Rotor and Flux Position Estimation in Delta-Connected ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 2, MARCH/APRIL 2006

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Rotor and Flux Position Estimation in Delta-Connected AC Machines Using the Zero-Sequence Carrier-Signal Current Fernando Briz, Member, IEEE, Michael W. Degner, Senior Member, IEEE, Pablo García Fernández, Student Member, IEEE, and Alberto B. Diez, Member, IEEE

Abstract—This paper analyzes carrier-signal voltage injection zero-sequence current-based sensorless control techniques for delta-connected three-phase ac machines. The analysis will focus on rotor position estimation (tracking of rotor-position-dependent saliencies), but the method applies equally well to flux position estimation (tracking of flux-dependent saliencies). The paper first develops a theoretical model and then provides analysis of relevant implementation aspects, such as selection of carrier-signal frequency and voltage magnitude, measurement of the zerosequence carrier-signal current, measurement and compensation of saturation-induced saliencies, and the signal processing needed for position/flux angle estimation. A similar implementation to that proposed in this paper, and with practically the same performance in terms of accuracy and estimation bandwidth, can be obtained for the case of wye-connected machines using the zero-sequence carrier-signal voltage, as shown in IEEE Trans. Ind. Appl., vol. 41, no. 6 pp. 1637–1646, Nov./Dec. 2005. Index Terms—Rotor position estimation, sensorless control, zero-sequence current.

I. I NTRODUCTION

S

ENSORLESS control techniques for ac machines that rely on the fundamental excitation have been shown to be capable of providing high-performance field-oriented control in the medium-speed to high-speed range [1], [2]. Such techniques, however, fail in the low-speed range or for dc excitation due to the lack of observability for rotor quantities. To overcome this limitation, sensorless control methods based on tracking the spatial position of saliencies in electric machines have been proposed [3]–[15]. Carrier-signal-based sensorless methods inject a highfrequency excitation signal (voltage [3]–[10] or current [11]) Paper IPCSD-05-105, presented at the 2005 Industry Applications Society Annual Meeting, Hong Kong, October 2–6, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Industrial Drives Committee of the IEEE Industry Applications Society. Manuscript submitted for review October 15, 2005 and released for publication December 19, 2005. This work was supported in part by the Research, Technological Development and Innovation Programs of the Principado of Asturias-ERDF under Grant PB02-055 and in part by the Spanish Ministry of Science and Education-ERDF under Grant MEC-04-DPI2004-00527. F. Briz, P. García Fernández, and A. B. Diez are with the Department of Electrical, Computer, and Systems Engineering, University of Oviedo, Gijón E-33204, Spain (e-mail: [email protected]; [email protected]; [email protected]). M. W. Degner is with the Electric Machine Drive Systems Department, Sustainable Mobility Technologies, Ford Motor Company, Dearborn, MI 48121-2053 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TIA.2006.870046

Fig. 1. Injection of the carrier-signal voltage.

that interacts with the machine saliencies to produce components in measurable electrical variables. When a carriersignal voltage is injected into a three-phase delta-connected ac machine (see Fig. 1), saliency-related information can be obtained from the measured line currents, phase currents, and zero-sequence current. While all of these signals are governed by the same physical principles and can potentially be used for saliency position estimation, choosing one option over the others depends on the application and involves tradeoffs between practical issues like the number of sensors, the amount of cabling needed, the effects caused by the nonideal behavior of the inverter, measurement resolution, estimation bandwidth, and signal-to-noise ratio. This paper analyzes the use of carrier-signal injection for rotor and flux position estimation in delta-connected threephase ac machines. The paper includes analysis of the physical principles of the method as well as mathematical modeling. Advantages and disadvantages of using the zero-sequence carriersignal current with respect to using line or phase currents are discussed. Implementation issues covered include the selection of the frequency and magnitude of the carrier-signal voltage, measurement of the zero-sequence carrier-signal current, and decoupling of saturation-induced saliencies.

