A
Novel Resolver Converter Based on a Modified Tracking Method Mohieddine Benammar & Antonio S.P. Gonzales Jr. Department of Electrical Engineering College of Engineering, Qatar University Doha, Qatar
[email protected];
[email protected]
Abstract-Resolvers
are
robust
and
reliable
In the first mode of operation described above (Fig. 1), the excitation signal applied to the rotor winding of the resolver is of the form:
transducers,
working as rotary transformers, and are used for mechanical angle measurement. In the present work, the two stator windings of the resolver are excited with sine and cosine signals; the rotor winding produces a sinusoidal signal that presents a phase shift
VexR(t)
equal to the mechanical angle of the rotor. A resolver converter is
=
A sin(OJt).
(1)
developed to extract this mechanical angle in a novel way. The
where A and OJ are the maximum amplitude and angular frequency, typically a few Volts and 8000n rad/s. As a result and provided that the angular speed of the rotor is much smaller than OJ, the two stator windings located at right angle produce signals, typically of the fonn:
proposed method is based on a phase-locked loop technique that produces a measure of the angle that tracks the true mechanical angle
of
the
rotor.
The
paper
presents
full
details
of
the
converter, Matlab simulation and experimental results.
Keywords-resolver converter; mechanical angle measurement
I.
PLL;
tracking
method;
{
INTRODUCTION
Resolvers are robust angle transducers suited to application in harsh environments, and have been used for decades in positioning applications such as tanks, aircrafts, satellite antennas, radars, brake-by-wire, etc. The resolver (Fig. 1) is basically a rotary transfonner [1], usually with an annature on the rotating shaft and two stator windings which are placed at right angles to one another; the coefficient of coupling between their rotor and stator windings depends on the mechanical angle of the shaft. The resolver may be operated in one of two modes: (i) either the rotor winding is used as primary and is supplied with a sinusoidal excitation signal and the induced stator signals are used to extract the mechanical angle, or (ii) the two stator windings are used as primaries and are supplied with sine and cosine signals which results in a rotor winding signal indicative of the mechanical angle. In both modes of operation, the resolver requires a converter to extract the mechanical angle from the resolver output(s).
IEEE
=
,B x sin (e) x VexR(t)
vcos(t,e)
=
,B x cos (e) x VexR(t)
(2)
where () is the unknown shaft angle, and fJ is a constant representing the transformation ratio between stator and rotor windings. The resolver converter (Fig. 1) is used to process the resolver signals and provides a measure tp of the rotor angle. Various resolver conversion schemes [2]-[20] have been reported for the fust mode of operation; some are implemented using analog electronics or mixed analog-digital techniques, while others are software-based. Commercially available closed-loop converters employ the classical phase-locked loop (PLL) technique [6]-[8]. These are closed-loop converters (Fig. 2) in which an estimated measure tp of the angle is made to track the changes of the shaft angle B. The implementation of these converters requires the computation of the sine and cosine of tp, usually using LUT. References [9] and [10] propose PLL methods that employ observer techniques in order to improve the transient response of the converter.
Fig. 2. Simplified diagram of the conventional PLL resolver converter.
Fig. 1. Resolver operated with its rotor winding as primary.
978-1-4673-5200-0/13/$31.00 ©2013
VSJn(t,e)
586
Simplification of the implementation of the classical PLL method has been reported in [11] where implementation was based on analog electronics only. Other converters are based upon the determination of the tangent/cotangent of the shaft angle where the signals in (2) are demodulated and their absolute values are determined; by appropriate processing the smaller of the two values is divided by the greater, providing either the tangent or cotangent of 8. The value of 8 is then either computed numerically or determined from a look-up table (LUT) [4], [5]. Other converters are based upon the linearization of the difference between the absolute values of the demodulated signals in (2) [12]-[14].
Fig. 3. Resolver operated in mode 2 and associated classical converter.
The time-dependent component of (6) is filtered out by a low pass filter which yields a signal proportional to sine8-'1'); this is the error signal in the closed loop arrangement. The multiplier and low pass filter constitute a phase-sensitive detector (PSD). The integrator produces the estimated measure IJI that tracks 8 such that the steady-state error converges to zero.
Schemes employing a technique that produces multiple phase-shifted sinewaves from the demodulated resolver signals have been reported; the angle is then estimated from the instantaneous amplitudes of the pseudo-linear segments of these multiple signals [15]. In the second mode of operation of the resolver the rotor winding is used as a secondary, both stator windings are used as primaries excited simultaneously with two quadrature waveforms:
{V V
E¥Csin(t) E¥Ccos(t)
=
=
A
x si
Ax
n (m!)
cos
(3)
(m!)
