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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 50, NO. 6, NOVEMBER/DECEMBER 2014

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Application of Linear-Phase Filters in Induction Motor Speed Detection Zhi Gao, Senior Member, IEEE, Larry Turner, Senior Member, IEEE, and Roy S. Colby, Senior Member, IEEE

Abstract—In rotor-slot-harmonic-based speed detection schemes, a grid-connected induction motor’s rotational speed is resolved from the phase of a rotor slot harmonic signal for condition monitoring purposes. In such schemes, frequencyselective digital filters are often applied to attenuate interferences in sampled motor currents before rotor slot harmonic signals are extracted. This paper demonstrates that the use of linear-phase filters, which is a family of digital filters with linear phase responses, helps preserve the rotor slot harmonic signal’s phase during the filtering operation. This paper also shows that, for grid-connected induction motors with periodically time-varying loads, the rotor slot harmonic signal’s frequency is modulated by periodic speed oscillations and hence contains multiple sidebands in the frequency spectrum. Experimental results further show that linear-phase filters allow for a more accurate detection of the induction motor speed than filters with nonlinear phase responses. Index Terms—Condition monitoring, digital filters, frequency modulation (FM), group delay, induction motor, linear phase, rotor slot harmonics, speed detection.

N OMENCLATURE ia , ib , ic iC (t), iC [n] IC (ejω ) ish (t),ish [n] Ish (ejω ) θsh (t),θsh [n]

fsh (t),fsh [n] Jk (·)

Phase a, b, and c current measurements, respectively. Complex current vector in the continuous- and discrete-time domains, respectively. Frequency spectrum of iC [n]. Rotor slot harmonic signal in the continuousand discrete-time domains, respectively. Frequency spectrum of ish [n]. Rotor slot harmonic signal’s phase in the continuous- and discrete-time domains, respectively. Rotor slot harmonic frequency in the continuousand discrete-time domains, respectively. kth-order Bessel function of the first kind.

Manuscript received April 10, 2013; revised July 17, 2013; accepted September 2, 2013. Date of publication August 8, 2014; date of current version November 18, 2014. Paper 2013-EMC-186.R1, presented at the 2012 IEEE Energy Conversion Congress and Exposition, Raleigh, NC, USA, September 15–20, and approved for publication in the IEEE T RANSACTIONS ON I NDUSTRY A PPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications Society. Z. Gao was with Strategy and Innovation, Schneider Electric, Knightdale, NC 27545 USA. He is currently with Eaton Corporation, Moon Township, PA 15108 USA (e-mail: zhigao@ieee.org). L. Turner and R. S. Colby are with Strategy and Innovation, Schneider Electric, Knightdale, NC 27545 USA (e-mail: larry.turner@us.schneiderelectric.com; steve.colby@us.schneider-electric.com). 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.2014.2346710

β δ(·) BW s s0 s1 ωm f0 R P nW h[n] H(ejω ) A(ω) φ(ω) M τg

Frequency modulation (FM) index. Dirac delta function. Filter bandwidth. Induction motor per-unit slip. Slip corresponding to the average motor speed. Slip corresponding to the amplitude of the first speed harmonic. Motor speed oscillation’s angular frequency. Supply frequency. Number of rotor slots (bars). Number of pole pairs. Odd integer with possible value of ±1, ±3, . . . Filter impulse response in the discrete-time domain. Frequency response of h[n]. Amplitude response of h[n]. Phase response of h[n]. Order of a linear-phase finite impulse response (FIR) filter (positive even integer). Filter time delay. Also known as filter group delay. I. I NTRODUCTION

W

ITH THE advent of microprocessors that are optimized with fast digital signal processing capabilities and low power consumptions for industrial and commercial applications, more and more induction motor condition monitoring and diagnosis tasks are executed on microprocessor-based hardware platforms [1], [2]. Frequency-selective digital filters are widely used on such hardware platforms. By performing filtering operations on discrete-time current or voltage signals, those digital filters enable sophisticated algorithms that extract operatingcondition-related information efficiently at a reduced cost for motor condition monitoring and diagnosis applications, such as in [3]. The digital filters are nothing more than a set of predetermined coefficients stored in the microprocessor’s memory, and they are rarely subject to component nonlinearities or imperfections commonly found in their analog counterparts, which are made from electronic components with temperatureor input-dependent properties. In rotor-slot-harmonic-based speed detection schemes, the grid-connected squirrel-cage induction motor’s rotational speed is estimated from the rotor slot harmonics for monitoring purposes [4]–[11]. Digital filters are frequently used to extract rotor slot harmonics from sampled motor currents before the motor speed is resolved [10]. An accurate estimate of the motor speed provides critical information to many motor-current-based

