*2.2. Mathematical Modelling*

In order to explain the operation of the proposed high-harmonic injection-based field excitation scheme for the brushless operation of WFSMs, a mathematical model is presented in this sub-section.

For the typical WFSM topology employing *3r<sup>d</sup>* harmonic injection-based field excitation scheme [12], the magnetomotive force (MMF) for each phase is expressed as:

$$\begin{aligned}MMF\_A &= i\_A N\_{\varphi1} \left(\sin\theta\_s + \frac{1}{3}\sin 3\theta\_s\right) \\ MMT\_B &= i\_B N\_{\varphi1} \left\{\sin\left(\theta\_s - \frac{2\pi}{3}\right) + \frac{1}{3}\sin 3\theta\_s\right\} \\MMF\_C &= i\_C N\_{\varphi1} \left\{\sin\left(\theta\_s + \frac{2\pi}{3}\right) + \frac{1}{3}\sin 3\theta\_s\right\} \end{aligned} \tag{1}$$

where,

$$N\_{\wp1} = \frac{2}{\pi} (per\ phase\ number\ of\ turns)$$

*θs* = electrical angle (spatial), and

*ω* = angular frequency (electrical).

The armature winding currents can be expressed as:

$$\begin{array}{l} i\_A = I\_1 \sin(\omega t) + I\_H\\ i\_B = I\_1 \sin(\omega t - \frac{2\pi}{3}) + I\_H\\ i\_C = I\_1 \sin(\omega t + \frac{2\pi}{3}) + I\_H \end{array} \tag{2}$$

where *I*1 is the fundamental and *IH* is the supplied single-phase harmonic current into the armature winding.

When you bring Equation (2) to Equation (1), and add the MMF of three-phase armature windings *A*, *B*, and *C*, the net MMF of the armature winding is expressed as:

$$MMF\_{ABC}(\theta\_{\mathfrak{s}}, i) = MMF\_{\mathcal{A}} + MMF\_{\mathcal{B}} + MMF\_{\mathbb{C}} \tag{3}$$

$$MMF\_{ABC}(\theta\_s, i) = \begin{bmatrix} N\_{\rho1} \left\{ \sin(\theta\_s) + \frac{1}{3} \sin 3\theta\_s \right\} \left\{ I\_1 \sin(\omega t) + I\_H \right\} \\ + N\_{\rho1} \left\{ \sin(\theta\_s - \frac{2\pi}{3}) + \frac{1}{3} \sin 3\theta\_s \right\} \left\{ I\_1 \sin(\omega t - \frac{2\pi}{3}) + I\_H \right\} \end{bmatrix} \tag{4}$$

$$F\_{ABC}(\theta\_{s},i) = \left\{ \begin{array}{c} +N\_{\theta 1} \left\{ \sin(\theta\_{s} - \frac{2\pi}{3}) + \frac{1}{3} \sin 3\theta\_{s} \right\} \left\{ I\_{1} \sin(\omega t - \frac{2\pi}{3}) + I\_{H} \right\} \\ + N\_{\theta 1} \left\{ \sin(\theta\_{s} + \frac{2\pi}{3}) + \frac{1}{3} \sin 3\theta\_{s} \right\} \left\{ I\_{1} \sin(\omega t + \frac{2\pi}{3}) + I\_{H} \right\} \end{array} \right\} \tag{4}$$

$$\begin{aligned} \text{MMF}\_{ABC}(\theta\_{\theta}, i) &= N\_{\varphi 1} \left[ \begin{array}{c} I\_1 \left\{ \begin{array}{c} \sin(\omega t) \sin(\theta\_s) + \sin(\omega t - \frac{2\pi}{3}) \sin(\theta\_s - \frac{2\pi}{3}) \\ + \sin(\omega t + \frac{2\pi}{3}) \sin(\theta\_s + \frac{2\pi}{3}) \end{array} \right\} \\ + I\_H \sin 3\theta\_s \end{array} \right] \end{aligned} \tag{5}$$

$$MMF\_{ABC}(\theta\_{s\prime}, i) = \frac{3}{2} I\_1 N\_{\phi 1} \cos(\omega t - \theta\_s) + I\_H N\_{\phi 1} \sin 3\theta\_s \tag{6}$$

