Squirrel-Cage Solid Rotor: Leakage Circuit Loops

Abstract

In squirrel-cage solid rotor, leakage fields are created by eddy currents induced in the slot bars, teeth crowns, teeth and on the bottom of the rotor slots (rotor yoke region). At the strong skin effect, we assume the leakage field in the body of the solid rotor is distributed along the periphery of its teeth and bottom of the rotor slots. Therefore, a “peripheral” model can be used to describe the field distribution in squirrel-cage solid rotor. On the basis of such a model, we consider below the circuit loops of the eddy currents induced in squirrel-cage solid rotors and determine their parameters at the strong skin effect.

Appendix A.19 Transformations

19.1.1 A.19.1 Expression [Z _cz2(Z _a  + Z _τcz2)c ² _cz2 ]/[Z _cz2 + (Z _a  + Z _τcz2)c ² _cz2 ]: Real and Imaginary Components

We consider the real and imaginary components of expression (19.70)

$$ \frac{Z_{cz2}\left({Z}_a+{Z}_{\tau cz2}\right){c}_{cz2}^2}{Z_{cz2}+\left({Z}_a+{Z}_{\tau cz2}\right){c}_{cz2}^2} $$

(A.19.1)

For this purpose, we first define the real and imaginary components of expression (Z _a  + Z _τcz2)c ² _cz2 used in (A.19.1). Considering the conditions given in (19.98), the value of (Z _a  + Z _τcz2)c ² _cz2 can be presented as

$$ \begin{array}{l}\left({Z}_a+{Z}_{\tau cz2}\right){c}_{cz2}^2=\left[\left({r}_{ca}/s+j{x}_{ca\sigma}\right)+\left({r}_{\tau cz2}/s+j{x}_{\tau cz2}\right)\right]\left({k}_{cz2r}+j{k}_{cz2x}\right)\hfill \\ {}\kern6.24em =\left[\left({r}_{ca}/s\right){k}_{cz2r}-{x}_{ca\sigma}{k}_{cz2x}+\left({r}_{\tau cz2}/s\right){k}_{cz2r}-{x}_{\tau cz2}{k}_{cz2x}\right]+j\Big[{x}_{ca\sigma}{k}_{cz2r}\hfill \\ {}\kern7.24em +\left({r}_{ca}/s\right){k}_{cz2x}\hfill \\ {}+{x}_{\tau cz2}{k}_{cz2r}+\left({r}_{\tau cz2}/s\right){k}_{cz2x}\Big]=\left[\frac{r_{ca}}{s}\left(1+\frac{r_{\tau cz2}}{r_{ca}}\right){k}_{cz2r}-{x}_{ca\sigma}\left(1+\frac{x_{\tau cz2}}{x_{ca\sigma}}\right){k}_{cz2x}\right]\hfill \\ {}\kern-.28em +j\left[{x}_{ca\sigma}\left({k}_{cz2r}+\frac{r_{ca}/s}{x_{ca\sigma}}{k}_{cz2x}\right)+{x}_{\tau cz2}\left({k}_{cz2r}+\frac{r_{\tau cz2}/s}{x_{\tau cz2}}{k}_{cz2x}\right)\right]=\frac{r_{ca}^{{\prime\prime} }}{s}+j{x}_{ca\sigma}^{{\prime\prime}}\hfill \end{array} $$

(A.19.2)

where $\begin{array}{l}\frac{r_{ca}^{{\prime\prime} }}{s}=\left[\frac{r_{ca}}{s}\left(1+\frac{r_{\tau cz2}}{r_{ca}}\right){k}_{cz2r}-{x}_{ca\sigma}\left(1+\frac{x_{\tau cz2}}{x_{ca\sigma }}\right){k}_{cz2x}\right];\hfill \\ {}{x}_{ca\sigma}^{{\prime\prime} }={x}_{ca\sigma}\left({k}_{cz2r}+\frac{r_{ca}/s}{x_{ca\sigma }}{k}_{cz2x}\right)+{x}_{\tau cz2}\left({k}_{cz2r}+\frac{r_{\tau cz2}/s}{x_{\tau cz2}}{k}_{cz2x}\right)\hfill \end{array}$