0093-9994 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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monitoring or control applications, such as nonintrusive efficiency estimation for in-service induction motors [12] and realtime monitoring and protection against rotor overheating [13]. The design of frequency-selective digital filters involves choosing filters that meet a set of specified criteria. A major part of the design effort is focused on selecting filters with desired amplitude responses. However, given a digital filter with an appropriate amplitude response, it may still distort motor speed estimates if it has a considerably nonlinear phase response. Linear-phase digital filters are introduced and analyzed in this paper for rotor-slot-harmonic-based motor speed detection schemes. Compared to digital filters with nonlinear phase responses, linear-phase filters do not produce phase distortions in the extracted rotor slot harmonic signals and hence allow for an accurate estimate of the motor speed. This family of filters may also be used in other applications, such as precise motor electrical parameter estimations in motor drives [14], [15]. The time delay introduced by the linear-phase filter is further analyzed and quantified. The analysis provides a basis for latency compensation, in which various signals are temporally synchronized in motor condition monitoring or control tasks, such as in high-performance motor drive systems [16], [17]. Another major contribution of this paper is an empirical model that characterizes the rotor slot harmonic behavior for induction motors with periodically time-varying speed profiles. The characterization is formulated on the basis of an FM concept and hence enables a straightforward interpretation of multiple rotor slot harmonic sidebands that are found in the motor current’s frequency spectrum. The rotor slot harmonic sidebands are directly related to the induction motor’s periodic speed oscillations. This paper is organized as follows. Section II uses a complex vector notation to empirically characterize the behavior of rotor slot harmonics in grid-connected squirrel-cage induction motors with periodically time-varying speed profiles. Following this discussion, linear-phase filters are introduced in Section III to extract rotor slot harmonics for the purpose of speed detection and condition monitoring in grid-connected motors. Filter design and implementation techniques are also given in this section. Experimental results that validate the use of linearphase filters for grid-connected motors are given in Section IV, and the conclusions are stated in Section V. II. F REQUENCY M ODULATIONS IN ROTOR S LOT H ARMONICS In squirrel-cage induction motors, rotor slot harmonics are caused primarily by discrete conductor bars in the rotor slots when excited by a flux wave at the supply frequency f0 (in hertz) from the stator [18]–[22]. The rotor slot harmonic frequency provides critical information for the detection of an induction motor’s rotational speed [4]–[11]. For grid-connected induction motors that are coupled to periodically time-varying loads, such as reciprocating compressors [23], load torque and speed oscillations lead to modulations in the rotor slot harmonic frequency. This section characterizes the FMs found in the rotor slot harmonics using a complex vector notation.

Fig. 1. Frequency spectrum of a complex current vector from a 20-hp 4-pole induction motor with 40 rotor slots. (a) Frequency band between 735 and 795 Hz. (b) Frequency band between 1095 and 1155 Hz.

A. Rotor Slot Harmonics in Complex Current Vectors The complex current vector offers a convenient way to visualize and analyze FMs in rotor slot harmonics. The relationship between the motor speed and rotor slot harmonic frequency is also outlined for subsequent discussions on FMs in rotor slot harmonics. 1) Complex Current Vector: For a polyphase induction motor with a floating neutral point, given its phase a, b, and c current measurements ia , ib , and ic , respectively, the complex current vector iC (in amperes) is defined as iC =

2 (ia + α · ib + α2 · ic ) 3

(1)

where α = ej·2π/3 , where j is the imaginary unit. In case currents from only two phases are measured, iC may be calculated from (1) by noting that ia + ib + ic = 0 [11], [24]. For a grid-connected squirrel-cage polyphase induction motor, the complex current vector constructed according to (1) contains not only a fundamental current component at the supply frequency but also other motor-operating-condition-related information, such as rotor slot harmonics. Fig. 1(a) shows a segment of a complex current vector’s frequency spectrum. The

GAO et al.: APPLICATION OF LINEAR-PHASE FILTERS IN INDUCTION MOTOR SPEED DETECTION

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TABLE I PARAMETERS OF THE 20-hp T EST M OTOR

Fig. 2. Complex vector diagram of a single rotor slot harmonic signal.

motor is a 20-hp 4-pole induction motor with 40 rotor slots. It is connected to a power grid with f0 = 60 Hz when the complex current vector is acquired. Complete motor ratings and parameters are listed in Table I. During its operation, the motor experiences a roughly 5-Hz periodically time-varying load torque oscillation and hence leads to a 5-Hz oscillation in the motor speed. In Fig. 1(a), frequency components associated with the rotor slot harmonics are marked by “•” symbols. In this figure, the rotor slot harmonic component at 761 Hz has the largest amplitude. Two pairs of sidebands are also observed in Fig. 1(a): The first pair of sidebands is located at 761 ± 5 Hz, while the second pair is located at 761 ± 10 Hz. Fig. 1(b) shows another segment of the same complex current vector’s frequency spectrum between 1095 and 1155 Hz. The rotor slot harmonic component and its major sidebands that are centered on 1121 Hz demonstrate a similar distribution when compared to their counterparts in Fig. 1(a). As will be explained in Section II-B, those rotor slot harmonic sidebands are directly related to the 5-Hz speed oscillation. 2) Rotor Slot Harmonic Frequency: According to the authors of [8], [11], and [19]–[22], for a typical grid-connected squirrelcage induction motor fed by a balanced sinusoidal supply at a fixed supply frequency f0 , its rotor slot harmonic frequency, fsh (in hertz), is correlated to the motor’s per-unit slip, s, via 1−s fsh = ±f0 · R · + nW (2) P where the use of the “±” sign depends on the location of the rotor slot harmonic in the frequency spectrum. A “+” sign corresponds to a rotor slot harmonic with a positive frequency, while a “−” sign corresponds to a rotor slot harmonic with a negative frequency [11]. In (2), R is the number of rotor slots, P is the number of pole pairs, and nW = ±1, ±3, . . ., is an odd integer [8], [11]. Depending on the motor design, the principle slot harmonic, i.e., the rotor slot harmonic typically with the largest amplitude, corresponds to either nW = +1 or nW = −1 [19], [20]. For example, in Fig. 1(a), the test motor’s rotor slot harmonic spectral components are centered on 761 Hz; hence, the “+” sign is used in front of f0 in (2). In addition, according to Table I, R = 40, and P = 2; hence, nW = −7. Likewise, in Fig. 1(b), the rotor slot harmonic spectral components are centered on 1121 Hz; hence, the “+” sign is used in front of f0 , and nW = −1.

