Solid Rotor with Conducting Slot Wedges: Leakage Circuit Loops

Abstract

At the strong skin effect, leakage fields in solid rotor equipped with conducting slot wedges can be described using a “peripheral” rotor model. The “peripheral” model was used in Chaps. 17 and 19 to describe leakage fields in slotted and squirrel-cage solid rotors and also to establish the eddy current circuit loops induced in solid rotors of such construction. In this chapter, results obtained in Chaps. 17 and 19 are used to establish the eddy current circuit loops induced in a solid rotor with conducting slot wedges at the strong skin effect.

Appendix A.21 Transformations

21.1.1 A.21.1 Factors c_kz, (c² _klc _kz ) and (c² _klc² _Πzc² _kz): Real and Imaginary Components

In expression (21.41), factors c _kz , (c ² _kl c _kz ) and (c ² _kl c ²_Πz c ² _kz ) are used. These factors can be represented as the sum of the real and imaginary components. Taking into account (21.36), (21.37), (21.38) and (21.39), we have from (21.18) for factor c _kz

$$ {c}_{kz}=1+\frac{Z_{\tau kl}}{Z_{\Pi z}}=1+\frac{r_{\tau kl}/s+j{x}_{\tau kl}}{r_{cz}/s+j\left({x}_{cz\sigma }+{x}_{\Pi z}\right)} $$

(A.21.1)

In (A.21.1), the following non-dimensional values can be used:

$$ {\alpha}_{kz}=\frac{r_{\tau kl}}{r_{cz}};{\beta}_{kz}=\frac{x_{cz\sigma }}{r_{cz}/s};{\gamma}_{cz}=\frac{x_{\Pi z}}{r_{cz}/s};{\upsilon}_{kz}=\frac{x_{\tau kl}}{r_{cz}/s} $$

(A.21.2)

Then, expression (A.21.1) takes the form

$$ \begin{array}{l}{c}_{kz}=1+\frac{r_{\tau kl}/s+j{x}_{\tau kl}}{r_{cz}/s+j\left({x}_{cz\sigma}+{x}_{\Pi z}\right)}=1+\frac{\alpha_{kz}+j{\upsilon}_{kz}}{1+j\left({\beta}_{kz}+{\gamma}_{kz}\right)}\hfill \\ {}\kern1.1em =1+\frac{\left[{\alpha}_{kz}+{\upsilon}_{kz}\left({\beta}_{kz}+{\gamma}_{kz}\right)\left]+j\right[{\upsilon}_{kz}-{\alpha}_{kz}\left({\beta}_{kz}+{\gamma}_{kz}\right)\right]}{1+{\left({\beta}_{kz}+{\gamma}_{kz}\right)}^2}={c}_{kkr}+j{c}_{kkx}\hfill \end{array} $$

(A.21.3)

where ${c}_{kkr}=1+\frac{\alpha_{kz}+{\upsilon}_{kz}\left({\beta}_{kz}+{\gamma}_{kz}\right)}{1+{\left({\beta}_{kz}+{\gamma}_{kz}\right)}^2};{c}_{kkx}=\frac{\upsilon_{kz}-{\alpha}_{kz}\left({\beta}_{kz}+{\gamma}_{kz}\right)}{1+{\left({\beta}_{kz}+{\gamma}_{kz}\right)}^2}$.

Using expressions shown in (A.21.3) and (21.33), (21.34) and (21.35), we have for the product of the factors (c ² _kl c _kz )

$$ \begin{array}{l}\left({c}_{kl}^2{c}_{kz}\right)=\left({k}_{kzr}+j{k}_{kzx}\right)\left({c}_{kkr}+j{c}_{kkx}\right)\hfill \\ {}\kern2.95em =\left({k}_{kzr}{c}_{kkr}-{k}_{kzx}{c}_{kkx}\right)+j\left({k}_{kzx}{c}_{kkr}+{k}_{kzr}{c}_{kkx}\right)={k}_{kkzr}+j{k}_{kkzx}\hfill \end{array} $$

(A.21.4)

where k _kkzr  = k _kzr c _kkr  − k _kzx c _kkx ; k _kkzx  = k _kzx c _kkr  + k _kzr c _kkx .

