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

In this animation, the rotating air-gap mmf (Tesla’s rotating magnetic field) in an elementary two-pole, two-phase, symmetrical stator is explained. First, we create the rotating air-gap mmf field by applying balanced two-phase voltages to balanced stator windings, which causes symmetrical two-phase currents to flow. Next, we view this rotating field from several frames of reference which are first stationary and then moving, respectively, slower, equal to, and faster than the synchronous speed. By studying this animation, you will gain a solid understanding of Tesla’s rotating magnetic field and develop your intuition for comprehending concepts of advanced reference frame theory. The animated figures can be paused/resumed by clicking on them.

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The figure to the left shows a two-pole, two-phase stator. Its windings are orthogonal and identical in distribution and parameters. Here, and throughout this animation, the blue color is associated with phase a and the red color is associated with phase b. Each stator circle represents the location of peak winding density for a sinusoidally-distributed winding. A [put circle with dot here] or [put circle with cross here] indicates that the instantaneous current is, respectively, positive into the page or negative out of the page. Balanced two-phase voltages are applied to the as- and bs-windings, where the negative terminal is the primed

$$V_{as} = \sqrt{2}V_s\cos\omega_et~~~~~~~~~~\text{(1)}$$ $$V_{bs} = \sqrt{2}V_s\sin\omega_et~~~~~~~~~~\text{(2)}$$ where $V_s$ is the rms voltage and $\omega_e$ is the radian frequency. The capitalized variables indicate that these voltages are in the steady state. According to Faraday’s law, a positive current in the as winding creates flux that links the winding along the as axis in the direction found via the right-hand rule and indicated by the blue arrow; likewise a positive current in the bs winding creates flux linking it as indicated by the red arrow.

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The stator windings are symmetrical rL circuits; thus, balanced two-phase currents flow in the windings are $$I_{as} = \sqrt{2}I_s\cos\big(\omega_et+\theta_{esi}(0)\big)~~~~~~~~~~\text{(3)}$$ $$I_{bs} = \sqrt{2}I_s\sin\big(\omega_et+\theta_{esi}(0)\big)~~~~~~~~~~\text{(4)}$$ where $I_s$ is the rms current and $\theta_{esi}(0)$ is the current phase angle at time zero (with $\theta_{esv}(0) = 0$ as in the case with (1) and (2)).

The stator currents given by (3) and (4) create mmfs in the air-gap expressed, respectively, as $$\text{mmf}_{as} = \frac{N_s}{2}I_{as}\cos\phi_s~~~~~~~~~~\text{(5)}$$ $$\text{mmf}_{bs} = \frac{N_s}{2}I_{bs}\sin\phi_s~~~~~~~~~~\text{(6)}$$ where $\phi_s$ is the counterclockwise (ccw) angle around the stator starting at the as axis.

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The currents in (3) and (4) and mmfs in (5) and (6) are animated in the figure at left. The size of the [put circle with dot here] or [put circle with cross here] indicates the instantaneous assumed positive direction of the current, with a larger symbol indicating greater magnitude. These are time-synchronized with the animated current traces in the figure above.

For visual clarity, only the air gap mmfs along the as and bs axes are shown. The instantaneous value of $\text{mmf}_{as}\big|_{\phi_s=0}$ is the magnitude of the horizontal blue arrow, which points in the direction of positive mmf. Likewise, the instantaneous value of $\text{mmf}_{bs}\big|_{\phi_s=\pi/2}$ is represented by the vertical red arrow. The sum of these two mmfs, which are displaced from each other $\pi/2$ radians in both space for a two-phase machine and in time phase, is a constant magnitude ccw-rotating mmf – Tesla's Rotating Magnetic Field.

We have also introduced a third axis, the q axis. As shown, it is simply a stationary axis aligned with the as axis; however, we will now allow the $q$ axis to rotate at various speeds and view Tesla's rotating field from it.