Accuracy, Bandwidth, and Stability Limits of Carrier ... - IEEE Xplore

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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 43, NO. 4, JULY/AUGUST 2007

Accuracy, Bandwidth, and Stability Limits of Carrier-Signal-Injection-Based Sensorless Control Methods Pablo García, Member, IEEE, Fernando Briz, Senior Member, IEEE, Michael W. Degner, Senior Member, IEEE, and David Díaz-Reigosa

Abstract—This paper studies the performance limits of carrier-signal-injection-based sensorless control methods. When a high-frequency carrier-signal voltage is injected into a salient ac machine, two types of signals are created that can be used to estimate the rotor position: 1) the negative-sequence carrier-signal current and 2) the zero-sequence carrier-signal components. The limitations of ac machine sensorless control using each of these signals are analyzed in this paper. The analysis focuses on three key performance metrics: 1) the accuracy; 2) the bandwidth; and 3) the stability of the estimated position.

s vqds _c V c , ωc , θ c

Index Terms—Rotor position estimation, saliency-based sensorless control.

c v0qd _c

N OMENCLATURE Lσs , ∆Lσs

Average and differential stator transient inductances. h Harmonic order of the saliency relative to electrical angular units. Angular position of the saliency in electrical θe radians. Mechanical speed and position, ωrm , θrm respectively. ∗ Commanded variables. ∧ Estimated variables. S, R Number of stator and rotor slots, respectively. Superscript “s ” Variables shown in a stationary reference frame Superscript “cn ” Variables shown in a negative-sequence carrier-signal reference frame (i.e., rotating at −ωc ). Paper IPCSD-07-010, presented at the 2006 Industry Applications Society Annual Meeting, Tampa, FL, October 8–12, 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, 2006 and released for publication February 15, 2007. This work was supported in part by the Spanish Ministry of Science and Education–European Regional Development Fund under Grant MEC-04DPI2004-00527. P. García, F. Briz, and D. Díaz-Reigosa are with the Department of Electrical, Computer and Systems Engineering, University of Oviedo, 33204 Gijón, Spain (e-mail: [email protected]; [email protected]; [email protected]). M. W. Degner is with Research and Advanced Engineering, Ford Motor Company, Dearborn, MI 48121-2053 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIA.2007.900460

van , vbn , vcn s v0sc V0ch , V0c2h

ia , ib , ic iqds_c iqds_cp iqds_cn Icp , Icn

ιcn , υc0

Injected carrier-signal voltage. Magnitude, frequency, and phase angle of the injected carrier-signal voltage, respectively. Phase to neutral voltages. Zero-sequence carrier-signal voltage. Magnitude of the hθe and −2hθe components of the zero-sequence carrier-signal voltage, respectively. Zero-sequence carrier-signal voltage vector in a carrier-signal synchronous reference frame. Phase currents. Carrier-signal current. Positive-sequence component of the carriersignal current. Negative-sequence component of the carrier-signal current. Magnitude of the positive-sequence and negative-sequence components of the carrier-signal current, respectively. Fast Fourier transform (FFT) of the negative-sequence carrier-signal current and the zero-sequence carrier-signal voltage vector, respectively. I. I NTRODUCTION

S

ENSORLESS methods based on the tracking of saliencies have been demonstrated to allow position/speed estimation at very low and zero speeds [1]–[13]. These methods use various forms of high-frequency excitation to estimate the spatial location of asymmetries. The asymmetries can be created specifically for the purpose of sensorless control but are often intrinsic to the magnetic/mechanical design of the machine [1]–[13]. 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. The types of excitation that have been proposed can be classified into two main categories: 1) injection of a carrier signal superimposed on the fundamental excitation [1], [2], [5]–[13] and 2) excitation created by the pulsewidth modulation (PWM) of the inverter switches, commonly using modified forms of PWM [3], [4]. This paper studies the performance limits of carriersignal injection-based sensorless control methods. When a

0093-9994/$25.00 © 2007 IEEE

GARCÍA et al.: PERFORMANCE LIMITS OF CARRIER-SIGNAL-INJECTION-BASED SENSORLESS CONTROL METHOD

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harmonic order and the angular position of the saliency in elec Lσs , which trical radians, respectively [5]. When ∆Lσs is typically the case, the magnitude of the second component V0c2h in (3) is significantly smaller than the magnitude of the first component V0ch and can be safely neglected [6]. Thus s ∼ v0sc = V0ch cos(ωc t ± hθe ).

Fig. 1.