From (19.98), we have for the impedance Z _cz2 used in (A.19.1) the condition Z _cz2 = r _c2/s + jx _c2. Taking into account the condition Z _cz2 = r _c2/s + jx _c2 and expression (A.19.2), we can use in (A.19.1) the following non-dimensional factors:

$$ {\alpha}_{cz2}=\frac{r_{ca}^{{\prime\prime} }}{r_{c2}};{\beta}_{cz2}=\frac{x_{c2}}{r_{c2}/s};{\gamma}_{cz2}=\frac{x_{ca\sigma}^{{\prime\prime} }}{r_{c2}/s} $$

(A.19.3)

On the basis of expressions (A.19.1), (A.19.2) and (A.19.3), it follows that

$$ \begin{array}{l}\frac{Z_{cz2}\left({Z}_a+{Z}_{\tau cz2}\right){c}_{cz2}^2}{Z_{cz2}+\left({Z}_a+{Z}_{\tau cz2}\right){c}_{cz2}^2}=\frac{\left({r}_{c2}/s+j{x}_{c2}\right)\left({r}_{ca}^{{\prime\prime} }/s+j{x}_{ca\sigma}^{{\prime\prime}}\right)}{\left({r}_{c2}/s+{r}_{ca}^{{\prime\prime} }/s\right)+j\left({x}_{c2}+j{x}_{ca\sigma}^{{\prime\prime}}\right)}\hfill \\ {}\kern9em =\frac{r_{c2}}{s}\kern0.14em \frac{\left(1+j{\beta}_{cz2}\right)\left({\alpha}_{cz2}+j{\gamma}_{cz2}\right)}{\left(1+{\alpha}_{cz2}\right)+j\left({\beta}_{cz2}+{\gamma}_{cz2}\right)}\hfill \\ {}\kern9em =\frac{r_{c2}}{s}\kern0.14em \frac{\alpha_{cz2}\left(1+{\beta}_{cz2}^2\right)+{\alpha}_{cz2}^2+{\gamma}_{cz2}^2}{{\left(1+{\alpha}_{cz2}\right)}^2+{\left({\beta}_{cz2}+{\gamma}_{cz2}\right)}^2}\hfill \\ {}\kern10em +j{x}_{c2}\frac{\left({\gamma}_{cz2}/{\beta}_{cz2}\right)\left(1+{\beta}_{cz2}^2\right)+{\alpha}_{cz2}^2+{\gamma}_{cz2}^2}{{\left(1+{\alpha}_{cz2}\right)}^2+{\left({\beta}_{cz2}+{\gamma}_{cz2}\right)}^2}\hfill \\ {}\kern9em =\frac{r_{c2}}{s}{k}_{c2r}^{{\prime\prime} }+j{x}_{c2}{k}_{c2x}^{{\prime\prime}}\hfill \end{array} $$

(A.19.4)

where ${k}_{c2r}^{{\prime\prime} }=\frac{\alpha_{cz2}\left(1+{\beta}_{cz2}^2\right)+{\alpha}_{cz2}^2+{\gamma}_{cz2}^2}{{\left(1+{\alpha}_{cz2}\right)}^2+{\left({\beta}_{cz2}+{\gamma}_{cz2}\right)}^2};{k}_{c2x}^{{\prime\prime} }=\frac{\left({\gamma}_{cz2}/{\beta}_{cz2}\right)\left(1+{\beta}_{cz2}^2\right)+{\alpha}_{cz2}^2+{\gamma}_{cz2}^2}{{\left(1+{\alpha}_{cz2}\right)}^2+{\left({\beta}_{cz2}+{\gamma}_{cz2}\right)}^2}$.