In this animation, we illustrate the three common brushless dc (BLDC) motor operating modes that are described in Chapter 6: Normal, $\phi_v=0$; Maximum-Torque per Volt, $\phi_v=\phi_{vMT/V}$; and Maximum-Torque per Ampere, $\phi_v=\phi_{vMT/A}$. To illustrate these three modes, we will view relevant machine variables in both the stationary and synchronously rotating reference frames for both free acceleration and under a load torque step for each mode.
Permanent-Magnet AC Machine
There are a few concepts and notations that will help us to understand the diagrams and transient waveforms. First, we will use the three-phase, four-pole permanent-magnet (PM) ac machine considered in Example 2F and throughout Chapter 4. In particular, $r_s=3.4~\Omega$, $L_{ls}=1.1~$mH, $L_{ms}=11~$mH (recall that $L_{ms}$ must be multiplied by 1.5 for a three-phase machine), and $\lambda'_m=0.0826~\text{V}\cdot\text{s/rad}$. Note that although we are using numerical values for all of the machine parameters in order to simulate the machine and generate the plots below, the concepts are general and independent of these parameters. We will control this machine with a three-phase inverter, such as the one described in Chapter 6, to control the drive. Also note that, although we are analyzing a four-pole machine, our animation will follow standard convention and display a two-pole machine cross-section. It is understood that all values must be adjusted to account for the number of poles.
The three-phase inverter controlling the machine allows us to apply to the stator windings balanced, sinusoidal, three-phase voltages $V_{as}$, $V_{bs}$, and $V_{cs}$ whose frequency $\omega_e$ matches the rotor speed $\omega_r$ (i.e., $\omega_e=\omega_r$) and whose instantaneous phase can be shifted dynamically via the variable $\phi_v$. Thus, $$V_{as}(t) = \sqrt{2}V_s\cos(\omega_et+\phi_v)~~~~~~~~~~~~~~~~~~~~~~~\text{(1)}$$ $$V_{bs}(t) = \sqrt{2}V_s\cos(\omega_et-2\pi/3+\phi_v)~~~~~~~~~~\text{(2)}$$ $$V_{cs}(t) = \sqrt{2}V_s\cos(\omega_et+2\pi/3+\phi_v)~~~~~~~~~~\text{(3)}$$ where $$\phi_v=\theta_{esv}(0)-\theta_r(0)~~~~~~~~~~\text{(4)}$$ and since $\theta_r(0)$ is generally selected to be zero, $\phi_v=\theta_{esv}(0)$. It is important to realize that $\phi_v$ can be adjusted dynamically; it is not necessarily constant.
When analyzing the brushless dc drive in the steady state, the rotor reference frame is equivalent to the synchronously rotating reference frame (i.e., $\omega=\omega_r=\omega_e$). Under transient conditions, the rotor speed may deviate slightly from the synchronous speed; however, today’s fast inverters can quickly sense rotor speed changes and compensate by adjusting $\omega_e$ to match the transient $\omega_r$. Thus, our assumption that the rotor reference frame is equivalent to the synchronously rotating frame will typically be valid on millisecond timescales.
It is convenient to analyze the brushless dc drive in the rotor reference frame, aligning the d axis with the rotor north magnetic pole and the q axis $\pi/2$ radians counterclockwise ahead. The rotor position (i.e., the position of the q axis) is $\theta_r(t)=\omega_rt+\theta_r(0)$, and we are free to set the initial position $\theta_r(0) = 0$ for convenience, so that $\theta_r(t)=\omega_rt$. We see that the instantaneous phase of $V_{as}$, which is $\omega_rt+\phi_v$, can be written $\theta_r(t)+\phi_v$. This shows that $\phi_v$ orients the stator voltage with respect to the rotor position.
The essence of the Field Orientation is to vary the position and strength of the stator magnetic poles with respect to the permanent magnet in order to achieve a desired response. A steady-state phasor diagram is shown below, where $$\tilde{V}_{as} = \left(r_a+j\omega_rL_{ss}\right)\tilde{I}_{as}+\tilde{E}_{a}~~~~~~~~~~\text{(5)}$$ $$\tilde{E}_a = \frac{\omega_r\lambda'^r_n}{\sqrt{2}}\angle{0^\circ}~~~~~~~~~~\text{(6)}$$
