Rotor Position Estimation of AC Machines Using the ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 6, NOVEMBER/DECEMBER 2005

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Rotor Position Estimation of AC Machines Using the Zero-Sequence Carrier-Signal Voltage Fernando Briz, Member, IEEE, Michael W. Degner, Senior Member, IEEE, Pablo García, Student Member, IEEE, and Juan Manuel Guerrero, Member, IEEE

Abstract—This paper analyzes the sensorless control of ac machines based on the zero-sequence voltage generated by the injection of a carrier-signal voltage. The analysis focuses on rotor position estimation (tracking of rotor-position-dependent saliencies), but the methods discussed apply equally to flux position estimation (tracking of flux-dependent saliencies). Analyses of relevant aspects like selection of the carrier-signal frequency, measurement of the zero-sequence carrier-signal voltage, and decoupling of saturation-induced saliencies are included. Index Terms—Rotor position estimation, saliency-based sensorless control, zero-sequence voltage.

I. I NTRODUCTION

N

UMEROUS sensorless methods, based on machine models and measured electrical variables (currents and voltages), for estimating the rotor velocity of ac machines have been proposed [1], [2]. While such methods provide adequate performance in the medium- and high-speed range, they suffer from increased parameter sensitivity in the low-speed range. In addition, for dc excitation, they are incapable of providing estimates due to the unobservable nature of rotor quantities for that type of excitation. To overcome these limitations, saliency (asymmetry) tracking-based methods have recently been developed [3]–[8]. Since the spatial saliencies tracked by these methods are intrinsic to the magnetic/mechanical design of the machine, their location (phase angle) does not change significantly with respect to their source (flux, rotor/stator structure) as a function of operating point. Estimating the position of these saliencies, and hence the rotor position or flux angle, is made possible at low and zero speeds by the addition of a high-frequency

Paper IPCSD-05-055, presented at the 2004 Industry Applications Society Annual Meeting, Seattle, WA, October 3–7, 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 February 14, 2005 and released for publication August 8, 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 by the Spanish Ministry of Science and Technology-ERDF under Grant DPI2001-3815. F. Briz, P. García, and J. M. Guerrero are with the Department of Electrical, Computer, and Systems Engineering, University of Oviedo, Gijón 33204, Spain (e-mail: [email protected]; [email protected]; guerrero@isa. uniovi.es). M. W. Degner is with the Electric Machine Drive Systems Department, Sustainable Mobility Technologies, Ford Motor Company, Dearborn, MI 481212053 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TIA.2005.857469

excitation superimposed on the fundamental excitation. The major differences between these saliency-based methods are the type of high-frequency excitation and the signal processing used for estimating the rotor position. This paper analyzes the use of zero-sequence carrier-signal voltage for rotor position estimation in induction machines when a balanced and symmetric high-frequency carrier-signal voltage (positive sequence) is applied to the stator windings. zero-sequence carrier-signal voltage techniques were initially proposed for flux angle estimation [9] and only recently have been proposed for rotor position estimation [10]. A difficulty in using the zero-sequence carrier-signal voltage is its measurement, which may make it less appealing than methods utilizing current sensors already present in the drive [3]–[6]. However, there are two major advantages to the use of zerosequence carrier-signal voltage. First, as observed in [10], the magnitude of the zero-sequence carrier-signal voltage does not depend on the frequency of the carrier-signal. This enables the use of higher carrier-signal frequencies, reducing the resulting carrier-signal current, the associated torque ripple, and acoustic noise. Second, harmonics in the carrier-signal voltage due to the nonideal behavior of the inverter [4], [11] are easier to decouple from the zero-sequence carrier-signal voltage, increasing the robustness and accuracy of position estimation. This paper includes analysis of the physical principles and mathematical modeling of the proposed method. It also discusses implementation issues, including the selection of carriersignal frequency, the effects caused by the nonideal behavior of the inverter, measurement methods for the zero-sequence carrier-signal voltage, and decoupling of saturation-induced saliencies.

