Akira Chiba

13.1 Rotor structure and suspension force

Figure 13.1(a) shows the typical rotor lamination of a cage induction motor. The circular silicon steel sheet has a centre hole punched for the shaft, and the small holes (or slots) around the circumference are to accommodate the rotor bars for construction of the cage. The rotor slots can be open to the rotor surface or closed depending on the design. Open slots reduce leakage inductance whereas closed slots make casting an aluminium cage more straightforward. For high speed applications the slots are circular and closed for improved rotor robustness. Lamination thickness is typically 0.5, 0.35 mm or thinner. Thin sheets reduce the iron core loss although the manufacturing cost is increased. They form a cylindrical iron stack for the rotor core. In super-high-speed machines the rotor iron core is made from a solid cylindrical piece of iron rather than laminated sheets to enhance shaft stiffness with the sacrifice of rotor losses and performance.

Figure 13.1(b) shows the typical short-circuit conductor arrangement of the squirrel cage rotor. The bars are located in the rotor slots with an axial core length of l_stk. Two end-rings connect the rotor bars together. The cage, as previously mentioned, is fabricated from copper bars and rings or cast in aluminium. In super-high-speed applications, a copper compound metal is employed to improve robustness but with a conductivity reduction. In most bearingless induction motors the rotor structure is more complex than the simple cage arrangement shown in Figure 13.1(b) and, since they are still prototype machines, the rotor cage is fabricated from copper bars, inserted into the core and connected by copper end-pieces (end-ring sections), which are braised or soldered on. This is the easiest way to fabricate a prototype rotor.

Suppose that a 2-pole revolving magnetic field is applied to a squirrel cage rotor. Electromotive forces (EMFs) are induced so that a 2-pole current distribution is generated in the rotor short circuit (the cage). The arrows in Figure 13.1(b) show the direction of the current flow at an instance in time. The current in the short circuits generate a 2-pole flux wave as indicated by N and S in the figure. The 2-pole flux rotates at the slip frequency with respect to the rotor reference frame (i.e., a reference frame rotating at rotor speed) in steady-state conditions where slip frequency = slip × supply frequency. Torque is generated as a product of the rotor current and magnetic field. To enhance torque, low resistance in the rotor circuit is necessary. If a 4-pole revolving magnetic field is also applied to the squirrel cage rotor then a 4-pole current distribution flows in the rotor short circuits. Hence the squirrel cage rotor can generate torque for both the 4-pole and 2-pole revolving magnetic fields (and even higher pole numbers) and it is not pole-specific.

A standard squirrel cage rotor causes problems when used in a bearingless induction machine to generate a suspension force. Suppose that an additional 2-pole winding is used for suspension force generation. For a step change in suspension force there has to be a step change in the 2-pole stator flux. Because voltage is induced into the rotor circuit, which generates rotor current, the rotor flux does not change suddenly. Hence the interaction of stator (primary) and rotor (secondary) circuits produces a delay in the 2-pole flux which in turn delays the suspension force response. In addition to the response delay, the rotor current also makes alignment of the flux – hence the force is also difficult to orientate.

To overcome the problems associated with the squirrel cage rotor, a pole-specific short circuit can be used. Figure 13.2(a) shows a pole-specific rotor short circuit for a 2-pole induction motor. Two rotor bars are connected by two end-rings. If a rotating 2-pole magnetic field is generated in the airgap by the stator winding currents, as indicated by N and S, EMFs are induced and current is generated in the rotor; with the flow shown by the arrows. If a 4-pole revolving magnetic field is applied, rotor current does not flow because the net rotor flux linkage between the bars is zero since the N and S poles linking to the short circuit cancel each other out, and the EMFs induced into the short circuits sum to zero. Hence the short circuit is 2-pole pole-specific. If a rotor has 16 slots then eight short circuits should be constructed so that eight end-rings are required at each end. These eight short circuits need to be isolated.