II. S ALIENCY -T RACKING -B ASED S ENSORLESS C ONTROL IN D ELTA -C ONNECTED M ACHINES When a three-phase delta-connected ac machine is excited with a high-frequency signal, its response can be modeled using a simplified model based on stator transient inductance provided that the excitation frequency is substantially faster than the stator transient time constant [3], [7]. When the machine is salient, assuming that saliency varies sinusoidally, the stator transient inductance can be modeled as consisting of a constant

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of requiring at least one additional current sensor, with the corresponding additional cabling and access to the motor terminal box. Due to this, this option will not be discussed further. A general solution for the line (6) and the zero-sequence (7) carrier-signal currents can be obtained from (1)–(7). These solutions are a function of the saliency harmonic order h and have the form √ Vc is0sc = 3 3 ωc Fig. 2. Zero-sequence current measurement in a delta-connected machine.

term and a sinusoidally varying saliency-position-dependent term [7] diab vab = (ΣLσs + 2∆Lσs cos(hθe )) dt dibc 2π vbc = ΣLσs +2∆Lσs cos h θe − 3 dt dica 4π vca = ΣLσs +2∆Lσs cos h θe − 3 dt

(1) (2) (3)

where ΣLσs and ∆Lσs are the average and differential stator transient inductances, h is the harmonic order of the saliency relative to electrical angular units, and θe is the angular position of the saliency in electrical radians. While the model shown in (1)–(3) is valid for any form of high-frequency excitation, it can be solved for the particular case of injecting a rotating high-frequency carrier-signal voltage (4) into the machine. For that case, the carrier-signal current vector formed by the phase currents (5), the carriersignal current vector formed by the line currents (6), and the zero-sequence carrier-signal current (7) (see Fig. 2) all contain saliency-position-related information and can be used for sensorless purposes, i.e., s jωc t vqds _c = Vc e j2π j4π 2 isqds∆_c = iab + ibc e 3 + ica e 3 3 j2π j4π 2 ia + ib e 3 + ic e 3 isqds_c = 3

(4)

×

∆Lσs ΣLσs cos (ωc t ± hθe ) − ∆Lσs2 sin (ωc t ± 2hθe ) ΣLσs (ΣLσs2 − 3∆Lσs2 ) − 2∆Lσs3 sin (3hθe ) (8)

for the zero-sequence carrier-signal current. The third order term ∆Lσs3 in the denominator can be neglected when ∆Lσs ΣLσs , which is typically the case. After this simplification, and by considering specific values for h, simple and insightful solutions of the zero-sequence and line carrier-signal currents can be obtained as follows. For h = 1, 4, 7, . . . , isqds_c = isqds_cp + isqds_cn = − jIcp ejωc t − jIcnh ej(−hθe −ωc t) − jIcn2h ej(2hθe −ωc t) is0sc

= I0ch cos (ωc t − hθe )+ I0c2h cos (ωc t+ 2hθe )). (10)

For h = 2, 5, 8, . . . , isqds_c = isqds_cp + isqds_cn = − jIcp ejωc t − jIcnh ej(hθe −ωc t) − jIcn2h ej(−2hθe −ωc t)

(12) where Icp

with ia = iab − ica

Icnh

ib = ibc − iab ic = ica − ibc is0sc = iab + ibc + ica .

Icn2h (7)

The carrier-signal current vector formed by the phase currents (5) and by the line currents (6) can mathematically be shown to be the √ same, with the only difference being the addition of a 1/ 3 scaling factor for the phase current. Use of the carrier-signal current formed by the line currents (6) has the advantage that drives typically incorporate line current sensors and, therefore, no additional sensors or cabling is needed. Conversely, use of the phase currents (5) has the disadvantage

(11)

is0sc = I0ch cos (ωc t + hθe ) + I0c2h cos (ωc t − 2hθe ))

(5) (6)

(9)

I0ch

I0c2h

= (Vc /ωc )(ΣLσs /(ΣLσs2 − 3∆Lσs2 )), magnitude of the positive-sequence carrier-signal current; = 3(Vc /ωc )(∆Lσs /(ΣLσs2 − 3∆Lσs2 )), magnitude of the hθe component of the negative-sequence carrier-signal current; = Icnh (∆Lσs /ΣLσs ), magnitude of the −2hθe component of the negative-sequence carrier-signal current; √ = 3 3(Vc /ωc )(∆Lσs /(ΣLσs2 − 3∆Lσs2 )), magnitude of the hθe component of the zero-sequence carrier-signal current; = I0ch(∆Lσs /ΣLσs ), magnitude of the −2hθe component of the zero-sequence carrier signal current.