II. S ALIENCY -T RACKING -B ASED S ENSORLESS T ECHNIQUES U SING THE Z ERO -S EQUENCE C ARRIER -S IGNAL V OLTAGE Carrier-signal based sensorless methods inject a highfrequency excitation signal (current or voltage) that interacts with the machine saliencies to produce components in measurable electrical variables. These components are then used to estimate the position of the saliency. When a machine is excited with a high-frequency signal, its response can be modeled using the stator transient inductance provided that the excitation frequency is substantially faster than the stator transient time constant [3], [7], [10]. When the machine is salient, assuming the saliency varies sinusoidally, the stator transient inductance

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Using (4) to calculate the resulting phase voltages va = Vc cos(ωc t) 2π vb = Vc cos ωc t − 3 4π vc = Vc cos ωc t − 3

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

can be modeled as consisting of a constant term and a varying saliency-position-dependent term (1)–(3) [10] dia van = (ΣLσs +2∆Lσs cos(h θe )) dt dib 2π vbn = ΣLσs +2∆Lσs cos h θe − 3 dt dic 4π vcn = ΣLσs +2∆Lσs cos h θe − 3 dt

(1) (2)

= Vc e

jθc

dθc . ωc = dt

1 (van + vbn + vcn ) 3

(4)

(5)

can be used to extract the saliency position information. Using (1)–(3) and provided that no zero-sequence currents exist (6), the zero-sequence voltage can be calculated as a function of the phase voltages va , vb , and vc , and phase inductances (7), i.e., dia dib dic + + =0 dt dt dt va Lb Lc + vb La Lc + vc La Lb s v0s =− . Lb Lc + La Lc + La Lb

(6) (7)

Note that the inductances La , Lb , and Lc in (7) stand for the inductance terms in (1)–(3), respectively.

s v0sc

cos

= Vc

(10)

substituting these as well as the inductance terms La , Lb , and Lc from (1)–(3) into (7) allows an analytical solution for the zero-sequence carrier-signal voltage to be obtained as that shown in (11) at the bottom of the page. While (11) is the general solution of the zero-sequence carrier-signal voltage, it does not provide much insight about its nature. Much simpler and more insightful solutions of (11) are obtained by considering specific values for h: 1) for h = 1, 4, 7, . . . s = V0ch cos(ωc t−hθe )−V0c2h cos(ωc t+2hθe ) (12) v0sc

The resulting zero-sequence voltage s v0s =

(9)

(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 (1)–(3) are valid for any form of high-frequency excitation, this paper focuses on the particular case of a balanced and symmetric high-frequency voltage excitation (Fig. 1) s vqds _c

(8)

2) for h = 2, 5, 8, . . . s = V0ch cos(ωc t+hθe )−V0c2h cos(ωc t−2hθe ) (13) v0sc

3) for h = 3, 6, 9, . . . s =0 v0sc

(14)

where V0ch = (Vc ΣLσs ∆Lσs )/(ΣL2σs − ∆L2σs ) is the magnitude of the hθe component of the zero-sequence carrier-signal voltage and V0c2h =(Vc ∆L2σs )/(ΣL2σs − ∆L2σs ) is the magnitude of the −2hθe component of the zero-sequence carrier-signal voltage. Some interesting conclusions can be obtained from the analysis of (12)–(14). 1) For values of h that are not integer multiples of three, the zero-sequence voltage is seen to consist of two phasemodulated components at hθe and −2hθe , respectively. 2) The magnitude of the second component V0c2h is negligible when compared to the magnitude of the first component V0ch when ∆Lσs ΣLσs , which is typically the case. 3) The magnitudes of the two components are independent of the carrier-signal frequency assuming that the magnitudes of the average and differential stator transient inductances are not a function of the carrier-signal frequency. 4) For h = 3, 6, . . . (triplen harmonics), no zero-sequence carrier-signal voltage is produced.

√ −1 cos(θc ) 2∆Lσs sin2 (hθe )+ΣLσs cos(hθe ) −(ΣLσs +2∆Lσs cos(hθe )) 3 sin(θc ) sin(hθe ) sin 23 hπ 2 3ΣL2σs +4ΣLσs ∆Lσs cos(hθe ) 2 cos 23 hπ +1 +4∆L2σs cos(hθe )2 2 cos 23 hπ +1 − sin 23 hπ

2

3 hπ

(11)

BRIZ et al.: ROTOR POSITION ESTIMATION OF AC MACHINES USING ZERO-SEQUENCE CARRIER-SIGNAL VOLTAGE

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A. Carrier-Signal Current Injection of the carrier-signal voltage produces a carriersignal current. Its analytical form isqds_c = −jIcp ejωc t − jIcn ej(± hθe −ωc t)