Carrier-signal voltage injection.

high-frequency carrier-signal voltage is injected into a salient ac machine, two types of signals are created that can be used to estimate the rotor position: 1) the negative-sequence carrier-signal current and 2) the zero-sequence carrier-signal components (zero-sequence voltage and zero-sequence current for the case of wye-connected and delta-connected machines, respectively). The limitations of ac machine sensorless control using each of these signals are analyzed, focusing on two key performance metrics: 1) the accuracy and 2) the bandwidth of the estimated position. Although the analysis presented in this paper is based on induction machines, the concepts and conclusions presented can be applied to other types of ac machines, including permanentmagnet and synchronous reluctance machines. II. M ODELING OF S ALIENT M ACHINES AND C ARRIER -S IGNAL I NJECTION U SING A R OTATING V OLTAGE V ECTOR When a balanced polyphase high-frequency carrier-signal voltage (1) is applied to a machine (Fig. 1), its dynamic response can be modeled using the stator transient inductance (1), provided that the carrier-signal frequency ωc is substantially faster than the stator transient time constant [1], i.e., s jωc t ∼ vqds = jωc Lsσs isqds_c . _c = Vc e

(1)

In wye-connected machines, the carrier-signal voltage interacts with stator transient inductance saliencies to produce a carrier-signal current (2) and a zero-sequence carrier-signal voltage (3), both shown for the case of a single saliency [5], [6]. Thus, 2 ia + ib ej2π/3 + ic ej4π/3 = isqds_cn + isqds_cn isqds_c = 3

Icp

= − jIcp ejωc t − jIcn ej(−ωc t ± hθe ) (2) L ∆L Vc V 2 σs 2 and Icn = c 2 σs 2 = Lσs − ∆Lσs Lσs − ∆Lσs ωc ωc

1 s v0sc = (van + vbn + vcn ) 3 = V0ch cos(ωc t ± hθe ) − V0c2h cos(ωc t ± 2hθe )) (3) ∆Lσs Lσs ∆Lσs V0ch = Vc 2 and V0c2h = V0ch Lσs ∆L2σs Lσs where Lσs and ∆Lσs are the average and differential stator transient inductances, respectively, and h and θe are the

(4)

It can be observed from (2) and (4) that the negativesequence carrier-signal current and the zero-sequence carriersignal voltage each contain similar information with respect to the saliency’s position and are therefore useful for saliency position estimation. In delta-connected machines, the interaction between the carrier-signal voltage vector and the saliency produces a carriersignal current with a form similar to (2) and a zero-sequence carrier-signal current with a form similar to (3) (substituting the voltages for currents) [7]. Further discussion on deltaconnected machines is omitted due to space constraints, but all of the conclusions reached are applicable to this class of machines. III. R OTOR P OSITION E STIMATION 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 created by the stator and rotor slotting was used in this paper [5]–[8], [11]. The combination of rotor and stator slotting produce a permeance waveform (enhanced by rotor designs with open or semiopen rotor slots) that has a fundamental spatial harmonic given by (5), 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. For the permeance waveform to couple with the stator windings, its harmonic order must be an integer multiple of the number of poles p (6), assuming that the machine has an integer number of slots per pole per phase. The fundamental of this permeance waveform rotates at the mechanical speed shown in (7), where ωrm is the rotor speed in mechanical units [8], i.e., hsp = |R − S| hsp = n · p, ωp =

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

R ωrm . (R − S)

(6) (7)

Fig. 2 shows the experimentally measured spectrum of the negative-sequence carrier-signal current and the zero-sequence carrier-signal voltage. The parameters of the test machine are shown in Table I. The rotor–stator slotting produces a component at R · ωrm (14 ωr ) in both signals, with respect to the carrier-signal frequency. One important difference between the negative-sequence carrier-signal current and the zero-sequence carrier-signal voltage is that the first is a complex vector signal and the second is a scalar signal. To simplify the analysis, as well as the practical implementation of the methods, a similar coordinate transformation is applied to both signals, as shown in Fig. 3

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Fig. 2. Experimentally measured spectrum of the carrier signals using a carrier-signal voltage of ωc = 500 Hz and Vc = 20 V (peak). The motor was operated at rated flux and load (ωe = 4 Hz and ωr = 1 Hz). (a) Negative-sequence carrier-signal current. (b) Zero-sequence carrier-signal voltage. TABLE I INDUCTION MOTOR PARAMETERS

Fig. 4. (left) Schematic representation and (right) measured estimation error resulting from inexact decoupling of secondary saliencies in the negativesequence carrier-signal current.

minated by additional components Icnhi(8). These additional components form a disturbing Icnhi , which has a signal maximum potential value of | Icnhi |max . Thus, jRθrm icn + Icnhi ejhi θi . (8) qds_cn = IcnR e Fig. 3. Schematic representation of the signal processing used to estimate the rotor position using (a) negative-sequence carrier-signal current and (b) zerosequence carrier-signal voltage.

