*3.3. Calculation of Excitation Winding Parameters*

Another parameter to be calculated is the excitation time constant, which is closely related to the resistance and inductance of the excitation winding. Figure 9 shows the excitation window and its size, which is used for the assembly of excitation windings.

**Figure 9.** Excitation window and excitation winding. (**a**) Excitation window and its size. (**b**) Excitation winding.

The resistance of the excitation winding can be expressed as:

$$r\_f = \frac{\rho\_{\text{Cu}} N\_f \pi D\_f}{S\_{\text{cf}}} \tag{19}$$

where *ρCu* is the conductivity of copper, *Nf* is the turn of excitation winding, *Df* is the excitation coil diameter and *Scf* is the cross-sectional area of the excitation coil.

Under the excitation current alone, the magnetic density of a one-sided air gap between the HTR and the squirrel-cage rotor can be expressed as:

$$B\_{dcl}(\theta) = \frac{N\_f I\_f}{2} k\_m \sum\_{n=0}^{\infty} \lambda\_n \cos \, n\theta \tag{20}$$

Therefore, the average air gap flux density at one side of the machine can be expressed as:

$$B\_{\rm av} = \frac{1}{2\pi} \int\_0^{2\pi} B\_{\rm dcl}(\theta) d\theta = \frac{N\_f I\_f}{2} k\_{\rm m} \lambda\_{\rm dcl}^\* \Lambda\_B \tag{21}$$

The main flux linkage of excitation winding is:

$$
\psi\_{\mathcal{Y}} = N\_f B\_{\text{av}} S\_\delta = I\_f N\_f^2 \pi R\_H (I\_1 + 2\delta) k\_m \lambda\_{dc0}^\* \Lambda\_B \tag{22}
$$

where *S<sup>δ</sup>* represents the total area of a one-sided air gap between HTR and the squirrel-cage rotor, *RH* is the radius of the HTR, and *l*<sup>1</sup> is the single length of the HTR.

The main inductance of the excitation winding is:

$$L\_{\sharp\overline{f}} = \frac{\Psi\_{\overline{f}\overline{f}}}{I\_f} = N\_f^2 \pi R\_H (l\_{\perp 1} + 2\delta) k\_m \lambda\_{dc0}^\* \Lambda\_B \tag{23}$$

Moreover, (23) shows that the main self-inductance of the excitation winding only contains the dc component, because in the same pole magnetic field structure, the selfinductance parameters of the excitation winding are determined by the overall magnetic circuit state and have nothing to do with the details of the air gap flux density waveform.

The excitation time constant can be expressed by:

$$\mathbf{t}\_f = \frac{\mathbf{L}\_{\text{ff}}}{r\_f} = \frac{N\_f \pi R\_H (l\_1 + 2\delta) k\_m \lambda\_{dc0}^\* \Lambda\_B}{\rho\_{\text{Cu}} \pi D\_f} S\_{cf} \tag{24}$$

In the design of an HTR-ASD, the excitation winding time constant *tf* is expected to be as small as possible. On the one hand, it can reduce the excitation establishment time and improve the system mobility. On the other hand, it can make it easier to adjust the flux linkage of the excitation winding in the process of speed regulation, in order to strengthen the dynamic response ability of the system in the process of speed regulation.