(15)

where Icp = (Vc /ωc )(ΣLσs /(ΣL2σs − ∆L2σs )) and Icn = (Vc /ωc )(∆Lσs /(ΣL2σs − ∆L2σs )), can be determined from the same set of equations used to obtain the zero-sequence carriersignal voltage [10]. The use of the negative sequence carrier-signal current for saliency position estimation has been widely studied [3]–[5] and is not covered in this paper. Since the sensorless method analyzed in this paper only uses the zero-sequence voltage, the presence of the carrier-signal current is an unwanted effect as it produces noise, vibration, and additional losses. One interesting fact that can be observed from (15) is that the magnitude of the carrier-signal current is inversely proportional to the carriersignal frequency. This implies that increasing the carrier-signal frequency, for a constant carrier-signal voltage, reduces the carrier-signal current and its adverse effects. B. Rotor Position Estimation Using the Zero-Sequence Carrier-Signal Voltage Using high-frequency excitation for rotor position estimation requires a rotor position-dependent saliency that couples with the stator windings to produce a measurable signal. Although rotor position-dependent saliencies are sometimes present in standard machine designs due to the combined effect of rotor and stator slotting [4], [5], [7], [10], they can also be specifically created in several ways [3], [6]. In all cases, rotor designs with open or semi-open slots are needed. Independent of how a saliency is created, in order for it to couple with the stator windings and produce a zero-sequence carrier-signal voltage, the expression hsp = n · p,

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

(16)

needs to be satisfied, assuming the machine has an integer number of slots per pole per phase, where p is the number of poles and hsp is the saliency harmonic order relative to 360 mechanical degrees. 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|

(17)

where S and R are equal to the number of stator and rotor slots, respectively. The fundamental of this permeance waveform rotates at the mechanical speed shown in [5], [10] ωp =

R ωrm . (R − S)

Fig. 2. Frequency spectrum of the zero-sequence carrier-signal voltage ωc = 500 Hz, Vc = 20 V (peak). (a) No fundamental current, ωr = 1 Hz. (b) Rated flux-rated load, ωe = 4 Hz, ωr = 1 Hz. TABLE I INDUCTION MOTOR PARAMETERS

of this machine are shown in Table I. There are two important observations that can be seen in Fig. 2. First, rotor–stator slotting produces a rotor-position-dependent modulation of the zero-sequence carrier-signal voltage of the form s v0sc = V0ch cos(ωc t + hθr )

(19)

with h being the number of rotor slots per pole pair (14 for this case). Second, when fundamental current exists, saturation-induced saliencies are also created [harmonics in Fig. 2(b) that are functions of the fundamental excitation frequency ωe ]. While those saliencies may be useful for flux angle estimation, they are a disturbance to rotor position estimation and need to be decoupled or otherwise compensated for in order to minimize the estimation error. In addition to the rotor–stator slotting saliency and saturation-induced saliencies, some level of asymmetry in the stator windings due to the manufacturing process usually exists. This produces a stationary saliency, which can give rise to a measurable component of the zero-sequence carrier-signal voltage. However, such a component would be expected to be similar from machine-to-machine with the same design, being relatively easy to decouple. Additional asymmetries, e.g., eccentricities in the rotor, have not been observed to cause any measurable effect in healthy machines. Other potential sources of saliencies would be those caused by faults in the machine, including turn-to-turn faults in the stator windings, and damaged rotor bars. The actual impact of such saliencies on the zero-sequence carrier-signal voltage, and therefore on the estimated position, would strongly depend on the level of fault.

(18)

Fig. 2 shows the frequency spectrum of the zero-sequence carrier-signal voltage (5) in a test machine, for both no fundamental excitation and rated flux, rated load. The parameters

III. S ELECTION OF THE C ARRIER -S IGNAL F REQUENCY Selection of the carrier-signal frequency involves several tradeoffs. Increasing the carrier-signal frequency has two major advantages: 1) the carrier-signal current magnitude is reduced

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Fig. 3. Experimentally measured magnitude of the 2ωe and 14ωr components of the zero-sequence carrier-signal voltage as a function of the carrier-signal frequency for the motor in Table I. The motor was operated at rated flux, rated load. A carrier-signal voltage of 20 V (peak) was used.

and 2) the spectral separation with respect to the fundamental excitation is increased, which reduces the interaction between the fundamental excitation and the carrier-signal excitation [12]. However, increasing the carrier-signal frequency has some detrimental effects, most of them related to the nonideal behavior of the inverter. These effects are analyzed in the following section. The theoretical model developed in Section II was based on the high-frequency behavior of the machine. One of the more interesting conclusions reached from this model was that the zero-sequence carrier-signal voltage is independent of the carrier-signal frequency. This conclusion is only valid if the average and differential stator transient inductances are not functions of the carrier-signal frequency. Fig. 3 shows the magnitudes of the rotor–stator slotting and the saturationinduced components of the zero-sequence carrier-signal voltage for several carrier-signal frequencies. From Fig. 3, it can be seen that the rotor-position-dependent component’s magnitude is almost constant, independent of the carrier-signal frequency, while the saturation-induced component’s magnitude decreases as ωc increases. This suggests that increased carrier-signal frequencies would result in a slight reduction in the sensitivity to the presence of saturation-induced 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 = 2500 Hz and a carrier-signal voltage magnitude of Vc = 20 V (peak). This resulted in a carrier-signal current (15) with a magnitude of 48 mA (peak), which corresponds to 1.2% of the rated current. The inverter switching frequency was ωs = 15 kHz, with a dead time of 2 µs. No compensation of the dead time was implemented. IV. I MPLEMENTATION I SSUES A FFECTING THE C ARRIER -S IGNAL V OLTAGE E XCITATION AND M EASUREMENT OF THE Z ERO -S EQUENCE C ARRIER -S IGNAL V OLTAGE The injection of the carrier-signal voltage and the measurement of the resulting zero-sequence carrier-signal voltage are simple in principle (Fig. 1). There are, however, a number of implementation issues that can strongly influence the results. These issues are analyzed in this section.