[6], [7]. This transformation has two effects: 1) it transforms the carrier signals to be near zero frequency, making further signal processing easier; and 2) it converts the zero-sequence voltage into a complex signal. This allows the subsequent signal processing steps shown in Fig. 3 to have the same form for both types of signals, which simplifies the comparative analysis. The signal processing consists of four steps: 1) coordinate transformation; 2) separation of the desired carrier-signal component from the overall signal using bandpass filtering; 3) compensation of any remaining undesired deterministic content, i.e., secondary saliencies; and 4) estimation of the saliency position. IV. S IGNAL - TO -N OISE R ATIO OF THE C ARRIER S IGNALS : T OTAL H ARMONIC D ISTORTION (THD) Ideally, the carrier signals entering the “saliency position estimation” block shown in Fig. 3 would only consist of rotorposition-related components. However, this is never achieved in practice due to both incorrect decoupling of secondary saliencies and the presence of other “noise” in the signals. Fig. 4 graphically shows the effect on rotor position estimation using the negative-sequence carrier-signal current when the component containing the desired information IcnR is conta-

A key parameter determining the accuracy and robustness of the estimated position is the magnitude of the rotor-positiondependent component IcnR (signal), which is relative to the other components Icnhi (noise). The THD caused by the undesired “noise” was found to be an insightful metric for quantifying this relationship [7]. The THD of the negative-sequence carrier-signal current is calculated using cn ι2cn − ιcn (R · ωrm )2 2 (9) THD iqds_cn = ιcn 2 2 ι where ιcn = FFT(icn ιcn = +bw qds_cn ) and n=−bw cn(n) , with ιcn(R·ωrm ) being the desired signal IcnR in Fig. 4. Similarly, the THD of the zero-sequence carrier-signal voltage can be calculated using 2 − υ (R · ω 2 c υc0 rm ) c0 2 THD v0qds_c = (10) υc0 2 +bw υ c 2 where υc0 = FFT(v0qds _c ) and υc0 = n=−bw c0(n) . It should be noted that calculation of the THD using (9) and (10) is defined using the input signals to the “saliency position estimation” block in Fig. 3, after the decoupling of any secondary saliencies, and assumes that the filtering removed all content outside the filter bandwidth. Fig. 5 shows the THD of the negative-sequence carriersignal current and the zero-sequence carrier-signal voltage as a


Fig. 5. Experimentally measured THD of the negative-sequence carrier-signal current and the zero-sequence carrier-signal voltage as a function of the carrier frequency after almost perfect decoupling of saturation-induced components [() Vc = 20 V and () Vc = 40 V] and with an error of 20% in decoupling [() Vc = 20 V]. The motor was operated at rated flux and load, with asymmetric regular sampling PWM and filter bandwidths of bw = 100 Hz.

function of the carrier frequency and voltage magnitude. The figure includes examples of both almost perfect decoupling of the saturation-induced components at 2ωe and −4ωe and with 20% error in the decoupling. To obtain those results, the following were done: 1) asymmetric regular sampling PWM was used in the inverter; 2) a bandwidth bw of 100 Hz was used in the bandpass filters, and 3) 12-bit analog-to-digital (A/D) converters were used to capture the stator currents and the zerosequence voltage. Several conclusions can be reached from Fig. 5. First, the THD of the negative-sequence carrier-signal current increases slightly as the carrier frequency increases, whereas the opposite is true for the zero-sequence carrier-signal voltage. The THD of the zero-sequence carrier-signal voltage is usually smaller than that of the negative-sequence carrier-signal current, with the difference increasing as the carrier frequency increases. It can also be noted from Fig. 5 that errors in decoupling cause similar values of THD in both signals. This result is not unexpected since the error in decoupling dominates the “noise” contribution in the THD calculation and therefore results in almost the same value for both signals. The exact relationship between the THD of the carrier signals and the overall accuracy of the estimated position can be obtained for the case of a single known disturbing component Icnhi [see Fig. 4 and (8)]. The curve for this exact relationship is shown in Fig. 6 (solid lines). The maximum error with a single secondary saliency is limited to ≈12.9 mechanical degrees, which occurs when Icnhi ≥ IcnR and for a THD of √ 1/ 2. This value corresponds to a rotor slot pitch and means, in practice, that a jump in the estimated saliency position to the next rotor bar occurred. This represents instability in the sensorless estimate [13]. In practice, the carrier signal contains multiple disturbing components. Some of these are related to specific frequencies (e.g., fundamental excitation frequency) and are therefore a form of “colored noise.” These are in addition to the “white noise” that can exist in any measured signal. Since this content is not deterministic, the relationship between the THD of the signals and the estimation error cannot be calculated analytically but requires the use of numerical methods and statistical

