The Triple-Cage Rotor: The Slot Leakage Circuit Loops

Abstract

In Chap. 14, we considered the circuit loops of a triple-cage rotor on the basis of its layered model arising as a result of using the average values of electromagnetic parameters ρ and μ for the corresponding rotor teeth regions. At a strong skin effect, the rotor tooth regions is characterized by the presence of a tangential component of the magnetic field (leakage field), which is localized mainly in the rotor slots, and it is caused by the currents of the rotor windings. Therefore, we consider below the circuit loops of the triple-cage rotor on the basis of describing the rotor slot leakage field.

Appendix A.15 Transformations and Results of Calculations

15.1.1 A.15.1 Factors (c₂ ²c _τ3) and (c₂ ²c₃ ²c² _τ3): Real and Imaginary Components at a Weak Skin Effect

The products of the factors (c ²₂ c _τ3) and (c ²₂ c ²₃ c ² _τ3 ) used in (15.23) can be presented as the sum of the real and imaginary components at a weak skin effect. Taking into account (15.24), (15.25), and (15.26), the product of the factors (c ²₂ c _τ3) can be written as

$$ \begin{array}{l}{c}_2^2{c}_{\tau 3}=\left(1+j2{\varepsilon}_2^2\right)\left(1+j2{k}_{z23}{\varepsilon}_2{\varepsilon}_3\right)=1+j2\left({\varepsilon}_2^2+{k}_{z23}{\varepsilon}_2{\varepsilon}_3\right)\hfill \\ {}\kern2em =1+j2{\varepsilon}_2^2\left(1+{k}_{z23}\frac{\varepsilon_3}{\varepsilon_2}\right)=1+j2{k}_{\tau 23}{\varepsilon}_2^2\hfill \end{array} $$

(A.15.1)

where k _τ23 = 1 + k _z23(ε ₃/ε ₂).

With consideration for (15.24), (15.25), and (15.26), for the product of the factors (c ²₂ c ²₃ c ² _τ3 ) we have

$$ \begin{array}{l}{c}_2^2{c}_3^2{c}_{\tau 3}^2=\left(1+j2{\varepsilon}_2^2\right)\left(1+j2{\varepsilon}_3^2\right)\left(1+j4{k}_{z23}{\varepsilon}_2{\varepsilon}_3\right)\hfill \\ {}\kern2.85em =\left(1+j2{\varepsilon}_2^2+j2{\varepsilon}_3^2\right)\left(1+j4{k}_{z23}{\varepsilon}_2{\varepsilon}_3\right)=1+j2\left({\varepsilon}_2^2+{\varepsilon}_3^2+2{k}_{z23}{\varepsilon}_2{\varepsilon}_3\right)\hfill \\ {}\kern2.85em =1+j2{\varepsilon}_2^2\left(1+\frac{\varepsilon_3^2}{\varepsilon_2^2}+2{k}_{z23}\frac{\varepsilon_3}{\varepsilon_2}\right)=1+j2{k}_{c23}{\varepsilon}_2^2\hfill \end{array} $$

(A.15.2)

where k _c23 = 1 + ε ²₃ /ε ²₂  + 2k _z23(ε ₃/ε ₂).

15.1.2 A.15.2 Factors (c₂ ²c _τ3) and (c₂ ²c₃ ²c² _τ3): Real and Imaginary Components at a Strong Skin Effect

The products of the factors (c ²₂ c _τ3) and (c ²₂ c ²₃ c ² _τ3 ) used in (15.23) can be presented as the sum of the real and imaginary components at a strong skin effect. For this purpose, we use the following conditions

$$ {c}_2^2={k}_{c2r}+j{k}_{c2x};{c}_3^2={k}_{c3r}+j{k}_{c3x};{c}_{\tau 3}={c}_{\tau 3r}+j{c}_{\tau 3x} $$

arising from the expressions obtained in (15.4), (15.9), (15.10), and (13.59).

