Stator Current Spectral Content of Inverter-Fed Cage Rotor Induction Motor

The paper analyzes the influence of the number of rotor bars on the stator current spectral content in a three-phase cage induction motor fed by a pulse-width modulated (PWM) inverter. It is shown that each of the higher-order time harmonics in the supply voltage produces space harmonics in a rotating magnetic flux density wave, which results in induced rotor slot harmonics (RSHs) in the stator current spectrum. The conditions for the existence of these space harmonics are identical to those applying to a mains-fed motor. In other words, the number of rotor bars of a mains-fed motor yielding an RSH-free stator current spectrum produces the same stator current spectrum even in case the motor is inverter-fed. Additionally, to minimize the adverse effects of RSHs in the stator current spectrum, one must consider not only the number of rotor bars, but also its relationship with the frequency modulation ratio of the PWM inverter. Analytical predictions are presented to illustrate these results supported both by numerical simulations of the induction motor modelled through the winding function theory and experimentally taking the case a two-pole cage induction machine as a case study.


I. INTRODUCTION
After more than a century since their invention, induction motors still play a paramount importance in industry and vehicle traction applications over a huge variety of ratings, thanks to their cheapness, ruggedness and reliability [1]- [3]. The ongoing need to limit the use rare earths due to fluctuating cost, environmental and availability issues is promoting a renewed interest in the exploitation of induction motors [4] as preferred alternative to permanent-magnet machines whenever possible. For sure, more and more efforts are being placed on the attempt to improve the performance of induction motors under various respects, such as increasing their efficiency to meet present energy saving standards and targets [5] as well as reducing noise and vibrations [6]. These The associate editor coordinating the review of this manuscript and approving it for publication was Qinfen Lu . targets apply to traditional mains-fed motor but are even more relevant to inverter-fed motors, which account for the vast majority today thanks to their capability of operating at variable speed. Several approaches can be pursued to improve the performance of inverter-fed induction motors. Recently interesting approaches have been proposed based on advanced and smartly designed supply, automation and management techniques where the main focus is on the inverter technology and control [2], [3], [5], [6]. In contrast, an apparent lack appears in the literature regarding the potentials of acting on the induction motor design to optimize its operation when supplied from a PWM inverter.
Recent research, conducted on mains-fed induction motors, has highlighted the importance of an appropriate selection of the number of rotor bars, for example, to minimize Rotor Slot Harmonics (RSHs) and their relevant parasitic effect under sinusoidal supply [7], [8]. The main goal of the paper is to extend the discussion and results presented in [7] and [8] covering the case of inverter-fed motors. The main problem being considered is how some key design choices such as the selection of the number of stator slots and rotor bars, for a given number of poles, can affect the performance of the induction motor when it is fed from a PWM inverter. To this end, a preliminary analysis will be firstly presented of the supply voltage spectrum from an inverter-fed induction motor in order to identify a simple but comprehensive model for the time harmonics produced at the inverter output. As a further step, the air-gap magnetic fields caused by PWM-related time harmonics will be identified along with the generation mechanism of RSHs. Conditions will be then derived for the occurrence of RSHs of different possible orders in the stator current spectrum. This will finally lead to identify some practical rules to avoid and minimize RSHs, with special regard to low-frequency low-order ones, which can have the worst impact on motor performance. It will be shown how in case of inverter-fed motors, the selection of the number of stator and rotor slots, for any given number of pole pairs, is not the only useful criterion to follow, as an essential role is played by the PWM modulation ratio. It will be thus shown how the inverter-fed induction motor needs to be optimized not as a stand-alone component, but together with the particular inverter which supplies it and depending on the particular PWM modulation ratio to be used.
The main results presented will be illustrated through numerical simulations and through measurements on a dedicated experimental setup.