The rotor slot harmonics are determined by the number of pole pairs P , the number of rotor slots R, and the relationship between them [11]. Certain combinations of P and R may not yield observable rotor slot harmonics [19], [20]. However, the number of rotor slots is usually chosen by motor designers based on certain conventions to achieve the desired motor performance, and there are a limited number of choices for R in practice [22]. Hence, the rotor slot harmonics are present for a large class of motors. For the sake of discussion, only rotor slot harmonics with positive frequencies are considered in the following sections. Analysis and discussions made in this paper may be easily extended to rotor slot harmonics with negative frequencies. B. Frequency Modulations in Rotor Slot Harmonics The current vector defined in (1) is a complex quantity that varies over time t. Using this notation, a current vector that only contains a single rotor slot harmonic signal may be expressed by a time-varying complex quantity iC (t) = Ash (t) · ej·θsh (t)

(3)

where Ash (t) (in amperes) and θsh (t) (in radians) are the rotor slot harmonic signal’s amplitude and phase, respectively. Fig. 2 shows a complex current vector iC (t) that has a positive fsh in a right-handed coordinate system. 1) Instantaneous Rotor Slot Harmonic Frequency: For induction motors that drive periodically time-varying loads like reciprocating compressors, they often experience periodic motor speed oscillations [14], [23]. In this case, a first-order approximation of the motor’s per-unit slip is s(t) = s0 − s1 · cos(ωm t)

(4)

where s0 is the slip that corresponds to the average motor speed, s1 is the slip that corresponds to the amplitude of the first speed harmonic, and ωm (in radians per second) is the angular frequency of the motor speed oscillation. With a 5-Hz motor speed oscillation, ωm = 2π × 5 ≈ 31.4 rad/s. Substituting (4) into (2), the rotor slot harmonic frequency becomes 1−s0 R +nW + · s1 · f0 · cos(ωm t). (5) fsh (t) = f0 · R · P P

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Fig. 4. Amplitude spectrum of a single-tone frequency-modulated complex current vector (k = −3 to 3). Fig. 3. Bessel functions of the first kind for k = 0, 1, 2, and 3.

2) Frequency Modulation: It is useful to choose a starting time such that the rotor slot harmonic signal’s phase is zero at t = 0. In this case, θsh (t) is obtained by integrating fsh (t) in (5) t θsh (t) = 2π ·

fsh (τ ) dτ = ωc t + β sin(ωm t)

(6)

0

where ωc = 2πf0 · [(1 − s0 ) · R/P + nW ] and β = 2πf0 · (R/P ) · (s1 /ωm ), as shown in Fig. 2. The rotor slot harmonic signal’s phase θsh (t) is a function of motor slip quantities s0 and s1 and hence carries critical information of the motor speed. For the sake of discussion, assuming a constant amplitude Ash in the rotor slot harmonic signal, the complex current vector iC (t) is characterized by a single-tone frequency-modulated signal after (6) is substituted into (3) iC (t) = Ash (t) · ejωc t · g(t)

(7)

where g(t) = ej·β sin(ωm t) . The motor-slip-dependent quantity β is referred to as the frequency modulation index, and β sin(ωm t) is a modulating signal in g(t). In a discrete-time system sampled at fS (in hertz), the nth sample of the complex current vector is obtained by replacing t in (7) with n/fS

iC [n] = Ash · ejωc n · g[n]

(8)

= ωm /fS . where g[n] = ej·β sin(ωm n) , ωc = ωc /fS , and ωm 3) Rotor Slot Harmonic Frequency Spectrum: The rotor slot harmonic signal’s frequency spectrum may be obtained by first considering the discrete-time Fourier transform of g[n] and then applying the Fourier transform’s frequency-shift property to obtain the final result. To start with, note that the expression g[n] in (8) is periodic . Using the Jacobi–Anger expansion, the Fourier series with ωm of g[n] is

g[n] =

∞

Jk (β) · ej·kωm n

(9)

k=−∞

where Jk (β) is the kth-order Bessel function of the first kind with an argument β. Note that J−k (β) = (−1)k · Jk (β) [25]. Fig. 3 plots the Bessel function of the first kind for k = 0 to 3 with varying β. Given a small frequency modulation index β, i.e., β ≤ 1, high-order Bessel functions are usually much smaller than their low-order counterparts according to Fig. 3.

Applying the Fourier transform to the g[n] in (9) yields ∞ ∞ G(ejω ) = 2π δ (ω − kωm + 2πl) (10) Jk (β) · k=−∞

l=−∞

where δ(·) is the Dirac delta function. The frequency spectrum of the complex current vector iC [n] in (8) is denoted as IC (ejω ). According to the Fourier transform’s frequency-shift property [26], IC (ejω ) is G(ejω ) shifted by ωc in frequency and scaled by Ash in amplitude, as shown in ∞ ∞ jω IC (e ) = 2πAsh δ(ω−ωc −kωm +2πl) . Jk (β) · k=−∞

l=−∞

(11) Note that, because of the discrete-time Fourier transform’s periodicity property, IC (ejω ) is periodic with 2π. Hence, it is sufficient to consider the part of IC (ejω ) with angular frequency ω between −π and π, i.e., l = 0. Theoretically, the spectrum IC (ejω ) is centered on ωc (k = 0) and contains an infinite number of sidebands at frequencies (k = ±1, ±2, . . .) according to the empirical model ωc + kωm developed in Section II-B2). In practice, however, only a limited number of rotor slot harmonic sidebands can be observed by current measurement devices due to the devices’ finite resolutions. For example, only two pairs of sidebands are found in Fig. 1(a). In Fig. 1(b), the amplitude of the third upper sideband at 1136 Hz is too small to be plotted. Based on the discussion that follows (9), it is often sufficient to consider rotor slot harmonic components and their major sidebands in rotor-slotharmonic-based motor speed detection schemes. In this case, (11) is reduced to IC (ejω ) = 2πAsh