Then for the product of the factor (c ²₂ c _τ3) used in (15.23), we have

$$ \begin{array}{l}\left({c}_2^2{c}_{\tau 3}\right)=\left({k}_{c2r}+j{k}_{c2x}\right)\left({c}_{\tau 3r}+j{c}_{\tau 3x}\right)\hfill \\ {}\kern2.9em =\left({k}_{c2r}{c}_{\tau 3r}-{k}_{c2x}{c}_{\tau 3x}\right)+j\left({k}_{c2x}{c}_{\tau 3r}+{k}_{c2r}{c}_{\tau 3x}\right)={k}_{c3r}+j{k}_{c3x}\hfill \end{array} $$

(A.15.3)

where k _c3r  = k _c2r c _τ3r  − k _c2x c _τ3x ; k _c3x  = k _c2x c _τ3r  + k _c2r c _τ3x .

For the product of the factor (c ²₂ c ²₃ c ² _τ3 ) used in (15.23), it follows that

$$ \begin{array}{l}\left({c}_2^2{c}_3^2\right){c}_{\tau 3}^2=\left({k}_{c2r}+j{k}_{c2x}\right)\left({k}_{c3r}+j{k}_{c3x}\right){\left({c}_{\tau 3r}+j{c}_{\tau 3x}\right)}^2\hfill \\ {}\kern3.7em =\left[{k}_{c2r}{k}_{c3r}-{k}_{c2x}{k}_{c3x}+j\left({k}_{c2x}{k}_{c3r}+{k}_{c2r}{k}_{c3x}\right)\right]\left[\left({c}_{\tau 3r}^2-{c}_{\tau 3x}^2\right)+j2{c}_{\tau 3r}{c}_{\tau 3x}\right]\hfill \\ {}\kern3.7em =\left[\left({k}_{c2r}{k}_{c3r}-{k}_{c2x}{k}_{c3x}\right)\left({c}_{\tau 3r}^2-{c}_{\tau 3x}^2\right)-2\left({k}_{c2x}{k}_{c3r}+{k}_{c2r}{k}_{c3x}\right){c}_{\tau 3r}{c}_{\tau 3x}\right]\hfill \\ {}\kern4.95em +j\left[\left({k}_{c2x}{k}_{c3r}+{k}_{c2r}{k}_{c3x}\right)\left({c}_{\tau 3r}^2-{c}_{\tau 3x}^2\right)+2\left({k}_{c2r}{k}_{c3r}-{k}_{c2x}{k}_{c3x}\right){c}_{\tau 3r}{c}_{\tau 3x}\right]\hfill \\ {}\kern3.7em ={k}_{c23r}+j{k}_{c23x}\hfill \end{array} $$

(A.15.4)

where $\begin{array}{ll}{k}_{c23r}\hfill & \kern-1em =\left({k}_{c2r}{k}_{c3r}-{k}_{c2x}{k}_{c3x}\right)\left({c}_{\tau 3r}^2-{c}_{\tau 3x}^2\right)-2\left({k}_{c2x}{k}_{c3r}+{k}_{c2r}{k}_{c3x}\right){c}_{\tau 3r}{c}_{\tau 3x}\hfill \\ {}{k}_{c23x}\hfill & \kern-1em =\left({k}_{c2x}{k}_{c3r}+{k}_{c2r}{k}_{c3x}\right)\left({c}_{\tau 3r}^2-{c}_{\tau 3x}^2\right)+2\left({k}_{c2r}{k}_{c3r}-{k}_{c2x}{k}_{c3x}\right){c}_{\tau 3r}{c}_{\tau 3x}.\hfill \end{array}$

15.1.3 A.15.3 Impact Factors k′ _r3 and k′ _r4

The calculations of factors k ^′ _r3 and k ^′ _r4 were implemented using the expressions shown in (15.35) and (15.39). The results of the calculations are shown in Table A.15.1.