II. ANALYSIS OF THE VOLTAGE SPECTRAL CONTENT AT THE PWM-INVERTER OUTPUT
Here, a detailed analysis of the voltage spectral content at the output of a carrier-based PWM with min-max sequence injection is performed. It is worth mentioning that bipolar PWM with min-max sequence injection produces the same pulse pattern as that of the space vector modulation (SVM), with the center-weighted seven-segment vector sequence [9]. However, considering the easy implementation of carrierbased PWM, and the availability of PWM outputs in most digital platforms, the slight modification of the reference signal using min-max is a straightforward way of achieving to center-weighted SVM equivalent operation without complex algorithms, calculations, and vector generation sequence. For that reason, carrier-based PWM with min-max can be considered for spectrum derivation. As a result of this analysis, approximate analytical expressions that describe the line-toline voltage at the input of the three-phase induction motor are illustrated.
As it is well known, the use of PWM inverters allows the simultaneous change of the motor supply voltage both in terms of the rms value and frequency. In this paper, a PWM frequency converter in synchronous mode is analyzed, i.e., in the case where the frequency modulation ratio m f between the carrier frequency f c and the modulation frequency f m , is an integer [10], [11]. The carrier frequency is the frequency of the triangular wave, while the modulation frequency is that of the desired voltage fundamental at the inverter output.
To eliminate the even harmonics at low values of m f , an odd integer number is preferred for m f [10]. In this particular case, the spectral content of the voltage signal at the PWM inverter output will be analyzed for m f = 15. This case corresponds to a triangular wave signal with a carrier frequency f c = 750 Hz and a modified sinusoidal signal with a modulation frequency f m = 50 Hz. Another critical parameter in PWM is the modulation index m a , which is the ratio between the amplitude of the modulation wave A m and the amplitude of the carrier wave A c : By analyzing the output phase voltage waveform, obtained by comparison of carrier and modulating wave through minmax sequence injection, Fig. 1, it can be easily shown that, apart from the fundamental frequency wave, phase voltage contains a dc component, together with higher-order time harmonics of the following orders: The order of these harmonics is generally described by the following expressions [10], [11], where for odd values of j the harmonics exist only for even values of k. For even values of j, the harmonics exist only for odd values of k. In contrast, for a frequency modulation ratio m f ≥ 9, the harmonic amplitudes are almost independent of m f , though m f defines their occurrence frequencies [10]. In addition, some of the harmonics i.e. of m f , 2m f ± 3, 3m f orders are absent in the line-to-line voltages, as the phase voltages in all three phases are in-phase, thus cancelling their impact as it is depicted in Fig. 2 and Fig. 3, for two different modulation indexes.    Accordingly, in the line-to-line voltages at the output of the converter, apart from the fundamental, higher-order time harmonics also appear, the most prominent ones having orders 2m f ± 1, 4m f ± 1, 3m f ± 2, 3m f ± 4, m f ± 2. Which group would be particularly prominent depends on the value of the modulation index m a , Fig. 4.
Furthermore, harmonic analysis shows that time harmonics of orders m f − 2, 2m f + 1, 3m f − 2, 3m f + 4, 4m f + 1 form a positive sequence of voltages, while those of the orders i.e., negative sequence of line-to-line three-phase voltages. It is also interesting to note that the harmonics from the same group have identical amplitudes.
As shown in Fig. 4, the fundamental increases linearly with the modulation index as long as the value of the modulation index is less than 1.1547 [9]. In the following, only the influence of harmonics from the first five groups will be considered, namely harmonics of orders m f ± 2, 2m f ± 1, 3m f ± 2, 3m f ± 4 and 4m f ± 1, respectively. Their normalized amplitudes can be analytically interpolated: (10) where U mn indicates the amplitude of the n-th group of harmonics. In a real three-phase bridge inverter, all of the above amplitudes, (5)-(10), should be multiplied by U dc equal to (3 √ 2/π)U LL . The harmonic analysis leads to the following conclusion: the phase shift of harmonic voltages in relation to the fundamental is independent of the modulation index m a , except for the fifth set of higher-order harmonics, where it shifts by ±π when the modulation index is higher than m a = 0.62. However, this coincides in all three phases, so the voltage sequence remains the same -positive sequence for 4m f + 1 and negative for 4m f − 1.
Therefore, the input line-to-line voltage U ab of a motor fed by a PWM inverter can be modeled by (11) and (12), as shown at the bottom of the next page, where α = π/3. The amplitudes of the voltage components depend on the modulation index m a , according to (5)- (10).
Similar expressions for the other two line-to-line voltages can be derived from previous conclusions considering a specific harmonic component positive or negative voltage sequence. if m a < 0.62, otherwise:

III. ROTOR SLOT HARMONICS IN THE STATOR CURRENT SPECTRUM-MAINS-FED MOTOR
In the case of a symmetrical three-phase cage induction motor, fed by a symmetrical three-phase AC voltage, the existence of rotor slot harmonics (RSHs) of order λ in the stator current spectrum is directly related to the number of rotor bars R and the number of pole pairs p [7], [8]. For λ = 1, we have rotor slot harmonics of the first order, the so-called principal slot harmonics, PSHs.
The condition for the existence of the lower RSH of order λ in the stator current spectrum, at the frequency where f 1 indicates the mains frequency and s the slip, is given by the following relationship between the number of rotor bars and the number of pole pairs, where z is a theoretically arbitrary integer. In real situations, the number of rotor bars is usually in the range of 0.5S ≤ R ≤ 1.5S, where S is the number of stator slots. Hence, the range of the integer z is 1 ≤ z ≤ 3λS/(4mp) where m is number of phases.
The condition for the existence of the upper RSH of order λ in the stator current spectrum, at the frequency is the following relationship between the number of rotor bars and the number of pole pairs: In particular, if the number of rotor bars is then in the stator current spectrum, both RSHs of order λ at previously defined frequencies (13) and (15) appear. These equations show that the position of the RSHs in the stator current spectrum depends on the motor load, i.e., of its slip. This could be a basis for sensor-less speed estimation of a cage rotor induction motor [12], [13].

IV. ROTOR SLOT HARMONICS IN THE STATOR CURRENT SPECTRUM-INVERTER-FED MOTOR
A positive sequence voltage time harmonic of order h, whose frequency is f h = hf m , produces the magnetomotive force (mmf) waves in the motor air-gap, through the current harmonic it excites. Its distribution over space and time t can be described as follows: Similarly, the negative sequence voltage time harmonic of order h produces the following mmf wave: Assuming a uniform air gap width, i.e., by neglecting the slotting effect, the magnetic flux density waves are defined as: We can now introduce the following transformation of variables, where θ r is the position along the air-gap circumference measured in the rotor reference frame. Substituting VOLUME 10, 2022 (22) into (20) and (21), the magnetic flux density waves viewed from the rotor side are B a shν (t, θ r ) = B a shνm cos s a hν ω m t − νpθ r , where the following slip values are used: Additionally, in symmetrical three-phase induction motors, fed by a symmetrical system of three-phase voltages, the following spatial harmonic orders exist [14]: The corresponding magnetic flux density waves induce electromotive forces (emfs) and thus currents of appropriate frequencies in the short-circuited squirrel-cage rotor winding.

A. RSHS DUE TO DIRECT SEQUENCE PWM TIME HARMONICS
It is common to model the rotor cage as a set of adjacent loops, each embracing two close bars and the end-ring portions connecting them, as explained in [15]. The current in the first rotor loop, arising as a consequence of the magnetic flux density wave (20), produces the following mmf [16], sin µ π R I r max cos s a hν ω m t cos (µθ r ) (28) where µ is the spatial harmonic order (µ = 1, 2, 3, . . .) of rotor loop winding function, assuming that the origin is placed in the center of the first rotor loop, and I rmax is the peak value of the current induced in the loop. The previous expression is decomposed as follows, with a suitable definition of the constant K µ . In the adjacent rotor loop, displaced by 2π/R mechanical radians apart from the first, a current of the same intensity and frequency but phase-shifted by p · ν·2π/R flows. This current produces the following mmf: By summing the mmf of all rotor loops, we obtain: The previous sum is different from zero in the following cases, [16]: a) when the condition µ = −νp or µ = νp is satisfied; in such case (31) becomes: b) when condition µ + νp = λR (λ = 1, 2, 3 . . .) is satisfied; in such case (31) becomes: c) when condition µ − νp = λR (λ = 1, 2, 3 . . .) is satisfied; in such case (31) becomes: Based on the variable transformation, (22), these mmf waves, through the assumed uniform air gap, are viewed on the stator side as the following magnetic flux density waves: Thus, all the magnetic flux density waves resulting from a given time harmonic (of a frequency hf m ) on the stator side are reflected by the rotor back to the stator at the same frequency and two additional, slip-dependent frequencies.
The two rotational waves of magnetic flux density given by expressions (36) and (37) can induce emfs in the stator windings and thus excite stator currents with frequencies not present in the voltage source. Of course, this is possible only if the number of pole pairs of the flux density waves (36) and (37) is the same as the number of pole pairs of the magnetic field waves produced by the stator winding itself [17]. By comparing the obtained expressions with the case of a mains-fed induction motor, it is demonstrated that the condition for the existence of RSHs in the stator current spectrum remains unchanged [7], [8]. In contrast, the set of frequencies at which these harmonics occur, changes due to higher-order time harmonics in the supply voltage.
Thus, the lower RSH of order λ exists in the stator current spectrum at the frequency when the following relationship between the number of rotor bars and the number of pole pairs is met, where z is an integer. The condition for the existence of the upper RSH in the stator current spectrum of the order λ, at the frequency is the following relationship between the number of rotor bars and the number of machine pole pairs: If the number of rotor bars is equal to then both of the RSHs of order λ with frequencies (38) and (40) appear in the stator current spectrum. For the most prominent positive sequence higher-order time harmonics in PWM voltage supply, h takes the value h = 2m f + 1. As a consequence, these are the expected frequencies in the stator current spectrum -space harmonics excited by time harmonic of order 2m f + 1, Obviously, the former of these frequencies may be relatively low. This is especially true when 2m f is close to the ratio between the number of rotor bars and pole pairs -R/p or its multiple. Similar is true also for the time harmonic of order m f − 2, which is prominent for higher values of modulation index m a , Fig. 4.

