Machine Structure

34.10.3 Machine Structure

The essential difference between the Syncrel and, say, the induction machine is the design of the rotor. Most experi- mental Syncrel systems that have been built use the stator of

FIGURE 34.110 Cross section of a radially laminated Syncrel. an induction machine, including the same windings. The rotor

designs can take on a number of different forms, from the very simple and basic dumbbell-shaped rotor (such as that sketched

Q-axis

in Fig. 34.109) to more complex designs. Unfortunately, the Pole-piece designs that are simple to manufacture (such as the dumb- Lamination

Inter-lamination

bell design) do not give good performance; therefore, one is

D-axis forced into more complex designs. The design of the rotor in

space

a Syncrel is the key to whether it is economic to manufacture and has competitive performance with similar machines. The design of Syncrel rotors fall into four main categories of increasing manufacturing complexity and performance:

Shaft dumbbell or higher pole-number equivalent designs, flux-

barrier designs, radially laminated flux-barrier designs, and axially laminated designs. The first two of these design methodologies are old and lead to designs with poor to mod- est performance. Therefore, they will not be considered any further. The latter two, however, lead to machine designs with performance comparable to that of the induction machine.

Figures 34.110 and 34.111 show the cross section of a four- pole machine with a radial lamination flux barrier designed rotor and an axially laminated rotor. The radial lamination design allows the rotor to be built using similar techniques to standard radial laminations for other machines. The flux barriers can be punched for mass production, or wire-eroded

FIGURE 34.111 Cross section of an axially laminated Syncrel. for low production numbers. These laminations are simply

stacked onto the shaft to form the rotor. The punched areas can be filled with plastic or epoxy materials for extra strength, if required. The iron bridges at the outside of the rotor are

The axially laminated rotor is constructed with lamina- designed to saturate under normal flux levels and therefore do tions running the length of the rotor (i.e. into the page not adversely affect the performance of the machine. They are on Fig. 34.111). In between the laminations, a nonmagnetic there to provide mechanical strength.

packing material is used. This can be aluminum or bronze,

34 Motor Drives 987 for example, but a nonconductive material such as slot insula- major part of the modeling process is the conversion from a

tion is better since eddy currents can be induced in conductive three-phase model to a two-phase model. This is a process that materials. The ratio of the steel laminations to nonmagnetic is carried out for most sinusoidally wound machines, since it material is usually about 1:1. The axial laminations are all allows a variety of machines to be represented by very similar stacked on top of each other, and a nonmagnetic pole piece models. is bolted on top of the stack to hold the laminations to the

A further complication in this process is that the two-phase shaft. The strength of these bolts is usually the main limitation model is derived in a “rotating reference frame,” as opposed on the mechanical strength and hence the speed of rotation of to a stationary reference frame. Developing the equations in this rotor. If more or thicker bolts are used to increase strength,

a rotating reference frame has the advantage that the normal the magnetic properties of the rotor are compromised because sinusoidal currents feeding the machine are transformed into of the amount of lamination that has to be cut out to make

dc currents in steady state, and the angular dependence of the room for them.

machine’s inductances disappear.

Radial and axial laminated rotors are usually limited to One way of heuristically understanding the effect of the four-pole or higher machines because of the difficulty of rotating frame transformation is to imagine that we are observ- accommodating the shaft in two-pole designs. An axially lam- ing the machine’s behavior from the vantage point of the inated two-pole rotor has been built with the shafts effectively rotor. Because the sinusoidal flux density waveform is rotating bonded onto the end of the rotor. Another design was con- around the machine in synchronism with the rotor, it appears structed of a block of alternating steel and bronze laminations, from the rotor that the flux density is not changing with time – the whole structure being brazed together and the resultant

i.e. it is a flux density created by dc currents flowing in a single stack then being machined into a round rotor and shafts (this sinusoidally distributed winding. This single sinusoidal wind- rotor was used for high-speed generator applications).

ing is effectively rotating with the rotor. It should be noted Of the two rotor designs, the radially laminated one has that the transformation process of the fluxes, currents, volt- the best potential for economic production. The axially lami- ages, and machine parameters to the two-phase rotating frame nated rotor in general gives the best performance, but the mass is an invertible process; therefore one can apply the inverse production difficulties with folding and assembling the lami- transformation to ascertain what is happening in the original nations make its adoption by industry unlikely. On the other three-phase machine. hand, improved designs for radially laminated rotors mean

The models derived using the three-phase to two-phase that they can now produce performance very close to that transformations are known as dq models, the d and q referring of the axial-laminated designs, and the ease of manufacture to the two axes of the machine in the two-phase model (both would indicate that these rotors are the future of Syncrel stationary and rotating frame dq axes are shown in Fig. 34.109). rotors.