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analysis. Fig. 6 (dots) shows the THD for the more general case of multiple secondary saliencies with white noise. To obtain the results shown by the dots, an equation of the form (8) was used, with Icnhi including terms that represent incorrectly decoupled secondary saliencies (“colored noise”) and different levels of white noise. Two additional facts can be observed from Fig. 6: 1) there is a deterministic relationship between the THD and the estimation error for THD values less than ∼0.5; and 2) for a given THD value, the resulting estimation error is generally smaller for the case of the carrier signals contaminated with both colored noise and white noise than for the case of a single secondary saliency. This implies that a single disturbing component of relatively large magnitude, as is typically the case with saturation-induced components, has a more severe impact on the estimation accuracy than multiple components with smaller individual magnitudes but larger combined magnitude. It is finally noted that although the results shown in Fig. 6 were calculated for the negative-sequence carrier-signal current, the exact same results would be obtained if the zero-sequence carrier-signal voltage was used for the calculations, i.e., substituting voltage for current in (8). Understanding the sources of noise that contribute to the carrier-signal distortion is necessary to increase the performance of the sensorless control, i.e., to reduce the estimation error and to increase the estimation bandwidth. This is analyzed in the following sections, with the analysis targeting two sources of distortion: 1) errors in the injected carrier-signal voltage and 2) errors in the carrier-signal measurement. V. E FFECT AND S OURCES OF D ISTORTION IN THE C ARRIER -S IGNAL V OLTAGE A. Effect of Distortion of the Carrier-Signal Voltage The effects of distortion in the carrier-signal voltage (1) on the resulting carrier signals (2)–(4) are intrinsically linked to the magnitude of the carrier-signal voltage itself, i.e., larger carriersignal voltage magnitudes are less sensitive to a given amount of distortion than smaller magnitudes. Arbitrarily increasing the carrier-signal voltage magnitude, however, has several drawbacks relative to the normal operation of the drive that limits how far it can be increased in practice (noise, vibration, losses, etc.). Increasing the carrier frequency mitigates these drawbacks to some effect, with the added advantage of increased spectral separation from the fundamental frequency. The maximum value is also limited, however, by several practical issues, including the inverter switching frequency, microprocessor capability, etc. Distortion of the injected carrier-signal voltage is primarily of interest for components near the positive-sequence and negative-sequence carrier frequencies ±ωc . These regions of the spectrum are shown in Fig. 7 for the experimental motor. It can be observed from Fig. 7(a) that no significant content exists at frequencies near the positive-sequence component. The carrier-signal voltage spectrum at frequencies near −ωc , on the other hand, contains several components where ideally none would exist, as can be seen in Fig. 7(b). The effect

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Fig. 6. Standard deviation and maximum error of the estimated position as a function of the carrier-signal THD. The continuous lines represent the analytically calculated result for a single secondary component (no white noise), and the dots represent the numerically calculated results for the case of secondary saliencies, including both colored noise and white noise. (a) Standard deviation of the error. (b) Maximum error.

Fig. 7. Spectrum of the inverter output voltage at frequencies near (a) positive-sequence carrier frequency and (b) negative-sequence carrier frequency. A carrier signal of ωc = 500 Hz and Vc = 20 V (peak) was used. The motor was operated at rated flux and rated load (ωe = 4 Hz and ωr = 1 Hz). The current regulator was disabled, and asymmetric regularly sampled PWM was used.

of these on the desired carrier signals can be deduced from (1)–(3). Analyzing the negative-sequence carrier-signal current first, a simple model for distortion of the carrier-signal voltage is shown in (11), where an additional component of magnitude ∆Vc , at a frequency of −ωc + ∆ω, was added, with ∆ω being smaller than the bandwidth of the bandpass filter (|∆ω| < bw). The resulting carrier-signal current has the following form: s jωc t + ∆Vc ej(−ωc +∆ω)t vqds _c = Vc e

(11)

isqds_c = − jIcp ejωc t − jIcn ej(−ωc t±hθe )

∆Icp

− j∆Icp ej(−ωc +∆ω)t − j∆Icn ej((−ωc +∆ω)t±hθe ) Lσs ∆Vc = ωc − ∆ω L2σs − ∆L2σs

∆Icn =

∆L ∆Vc 2 σs 2 . ωc − ∆ω Lσs − ∆Lσs

(12)

The resulting carrier-signal current is seen to consist of four components. The terms Icp and Icn result from the ideal (commanded) carrier-signal voltage vector (2), whereas the terms ∆Icp and ∆Icn result from the distortion. The critical term in (12) is ∆Icp . This term rotates in the same direction as its source voltage, being a positive-sequence component, therefore. However, it is spectrally close to the negative-sequence carrier-signal current. The relationship between ∆Icp and Icn ,

Fig. 8. Calculated THD of the (a) negative-sequence carrier-signal current and (b) zero-sequence carrier-signal voltage caused by distortion of the injected carrier-signal voltage. ∆Lσs / Lσs for the experimental machine is marked with a gray line.

i.e., noise-to-signal ratio, is given by (13), with the resulting THD being (14). From (13), it can be observed that the distortion of the injected carrier-signal voltage ∆Vc is ampliﬁed by a gain Lσs /∆Lσs , which means that even modest values


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Fig. 9. Spectrum of the (a) commanded and (b) inverter output voltages at frequencies near the negative-sequence carrier frequency, with the current regulator enabled. The current regulator was tuned for a 250-Hz bandwidth. A carrier signal of ωc = 500 Hz and Vc = 20 V (peak) was used. The motor was operated at s∗ )|. (b) |FFT(v s )|. rated flux and rated load (ωe = 4 Hz and ωr = 1 Hz). (a) |FFT(vqds qds

of ∆Vc [like those observed in Fig. 7(b)] can give rise to significant values of ∆Icp as compared to Icn . Thus, ∆Icp ∆Vc Lσs = · (13) Icn Vc ∆Lσs 2 cn ∆Icp . (14) THD iqds_cn = 2 + I2 ∆Icp cn A similar analysis can be used to assess the effect that distortion of the injected carrier-signal voltage of the form (11) has on the zero-sequence carrier-signal voltage (15). The term ∆V0ch is the resulting disturbance, with the relationship between ∆V0ch and V0ch , i.e., noise-to-signal ratio, being (16), with the corresponding THD being (17). Thus, s = V0ch cos(ωc t ± hθe ) v0sc