Fig. 4. Zero-sequence voltage measurement using a balanced resistor network.

A. Zero-Sequence Voltage Measurement Using Phase-to-Neutral Voltages Calculating the zero-sequence voltage using (5) implies measuring the three phase-to-neutral voltages. One advantage of measuring the phase-to-neutral voltages is that zero-sequence components present in the phase voltages va , vb , and vc are automatically decoupled, with the measured zero-sequence voltage only consisting of the components induced by the interaction between the carrier-signal excitation and the asymmetries in the machine. This requires the use of three voltage sensors, which increases the cost of the system. Methods that use a single voltage sensor are analyzed next. B. Zero-Sequence Voltage Measurement Using the Neutral Voltage and the Midpoint of the DC Bus An alternative to measuring the phase-to-neutral voltages directly is to measure the neutral voltage of the electric machine with respect to the midpoint of the dc bus vn0 (Fig. 4). This requires only a single voltage sensor. With the phase voltages va , vb , and vc referred to the midpoint of the dc-bus voltage, (5) can be rewritten as vn0 =

1 s (va + vb + vc ) − v0s . 3

(20)

It can be observed from (20) that the zero-sequence voltage measured with respect to the midpoint of the dc bus contains not only the components induced by the interaction of the carrier-signal voltage with the saliency (5) but also components injected by the inverter s = v0abc

1 (va + vb + vc ). 3

(21)

Although these inverter-generated components make the task more difficult, it does not mean that vn0 cannot still provide useful information. Since the proposed technique is based on the tracking of specific components, i.e., those around the carrier-signal frequency, the zero-sequence voltage vn0 can be used instead of (5) provided that the zero-sequence voltage injected from the inverter (21) is spectrally separated from the carrier-signal frequency. Fig. 5 shows the spectrum of the zero-sequence voltage vn0 . The frequencies corresponding to the most relevant components are labeled. The components at 3ωe and near ωc are induced


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Fig. 5. Experimentally measured spectrum of the zero-sequence voltage vn0 . The motor was operated at rated flux, rated load.

by saliencies. The first is caused by the interaction of the fundamental excitation and saturation. The second is caused by interaction between the carrier-signal voltage and either rotor–stator slotting or saturation-induced saliencies. Most of the remaining components are produced by the inverter. Although not seen in the figure due to the limited frequency scale, additional high-frequency zero-sequence voltage components exist, including higher-order harmonics of the switching frequency and components caused by the combined effect of fast switching transients at the inverter output and parasitics in the motor windings and inverter to motor cabling. Separating the saliency position information carrying components in vn0 from the other components injected by the inverter can be done using filters. Analog anti-aliasing filters are necessary to filter-off frequencies above the Nyquist frequency, i.e., half of the sampling frequency (typically equal to the switching frequency). The components eliminated by this filtering would include components at the switching frequency, components at ωs ± ωc , and components at higher frequencies not shown in the figure. Components at lower frequencies that have a large spectral separation from the carrier-signal frequency are also easy to filter. This is the case for the component at 3ωe . The filtering of zero-sequence voltage components with reduced spectral separation from the desired carrier-signal components can present more problems. This is analyzed next. Components Near the Carrier-Signal Frequency Caused by Intermodulation: The injection of the carrier-signal voltage using pulsewidth modulation produces a group of intermodulation components in the zero-sequence voltage at frequencies that are a combination of the fundamental excitation frequency and the carrier-signal frequency. Several implementation issues have been observed to influence these components at frequencies near the carrier-signal frequency. Examples of this are observed in Fig. 6. In Fig. 6 (left column), the spectra of the zero-sequence voltage vn0 for two different modulation methods (sine triangle and space vector modulation) and two different carrier-signal frequencies are shown. From Fig. 6, several conclusions can be reached. 1) For relatively low carrier-signal frequencies with sinetriangle modulation, the intermodulation components of the zero-sequence voltage near the carrier-signal frequency were found to be relatively minor [Fig. 6(a)]. Other implementation issues like the dead time of the inverter were found to influence these components, so this conclusion may not be general in nature.

