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Analysis of Multiphase Induction Machines With Winding Faults Judith Apsley, Member, IEEE, and Steve Williamson, Fellow, IEEE
Abstract—The paper shows how the techniques of generalized harmonic analysis may be used to simulate the steady-state behavior of a multiphase cage induction motor with any form of open-circuit or short-circuit fault in the stator winding. The analytical model is verified using a four-pole machine with a 48-slot stator. Each coil of the stator winding of this machine is brought out to a patchboard that enables the stator to be configured for single-phase, two-phase, three-phase, four-phase, six-phase, or 12-phase excitation. Experimental results are compared with computer predictions for a six-phase machine with both open-circuit and short-circuit faults. Index Terms—Induction machines, multiphase machines, open circuit, short circuit, winding faults.
I. I NTRODUCTION
M
ULTIPHASE machines are finding favor in some variable-speed drive applications because they may have higher efficiency, produce lower magnitude pulsating torques, and be acoustically quieter than their three-phase counterparts [1]–[4]. One further advantage is that multiphase machines are said to be fault-tolerant, offering greater security in missioncritical applications. They appear to be particularly suited as the main propulsion motors for future marine applications. Several authors have addressed the issue of fault tolerance [2], [3], [5], [6], focusing for the large part on the behavior of machines in which a single phase is open circuited and on compensation strategies for open-circuit faults [6]–[8]. The compensation strategies in [7] and [8] assume sinusoidally distributed windings. This approximation is valid for a low number of phases, and is a useful simplification for control. In [9], the third harmonic of the winding distribution is also included. However, as the number of phases increases to a limit where there is only one slot per pole per phase, the magnetomotive force (MMF) approaches a rectangular waveform rather than a sinusoid. This is also true for the MMF contribution from a single faulty coil. The concentrated winding model [10], allows for the rectangular MMF in the calculation of interphase inductances, and applies the resulting differential equation model to the transient analysis of winding faults [11]. The model predicts motor waveforms in the time domain but is Paper IPCSD-05-079, presented at the 2005 IEEE International Electric Machines and Drives Conference, San Antonio, TX, May 15–18, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications Society. Manuscript submitted for review May 1, 2005 and released for publication November 19, 2005. The authors are with the School of Electrical and Electronic Engineering, University of Manchester, Manchester M60 1QD, U.K. (e-mail: j.apsley@ manchester.ac.uk; [email protected]). Digital Object Identifier 10.1109/TIA.2005.863915
computationally intense and does not readily give information about the magnitude and frequency of torque pulsations. An alternative steady-state approach is to use Fourier series expansions of the air-gap field spatial distribution and supply time variation [4], [12]. In this paper the authors will describe a form of analysis that is readily able to model the effect that more complex forms of fault have on machine behavior, including intraphase short circuits and open circuits in machines with parallel paths. The purpose of the paper is to describe the use of a powerful and versatile technique, rather than to give a comprehensive exposition of the consequences of winding faults. For that reason, all of the results presented in this paper refer to a sixphase winding, although the authors have carried out extensive validation at other phase numbers. The technique is verified by experimental measurements. II. A NALYTICAL M ETHOD A. Generalized Harmonic Analysis The analytical model is based on the techniques of generalized harmonic analysis, which has been described in detail in the literature [4], [13] and will, therefore, only be summarized in this paper. The model assumes sinusoidal excitation and constant rotor speed. Converter-fed machines are dealt with using frequency domain analysis and superposing timeharmonic solutions. The method involves calculating coupling impedances which relate the back electromotive forces (EMFs) induced in each circuit of the machine to the currents flowing in those same circuits. A stator circuit is defined as any seriesconnected set of coils. It might consist of an entire phase winding, if all the coils in that winding are connected in series. Alternatively it might be a single turn of a coil if that turn is short circuited. The definition of a “circuit” on the rotor differs from that used for the stator. Here, a “circuit” is a rotor current distribution of known frequency, pole number, and sense of rotation. We will, for example, refer to the ν th harmonic rotor current distribution. This will indicate that a particular rotor bar (the “reference” bar) carries a current given by iνR (t) = e
√ ν jsν ωt . 2I¯R e
(1)
All other bars carry currents of the same magnitude and frequency, with a phase progression of 2νπ/Nb radians from bar to bar: Nb number of rotor bars; rotor slip with respect to 2ν-pole wave. sν
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There are four different types of coupling impedance: Z m,n coupling impedance which relates the EMF induced in the mth stator circuit to the current flowing in the nth stator circuit; ν Z R,m coupling impedance relating the ν th harmonic EMF induced in the reference rotor bar to the current flowing in the mth stator circuit; ν Z m,R coupling impedance relating the EMF induced in the mth stator circuit to the ν th space-harmonic current distribution in the rotor; ν ZR ν th -harmonic impedance per bar of the rotor. A summary of the way in which all of these impedances may be calculated is given in the Appendix. The voltage balance equation for the mth stator circuit, is given by the general equation [13] Vm =
M
I¯n Z m,n +
ν ν I¯R Z m,R
(2)
Fig. 1. Loop diagram for one phase of the six-phase motor (a) prefault and (b) following a short circuit in one coil.
ν
n=1
where V m voltage applied to mth stator circuit; In current flowing in the nth stator circuit; M number of stator circuits. The range of possible values for the harmonic index is not specified in (2). The number of harmonic rotor current distributions that must be taken into account will depend on the operating condition of the machine. Stator winding faults introduce electromagnetic asymmetry, so the air-gap field tends to be rich in harmonics. In the work presented in this paper the authors found is necessary to consider all positive and negative values of ν in the range up to 4 times the number of stator slots. The voltage balance equation for the ν th harmonic rotor current distribution takes a similar form to (2) ν ν ZR + 0 = I¯R
M
ν I¯m Z R,m .
(3)
m=1
Substitution from (3) into (2) enables the stator voltage balance equation to be expressed in terms of the stator currents only M Z νm,R Z νR,n ¯ In Z m,n − . (4) Vm = ν ZR ν n=1 There are M such equations, which may be written in compact form V = ZI.
(5)
Z is an M × M impedance matrix, and V and I are voltage and current vectors of dimension M . If the voltages applied to all stator circuits are known, (5) may be solved to determine the currents flowing in each stator circuit, without further manipulation. Commonly, however, the stator circuits are interconnected in some way (e.g., in star) so that the voltages applied to each circuit may not necessarily be specified. Under such circumstances, (5) must be transformed from circuit (i.e., branch) variables
to loop variables using the connection matrix method [14]. Once all the stator currents have been determined, the rotor harmonic current distributions can be recovered via (3), and the solution completed by calculating torques, air-gap fields, and losses. B. Representation of Winding Faults The formal structure of the method outlined above enables a single turn, a coil, or a set of coils to be defined as a circuit, and the corresponding coupling impedances to be determined. One or more open-circuited windings are accommodated in the model by setting the relevant circuit currents to 0, and deleting the corresponding voltage equations. Each open circuit reduces the dimension of the voltage balance equation [i.e., (5)] by 1. Short circuits, on the other hand tend to increase the dimension of the voltage balance equation. Fig. 1(a) for example, represents one phase of a series-connected winding in its prefault configuration, in which the whole of the phase winding is regarded as a single circuit, as defined above. Fig. 1(b) shows the same phase winding with a short-circuit fault. It is now regarded as two independent circuits, corresponding to the shorted and the unshorted parts of the winding. The currents flowing in these two circuits will differ from each other, increasing the dimension of the voltage balance equation by 1. For the purposes of illustration, the short circuit is shown at one end of the phase, but it could equally well have been positioned at any point in the winding. Had the whole of the phase winding been short circuited, it would have been necessary only to set the voltage applied to that winding to 0. C. Test Rig The experimental results used in this paper were verified on an experimental test rig, based on a rewound D180 frame 50 Hz four-pole squirrel-cage motor. The 48-slot stator was fitted with a double-layer mush winding, with both ends of every stator coil brought out to an external patchboard. This enabled the windings to be reconfigured readily for 1, 2, 3, 4, 6, or
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Fig. 2. Predicted and measured values of (a) supply current (A rms), (b) input active power (W), and (c) input reactive power (VAR) for the six-phase, nominally balanced machine at four load points.