∆V0ch V0ch

+ ∆V0ch cos ((−ωc + ∆ω)t ± hθe ) (15) Lσs ∆Lσs ∆Vc = , with ∆V0ch = ∆Vc 2 Lσs ∆L2σs Vc

c THD v0qds _c =

(16) 2 ∆V0ch 2 2 . ∆V0ch + V0ch

(17)

Equations (14) and (17) are graphically represented in Fig. 8, with the THD of the injected carrier-signal voltage being defined as (18). From the figure, the increased sensitivity of the negative-sequence carrier-signal current, as compared to the zero-sequence carrier-signal voltage, to the distortion of the injected carrier-signal voltage can be seen, i.e., ∆Vc2 . (18) THD(Vc ) = 2 Vc + ∆Vc2 Distortion of the injected carrier-signal voltage can be produced by several different sources, including: 1) the current regulator reaction to the negative-sequence carrier-signal currents; 2) the PWM strategy; and 3) the nonideal behavior of the inverter. These are analyzed in the following sections. B. Distortion Caused by the Current Regulator The reaction of the current regulator to the carrier-signal components of the current is a source of carrier-signal voltage

Fig. 10. PWM of the voltage command. The carrier frequency is 1/4 of the switching frequency. Note: The carrier-signal voltage magnitude, which is relative to the triangle magnitude, has been exaggerated for illustrative purposes. (a) Symmetric regular sampling. (b) Asymmetric regular sampling.

distortion. Fig. 2(a) shows the spectrum of the stator current near the negative-sequence carrier frequency. Since only the fundamental current was being commanded, these currents are seen as a disturbance by the current regulator, which tries to compensate for them by adding terms to the voltage command, as shown in Fig. 9(a). Many of these components can also be seen in the output of the inverter, as shown in Fig. 9(b). Since these terms are similar to those modeled by (11), they are a significant source of distortion. A second mechanism for interference between the current regulator and the carrier signal is the fast transients in the current regulator output voltage due to torque (or current) steplike commands [9]. The frequency content of those transients is spread over a wide range of frequencies. A portion of these coincides with the negative-sequence carrier frequency and is, therefore, a distortion of the form shown by (11). The easiest and most effective way to reduce the current regulator interference with the carrier signals is to ensure spectral separation between the carrier frequency and the current regulator bandwidth, i.e., increase the carrier frequency or reduce the current regulator bandwidth. In general, reducing

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Fig. 11. Spectrum of the inverter output voltage and the stator current at frequencies near the negative-sequence carrier frequency for (left) symmetric and (right) asymmetric regularly sampled PWM. Two values of carrier frequency were used. (a) ωc = 500 Hz. (b) ωc = 5000 Hz. The motor was operated at rated flux and 66% of rated load. The current regulator was disabled, and the carrier-signal voltage magnitude was Vc = 20 V (peak).

the current regulator bandwidth is not desirable since it also reduces the overall performance of the drive. Increasing the carrier frequency also has its limitations, as was discussed in Section IV. Band-rejection filters in the current feedback path (see Fig. 1) can also be used to reduce the magnitude of the carrier-signal components reaching the current regulator. While the design of such filters is relatively straightforward for the positivesequence carrier-signal current component since it is at a constant well-defined frequency, it is much more complicated for the case of the negative-sequence carrier-signal current, which consists of several components [see Fig. 2(a)] with variable frequency. More sophisticated filtering methods can be used but at a price of increased complexity [9]. C. Distortion Caused by the PWM Strategy The modulation strategy can also strongly influence the quality of the injected carrier-signal voltage. A sine-triangle-type modulation is used to illustrate this in the following discussion. All of the ideas and conclusions presented are also applicable to space vector modulation. Symmetric regular sampling is one of the most common forms of PWM. This process is illustrated in Fig. 10(a). In the regular sampling method, voltage commands are updated once each switching period. When a carrier-signal voltage is