From (A.21.3), (21.33), (21.34), (21.35) and (17.75), it follows for the product of the factors (c ² _kl c ²_Πz c ² _kz ) that

$$ \begin{array}{l}\left({c}_{kl}^2{c}_{\Pi z}^2{c}_{kz}^2\right)=\left({k}_{kzr}+j{k}_{kzx}\right)\left({k}_{\Pi zr}+j{k}_{\Pi zx}\right){\left({c}_{kkr}+j{c}_{kkx}\right)}^2\hfill \\ {}\kern4.3em =\left[\left({k}_{kzr}{k}_{\Pi zr}-{k}_{kzx}{k}_{\Pi zx}\right)+j\left({k}_{kzx}{k}_{\Pi zr}+{k}_{kzr}{k}_{\Pi zx}\right)\right]\hfill \\ {}\kern5.3em \times \left[\left({c}_{kkr}^2-{c}_{kkx}^2\right)+j2{c}_{kkr}{c}_{kkx}\right]\hfill \\ {}\kern4.3em =\left[\left({k}_{kzr}{k}_{\Pi zr}-{k}_{kzx}{k}_{\Pi zx}\right)\left({c}_{kkr}^2-{c}_{kkx}^2\right)-2{c}_{kkr}{c}_{kkx}\left({k}_{kzx}{k}_{\Pi zr}+{k}_{kzr}{k}_{\Pi zx}\right)\right]\hfill \\ {}\kern5.3em +j\left[\left({k}_{kzx}{k}_{\Pi zr}+{k}_{kzr}{k}_{\Pi zx}\right)\left({c}_{kkr}^2-{c}_{kkx}^2\right)+2{c}_{kkr}{c}_{kkx}\left({k}_{kzr}{k}_{\Pi zr}-{k}_{kzx}{k}_{\Pi zx}\right)\right]\hfill \\ {}\kern4.3em ={k}_{k\Pi r}+j{k}_{k\Pi x},\hfill \end{array} $$

(A.21.5)

where $\begin{array}{ll}{k}_{k\Pi r}\hfill & \kern-1.em =\left({k}_{kzr}{k}_{\Pi zr}-{k}_{kzx}{k}_{\Pi zx}\right)\left({c}_{kkr}^2-{c}_{kkx}^2\right)-2{c}_{kkr}{c}_{kkx}\left({k}_{kzx}{k}_{\Pi zr}+{k}_{kzr}{k}_{\Pi zx}\right);\hfill \\ {}{k}_{k\Pi x}\hfill & \kern-1.em =\left({k}_{kzx}{k}_{\Pi zr}+{k}_{kzr}{k}_{\Pi zx}\right)\left({c}_{kkr}^2-{c}_{kkx}^2\right)+2{c}_{kkr}{c}_{kkx}\left({k}_{kzr}{k}_{\Pi zr}-{k}_{kzx}{k}_{\Pi zx}\right).\hfill \end{array}$

21.1.2 A.21.2 Factors c _kz1, (c ² _kl c _kz1) and (c ² _kl c ²_Πz1 c ² _kz1 ): Real and Imaginary Components

We consider factors c _kz1, (c ² _kl c _kz1) and (c ² _kl c ²_Πz1 c ² _kz1 ) used in expression (21.77). Factor c _kz1 is determined by the expression given in (21.76) and by using (21.72) it can be reduced to the form

$$ {c}_{kz1}=1+\frac{Z_{\tau kl}^{**}}{Z_{\Pi z1}^{**}}=1+\frac{Z_{\tau kl}}{Z_{\Pi z1}}=1+\frac{r_{\tau kl}/s+j{x}_{\tau kl}}{r_{cz1}/s+j\left({x}_{cz1\sigma }+{x}_{\Pi z1}\right)} $$

(A.21.6)

In (A.21.6), we use the following non-dimensional values:

$$ {\alpha}_{kz1}=\frac{r_{\tau kl}}{r_{cz1}};{\beta}_{kz1}=\frac{x_{cz1\sigma }}{r_{cz1}/s};{\gamma}_{cz1}=\frac{x_{\Pi z1}}{r_{cz1}/s};{\upsilon}_{kz1}=\frac{x_{\tau kl}}{r_{cz1}/s} $$

(A.21.7)

Then, expression (A.21.6) acquires the form

$$ \begin{array}{l}{c}_{kz1}=1+\frac{r_{\tau kl}/s+j{x}_{\tau kl}}{r_{cz1}/s+j\left({x}_{cz1\sigma }+{x}_{\varPi z1}\right)}=1+\frac{\alpha_{kz1}+j{\upsilon}_{kz1}}{1+j\left({\beta}_{kz1}+{\gamma}_{kz1}\right)}\hfill \\ {}\kern1.49em =1+\frac{\left[{\alpha}_{kz1}+{\upsilon}_{kz1}\left({\beta}_{kz1}+{\gamma}_{kz1}\right)\left]+j\right[{\upsilon}_{kz1}-{\alpha}_{kz1}\left({\beta}_{kz1}+{\gamma}_{kz1}\right)\right]}{1+{\left({\beta}_{kz1}+{\gamma}_{kz1}\right)}^2}={c}_{kk1r}+j{c}_{kk1x}\hfill \end{array} $$

(A.21.8)

where ${c}_{kk1r}=1+\frac{\alpha_{kz1}+{\upsilon}_{kz1}\left({\beta}_{kz1}+{\gamma}_{kz1}\right)}{1+{\left({\beta}_{kz1}+{\gamma}_{kz1}\right)}^2};{c}_{kk1x}=\frac{\upsilon_{kz1}-{\alpha}_{kz1}\left({\beta}_{kz1}+{\gamma}_{kz1}\right)}{1+{\left({\beta}_{kz1}+{\gamma}_{kz1}\right)}^2}$.