12 phase excitation. The stator of the test machine was excited from a specially constructed converter, consisting of 12 independent H-bridges. These provided open-loop voltage-forced excitation of the appropriate phase number. As is common with multiphase machines, each phase was separately excited. There were no star points. For all of the results presented in this paper the machine was connected in six-phase mode, i.e., with two slots per pole and phase. This gave a good compromise between the number of measurement channels required, and the different ways of connecting coils within a winding. The motor was mounted on a three-axis piezoelectric force table, for measuring transient and pulsating torques, as described in [15], enabling validation of predicted torque ripple both pre and post fault. The force table has a low-frequency limit of 1 Hz, so dc torque was measured with an in-line shaft transducer. Motor terminal quantities were measured with a Norma D6000 power analyser. Measurements were made for a harmonic sweep up to the 15th harmonic of the fundamental (50 Hz) excitation and for total quantities. Results for current and input active and reactive power show the fundamental only, but results for torque include harmonics up to the 15th.
III. R ESULTS A. Balanced Machine In order to check the validity of the machine parameters used in the model, the first set of simulations and experiments were carried out with the stator winding in the unfaulted condition. The results obtained are summarized in Fig. 2(a)–(c), which give fundamental input current, active power and reactive power or volt-ampere reactives (VARs) to each of the six phases, for full-load, no-load, and for two part-loaded conditions. The results refer to the fundamental-frequency (50 Hz) component of the excitation. Imbalances are clearly evident even in these prefault results, arising from small asymmetries in the individual H-bridges of the inverter. These are manifested as small differences between the voltages applied to the six phases, which produce differences in the phase currents. The simulation results use the measured values of the applied phase voltages as input data, and so the calculated results also reflect these imbalances. The ability to handle unbalanced voltages with ease is one of the advantages of generalized harmonic analysis. The results presented in Fig. 2(a)–(c) show good agreement
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Fig. 4. Predicted and measured torque ripple in the prefault mode (gray) and with phase #3 open circuit (black), for a six-phase motor at full load (1475 r/min).
Fig. 3. Predicted and measured values of (a) supply current, (b) input active power and (c) input reactive power for the six-phase machine with phase #3 open circuit at full load (1475 r/min).
between experimental measurement and calculation, indicating that the machine parameters used in the simulation work are of sufficient accuracy. B. Open-Circuit Fault: Series-Connected Winding The single open-circuited phase winding is arguably the commonest form of winding fault, arising from either a fault in the winding itself or in the converter leg that it is connected to. Fig. 3(a)–(c) shows the current and input active and reactive power at full load (1475 r/min) in the six-phase machine with one phase (#3) open circuited. Again, good agreement is obtained between the measurement and calculation. The results also show that the greatest impact is seen in the two phases that are physically located adjacent to the faulted phase in the stator winding, which exhibit an increase in input current in an attempt to provide the air-gap MMF that is missing in their locality. Fig. 4 shows the predicted and measured magnitudes of the torque ripple obtained at full load, over the frequency range 0–700 Hz. The results show distinct components at multiples of twice supply frequency, as would be expected in any machine with a winding imbalance. In order to show the effect of the open-circuited winding, the calculated magnitudes of the torques produced at these frequencies are also marked on Fig. 4. These show that the open circuit produces, for example, a 25 dB (i.e., 18-fold) increase in the predicted 100-Hz torque pulsation. Where predicted values for the prefault condition are not marked on Fig. 4, they are less than the noise floor of 0.1 Nm.
Fig. 5. Comparison of calculated (a) steady torque and (b) 100 Hz torque ripple, before (solid) and after (dashed) an open circuit fault, and using fault compensation (dotted).
Fig. 5(a) and (b) presents calculated curves to illustrate the performance of the machine before and after a single-phase open-circuit fault. In the postfault curves it is assumed that the excitation to the unfaulted phases is not changed when the fault occurs. Fig. 5(a) and (b) shows that this will produce a small increase in slip, but a significant increase in torque ripple, as already observed. If the current in the remaining phases is increased to obtain the same fundamental (i.e., 4-pole) component of MMF, then the curves labeled “compensated” are produced, increasing the steady torque to its prefault level and significantly attenuating the magnitude of the pulsating torque [16]. C. Open-Circuit Faults: Parallel-Connected Winding In the six-phase configuration the stator winding of the experimental motor has eight coils per phase. These may all be connected in series (as above) or in two, four, or eight parallel
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Fig. 6. Prefault and open-circuit fault representations of a parallel-connected winding, showing coil locations.
paths. Parallel paths ameliorate the effects of open circuit faults by allowing currents to flow in some of the coils in the faulted phase, thereby providing a measure of fault tolerance. In this section of the paper, we explore the effect of an open circuit in the case of a stator winding with two parallel paths, as shown in Fig. 6(a), prior to the fault and Fig. 6(b) with an opencircuit fault. Fig. 7(a)–(c) compares experimentally measured and calculated values of the input current, power, and VARs in each of the six phases of the test machine at 1490 r/min, with one of the parallel paths in phase #3 open circuited. As before, the agreement between calculated and experimental results is seen to be good. Fig. 7(a)–(c) also includes the calculated values of the corresponding inputs that would be obtained at 1490 r/min in the series connected machine with an open-circuit fault on phase #3. For the “series-connected” calculations the applied voltage is twice that used for the “parallel-connected” experiment and calculation, and the line current has been scaled by a factor of 2 in order to give a meaningful comparison. These results clearly show that the parallel connection does indeed reduce the imbalance produced by the open-circuit fault. In a further attempt to reduce the disturbance produced by an open-circuit fault, more parallel paths may be used. Alternatively, the eight coils of each phase may be connected in a series/parallel arrangement, as shown in Fig. 8(a). When an open-circuit fault occurs on one of the coils, the connection diagram for the faulted phase becomes that shown in Fig. 8(b), which shows that current will still be able to flow in six of the eight coils of this phase. Fig. 9(a)–(c) compares experimentally measured and calculated values of the input current, power, and VARs in each of the six phases of the test machine at 1490 r/min, with the stator windings in series-parallel connection. One of the parallel paths in phase #3 is open-circuited, and the inputs to the remaining two parallel paths are labeled 3A and 3B on the diagrams. The agreement between calculated and experimental results is again seen to be good, and the balance between phases is better than that obtained with the parallel connection, given in Fig. 7(a)–(c).