B. RSHS DUE TO INVERSE SEQUENCE PWM TIME HARMONICS
Similar analysis can be performed for flux density waves caused by inverse-sequence higher-order time harmonics in the supply voltage. It can be shown that the conditions for the occurrence of RSHs in the stator current spectrum are now inverted compared to the positive sequence. More precisely, the condition for the existence of the lower RSH of order λ, at frequency (38), is given by expression (41). Similarly, the condition (39) should be met for the presence of the upper RSH of order λ, at frequency (40). If the number of rotor bars is given by the expression (42), then both of RSHs appear in the stator current spectrum. Now, inverse-sequence time harmonics of the order 2m f −1 and m f + 2, can induce RSHs at rather low frequencies.

V. EXPERIMENTAL SETUP
In order to confirm the analytically obtained results, measurements of the stator current spectrum were performed on a three-phase two-pole squirrel cage induction motor with the following data: 2.2 kW, 400 V, wye connection, 50 Hz, 4.48 A, 2830 rpm. The stator and rotor of the motor have S = 24 slots and R = 18 bars, respectively. The rotor bars are skewed for one stator slot pitch.
The motor was supplied with a laboratory-built three phase IGBT (Semikron SKM40GD123D) inverter with a DC link voltage of approximately 560 V (diode-rectified three-phase grid voltage). A C2000 Delfino MCU (LaunchXL-F28379D, Texas Instruments) was used for controlling the motor with open-loop control by conventional SVM pulse-width modulation. The launchpad enabled potentiometer-based setting of output and carrier frequencies, and amplitude modulation factors m a . The motor shaft was coupled to another similar induction motor, operating in generator mode, being supplied by a Danfoss FC-302 frequency converter in an open-loop operating condition. Stator current measurements were performed with Tektronix current probes (A6302 with a AM503 amplifier) and a Tektronix DPO4034B 350 MHz oscilloscope. Data were sampled with 1 MS/s for 20 seconds and were stored for post-processing and spectral analysis. The shaft speed was measured with a Chauvin Arnoux C.A 27 tachometer.

