*A. The Induction Motor*

The theory of the induction motor is well known [10], so only the basics will be described here. Fig. IM-1 shows a cross-section view of a three-phase induction motor, with the stator and rotor coils represented by concentrated windings. Voltage equations can be written for the stator and rotor phases in terms of self and mutual-inductances. As the rotor moves in figure IM-1, the mutual inductances between the rotor and stator coils will change, because the angle between the axes of the rotor and stator changes. To eliminate the time-varying inductances, the equations are frequently transformed to q-d-0 variables in the arbitrary reference frame. For this simulation, we used a stationary reference frame, which has the advantage of eliminating some terms from the voltage equations.

The simulation of the induction motor, is conveniently accomplished by solving for the flux linkages per second in terms of the voltages applied to the machine. The derivatives of the stator flux linkages are given by equations IM1 to IM3. In these equations and the following equations, the superscript "s" indicates the stationary reference frame. The subscript "s" indicates stator quantities, and omega sub b is the base radian electrical frequency.

Likewise, the derivatives of the rotor flux linkages (per second) are given by equations IM-4 to IM-6, wherein the subscript "r" indicates rotor quantities:

The mutual flux linkages, denoted by Psi sub md and mq, are given by equations, IM-7 to IM-9.

The stator and rotor currents (in the stationary reference) frame can then be found using equations IM-10 to IM-15.

Equations IM-1 to IM-15 provide electrical quantities. The induction motor is, of course, an electromechanical device so the model also requires expressions for the electromagnetic torque and the speed of the machine. Equation IM-16 expresses the electromagnetic torque in terms of the flux linkages, and equation IM-17 determines the rotational speed from the machine torque, load torque, and moment of inertia. In both of the following equations, P is the number of poles. It should be noted that the model neglects core loss as well as friction and windage loss.

Fig. IM-2 shows the Graphic Modeller simulation of the induction motor.
As noted above, the model for the induction motor requires voltages as
inputs. Thus one block consists of a three-phase source that provides
a balanced set of three-phase voltages. The induction motor block
is a compound block that contains another level, and will be described
in the next paragraph. The final block in the model is the load torque,
which by suitable choice of constants allows constant power, constant torque,
horsepower squared, and horsepower cubed loads. For convenience there
are also two strip plot recorders that plot the stator and rotor phase
currents. Double clicking them, after a simulation run, will plot
the appropriate variables.

Double clicking on the induction motor block reveals the next level of detail as shown in Fig. IM-3. Compound blocks can be used to allow multiple levels in a model. That has the advantage of keeping the amount of blocks to a reasonable number at any given level of the model. In this case, the compound block was used so the model could be used as a tool by undergraduates who are not concerned with the simulation equations. More advanced undergraduate and graduate students, on the other hand can go down a level to understand the theory behind the simulation.

Since the inputs and outputs to the compound block are phase voltages
and currents, they must be transformed to the stationary reference frame.
Thus, the leftmost blue block contains the equations to transform the phase
voltages to the stationary reference frame, and the q-d-0 stationary reference
frame currents to phase currents. The center green block contains
ACSL code representing equations IM-1 through IM-16, and is thus the actual
simulation of the electrical portion of the induction motor. This
block also contains constants for the parameters of the machine, which
can be changed by the user to represent other machines. The purple
box contains the code for equation IM-17 and determines the speed and position
of the rotor as a function of time. The last block (the right blue
one) is another transformation. In this case the rotor q-d-0 currents
are transformed to phase currents in the rotor reference frame.

While the model is relatively simple, it can be very useful. By starting from zero speed and applying the voltages, the acceleration of the machine can be obtained either with or without a load on the shaft. When there is no load, it is called the free acceleration. Fig. IM-4 shows the free acceleration of a 220 volt, three-horsepower motor. Since friction and windage were not included, the motor reaches synchronous speed (1800 rpm for a four-pole motor). Most textbooks and manufacturers' handouts show the steady-state torque-speed characteristic, but the reality is the motor is subjected to pulsating torques during the startup. It is interesting to note that a four-pole, three-horsepower motor would have a rated torque of about nine lb-ft or 12.2 nt-m. The oscillating torques, however, reach a peak of about 130 nt-m.

In Fig. IM-4, the torque was shown as a function of speed; however it could also be plotted as a function of time. Doing that, it was found that it took approximately one-half second to accelerate the motor to synchronous speed. Then the moment of inertia can be changed to demonstrate the effect of starting high inertia loads. Similarly, a load torque can be placed on the machine to determine the effect on the acceleration time.

As mentioned above, the parameters of the machine can be changed to compare different machines. Fig. IM-5 shows the free acceleration of a 460 volt, four-pole, 50 hp motor. The most obvious difference when compared to the three-horsepower motor is that the number of torque oscillations has increased substantially. Of course, since the machine has a higher power rating, the amplitude of the oscillations is also much larger. The time required to accelerate the unloaded machine was not much larger, however, being about 0.7 seconds.

Fig. IM-5: Free acceleration of a 50 horsepower motor

One advantage of a model for the motor is the ability to look at variables that would be difficult or impossible to measure. Fig. IM-6 shows the stator phase currents as a function of time during the free acceleration of the three-horsepower motor. Their frequency is essentially constant at 60 hz, but the amplitude is much larger than rated current until the machine reaches breakdown torque. Once the machine reaches synchronous speed, the motor draws only a small current to provide the excitation and the losses of the stator and rotor windings.

It would be difficult, if not impossible, to measure the rotor currents
in a squirrel-cage induction motor, but they are obtainable (referenced
to the stator) from the model. Fig. IM-7 shows the rotor phase
currents. These are particularly instructive, as they clearly show
that the frequency of the rotor currents changes with the speed of the
machine. At start, the rotor currents have a 60 hz frequency, but
the frequency drops as the motor accelerates, reaching very low frequencies
as the motor nears synchronous speed. Of course, once the motor reaches
synchronous speed there is no relative motion between the rotor squirrel-cage
bars and the rotating magnetic field. Thus the current in the rotor
bars drops to zero as shown in the figure.

Those readers who choose to install the Graphic Modeller Reader and
investigate the model further will find they can run a variety of studies
with it. As an example, one could vary the load torque and observe the
steady-state speed of the machine. The readings could then be plotted,
manually or in a spreadsheet, to provide the torque speed curve from zero
to full-load.