Fig. 7. Predicted and measured values of (a) current, (b) input active power, and (c) input reactive power for a parallel winding with an open-circuit fault in one leg and predicted values for the equivalent series-connected winding.
Fig. 8. Prefault and open-circuit fault representations of a seriesparallel-connected winding, showing coil locations.
Table I shows the calculated values of the principle torque ripple components for an unfaulted machine, and for a machine with an open-circuit fault with series-connected windings and for both of the parallel connections discussed in this section. These data pertain to a speed of 1490 r/min, with the same “volts per coil” excitation. These data once again indicate that the presence of parallel paths has a general mitigating effect on the unwanted consequences of a winding open circuit.
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Fig. 10. Predicted and measured values of (a) supply current, (b) input active power, and (c) input reactive power for the six phase machine with phase #3 short circuit at full load (1464 r/min) and reduced volts (13.5% of rated value).
Fig. 9. Predicted and measured values of (a) current, (b) input active power, and (c) input reactive power for a series-parallel winding with an open circuit fault in leg #3. 3A and 3B are the parallel paths within the faulted leg. TABLE I CALCULATED TORQUE RIPPLE WITH AN OPEN -CIRCUIT FAULT FOR D IFFFERENT W INDING C ONFIGURATIONS
D. Short-Circuit Faults Short-circuit faults may occur in the slots or end windings, within a phase, between phases, or between phase and frame. The results below investigate short circuits within a phase, although the technique can also model phase-to-phase faults and phase-to-frame faults, provided that the return path from frame to inverter can be defined. The results are for a six-phase motor with eight series-connected coils per phase. As all coil terminations are brought out to a patch panel, a shorting link could be added to short out one or more complete coils. Fig. 10(a)–(c) compares experimentally measured and calculated values of the input current, power, and VARs in each of the six phases of the test machine at 1464 r/min, with one phase (#3) short circuit. The voltage has been reduced to 13.5% of rated line voltage for this test to keep the fault current within its rated value.
Fig. 11. Predicted and measured values of (a) supply current, (b) input active power, and (c) input reactive power for the six-phase machine with a single coil (3A) short circuit at full load (1464 r/min) and reduced volts (7.5% of rated value). 3B shows the rest of phase #3.
Fig. 11(a)–(c) repeats the comparison with a single coil from the phase (#3) short circuit. The short-circuited coil is labeled “3A” and the remainder of the phase as “3B.” The inverter voltage was applied across the complete phase. The voltage has been further reduced to 7.5% of rated line voltage for this test to keep the fault current within its rated value.
APSLEY AND WILLIAMSON: ANALYSIS OF MULTIPHASE INDUCTION MACHINES WITH WINDING FAULTS
The results show good agreement between the predicted and measured values on the unfaulted phases. Predicted shortcircuit current for the complete phase winding agrees well with the measured value. The agreement is less good for the short circuits on a single coil. At this low value of voltage endwinding and slot leakage effects become significant and limit the accuracy of the model. A developing short-circuit fault first appears at the motor terminals as an excessive current in the faulted phase. As the short-circuit propagates between turns, the two phases that are physically located adjacent to the faulted phase in the stator winding also exhibit an increase in input current. IV. C ONCLUSION The techniques of generalized harmonic analysis have been shown to provide accurate predictions of performance in multiphase cage induction machines with any form of open-circuit or short-circuit fault on the stator side. The model has been used to evaluate simple motor design strategies to enhance fault tolerance. For an open-circuit fault the inherent fault tolerance of the multiphase motor is illustrated, as the current in adjacent phases increases to compensate for the MMF that should have been provided by the faulted phase. The configuration of the stator windings (with series-parallel connections) can further mitigate the effect of an open-circuit fault. The model is able to predict the more demanding conditions of short-circuit faults. A PPENDIX C ALCULATION OF I MPEDANCES Fourier analysis is used to resolve each stator circuit into a harmonic conductor distribution. The conductor distribution for the mth stator circuit, for example, may be written in the form ν cm (y) = C m e−jνθ (6) ν
where ν takes all nonzero positive and negative values, and θ is ν an angular coordinate in the stator reference frame. C m is the ν th harmonic complex conductor distribution of the mth stator circuit, obtained by summing the contributions of all of its coils. m , and the If the number of coils in the mth stator circuit is Ncoils th i coil has Ni turns, then i i m Ncoils θ +θ 1 2 jν 2 ν 2 Kpν Kbνs Cm = Ni e (7) jπd i=1 where d θ1 , θ2 Kpν Kbνs
mean diameter of the air gap; angular coordinates of the slots carrying ith coil; ν th harmonic pitch factor; ν th harmonic stator slot mouth width factor; Kbνs
=
sin νd bs νd bs
where bs is the width of the stator slot mouth (m).
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If the mth stator circuit is excited with current √ im (t) = e 2I¯m ejωt
(9)
it produces an air-gap field that consists of a set of rotating harmonic waves √ jµ0 dI¯m C νm j(ωt−νθ) e 2 (10) Bg,m (θ, t) = e 2νg ν where g is the effective length of the air gap. The (forwards-acting) EMF which this induces in a stator circuit (say the nth ) is readily obtained from the flux cutting rule ∗ √ jµ0 ω πd3 I¯m C νm C νn jωt e (11) Un,m (t) = −e 2 2g 4ν ν where is the effective axial length of the core. Equation (11) may now be used to determine the coupling impedance which relates the current flowing in the mth stator circuit to the back EMF it induces in the nth ν
Z n,m
∗ ν
jµ0 ω πd3 C m C n . = 4ν 2 g
(12)
Substituting n = m into (12) gives the self-impedance of the mth stator circuit due to flux which crosses the air-gap. This must be augmented by the circuit resistance and the leakage reactance due to end turn and slot leakage in order to give the complete self-impedance of that circuit. In a reference frame fixed to the rotor, the air-gap flux given by (10) takes the form jµ0 dI¯m C ν √ m j(sν ωt−νθ ) e 2 (13) Bg,m (θ , t) = e 2νg ν where sν is the rotor slip with respect to the ν th harmonic and θ is the angular coordinate system fixed to rotor. The EMF that this flux distribution induces in a rotor located at the origin of the rotor coordinate system, consists of a set of time-harmonic components ν ν ¯ √ jµ0 sν ω d2 Ksk Im C m jsν ωt e UR,m (t) = −e 2 4ν 2 g ν (14) ν is the ν th harmonic skew factor. where Ksk Equation (14) relates the harmonic EMFs induced in the reference rotor bar to the current flowing in the mth stator circuit, and therefore enables the corresponding harmonic statorto-rotor coupling impedance to be defined ν