K2

[Jk (β) · δ (ω − ωc − kωm )]

(12)

k=K1

where the rotor slot harmonic components and their major sidebands are included between integers K1 and K2 inclusive. Fig. 4 shows a typical amplitude spectrum of a single-tone frequency-modulated signal with k between −3 and 3. The spectral components are functions of the frequency modulation index β. It is worth noting that, because β is a function of the motor slip, those spectral components are hence also functions of the motor slip. The complex current vector’s frequency spectrum shown in Fig. 1(a) corresponds to ωc = 2π × 761 rad/s, ωm = 2π × 5 rad/s, K1 = −2, and K2 = 2. Likewise, the frequency spectrum shown in Fig. 1(b) corresponds to ωc = 2π × 1121 rad/s, ωm = 2π × 5 rad/s, K1 = −3, and K2 = 2.


Fig. 5. Use of a linear-phase filter in a rotor-slot-harmonic-based motor speed detection scheme.

It is also worth noting that the sampling frequency fS is usually chosen to be sufficiently high so that −π < ωc + ≤ π for any k between K1 and K2 inclusive. For examkωm = 2π × (1121 + ple, in the second example above, ωc + kωm 5k)/fS . Given K1 = −3, and K2 = 2, a sampling frequency fS of greater or equal to 2 × (1121 + 5 × 2) = 2262 Hz must be selected to yield meaning results in Fig. 1(b). III. L INEAR -P HASE F ILTERS Following the discussions in Section II, the phase of a rotor slot harmonic signal carries critical information of the motor speed. By differentiating this phase quantity, as shown in the Appendix, the induction motor’s rotational speed can be resolved from the rotor slot harmonics [11]. In practice, the amplitude of a rotor slot harmonic signal is far smaller than the fundamental current component. Furthermore, as shown in Fig. 1(a), rotor slot harmonic components are present along with harmonics that are at integer multiples of f0 [6], [8]. Due to the presence of harmonics of f0 and the fundamental current component, the motor speed information cannot be directly computed from the complex current vector. Frequency-selective digital filters are often applied to suppress the harmonics of f0 , the fundamental current component, and other interferences before the motor speed can be resolved from the filtered rotor slot harmonic signal. Fig. 5 shows the use of a digital filter in a rotor-slotharmonic-based motor speed detection scheme. It applies a frequency-selective digital filter, i.e., the first block in Fig. 5, to a complex current vector iC [n] constructed from the sampled motor currents and produces a rotor slot harmonic signal ish [n] in the discrete-time domain. A frequency demodulator then resolves an instantaneous rotor slot harmonic frequency fsh [n] (in hertz) from ish [n]. Finally, a rotor speed detector derives the motor’s per-unit slip s[n] from fsh [n] on a sample-by-sample basis. The design and implementation of the frequency demodulator and the rotor speed detector are straightforward and hence are given in the Appendix. The design and realization of appropriate frequency-selective digital filters, however, are not trivial tasks. The filter has to retain the desired rotor slot harmonic component and its major sidebands while suppressing interferences. In addition, because the motor speed is ultimately resolved from the rotor slot harmonic signal’s phase, the filter has to preserve the signal’s phase. The phase distortion introduced to ish [n] by filtering may translate to errors in the subsequent motor speed estimates. Therefore, particular attention must be paid toward the filter’s phase response, structure, and realization. Linear-phase filters, a family of filters with linear phase response and, hence, constant time delay in the passband, are chosen as a preferred choice when implementing the first block

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in Fig. 5. Compared to filters with nonlinear phase responses, they ensure that no phase distortions are introduced when filtering iC [n] to produce the rotor slot harmonic signal ish [n]. This section explains a linear-phase filter’s ability to retain phase information during filtering and provides discussions on the filter’s structure and implementation. A. Digital Filters With Linear Phase Responses Given a positive even integer M , a typical linear-phase filter used in Fig. 5 takes the form of a symmetric FIR system [26] h[M/2 − k] = h[M/2 + k],

0 ≤ k ≤ M/2.

(13)

The FIR system’s frequency response H(ejω ), according to the discrete-time Fourier transform, is H(ejω ) =

M

h[n] · e−jωn

n=0 M/2−1

=

n=0

+

h[n] · e−jωn + h[M/2] · e−jωM/2 M

h[n] · e−jωn .

(14)

n=M/2+1

Equation (14) may be rewritten as

M/2 jω

H(e ) =

h[M/2 − k] · e−jω(M/2−k) + h[M/2]

k=1

M/2

· e−jωM/2 +

h[M/2 + k] · e−jω(M/2+k) .