From (21.33), (21.34), (21.35) and (A.21.8), the product of the factors (c ² _kl c _kz1) can be expressed as

$$ \begin{array}{l}\left({c}_{kl}^2{c}_{kz1}\right)=\left({k}_{kzr}+j{k}_{kzx}\right)\left({c}_{kk1r}+j{c}_{kk1x}\right)=\left({k}_{kzr}{c}_{kk1r}-{k}_{kzx}{c}_{kk1x}\right)\hfill \\ {}\kern4.5em +j\left({k}_{kzx}{c}_{kk1r}+{k}_{kzr}{c}_{kk1x}\right)={c}_{k\Pi zr}+j{c}_{k\Pi zx}\hfill \end{array} $$

(A.21.9)

where c _kΠzr  = k _kzr c _kk1r  − k _kzx c _kk1x ; c _kΠzx  = k _kzx c _kk1r  + k _kzr c _kk1x .

Considering (21.33), (21.34), (21.35), (17.113) and (A.21.8), we have for the product of the factors (c ² _kl c ² _Πz1 c ² _kz1 )

$$ \begin{array}{l}\left({c}_{kl}^2{c}_{\Pi z1}^2{c}_{kz1}^2\right)=\left({k}_{kz r}+j{k}_{kz x}\right)\left({k}_{\Pi z1r}+j{k}_{\Pi z1x}\right){\left({c}_{kk1r}+j{c}_{kk1x}\right)}^2\hfill \\ {}\kern4.99em =\left[{k}_{\Pi z1r}{k}_{kz r}-{k}_{\Pi z1x}{k}_{kz x}+j\left({k}_{\Pi z1x}{k}_{kz r}+{k}_{\Pi z1r}{k}_{kz x}\right)\right]\left[\left({c}_{kk1r}^2-{c}_{kk1x}^2\right)+j2{c}_{kk1r}{c}_{kk1x}\right]\hfill \\ {}\kern4.99em =\left[\left({k}_{\Pi z1r}{k}_{kz r}-{k}_{\Pi z1x}{k}_{kz x}\right)\left({c}_{kk1r}^2-{c}_{kk1x}^2\right)-2\left({k}_{\Pi z1x}{k}_{kz r}+{k}_{\Pi z1r}{k}_{kz x}\right){c}_{kk1r}{c}_{kk1x}\right]\hfill \\ {}\kern5.99em +j\left[\left({k}_{\Pi z1x}{k}_{kz r}+{k}_{\Pi z1r}{k}_{kz x}\right)\left({c}_{kk1r}^2-{c}_{kk1x}^2\right)+2\left({k}_{\Pi z1r}{k}_{kz r}-{k}_{\Pi z1x}{k}_{kz x}\right){c}_{kk1r}{c}_{kk1x}\right]\hfill \\ {}\kern4.99em ={k}_{k\Pi zr}+j{k}_{k\Pi zx}\hfill \end{array} $$

(A.21.10)

where $\begin{array}{l}{k}_{k\Pi zr}=\left({k}_{\Pi z1r}{k}_{kzr}-{k}_{\Pi z1x}{k}_{kzx}\right)\left({c}_{kk1r}^2-{c}_{kk1x}^2\right)-2\left({k}_{\Pi z1x}{k}_{kzr}+{k}_{\Pi z1r}{k}_{kzx}\right){c}_{kk1r}{c}_{kk1x};\hfill \\ {}{k}_{k\Pi zx}=\left({k}_{\Pi z1x}{k}_{kzr}+{k}_{\Pi z1r}{k}_{kzx}\right)\left({c}_{kk1r}^2-{c}_{kk1x}^2\right)+2\left({k}_{\Pi z1r}{k}_{kzr}-{k}_{\Pi z1x}{k}_{kzx}\right){c}_{kk1r}{c}_{kk1x}.\hfill \end{array}$

21.1.3 A.21.3 Expression [Z_kl2(Z_τkl2 + Z_Πz)c² _kl2]/[Z_kl2 + (Z_τkl2 + Z_Πz)c² _kl2] and Impedance Z_kl2: Real and Imaginary Components