ν
Z R,m = (8)
ν Cm jµ0 sν ω d2 Ksk . 4ν 2 g
(15)
The same flux wave [i.e., that given by (13)] also induces EMFs in the other bars of the cage. These EMFs will have the same form as (14), but with phase shifts that depend on the spatial displacement of the bar concerned, and the harmonic
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number of the inducing wave. A 2ν-pole stator-driven flux wave will induce a 2ν-pole distribution of EMFs in the cage, and this in turn will produce a 2ν-pole distribution of rotor currents. The ν th harmonic current in the reference rotor bar is given by (1). This current distribution will produce a skewed air-gap field, which in the rotor reference frame is given by +∞ ν √ jµ0 Nb I¯R ν 2 Bg,R (θ , t) = e 2πg(ν + qNb ) q=−∞ j sν ωt−(ν+qNb ) θ − 2πλx Nb ×e . (16) Nonzero values of q in (16) give the rotor slot harmonic fields produced by the ν th harmonic rotor current distribution. x is the axial distance from the center of the core and λ is the rotor skew, in rotor slot pitches. The EMF that this air-gap flux distribution induces in the reference rotor bar may be used to determine a ν th harmonic self-impedance of the bar. This must be augmented by the effective resistance and leakage reactance per bar, allowing for deep bar effect at the appropriate frequency and for the impedance of the end rings. The final result is ν
ZR =
jµ0 sν ωd Nb + Rνb + jsν Xνb 4πν 2 g
(17)
where Rbν is the effective ν th harmonic bar resistance and Xbν is the effective ν th harmonic bar leakage reactance. The the rotor slot harmonics induce ripple-frequency EMFs in the stator circuits, which, in the context of the work of this paper, may be ignored. Only the ν th harmonic component of the distribution given by (16) induces supply frequency EMFs in the stator. When transformed into the stator reference frame this component is given by the expression ν √ jµ0 Nb I¯R j ωt−ν θ− 2πλx ν Nb e Bg,R (θ, t) = e 2 . (18) 2πνg As before, the EMF which this flux distribution induces in the mth stator circuit is readily determined from the flux-cutting rule. This may then be used to define the coupling impedance which relates the ν th harmonic distribution of currents in the rotor cage to the back EMF that it induces in the mth stator circuit ∗ ν
ν Z m,R
ν Cm jµ0 ω d2 Nb Ksk . = 2 4ν g
[5] G. K. Singh and V. Pant, “Analysis of a multiphase induction machine under fault condition in a phase-redundant AC drive system,” Electr. Mach. Power Syst., vol. 28, no. 6, pp. 577–590, Jun. 2000. [6] H. A. Toliyat, “Analysis and simulation of five-phase variable-speed induction motor drives under asymmetrical connections,” IEEE Trans. Power Electron., vol. 13, no. 4, pp. 748–756, Jul. 1998. [7] J.-R. Fu and T. A. Lipo, “Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase,” IEEE Trans. Ind. Appl., vol. 30, no. 5, pp. 1267–1274, Sep./Oct. 1994. [8] Y. Zhao and T. A. Lipo, “Modeling and control of a multi-phase induction machine with structural unbalance. Part I. Machine modeling and multidimensional current regulation,” IEEE Trans. Energy Convers., vol. 11, no. 3, pp. 570–577, Sep. 1996. [9] R. O. C. Lyra and T. A. Lipo, “Torque density improvement in a six-phase induction machine with third harmonic current injection,” IEEE Trans. Ind. Appl., vol. 38, no. 5, pp. 1351–1360, Sep./Oct. 2002. [10] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a concentrated winding induction machine for adjustable speed drive applications Part 1 (Motor Analysis),” IEEE Trans. Energy Convers., vol. 6, no. 4, pp. 679–683, Dec. 1991. [11] H. A. Toliyat and T. A. Lipo, “Transient analysis of cage induction machines under stator, rotor bar and end ring faults,” IEEE Trans. Energy Convers., vol. 10, no. 2, pp. 241–247, Jun. 1995. [12] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a concentrated winding induction machine for adjustable speed drive applications Part 2 (Motor Design and Performance),” IEEE Trans. Energy Convers., vol. 6, no. 4, pp. 684–692, Dec. 1991. [13] S. Williamson and E. R. Laithwaite, “Generalised harmonic analysis for the steady-state performance of sinusoidally-excited cage induction motors,” Proc. Inst. Elect. Eng., vol. 132, pt. B, no. 3, pp. 157–163, May 1985. [14] S. Williamson and A. C. Smith, “A unified approach to the analysis of single-phase induction motors,” IEEE Trans. Ind. Appl., vol. 35, no. 4, pp. 837–843, Jul./Aug. 1999. [15] R. C. Healey, S. Lesley, S. Williamson, and P. R. Palmer, “The measurement of transient electromagnetic torque in high performance electrical drives,” in Proc. 6th Int. Conf. Power Electronics and Variable Speed Drives, Nottingham, U.K., 1996, pp. 226–229. [16] S. Williamson, A. C. Smith, and C. Hodge, “Fault tolerance in multiphase propulsion motors,” in Proc. Inst. Marine Engineering, Science and Technology, Journal of Marine Engineering Technology, Edinburgh, U.K., 2004, pp. 3–7.
Judith Apsley (M’05) received the B.A. degree in electrical sciences from Cambridge University, Cambridge, U.K., in 1986, and the Ph.D. degree in electrical engineering from the University of Surrey, Guildford, U.K. in 1996 She started her career as a Technologist Apprentice with Westland Helicopters in 1982, moving to the Machines and Drives Group at ERA Technology from 1987 to 1995. Following a career break to raise a family, she joined the Power Conversion Group at the University of Manchester, Manchester, U.K., in 2001.
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R EFERENCES [1] E. E. Ward and H. Harer, “Preliminary investigation of an invertorfed 5-phase induction motor,” Proc. Inst. Elect. Eng., vol. 116, no. 6, pp. 980–984, Jun. 1969. [2] T. M. Jahns, “Improved reliability in solid state AC drives by means of multiple independent phase drive units,” IEEE Trans. Ind. Appl., vol. IA-16, no. 3, pp. 321–331, May/Jun. 1980. [3] E. A. Klingshirn, “High phase order induction motors. II. Experimental results,” IEEE Trans. Power App. Syst., vol. PAS-102, no. 1, pp. 54–59, Jan. 1983. [4] S. Williamson and A. C. Smith, “Pulsating torque and losses in multiphase induction machines,” IEEE Trans. Ind. Appl., vol. 39, no. 4, pp. 986–993, Jul./Aug. 2003.
Steve Williamson (M’81–SM’88–F’95) received the first degree and Ph.D. degree in electrical engineering from Imperial College, London, U.K. He first took up a lectureship at Aberdeen University. Leaving Aberdeen in 1981, he moved back to Imperial College, first as a Senior Lecturer then as a Reader. In 1989, he became a Professor in the Department of Engineering at Cambridge University, Cambridge, U.K., and shortly afterwards a Fellow at St. Johns College. In 1997, after 24 years as an Academic, he left the university system, to take up the post of Group Technical director for Brook Hansen, which was then part of the BTR group. Several takeovers and reorganizations saw him back in academia, this time at the University of Manchester, Manchester, U.K., where he is currently Head of the School of Electrical and Electronic Engineering. Dr. Williamson is a Fellow of the Royal Academy of Engineering.