From (9) and (11), it can be observed that the carrier-signal line current consists of a positive-sequence component that does not contain saliency position information and two

BRIZ et al.: ROTOR AND FLUX POSITION ESTIMATION IN DELTA-CONNECTED AC MACHINES

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TABLE I INDUCTION MOTOR PARAMETERS

The fundamental of this permeance waveform rotates at the mechanical speed [5], [10] ωp =

Fig. 3. Frequency spectrum of the zero-sequence carrier-signal current and the negative-sequence carrier-signal current under different working conditions and carrier-signal frequencies. A carrier-signal voltage Vc = 15 V (peak) was used, ωe = 4 Hz, ωr = 1 Hz. (a) ωc = 500 Hz, no fundamental current. (b) ωc = 500 Hz, rated-flux, rated load. (c) ωc = 3750 Hz rated-flux, rated load.

negative-sequence phase-modulated components of magnitude Icnh and Icn2h . From (10) and (12), the zero-sequence carrier-signal current is seen to consist of two phase-modulated components of magnitude I0ch and I0c2h . When ∆Lσs ΣLσs , which is typically the case, then Icn2h Icnh and I0c2h I0ch . Finally, it can be shown that for h = 3, 6, . . ., no zero-sequence carrier-signal current is produced and the carrier-signal current contains just a magnitude-modulated positive-sequence component. Rotor position estimation using carrier-signal excitation requires a rotor-position-dependent saliency that couples with the stator windings to produce a measurable signal. The saliency due to the combined effect of stator and rotor slotting was used in this paper [5], [8]. Saliencies caused by the combination of rotor and stator slotting produce a permeance waveform that has a fundamental spatial harmonic given by hsp = |R − S|

(13)

where hsp is the saliency harmonic order relative to 360 mechanical degrees, and S and R are equal to the number of stator and rotor slots, respectively. In order for it to couple with the stator windings and produce a zero-sequence carrier-signal current, the expression hsp = n × p,

n = 1, 2, 4, 5, 7, 8, 10, . . .

(14)

needs to be satisfied, assuming the machine has an integer number of slots per pole per phase, where p is the number of poles.

R ωrm . (R − S)

(15)

Fig. 3(a)–(c) shows the frequency spectrum of the negativesequence carrier-signal current (a complex vector signal) and of the zero-sequence carrier-signal current (a scalar signal) from the test machine. Its parameters are shown in Table I. With the machine operated at constant rotor speed, the rotor–stator slotting produces a modulation of both signals at a frequency equal to R × ωrm with respect to the carrier-signal frequency, with ωrm being the rotor speed in mechanical units. This phase modulation of both the negative-sequence carrier-signal current and the zero-sequence carrier-signal current by the rotor position is used for rotor position estimation. III. Z ERO -S EQUENCE C ARRIER -S IGNAL C URRENT V ERSUS N EGATIVE -S EQUENCE C ARRIER -S IGNAL C URRENT According to the previous analysis, the negative-sequence carrier-signal current and the zero-sequence carrier-signal current exhibit similar content relative to saliency position. Deciding which to use depends on the analysis of some key implementation issues. Using the zero-sequence carrier-signal current has the penalty of requiring one additional sensor, additional cabling, and access to the motor terminal box. While these make this option less appealing, it provides some interesting advantages that make it attractive for certain applications. The most relevant are analyzed in this section. A. Current Sensors Scaling and A/D Converter Resolution Fig. 4(a) shows the zero-sequence current, and Fig. 4(b) the corresponding frequency spectrum, for the case of the machine in Table I operated at rated flux, rated load when a carriersignal voltage is injected. It can be observed from the figure that the magnitude of the zero-sequence current is relatively small (in the range of 5%) compared to the magnitude of the phase current. The current sensor used to measure the zero-sequence current (see Fig. 2) can be scaled for this relatively small current. This improves the signal-to-noise ratio and allows for higher resolution in the A/D converter sampling the zero-sequence current, resulting in increased accuracy for extracting the carrier signal portion of that current. Opposite this, the line current sensors and their associated A/D converters need to be scaled to accommodate the fundamental current magnitude, which results in a reduced sensitivity for measuring the negative-sequence carrier-signal current.

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Fig. 5. Experimentally measured magnitude of the positive-sequence carriersignal current Icp relative to the rated current as a function of the carrier-signal frequency for the cases of a carrier-signal voltage magnitude equal to Vc = 30 V (peak) (∇), 15 V (), and 7.5 V (), respectively. The motor was operated at rated flux, rated load.