Figure 13.2(b) shows a 4-pole pole-specific short circuit. The four rotor bars are connected by four end-ring segments (rather than complete rings) – two at each end – to form one short circuit. If the current flows in the direction indicated by the arrows then a 4-pole magnetic field is generated in the airgap. If there are 16 slots in the rotor core then 4 short circuits should be constructed with a mechanical phase-shift angle of 22.5 deg. If a 4-pole revolving magnetic field is applied then current is generated in the short circuits. However, there is no current when a 2-pole revolving magnetic field is applied because the EMFs induced into the circuits sum to zero.

Now, let us go back and consider a standard cage rotor to understand why it is impractical in a bearingless induction machine application. Figure 13.3 shows equivalent circuits for a bearingless induction motor with both 4-pole and 2-pole 3-phase windings. R_s and R_r are the stator and rotor resistances, l_s and l_r are stator and rotor leakage inductances and L_m and R_m are the magnetizing inductance and iron loss resistance. The subscripts 2 and 4 indicate the 2-pole and 4-pole windings. The slip is s₂ for the 2-pole and s₄ for the 4-pole winding current. These equivalent circuits are usually used for steady-state characteristics calculation in conventional induction motors driven with sinusoidal voltage and current.

Let us suppose that the 4-pole and 2-pole windings are assigned as the motor and suspension force windings respectively. When no torque is generated, s₄ is zero resulting in a high value (infinity) of the effective resistance R_r4/s₄ so that I_r4 is zero (i.e., there is no rotor current) and the terminal current I_t4 is equal to excitation current I_m4, assuming that iron loss is negligible. Note that the excitation current I_m4 generates the airgap flux.

When torque is required, s₄ increases and R_r4/s₄ decreases so that we now get a rotor current I_r4. Torque is the product this rotor current and the stator component of the airgap flux. The airgap flux is mostly determined by the airgap length as well as the saturation of the rotor and stator iron material. R_r4/s₄ can be split into R_r4 (rotor resistance, i.e., rotor copper loss = I²_t4R_r4) and R_r4 × (1 − s₄)/s₄ (electromechanical energy conversion = I²_t4R_r4 × (1 − s₄)/s₄). Therefore R_r4 and s₄ are normally designed to be as small as possible at rated power for efficient operation (though there is often a compromise with the starting torque requirement, which requires a higher rotor resistance). For general-purpose 3-phase induction motors with ratings of a few kW (connected to a standard fixed-frequency supply), the rated slip s₄ is in the range of 0.02 (standard cage-rotor motors) to 0.1 (wound-rotor and specialized machines). For high speed or large motors (which are usually higher in efficiency), s₄ decreases to perhaps 0.005 or even less. If the rotor shaft rotational speed is ω_rm then the slip for the 4-pole winding is

(13.1)

where ω₄ is the frequency (in rad/s) of the 4-pole winding voltage and current. If a squirrel cage rotor is employed, the rotor bars and end-rings are optimized to make R_r4 (and also R_r2) as small as possible. For the 2-pole winding, the slip is

(13.2)

where ω₂ is frequency of the 2-pole winding voltage and current. The 4-pole winding is assigned as the motor winding so that, at low slip, ω_rm ≈ ω₄/2. For static suspension force generation, the current frequency of the windings should be equal so that ω₂ = ω₄, hence s₂ = 0.5. This is a high slip so that the rotor current is the dominant component of the terminal current I_t2 rather than the excitation current I_m2. This leads to a very inefficient machine and problems with the suspension force generation.

The 4-pole and 2-pole winding fluxes are generated by the excitation currents I_m4 and I_m2. I_m4 is usually kept constant to provide excitation for torque generation so that I_m2 is proportional to the suspension force. To regulate the excitation current, the terminal current I_t2 is controlled by a regulator. The current transfer function Im2/It2 is

(13.3)

where

and

This transfer function illustrates that the first pole frequency is ω_a2 and the amplitude starts to role off at a rate of −20 dB/dec. The second pole frequency ω_b2 is between 12 and 20 times higher than the first pole frequency since l_r2/L_m2 is in the range of 0.05–0.08. The suspension force response is influenced by this current transfer function.

