveforms by utilizing the phase relationship of
M. Caruso, A. O. Di Tommaso, F. Genduso R. Miceli and
G. Ricco Galluzzo, are with the DEIM (Dept. of Energy, the integrated waveforms with the delayed carrier so that noise
Information Engineering and Mathematical Model, Polytechnic and disturbances are reduced without applying filters.
School, University of Palermo. Viale delle Scienze, Block 9, zip The accuracy of the shaft angular speed and position mea-
code 90128, Palermo, Italy. (email: massimo.caruso14@unipa.it,
antoninooscar.ditommaso@unipa.it, fabio.genduso@unipa.it, surement is fundamentally based on the quality of resolvers
rosario.miceli@unipa.it, giuseppe.riccogalluzzo@unipa.it). and on high resolution RDCs. Unfortunately the latter are

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expensive and many efforts were made in past works in order demonstrate the effectiveness of the proposed solution. Finally
to simplify the RDC topologies and to reduce their costs [12], Section VI summarizes conclusions.
[18].
In this paper the Authors focus mainly on high accuracy II. P ROBLEM S TATEMENT
rotor position estimation of electrical drives at the lowest A typical resolver consists of a rotating excitation winding
possible cost, trying to reduce overburdens as much as possible supplied by AC signal and two static windings whose magnetic
without loss of accuracy and dynamic performance. In this axes are 90° displaced each other. A coupling transformer is
context an alternative solution for a RDC, based on a software used to supply (from an external power source) the resolver
approach, is proposed in this paper. rotor winding with a sinusoidal excitation current. Its output
The software approach is useful for a significant reduction consists of two sine wave voltage signals, whose amplitudes
of hardware components with respect to a commercial so- are modulated according to the sine and cosine of the shaft
lution. A smaller number of components, in fact, implies a absolute position. The two signals are expressed in the time
greater economy and a significant increase of the reliability domain [21]–[23] as follows:
due to the reduction of the fault rates of the system: since the
drive system became fully operational (now 5 years in witch it  h
 vsin = kve sin θ cos (ωr t) + 1 dθ cos θ sin (ωr t)
i
ran at most for 6 hours a day) there has never been a fault in its h ωr dt i (1)
speed/position estimation part. It reduces also the susceptibility  vcos = kve cos θ cos (ωr t) + 1 dθ sin θ sin (ωr t)
ωr dt
to electromagnetic interference (EMI) and restrain calibration
errors due to temperature drift or components aging. Moreover, If ωr is high enough with respect to the maximum rotor
the RDC tracking loop contains only a PI controller, reducing angular speed dθ/dt (in our case for ωr = 2πf = 2π5000 =
significantly the computational level (freeing DSP resources) 31415 rad/s and dθ
dt max = 400 rad/s), the second term of sum
and the set-up time by using a simple set of equations. in expressions (1) can be neglected, leading to the following
A cost analysis has been made concluding that the cost of approximated equation:
the presented solution is less than 25% of the lowest cost of 
vsin = kve [sin θ cos (ωr t)]
a RDC chip today available on the market. Although in the (2)
vcos = kve [cos θ cos (ωr t)]
scientific literature many different low cost classified RDC
are presented [12], [16], [19], [20], nobody has detailed these The traditional solution, consisting in a RDC chip, results
costs. expensive being its cost not less than 68 USD (according to
The software realization of the RDC functions is carried out Farnell, http://uk.farnell.com), which is almost equal to half
with much ease, allowing, moreover, its integration in the main the cost of a resolver (e.g. Resolver Tamagawa, 133 USD).
control algorithm of the electric drive and its implementation Furthermore, the dynamic and static performances of such
on the same DSP as proposed in [9]. The low weight and the RDC Integrated Circuits (ICs) are fixed or can be set up by
portability of the implemented code allow the use of low cost using additional passive components, such as resistors and
DSP contributing, thereby, to the cost reduction of the whole capacitors. As a consequence, the reproducibility of outputs
electrical drive. and performance of the position tracking loop depend on the
parametric variability of these components (due, for example,
Here a dSPACE® board has been used to implement both
to changes in temperature, aging, etc.).
the RDC and the vector control of the PMSM (Permanent
Magnet Synchronous Motor), in order to speed up realization
III. D ESCRIPTION OF THE S OFTWARE BASED R ESOLVER
and testing; the necessary hardware is limited to a few low-
TO D IGITAL C ONVERTER
cost components that can be easily located on the power
converter control board. Obviously the dSPACE® board has In this section, the hardware, the software RDC system
been used only as a rapid prototyping system. At the end of the and its operation are presented and extensively described.
experimental campaign, the software previously implemented In addition, the mathematical model of the RDC and the
on dSPACE® is then rewritten in order to match a commercial equations used to set it up are hereafter reported and discussed.
DSP.
Main features of the software based RDC here presented A. Analog Signal Conditioning Network
are summarized as follows: high accuracy, simple set up, high The electronic analog interface scheme and the block dia-
reliability and stability, good performance, low software and gram of the proposed software based RDC are shown in Fig.
hardware complexity. 1 and Fig. 2, respectively. The 5 kHz sinusoidal excitation
The experimental results and the performance comparison (carrier) signal for the resolver is generated by the dSPACE®
between the software based RDC and that of an encoder have board through a digital-to-analog converter (DAC) operating
shown that the digital speed and position measurements can at 40 kHz. The carrier signal is then sent to an electronic
reach high accuracy at lower costs. conditioning network which is composed by a 2nd order
This paper is structured as follows: Section II summarizes analog Band-Pass Filter (BPF). The BPF is centered at 5 kHz
the problem statement. Section III describes the RDC software with a bandwidth of 2.5 kHz being able to filter the higher
application with details about the control loop design. Sec- harmonics of the signal generated by the dSPACE® board. In
tion V presents both simulations and experimental results to addition, a power amplifier is used to amplify the sinusoidal