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Analysis of Multiphase Induction Machines With Winding Faults Judith Apsley, Member, IEEE, and Steve Williamson, Fellow, IEEE
Abstract—The paper shows how the techniques of generalized harmonic analysis may be used to simulate the steady-state behavior of a multiphase cage induction motor with any form of open-circuit or short-circuit fault in the stator winding. The analytical model is verified using a four-pole machine with a 48-slot stator. Each coil of the stator winding of this machine is brought out to a patchboard that enables the stator to be configured for single-phase, two-phase, three-phase, four-phase, six-phase, or 12-phase excitation. Experimental results are compared with computer predictions for a six-phase machine with both open-circuit and short-circuit faults. Index Terms—Induction machines, multiphase machines, open circuit, short circuit, winding faults.
I. I NTRODUCTION
M
ULTIPHASE machines are finding favor in some variable-speed drive applications because they may have higher efficiency, produce lower magnitude pulsating torques, and be acoustically quieter than their three-phase counterparts [1]–[4]. One further advantage is that multiphase machines are said to be fault-tolerant, offering greater security in missioncritical applications. They appear to be particularly suited as the main propulsion motors for future marine applications. Several authors have addressed the issue of fault tolerance [2], [3], [5], [6], focusing for the large part on the behavior of machines in which a single phase is open circuited and on compensation strategies for open-circuit faults [6]–[8]. The compensation strategies in [7] and [8] assume sinusoidally distributed windings. This approximation is valid for a low number of phases, and is a useful simplification for control. In [9], the third harmonic of the winding distribution is also included. However, as the number of phases increases to a limit where there is only one slot per pole per phase, the magnetomotive force (MMF) approaches a rectangular waveform rather than a sinusoid. This is also true for the MMF contribution from a single faulty coil. The concentrated winding model [10], allows for the rectangular MMF in the calculation of interphase inductances, and applies the resulting differential equation model to the transient analysis of winding faults [11]. The model predicts motor waveforms in the time domain but is Paper IPCSD-05-079, presented at the 2005 IEEE International Electric Machines and Drives Conference, San Antonio, TX, May 15–18, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications Society. Manuscript submitted for review May 1, 2005 and released for publication November 19, 2005. The authors are with the School of Electrical and Electronic Engineering, University of Manchester, Manchester M60 1QD, U.K. (e-mail: j.apsley@ manchester.ac.uk; [email protected]). Digital Object Identifier 10.1109/TIA.2005.863915
computationally intense and does not readily give information about the magnitude and frequency of torque pulsations. An alternative steady-state approach is to use Fourier series expansions of the air-gap field spatial distribution and supply time variation [4], [12]. In this paper the authors will describe a form of analysis that is readily able to model the effect that more complex forms of fault have on machine behavior, including intraphase short circuits and open circuits in machines with parallel paths. The purpose of the paper is to describe the use of a powerful and versatile technique, rather than to give a comprehensive exposition of the consequences of winding faults. For that reason, all of the results presented in this paper refer to a sixphase winding, although the authors have carried out extensive validation at other phase numbers. The technique is verified by experimental measurements. II. A NALYTICAL M ETHOD A. Generalized Harmonic Analysis The analytical model is based on the techniques of generalized harmonic analysis, which has been described in detail in the literature [4], [13] and will, therefore, only be summarized in this paper. The model assumes sinusoidal excitation and constant rotor speed. Converter-fed machines are dealt with using frequency domain analysis and superposing timeharmonic solutions. The method involves calculating coupling impedances which relate the back electromotive forces (EMFs) induced in each circuit of the machine to the currents flowing in those same circuits. A stator circuit is defined as any seriesconnected set of coils. It might consist of an entire phase winding, if all the coils in that winding are connected in series. Alternatively it might be a single turn of a coil if that turn is short circuited. The definition of a “circuit” on the rotor differs from that used for the stator. Here, a “circuit” is a rotor current distribution of known frequency, pole number, and sense of rotation. We will, for example, refer to the ν th harmonic rotor current distribution. This will indicate that a particular rotor bar (the “reference” bar) carries a current given by iνR (t) = e
√ ν jsν ωt . 2I¯R e
(1)
All other bars carry currents of the same magnitude and frequency, with a phase progression of 2νπ/Nb radians from bar to bar: Nb number of rotor bars; rotor slip with respect to 2ν-pole wave. sν
0093-9994/$20.00 © 2006 IEEE
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There are four different types of coupling impedance: Z m,n coupling impedance which relates the EMF induced in the mth stator circuit to the current flowing in the nth stator circuit; ν Z R,m coupling impedance relating the ν th harmonic EMF induced in the reference rotor bar to the current flowing in the mth stator circuit; ν Z m,R coupling impedance relating the EMF induced in the mth stator circuit to the ν th space-harmonic current distribution in the rotor; ν ZR ν th -harmonic impedance per bar of the rotor. A summary of the way in which all of these impedances may be calculated is given in the Appendix. The voltage balance equation for the mth stator circuit, is given by the general equation [13] Vm =
M
I¯n Z m,n +
ν ν I¯R Z m,R
(2)
Fig. 1. Loop diagram for one phase of the six-phase motor (a) prefault and (b) following a short circuit in one coil.
ν
n=1
where V m voltage applied to mth stator circuit; In current flowing in the nth stator circuit; M number of stator circuits. The range of possible values for the harmonic index is not specified in (2). The number of harmonic rotor current distributions that must be taken into account will depend on the operating condition of the machine. Stator winding faults introduce electromagnetic asymmetry, so the air-gap field tends to be rich in harmonics. In the work presented in this paper the authors found is necessary to consider all positive and negative values of ν in the range up to 4 times the number of stator slots. The voltage balance equation for the ν th harmonic rotor current distribution takes a similar form to (2) ν ν ZR + 0 = I¯R
M
ν I¯m Z R,m .
(3)
m=1
Substitution from (3) into (2) enables the stator voltage balance equation to be expressed in terms of the stator currents only M Z νm,R Z νR,n ¯ In Z m,n − . (4) Vm = ν ZR ν n=1 There are M such equations, which may be written in compact form V = ZI.