Next, let us consider the case when a pole-specific rotor is employed. If a 4-pole arrangement is employed then R_r2 is infinite because there is no 2-pole rotor current due to the 2-pole revolving magnetic field. Hence the excitation current is proportional to the terminal current I_t2 (neglecting iron loss). Therefore the transfer function I_m2/I_t2 is unity. In practice I_m2/I_t2 decreases at high frequency because of the iron loss. However, the frequency at this point is usually high enough to not affect the mechanical dynamics.

In Figure 13.4, the curves show the phase angle characteristics. The curve for the squirrel cage rotor is calculated using (13.3) while the curve for the pole-specific rotor includes the iron loss resistance. Note that the pole-specific rotor exhibits a better transfer function. The data points indicate experimental results for verification. A static suspension force is applied to the test machine and the direction of the suspension force command vector is obtained. At zero speed, the suspension force command is exactly opposite to the applied suspension force. As the speed increases, the suspension force command is automatically generated with a phase-lead angle to compensate for the phase lag of the generated suspension force. The command amplitude also increases to compensate for the decrease in gain. For the cage rotor, at a speed of about 5 rev/s, the phase lag angle is too large to maintain magnetic suspension. However, the pole-specific rotor provides a good frequency characteristic.

When the roles of the 4-pole and 2-pole windings are reversed, i.e., when the 2-pole winding is assigned as the motor winding, the transfer function of the squirrel cage improves but the pole-specific rotor is still superior.

13.2 Indirect type drives

In this section the indirect type of bearingless induction motor drive is introduced. A primitive controller structure is first considered for ease of understanding. The controller structure is almost the same as that used for the primitive bearingless motor except for amplitude and phase angle compensation of the motor winding current. This controller is valid under steady-state conditions. For fast response, an airgap field-oriented controller is then introduced. It is found that the airgap flux linkage is regulated even under fast acceleration during a short transient over-current period.

Figure 13.5 shows the block diagram of a primitive controller. The controller is divided into two parts: one is a radial position controller and the other is a motor controller. The motor controller structure is slightly different from that of the primitive bearingless motor although the radial position controller has the same structure. The bearingless machine has 4-pole and 2-pole windings designated as motor and suspension force windings respectively. The mechanical synchronous speed command ω*_4m is doubled to generate an electrical synchronous speed command 2ω*_4m and the cosine and sine functions of 2ω*_4mt are obtained. These functions are the airgap flux linkages and hence they are the commands used for the radial displacement derivatives of airgap flux linkages λ′_gc and λ′_gs, where λ′_gc and λ′_gs are proportional to airgap flux linkages λ_gc and λ_gs. This assumes that the mutual inductance between the 4-pole and 2-pole windings is proportional to the rotor displacement. Since the amplitudes of the cosine and sine functions are unity, the amplitude of the airgap flux linkage is unity. The orientation of the airgap flux linkage is given by 2ω*_4mt. The amplitude and direction of the airgap flux linkage is independently generated and the current amplitude I*_mp and phase angle θ* are adjusted so that the actual airgap flux linkage follows the commands λ_gc and λ_gs. In the modulation block, suspension currents are generated from suspension force commands, and the amplitudes of flux linkage derivatives are taken to be unity in the calculation.

Current commands are generated with an amplitude of I*_mp and phase-lead angle of θ*. For example, the u-phase motor current is

(13.4)

The amplitude and phase-lead angle should be adjusted so that the motor airgap flux linkage corresponds to the flux commands λ_gc and λ_gs. Since the airgap flux linkage is generated by the excitation current I_m4 in Figure 13.3, the line current is obtained from (13.3). Substituting 4 for 2 and solving for I_t4/I_m4 yields

(13.5)

where

and

The amplitude and phase angle of the transfer function of I_t4/I_m4 is shown in Figure 13.6. Note that correspondence of the motor airgap flux linkage with respect to the flux command is realized when the amplitude and phase-lead angle are gradually increased as a function of slip. This fact can be explained from Figure 13.3(a) as follows:

To see the effectiveness of the primitive controller several tests can be carried out. The primitive controller is effective when the motor is generating torque in the following tests [3]:

These facts indicate that airgap flux linkage components in the bearingless motor correspond to the airgap flux linkage commands.