Fig. 6. Frequency spectrum of the zero-sequence carrier-signal voltage measured in the neutral connection of the motor (left) and zero-sequence carriersignal voltage (right) for different modulation techniques and carrier-signal frequencies. A carrier-signal voltage of 20 V (peak) was used, and the motor was operated at rated flux, rated load ωe = 4 Hz, ωr = 1 Hz. (a) Sinetriangle modulation ωc = 500 Hz. (b) Space-vector modulation, ωc = 500 Hz. (c) Sine-triangle modulation, ωc = 500 Hz.

2) Standard space-vector modulation produces large amounts of zero-sequence components near the carriersignal frequency that cause a tremendous distortion in the neutral voltage vn0 [Fig. 6(b)]. This effect is caused by the intermodulation of the zero-sequence components from standard space vector modulation with carrier-signal excitation. 3) A zero-sequence voltage component at ωc −ωe was found for the case of very high carrier-signal frequencies. An example of this can be observed in Fig. 6(c). The intermodulation components that are near the carriersignal frequency are very difficult to compensate due to the lack of spectral separation from the signals of interest. Although this does not preclude using the voltage vn0 instead of the three phase-to-neutral voltages for measuring the zero-sequence carrier-signal voltage, it does mean that its use should be carefully verified for each particular implementation. An alternative method for the measurement of the zero-sequence voltage, which requires a single voltage sensor and eliminates most of the problems detailed up to this point in this section, is presented in the following section. C. Zero-Sequence Voltage Measurement Using an Auxiliary Resistor Network A method of mitigating the effects created by invertergenerated zero-sequence voltages is to measure them directly and then decouple them. The measurement of these voltages can be made by connecting a balanced three-phase load in parallel with the motor, as shown in Fig. 4 for the case of a balanced resistor network [13]. In this case, it can be shown that

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Fig. 7. Experimentally measured spectrum of the zero-sequence voltage vn_nR . The motor was operated at rated flux, rated load.

the resistor network’s common terminal voltage, referred to the midpoint of the dc bus, is equal to the zero-sequence voltage produced by the inverter, i.e., vnR = v0abc .

(22)

This assumes that the parasitic components of the resistor network are negligible in the frequency range of interest (< 5 kHz). In practice, this was found to be a good assumption. From (20)–(22), the machine’s zero-sequence voltage can then be measured as the difference of the voltage at the machine’s neutral point and the resistor network’s neutral point (see Fig. 4), i.e., vn_nR = vnR − vn .

(23)

Fig. 7 shows the frequency spectrum of the zero-sequence voltage measured using (23). By comparison with Fig. 5, it can be observed that a significant portion of the zero-sequence voltage generated by the inverter has been effectively decoupled. The advantages of using the auxiliary resistor network with respect to components generated by the inverter can be seen in Fig. 6 (right column). Resistors of 15 kΩ were used. From this figure, it can be observed that the resistor network removed the inverter-generated components, with the resulting frequency spectra being almost identical, independent of the modulation method or carrier-signal frequency. The use of the auxiliary resistor network has proven to be effective in support of the experiments carried out as part of this work. Due to this, all measurements of the zero-sequence voltage presented hereafter were obtained using it. A hall effect voltage sensor was used, the measured signal being sampled using a 10-bit A/D converter. The experiments carried out to verify its effectiveness included using different cable lengths, ranging from 2 to 50 m, both shielded and unshielded.

V. F ILTERING OF THE Z ERO -S EQUENCE C ARRIER -S IGNAL V OLTAGE The zero-sequence voltage measured using (23) still contains several components unrelated to the desired saliency position, including triplen harmonics of the fundamental excitation and inverter switching-related harmonics (see Fig. 7). There are several options for implementing filters to separate the zero-

sequence carrier-signal voltage from the voltage vn_nR . Ideally, these filters should cause minimal distortion, especially in the phase of the zero-sequence carrier-signal voltage since it contains the desired saliency position information. Otherwise, compensation of the phase distortion is required. The separation of specific components from the zero-sequence voltage can be accomplished using an analog bandpass filter. The design of this filter needs to cutoff high-frequency components that could cause aliasing, mostly switching related, and to reject the low-frequency components, mainly triplen harmonics of the fundamental excitation. A resonant filter is one option to achieve these goals. This type of filter has the disadvantage, however, that its maximum phase slew rate (phase versus frequency variation) occurs in its pass frequency. This makes the compensation of the phase very sensitive to errors in the signal frequency. These drawbacks can be overcome by using serially connected notch and low-pass filters. The block diagram for this form of filtering is shown in Fig. 8(a). The notch filter was tuned to reject the components at the switching frequency while the low-pass filter was tuned to reject higher frequencies. 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-signal frequency. This allows the phase shift caused by the filter to be easily compensated. The Bode diagram of the filter is shown in Fig. 8(b). The zero-sequence voltage sampled by the analog-to-digital converter shown in Fig. 8(a) mainly contains components near the carrier-signal frequency and triplen harmonics of the fundamental excitation. These have the form v0 = V0ch sin(ωc t + hθr ) + V03ωe sin(3ωe t + θe )