Fig. 4. (a) Zero-sequence current (zoomed to show the component caused by the carrier voltage injection) and (b) frequency spectrum (zoomed to better show the components around the carrier frequency). A carrier-signal voltage of ωc = 500 Hz and Vc = 20 V (peak) was used, and the motor was operated at rated flux, rated load (ωe = 4 Hz, ωr = 1 Hz).

B. Selection of the Carrier-Signal Frequency and Voltage Magnitude There are several issues when selecting the carrier-signal frequency that favors a higher frequency: 1) it reduces the resulting carrier-signal current; 2) it allows for an increased estimation bandwidth; and 3) it increases the spectral separation with respect to the fundamental excitation, making filtering easier [7]. Because of the inductive behavior of the machine at frequencies of interest, the magnitudes of the carrier-signal currents, including positive-sequence Icp , negative-sequence Icnih , and zero-sequence I0cih components, decrease proportionally with increasing carrier-signal frequency. Reduction of Icp is always desirable, as it does not contain useful information, but is responsible for a major portion of the noise, losses, and vibration caused by the injection of the carrier-signal voltage, since usually Icp Icnih . Fig. 5 shows the measured magnitude of Icp relative to rated current as a function of the carrier-signal frequency and voltage magnitude. Benefits of selecting a highfrequency small-magnitude carrier-signal voltage are evident from this figure. However, high-frequency small-magnitude carrier-signal voltages also result in reduced magnitudes for the negative-sequence and zero-sequence carrier-signal currents. This may have adverse effects, like a smaller signal-tonoise ratio in the measured signals, or increased sensitivity to the distortion of the injected carrier-signal voltage caused by the nonideal behavior of the inverter. The influence of the carrier-signal voltage magnitude and frequency on the negativesequence and zero-sequence carrier-signal currents is analyzed in the following two sections. Negative-Sequence Carrier-Signal Current: The components of the negative-sequence carrier-signal current for two different carrier-signal frequencies are shown in the frequency spectrums in Fig. 3(b) and (c) (left column). In addition to the rotor–stator slotting component at −ωc + 14ωr , which con-

tains rotor position information, the fundamental excitationfrequency-dependent components of relatively large magnitude are also present, the most relevant being at −ωc + 2ωe . Saturation of the magnetic paths and additional components of the injected carrier-signal voltage (4) due to the nonideal behavior of the inverter have been reported to cause these additional components in the negative-sequence carrier-signal current [4], [7]. Significant differences, both in the frequencies present and in their relative magnitude, are seen between Fig. 3(b) and (c). A deterioration in the quality of the negative-sequence carriersignal current for the case of ωc = 3750 Hz is apparent. The deterioration seen can be grouped into two categories: 1) the reduced magnitude of the negative-sequence carrier-signal current, which has an adverse impact on the signal-to-noise ratio, and 2) the additional components in the injected carrier-signal voltage caused by the inverter. A key parameter determining the accuracy and robustness of the sensorless method is the magnitude of the rotor positiondependent component (signal) relative to the other components in the negative-sequence carrier-signal current (noise). The total harmonic distortion (THD) caused by the undesired “noise” was found to be an insightful metric for quantifying this relationship. To obtain the THD, the frequency spectrum of the negative-sequence carrier-signal current, denoted as ιcn , is first calculated using ιcn = FFT isqds_cn

(16)

with Σιcn2 =

−ω c +bw

ιcn(i)2

(17)

i=−ωc −bw

being the power associated with the negative-sequence carriersignal current. It is noted that (17) includes a range of frequencies +/ − bw of the negative-sequence carrier-signal frequency. While this range of frequencies would ideally be +/− infinity, which would include all of the content in the negative-sequence carrier-signal current spectrum, it is limited in practice in order to maintain reasonable signal processing. The limit chosen corresponds to the bandwidth of the bandpass filter (BPF) used to separate the negative-sequence current from the overall stator current, as schematically represented in Fig. 6(a). Selection of


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Fig. 6. Schematic representation of the BPFs used to separate (a) negativesequence and (b) zero-sequence carrier-signal currents.