However, there is a problem if there is a sudden change in rotational speed command 2ω*_4m since the primitive controller is only valid under steady-state conditions. In order to realize airgap flux linkage steering during fast acceleration and deceleration in an application, field-oriented control theory is required (which is often referred to as vector control theory).

Figure 13.7 shows an example of the indirect type of a vector controller [8,9]. The controller regulates the airgap flux linkage vector even during transient conditions. The angular speed of the shaft ω_rm is detected by a rotary encoder. The shaft speed is multiplied by the number of pole pairs, i.e., 2 for a 4-pole motor winding. Then 2 ω_rm is compared with the rotational speed command 2ω*_rm and a torque current command i_qs is generated through the PI speed controller. An excitation current command i_dm is given as a constant. The slip frequency 2ω_s and d-axis current are generated using the following equations

(13.6)

(13.7)

One can see a derivative operator in the numerator of (13.6). However, in a practical system, an integration function is implemented in a later block to counteract this derivative operation. Note that the induction machine parameters, such as the rotor resistance and the mutual and leakage inductances, are required for the indirect vector controller. These parameters may vary with temperature, magnetic saturation or operation point. On-line identification of the machine parameters is possible using several methods, such as a model reference adaptive system, observers and so on. Advanced readers are encouraged to study papers on induction motor parameter identification.

The slip frequency is added to 2ω_rm so that the instantaneous speed of the airgap flux linkage 2ω₀ is obtained. Integration of the angular speed, i.e.,

(13.8)

provides the instantaneous angular position of airgap flux linkage. The sinusoidal and cosinusoidal functions of 2φ₀ provide the components of the airgap flux linkage vectors λ_gc and λ_gs respectively. These components are used in the modulation block.

In the current generator, 3-phase symmetrical current commands are generated using the d- and q-axis stator currents i_ds and i_qs, as well as the airgap flux linkage position φ₀. For an example, the u-phase motor current is

(13.9)

The v- and w-phase currents have phase shifts of 2π/3. The motor currents are regulated using these instantaneous currents. Hence the rotational angular position of airgap flux linkage in an actual bearingless induction motor should correspond to φ₀ and the amplitude of the airgap flux linkage is constant, which is determined by the current command i_dm. Even in transient conditions, such as during fast acceleration and deceleration, the control is valid.

Figure 13.8 shows the performance during a fast acceleration. A step in speed command 2ω*_rm from 1200 to 2700 r/min is given to the bearingless induction machine drive. The torque current command i_qs increases as a step function to the maximum value of 10 A in this case. The speed increases as a ramp function. The dynamic response is determined by the rotational inertia of the system. If the vector controller is not used then significant fluctuations of 100 μm are seen in the radial displacements x and y during the period from 20 ms after the acceleration start-up to the end of the acceleration. The fluctuation is delayed for 20 ms because of a delay in the change of airgap flux linkage.

In the radial displacement waveforms, there is no serious fluctuation when the airgap-oriented vector controller is employed. Only a small variation in x and y, caused by shaft unbalance and mechanical tolerance, is seen. The frequency of these small radial vibrations gradually increases as the speed increases.

The cosinusoidal waveform of airgap flux linkage λ_gc and detected flux waveform Ψ_t are also shown. Ψ_t is detected by a search coil wound around the stator teeth which corresponds to the position of cosinusoidal flux component. If a primitive controller is employed, a phase-shift and amplitude difference is seen in Ψ_t after 20 ms of the step in speed command. However, good correspondence is seen in this figure. This result indicates that the airgap flux linkage in the actual bearingless induction machine corresponds to the flux command [4].

13.3 Direct type drives

In the previous section an indirect vector controller was described. The indirect vector controller is cost-effective because it does not require flux sensors. However, it does require the machine parameters, which may vary with operational conditions. On the other hand a direct vector controller does not need the machine parameters because the revolving magnetic field is detected by flux sensors. Thus the direct vector controller is robust with respect to temperature variation. Some recent direct vector controllers do not need flux sensors, with the fluxes being estimated from the terminal voltage and line current, as well as the machine parameters; and a combination of indirect and direct control has also been recently proposed.