Table A.15.1 Values of impact factors k ^′ _r3 and k ^′ _r4 depending on the relative depth of field penetration in the starting winding slot bar

15.1.4 A.15.4 Factors (c₂ ²c _τ31) and (c₂ ²c₃₁ ² c² _τ31): Real and Imaginary Components

The products of factors (c ²₂ c _τ31) and (c ²₂ c ²₃₁ c ² _τ31 ) used in (15.74) can be presented as the sum of the real and imaginary components. Taking into account (13.59), (15.52), (15.77), and (15.78), for the product of factor (c ²₂ c _τ31), we have

$$ \begin{array}{l}\left({c}_2^2{c}_{\tau 31}\right)=\left({k}_{c2r}+j{k}_{c2x}\right)\left({c}_{\tau 31r}+j{c}_{\tau 31x}\right)\hfill \\ {}\kern3.25em =\left({k}_{c2r}{c}_{\tau 31r}-{k}_{c2x}{c}_{\tau 31x}\right)+j\left({k}_{c2x}{c}_{\tau 31r}+{k}_{c2r}{c}_{\tau 31x}\right)={k}_{c31r}+j{k}_{c31x}\hfill \end{array} $$

(A.15.5)

where k _c31r  = k _c2r c _τ31r  − k _c2x c _τ31x ; k _c31x  = k _c2x c _τ31r  + k _c2r c _τ31x .

From (13.59), (15.52), (15.77), and (15.78), it follows for the product of factor (c ²₂ c ²₃₁ )c ² _τ31 that

$$ \begin{array}{l}\left({c}_2^2{c}_{31}^2\right){c}_{\tau 31}^2=\left({k}_{c2r}+j{k}_{c2x}\right)\left({k}_{c31r}+j{k}_{c31x}\right){\left({c}_{\tau 31r}+j{c}_{\tau 31x}\right)}^2\hfill \\ {}\kern4.45em =\left[{k}_{c2r}{k}_{c31r}-{k}_{c2x}{k}_{c31x}+j\left({k}_{c2x}{k}_{c31r}+{k}_{c2r}{k}_{c31x}\right)\right]\left[\left({c}_{\tau 31r}^2-{c}_{\tau 31x}^2\right)\right.\hfill \\ {}\kern5.7em \left.+j2{c}_{\tau 31r}{c}_{\tau 31x}\right]\hfill \\ {}\kern4.45em =\left[\left({k}_{c2r}{k}_{c31r}-{k}_{c2x}{k}_{c31x}\right)\left({c}_{\tau 31r}^2-{c}_{\tau 31x}^2\right)-2\left({k}_{c2x}{k}_{c31r}+{k}_{c2r}{k}_{c31x}\right)\right.\hfill \\ {}\kern5.7em \left.\times {c}_{\tau 31r}{c}_{\tau 31x}\right]\hfill \\ {}\kern5.6em +j\left[\left({k}_{c2x}{k}_{c31r}+{k}_{c2r}{k}_{c31x}\right)\left({c}_{\tau 31r}^2-{c}_{\tau 31x}^2\right)+2\left({k}_{c2r}{k}_{c31r}-{k}_{c2x}{k}_{c31x}\right)\right.\hfill \\ {}\kern5.6em \left.\times {c}_{\tau 31r}{c}_{\tau 31x}\right]\hfill \\ {}\kern4.45em ={k}_{c231r}+j{k}_{c231x}\hfill \end{array} $$

(A.15.6)

where $\begin{array}{ll}{k}_{c231r}\hfill & \kern-1em =\left({k}_{c2r}{k}_{c31r}-{k}_{c2x}{k}_{c31x}\right)\left({c}_{\tau 31r}^2-{c}_{\tau 31x}^2\right)-2\left({k}_{c2x}{k}_{c31r}+{k}_{c2r}{k}_{c31x}\right){c}_{\tau 31r}{c}_{\tau 31x}\hfill \\ {}{k}_{c231x}\hfill & \kern-1em =\left({k}_{c2x}{k}_{c31r}+{k}_{c2r}{k}_{c31x}\right)\left({c}_{\tau 31r}^2-{c}_{\tau 31x}^2\right)+2\left({k}_{c2r}{k}_{c31r}-{k}_{c2x}{k}_{c31x}\right){c}_{\tau 31r}{c}_{\tau 31x}.\hfill \end{array}$