this limit was based on the dynamic content in the negativesequence carrier-signal current and is ultimately determined by the desired position estimation bandwidth. The choice of +/ − bw in (17) implies that an ideal, i.e., square shaped magnitude, BPF is used, which is not possible in practice. Even so, the nonideal behavior of the filter did not have a significant influence on the results presented in this paper. Assuming steady-state operation, the THD of ιcn(−ωc +14ωr ) , the component of the negative-sequence carrier-signal current containing the rotor position information, can be defined as [16] s Σιcn2 − ιcn(−ωc +14ωr )2 . (18) THD iqds_cn = Σιcn2 Fig. 7(a) (white symbols) shows the THD of the negativesequence carrier-signal current as a function of the carriersignal frequency and voltage magnitude. A value of THD close to unity means that a majority of the power in the negativesequence carrier-signal current corresponds to nonuseful content (noise). This was an expected result due to the large magnitude of the fundamental excitation-dependent components present in Fig. 3(b) and (c). Since decoupling for the more significant of these components is usually needed in practice to obtain reliable position estimation, it is interesting to recalculate the THD with the fundamental excitation-dependent components removed. An example of this is shown in Fig. 7(a) (gray symbols), where the −ωc + 2ωe and −ωc − 4ωe components of the negative-sequence carrier-signal current have been removed and Σιcn2 =

−ω c +bw

ιcn(i)2 − ιcn(−ωc +2ωe )2 − ιcn(−ωc −4ωe )2

i=−ωc −bw

(19) instead of (17) was used for the calculation of the THD (18). It can be seen from Fig. 7(a) that compensation of fundamental excitation-dependent components reduces the THD. Larger carrier-signal voltage magnitudes are also observed to decrease the THD. According to Fig. 7(a), carrier signals in the range of 1000–1500 Hz and 30 V in magnitude would provide the lowest THD (for the voltage levels shown in the figure). However, from Fig. 5, it can be observed that such carrier-signal magnitudes produce positive-sequence carrier currents in the range of 5%–10% of rated current, causing noticeable levels of noise, vibration, and additional losses. Zero-Sequence Carrier-Signal Current: Fig. 3(b) and (c) (right column) shows the frequency spectrum of the zerosequence carrier-signal current for two different carriersignal frequencies. A significant difference with respect to the negative-sequence carrier-signal current is that the zero-

Fig. 7. Experimentally measured THD of (a) negative-sequence carrier-signal current and (b) zero-sequence carrier-signal current as a function of the carrier frequency for the case of a carrier-signal voltage magnitude Vc = 7.5 V (peak) (), 15 V (), and 30 V (∇), respectively. The motor was operated at rated flux, rated load. A bandwidth bw = 100 Hz was used.

sequence components, and their relative magnitude, are barely affected by the carrier-signal frequency. One explanation for this is that inverter distortion of the injected carrier-signal voltage does not have a significant influence on the zerosequence carrier-signal current. The zero-sequence carriersignal current was systematically studied for this paper (i.e., for multiple carrier-signal frequencies and working conditions) and was found to have significantly fewer fundamental frequencydependent components compared with the negative-sequence carrier-signal current. Those present were found to be easier to model and, therefore, to decouple. As was done for the negative-sequence carrier-signal current, THD was used to quantify the magnitude of the rotor position-dependent component (signal) of the zero-sequence carrier-signal current relative to the rest of components (noise) [see Fig. 6(b)]. Fig. 7(b) (white symbols) shows the THD calculated using ιc0 = FFT (is0sc ) Σιc02 =

ω c +bw

ιc0(i)2

(20) (21)

i=ωc −bw

THD (is0sc )

=

Σιc02 − ιc0(ωc +14ωr )2 . Σιc02

(22)

500


Fig. 8. Experimentally measured ratio of the ωc + 14ωr and ωc + 2ωe components of the zero-sequence carrier-signal current as a function of the carrier frequency. The motor was operated at rated flux, rated load (Vc = 15 V).