In the bearingless induction motor, instantaneous rotational position and amplitude of the airgap flux linkage vector are required for the radial position controller. These values are given by the motor vector controller described in the previous section. In this section, the airgap flux linkage is detected by flux sensors and sampling of the terminal voltage and current. There are advantages to direct detection of airgap flux linkage in bearingless induction motor drives:

In this section two types of direct flux detection are introduced.

13.4 Two-pole motor drive

Since the induction motor has a cylindrical rotor, either of the 4-pole and 2-pole windings can be assigned as the motor drive winding. In this section a comparison of winding assignment is made.

A comparison of a machine with a 2-pole motor drive with 4-pole suspension winding with respect to the 4-pole motor winding arrangement is summarized below:

Characteristic (a) is an advantage of the 2-pole motor drive. The transfer function of the suspension force to the suspension force winding current, shown in the first section of this chapter, has a phase-lag and amplitude reduction caused by the rotor short circuit. The phase-lag causes interference of the two perpendicular suspension forces and a decrease in phase margin of the radial position controller. Calculation of the transfer function of the 4-pole suspension winding should be based on an instantaneous flux representation rather than the equivalent circuit. The analysis should start from the flux linkage to current relationships, with a12 × 12 matrix to determine the flux linkages and currents of the rotor and both 3-phase stator winding sets, including a consideration of the radial rotor displacement [6]. The matrix relationship is simplified by a 3-phase to 2-phase transformation which results in an 8 × 8 matrix and then the stored magnetic energy is calculated. Suspension forces are derived from the partial derivatives of the stored magnetic energy so that the suspension force is a product of the airgap flux currents in the 2-pole and 4-pole windings as well as a derivative of mutual inductance with respect to a radial displacement. A state-space equation with eight states, i.e., 4-pole and 2-pole rotor and stator flux linkages in two perpendicular axes, is set up and the airgap flux currents and suspension forces are calculated. The suspension force and current relationship is derived through simulation to obtain the frequency characteristics. For low-power bearingless machines, squirrel cage circuits are acceptable because of the short time constant.

Characteristic (b) is also an advantage of the 2-pole motor drive. The frequency of a 2-pole motor drive is about a half of the 4-pole motor drive. In addition, the frequency of the suspension force winding current in the 2-pole motor is halved for static suspension force generation because the suspension current frequency is equal to the motor current frequency. Since the current frequency is halved, only half the terminal voltage is required so that the VA requirement, i.e., the product of voltage and current, is reduced.

Characteristic (c) can be an advantage in some applications. In generating a static suspension force for suspending the shaft weight, ω₄ and ω₂ are the same value. In a 2-pole motor drive, ω₂ is nearly equal to the shaft speed ω_rm so that the slip s₄ is almost −0.5. A negative slip indicates generator operation, hence real power is generated. The output power is sufficient to supply the winding copper losses and inverter losses if the speed and suspension force are high enough so that the suspension power is provided by self-excitation [7].

An effective application can be vibration suppression at high speeds. A shaft is suspended by external bearings. As the speed increases, self-excitation occurs. Then a damping force is generated by the bearingless induction rotor which is located at the centre of the shaft length so that the shaft vibration is suppressed effectively.

Characteristic (d) is a disadvantage of the 2-pole motor drive. As shown in the first section of this chapter, a pole-specific short circuit can be constructed for better suspension force generation. The number of end-ring segments at each end is equal to the number of rotor bars. For example, 16 rotor bars require eight end-ring segment layers at each end. For the 4-pole motor drive, only 4 coil end-ring layers are necessary. Reduction of end-ring segment layers is possible if more than two rotor bars are connected to an end-ring segment. However, the radial force transfer characteristics are inferior because of the rotor short circuits between adjacent bars which produce non-pole-specific current [1].

References