As a result, the resolver rotor winding generates a signal of the form: VR (
t B) 13-1 X [sine B) x VE¥Csin (t) + cos(B) x VEXCcos(t)] .(4) 13-1 A x cos(OJt - B) ,
=
=
Thus the stator and rotor windings exchange roles when compared to the previous mode of operation (Fig. 3). The classical converters based on this method aim at measuring the phase shift between the signal in (4) and the second signal in (3); this phase shift represents a measure of 8. The objective of this paper is to present a new PLL-type converter for the resolver operated in this second mode. II.
The implementation of the proposed converter of Fig. 4 using analog electronics is complicated by the presence of the sine function block that determines sin(wt-'1'). Additionally, analog multipliers are expensive and are associated with linearity and offset problems. Hence, the topology of the converter has been modified as shown in Fig. 5, for which the general concept explained above is still valid. The sine function block of Fig. 4 is replaced by a window comparator that detects the window at which sin(wt-'1') is positive. The multiplier of the PSD is replaced by an amplifier with a gain of +1/-1 controlled by the output of the window comparator; the general concept explained above is still valid. The controlled amplifier produces a signal of the form:
t B) 13-1 A x cos(at - B) x sgn [sin(at - T)].
VA (
,
=
A simple analysis shows that for the diagram of Fig. 5, and provided that the time constant t of the low pass filter is chosen to give the average (DC component) of (7), the error signal may be expressed as (2A/lhr)xsin(8 - 'P).
PROPOSED METHOD c: 0 0-
Fig. 4 depicts the general concept of the proposed converter. The basic idea of the converter is to create a closed loop system in which an estimated measure ('1') is made to track the value of the input angle 8. The converter uses three synchronized signals: the excitation signals (3) and an additional sawtooth reference signal of the form:
:::: I.! u c: �
" c: "
Error signal sin(B-'f?
11.,-,
Fig. 4. Basic concept of the proposed Resolver converter topology.
(5)
O�t�2mOJ.
In this concept, a multiplier is used to determine a voltage VAAt, 8): VM (t, 8)
=
=
13-1 A x cos(a>t - B) x sin(a>t - IJI) 0.513-1 A x [sin(2a>t - 8
-
IJI) + sine 8 IJI)]
(7)
(6)
-
Fig. 5. Modified topology of the proposed Resolver converter.
587
III.
The window comparator used to detect the intervals at which sin(ffit-'P) is positive is made up of comparators CMPl CMP5 and the NOR gate. The monostable used to reset the integrator, when 'P crosses ±360°, produces reset pulses of duration of l.lms; a reasonable duration when the resolver rotates at low speeds which is often the case as these transducers are used mostly in positioning applications.
SIMULATION
The simulation of the proposed converter perfonnance has been carried out using Matlab/Simulink software. The following practical values have been used for the converter parameters: AI{J=6.5V; angular frequency of the excitation signals (3) and sawtooth signal (5) was w= 8000n rad/s; the reference sawtooth signal (5) has an amplitude varying from 0 to 5V representing an angle varying from 0 to 2n; the time constant of the low-pass filter was T=0.0015s. The choice of this time constant was made on the following basis: the frequency of the signal at the input of the filter is double that of the excitation signals (i.e. 8kHz) and the choice of T results in an adequate cutoff frequency of the filter of 106Hz that guarantees removal of the AC component of (7). Fig. 6 shows a sample of the results of the transient perfonnance using various loop gains (2AK/{J7T:), K is the gain shown in Fig. 5. As expected, with the presence of the first-order low pass filter followed by the integrator, the converter behaves as a second order system. For a reasonable response with low overshoot, the response time is about 10ms. Fig. 7 shows a sample of the performance of the converter by simulating clockwise rotation of the resolver at 300 rpm; the loop gain was 350. It is clear that the output angle 'P closely tracks the input angle 8 even at this speed. IV.
Fig. 9 shows the excitation signals of the resolver, the sawtooth reference signal and the output of the rotor at an input angle 8=45°; these results have been generated using the real resolver. Fig. lO shows the experimental transient response of the converter with various loop gains and step input angle of 150°. These experimental results are in agreement with simulation at the same loop gain values (compare Fig. 6 to Fig. 10). Note that given the impossibility of generating a step input angle using a real resolver, the results of Fig. lO have been obtained by generating the converter input signal (rotor output) from a virtual resolver built under LabVIEW environment and using a 12-bit resolution data acquisition card.
e
3 kCl
s
U
Gi
::::..