Although some differences are observed comparing this THD with the case of the negative-sequence carrier-signal current, more interesting are the results when the ωc + 2ωe and ωc − 4ωe components of the fundamental excitation frequency are removed. Fig. 7(b) (gray symbols) shows the THD (22) being calculated using Σιc02 =

ω c +bw

ιc0(i)2 − ιc0(ωc +2ωe )2 − ιc0(ωc −4ωe )2 (23)

i=ωc −bw

instead of (21). Comparing Fig. 7(a) and (b) (gray symbols), the benefits of using the zero-sequence carrier-signal current instead of the negative-sequence carrier-signal current become evident. First, the THD of the zero-sequence carrier-signal current is always significantly smaller than the THD of the negativesequence carrier-signal current. Second, increasing the carriersignal frequency is observed to have practically no influence on the THD of the zero-sequence carrier-signal current, even for small carrier-signal voltages. This allows the use of a higher frequency small-magnitude carrier-signal voltage, with the resulting losses, vibration, and noise caused by the carrier current being reduced. In addition, high carrier-signal frequencies allow for higher bandwidth position estimation. An additional advantage of using large carrier-signal frequencies can be observed from Fig. 8. The figure shows the magnitude of the rotor–stator slotting component normalized by the magnitude of the ωc + 2ωe component of the zerosequence carrier-signal current for several carrier-signal frequencies. From Fig. 8, it can be seen that this ratio increases with carrier-signal frequency. This suggests that increased carrier-signal frequencies will result in a reduction in the sensitivity to the presence of fundamental excitation-dependent components for detecting rotor position-dependent components. Unless otherwise stated, all the experimental results presented in this paper used a carrier-signal frequency of ωc = 3750 Hz and a carrier-signal voltage magnitude of Vc = 15 V (peak). From Fig. 5, it is observed that this results in a carriersignal current of about 1% of the rated current. The switching frequency was ωs = 15 kHz. IV. F ILTERING OF THE Z ERO -S EQUENCE C URRENT To estimate the rotor position from the zero-sequence carrier signal current, first the carrier signal portion of the zero-sequence current needs to be separated from the overall

Fig. 9. Block diagram and frequency response of the filtering implemented to isolate the zero-sequence carrier-signal current. The fourth-order low-pass Butterworth filter was designed for a cutoff frequency of 6 kHz, and the secondorder 15-kHz notch filter was designed for a bandwidth of 1 kHz. (a) Signal processing block diagram. (b) Bode diagram of the analog filtering (carrier frequency of 3750 Hz; the phase around the carrier frequency is zoomed).

zero-sequence current. The signal processing used is almost identical to that reported in [8] for the case of the zero-sequence carrier-signal voltage. It is accomplished using a combination of a band-stop (notch) filter to reject harmonics at the switching frequency and a low-pass filter to remove higher harmonics that could cause aliasing. The block diagram for this form of filtering is shown in Fig. 9(a) [8]. The low-pass filter cutoff frequency was chosen to be high enough that an almost linear phase and a reduced phase slew rate existed near the carrier frequency. This allows the phase shift caused by the filter to be easily compensated. The Bode diagram of the combined filters is shown in Fig. 9(b). Prior to extraction of the rotor position information from the zero-sequence carrier-signal current, the sampled zerosequence current is converted into a carrier signal synchronous complex current vector [7], [8] ic0qd = i0 e−jωc .

(24)

The resulting complex current vector (24) is then low-pass filtered [see Fig. 9(a)]. It is noted that the digital low-pass filter in Fig. 9(a) corresponds to the BPF shown in Fig. 6(b), and is a fourth-order Butterworth filter, tuned for a bandwidth bw = 100 Hz. Fig. 10(a) shows the zero-sequence carrier signal complex current vector obtained after filtering, and Fig. 10(b) show its frequency spectrum. In addition to the rotor position-dependent component 14ωr , saturation-induced saliencies produce components, the most relevant being at 2ωe and −4ωe . An intermodulation component at a frequency of −2ωe − 14ωr can


501

Fig. 10. (a) Experimentally measured real and imaginary parts of the zerosequence carrier-signal current vector. (b) Frequency spectrum. The motor was operated at rated flux, rated load (ωr = 1 Hz, ωe = 4 Hz).

also be observed. A model of the zero-sequence carrier signal complex vector can be written as ic0qd_c = I014ωr ej14θr t + I02ωe ej (2θe t+θ0_2ωe ) + I04ωe e−j (4θe t+θ0_4ωe ) + I0ωre e−j (2θe t+14θr t+θ0_ωre ) .

Fig. 11. (a) Magnitude and (b) phase, relative to the stator current vector, of the saturation and intermodulation components of the zero-sequence carriersignal current vector [see (25)]. The motor was operated at rated flux, the load varied with the slip. A carrier-signal voltage of ωc = 3750 Hz and Vc = 15 V (peak) were used.

