On this tab, we will view Tesla’s rotating magnetic field from the stationary q axis aligned with the as axis. We will also introduce the concept of the developed view of the machine to facilitate better visualization of the fields.
| Paul Krause | Oleg Wasynczuk | Steven Pekarek | Timothy O'Connell |
On this tab, we will view Tesla’s rotating magnetic field from the stationary q axis aligned with the as axis. We will also introduce the concept of the developed view of the machine to facilitate better visualization of the fields.
The total air-gap mmf, $\text{mmf}_s$, due to the two-pole stator windings is
$$\begin{split}
\text{mmf}_s & = \text{mmf}_{as}+\text{mmf}_{bs} \\
& = \frac{N_s}{2}\big(I_{as}\cos\phi_s+I_{bs}\sin\phi_s\big)
\end{split}~~~~~~~~~~\text{(7)}$$
If we plug (3) and (4) from the previous tab into (7), after some algebraic manipulation the result is the equation for Tesla’s rotating magnetic field, as viewed from the stationary q axis positioned at $\phi_s=0$ (the as axis):
$$\text{mmf}_s^s = \frac{\sqrt{2}N_s}{2}I_s\cos\big[\omega_et+\theta_{esi}(0)-\phi_s\big]~~~~~~~~~~\text{(8)}$$
Here, we have added the superscript $s$ to indicate that we are viewing the rotating field from a stationary q axis.
Equation (8) is viewed as the rotating purple arrow in the figure below. The a phase and b phase mmfs can be toggled on and off by clicking the “Phase mmf” or “Net mmf” button, respectively; which clearly shows their contributions to the net mmf. To view the rotating magnetic field on a flat plane, we can “unroll” the machine and create the developed view. To create this view, imagine making a horizontal slice at $\phi_s=\pi$ from the stator’s outer edge to the machine’s center, which will bisect the bs’ winding. Then, unroll and flatten the machine from that cut, centering the view on the q axis at $\phi_s=0$. In the developed view, the stator is at the top and the rotor is at the bottom, thus ccw is right-to-left and Tesla’s rotating magnetic field is an arrow moving in that direction. You can toggle between the machine and developed views by clicking the corresponding buttons. Please be aware that in the text $\omega_e$ cut at $\phi_s = 0$ to create the developed winding. It is a manner of choise, the results are the same.
Above, we are viewing Tesla’s rotating magnetic field (8) with 360° $\phi_s$-vision while we sit stationary in the air gap. As above, the phase mmfs can be toggled on and off by clicking the “Phase mmf” or “Net mmf” button, respectively. The net mmf moves from right to left (ccw) at a rate of $\omega_e$ rad/s. This is known as the synchronous speed. If we choose to sit on the stationary q-axis at $\phi_s=0$, the net mmf at our location pulses up and down sinusoidally, reaching its first peak at $t = -\theta_{esi}(0)/\omega_e$, and then periodically thereafter at a frequency of $\omega_e$. If we choose to sit at any other stationary location along the air gap, we will see the same thing, but with the time of the first mmf peak shifted by $\phi_s/\omega_e$.
At this point, it is worth noting that all of the above can be applied to an elementary two-pole, three-phase symmetrical stator, too. For the three-phase case, (1) and (2) and (3) and (4) are replaced with balanced three-phase stator voltages and currents, respectively. These create balanced a-, b-, and c-phase mmfs spaced 120$^\circ$ apart and aligned with the a-, b-, and c-axes, respectively. The sum of these three mmfs produces Tesla’s rotating magnetic field as expressed by (8), but with a 3/2 coefficient in front. Thus, all of the animations of Tesla’s rotating magnetic field presented thus far and on the following tab are representative of a three-phase stator, too, if the arrow magnitudes are scaled by 3/2.
Tesla’s magnetic field, as created by an elementary two-pole three-phase symmetrical stator, is animated below. The three individual phase mmfs and their sum are shown. This animation should be compared to the animation of the two-phase machine above. We introduce this here as a brief look ahead to Animation B, which depicts a two-pole three-phase permanent magnet synchronous machine.