(24)

that is consistent with the content seen in Fig. 7 after the analog filtering from Fig. 8(a). Prior to extraction of the rotor position information from the zero-sequence voltage, the sampled zero-sequence voltage is converted into a carrier-signal synchronous complex voltage vector c = v0 e−jωc t . v0qd

(25)

The resulting complex voltage vector is given by [10], [12] V0c jhθr − e−j(hθr +2ωc t) e 2 V03ωe j((ωc +3ωe )t+θe ) −j − e−j(−ωc +3ωe )t+θe ) . (26) e 2

c v0qd = −j

Low-pass filtering this signal produces the zero-sequence carrier-signal voltage vector c jhθr v0qd _c = −jV0ch e

(27)

that has its phase angle modulated by the saliency position.


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Fig. 8. Block diagram and frequency response of the filtering implemented to isolate the zero-sequence carrier-signal voltage. The fourth-order low-pass Butterworth filter was designed for a cutoff frequency of 6 kHz, and the second-order 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-signal frequency of 2500 Hz, the phase around the carrier-signal frequency is zoomed).

While not strictly necessary, this transformation was found to be convenient as it enables the use of filtering methods specific to complex vectors to obtain the saliency position. VI. M ODELING AND D ECOUPLING OF S ATURATION -I NDUCED S ALIENCIES In addition to the components shown in (24)–(27), saturationinduced saliencies often produce additional components in the zero-sequence carrier-signal voltage vector [Fig. 2(b)]. This can be seen in Fig. 9(b). In this figure, the zero-sequence carriersignal voltage vector contains a rotor-dependent component rotating at 14ωr and saturation-induced components, the most significant at 2ωe and −4ωe . Accurate rotor position estimation requires that these saturation-induced components be compensated for or decoupled. One method of doing this is to first measure and store the saturation-induced components as a function of 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. Fig. 9(c) shows the 2ωe component of the zero-sequence carrier-signal voltage vector from Fig. 9(b), isolated using off-line digital signal processing. To analyze the influence of saturation, the magnitudes and phase angles, relative to the stator current [Fig. 9(a)], of the 2ωe and −4ωe components of the zero-sequence carrier-signal voltage vector (V02ωe , V04ωe , θ0_2ωe , and θ0_4ωe , respectively) were measured for different operating conditions (levels of flux and load). Since saturation-

induced components of the zero-sequence carrier-signal voltage vector do not rotate at the same frequency as the stator current, the measured phase angles θ0_2ωe and θ0_4ωe correspond to the instant when the angle of the stator current vector was equal to zero. This is illustrated in Fig. 10, where the complex vector representation of the stator current in Fig. 9(a) and of the 2ωe component of the zero-sequence carrier-signal voltage vector in Fig. 9(c) are shown. Fig. 11 shows the magnitude and phase, relative to the stator current angle, of the 2ωe component of the zero-sequence carrier-signal voltage as a function of the slip frequency and fundamental current level. The estimated trajectory of the stator current magnitude as a function of the slip frequency, when the machine is operated under field oriented control and constant rotor flux, is marked on top of the plots. From that figure, some interesting conclusions can be reached. 1) Both magnitude and phase are smooth functions. In addition, a noticeable symmetry is observed between motoring and generating operation modes. 2) The phase depends strongly on the slip frequency and is barely affected by the stator current magnitude. This suggests that it is mostly related to the main flux of the machine [10]. 3) The magnitude is almost constant when the machine is operated with constant rotor flux. It also decreases for large slips and small current magnitudes, which correspond to the machine working with reduced levels of flux. While data similar to that shown in Fig. 11 would be necessary when a machine is operated with arbitrary levels of rotor

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Fig. 9. Measurement and compensation of saturation-induced harmonics of c c the zero-sequence carrier-signal voltage vector. (a) isqs , isds . (b) v0q _c , v0d_c . c c c c (c) v0q_2ωe , v0d_2ωe . (d) v0q_14ωr , v0d_14ωr .