15.1.5 A.15.5 The Expression [Z ₃₂(Z ₄₁ + Z _τ32)c² ₃₂]/[Z ₃₂ + (Z ₄₁ + Z _τ32)c² ₃₂]: Real and Imaginary Components

We now consider the second term of expression (15.74)

$$ \frac{Z_{32}\left({Z}_{41}+{Z}_{\tau 32}\right){c}_{32}^2}{Z_{32}+\left({Z}_{41}+{Z}_{\tau 32}\right){c}_{32}^2} $$

(A.15.7)

This expression can be presented as the sum of the real and imaginary components. We first express the value of (Z ₄₁ + Z _τ32)c ²₃₂ used in (A.15.7) as the sum of the real and imaginary components. Then, taking into account (15.75), the value of (Z ₄₁ + Z _τ32)c ²₃₂ can be presented as

$$ \begin{array}{l}\left({Z}_{41}+{Z}_{\tau 32}\right){c}_{32}^2=\left[\left({r}_{c4}/s\right)+j{x}_{\Pi 4\sigma }+\left({r}_{\tau 32}/s\right)+j{x}_{\tau 32}\right]\left({k}_{c32r}+j{k}_{c32x}\right)\hfill \\ {}\kern6.1em =\left[\left({r}_{c4}/s\right){k}_{c32r}-{x}_{\Pi 4\sigma }{k}_{c32x}+\left({r}_{\tau 32}/s\right){k}_{c32r}-{x}_{\tau 32}{k}_{c32x}\right]\hfill \\ {}\kern7.3em +j\left[{x}_{\Pi 4\sigma }{k}_{c32r}+\left({r}_{c4}/s\right){k}_{c32x}+{x}_{\tau 32}{k}_{c32r}+\left({r}_{\tau 32}/s\right){k}_{c32x}\right]\hfill \\ {}\kern6.1em =\left[\frac{r_{c4}}{s}\left(1+\frac{r_{\tau 32}}{r_{c4}}\right){k}_{c32r}-{x}_{\Pi 4\sigma}\left(1+\frac{x_{\tau 32}}{x_{\Pi 4\sigma }}\right){k}_{c32x}\right]\hfill \\ {}\kern7.3em +j\left[{x}_{\Pi 4\sigma}\left({k}_{c32r}+\frac{r_{c4}/s}{x_{\Pi 4\sigma }}{k}_{c32x}\right)+{x}_{\tau 32}\left({k}_{c32r}+\frac{r_{\tau 32}/s}{x_{\tau 32}}{k}_{c32x}\right)\right]\hfill \\ {}\kern6.1em =\frac{r_{c4}}{s}\left[\left(1+\frac{r_{\tau 32}}{r_{c4}}\right){k}_{c32r}-\frac{x_{\Pi 4\sigma }}{r_{c4}/s}\left(1+\frac{x_{\tau 32}}{x_{\Pi 4\sigma }}\right){k}_{c32x}\right]\hfill \\ {}\kern7.3em +j\left({x}_{\Pi 4\sigma }{k}_{c324x}+{x}_{\tau 32}{k}_{\tau 324x}\right)={r}_{c4}^{{\prime\prime} }/s+j{x}_{\Pi 4\sigma}^{{\prime\prime}}\hfill \end{array} $$

(A.15.8)

where $\begin{array}{l}\frac{r_{c4}^{{\prime\prime} }}{s}=\frac{r_{c4}}{s}\left[\left(1+\frac{r_{\tau 32}}{r_{c4}}\right){k}_{c32r}-\frac{x_{\Pi 4\sigma }}{r_{c4}/s}\left(1+\frac{x_{\tau 32}}{x_{\Pi 4\sigma }}\right){k}_{c32x}\right];{x}_{\Pi 4\sigma}^{{\prime\prime} }={x}_{\Pi 4\sigma }{k}_{c324x}\hfill \\ {}\kern2em +{x}_{\tau 32}{k}_{\tau 324x};\hfill \\ {}{k}_{c324x}={k}_{c32r}+\frac{r_{c4}/s}{x_{\Pi 4\sigma }}{k}_{c32x};{k}_{\tau 324x}={k}_{c32r}+\frac{r_{\tau 32}/s}{x_{\tau 32}}{k}_{c32x}.\hfill \end{array}$