VI. RESULTS AND DISCUSSION
A detailed dynamic mathematical model of the selected induction motor was prepared, implementing the winding function theory. It uses the parameterized winding function (PWF) model, in which the number of rotor bars and its skewing angle appear as free parameters [7], [8], [18], [19]. The adoption of such a numerical model, in combination with and independently of experimental results, is deemed to be very effective in order to make the validation complete and  explanatory, also given the high amount of parasitic effects which may affect measurements.
Just for the sake of illustration, Fig. 8 shows one period of stator phase currents recorded from experiments and obtained from the mathematical model based on the derived analytical expressions (11), (12) for converter voltages.
As a consequence of an unfortunate choice of the number of rotor bars, the motor under test (designed for mains-fed operation) has both of RSHs of each order in the stator current spectrum, at frequencies as the condition (17) is satisfied for λ = 1 and z = 3, λ = 2 and z = 6, λ = 3 and z = 9 etc.
In the case of a mains-fed motor with a frequency of f m = 25 Hz, at its rated load, the stator current spectrum obtained from the simplified mathematical model considering an unskewed rotor is shown in Fig. 9 For the rated-loaded, inverter-fed motor, supplied with a fundamental frequency of 25 Hz and a carrier signal   frequency of 625 Hz and a modulation factor m a = 0.45, with unskewed rotor bars, the stator current spectrum is obtained as shown in Fig. 10. Almost all of the previously predicted components in the stator current spectrum are clearly visible. Their exact frequencies and number designation are given in detail by Table 1 in the Appendix.
In order to observe the effect of skewing of rotor bars by one stator slot pitch, the mathematical model of the motor was modified accordingly [20]. The resulting current spectrum is given in Fig. 11. By comparing this stator current spectrum with the previous one presented in Fig. 10, it is evident that space harmonics are significantly attenuated or even missing for higher orders. This is not the case with time harmonics, instead, as one could expect.
For the case of an inverter-fed motor at rated load, supplied by a fundamental frequency of 50 Hz and a carrier signal frequency of 750 Hz, with a modulation factor m a = 0.9,   the stator current spectrum is obtained from the mathematical model as shown in Fig. 12.
For the same conditions as above, but with a carrier signal frequency of 1250 Hz as well as with a modulation factor m a = 0.9, the stator current spectrum is obtained from the mathematical model, as shown in Fig. 13.
Experimentally recorded stator current spectra for three different modulation frequencies, carrier frequencies, and modulation indexes are given in Figs. 14 -16. They show a plenty of higher frequency components, predominantly time harmonics due to inverter output. However, the predicted space harmonics exist and are clearly visible, too. All are marked in the figures according to the indexes introduced in Table 1, in the Appendix.
As all of the even number of rotor bars in two-pole cage induction motor produce at least one of the first order RSHs, lower or upper or both of them [8], in order to analyze  the influence of the number of rotor bars on stator current spectrum, an example with an odd number of rotor bars, i.e., R = 27, is given in Fig. 17.
According to conditions (39) and (41) neither of the first order RSHs exist in this case. Thus the stator current spectrum contains only time harmonics resulting from the non-sinusoidal inverter output voltage. In addition, the skewing for one stator slot pitch shows similar results as obtained for unskewed rotor -demonstrating only marginal effects [19].

VII. CONCLUSION
The paper initially gives a brief overview of the voltage spectral composition at the output of a carrier-based PWM inverter with min-max sequence injection and presents approximate analytical expressions for line-to-line voltages.
It further shows that the conditions for RSHs in the stator current spectrum of an inverter-fed cage induction motor are identical to the case of mains-fed motors. However, in the former case, each of the time harmonics in line voltages produces its pair of space harmonics, known as RSHs. It is a matter of fact that, as experiments clearly show, the stator current spectrum of an inverter-fed motor includes a multitude of extra harmonics. Some of them directly relate to harmonics present in DC voltage of the inverter whose effect is evident in stator current spectrum when the motor is driven in openloop control mode. Still, it is undoubtedly desirable not to associate them with harmonics resulting from an unfortunate choice of the number of rotor bars. Thus, it can be said that the choice of the appropriate number of rotor bars in the case of an inverter-fed motor is crucialmuch more than in the case of mains-fed machines. In the case of three-phase, two-pole motors, this means targeting an adequate number of rotor bars among the odd numbers. Namely, any of the even numbers of rotor bars produces at least one of the PSHs of all orders. In the case of a higher number of pole pairs (p > 1), such a number of rotor bars exists even among the even numbers. In fact, according to recently published research, it has been shown that the degree of freedom to choose the appropriate number of rotor bars rises as the number of pole pairs increases.
Additionally, it has been demonstrated that an especially unfavorable case occurs when the frequency modulation ratio, or its multiple, is close to the ratio of the number of rotor bars and pole pairs, or multiple of this ratio. In that case, RSHs in the stator current spectrum can occur at low frequencies, which are therefore more pronounced. The undoubted consequences are increased losses in the motor windings, as well as undesirable noise, vibration, and harshness (NVH) issues.