Fig. 11. Magnitude and phase of the 2ωe harmonics of the zero-sequence carrier-signal voltage vector as a function of slip and fundamental current level. (a) V02ωe . (b) θ0_2ωe .

Fig. 10. Stator current complex vector and 2ωe component of the zerosequence carrier-signal voltage vector from Fig. 9(a) and (c), respectively. Measurement of the compensating angle θ0_2ωe of the component at 2ωe of the zero-sequence carrier-signal voltage vector is made when the stator current angle is 0 (t = 0 in the figure).

flux, simplifications can be made when it is operated with a constant rotor flux (field weakening above rated speed is not considered since the proposed method is intended to be used only at low or zero speeds). In this case, the slip frequency and stator current have a defined relationship (constant rotor flux curves shown in Fig. 11). Fig. 12 shows the magnitude and phase of the 2ωe and −4ωe saturation-induced components of the zero-sequence carrier-signal voltage vector when the machine is operated with constant rotor flux. These values were stored in look-up 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. 13. Linear interpolation was used when the q-axis current did not exactly correspond to an entry in the table.

Fig. 12. Magnitude and phase of the 2ωe and −4ωe harmonics of the zero-sequence carrier-signal voltage vector as a function of the slip when the machine is operated with constant rotor flux. (a) V02ωe (◦), V04ωe (). (b) θ0_2ωe (◦), θ0_4ωe ().

Fig. 9(d) shows the resulting zero-sequence carrier-signal voltage vector obtained from the signal shown in Fig. 9(b) using the data from Fig. 12 and the decoupling shown in Fig. 13. The estimated phase angle of the 2ωe saturation-induced component is calculated using (see Fig. 10) θ2ωe = 2ϕe + θ0_2ωe .

(28)


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Fig. 13. Decoupling of saturation-induced components of the zero-sequence carrier-signal voltage vector and rotor position and velocity estimation using a tracking observer.

Fig. 15. Sensorless position control when a position step from 0◦ to 90◦ is commanded. The machine was operated at rated flux and 80% rated load. (a) θˆr . (b) θr − θˆr . (c) θˆrf .

Fig. 16. Estimated rotor position error at rated flux, rated load. Fig. 14. Sensorless velocity control when rated load is applied to the machine. The machine was operated at rated flux. (a) ieqs . (b) θˆr . (c) ω ˆr .

The resulting compensating 2ωe zero-sequence complex voltage vector is calculated using c v0qd _2ωe = V02ωe e

jθ2ωt

e

.

(29)

A similar procedure and equations were used for the −4ωe component. In Fig. 12, only 13 entries per parameter were used in the look-up table (52 total). This means that both memory and computational requirements for its on-line implementation are minimal.

that the speed briefly reverses after the torque step is applied. This response depends, in addition to the performance of the sensorless method, on issues like the mechanical inertia, and the bandwidth of the current and speed regulators. The same was obtained when the machine was operated with sensored control using the same bandwidth controllers. Fig. 15 shows the estimated rotor position, the estimation error, and the estimated rotor flux angle when the machine is operated in sensorless position control and a position step is commanded. A constant load torque of 80% of its rated value was applied. Stable operation and good dynamic response are observed during both speed and position sensorless control. Fig. 16 shows the estimated rotor position error when the machine is operated at standstill in sensorless position control at rated flux, rated load.

VII. S ENSORLESS V ELOCITY AND P OSITION C ONTROL Sensorless field orientation, velocity, 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. 13. The rotor angle and velocity were estimated using a tracking observer [3], [5]. Fig. 14 shows the transient response during velocity control when rated load is applied and then removed. The speed reference was held constant at 30 r/min. It is observed from Fig. 14

VIII. C ONCLUSION The use of the zero-sequence carrier-signal voltage for rotor position estimation of induction machines has been analyzed in this paper. While sharing some properties with negativesequence carrier-signal current-based techniques, it provides two major advantages: 1) the magnitude of the zero-sequence voltage does not depend on the carrier-signal frequency and