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Fig. 1. The resolver analog interface.

cos 10

ω 8
vsin LPF
ZOH BPF DS 8 X 6
verr φ
+
-
PI ∫ z -6 4

vcos 2
vsin, vcos (V)

ZOH BPF DS 8 X
0

−2
sin
−4

Fig. 2. Block scheme of the software based RDC. ZOH=Zero-Order Hold, −6

BPF=Band-Pass Filter, DS=Down Sample, LPF=Low-Pass Filter. −8

−10

0.05 0.1 0.15 0.2 0.25 0.3 0.35

signal and to supply the rotor excitation winding. The BPF time (s)

has been introduced in order to cancel possible DC offsets on

Fig. 3. The demodulated sine and cosine signals during an acceleration from
the resolver excitation signal, before the power amplifier (see 0 to 4000 rpm.
Fig. 1).
The sine and cosine output signals are delivered to a
differential amplifier and passed through an anti-aliasing RC signal for the resolver, by using the same interrupt of the
filter. Finally, the signals are fed to the two 12 bit ADC of the dSPACE® board (see Fig. 2). Each of the digitized sine and
dSPACE® board (see Appendix for characteristics). cosine modulated signals are then filtered through a digital
The overall cost of the analog conditioning network is about 16th order FIR (Finite Impulse Response) anti-aliasing band-
13 USD taking as reference for each component an average pass filter, reducing the signal bandwidth to 5000±1250 Hz.
rate extracted from main catalogues about electronic devices. The advantage of using FIR filters relies on the fact that they
The cost of a DSP available on market able to integrate the have a linear phase with respect to frequency, meaning that
functions here required (100 MHz clock frequency should be they introduce a constant group delay which can be easily
enough) ranges from a minimum of 1.95 USD to a maximum compensated in steady state as will be shown next in Section
of 4.70. Therefore, the whole industrial cost of the RDC can V. Afterwards, the downsampling technique is used reducing
be estimated at around 17 USD. the sampling rate to 5 kHz, which is achieved by selecting
only one sample from a group of consecutive 8 samples of
B. Closed Loop Position Tracking System the sine - cosine signals (decimation). This technique allows
The simplified block diagram of the closed loop position to demodulate the signals, enabling the sine and cosine for
tracking implemented on the dSPACE® control board, is the position and speed closed loop tracking system. With this
shown in Fig. 2. This algorithm is based on the oversampling decimation, the resolution has been improved theoretically
technique. As a matter of fact, the sine and cosine modulated by 1.5 bits [24], [25]. Figure 3 shows an example of the
resolver outputs are sampled at 8-times the excitation signal two digitally demodulated sine and cosine signals during an
frequency (i.e. 40 kHz). The ADC sampling is also synchro- acceleration test from 0 to 4000 rpm.
nized to the DAC which generates the sinusoidal excitation In particular, with the previously mentioned synchronization