(15)

k=1

The first and the last term on the right side of (15) may be consolidated if the FIR system’s symmetry (13) is considered H(ejω ) = A(ω)·ejφ(ω) −jωM/2 = h[M/2]·e ⎧ ⎫ M/2 ⎨ ⎬ + h[M/2−k]·(ejωk +e−jωk) ·e−jωM/2 . (16) ⎩ ⎭ k=1

In (16), the FIR system’s frequency response may be split M/2 into two parts: an amplitude response A(ω) = h[M/2]+ k=1 h[(M/2)−k]·2 cos(ωk) and a phase response φ(ω) = −ωM/2. Given a fixed M in (16), the FIR filter’s phase response φ(ω) varies linearly with respect to the angular frequency ω. Fig. 6 shows the frequency response of a type I linear-phase FIR filter with a passband between 1106 and 1136 Hz. The filter has a bandwidth of BW = 30 Hz. As indicated in Fig. 6(b), the filter’s phase response varies linearly with respect to the frequency within its passband. The discontinuities beyond the filter’s passband are caused by a phase wrapping effect between 0 and 2π. Because frequency components beyond the filter’s passband are sufficiently attenuated, such phase wrapping effect has negligible influence on the filter’s output. When the linear-phase filter shown in Fig. 6 is used in the rotor-slot-harmonic-based motor speed detection scheme (see Fig. 5), the filter’s flat amplitude response within the passband

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B. Design Considerations of Linear-Phase FIR Filters

Fig. 6. Frequency response of a linear-phase filter. (a) Amplitude response. (b) Phase response.

[see Fig. 6(a)] allows the rotor slot harmonic component and its sidebands between 1106 and 1136 Hz to pass through without significant amplitude distortion. This helps attenuate the fundamental current component located at f0 = 60 Hz and the 19th harmonic at 19 · f0 = 1140 Hz, as well as other interferences in the complex current vector iC [n]. When the rotor slot harmonic component and its major sidebands (12) are within the linear-phase filter’s passband, the frequency spectrum of the rotor slot harmonic signal ish [n] at the filter’s output in Fig. 5, denoted as Ish (ejω ), is Ish (ejω ) = A(ω) · e−jωM/2 · IC (ejω ) K2 = Ish e−jωM/2 · [Jk (β) · δ(ω−ωc −kωm )] (17) k=K1

where Ish = 2πAsh · A(ω). Considering that A(ω) = 1 for frequency components within the linear-phase filter’s passband in Figs. 1(b) and 6(a), the coefficient e−jωM/2 translates (17) from the frequency domain to the discrete-time domain according to the Fourier transform’s time-shift property ish [n − M/2] = iC [n].

(18)

Considering only rotor slot harmonic components and their major sidebands, the phase of ish [n] in (18) is identical to the phase of iC [n] in (8), except for an (M/2)-sample time delay induced by the linear-phase filter, i.e., the filter entails an (M/2)-sample time delay between its input and output. All frequency components within the filter’s passband are uniformly delayed by M/2 samples. As a result, the filter does not introduce distortions to the rotor slot harmonic signal’s phase θsh [n], and the accuracy of the final estimated motor speed is hence guaranteed. In contrast, filters with nonlinear phase responses may produce rotor slot harmonic signals with distorted phases due to their frequency-dependent phase responses and consequently lead to filter-induced artifacts in the final estimated motor speed.

Linear-phase FIR filters are typically designed by first computing an impulse response from the desired frequency response using a least squares approach and then multiplying the impulse response with a finite-duration window to achieve design objectives [26], [27]. Carefully designed infinite impulse response (IIR) Bessel filters with approximately linear phase response in its passband may also be used in implementation. Compared to other IIR filters, the Bessel filter has a flat amplitude response in the passband, a slow rate of amplitude attenuation in the transition band, and no ringing in the stopband. The design and realization of IIR filters may be found in [26]. This section, however, highlights major issues when designing and implementing linear-phase FIR filters for rotor-slot-harmonicbased motor speed detection schemes. 1) Filter Bandwidth: The linear-phase filter has a rather narrow bandwidth because it needs to suppress interferences that are often found in the neighborhood of the rotor slot harmonics. Referring to (2), given an integer R/P , the rotor slot harmonic at steady-state motor operation is separated from the nearest harmonic of f0 by sf0 · R/P Hz. For the 20-hp test motor at rated condition, this translates to 23.33 Hz. When a motor experiences periodically time-varying torque and speed oscillations, the rotor slot harmonic sidebands may spread to places even closer to the harmonics of f0 . For example, in Fig. 1(a), the rotor slot harmonic sideband at 771 Hz is only 9 Hz away from the 13th harmonic at 780 Hz. As an empirical rule, it is often useful to choose the filter bandwidth BW using Carson’s rule when designing an appropriate linear-phase filter in the presence of narrow-band rotor slot harmonic FMs due to the motor’s time-varying loads [28]. A good choice of linear-phase filter’s bandwidth allows the extraction of rotor slot harmonics and their major sidebands in the presence of significant motor speed oscillations. In case of significant FMs in rotor slot harmonics due to large load torque variations, the filter’s bandwidth may be chosen to accommodate the rotor slot harmonic component and its most significant sidebands. In summary, a precise quantification of the bandwidth requires application-specific information and is contingent upon the amount of filter distortion allowed. 2) Filter Structure: In practice, the symmetry shown in (13) may be exploited to reduce the linear-phase FIR filter’s complexity in implementation. The convolution of the filter coefficients h[n] with the complex current vector iC [n] leads to ish [n] =

M

h[k] · iC [n − k]

k=0 M/2

=

h[M/2 − k] · iC [n − (M/2 − k)]

k=1

+ h[M/2] · iC [n − M/2] M/2 + h[M/2 + k] · iC [n − (M/2 + k)] k=1 M/2

=

k=1

h[M/2 − k] · {iC [n − (M/2 − k)]

+ iC [n−(M/2+k)]}+h[M/2] · iC [n−M/2]. (19)


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Fig. 7. Signal flow graph of a linear-phase FIR filter in a direct-form realization. (a) Conventional FIR filter with M + 1 coefficients. (b) Optimized FIR filter with M/2 + 1 coefficients.