(5)
Z is an M × M impedance matrix, and V and I are voltage and current vectors of dimension M . If the voltages applied to all stator circuits are known, (5) may be solved to determine the currents flowing in each stator circuit, without further manipulation. Commonly, however, the stator circuits are interconnected in some way (e.g., in star) so that the voltages applied to each circuit may not necessarily be specified. Under such circumstances, (5) must be transformed from circuit (i.e., branch) variables
to loop variables using the connection matrix method [14]. Once all the stator currents have been determined, the rotor harmonic current distributions can be recovered via (3), and the solution completed by calculating torques, air-gap fields, and losses. B. Representation of Winding Faults The formal structure of the method outlined above enables a single turn, a coil, or a set of coils to be defined as a circuit, and the corresponding coupling impedances to be determined. One or more open-circuited windings are accommodated in the model by setting the relevant circuit currents to 0, and deleting the corresponding voltage equations. Each open circuit reduces the dimension of the voltage balance equation [i.e., (5)] by 1. Short circuits, on the other hand tend to increase the dimension of the voltage balance equation. Fig. 1(a) for example, represents one phase of a series-connected winding in its prefault configuration, in which the whole of the phase winding is regarded as a single circuit, as defined above. Fig. 1(b) shows the same phase winding with a short-circuit fault. It is now regarded as two independent circuits, corresponding to the shorted and the unshorted parts of the winding. The currents flowing in these two circuits will differ from each other, increasing the dimension of the voltage balance equation by 1. For the purposes of illustration, the short circuit is shown at one end of the phase, but it could equally well have been positioned at any point in the winding. Had the whole of the phase winding been short circuited, it would have been necessary only to set the voltage applied to that winding to 0. C. Test Rig The experimental results used in this paper were verified on an experimental test rig, based on a rewound D180 frame 50 Hz four-pole squirrel-cage motor. The 48-slot stator was fitted with a double-layer mush winding, with both ends of every stator coil brought out to an external patchboard. This enabled the windings to be reconfigured readily for 1, 2, 3, 4, 6, or
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Fig. 2. Predicted and measured values of (a) supply current (A rms), (b) input active power (W), and (c) input reactive power (VAR) for the six-phase, nominally balanced machine at four load points.
12 phase excitation. The stator of the test machine was excited from a specially constructed converter, consisting of 12 independent H-bridges. These provided open-loop voltage-forced excitation of the appropriate phase number. As is common with multiphase machines, each phase was separately excited. There were no star points. For all of the results presented in this paper the machine was connected in six-phase mode, i.e., with two slots per pole and phase. This gave a good compromise between the number of measurement channels required, and the different ways of connecting coils within a winding. The motor was mounted on a three-axis piezoelectric force table, for measuring transient and pulsating torques, as described in [15], enabling validation of predicted torque ripple both pre and post fault. The force table has a low-frequency limit of 1 Hz, so dc torque was measured with an in-line shaft transducer. Motor terminal quantities were measured with a Norma D6000 power analyser. Measurements were made for a harmonic sweep up to the 15th harmonic of the fundamental (50 Hz) excitation and for total quantities. Results for current and input active and reactive power show the fundamental only, but results for torque include harmonics up to the 15th.
III. R ESULTS A. Balanced Machine In order to check the validity of the machine parameters used in the model, the first set of simulations and experiments were carried out with the stator winding in the unfaulted condition. The results obtained are summarized in Fig. 2(a)–(c), which give fundamental input current, active power and reactive power or volt-ampere reactives (VARs) to each of the six phases, for full-load, no-load, and for two part-loaded conditions. The results refer to the fundamental-frequency (50 Hz) component of the excitation. Imbalances are clearly evident even in these prefault results, arising from small asymmetries in the individual H-bridges of the inverter. These are manifested as small differences between the voltages applied to the six phases, which produce differences in the phase currents. The simulation results use the measured values of the applied phase voltages as input data, and so the calculated results also reflect these imbalances. The ability to handle unbalanced voltages with ease is one of the advantages of generalized harmonic analysis. The results presented in Fig. 2(a)–(c) show good agreement
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Fig. 4. Predicted and measured torque ripple in the prefault mode (gray) and with phase #3 open circuit (black), for a six-phase motor at full load (1475 r/min).
Fig. 3. Predicted and measured values of (a) supply current, (b) input active power and (c) input reactive power for the six-phase machine with phase #3 open circuit at full load (1475 r/min).
between experimental measurement and calculation, indicating that the machine parameters used in the simulation work are of sufficient accuracy. B. Open-Circuit Fault: Series-Connected Winding The single open-circuited phase winding is arguably the commonest form of winding fault, arising from either a fault in the winding itself or in the converter leg that it is connected to. Fig. 3(a)–(c) shows the current and input active and reactive power at full load (1475 r/min) in the six-phase machine with one phase (#3) open circuited. Again, good agreement is obtained between the measurement and calculation. The results also show that the greatest impact is seen in the two phases that are physically located adjacent to the faulted phase in the stator winding, which exhibit an increase in input current in an attempt to provide the air-gap MMF that is missing in their locality. Fig. 4 shows the predicted and measured magnitudes of the torque ripple obtained at full load, over the frequency range 0–700 Hz. The results show distinct components at multiples of twice supply frequency, as would be expected in any machine with a winding imbalance. In order to show the effect of the open-circuited winding, the calculated magnitudes of the torques produced at these frequencies are also marked on Fig. 4. These show that the open circuit produces, for example, a 25 dB (i.e., 18-fold) increase in the predicted 100-Hz torque pulsation. Where predicted values for the prefault condition are not marked on Fig. 4, they are less than the noise floor of 0.1 Nm.
Fig. 5. Comparison of calculated (a) steady torque and (b) 100 Hz torque ripple, before (solid) and after (dashed) an open circuit fault, and using fault compensation (dotted).
Fig. 5(a) and (b) presents calculated curves to illustrate the performance of the machine before and after a single-phase open-circuit fault. In the postfault curves it is assumed that the excitation to the unfaulted phases is not changed when the fault occurs. Fig. 5(a) and (b) shows that this will produce a small increase in slip, but a significant increase in torque ripple, as already observed. If the current in the remaining phases is increased to obtain the same fundamental (i.e., 4-pole) component of MMF, then the curves labeled “compensated” are produced, increasing the steady torque to its prefault level and significantly attenuating the magnitude of the pulsating torque [16]. C. Open-Circuit Faults: Parallel-Connected Winding In the six-phase configuration the stator winding of the experimental motor has eight coils per phase. These may all be connected in series (as above) or in two, four, or eight parallel
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Fig. 6. Prefault and open-circuit fault representations of a parallel-connected winding, showing coil locations.
paths. Parallel paths ameliorate the effects of open circuit faults by allowing currents to flow in some of the coils in the faulted phase, thereby providing a measure of fault tolerance. In this section of the paper, we explore the effect of an open circuit in the case of a stator winding with two parallel paths, as shown in Fig. 6(a), prior to the fault and Fig. 6(b) with an opencircuit fault. Fig. 7(a)–(c) compares experimentally measured and calculated values of the input current, power, and VARs in each of the six phases of the test machine at 1490 r/min, with one of the parallel paths in phase #3 open circuited. As before, the agreement between calculated and experimental results is seen to be good. Fig. 7(a)–(c) also includes the calculated values of the corresponding inputs that would be obtained at 1490 r/min in the series connected machine with an open-circuit fault on phase #3. For the “series-connected” calculations the applied voltage is twice that used for the “parallel-connected” experiment and calculation, and the line current has been scaled by a factor of 2 in order to give a meaningful comparison. These results clearly show that the parallel connection does indeed reduce the imbalance produced by the open-circuit fault. In a further attempt to reduce the disturbance produced by an open-circuit fault, more parallel paths may be used. Alternatively, the eight coils of each phase may be connected in a series/parallel arrangement, as shown in Fig. 8(a). When an open-circuit fault occurs on one of the coils, the connection diagram for the faulted phase becomes that shown in Fig. 8(b), which shows that current will still be able to flow in six of the eight coils of this phase. Fig. 9(a)–(c) compares experimentally measured and calculated values of the input current, power, and VARs in each of the six phases of the test machine at 1490 r/min, with the stator windings in series-parallel connection. One of the parallel paths in phase #3 is open-circuited, and the inputs to the remaining two parallel paths are labeled 3A and 3B on the diagrams. The agreement between calculated and experimental results is again seen to be good, and the balance between phases is better than that obtained with the parallel connection, given in Fig. 7(a)–(c).