(25)

V. M EASUREMENT AND D ECOUPLING OF S ATURATION -I NDUCED S ALIENCIES Accurate rotor position estimation requires that saturationinduced saliency components of the zero-sequence carriersignal current be compensated for or decoupled. One method of doing this is to first measure and store the saturation-induced saliency components as a function of the operating condition during an offline commissioning process. The stored values can then be used for online decoupling during the regular sensorless operation of the drive. Determining the number of components of the zero-sequence carrier-signal current vector to be decoupled involves a tradeoff between the desired accuracy of the estimated rotor position and computational requirements. Decoupling −2ωe is required for most machine designs since the magnitude of the −2ωe component is usually comparable to the magnitude of the rotor position related saliencies. Without decoupling, inadmissible estimation errors and even stability problems can exist. The remaining undesired components of the zero-sequence carriersignal current vector (components at −4ωe and −2ωe − 14ωr in Fig. 11) are typically relatively small in magnitude and do not compromise the stability of the method. However, their compensation can increase the accuracy of the estimated position. To analyze the behavior of saturation and intermodulation saliencies, the magnitudes and phase angles, relative to the stator current, of the 2ωe , −4ωe and −2ωe − 14ωr components in the zero-sequence carrier-signal current vector (25) were measured for different working conditions. Since saturationinduced components of the zero-sequence carrier-signal voltage vector do not rotate at the same frequency as the stator current, the phase angles θ0_2ωe , θ0_4ωe , and θ0_ωre [see (25)] correspond to the instant when the angle of the stator current vector was equal to zero. Fig. 11 shows the measured magnitudes and phases of the 2ωe , −4ωe , and −2ωe − 14ωr components of the zero-sequence carrier-signal current vector when the machine is operated with constant rotor flux. From Fig. 11, it can be observed that the magnitudes and phases of saturation-induced

Fig. 12. Decoupling of saturation-induced components of the zero-sequence carrier-signal current vector and rotor position and velocity estimation using a tracking observer.

components are smooth functions, allowing a reduced number of operating points to be stored while maintaining an accurate model of the behavior. These values can be stored in lookup tables during a commissioning process and later accessed, according to the q-axis (torque producing) current, during the sensorless operation of the drive, as shown in Fig. 12. This commissioning process would be done once during the initial commissioning of each inverter–motor configuration. Linear interpolation was used when the q-axis current did not exactly correspond to an entry in the table. It is noted that the phase angle of the intermodulation component θ0_ωre depends both on the fundamental current angle and on the rotor angle. Its decoupling, therefore, requires feedback of the estimated rotor position, as shown in Fig. 12. Fig. 13 shows the result of the saturation-induced harmonics decoupling process. Fig. 13(a) shows the q-axis and d-axis components of the stator current, and Fig. 13(b) shows the measured zero-sequence carrier signal complex vector after the filtering shown in Fig. 9(a). Fig. 13(c) shows the estimated 2ωe component of the zero-sequence carrier-signal current complex vector, obtained from the stored data shown in Fig. 11. Fig. 13(d) shows the resulting zero-sequence carrier signal complex vector once the 2ωe , −4ωe , and −2ωe − 14ωr components have been decoupled as shown in Fig. 12.

502


Fig. 13. Measurement and compensation of saturation-induced harmonics in the zero-sequence carrier-signal current. (a) isqs , isds , (b) ic0q_c , ic0d_c , (c) ic0q_2ωe , ic0d_2ωe , and (d) ic0q_14ωr , ic0d_14ωr .

VI. S ENSORLESS C ONTROL Sensorless field orientation and position control were implemented using the proposed method. The motor was operated at rated flux and saturation-induced saliencies were decoupled using the scheme shown in Fig. 12. Rotor angle and velocity were estimated using a tracking observer [3]. Fig. 14(a)–(c) shows sensorless position control when a step is commanded. Stable operation and good dynamic response can be observed. The influence that the number of components decoupled from the zero-sequence carrier current vector has on the estimated rotor position error can be observed comparing Fig. 14(c) and (d). In Fig. 14(c), only the component at 2ωe was decoupled, while in Fig. 14(d), the −4ωe and −2ωe − 14ωr components were also decoupled, a slight decrease of the estimation error is seen with respect to Fig. 14(c). The standard deviation of the error was 0.66◦ for the case shown in Fig. 14(c) and 0.47◦ for the case shown in Fig. 14(d). Fig. 15 shows the transient response during velocity control when a step is commanded, with the machine being operated at constant load. Stable and accurate control is observed from the figure.