According to (19), the number of filter coefficients in Fig. 7(a) can be halved [26]. The simplified filter structure is shown in Fig. 7(b) in light of (19). It is worth noting that z −1 in Fig. 7 denotes a one-sample delay in the discrete-time domain. Depending on the value of nW in (2), the rotor slot harmonic spectral components may be found at different frequency bands in the complex current vector’s frequency spectrum, as shown in Figs. 1(a) and (b). Correspondingly, the passband of a linearphase filter must be selected to match the desired rotor slot harmonic spectral components. Frequency shifting technique, such as the one used by the superheterodyne receivers in telecommunication radio broadcast applications, may also be used to avoid designing multiple bandpass filters in rotor-slotharmonics-based motor speed detection applications [11], [29]. IV. E XPERIMENTAL VALIDATION AND D ISCUSSIONS To validate that linear-phase filters allow for more accurate speed estimates than their nonlinear-phase counterparts in rotor-slot-harmonic-based motor speed detection schemes, experiments have been performed on a 20-hp 4-pole induction motor with 40 rotor slots. The motor is connected to a 60-Hz power grid. During the experiment, three-phase currents are collected from clamp-on current probes, and a data acquisition system samples the currents at a sampling frequency of 5 kHz. A dc machine is coupled to the induction motor as its load. It is controlled by a programmable logic controller through a dc motor drive to execute load profiles that emulate dynamic motor operations. The motor speed changes rapidly during the emulated dynamic motor operations. The motor’s speed is measured by a dc tachometer mounted on the motor shaft. Calibrations have been performed on the dc tachometer to determine the exact relationship between the motor speed and the dc output. The calibrations help compensate for thermal drifts and offset in the dc tachometer. A. Experimental Results Fig. 8(a) shows the motor speed detected during a 1-s interval after a linear-phase filter shown in Fig. 6 is applied to the complex current vector. For the sake of discussion, Fig. 8(b) shows

Fig. 8. Induction motor speed detected from a rotor slot harmonic signal for the 20-hp 4-pole induction motor with 40 rotor slots during dynamic motor operations. (a) Motor speed detected using a linear-phase FIR filter. (b) Motor speed detected using a nonlinear-phase Chebyshev filter. (c) Motor speed detected using a Bessel filter.

the motor speed detected from the same complex current vector using a type I Chebyshev filter in the discrete-time domain with a grossly nonlinear phase response. Figs. 9(a) and (b) give the amplitude and phase responses of this nonlinear-phase filter. As shown in Fig. 9(a), this type of filter has a fast rolloff in its transition band. Both the linear- and nonlinear-phase filters

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Fig. 9. Frequency response of a nonlinear-phase filter. (a) Amplitude response. (b) Phase response.

have the same cutoff frequencies within their passbands. As will be elaborated in Section IV-B, a first-in first-out queue is used to compensate for the linear-phase filter’s delay when plotting Fig. 8(a), while an approach similar to the one adopted in [16] is used to empirically compensate for the nonlinear-phase filter’s delay in Fig. 8(b). As shown in Fig. 8, different filter phase responses lead to dissimilar motor speed estimates. If an error metric is defined as the absolute difference between the speed estimated from the rotor slot harmonic and the speed measured from the tachometer output, then the average error in Fig. 8(a) is 0.28 r/min. In Fig. 8(b), this average error increases to 0.65 r/min. Generally speaking, in rotor-slot-harmonic-based induction motor speed detection schemes, the use of linear-phase filters helps preserve the shape of the filtered rotor slot harmonic signal and thus allows for a more accurate estimate of the induction motor speed on a sample-by-sample basis. It is worth noting that the linear-phase FIR filter used in the rotor-slot-harmonic-based motor speed detection scheme has a relatively large M associated with it; therefore, it usually has a start-up transient caused by a zero initial condition in the filter’s internal states. To address this issue, one solution is to discard the filter’s output that corresponds to its start-up transient. B. Further Discussions The linear-phase FIR filter’s relatively large M allows for a fast rolloff in its transition band, i.e., the filter’s amplitude response has a steep transition from its passband to stopband. Consequently, interferences like harmonics of f0 are successfully attenuated. In practice, factors such as run-time complexity need to be carefully considered. 1) Run-Time Complexity: The run-time complexity of the rotor-slot-harmonic-based motor speed detection scheme is determined by its three major algorithmic components shown in Fig. 5. Among them, the operation of the rotor speed detector, as described by (24) in the Appendix, is trivial and hence is not discussed here.