Fig. 7. Predicted and measured values of (a) current, (b) input active power, and (c) input reactive power for a parallel winding with an open-circuit fault in one leg and predicted values for the equivalent series-connected winding.
Fig. 8. Prefault and open-circuit fault representations of a seriesparallel-connected winding, showing coil locations.
Table I shows the calculated values of the principle torque ripple components for an unfaulted machine, and for a machine with an open-circuit fault with series-connected windings and for both of the parallel connections discussed in this section. These data pertain to a speed of 1490 r/min, with the same “volts per coil” excitation. These data once again indicate that the presence of parallel paths has a general mitigating effect on the unwanted consequences of a winding open circuit.
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Fig. 10. Predicted and measured values of (a) supply current, (b) input active power, and (c) input reactive power for the six phase machine with phase #3 short circuit at full load (1464 r/min) and reduced volts (13.5% of rated value).
Fig. 9. Predicted and measured values of (a) current, (b) input active power, and (c) input reactive power for a series-parallel winding with an open circuit fault in leg #3. 3A and 3B are the parallel paths within the faulted leg. TABLE I CALCULATED TORQUE RIPPLE WITH AN OPEN -CIRCUIT FAULT FOR D IFFFERENT W INDING C ONFIGURATIONS
D. Short-Circuit Faults Short-circuit faults may occur in the slots or end windings, within a phase, between phases, or between phase and frame. The results below investigate short circuits within a phase, although the technique can also model phase-to-phase faults and phase-to-frame faults, provided that the return path from frame to inverter can be defined. The results are for a six-phase motor with eight series-connected coils per phase. As all coil terminations are brought out to a patch panel, a shorting link could be added to short out one or more complete coils. Fig. 10(a)–(c) compares experimentally measured and calculated values of the input current, power, and VARs in each of the six phases of the test machine at 1464 r/min, with one phase (#3) short circuit. The voltage has been reduced to 13.5% of rated line voltage for this test to keep the fault current within its rated value.
Fig. 11. Predicted and measured values of (a) supply current, (b) input active power, and (c) input reactive power for the six-phase machine with a single coil (3A) short circuit at full load (1464 r/min) and reduced volts (7.5% of rated value). 3B shows the rest of phase #3.
Fig. 11(a)–(c) repeats the comparison with a single coil from the phase (#3) short circuit. The short-circuited coil is labeled “3A” and the remainder of the phase as “3B.” The inverter voltage was applied across the complete phase. The voltage has been further reduced to 7.5% of rated line voltage for this test to keep the fault current within its rated value.
APSLEY AND WILLIAMSON: ANALYSIS OF MULTIPHASE INDUCTION MACHINES WITH WINDING FAULTS
The results show good agreement between the predicted and measured values on the unfaulted phases. Predicted shortcircuit current for the complete phase winding agrees well with the measured value. The agreement is less good for the short circuits on a single coil. At this low value of voltage endwinding and slot leakage effects become significant and limit the accuracy of the model. A developing short-circuit fault first appears at the motor terminals as an excessive current in the faulted phase. As the short-circuit propagates between turns, the two phases that are physically located adjacent to the faulted phase in the stator winding also exhibit an increase in input current. IV. C ONCLUSION The techniques of generalized harmonic analysis have been shown to provide accurate predictions of performance in multiphase cage induction machines with any form of open-circuit or short-circuit fault on the stator side. The model has been used to evaluate simple motor design strategies to enhance fault tolerance. For an open-circuit fault the inherent fault tolerance of the multiphase motor is illustrated, as the current in adjacent phases increases to compensate for the MMF that should have been provided by the faulted phase. The configuration of the stator windings (with series-parallel connections) can further mitigate the effect of an open-circuit fault. The model is able to predict the more demanding conditions of short-circuit faults. A PPENDIX C ALCULATION OF I MPEDANCES Fourier analysis is used to resolve each stator circuit into a harmonic conductor distribution. The conductor distribution for the mth stator circuit, for example, may be written in the form ν cm (y) = C m e−jνθ (6) ν
where ν takes all nonzero positive and negative values, and θ is ν an angular coordinate in the stator reference frame. C m is the ν th harmonic complex conductor distribution of the mth stator circuit, obtained by summing the contributions of all of its coils. m , and the If the number of coils in the mth stator circuit is Ncoils th i coil has Ni turns, then i i m Ncoils θ +θ 1 2 jν 2 ν 2 Kpν Kbνs Cm = Ni e (7) jπd i=1 where d θ1 , θ2 Kpν Kbνs
mean diameter of the air gap; angular coordinates of the slots carrying ith coil; ν th harmonic pitch factor; ν th harmonic stator slot mouth width factor; Kbνs
=
sin νd bs νd bs
where bs is the width of the stator slot mouth (m).
