**3. The Equivalent Circuit for Analysis and the Analytical Calculation of Key Parameters** *3.1. The Equivalent Circuit for Analysis and the Calculation of Output Torque*

Although the winding structure of the squirrel-cage rotor is a squirrel cage type, the winding can still be converted into equivalent three-phase winding to simplify the analysis. Therefore, the equation of the HTR-ASD in the rotor d-q frame can be expressed as:

$$
\begin{bmatrix} u\_{sd} \\ u\_{sq} \\ u\_f \end{bmatrix} = p \begin{bmatrix} L\_{sd} & 0 & M\_{\text{aff}} \\ 0 & L\_{sq} & 0 \\ \frac{3}{2} M\_{\text{aff}} & 0 & L\_f \end{bmatrix} \begin{bmatrix} i\_{sd} \\ i\_{sq} \\ i\_f \end{bmatrix} + \begin{bmatrix} r\_s & 0 & 0 \\ 0 & r\_s & 0 \\ 0 & 0 & r\_f \end{bmatrix} \begin{bmatrix} i\_{sd} \\ i\_{sq} \\ i\_f \end{bmatrix} + \begin{bmatrix} -\omega \psi\_{sq} \\ \omega \psi\_{sd} \\ 0 \end{bmatrix} \tag{2}
$$

where *ω* = *ωHTR* − *ωSCR*. *ωHTR* is the angular velocity of the homopolar-type rotor, and *ωSCR* is the angular velocity of the squirrel-cage rotor.

To obtain the equivalent circuit, the excitation winding is converted to the squirrel-cage rotor side. The flux linkage equation can be expressed as:

$$
\begin{bmatrix}
\psi\_{sd} \\
\psi'\_f
\end{bmatrix} = \begin{bmatrix}
L\_{sd} & L\_{md} \\
L\_{md} & L'\_f
\end{bmatrix} \begin{bmatrix}
i\_{sd} \\
i'\_f
\end{bmatrix} \tag{3}
$$

where *i <sup>f</sup>* = <sup>2</sup> <sup>3</sup> *if Maf* /*Lmd*.

As the squirrel cage is short circuited, there is *usd*= *usq*= 0. Therefore, the equivalent circuit is as shown in Figure 6.

**Figure 6.** The equivalent circuit of HTR-ASD.

When HTR-ASD operates in steady state, there is:

$$\begin{cases} \ I\_{s\eta} = -\frac{\omega \psi\_{sd}}{r\_s} \\\ I\_{sd} = \frac{\omega \psi\_{sq}}{r\_s} \end{cases} \tag{4}$$

The electromagnetic torque can be expressed as:

$$T\_{cm} = 1.5p \left(\psi\_{sd} I\_{sq} - \psi\_{sq} I\_{sd}\right) = -1.5p \frac{r\_s}{\omega} \left(I\_{sq}^2 + I\_{sd}^2\right) \tag{5}$$

Combined with Equations (3) and (4), *Isd* can be expressed as:

$$I\_{sd} = \frac{\omega^2 L\_{sq} L\_{ml} i\_f'}{r\_s^2 + \omega^2 L\_{sq} L\_{sd}} \tag{6}$$

The steady-state salient ratio *ρ* is defined as:

$$\rho = \frac{L\_{sq}}{L\_{sd}}\tag{7}$$

Combined with Equations (4)–(7), *Tem* can be expressed as:

$$T\_{cm} = -1.5 \rho r\_s i'^2\_f \frac{\omega L\_{md}^2 \left(r\_s^2 + \rho^2 \omega^2 L\_{sd}^2\right)}{\left(r\_s^2 + \rho \omega^2 L\_{sd}^2\right)^2} \tag{8}$$

When *ρ* = 1, *Tem* can be expressed as:

$$T\_{em} = -\frac{2}{3} pr\_s \dot{i}\_f^2 \frac{M\_{\text{af}}^2}{\frac{r\_s^2}{\omega} + \omega L\_{sd}^2} \tag{9}$$

From Equations (7)–(9), it can be seen the torque of the HTR-ASD is related to the excitation current and slip speed. When the excitation current is constant, its torque characteristics are similar to an asynchronous motor. When the load torque is constant, the speed can be adjusted by changing the excitation current. Generally, the difference between *Lsq* and *Lsd* is not too large, and Equation (9) can be used to reflect the torque of the HTR-ASD for simplifying the calculations.

#### *3.2. Calculation of Air Gap Magnetic Field Parameters*

Strictly speaking, because the homopolar-type rotor has a special three-dimensional magnetic circuit structure with the same pole, it is necessary to develop a 3D-FEM to accurately calculate its parameters. Nevertheless, since this method requires huge computation time, it is usually used for the final performance check of the design and is not suitable for analytical research and calculations.

To simplify the calculation, the air gap performance function can be considered. The authors of [27] use the rotor shape function to represent the air gap performance function. However, according to the hypothesis of the equal magnetic potential plane, the air gap magnetic field is perpendicular to the rotor surface. Figure 7a shows the axial view of the HTR-ASD, and Figure 7b shows the no-load magnetic density distribution within one rotor tooth pitch. Obviously, the air gap magnetic density waveform is very different from the rotor slot shape.