For an M th-order FIR linear-phase filter, it has M + 1 real coefficients. As shown in Fig. 7(a), to produce each rotor slot harmonic signal ish [n], a microprocessor needs to perform M + 1 multiplications between the complex current vector iC [n] and the filter’s real coefficients, as well as M complex additions. This is equivalent to 2 · (M + 1) real multiplications and 2M real additions. Given a symmetric FIR filter with an even order M in (13), the operation of the filter may be further reduced. As shown in Fig. 7(b), to produce each rotor slot harmonic signal ish [n], a microprocessor performs (M/2) + 1 multiplications between the complex current vector iC [n] and the filter’s real coefficients, as well as M complex additions. This is equivalent to M + 2 real multiplications and 2M real additions. The frequency demodulator involves first obtaining the phase of the rotor slot harmonic signal ish[n] and then passing it through an FIR filter described by (23) in the Appendix. When computing each instantaneous rotor slot harmonic frequency fsh [n], multiplications of phases with the coefficient “8” may be efficiently replaced by binary shifts if fixed-point arithmetic is used. Based on the aforementioned analysis, the algorithm’s runtime complexity between two successive samples is determined primarily by the filter’s order M . This run-time complexity may be further reduced if an appropriately designed IIR filter is used as the frequency-selective filter in Fig. 5. A Bessel IIR filter performs the filtering operation via N M ak ish [n − k] + bk iC [n − k] (20) ish [n] = k=1

k=0

where ak (k = 1, 2, . . . , N ) and bk (k = 0, 1, . . . , M ) are the Bessel filter’s coefficients. According to (20), to produce each rotor slot harmonic signal ish [n], a microprocessor needs to perform M + N + 1 multiplications between the complex current vector iC [n] and the filter’s real coefficients ak and bk , as well as M + N complex additions. This is equivalent to 2 · (M + N + 1) real multiplications and 2 · (M + N ) real additions. The actual computation time between two successive samples of ish [n] depends on filter realization and hardware platform. Fig. 8(c) shows the estimated motor speed using a sixth-order Bessel filter. Despite significant reductions in run-time complexity (M , N M ), Bessel filters have exceedingly slow rolloff in the transition band when compared to other IIR filters or appropriately designed FIR filters. The average speed error in Fig. 8(c) is 0.44 r/min. If strong interferences exist in the neighborhood of the rotor slot harmonics, they may be inadvertently included in the filtered rotor slot harmonic signal ish [n] and consequently induce distortions to the final motor speed estimates. The average speed error may increase in this case. 2) Speed Detection at Steady-State Motor Operation: The use of linear-phase filters allows for accurate estimates of motor speed at either dynamic or steady-state motor operations. However, at steady-state or quasi-steady-state motor operations, the frequency modulation index β becomes negligibly small as s1 approaches zero. According to Fig. 3, Jk (β) becomes sufficiently small for all k = 0. As a result, fewer rotor slot harmonic sidebands are observed in the complex current vector’s frequency spectrum following (12).


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Fig. 11. Bessel filter’s time delay.

passed through a nonlinear-phase filter, they experience less phase distortions compared to cases where the rotor slot harmonic component and its major sidebands are spaced far apart from each other, such as in Fig. 1. Consequently, a nonlinearphase filter is able to produce reasonably accurate motor speed estimates in those conditions. For example, Fig. 10(c) shows motor speeds estimated from both linear- and nonlinear-phase filters. For illustration purpose, the motor’s input power is calculated from its terminal voltage and current measurements and is also shown in Fig. 10(c). In this figure, the average error is 0.09 r/min when a linear-phase filter is used. This average error increases slightly to 0.11 r/min when a nonlinear-phase filter is used. Accuracy in the motor speed estimated from a nonlinear-phase filter deteriorates as motors experience more dynamic operations. 3) Filter Time Delay: The linear-phase FIR filter is a causal system. For such a causal system, there is usually a time delay associated with signals that propagate through from its input to output. For example, for the linear-phase FIR filter (13) with frequency response shown in Figs. 6(a) and (b), it has been established in (18) that the filter introduces an (M/2)-sample time delay between its input and output. Furthermore, because the FIR filter’s phase response is φ(ω) = −ωM/2, as shown in (16), the (M/2)-sample time delay actually corresponds to the negative slope of this phase response. Generally speaking, the time delay τg (ω) (in samples), which is the transit time of a signal at a particular frequency ω propagating from a causal system’s input to its output, corresponds to the negative derivative of the filter’s phase response φ(ω) with respect to the angular frequency ω τg (ω) = −

Fig. 10. Induction motor speed detected from a rotor slot harmonic signal for the 20-hp 4-pole induction motor with 40 rotor slots during a steady-state motor operation. (a) Frequency band between 735 and 795 Hz in the complex current vector’s frequency spectrum. (b) Frequency band between 1095 and 1155 Hz in the complex current vector’s frequency spectrum. (c) Motor speed and input power.

As shown in Figs. 10(a) and (b), at steady-state or quasisteady-state motor operations, the rotor slot harmonic component and its major sidebands are tightly packed together. When

dφ(ω) . dω

(21)

For a given filter, τg is also known as its group delay. Fig. 11 shows the time delay τg (ω) of the Bessel filter used in Fig. 8(c). Because the Bessel filter has a maximally linear phase response, Fig. 11 is characterized by an almost constant group delay of 179 samples across the filter’s passband. It follows from (21) that linear-phase filters with frequency responses (16) have constant group delays. In contrast, filters with nonlinear phase responses have frequency-dependent nonuniform group delays, and often result in phase distortion, or temporal smearing of the estimated motor speed. The frequency-dependent nonuniform group delays may further complicate latency compensation schemes, which synchronize various signals temporally to the same epoch in motor condition monitoring or control tasks. For linear-phase filters, the latency compensation scheme is usually implemented via a first-in