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Angle
verified that the use of this filter has significantly reduced
θ φ
disturbances over the speed signal.
high speed
The time delay introduced by the combination of FIR BPF
high
error and non-linear closed loop (see Fig. 2) determines an error on
the estimated angle ϕ with respect to θ. The presence of this
error, which is proportional to the resolver angular speed, can
low speed be graphically explained with the help of Fig. 4: by considering
a constant time delay of ϕ with respect to θ, at low speed the
error on angle θ−ϕ is relatively low, while at high speed (were
low
both signals have a more pronounced slope) the angle error
error is higher. In conclusion, this error is directly proportional to
speed. A compensation of this error is possible in steady state
O Time Time Time operation (i.e at constant speed), by delaying the angle ϕ signal
delay delay
by a specified amount of samples. This fact will be extensively
discussed and experimentally demonstrated in section V. In
Fig. 4. Combined effect of the FIR filter delay and the angular speed on the
angle error. addition, by referring to fig.4, the error ∆θ is given by the
following equation:

Fol
θ + Δθ φ ∆θ = ω(t + ∆t) − ωt = ω∆t (6)
PI P In order to compensate this error, the system is forced to
-
have an error proportional to the time delay, which is then
cancelled by the PI controller. This fact implies a delay of the
related signal, which has determined the adoption of the z −6
Fig. 5. Simplified block diagram of the closed loop position tracking system.
delay (as experimental validation, see Figs 9 and 10).

and decimation, the output signals of the resolver are always C. S ET- UP OF THE P OSITION T RACKING L OOP PARAME -
sampled at the same instant within a period of the carrier TERS
signal, i.e. at: When the position error ∆θ = θ − ϕ is small enough, then
sin (∆θ) ∼
= ∆θ and the block diagram of the position tracking
2π
ts = t0 + n, (3) loop represented in Fig. 2 can be redrawn in a more simple
ωr manner, as shown in Fig. 5.
where n is an integer number and t0 is a time instant at By considering the forward Euler integration method with
which cos(ωr t0 ) ≈ 1. sampling time period T it is possible to carry out the transfer
By taking (3) into account, (2) can then be rewritten as: function (TF) in the discrete-time domain (z domain) that
 characterize the RDC conversion procedure. With reference
vsin = kve [sin θ cos (ωr ts )] = to Fig. 5 the transfer function referred to the process P is
vcos = kve [cos θ cos (ωr ts )] = given by, :
(4)
= kve [sin θ cos (ωr t0 )] = k 0 sin θ


= kve [cos θ cos (ωr t0 )] = k 0 cos θ T

P (z) = k 0 (7)
z−1
with kve cos (ωr t0 ) = k 0 = const. By considering a proportional gain KP and an integral gain
The demodulated sine and cosine signals v sin and v cos are KI , the TF of the PI controller is expressed by:
multiplied by cosϕ and sinϕ, respectively, and then subtracted
each other. The obtained output signal is proportional to the KP (z − 1) + KI T
FP I (z) = (8)
sine of the error between the resolver shaft angle θ and the z−1
estimated angle ϕ, as shown by the following equation: The open loop TF of the proposed closed loop control
system is expressed in the z-domain as:
verr = k 0 (sin θ cos ϕ − cos θ sin ϕ) = k 0 sin (θ − ϕ) (5) KP (z − 1) + KI T
Fol (z) = k 0 T · (9)
(z − 1)2
The error signal verr is then processed by the PI controller,
in order to minimize it, obtaining the speed signal. The final and the input/output error TF is given by:
operation, which closes the control loop, consists into the
integration of the PI output signal, in order to make the Θ−Φ (z − 1)2
estimated angle ϕ available at the output of the RDC. In Fe (z) = =
Θ (z − 1)2 + k 0 T KP (z − 1) + k 0 KI T 2
addition, an IIR 1st order Butterworth low-pass filter, shown in (10)
Fig. 2, is used to improve the speed signal trend coming from If the input signal is a parabolic ramp, i.e. during acceler-
the PI output. As a matter of fact, it has been experimentally ation, the steady-state error is:

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TABLE I
M AIN DATA OF THE RESOLVER ARTUS S26SM19RX452CO1F.
aT 2
e = limz→1 Fe (z) · (z−1)2
=
2 (11) Input Voltage/Frequency 10V @ 5kHz
limz→1 (z−1)2 +k0 T KaT 0 2
P (z−1)+k T KI
= a
k0 KI Transformation Ratio 0.5 ± 10%
Max. Electrical Error ± 10’
If KI is high enough with respect to a (acceleration con- Max. Null Voltage 20mVrms
stant), then the related error can be minimized. Therefore, for Operating Temperature – 55 to +155 °C
a fixed error the maximum acceleration can be calculated by Max. Operating Speed 10 000 rpm
Mass 0.34 kg
(11). For example, during a constant acceleration if a specified
position error e∗ has to be kept within 1 degree, the maximum 0.54

acceleration will be: 0.52

Tranformation ratio
0.5

0.48
π
amax = e∗ k 0 KI = · 1 · k 0 · KI = 261 [rad/s2 ] (12) 0.46
180 0.44

If the input signal is a linear ramp, i.e. when the speed ω 0.42 0
10
1
10
assumes a constant value, the position error is: 10
5
0

Phase (deg)
ωT
e = limz→1 F e (z) · z−1 = −5
ωT (z−1) (13) −10
limz→1 (z−1)2 +k0 T KP (z−1)+k0 T 2 KI
=0 −15
−20
Thus, this error is theoretically null. However, the experi- −25 0 1
10 10
mental tests of section V will demonstrate that this error is Frequency (kHz)

not null.
Fig. 6. Frequency response of the ARTUS resolver.
The closed-loop RDC TF related to the simplified block of
Fig. 5 is given by:
The DPS 30 is connected to a dSPACE® rapid prototyping
Φ T KP (z − 1) + KI T 2 control board to drive the IGBT bridge. The RDC algorithm is
Fcl (z) = = (14) implemented on the same control board and its PI parameters
Θ (z − 1)2 + k 0 KP T (z − 1) + k 0 KI T 2
have been chosen to reach a 250 Hz closed loop bandwidth
By assuming two complex conjugate dominant poles, the (see Appendix). Figure 8 shows a simplified block diagram of
following PI parameters approximated set-up results: the proposed RDC.
2ζωn
KP = (15) V. S IMULATIONS , E XPERIMENTAL T ESTS AND R ESULTS
k0
In this section several simulation and experimental tests for
ωn2 the validation of the proposed system are summarized and
KI = (16) discussed. These simulation tests were performed in order
k0
Therefore, by selecting the most satisfactory values for both to practically determine the frequency response of the RDC,
the natural frequency and the damping factor, the RDC can be whose block diagram is represented in Fig. 2, and to compare
set in a very easy and quick way. it with the response obtained from the simplified mathematical
model represented in Fig. 5.
For this purpose a sinusoidal time-varying reference position
IV. R ESOLVER AND RDC T EST B ENCH
signal was fixed, according to the following equation:
An experimental test bench for a PMSM drive has been
set up. The PMSM under test is driven by a three-phase θref (t) = θx sin(2πfref t) (17)
voltage source inverter (VSI) (Automotion Inc., type DPS
30-A) and it is connected to a hysteresis brake (Magtrol
Inc., type DSP6001), in order to perform experimental tests
at different speeds and load conditions. In order to mea-
sure the motor speed, a two-pole ARTUS resolver (type
26SM19RX452CO1F/00) and an optical incremental encoder
(LTN Servotechnik, 1024 ppr, type G36 W) are connected to
the PMSM shaft, as shown in Fig. 7. The main characteristics
and parameters of the ARTUS resolver are listed in Table I,
while the measured frequency response [8], [26] is shown in
Fig. 6. The resolver supply frequency has been chosen equal to
5 kHz because in this case (as confirmed by the datasheet) the
transformation ratio is about 0.5 and the phase shift between Fig. 7. Optical encoder (E) and Artus resolver (R) used during the experi-
input and output voltages is near to zero. mental tests.