Now, using (A.15.8) and the following non-dimensional values

$$ {\alpha}_{324}=\frac{r_{c4}^{{\prime\prime} }}{r_{c32}};{\beta}_{324}=\frac{x_{\Pi 32\sigma }}{r_{c32}/s};{\gamma}_{324}=\frac{x_{\Pi 4\sigma}^{{\prime\prime} }}{\left({r}_{c32}/s\right)} $$

(A.15.9)

we can obtain

$$ \begin{array}{l}\frac{Z_{32}\left({Z}_{41}+{Z}_{\tau 32}\right){c}_{32}^2}{Z_{32}+\left({Z}_{41}+{Z}_{\tau 32}\right){c}_{32}^2}=\frac{\left({r}_{c32}/s+j{x}_{\Pi 32\sigma}\right)\left({r}_{c4}^{{\prime\prime} }/s+j{x}_{\Pi 4\sigma}^{{\prime\prime}}\right)}{\left({r}_{c32}/s+{r}_{c4}^{{\prime\prime} }/s\right)+j\left({x}_{\Pi 32\sigma }+{x}_{\Pi 4\sigma}^{{\prime\prime}}\right)}\hfill \\ {}\kern8.85em =\frac{r_{c32}}{s}\ \frac{\left(1+j{\beta}_{324}\right)\left({\alpha}_{324}+j{\gamma}_{324}\right)}{\left(1+{\alpha}_{324}\right)+j\left({\beta}_{324}+{\gamma}_{324}\right)}\hfill \\ {}\kern8.85em =\frac{r_{c32}}{s}\ \frac{\alpha_{324}\left(1+{\beta}_{324}^2\right)+{\alpha}_{324}^2{+}_{324}^2}{{\left(1+{\alpha}_{324}\right)}^2+{\left({\beta}_{324}+{\gamma}_{324}\right)}^2}\hfill \\ {}\kern9.85em +j{x}_{\Pi 32\sigma}\frac{\left({\gamma}_{324}/{\beta}_{324}\right)\left(1+{\beta}_{324}^2\right)+{\alpha}_{324}^2{+}_{324}^2}{{\left(1+{\alpha}_{324}\right)}^2+{\left({\beta}_{324}+{\gamma}_{324}\right)}^2}\hfill \\ {}\kern8.85em =\frac{r_{c32}}{s}{k}_{32r}^{{\prime\prime} }+j{x}_{\Pi 32\sigma }{k}_{32x}^{{\prime\prime}}\hfill \end{array} $$

(A.15.10)

where ${k}_{32r}^{{\prime\prime} }=\frac{\alpha_{324}\left(1+{\beta}_{324}^2\right)+{\alpha}_{324}^2{+}_{324}^2}{{\left(1+{\alpha}_{324}\right)}^2+{\left({\beta}_{324}+{\gamma}_{324}\right)}^2};{k}_{32x}^{{\prime\prime} }=\frac{\left({\gamma}_{324}/{\beta}_{324}\right)\left(1+{\beta}_{324}^2\right)+{\alpha}_{324}^2{+}_{324}^2}{{\left(1+{\alpha}_{324}\right)}^2+{\left({\beta}_{324}+{\gamma}_{324}\right)}^2}$.

15.1.6 A.15.6 The Additional Working Winding Slot Bar Sub-Layers: The Thicknesses

The values of h ₃₁ and h ₃₂ were calculated using the condition obtained in (15.83). The results of the calculations are shown in Table A.15.2.