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first-out queue with a memory depth τg that corresponds to the group delay determined at the filter’s design stage. However, for nonlinear-phase filters, such as the one shown in Fig. 9, the latency compensation is not a trivial task. In this paper, an aggregate delay that corresponds to the average group delay within the nonlinear-phase filter’s passband is used when plotting Figs. 8 and 10. In practice, a precise compensation of nonlinearphase filters’ nonuniform group delays requires applicationspecific knowledge and is contingent upon the amount of filter distortion allowed. Therefore, from a design perspective, linear-phase filters are preferred over their nonlinear-phase counterparts because they allow for straightforward latency compensation schemes. V. C ONCLUSION For grid-connected squirrel-cage induction motors that are connected to periodically time-varying loads, their rotor slot harmonic frequency is modulated by periodic speed oscillations. To resolve the motor speed on a sample-by-sample basis, frequency-selective linear-phase filters are applied to the complex current vector to attenuate interferences and to isolate the rotor slot harmonic signal. This type of filters extracts the desired rotor slot harmonic signal in its passband without introducing phase distortions and hence helps ensure accurate speed estimates for grid-connected induction motor condition monitoring applications. A PPENDIX ROTOR -S LOT-H ARMONIC -BASED M OTOR S PEED D ETECTION The frequency demodulator and the rotor speed detector, i.e., the second and third blocks illustrated in Fig. 5, may be implemented purely in the discrete-time domain. 1) Frequency Demodulator: Following the definition of the rotor slot harmonic signal in (3), its instantaneous frequency at time t is related to the time derivative of the phase θsh (t): fsh (t) =

1 dθsh (t) · . 2π dt

(22)

In a discrete-time domain, the time derivative (22) may be approximated by a fourth-order phase difference estimator derived from Lagrange polynomial approximation [30] fsh [n] =

fS · (−θsh [n] + 8 · θsh [n − 1] 24π − 8 · θsh [n − 3] + θsh [n − 4])

(23)

where θsh [n] = ∠ish [n]. 2) Rotor Speed Detector: Given a positive instantaneous rotor slot harmonic frequency fsh [n] at the frequency demodulator’s output, the rotor speed detector utilizes (2) to calculate the per-unit slip fsh [n] P s[n] = 1 − · − nW (24) R f0 on a sample-by-sample basis.

3) Commissioning: A successful commissioning of the motor speed detection requires knowledge of the motor’s supply frequency f0 , number of pole pairs P , number of rotor slots R, and odd integer nW . Among those parameters, the first two are usually available from the motor’s nameplate. The number of rotor slots can usually be identified after a few trials by applying spectral estimation techniques to signals acquired from the motor and then examining frequency bands that are associated with the rotor slot harmonics [8]. It may alternatively be obtained by comparing a slip quantity calculated from the motor’s input power against a separate slip quantity derived from the motor’s current frequency spectrum [31]. For the odd integer nW , Section II-A2 has shown that it takes a value of either +1 or −1 for the principal rotor slot harmonic. The exact value of nW is determined based on the relationship between the number of pole pairs P and the number of rotor slots R [11]. Once the quantities f0 , P , R, and nW are known, the rotor speed detector (24) is able to calculate the per-unit slip on a sample-by-sample basis accordingly.

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Zhi Gao (S’03–M’07–SM’11) received the M.S. and Ph.D. degrees in electrical engineering from the Georgia Institute of Technology, Atlanta, GA, USA, in 2004 and 2006, respectively. He was with Schneider Electric prior to joining Eaton Corporation, Moon Township, PA, USA, in 2012, where he is currently a Principal Engineer responsible for product development. His research interests include condition monitoring, diagnosis and protection of electric machines and power circuits based on digital signal processing theories and techniques, and the design and control of motor drives and electric machines for industrial automation applications. He is the holder of eight U.S. patents. Dr. Gao is an Associate Editor of the IEEE T RANSACTIONS ON I NDUSTRY A PPLICATIONS and the recipient of the 2012 IEEE Industry Applications Society Andrew W. Smith Outstanding Young Member Award. Larry Turner (M’09–SM’12) received the M.S. degree in electrical engineering (signal processing specialty) from the University of Florida, Gainesville, FL, USA, in 1993. He worked in the defense, communications, and semiconductor industries prior to joining Schneider Electric, Knightdale, NC, USA, in 2002, where he is currently a Senior Staff Engineer and a member of a global corporate research team. His research interests include adaptive systems, modeling, communications, operating systems, and embedded development, with applications to automation, wireless communications, power systems, and motor diagnostics. He is the holder of 14 U.S. and European patents and pending patent applications. Roy S. Colby (M’81–SM’95) received the B.S. degree in electrical engineering from the Massachusetts Institute of Technology, Cambridge, MA, USA, in 1978 and the M.S. and Ph.D. degrees in electrical engineering from the University of Wisconsin, Madison, WI, USA, in 1983 and 1987, respectively. He was with General Electric, North Carolina State University, and United Technologies Corporation prior to joining Schneider Electric, Knightdale, NC, USA, in 1997, where he is currently a Fellow Engineer responsible for managing a global team in the areas of machine diagnostics and motor protection. His entire career has revolved around electric machines and drives as a Researcher, an Individual Contributor, and a Technical Team Leader. His noteworthy activities have included permanent-magnet synchronous drive control, development of ac elevator drive autotuning algorithms, design of magnetic bearings, ac drive system thermal modeling, switched reluctance motor design and analysis, and ac motor diagnostics. He is the author or coauthor of ten refereed journal articles. He is the holder of 20 U.S. patents. Dr. Colby is a Cofounder of the symposium of the Wisconsin Electric Machines and Power Electronics Consortium. He was an Associate Editor for the IEEE T RANSACTIONS ON P OWER E LECTRONICS and is currently a member of the IEEE Industry Applications Society Industrial Drives and Electric Machines Committees.