Equation (6) gives the net MMF produced in the air gap, which consists of two parts: (1) the fundamental MMF component, which produces the main armature field of rotating nature, and (2) spatial-location-fixed MMF component that develops a harmonic field in the air gap. This MMF component is utilized by the injected single-phase harmonic current. Since the main armature and the harmonic fields are produced by the currents having different frequencies, both the fields are not coupled.

Assuming that *θ*0 is the rotor excitation winding initial position angle, the spatial position of the excitation winding can be calculated as under:

$$
\theta\_{\mathbb{S}} = \omega t + \theta\_0 \tag{7}
$$

The generated flux of each winding pole will be:

$$\begin{array}{l} \psi\_{E} = n\_{E}P\_{\mathcal{S}}N\_{\mathfrak{q}1}\left\{\frac{3}{2}I\_{1}\cos(\omega t - \theta\_{\mathfrak{s}}) + I\_{H}\sin 3\theta\_{\mathfrak{s}}\right\} \\ \psi\_{E} = n\_{E}P\_{\mathcal{S}}N\_{\mathfrak{q}1}\left\{\frac{3}{2}I\_{1}\cos\theta\_{0} + I\_{H}\sin(3\omega t + 3\theta\_{0})\right\} \end{array} \tag{8}$$

where *nE* is the rotor excitation winding number of turns, and *Pg* is the air gap permeance. When *IH* is the *3r<sup>d</sup>* harmonic current, as in the case of [12], we have:

$$I\_H = I\_3 \sin \mathfrak{Z}(\omega t) \tag{9}$$

and the magnitude of the induced EMF in the rotor excitation winding can be calculated as [22,23]:

$$\begin{array}{l} \mathcal{e}\_{E} = 6 \frac{d \psi\_{E}}{dt} \\ \mathcal{e}\_{E} = 6 n\_{E} P\_{\mathcal{g}} N\_{\varrho 1} \{ 0 + 3 I\_{3} \omega \cos \mathfrak{z} (\omega t) \sin(3 \omega t + 3 \theta\_{0}) \} \\ \mathcal{e}\_{E} = 18 n\_{E} P\_{\mathcal{g}} N\_{\varrho 1} I\_{3} \omega \sin(6 \omega t + 3 \theta\_{0}) \end{array} \tag{10}$$

As the proposed high-harmonic field excitation scheme is based on the injection of single-phase *6th* harmonic current, *IH* in this case will become:

$$I\_H = I\_3 \sin \theta (\omega t) \tag{11}$$

and the magnitude of the induced EMF in the rotor excitation winding for the proposed high-harmonic injection-based field excitation scheme can be calculated as:

$$\begin{aligned} \omega\_{E} &= 6 \frac{d \psi\_{E}}{dt} \\ \omega\_{E} &= 6 \pi\_{E} P\_{\mathcal{S}} N\_{\varphi 1} \left\{ \begin{array}{l} 0 + 6I\_{3} \omega \cos \theta (\omega t) \sin(3 \omega t + 3 \theta\_{0}) \\ + 3I\_{3} \omega \sin \theta (\omega t) \cos(3 \omega t + 3 \theta\_{0}) \end{array} \right\} \end{aligned} \tag{12}$$

$$\varepsilon\_{E} = 3n\_{E}P\_{\mathcal{R}}N\_{\Psi1}I\_{3}\omega\{\mathcal{9}\sin(\theta\omega t + 3\theta\_{0}) - 3\sin 3(\omega t - 3\theta\_{0})\}\tag{13}$$

The induced EMF in the rotor excitation winding is rectified by an uncontrolled rectifier in order to inject DC to the field winding of the rotor to create the main rotor field. This field, when interacting with the 4-pole armature field, generates torque and archives brushless operation for WFSM.