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carrier signal excitation signal 12

ω=ω
0
ω=0.5ω0
10 ω=0.1ω0
A Synchr. ω=0.01ω0
D
BPF (fc=5 kHz) 8
D sin sin

average error (deg)

A
5 kHz carrier RDC cos cos
Resolver 6
signal D
generator A
4

DSP Analog interface

(filters and amplifiers) 2
 
0
Fig. 8. The RDC simplified block diagram.
−2
0 1 2 3 4 5 6 7
delay (samples)
where θx has been set to 10°, while fref has been varied
from 0.01 Hz to 1000 Hz. The simulated output signals of the Fig. 9. Average position error as function of the number of delayed samples.
resolver became, therefore, equal to:

vsin = k 0 [sin θref cos (ωr ts )]

12

(18)
vcos = k 0 [cos θref cos (ωr ts )] 10
delay 0
delay 1
and were fed to the respective inputs of the RDC (vsin and delay 2
8
vcos ) shown in Fig. 2. delay 3

average error (deg)

delay 4
In addition, the proportional and integral PI gains were set 6 delay 5
delay 6
to 10 and 1500, respectively. The output position signal will delay 7
be approximatively given by: 4

ϕout ∼
= ϕx sin(2πfref t + ρ) (19) 2

where ρ is the phase-shift between the output and the input 0

signals. The ϕx/θx ratios and ρ have been evaluated for several
−2
frequency values and have been used in order to draw the 0 0.1 0.2 0.3 0.4 0.5
speed (p.u.)
0.6 0.7 0.8 0.9 1

frequency response of the non linear RDC closed-loop system.

However, due to the time delay produced by the FIR BPFs, Fig. 10. Average position error as function of the PMSM speed.
a relevant steady-state position error has occurred during the
experimental tests, produced by the closed-loop RDC (see Fig.
4). This error is strictly dependent on ω and on the number The introduction of the six-step delay has not affected the
of delayed samples. In order to determine the relationship set-up equations described in section III-C. As a matter of fact,
between the position error speed of the input signal and the by comparing the above determined frequency response with
delay, several tests have been carried out by varying the output the frequency response computed from the RDC linearized
delay from z 0 to z −7 at different percentages of the PMSM mathematical model represented in Fig. 5 (see Fig. 11), it can
rated speed. With evidence of Fig. 9, which shows the average be noticed that the two frequency responses have an almost
position error/delayed sample trend at 1%, 10%, 50% and similar shape. From the plotted characteristics it is evident that
100% of ω0 , by increasing the number of delayed samples the the two trends are very close to each other, except for the gap
position error decreases almost with a linear trend. However, detected in the frequency range from 20 Hz to 200 Hz, with a
for a z −7 integer delay the error turns into negative values. maximum phase difference gap of about 90° at 120 Hz, which
These data can be also analyzed from the graph of Fig. 10, is assumed to be not relevant for the RDC tracking efficiency.
which shows the error as function of the PMSM speed and The reliability of the proposed model is confirmed also
parametrized for the number of delayed samples. Thus, from for high values of the reference position signal amplitude. In
the reported results, a six-step (z −6 ) delay block has been fact, by setting θx = 90° in (17) and by varying the input
introduced into the RDC system, as shown in Fig. 2 and, position reference frequency from 0.01 Hz to 1000 Hz, the
therefore, the position error has been minimized. computed input/output response and the phase response have
The error minimization is due to the fact that the six-step been plotted. From the obtained results, shown in Fig. 12,
delay, introduced in the control loop, produces in closed-loop it can be noticed that the resonant peak is shifted towards
configuration a group delay which sign is opposite to the one higher frequencies and its value has been decreased, This
produced by the FIR BPFs. In other words, the output ϕ is difference between the two frequency responses (experimen-
delayed, with respect to θ, approximately by the same time tally obtained) is due to the non-linear behavior of the RDC
delay produced by the FIR BPF. Thanks to the feedback of closed-loop system, which has provided similar trends, but
the RDC an error appears which is canceled by the action of non identical. Thus, for high variations of the input signal, an
PI, i.e. ϕ is forced to assume the same delay of θ. improvement of the signal fidelity is achieved.

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1.5 350
amplitude

1
300

0.5
250

position (deg)
−2 −1 0 1 2 3
10 10 10 10 10 10 200

0
150

−200
phase (deg)

100
−400

50
−600

−800 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
−2 −1 0 1 2 3 time (s)
10 10 10 10 10 10
frequency (Hz)
Fig. 14. Resolver (red) and encoder (blue) position signals during the
Fig. 11. Frequency response comparison between the RDC shown in Fig. 2 acceleration test from 0 to 4000 rpm.
(red) and the simplified one shown in Fig. 5 (blue) at θx = 10°.