The linear3 dq equations for the Syncrel can be derived as [4]

di d

34.10.4 Basic Mathematical Modeling

v d = Ri d +L d − ωL q i q (34.99)

dt

In order to give a more quantitative understanding of the

di q

machine, a basic mathematical dynamic model of the machine

v q = Ri q +L q

− ωL d i d (34.100)

dt

will be introduced.1 This model will assume that the iron material in the machine does not saturate. This means that the where flux density and flux linkage of the windings in the machine

v d i d = the d-axis voltage and current, are linear functions of the currents in the machine.

v q i q = the q-axis voltage and current, To derive the electrical dynamic model of a machine, one

L d ,L q = the d-and q-axes inductances, respectively, usually uses Faraday’s flux linkage expressions. In the case of

ω = the electrical angular velocity of the rotor. the Syncrel, the self and mutual flux linkage between the phases

is obviously a function of the angular position of the rotor; Thus far we have concentrated on the electrical dynamics therefore, one needs to have expressions for these inductances of the machine. The other very important aspect is the torque in terms of rotor position. The fundamental assumption used produced by the machine. It is possible to derive the torque to make this mathematically tractable is that the inductances of the Syncrel using the principle of virtual work based on vary as a sinusoidal function of the rotor position.2 The other coenergy as:

1 Note that the model is not derived but instead just stated. The structure

2 −L

T e = p p (L d q )i d i q (34.101)

of the model will be heuristically explained.

2 The sinusoidal variation of inductance with rotor position turns out to be very accurate because of the fact that the stator windings are sinusoidally

3 Linear refers to the fact that the equations are derived assuming that the wound. This forces the flux linkage to behave in a sinusoidal fashion.

iron circuit behaves linearly in relation to applied mmf and the flux produced.

988 M. F. Rahman et al.

wL

+ wL

FIGURE 34.112 Two-phase equivalent circuit of the Syncrel in a rotating frame.

The 3/2 factor is to account for the fact that the two-phase

machine produces two-thirds the torque of the three-phase machine.4 v q r

The only other remaining equation is the mechanical equation for the system:

J ˙ω r + Dω r +T F =T e (34.102)

L d i d where

≡ the rotational inertia of the rotor/load,

D θ ≡ the friction coefficient for the load, i T F ≡ the fixed load torque of the load,

ω r ≡ the rotor mechanical angular velocity (= ω/p d p ). s

Rotor

Remark

FIGURE 34.113 Space phasor diagram of a Syncrel. Equation (34.101) shows that the machine must be designed so

that L d −L q is as large as possible. This will maximize the torque that is produced by the machine for given d- and q-axis currents. To lower L q , one must design the q-axis so that it has as much of these. The interested reader should consult references [4–6]

cited at the end of this chapter.

air obstructing the flow of flux as possible, and the d-axis must One of the most common control strategies for any elec-

be designed so that it has as much iron as possible. In practice trical machine is to maximize the torque per ampere of input these quantities cannot be varied independently. current. The following discussion should be considered in con-

Figure 34.112 shows the equivalent circuit for the Syncrel junction with Fig. 34.113, which shows the relationship of the corresponding to Eqs. (34.99) and (34.100). One can see that various vectors in the machine to the d r q r - and d s q s -axes. the dynamic equations for the Syncrel are intrinsically simple.

It turns out that one of the critical parameters for the control In contrast, the induction machine electrical equations consist of the Syncrel is the angle of the resultant current vector in the of a set of four complex coupled differential equations.

machine in relation to the d-axis of the machine. It is possible to write the torque expression for the Syncrel in terms of this as