Table A.15.2 Magnitudes of thicknesses of the additional working winding slot bar sub-layers for one value of the relative depth of field penetration in the starting winding slot bar (ε ₂ = 0.4)

15.1.7 A.15.7 The Expression [Z ₂₂(Z _τ22 + Z ₃)c₂ ²²]/[Z ₂₂ + (Z _τ22 + Z ₃)c ₂ ²²]: Real and Imaginary Components

We now consider the second term of expression (15.95)

$$ \frac{Z_{22}\left({Z}_{\tau 22}+{Z}_3\right){c}_{22}^2}{Z_{22}+\left({Z}_{\tau 22}+{Z}_3\right){c}_{22}^2} $$

(A.15.11)

This expression can be presented as the sum of the real and imaginary components. We first express the value of (Z _τ22 + Z ₃)c ²₂₂ used in (A.15.11) as the sum of the real and imaginary components. Then, taking into account (15.96), the value of (Z _τ22 + Z ₃)c ²₂₂ can be presented as

$$ \begin{array}{l}\left({Z}_{\tau 22}+{Z}_3\right){c}_{22}^2=\left[\left({r}_{c3}/s\right)+j{x}_{\Pi 3\sigma }+\left({r}_{\tau 22}/s\right)+j{x}_{\tau 22}\right]\left({k}_{c22r}+j{k}_{c22x}\right)\hfill \\ {}\kern5.8em =\left[\left({r}_{c3}/s\right){k}_{c22r}-{x}_{\Pi 3\sigma }{k}_{c22x}+\left({r}_{\tau 22}/s\right){k}_{c22r}-{x}_{\tau 22}{k}_{c22x}\right]\hfill \\ {}\kern6.9em +j\left[{x}_{\Pi 3\sigma }{k}_{c22r}+\left({r}_{c3}/s\right){k}_{c22x}+{x}_{\tau 22}{k}_{c22r}+\left({r}_{\tau 22}/s\right){k}_{c22x}\right]\hfill \\ {}\kern5.8em =\left[\frac{r_{c3}}{s}\left(1+\frac{r_{\tau 22}}{r_{c3}}\right){k}_{c22r}-{x}_{\Pi 3\sigma}\left(1+\frac{x_{\tau 22}}{x_{\Pi 3\sigma }}\right){k}_{c22x}\right]\hfill \\ {}\kern6.9em +j\left[{x}_{\Pi 3\sigma}\left({k}_{c22r}+\frac{r_{c3}/s}{x_{\Pi 3\sigma }}{k}_{c22x}\right)+{x}_{\tau 22}\left({k}_{c22r}+\frac{r_{\tau 22}/s}{x_{\tau 22}}{k}_{c22x}\right)\right]\hfill \\ {}\kern5.8em =\frac{r_{c3}}{s}\left[\left(1+\frac{r_{\tau 22}}{r_{c3}}\right){k}_{c22r}-\frac{x_{\Pi 3\sigma }}{r_{c3}}\left(1+\frac{x_{\tau 22}}{x_{\Pi 3\sigma }}\right){k}_{c22x}\right]\hfill \\ {}\kern6.9em +j\left({x}_{\Pi 3\sigma }{k}_{c223x}+{x}_{\tau 22}{k}_{\tau 223x}\right)\hfill \\ {}\kern5.8em ={r}_{c3}^{{\prime\prime} }/s+j{x}_{\Pi 3\sigma}^{{\prime\prime}}\hfill \end{array} $$

(A.15.12)

where $\begin{array}{l}\frac{r_{c3}^{{\prime\prime} }}{s}=\frac{r_{c3}}{s}\left[\left(1+\frac{r_{\tau 22}}{r_{c3}}\right){k}_{c22r}-\frac{x_{\Pi 3\sigma }}{r_{c3}}\left(1+\frac{x_{\tau 22}}{x_{\Pi 3\sigma }}\right){k}_{c22x}\right];{x}_{\Pi 3\sigma}^{{\prime\prime} }={x}_{\Pi 3\sigma }{k}_{c223x}\hfill \\ {}\kern2em +{x}_{\tau 22}{k}_{\tau 223x};\hfill \\ {}{k}_{c223x}={k}_{c22r}+\frac{r_{c3}/s}{x_{\Pi 3\sigma }}{k}_{c22x};{k}_{\tau 223x}={k}_{c22r}+\frac{r_{\tau 22}/s}{x_{\tau 22}}{k}_{c22x}.\hfill \end{array}$