33 nF
I
e 8 � �
EXPERIMENT
The proposed converter circuit has been made using low-cost analog components as shown in Fig. 8; some details such as hysteresis in comparators are not shown. The resolver used was model R71SSRE211AOI-023-07Rl from LTN Servotechnik GmbH. The various fixed parameters (except for K which is adjusted by varying the value of the capacitor C of the integrator) were adjusted as indicated in the simulation section.
�
{.·r
r
VDD VDD
14 kn
.�:.'
1.25V
6.Skn
2 kCl
220 nF
150
10 kf.! VDD
2 kCl
�� 100
lip Angle alp Angle alp Angle alp Angle alp Angle
" g
l , ,............i-.. 1 TC4066BP .�! l.....�I.� 1 kCl :
Voo= + 10V Vss=·10V
� �
10kCl
••••
200 . �
1kCl
50
'Mth W1th W1th 'Mth
Loop Loop Loop Loop
Gain Gain Gain Gain
c: 'i; Cl Q. 0
-1.25V
= 280 = 700 = 350 =170
10kCl 2 kCl
o.:--....-----;:OC--;----:o.:: o.:: o o2----co3---� :: :: 0."-c04---� 0.05
r·······j:·:······l
.2
i!'
Time(s)
10kn
co >
·2.5V
.s
"
Fig. 6. Simulation of the transient performance of the converter
"'�
�
·3.7SV
� :. .. ....
10 kn
.,D.. «:;; 00 I
360 W l270 • . i 180 � 90 .fO 0':-1 "'------"---c; --:OA:". 2 "------"o - --cO.'c-3 ----"----10.5 �0k .4 "-----'--.1----"-0"c0
:to:
v"
VDD
· 5V 10kCl
0:;; 0
O.1fJ.F
Time (s)
v�
Fig. 7. Simulation of the performance of the converter with the resolver rotating at 300 rpm.
Fig. 8. Basic analog implementation of the proposed converter.
588
III =
co .!!!
+
Vss
S
'5'
CMP5
�:5
..J�
10 kQ
2 kCl
en
Fig. 11 depicts the experimental results obtained by using the resolver driven clockwise at constant speeds of 30 rpm and 130 rpm. The results show that the output voltage of the converter varies linearly in the range 0 to 5V (corresponding to an angle from 0 to 360°) which is the same as the range of the sawtooth reference signal (Fig. 9). This analog output voltage provides a measure of the input angle over the full 360° range. Fig. 12 illustrates the performance of the converter by driving the resolver at a reverse speed of 70 rpm. The results show that the output voltage of the converter varies linearly in the range 0 to -5V; the negative sign is indicative of the anti-clockwise direction of rotation. These two last figures demonstrate that the converter produces an analog voltage proportional to the input angle. Additionally, the converter is capable of detecting the direction of rotation. As this paper reports work in progress, more experimental results and assessment of the converter precision will be reported in a future communication.
Fig. 9. Resolver Excitation signals, sawtooth reference signal, and rotor output at 8=45°.
Fig. II. Experimental results obtained with the resolver driven clockwise at various speeds (upper: 30 rpm; lower: 130 rpm).
Fig. 12. Experimental results obtained with the resolver driven anti-clockwise at 70 rpm. CONCLUSION
In this paper, a low-cost and simple-to-implement method for resolver to analog conversion was proposed. The scheme generates a voltage proportional to the mechanical input angle that represents the position of the rotor of the resolver. The converter is based on a closed loop arrangement in which an estimated measure of the input angle is made to track the actual angle, in a similar way to conventional PLL resolver converter
Fig. 10. Experimental transient response of the converter at various values of the loop gain.
589
[10] R. Hoseinnezhad, "Position sensing in by-wire brake callipers using resolvers," IEEE Trans. Veh. Technol., vol. 55, no. 3, pp. 924-932.
but with much simpler implementation as the present method does not require LUT, yeO, analog multiplier, counter and DAC. The proposed method has been successfully implemented and tested. The results obtained show satisfactory performance. The characterization of the precision of the converter will be reported in a future communication.
[II] Lazhar Ben-Brahim, M. Benammar, A. Mahran, A. A1homsi, "A new closed-loop converter for sinusoidal encoders - Preliminary results," Proc. 13th European Conference on Power Electronics and Applications, 2009, 8-10 Sept. 2009. [12] M. Benammar, L. Ben-Brahim, and M. A. Alhamadi, "A novel resolver to-3600 linearized converter," IEEE Sensors Journal., vol. 4, no. I, pp. 96-101, Feb. 2004.
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