ELECTROMECHANICAL MOTION DEVICES
Rotating Magnetic Field Based Analysis
3^rd Edition

On this tab, we will view Tesla’s rotating magnetic field from the q-axis, as before, but we will now allow the q-axis to rotate at speeds slower to, equal to, and greater than the speed of Tesla’s field. This is the basis for Reference Frame Theory in advanced machines courses.

At left, the q-axis is initially stationary and aligned with the as-axis at $\phi_s=0$, and the figure is identical to the machine view on the previous tab. We will now allow the q-axis to rotate at a constant rate of $\omega$ rad/s and view Tesla’s rotating magnetic field (the net mmf) from this rotating axis. In addition to the stationary view, by clicking the various buttons below the figure, we can view the q-axis rotating at $\omega = \frac{1}{2}\omega_e$ (half synchronous speed), $\omega = \omega_e$ (synchronous speed), and $\omega = \frac{3}{2}\omega_e$ (greater than synchronous speed).

If we let the q-axis rotate ccw at constant speed $\omega$, its position $\theta$ with respect to the as-axis is $$\theta = \omega t+\theta(0)~~~~~~~~~~\text{(9)}$$ where $\theta(0)$ is the position at time zero. Although we are free to position it anywhere, for this animation we will set $\theta(0) = 0$, which aligns the q-axis with the as-axis at time zero. Define the ccw displacement from the q-axis as $\phi$, so that $$\phi_s = \theta+\phi~~~~~~~~~~\text{(10)}$$ If we substitute (3), (4), (9) and (10) into (7), after much algebraic manipulation we obtain $$\text{mmf}_s = \frac{\sqrt{2}N_s}{2}I_s\cos\big[(\omega_e-\omega)t+\theta_{esi}(0)-\phi\big]~~~~~~~~~~\text{(11)}$$ Notice the similarity between (11), which is Tesla’s rotating magnetic field viewed from a rotating q-axis and (8), which is Tesla’s field viewed from a stationary axis. In fact, (8) is obtained from (11) by setting $\omega = 0$ in the latter.

By selecting the various q-axis angular speeds we are viewing (11) and the q-axis, whose position is defined by (9), together on the same machine diagram.

Net mmf Phase mmf

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In the plot above, we are viewing Tesla’s rotating magnetic field from the (possibly rotating) q-axis with 360° $\phi$-vision. The phase mmfs can be toggled on and off as before. The developed view below it shows the machine as viewed from the q-axis. From this view, we see how the stator currents appear to an observer sitting on the q-axis at various q-axis speeds

• With $\omega = 0$, we are in the stationary reference frame and we have the same plot as at the bottom of the previous tab.
• With $\omega = \frac{1}{2}\omega_e$, the q-axis moves at half the synchronous speed and to an observer sitting on the q-axis, the net mmf wave appears to move from right to left at half the synchronous speed. Because the q-axis is moving away from the rotating magnetic field, the field appears slower than synchronous speed.
• With $\omega = \omega_e$, the q-axis moves at the synchronous speed and to an observer sitting on the q-axis Tesla’s rotating magnetic field appears not to rotate at all; it is stationary! In this reference frame, we often write $\phi = \phi_e$ to indicate that we are in the synchronously rotating reference frame.
• With $\omega = \frac{3}{2}\omega_e$, the q-axis rotates faster than Tesla’s rotating field, “catching up” to it every third rotation about the stator. To an observer sitting on the q-axis Tesla’s field appears to move clockwise, moving from left to right at half the synchronous speed.

One final important point is worth noting. In Chapter 2, we derived the q- and d-axis variables using a version of Park’s transformation. Traditionally, these transformed variables have been derived mathematically and given little intuitive physical meaning. However, now that we have animated Tesla’s rotating magnetic field and viewed it from multiple rotating reference frames, an intuitive understanding of q- and d-axis variables is evident. Choose a reference frame speed, $\omega$, and position yourself on its q-axis. The q- and d-axis variables in your frame are those variables necessary to produce Tesla’s rotating magnetic field as you view it in that reference frame. In other words, in the two animations immediately above, the q- and d-axis circuit variables (i.e., voltages and currents) in the selected reference frame are simply those that would produce the purple arrow (Tesla’s rotating magnetic field, or, the stator mmf) as viewed from that reference frame.

For example, in the synchronously rotating reference frame, with $\omega$ = $\omega_e$, Tesla’s rotating magnetic field is stationary; therefore, the q- and d-axis variables are constant. In other frames, Tesla’s rotating magnetic field appears to rotate at a speed dependent on the reference frame speed, and the q- and d-axis variables are sinusoids whose frequencies correspond to the apparent rotational speed of Tesla’s field. If Tesla’s field appears to rotate quickly in a given frame, then the frequency of the q- and d-axis variables in that frame will be relatively high, and vice versa.

It is worth spending the time to grasp this important concept. If you can visualize and remember it, reference frame theory and advanced electric power and drive system concepts will be much easier to master.