injected, it is added to the fundamental voltage command from the current regulator before being modulated (see Fig. 1). Asymmetric regularly sampled PWM can be used as an alternative. With this method, the voltage command is updated each half-cycle of the inverter, i.e., in each peak of the triangle waveform. This is schematically shown in Fig. 10(b). Fig. 11 shows the inverter output voltage and stator current spectrums for frequencies near the negative-sequence carrier frequency, using both symmetric (left) and asymmetric (right) regularly sampled PWM with carrier frequencies ωc equal to 500 and 5000 Hz (1/30 and 1/3 of the switching frequency, respectively). Symmetric regular sampling produces a component at −ωc − ωe that becomes significant as the carrier frequency increases [Fig. 11(b)]. All of the results shown so far have used a symmetric triangle wave for the PWM (see Fig. 10, bottom), an asymmetric triangle wave (sawtooth) could also be used. Although not presented here, an asymmetric triangle wave increases the magnitude of the component at −ωc − ωe in the injected carrier-signal voltage more than that produced using a symmetric wave. Fig. 12 summarizes the effect that the modulation strategy has on the THD of the injected carrier-signal voltage. It can be observed that for the case of the symmetric regular sampling method, the THD increases significantly with the carrier frequency. The component at −ωc − ωe is responsible for this behavior. This component also causes the large THD observed


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Fig. 13. Estimated magnitude of the () negative-sequence carrier-signal current and () zero-sequence carrier-signal voltage, as a percentage of the A/D converter’s full-scale value versus the carrier frequency, with a carriersignal voltage magnitude equal to Vc = 20 V (peak). Fig. 12. Experimentally measured distortion of the injected carrier-signal voltage as a function of the carrier frequency for a carrier-signal voltage magnitude of Vc = 20 V (peak) and using() symmetric regular sampling, () asymmetric regular sampling, and () sawtooth PWM. The motor was operated at rated flux and rated load. A bandwidth bw = 100 Hz was used.

for the sawtooth PWM method. Using Figs. 8 and 12 together, the effect that the carrier-signal voltage distortion has on the carrier signals of interest can be estimated. D. Distortion Caused by the Nonideal Behavior of the Inverter In the previous sections, it was demonstrated that the current regulator and the modulation strategy can produce distortion in the injected carrier-signal voltage. However, even if these sources of distortion are eliminated, distortion of the injected voltage is still observed. The nonlinear nature of the inverter has been reported as another potential source for this distortion [11]. The presence of this distortion can be observed in Fig. 11(a). In the figure, the current regulator was disabled, i.e., no reaction of the regulator to the presence of the carrier-signal current exists, and the inverter modulation strategy did not cause a significant distortion (symmetric and asymmetric regularly sampled spectrums are similar). The observed components can therefore be attributed to nonideal behavior of the inverter. Such components have systematically been observed to be at frequencies of the form −ωc + 2kωe , k = 1, −2, 4, −5, . . .. In spite of their relatively small magnitudes, these components can cause a serious deterioration of the negative-sequence carriersignal current, as explained in Section V-A. No methods have been proposed that totally eliminate these components, but they can be reduced by limiting the amount of deadtime and through careful design of the inverter switches and bus structure. VI. C ARRIER -S IGNAL M EASUREMENT AND P ROCESSING The sampling and processing of the carrier signals for estimating the rotor position can also influence their quality, i.e., THD. The effects of these are analyzed in this section. A. Sensor and A/D Converter Resolution Phase current measurement for control and protection purposes is a standard feature of most electric machine drives. No additional sensors, cabling, and A/D converters are therefore needed to measure the negative-sequence carrier-signal current. However, since the current sensors and A/D converters are

scaled to measure the maximum fundamental current magnitude, this means the full range of the sensors and A/D converters cannot be fully utilized for measuring the negative-sequence carrier-signal current. The zero-sequence carrier-signal voltage, on the other hand, requires access to the machine’s terminals and neutral point, with additional sensors, cabling, and A/D converters required when compared to a conventional drive. This is a significant drawback for this method in most applications. Using additional sensor(s) and A/D converter(s) has its advantages, however, since it allows the full range of the sensor(s) and A/D converter(s) to be used for sampling the zero-sequence voltage and improves the signal-to-noise ratio and how the zero-sequence voltage is measured. In this paper, an auxiliary balanced resistor network was used, which removes a major portion of the zerosequence voltage created by the normal inverter operation [6]. The zero-sequence voltage sensor and amplifier were scaled to be 100% of the A/D. For the experiments shown in this paper, the current sensors outputs were scaled to be 66% of the A/D converter input voltage limit with rated current in the motor. This allows 150% overcurrent in the drive before actually saturating the A/D converters. Scaling the zero-sequence voltage sensor and A/D converter is not as straightforward, since it strongly depends on converter input voltage limit with 100 V at the zero-sequence voltage sensor input. Fig. 13 shows the magnitude of the rotor-position-related components of the carrier signals as a percentage of the corresponding A/D converter’s full-scale value using the above-described scaling. Two conclusions can be observed from the figure: 1) the zerosequence carrier-signal voltage has better scaling; and 2) the negative-sequence carrier-signal current scaling gets worse as the carrier frequency increases. To evaluate the influence of the number of bits in the A/D converter, the signals were sampled using 16-bit A/D converters and then digitally processed to analyze the effects that result from a reduction in the number of bits. Fig. 14 shows the THD of the negative-sequence carrier-signal current and zerosequence carrier-signal voltage for different carrier frequencies and a simulated 8, 10, and 12 bits in the A/D converters. The THD for the case of 16 bits was very similar to the case of 12 bits and is not shown in the figure. It can be observed from the figure that with 8 bits of resolution, the THD of the negative-sequence carrier-signal current is too large to allow stable sensorless control, i.e., the THD is greater than the 0.5 threshold. Fortunately, modern electric machine drives typically