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If the mth stator circuit is excited with current √ im (t) = e 2I¯m ejωt
(9)
it produces an air-gap field that consists of a set of rotating harmonic waves √ jµ0 dI¯m C νm j(ωt−νθ) e 2 (10) Bg,m (θ, t) = e 2νg ν where g is the effective length of the air gap. The (forwards-acting) EMF which this induces in a stator circuit (say the nth ) is readily obtained from the flux cutting rule ∗ √ jµ0 ω πd3 I¯m C νm C νn jωt e (11) Un,m (t) = −e 2 2g 4ν ν where is the effective axial length of the core. Equation (11) may now be used to determine the coupling impedance which relates the current flowing in the mth stator circuit to the back EMF it induces in the nth ν
Z n,m
∗ ν
jµ0 ω πd3 C m C n . = 4ν 2 g
(12)
Substituting n = m into (12) gives the self-impedance of the mth stator circuit due to flux which crosses the air-gap. This must be augmented by the circuit resistance and the leakage reactance due to end turn and slot leakage in order to give the complete self-impedance of that circuit. In a reference frame fixed to the rotor, the air-gap flux given by (10) takes the form jµ0 dI¯m C ν √ m j(sν ωt−νθ ) e 2 (13) Bg,m (θ , t) = e 2νg ν where sν is the rotor slip with respect to the ν th harmonic and θ is the angular coordinate system fixed to rotor. The EMF that this flux distribution induces in a rotor located at the origin of the rotor coordinate system, consists of a set of time-harmonic components ν ν ¯ √ jµ0 sν ω d2 Ksk Im C m jsν ωt e UR,m (t) = −e 2 4ν 2 g ν (14) ν is the ν th harmonic skew factor. where Ksk Equation (14) relates the harmonic EMFs induced in the reference rotor bar to the current flowing in the mth stator circuit, and therefore enables the corresponding harmonic statorto-rotor coupling impedance to be defined ν
ν
Z R,m = (8)
ν Cm jµ0 sν ω d2 Ksk . 4ν 2 g
(15)
The same flux wave [i.e., that given by (13)] also induces EMFs in the other bars of the cage. These EMFs will have the same form as (14), but with phase shifts that depend on the spatial displacement of the bar concerned, and the harmonic
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number of the inducing wave. A 2ν-pole stator-driven flux wave will induce a 2ν-pole distribution of EMFs in the cage, and this in turn will produce a 2ν-pole distribution of rotor currents. The ν th harmonic current in the reference rotor bar is given by (1). This current distribution will produce a skewed air-gap field, which in the rotor reference frame is given by +∞ ν √ jµ0 Nb I¯R ν 2 Bg,R (θ , t) = e 2πg(ν + qNb ) q=−∞ j sν ωt−(ν+qNb ) θ − 2πλx Nb ×e . (16) Nonzero values of q in (16) give the rotor slot harmonic fields produced by the ν th harmonic rotor current distribution. x is the axial distance from the center of the core and λ is the rotor skew, in rotor slot pitches. The EMF that this air-gap flux distribution induces in the reference rotor bar may be used to determine a ν th harmonic self-impedance of the bar. This must be augmented by the effective resistance and leakage reactance per bar, allowing for deep bar effect at the appropriate frequency and for the impedance of the end rings. The final result is ν
ZR =
jµ0 sν ωd Nb + Rνb + jsν Xνb 4πν 2 g
(17)
where Rbν is the effective ν th harmonic bar resistance and Xbν is the effective ν th harmonic bar leakage reactance. The the rotor slot harmonics induce ripple-frequency EMFs in the stator circuits, which, in the context of the work of this paper, may be ignored. Only the ν th harmonic component of the distribution given by (16) induces supply frequency EMFs in the stator. When transformed into the stator reference frame this component is given by the expression ν √ jµ0 Nb I¯R j ωt−ν θ− 2πλx ν Nb e Bg,R (θ, t) = e 2 . (18) 2πνg As before, the EMF which this flux distribution induces in the mth stator circuit is readily determined from the flux-cutting rule. This may then be used to define the coupling impedance which relates the ν th harmonic distribution of currents in the rotor cage to the back EMF that it induces in the mth stator circuit ∗ ν
ν Z m,R
ν Cm jµ0 ω d2 Nb Ksk . = 2 4ν g
[5] G. K. Singh and V. Pant, “Analysis of a multiphase induction machine under fault condition in a phase-redundant AC drive system,” Electr. Mach. Power Syst., vol. 28, no. 6, pp. 577–590, Jun. 2000. [6] H. A. Toliyat, “Analysis and simulation of five-phase variable-speed induction motor drives under asymmetrical connections,” IEEE Trans. Power Electron., vol. 13, no. 4, pp. 748–756, Jul. 1998. [7] J.-R. Fu and T. A. Lipo, “Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase,” IEEE Trans. Ind. Appl., vol. 30, no. 5, pp. 1267–1274, Sep./Oct. 1994. [8] Y. Zhao and T. A. Lipo, “Modeling and control of a multi-phase induction machine with structural unbalance. Part I. Machine modeling and multidimensional current regulation,” IEEE Trans. Energy Convers., vol. 11, no. 3, pp. 570–577, Sep. 1996. [9] R. O. C. Lyra and T. A. Lipo, “Torque density improvement in a six-phase induction machine with third harmonic current injection,” IEEE Trans. Ind. Appl., vol. 38, no. 5, pp. 1351–1360, Sep./Oct. 2002. [10] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a concentrated winding induction machine for adjustable speed drive applications Part 1 (Motor Analysis),” IEEE Trans. Energy Convers., vol. 6, no. 4, pp. 679–683, Dec. 1991. [11] H. A. Toliyat and T. A. Lipo, “Transient analysis of cage induction machines under stator, rotor bar and end ring faults,” IEEE Trans. Energy Convers., vol. 10, no. 2, pp. 241–247, Jun. 1995. [12] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a concentrated winding induction machine for adjustable speed drive applications Part 2 (Motor Design and Performance),” IEEE Trans. Energy Convers., vol. 6, no. 4, pp. 684–692, Dec. 1991. [13] S. Williamson and E. R. Laithwaite, “Generalised harmonic analysis for the steady-state performance of sinusoidally-excited cage induction motors,” Proc. Inst. Elect. Eng., vol. 132, pt. B, no. 3, pp. 157–163, May 1985. [14] S. Williamson and A. C. Smith, “A unified approach to the analysis of single-phase induction motors,” IEEE Trans. Ind. Appl., vol. 35, no. 4, pp. 837–843, Jul./Aug. 1999. [15] R. C. Healey, S. Lesley, S. Williamson, and P. R. Palmer, “The measurement of transient electromagnetic torque in high performance electrical drives,” in Proc. 6th Int. Conf. Power Electronics and Variable Speed Drives, Nottingham, U.K., 1996, pp. 226–229. [16] S. Williamson, A. C. Smith, and C. Hodge, “Fault tolerance in multiphase propulsion motors,” in Proc. Inst. Marine Engineering, Science and Technology, Journal of Marine Engineering Technology, Edinburgh, U.K., 2004, pp. 3–7.
Judith Apsley (M’05) received the B.A. degree in electrical sciences from Cambridge University, Cambridge, U.K., in 1986, and the Ph.D. degree in electrical engineering from the University of Surrey, Guildford, U.K. in 1996 She started her career as a Technologist Apprentice with Westland Helicopters in 1982, moving to the Machines and Drives Group at ERA Technology from 1987 to 1995. Following a career break to raise a family, she joined the Power Conversion Group at the University of Manchester, Manchester, U.K., in 2001.
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R EFERENCES [1] E. E. Ward and H. Harer, “Preliminary investigation of an invertorfed 5-phase induction motor,” Proc. Inst. Elect. Eng., vol. 116, no. 6, pp. 980–984, Jun. 1969. [2] T. M. Jahns, “Improved reliability in solid state AC drives by means of multiple independent phase drive units,” IEEE Trans. Ind. Appl., vol. IA-16, no. 3, pp. 321–331, May/Jun. 1980. [3] E. A. Klingshirn, “High phase order induction motors. II. Experimental results,” IEEE Trans. Power App. Syst., vol. PAS-102, no. 1, pp. 54–59, Jan. 1983. [4] S. Williamson and A. C. Smith, “Pulsating torque and losses in multiphase induction machines,” IEEE Trans. Ind. Appl., vol. 39, no. 4, pp. 986–993, Jul./Aug. 2003.
Steve Williamson (M’81–SM’88–F’95) received the first degree and Ph.D. degree in electrical engineering from Imperial College, London, U.K. He first took up a lectureship at Aberdeen University. Leaving Aberdeen in 1981, he moved back to Imperial College, first as a Senior Lecturer then as a Reader. In 1989, he became a Professor in the Department of Engineering at Cambridge University, Cambridge, U.K., and shortly afterwards a Fellow at St. Johns College. In 1997, after 24 years as an Academic, he left the university system, to take up the post of Group Technical director for Brook Hansen, which was then part of the BTR group. Several takeovers and reorganizations saw him back in academia, this time at the University of Manchester, Manchester, U.K., where he is currently Head of the School of Electrical and Electronic Engineering. Dr. Williamson is a Fellow of the Royal Academy of Engineering.