**Figure 7.** The axial view of the HTR-ASD and no-load magnetic density distribution within one rotor tooth pitch. (**a**) The axial view of the HTR-ASD. (**b**) No-load magnetic density distribution within one rotor tooth pitch.

Before analyzing the composite magnetic flux density, it is necessary to analyze the waveform of single-side air gap magnetic flux density. Figure 8 shows the air gap permeance waveform and its components. To calculate different slot shapes, the per-unit values of the air gap performance function *λ*∗ *<sup>n</sup>* are adopted. *λ*∗(*θ*) is multiplied by the reference value Λ*<sup>B</sup>* to obtain the actual value, in which the reference value of specific permeability can be expressed by:

$$
\Lambda\_B = \mu\_0 / (k\_\delta \delta\_{\rm min}) \tag{10}
$$

where *μ*<sup>0</sup> and *k<sup>δ</sup>* denote the vacuum permeability and Carter's coefficient of slotting, respectively.

**Figure 8.** Air gap rate permeance waveform and each component.

The definition of the per-unit air gap permeance function is:

$$
\lambda^\* \_\delta (\theta) = \frac{B(\theta)}{\Lambda\_B F\_\delta} = \sum\_{n=0}^\infty \lambda\_n \cos n\theta \tag{11}
$$

where *F<sup>δ</sup>* is the magnetomotive force (MMF) of the air gap.

The air gap magnetic density on both sides during no-load excitation can be expressed as:

$$\begin{cases} \begin{aligned} B\_{dcl} &= F\_{F\delta} \sum\_{n=0}^{\infty} \lambda\_n \cos n\theta \\\ B\_{dcr} &= F\_{F\delta} \sum\_{n=0}^{\infty} (-1)^{n+1} \lambda\_n \cos n\theta \end{aligned} \tag{12}$$

The per-unit value of the dc air gap permeance function can be expressed as:

$$
\lambda\_{d\varepsilon}^\*(\theta) = \frac{B\_{d\varepsilon l} + B\_{d\varepsilon r}}{F\_{F\delta}\Lambda\_B} = 2 \sum\_{n=0}^{\infty(odd)} \lambda\_n \cos n\theta \tag{13}
$$

The amplitudes of armature winding MMF on the d-axis and q-axis can be expressed as:

$$\begin{cases} F\_{ad} = F\_{1m} \cos \theta\_1\\ F\_{aq} = F\_{1m} \sin \theta\_1 \end{cases} \tag{14}$$

The d-axis and q-axis components of air gap magnetic density on the left and right sides caused by armature reaction can be expressed as:

$$\begin{cases} \begin{array}{l} B\_{dl} = k\_{dm} F\_{1m} \cos \theta\_1 \sum\_{n=0}^{\infty} \lambda\_n \cos n\theta \\\ B\_{dr} = k\_{dm} F\_{1m} \cos \theta\_1 \sum\_{n=0}^{\infty} (-1)^{n+1} \lambda\_n \cos n\theta \end{array} \tag{15}$$

$$\begin{cases} \begin{array}{l} B\_{ql} := k\_{qm} F\_{1m} \sin \theta\_1 \sum\_{n=0}^{\infty} \lambda\_n \cos \eta \theta \\\ B\_{qr} := k\_{qm} F\_{1m} \sin \theta\_1 \sum\_{n=0}^{\infty} (-1) \lambda\_n \cos \eta \theta \end{array} \tag{16}$$

where *kdm* and *kqm* represent the proportion of air gap MMF in parallel d-axis and q-axis magnetic circuits on both sides.

The per unit values of the d-axis and q-axis air gap permeance functions can be expressed as:

$$\lambda\_d^\*(\theta) = \frac{B\_{dl} + B\_{dr}}{k\_{dm} F\_{1m} A\_B} = (2\lambda\_0 + \lambda\_2) \cos \theta + \sum\_{n=3}^{\infty(add)} (\lambda\_{n-1} + \lambda\_{n+1}) \cos n\theta \tag{17}$$

$$\lambda\_q^\*(\theta) = \frac{B\_{ql} + B\_{qr}}{k\_{qm} F\_{1m} \Lambda\_B} = (2\lambda\_0 - \lambda\_2) \sin \theta + \sum\_{n=3}^{\infty(odd)} (\lambda\_{n-1} - \lambda\_{n+1}) \sin n\theta \tag{18}$$

Therefore, *λ*∗ *dc*(*θ*), *λ*<sup>∗</sup> *<sup>d</sup>*(*θ*), and *λ*<sup>∗</sup> *<sup>q</sup>* (*θ*) can be regarded as a bridge between the rotor shape and the machine parameters. When the rotor shape is determined, their values can be obtained from a look-up table [31]. After that, they can be used to analytically calculate the air gap magnetic flux density and the winding inductance parameter.