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2) the distortion that the inverter has on the injected carriersignal voltage can be readily compensated. Implementation issues that affect the performance of the method were discussed, including selection of the carriersignal frequency, measurement of the zero-sequence carriersignal voltage, and decoupling of saturation-induced saliencies. Experimental results that confirm the capability of the method to provide accurate high-bandwidth position estimates at low and zero speed at rated flux and rated load were presented. ACKNOWLEDGMENT The authors wish to acknowledge the support and motivation provided by the University of Oviedo, Spain, and the 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 controlled induction machines,” in Conf. Rec. IEEE-IAS Annu. Meeting, San Diego, CA, Oct. 1996, pp. 257–261. [2] J. Holtz, “Sensorless control of induction motor drives,” Proc. IEEE, vol. 90, no. 8, pp. 1359–1394, Aug. 2002. [3] P. L. Jansen and R. D. Lorenz, “Transducerless position and velocity estimation in induction and salient AC machines,” IEEE Trans. Ind. Appl., vol. 31, no. 3, pp. 240–247, Mar./Apr. 1995. [4] N. Teske, M. Asher, K. J. Bradley, and G. M. Sumner, “Analysis and suppression of high-frequency inverter modulation in sensorless positioncontrolled induction machine drives,” IEEE Trans. Ind. Appl., vol. 39, no. 1, pp. 10–18, Jan./Feb. 2003. [5] M. W. Degner and R. D. Lorenz, “Position estimation in induction machines utilizing rotor bar slot harmonics and carrier frequency signal injection,” IEEE Trans. Ind. Appl., vol. 36, no. 3, pp. 736–742, May/Jun. 2000. [6] J. Cilia, G. M. Asher, and K. G. Bradley, “Sensorless position detection for vector controlled induction motor drives using an asymmetric outersection cage,” in Conf. Rec. IEEE-IAS Annu. Meeting, San Diego, CA, Oct. 1996, pp. 286–292. [7] J. Holtz and H. Pan, “Elimination of saturation effects in sensorless position controlled induction motors,” IEEE Trans. Ind. Appl., vol. 40, no. 2, pp. 623–631, Mar./Apr. 2004. [8] M. Schroedl, “Sensorless control of AC machines at low speed and standstill based on the inform method,” in Conf. Rec. IEEE-IAS Annu. Meeting, Chicago, IL, Sep./Oct. 2001, CD-ROM. [9] A. Consoli, G. Scarcella, and A. Testa, “A new zero-frequency fluxposition detection approach for direct-field-oriented-control drives,” IEEE Trans. Ind. Appl., vol. 36, no. 3, pp. 797–804, May/Jun. 2000. [10] F. Briz, M. W. Degner, P. García, and R. D. Lorenz, “Comparison of saliency-based sensorless control techniques for AC machines,” IEEE Trans. Ind. Appl., vol. 40, no. 4, pp. 1153–1161, Jul./Aug. 2004. [11] J. M. Guerrero, M. Leetmaa, F. Briz, A. Zamarron, and R. D. Lorenz, “Inverter nonlinearity effects in high frequency signal injection-based, sensorless control methods,” IEEE Trans. Ind. Appl., vol. 41, no. 2, pp. 618–626, Mar./Apr. 2005. [12] F. Briz, M. W. Degner, P. García, and A. Diez, “Transient operation of carrier-signal injection based sensorless techniques,” in Proc. IEEE IECON’03, Roanoke, VA, Nov. 2003, pp. 1466–1471. [13] J. X. Shen, Z. Q. Zhu, and D. Howe, “Sensorless flux-weakening control of permanent brushless machines using third-harmonic back-EMF,” in Proc. IEEE Int. Electric Machines and Drives Conf. (IEMDC), Madison, WI, Jun. 2003, CD-ROM.

Fernando Briz (A’96–M’99) received the M.S. and Ph.D. degrees 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 2005 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS Third Place Prize Paper Award and was the recipient of two IEEE Industry Applications Society Conference prize paper awards in 1997 and 2004, respectively.

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 joined the 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 of the 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 received the 2005 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS Third Place Prize Paper Award and has been the recipient of several IEEE Industry Applications Society Conference paper awards.

Pablo García (S’02) was born in Spain in 1975. He received the M.E. degree in industrial engineering from the University of Oviedo, Gijón, Spain, in 2001, and is currently working toward the Ph.D. degree in electrical engineering at the same university. His research interests include sensorless control and diagnosis of induction motors, neural networks, and digital signal processing. Mr. García was awarded a Fellowship of the Personnel Research Training Programme funded by the Spanish Ministry of Science and Technology in 2001.

Juan Manuel Guerrero (S’00–A’03–M’04) was born in Gijón, Spain, in 1973. He received the M.E. degree in industrial engineering and the Ph.D. degree in electrical and electronic engineering from the University of Oviedo, Gijón, Spain, in 1998 and 2003, respectively. Since 1999, he has been a Teaching Assistant in the Department of Electrical, Computer and Systems Engineering, University of Oviedo. From February to October 2002, he was a Visitor Scholar at the University of Wisconsin, Madison. His research interests include parallel-connected motors fed by one inverter, sensorless control of induction motors, control systems, and digital signal processing. Dr. Guerrero received an award from the College of Industrial Engineers of Asturias and León, Spain, for his M.E. thesis in 1999.