1.2
1
7
amplitude

0.8
6
0.6

0.4 5

0.2 4
0 −2 −1 0 1 2 3
10 10 10 10 10 10 3
error (deg)

2
0
1
phase (deg)

−100
0
−200
−1

−300 −2

−400 −2 −1 0 1 2 3 −3
10 10 10 10 10 10
frequency (Hz) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
time (s)

Fig. 12. Frequency response of the RDC shown in Fig. 2 at θx = 90°.

Fig. 15. Error between resolver and encoder position signals during the
acceleration test from 0 to 4000 rpm.

Root Locus Editor (C)

1 1.25e3
1.5e3 1e3

1.75e3 750
4500

2e3 500 4000

0.5
3500
2.25e3 250
3000
800
Imag Axis

2500
speed (rpm)

2.5e3
0 2.5e3 600
2000
speed (rpm)

0.9
400
2.25e3
0.8
250 1500
0.7
0.6 1000 200
0.5
−0.5 0.4
2e3 500 500 0
0.3
0.2 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
1.75e3 0.1 750 time (s)

1.5e3 1e3 −500

1.25e3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
−1 time (s)

−1 −0.5 0 0.5 1
Real Axis Fig. 16. Angular speed signals detected from the resolver (red) and encoder
(blue) during the acceleration test from 0 to 4000 rpm.
Fig. 13. Root locus in the z-domain of the proposed model.

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2016.2544242, IEEE
Transactions on Industrial Electronics
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

120 20

18
100

16
80
14
speed error (rpm)

60
12

speed (rpm)
40 10

8
20

6
0
4

−20
2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0

time (s) 0 0.5 1 1.5 2 2.5 3
time (s)

Fig. 17. Error (blue) and filtered error (red) between RDC and encoder speed Fig. 18. Comparison between the angular speed detected from the RDC (red)
signals during the acceleration test from 0 to 4000 rpm. and the optical encoder (blue) with ωref =10 rpm.

0.2

In addition, the root locus in the discrete-time domain

0.15
corresponding to the simplified model above reported is shown
in Fig. 13, which clearly demonstrates that there is a good 0.1

margin for the RDC open-loop gain increase, if a higher 0.05

tracking speed is needed; furthermore, the RDC is stable
because all poles are located within the unit circle shown in speed (rad/s) 0

Fig. 13. Therefore, the proposed simplified model has been −0.05
successfully validated as for (15) and (16), even with the
introduction of the six-step delay. −0.1

Moreover, a dynamic performance comparison between the −0.15

RDC and the commercial optical encoder described in sec. IV

−0.2
has been developed. In particular, Fig. 14 shows the trends 0 0.1 0.2 0.3 0.4 0.5
time (s)
0.6 0.7 0.8 0.9 1

of the position signals acquired with the resolver (and the

proposed software based resolver to digital converter) and with Fig. 19. RDC speed output signals at 0 rpm.
the 1024 point per revolution (ppr) optical encoder, whereas
Fig. 15 shows the error between the two position signals (blue
color) and the filtered one (red color) as function of time. specifically for relatively low speeds. For this purpose, the
These measurements were carried out during an acceleration motor has been driven with a reference speed of 10 rpm
test of the motor from 0 to 4000 rpm. By referring to Fig. and the speed output signals coming from both resolver and
15, it can be noticed that during the constant acceleration encoder have been plotted in Fig. 18. This figure demonstrates
phase the position error is confined between 6°, whereas in that the performance of the RDC is the same both at low and
almost steady-state condition it tends to zero (after 0.4 s). at high speed, while the encoder is affected by disturbances
In addition, the angular speed measured from both encoder due mainly to quantization, which leads to a change on its
and resolver has been acquired and plotted together in Fig. resolution.
16. From the details reported in the same figure, it can be
noticed that the two trends are similar, except for the fact
0.05
that the encoder output signal is affected by disturbances
more than the RDC output. These disturbances are due to 0.04

torsional vibrations produced by the hysteresis brake coupled 0.03

to the PMSM. Finally, the speed error between the RDC and
position error (deg)

