4.2.1 Faultless Rotor Cage Model
The SCIM analysed has no-integral number of rotor bars per pole. It is better to model the cage as having a number of identical magnetically coupled circuits (Monoz & Lipo, 1974). The rotor cage equivalent circuit representing the impedance nature of the circuit, with symmetric faultless rotor cage bars (Lipo, 1996) and (Wallace & Wright, 1974) is depicted in Fig. 4.1a, and its current phasor representation is shown in Fig. 4.1b. The rotor resistance and rotor inductance matrixes are well detailed in (Monoz & Lipo, 1974).
(a)
(b)
Figure 4.1: Model of the rotor cage (only three bars are shown); (a) Equivalent model, (b) Rotor current phasor representation
The corresponding bar currents can be expressed as (Boldea & Nasar, 2001) and (Boldea & Tutelea, 2009):
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πΌππ+1 = ππ π+2β ππ π+1 = 2πΌπ π+1sin (πππ2 ) (4.2)
πΌππ+2= ππ π+3β ππ π+2 = 2πΌπ π+2sin (πππ
2 ) (4.3)
Where Ξ΅er is the time phase shift between currents in adjacent end ring segments Isk and Isk+1, and Ibj is the current in the bar j, which corresponds to the difference between currents in adjacent ring segments isk and isk+1.
Several literature studies have been published on improving induction machine performance through geometry re-design, with much focus on stator slot opening, stator and rotor bar shape and use of magnetic and non-magnetic wedges (Jelassi, et al., 2013, Salon, et al., 2002, Appiah,
et al., 2013, Kappatou, et al., 2008, Madescu, et al., 2012 and Galindo, et al., 2002). No work
was dedicated to re-designing the geometry of the induction machine (particularly the rotor bar shape) in view of enhancing the performance of SCIMs when rotor bars are broken.
However, this section is focused on analysing the effect that different rotor bar shapes have upon performance of the SCIM under healthy and broken rotor bar operating conditions. In this regards, five different rotor bar shapes with the same cross-sectional area as shown in Fig. 4.2, including the original rotor shape in Table 4.1 are modelled and analysed. Since the results of the conventional induction machine (original rotor shape) are analysed and discussed in the previous chapter, this chapter focused mostly on the four different rotor bar shapes as shown in Fig. 4.2, whose performance indexes are analysed both under healthy and faulty operating conditions.
The behaviour of the machine for various shapes of the rotor bars under both healthy and faulty conditions is studied. In the case of broken rotor bars, all broken bars are adjacent. During induction motor start-up, the current migrates to the top of the bar (lowest impedance region) as a result of skin effect. The current migration results in temperature gradient as the top part of the bar heats-up faster than the lower part (lower impedance region). As the rotor accelerates, the
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current is evenly distributed throughout the entire bar, this phenomenon is called skin effect. Skin effect in rotor bars is intentionally used in design of SCIMs in order to develop a high starting current and a lower stator current (Coelingh, et al., 1998) and (Dey, et al., 2008).
According to (Coelingh, et al., 1998), the skin effect will vary as the shape of rotor and rotor bars change. Changes in rotor bar shapes will result in different resistances at different depths of the rotor bar, hence yield increased or decreased electromagnetic torque. With the five different rotor bar shapes, the area of the upper body of the bars will differ resulting in different effective resistance and different starting torque.
Finite element analysis of both magneto-static and transient analysis of the machine model are performed while comparing the different performance parameters of the machine. Table 4.1 presents the dimensions of the rotor slot for the conventional machine (with rotor bar T1), while Fig. 4.2 depicted rotor bars T2 to T5 which all were modelled with the same T1 induction motor specifications.
Table 4. 1: Rotor bar dimensions (T1)
Description
Parameter Value (mm)
Slot opening height_Hs0 1
Slot body height _Hs2 16.38
Slot opening width_Bs0 1.5
Slot wedge maximum width_Bs1 6.16
Slot maximum width_Bs2 2.12
(a) (b) (c) (d)
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4.3 Airgap Flux Density Distribution
Flux density distributions as mentioned earlier is influenced by rotor and stator MMF distribution and by magnetic saturation in the stator and rotor teeth slots (Boldea & Nasar, 2001). Finite element analysis (FEA) is used in the analysis of airgap flux density distribution behaviour of a different rotor geometry (T1 to T5) of a SCIM. Since the induction machine model is modelled in two-dimensional finite element method (2D FEM), the flux density is assumed to lie in the plane of the model (x, y).
The characteristic magnetic flux density distributions of the motor with different rotor bar types were computed and analysed. Fig. 4.3 depicts the effect of flux density distribution for different rotor bar type (T1 to T5) of a SCIM. It is observed that, change in rotor by shape results in change in flux density distribution and this is because the change in rotor bar shape consequently varies the rotor resistance and leakage reactance. The effect of flux density saturation is higher in rotor bar type T2, whose saturation occurs between the rotor bar slot and outer diameter of the rotor near the airgap, followed by rotor bar T4 which flux density saturates the rotor core in between the slot maximum width.
The 3rd higher rotor shape flux density saturation occur at rotor shape T1, which saturates the rotor core between the rotor bar slot and outer diameter of the rotor near the airgap, followed by rotor bar shape T5 and T3 respectively.
45 T1 T2 T3 T4 T5
Figure 4.3: Magnetic flux density distribution comparisons for healthy different rotor types.
For better understanding of how the behaviour of the flux density distribution is affected by different rotor bar shapes, the airgap flux density profile is illustrated in Fig. 4.4a and the corresponding FFT profile is shown in Fig. 4.4b below. The effect of magnetic saturation is observed in 9th order harmonic after the fundamental component, with rotor bar type T3 and T5 demonstrating the highest harmonic content, while T3 has the least. These harmonic components are caused by rotor bar shape that results in change in rotor resistance and leakage reactance.
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(a)
(b)
Figure 4.4: Magnetic flux density distribution comparison (a) airgap flux density profile and (b) FFT profile