Now, using (A.15.12) and the following non-dimensional values

$$ {\alpha}_{22}=\frac{r_{c3}^{{\prime\prime} }}{r_{c22}};{\beta}_{22}=\frac{x_{\Pi 22\sigma }}{r_{c22}/s};{\gamma}_{22}=\frac{x_{\Pi 3\sigma}^{{\prime\prime} }}{\left({r}_{c22}/s\right)} $$

(A.15.13)

we can receive

$$ \begin{array}{l}\frac{Z_{22}\left({Z}_{\tau 22}+{Z}_3\right){c}_{22}^2}{Z_{22}+\left({Z}_{\tau 22}+{Z}_3\right){c}_{22}^2}=\frac{\left({r}_{22}/s+j{x}_{\Pi 22\sigma}\right)\left({r}_{c3}^{{\prime\prime} }/s+j{x}_{\Pi 3\sigma}^{{\prime\prime}}\right)}{\left({r}_{22}/s+{r}_{c3}^{{\prime\prime} }/s\right)+j\left({x}_{\Pi 22\sigma }+{x}_{\Pi 3\sigma}^{{\prime\prime}}\right)}\\ {}\kern8.5em =\frac{r_{c22}}{s}\ \frac{\left(1+j{\beta}_{22}\right)\left({\alpha}_{22}+j{\gamma}_{22}\right)}{\left(1+{\alpha}_{22}\right)+j\left({\beta}_{22}+{\gamma}_{22}\right)}\\ {}\kern8.5em =\frac{r_{c22}}{s}\ \frac{\alpha_{22}\left(1+{\beta}_{22}^2\right)+{\alpha}_{22}^2+{\gamma}_{22}^2}{{\left(1+{\alpha}_{22}\right)}^2+{\left({\beta}_{22}+{\gamma}_{22}\right)}^2}\\ {}\kern9.5em +j{x}_{\Pi 22\sigma}\frac{\left({\gamma}_{22}/{\beta}_{22}\right)\left(1+{\beta}_{22}^2\right)+{\alpha}_{22}^2+{\gamma}_{22}^2}{{\left(1+{\alpha}_{22}\right)}^2+{\left({\beta}_{22}+{\gamma}_{22}\right)}^2}\\ {}\kern8.5em =\frac{r_{c22}}{s}{k}_{22r}^{{\prime\prime} }+j{x}_{\Pi 22\sigma }{k}_{22x}^{{\prime\prime}}\end{array} $$

(A.15.14)

where ${k}_{22r}^{{\prime\prime} }=\frac{\alpha_{22}\left(1+{\beta}_{22}^2\right)+{\alpha}_{22}^2{+}_{22}^2}{{\left(1+{\alpha}_{22}\right)}^2+{\left({\beta}_{22}+{\gamma}_{22}\right)}^2};{k}_{22x}^{{\prime\prime} }=\frac{\left({\gamma}_{22}/{\beta}_{22}\right)\left(1+{\beta}_{22}^2\right)+{\alpha}_{22}^2{+}_{22}^2}{{\left(1+{\alpha}_{22}\right)}^2+{\left({\beta}_{22}+{\gamma}_{22}\right)}^2}$.

15.1.8 A.15.8 The Starting Winding Slot Bar Sub-Layers: The Thickness

To define the values of h ₂₁ and h ₂₂, the condition obtained in (15.100) was used. The results of the calculations are given in Table A.15.3.

Table A.15.3 Magnitudes of thicknesses of the starting winding slot bar sub-layers depending on the relative depth of field penetration in the starting winding slot bar