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Fig. 14. Experimentally measured THD of the negative-sequence carriersignal current and the zero-sequence carrier-signal voltage with postprocessing to simulate () 8 bits, () 10 bits, and () 12 bits in the A/D converter as a function of the carrier frequency. A carrier-signal voltage of Vc = 20 V (peak) was used, and the motor was operated at rated flux and load (ωe = 4 Hz and ωr = 1 Hz).

Fig. 15. Experimentally measured THD of the negative-sequence carriersignal current and the zero-sequence carrier-signal voltage as a function of the carrier frequency for bandpass filter bandwidths of () 50 Hz, () 100 Hz, and () 200 Hz. A carrier-signal voltage of Vc = 20 V (peak) was used, and the motor was operated at rated flux and load (ωe = 4 Hz and ωr = 1 Hz).

have A/D converters for the current measurement with 10–12 bits. It can be observed from the figure that, although a small difference exists, 10- and 12-bit converters have practically the same THD. This THD stays below the 0.5 threshold for carrier frequencies less than ∼2500 Hz, which implies stable sensorless control for the conditions of the test. The THD of the zero-sequence carrier-signal voltage, on the other hand, shows almost no variation with the number of converter bits and is always significantly below the stable threshold for the range of carrier frequencies analyzed. B. Filter Bandwidth As stated previously, bandpass filtering is an effective means of separating the carrier signal containing the saliency-related information from the measured signals (see Fig. 3). The bandwidth of this filtering directly limits the estimation bandwidth. Large bandwidths not only result in better dynamics but also give rise to a larger THD since more undesired content is allowed to pass through. On the other hand, reduced bandwidths reduce the THD but have the unwanted effect of reducing the estimation bandwidth. The THD of the negative-sequence carrier-signal current and the zero-sequence carrier voltage is shown for different bandwidths in Fig. 15. It can be observed from the figure that bandwidths in the range of hundred hertz can be used for the case of the zero-sequence carrier-signal voltage, whereas more modest values are required for the case of the negative-sequence carrier-signal current to achieve a desired level of distortion (THD). The reduced THD of the zero-sequence carrier-signal voltage, as compared to the negative-sequence carrier-signal current, that is observed in Figs. 14 and 15 can be attributed to two reasons: 1) the reduced sensitivity of the zero-sequence carriersignal voltage to carrier-signal voltage distortion and 2) the better scaling of the sensor and A/D converter used to measure the zero-sequence carrier-signal voltage. VII. E XPERIMENTAL R ESULTS Sensorless position control was implemented using both the zero-sequence carrier-signal voltage and the negative-sequence carrier signal. A carrier signal of ωc = 1500 Hz and Vc = 20 V

Fig. 16. Sensorless position control using the zero-sequence carrier-signal voltage when a position step from 0◦ to 180◦ is commanded. A carrier signal of Vc = 20 V and ωc = 1500 Hz was used. The filters and PLL were tuned for a 200-Hz bandwidth in the estimated position. The machine was operated at rated flux and 80% of rated load.

(peak) was used. Saturation-induced components at ωc + 2ωe and ωc − 4ωe were decoupled. These components were measured during an offline commissioning process and stored in a lookup table, which was later accessed for real-time compensation [6], [7]. Other compensation methods could also be used with similar results [11], [12]. A phase-locked loop (PLL) was used to estimate the rotor position from the carrier signals (see Fig. 3). Fig. 16 shows the estimated and measured rotor positions and the estimation error when the machine was operated using the zero-sequence carrier-signal voltage, with the bandpass filters and the PLL tuned for a 200-Hz bandwidth. Fig. 17(a) and (b) shows the estimated and measured rotor positions and the estimation error for the case of sensorless control using the negative-sequence carrier-signal current. The large errors observed between the measured and estimated positions are actually caused by jumps to the next rotor bar in the estimated position. Since the estimated position was used for control, these jumps cause the measured position to deviate from the desired trajectory. It is interesting to note that there is a jump at time t = 0.1 s, which corresponds to when the torque (q-axis current) command was applied. For that case, high-frequency harmonics due to the fast transients in the q-axis current can contribute significantly to the overall distortion of the negative-sequence carrier-signal current, which


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Fig. 17. (a) and (b) Sensorless position control using the negative-sequence carrier-signal current for the same conditions as Fig. 16. (c) Estimated position error with the filters and PLL tuned for a bandwidth of 100 Hz.