Fig. 14. Sensorless position control when a position step from 0◦ to 180◦ is commanded. (a) Estimated rotor flux angle. (b) Estimated rotor position. Estimation error (c) when only the 2ωe component, and (d) when the 2ωe , −4ωe , and −2ωe − 14ωr components, of the zero-sequence carrier-signal current vector are decoupled. The machine was operated at rated flux and 80% rated load.

Fig. 15. Sensorless velocity control. The machine was operated at rated flux r , (b) ωr − ω r . and 80% rated load. (a) ω

injected carrier-signal voltage due to the nonideal behavior of the inverter, and 2) it allows better scaling of the current sensor and A/D converter. As a result of this, a low-magnitude highfrequency carrier-signal voltage can be used. This results in accurate high-bandwidth robust position estimation and reduces the adverse effects (vibration, noise, and losses) caused by the carrier-signal current.

VII. C ONCLUSION The use of the zero-sequence carrier-signal current has been shown to be a viable option for rotor position estimation in delta-connected three-phase ac machines. While sharing some properties with negative-sequence carrier-signal current based techniques, its major drawback is the need for an additional sensor, additional cabling, and access to the machine terminal box. In spite of these inconveniences, it provides two major advantages: 1) it is barely affected by the distortion of the

ACKNOWLEDGMENT The authors wish to acknowledge the support and motivation provided by the University of Oviedo (Spain) and Ford Motor Company. R EFERENCES [1] H. Kubota and K. Matsuse, “The improvement of performance at low speed by offset compensation of stator voltage in sensorless vector


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Fernando Briz (A’96–M’99) received the M.S. and Ph.D. degrees in electrical engineering and control from the University of Oviedo, Gijón, Spain, in 1990 and 1996, respectively. From June 1996 to March 1997, he was a Visiting Researcher at the University of Wisconsin, Madison. He is currently an Associate Professor in the Department of Electrical, Computer and Systems Engineering, University of Oviedo. His topics of interest include control systems, high-performance ac drives control, sensorless control, diagnostics, and digital signal processing. Dr. Briz received the IEEE Industry Applications Society Conference Prize Paper Award in 1997 and 2004, and the 2005 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS Third Place Prize Paper Award.

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Michael W. Degner (S’95–A’98–M’99–SM’05) received the B.S., M.S., and Ph.D. degrees in mechanical engineering from the University of Wisconsin, Madison, in 1991, 1993, and 1998, respectively, with focus on electric machines, power electronics, and control systems. His Ph.D. dissertation was on the estimation of rotor position and flux angle in electric machine drives. In 1998, he was with Ford Research Laboratory, Dearborn, MI, working on the application of electric machines and power electronics in the automotive industry. He is currently the Manager of the Electric Machine Drive Systems Department, Sustainable Mobility Technologies and Hybrid Programs Group, Ford Motor Company, where he is responsible for the development of all electric machines and their control systems for hybrid and fuel cell vehicle applications. His interests include control systems, machine drives, electric machines, power electronics, and mechatronics. Dr. Degner was the recipient of several IEEE Industry Applications Society Conference Paper Awards and received the 2005 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS Third Place Prize Paper Award.

Pablo García Fernández (S’02) was born in Spain in 1975. He received the M.E. degree in industrial engineering in 2001 from the University of Oviedo, Gijón, Spain, where he is currently working toward the Ph.D. degree in electrical engineering. Since 2001, he had been awarded with a fellowship by the Personnel Research Training Program funded by the Spanish Ministry of Science and Technology. In 2004, he was a Visitor Scholar at the Wisconsin Electric Machines and Power Electronics Consortium, University of Wisconsin, Madison. His research interests include sensorless control and diagnosis of induction motors, neural networks, and digital signal processing.

Alberto B. Diez (M’99) received the M.S. and Ph.D. degrees in electrical engineering from the University of Oviedo, Gijón, Spain, in 1983 and 1988, respectively. He was a member of the Executive Committee D2 “Rolling-Flat Products” of the European Commission for six years (1998–2004). He is currently an Associate Professor in the Electrical Engineering Department, University of Oviedo. His topics of interest include control systems, high-performance ac drives control, and industrial supervision and control processes.