Behavior of a Pentacle Connected Five-Phase IM Supplied by a Rectangular Voltage

The presented article deals with the steadystate analysis of five-phase Induction Motor (IM), which is supplied by rectangular voltage from a fiveleg Voltage Source Inverter (VSI). The new proposed approach is derived from complex Fourierseries using to express inverter terminal voltages. Based on mathematically described inverter output voltages, a formula for VSI output space phasor was derived. Assuming sinusoidal motor winding distribution and on the basis of motor parameters, stator and rotor current space phasors for defined motor load were determined. Finally, the electromagnetic torque ripple waveforms for different IM operation states were investigated.


Introduction
Variable speed electric drives, in general, preferentially utilize three-phase induction machines. Providing that the variable speed Alternating Current (AC) drives require a power electronic converter, the number of machine phases is practically unlimited. This fact led to an increasing interest for multiphase AC drives applications. Most often, they are five-phase induction machines that, by their nature, offer some advantages over their three-phase counterparts.
E. E. Ward and Harrer [1], for the first time in 1969, have presented the preliminary investigation on inverter fed five-phase IM and suggested that the am-plitude of torque pulsation could be reduced by increasing the number of phases [2].
Major advantages of using a five-phase machine instead of the three-phase one consist of its higher torque density, greater efficiency and fault tolerance [3], [4] and [5]. Other advantages include reduced electromagnetic torque pulsation and reduction in the required rating per inverter leg. Noise characteristics of the five-phase drives are better when compared with the three-phase ones. For this reason, it is expected to use them in residential areas and hospitals where noise presents an undesirable element.
In the majority of cases, the supply for a multiphase variable-speed AC drives is provided by a Voltage Source Inverter (VSI) [6]. Output voltage control has a significant effect on the electromagnetic torque ripple of the motor. Control of VSI is provided in prevail by Pulse Width Modulation (PWM) techniques. Space Vector PWM (SVPWM) has become the most popular because of its easy digital implementation. However, some industrial applications of the five-phase drives do not need to modulate the output voltage. Because the rectangular voltage supply causes the ripple in motor's electromagnetic torque, the inverter control is easier and reduces requirements on semiconductor switching units [7], [8], [9] and [10].

2.
Modeling of a Five-Phase VSI The power circuit topology of a five-phase source inverter is shown in Fig. 1. VSI consists of a parallel connection of five transistor legs denoted (a, b, . . . , e). It is supplied by a constant voltage source provided by an isolated Direct Current (DC)-source and a capacitive DC-link. Each leg is composed of two Insulated- Gate Bipolar Transistor (IGBT) transistors with antiparallel connected free-wheeling diodes used to ensure a negative current path through the switches. Inverters output terminals are numbered (1, 2, . . . , 5) [11].
To build a mathematical model of VSI, the complex Fourier series were used. They are extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be solved individually and then recombined to obtain the solution to the original problem.
In the next, we assume idealized semiconductor devices which satisfy the following properties: • Power switches can handle unlimited current, and they are able to block unlimited voltage.
• Voltage drop across the switch and leakage current are zero.
• The switches are turned on and off with no rise and fall times.
• Inverter input capacity is sufficiently high so we can suppose the converter input DC voltage constant for any output currents.
These assumptions simplify analysis of the power circuit and help to build against mathematical model.
The transistors in each leg are switched so that they form voltage impulses over the half period of the desired output frequency. The voltage impulse of the first transistors leg measured against the negative pole of the DC link can be expressed as [12], [13] and [14]: where u dc is DC and u 01 is AC voltage component, U e is DC link voltage and c k is Fourier coefficient defined as: (2) Figure 2 shows the first leg output voltage waveform calculated on the base of Eq. (1). This was drawn for DC supply U e = 350 V and output frequency f = 50 Hz. Voltages of the other legs are mutually displaced by the voltage shifting factor a = e j 2kπ 5 . Then, we can express them as: Motor phase voltages are given by the difference of voltages between two legs. Since a u dc = a 2 u dc = a 3 u dc = a 4 u dc = u dc (DC voltage shift), the phase voltages can be expressed as follows: c k e jkωt is the AC component of the voltage u a . Figure 3 shows a phasor diagram of the pentacle connected five-phase voltage system. The phase voltages are mutually shifted by 2 2π 5 . These form a secondorder voltage system.  To simplify the calculation of AC motor quantities, it is advantageous to employ space phasors. This transformation is very often used for the analysis of multiphase electric systems. The term "space" originally stands for the two-dimensional complex plane, in which the multi-phase quantities are transformed.
The transformation of space phasor is directly derived from the sum of voltage phase phasors. Based on the Eq. (4), the space phasor transformation is thus defined as published in [15], [16] and [17]: where a 1 = e j2 2π 5 is the space shifting factor for the second-order system.  The real and imaginary parts of the voltage space phasor can be separated and rewritten as: By using space phasor, the multi-phase components are transformed into a two-dimensional coordinate system. They are shown in Fig. 6. For further calculations, it is necessary to perform harmonic analysis of the two-phase voltage system. The following equations apply to the amplitudes of two-phase voltage harmonic components A αk and A βk , where k = 1, 2, 3, . . . : abs Re(u 1 ) + Re(u 2 )a 1 + Re(u 3 )a 2 1 + +Re(u 4 )a 3 1 + Re(u 5 )a 4 1 , abs Im(u 1 ) + Im(u 2 )a 1 + Im(u 3 )a 2 1 + +Im(u 4 )a 3 1 + Im(u 5 )a 4 1 .

Motor Currents Calculation
For the stator and rotor current space phasors calculation, a classical equivalent circuit of IM shown in Fig. 8 was used. The calculation was carried out used measured parameters of five-phase two poles IM. The primed quantities are rated to the effective number of turns of the stator winding (see App. A). Referred to the equivalent circuit above, the following equation for the stator current space vector is valid: where X 1 = X 1σ + X m , X 2 = X 2σ + X m and s k = kpω−ωm kpω -slip for the positive, -slip for the negative sequence components. ω m is the mechanical motor speed.
For the stator and rotor currents components: where, D = (R 1 + jX 1 ) R 2 s k + jR 1 X 1 − X 1 X 2 + X 2 m . The sum of current components defines the stator and rotor space phasor: Calculated stator and rotor current space phasors trajectories are shown in the Fig. 8. The computation was made for nominal motor load and speed (2850 rev·min −1 ). The motor parameters are listed in App. A.

Electromagnetic Torque Calculation
Electromagnetic torque of IM can be determined from the stator or rotor current values according to the formula [18] and [19]: By using phasors spatial relationship, the Eq (14) comes into the form: t (ms) → We can express the magnetic flux associated with the stator and rotor in terms of inductance and current. The relationship Eq. (15) changes into the following form: [18] The calculated time plot of the electromagnetic torque waveform of the five-phase IM pentacle connected and supplied by a rectangular voltage is shown in Fig. 10. The motor is loaded by a nominal load and works at 2850 rev·min −1 . In the electromagnetic torque waveform, the fifth harmonic component is well visible. The ripple presents about 20 % of the load torque. Torque magnitude is invariant with load change.

Conclusion
The article discusses the electromagnetic torque calculation of the five-phase IM which is pentacle connected. The motor is supplied by a five-phase VSI inverter with rectangular output voltage. Mathematical model using the space phasor theory in the complex plane was build.
In the calculated waveform of electromagnetic torque, a huge five harmonic component is visible. This one stays constant with a change in load. It varies only with the shape of the supply voltage.