0.02
encoder signals during the previously described acceleration
test is plotted in Fig. 17. It can be noticed that, after a 0.01

transient beginning at 0.05 s and lasting about 0.04 s, the

0
error (represented in blue color) returns within the +20/-20
rpm error band. Therefore, the settling time can be estimated −0.01

in about 0.04 s. The same figure shows also the averaged speed −0.02
error (red trend), demonstrating experimentally that the steady
−0.03
state error (at constant 4000 rpm) is almost equal to zero after 0 0.005 0.01 0.015 0.02 0.025 0.03
time (s)
0.035 0.04 0.045 0.05

0.6 s.
The accuracy of the proposed RDC can be appreciated more Fig. 20. RDC output position error at 0 rpm.

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2016.2544242, IEEE
Transactions on Industrial Electronics
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

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0278-0046 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2016.2544242, IEEE
Transactions on Industrial Electronics
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

Massimo Caruso received his M.S. Degree with Giuseppe Ricco Galluzzo received the B.S. de-
Cum Laude honors in Electrical Engineering gree in electrical engineering from the Univer- sity
fromăthe University of Palermo, Italy, in July 2008. of Palermo, Palermo, Italy. During 1992–2001, he
His thesis was focused on electric micromotors and was an Associate Professor of electrical drives with
theirămedical applications. the University of Palermo, where, since September
In 2009 he won the national competition for 2001, he has been a Full Professor of electrical
admission to the PhDăprogram in Electrical Engi- drives.
neering at the University of Palermo. His initial His research involves the fields of mathematical
research involved the design and simulation of elec- models of electrical machines and drive-system con-
trostatic micromotors and the control of single-phase trol and diagnostics.
inductionămotors.
During the 2011 he also joined the MSAL (MEMS Sensors and Actuators
Laboratory) group at the University of Maryland, College Park, focusing his
research on the design, microfabrication and testing of a SAW sensor and its
wireless power delivery system for biomass early detection and treatment.
He received his PhD in Electrical Engineering in March 2012.
His main research interests are electrical machines, drives, microsystems
and investigation of the magnetic field effects on biological essences.

Antonino Oscar Di Tommaso was born in Tübin-

gen (Germany) on June 5, 1972. He received the de-
gree in electrical engineering in 1999 and the Ph.D.
degree in 2004 from the University of Palermo,
Italy.
He was a post Ph.D. fellow in electrical machines
and drives at the Department of Electrical Engineer-
ing - University of Palermo, from 2004 to 2006.
Currently, he is a researcher and assistant professor
in Electrical Machines at the DEIM (Department of
Energy, Information engineering and Mathematical
models) - University of Palermo.
His main research interests deal with electrical machines, drives, diagnostics
on power converters, diagnostics and design of electrical rotating machines.

Fabio Genduso received the Ms.S. and the Ph.D.

Degree in Electrical Engineering in 1999 and 2004
respectively from the University of Palermo. In 2005
he joined the DEIM (Department of Energy, Infor-
mation engineering and Mathematical models) of the
University of Palermo as a post Ph.D. Fellow and an
assistant professor. He taught Power Electronics and
Drives, Electrical Machines, Network theory, and
Control of Electrical Drives in many undergraduate
and post graduate courses.
Actually he is a DEIM researcher and the respon-
sible of the research unit on electrical drives and Power Converters within the
Sustainable Development and Energy Saving Laboratory (SDESLAB) of the
University of Palermo.
His main research interests cover, power electronics, electrical drives and
control, Power system control, micro-grids, solar and wind energy.
Dr. Genduso is a reviewer of the IEEE TRANSACTIONS ON INDUS-
TRIAL ELECTRONICS and of other international technical journals.

Rosario Miceli (M’02) received the B.S. degree in

electrical engineering and the Ph.D. degree from the
University of Palermo, Palermo, Italy.
He is currently an Associate Professor of electrical
machines with the Polytechnic School, University of
Palermo. He is Personnel in Charge of the Sustain-
able Development and Energy Savings Laboratory
of the Palermo Athenaeum.
His main research interests include mathematical
models of electrical machines, drive system control,
and diagnostics, renewable energies, and energy
management.
Dr. Miceli is a reviewer for the IEEE TRANSACTIONS ON INDUSTRIAL
ELECTRONICS and the IEEE TRANSACTIONS ON INDUSTRY APPLI-
CATIONS.

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