increases the risk of instability. The stability of the sensorless control can be increased by reducing the bandwidth of both the bandpass filters and PLL. This is shown in Fig. 17(c). Another effective method for improving the stability of the sensorless control, which is not included in the results presented in this paper, is the use of a fundamental current observer, which can be used to decouple the stator current transients from the negative-sequence carrier-signal current [9]. VIII. C ONCLUSION Accuracy and bandwidth limits of carrier-signal voltage injection-based sensorless control methods have been explored in this paper. When a carrier-signal voltage is injected into a salient machine, two different types of signals contain saliency-position-related information: 1) the negative-sequence carrier-signal current and 2) the zero-sequence carrier-signal components. The THD has been shown to be an effective metric for quantifying the quality of these signals since a deterministic relationship between the THD and the estimation error, including the stability of the sensorless control, exists. Sources that cause distortion of the carrier signals have been identified and analyzed. It has also been demonstrated that the zerosequence carrier-signal voltage is less sensitive to the potential sources of distortion coming from the injection of the carriersignal voltage. This provides the following benefits: 1) the use of larger carrier frequencies; 2) larger estimation bandwidths; and 3) increased accuracy in the estimated position. On the other hand, the use of zero-sequence components requires one additional sensor, additional cabling, and access to the motor terminal box. ACKNOWLEDGMENT The authors would like to thank the University of Oviedo, Spain, and the Ford Motor Company for providing support and motivation.

[1] 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. [2] J. H. Jang, J. I. Ha, M. Ohto, K. Ide, and S. K. Sul, “Analysis of permanentmagnet machine for sensorless controls based on high-frequency signal injection,” IEEE Trans. Ind. Appl., vol. 40, no. 6, pp. 1595–1604, Nov./Dec. 2004. [3] 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. [4] M. Schroedl, “Sensorless control of AC machines at low speed and standstill based on the inform method,” in Proc. IEEE IAS Annu. Meeting, San Diego, CA, Oct. 1996, vol. 1, pp. 270–277. [5] 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. 1107–1115, Jul./Aug. 2004. [6] F. Briz, M. W. Degner, P. García, and J. M. Guerrero, “Rotor position estimation of AC machines using the zero sequence carrier signal voltage,” IEEE Trans. Ind. Appl., vol. 41, no. 6, pp. 1637–1646, Nov./Dec. 2005. [7] F. Briz, M. W. Degner, P. García, and A. B. Diez, “Rotor and flux position estimation in delta-connected AC machines using the zero sequence carrier signal current,” IEEE Trans. Ind. Appl., vol. 42, no. 2, pp. 495–503, Mar./Apr. 2006. [8] 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. [9] F. Briz, M. W. Degner, and A. B. Diez, “Dynamic operation of carrier signal injection based, sensorless, direct field controlled AC drives,” IEEE Trans. Ind. Appl., vol. 36, no. 5, pp. 1360–1368, Sep./Oct. 2000. [10] 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. [11] N. Teske, G. M. Asher, K. Sumner, and K. J. Bradley, “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. [12] P. García, F. Briz, D. Raca, and R. D. Lorenz, “Saliency tracking-based, sensorless control of AC machines using structured neural networks,” IEEE Trans. Ind. Appl., vol. 43, no. 1, pp. 77–86, Jan./Feb. 2007. [13] M. W. Degner and R. D. Lorenz, “Using multiple saliencies for the estimation of flux, position, and velocity in AC machines,” IEEE Trans. Ind. Appl., vol. 34, no. 5, pp. 1097–1104, Sep./Oct. 1998.

Pablo García (S’02–M’06) was born in Spain in 1975. He received the M.E. and Ph.D. degrees in electrical engineering from the University of Oviedo, Gijón, Spain, in 2001 and 2006, respectively. From 2001 to 2006, he was awarded a fellowship of the Personnel Research Training Program funded by the Spanish Ministry of Science and Technology. He was a Visitor Scholar at the Wisconsin Electric Machines and Power Electronics Consortium, University of Wisconsin, Madison, in 2004. He is currently with the Department of Electrical Engineering Department, University of Oviedo. His research interests include sensorless control and diagnosis of synchronous and asynchronous machines, adaptive parameter estimation, and digital signal processing.

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Fernando Briz (A’96–M’99–SM’06) 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 was a recipient of the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS Third Place Prize Paper Award in 2005 and of two IEEE Industry Applications Society Conference Prize Paper Awards in 1997 and 2004.

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, Hybrid Electric Vehicle and Fuel Cell Vehicle Laboratory, Research and Advanced Engineering, Ford Motor Company, Dearborn, MI, where he is responsible for the development of electric machines, power electronics, 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 a recipient of the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS Third Place Prize Paper Award in 2005 and of several IEEE Industry Applications Society Conference Paper Awards.

David Díaz-Reigosa was born in Spain in 1979. He received the M.E. degree in industrial engineering from the University of Oviedo, Gijón, Spain, in 2003. He is currently working toward the Ph.D. degree in electrical engineering at the same university. In 2003, he was awarded a fellowship of the Personnel Research Training Program funded by the Regional Ministry of Education and Science of the Principality of Asturias. His research interests include sensorless control of induction motors, permanent-magnet synchronous motors, and digital signal processing.