The expression for the impedance Z _kl2 was presented in (21.98). In (21.98), we use the following non-dimensional values:

$$ {\alpha}_{kl2}=\frac{r_{cck2}}{r_{ckl2}}\kern0.5em \mathrm{and}\kern0.5em {\gamma}_{kl2}=\frac{x_{cck2\sigma }}{r_{ckl2}/s} $$

(A.21.11)

Then, expression (21.98) obtains the form

$$ \begin{array}{l}{Z}_{kl2}=\frac{1}{\frac{1}{r_{cck2}/s+j{x}_{cck2\sigma }}+\frac{1}{r_{ckl2}/s}}+\frac{r_{\Pi k2}}{s}+j{x}_{\Pi k2}=\frac{r_{ckl2}}{s}\kern.18em \frac{\alpha_{kl2}+j{\gamma}_{kl2}}{\left(1+{\alpha}_{kl2}\right)+j{\gamma}_{kl2}}\hfill \\ {}\kern2.2em +\frac{r_{\Pi k2}}{s}+j{x}_{\Pi k2}=\frac{r_{ckl2}}{s}\kern.18em \frac{\left[{\alpha}_{kl2}\left(1+{\alpha}_{kl2}\right)+{\gamma}_{kl2}^2\right]+j{\gamma}_{kl2}}{{\left(1+{\alpha}_{kl2}\right)}^2+{\gamma}_{kl2}^2}+\frac{r_{\Pi k2}}{s}+j{x}_{\Pi k2}\hfill \\ {}\kern1.2em =\frac{r_{ckl2}}{s}\left[\frac{\alpha_{kl2}\left(1+{\alpha}_{kl2}\right)+{\gamma}_{kl2}^2}{{\left(1+{\alpha}_{kl2}\right)}^2+{\gamma}_{kl2}^2}+\frac{r_{\Pi k2}}{r_{ckl2}}\right]+j\left[{x}_{\Pi k2}+{x}_{cck2\sigma}\frac{1}{{\left(1+{\alpha}_{kl2}\right)}^2+{\gamma}_{kl2}^2}\right]\hfill \\ {}\kern1.2em =\frac{r_{kl2}}{s}+j\left({x}_{\Pi k2}+{x}_{cck2\sigma }{k}_{\Pi ck}\right)=\frac{r_{kl2}}{s}+j{x}_{kl2}\hfill \end{array} $$

(A.21.12)

where $\begin{array}{ll}\frac{r_{kl2}}{s}\hfill & \kern-1.2em =\frac{r_{ckl2}}{s}\left[\frac{\alpha_{kl2}\left(1+{\alpha}_{kl2}\right)+{\gamma}_{kl2}^2}{{\left(1+{\alpha}_{kl2}\right)}^2+{\gamma}_{kl2}^2}+\frac{r_{\varPi k2}}{r_{ckl2}}\right];{x}_{kl2}={x}_{\Pi k2}+{x}_{cck2\sigma }{k}_{\Pi ck}\hfill \\ {}{k}_{\Pi ck}\hfill & \kern-.92em ={\left[{\left(1+{\alpha}_{kl2}\right)}^2+{\gamma}_{kl2}^2\right]}^{-1}.\hfill \end{array}$

We consider the second term of expression (21.108) determined as

$$ \frac{Z_{kl2}\left({Z}_{\tau kl2}+{Z}_{\Pi z}\right){c}_{kl2}^2}{Z_{kl2}+\left({Z}_{\tau kl2}+{Z}_{\Pi z}\right){c}_{kl2}^2} $$

(A.21.13)

This expression can be presented as the sum of the real and imaginary components. In (A.21.13), we first consider the value of (Z _τkl2 + Z _Πz )c ² _kl2 . Using expressions given in (17.70) and (21.97), the value of (Z _τkl2 + Z _Πz )c ² _kl2 can be obtained as

$$ \begin{array}{l}\left({Z}_{\tau kl2}+{Z}_{\Pi z}\right){c}_{kl2}^2=\left[{r}_{\tau kl2}/s+j{x}_{\tau kl2}+{r}_z/s+j\left({x}_{cz\sigma}+{x}_{\Pi z}\right)\right]\left({k}_{kz2r}+j{k}_{kz2x}\right)\hfill \\ {}=\left[\left({r}_z/s\right){k}_{kz2r}-\left({x}_{cz\sigma}+{x}_{\Pi z}\right){k}_{kz2x}+\left({r}_{\tau kl2}/s\right){k}_{kz2r}-{x}_{\tau kl2}{k}_{kz2x}\right]\hfill \\ {}\kern.59em +j\left[\left({x}_{cz\sigma}+{x}_{\Pi z}\right){k}_{kz2r}+\left({r}_z/s\right){k}_{kz2x}+{x}_{\tau kl2}{k}_{kz2r}+\left({r}_{\tau kl2}/s\right){k}_{kz2x}\right]\hfill \\ {}=\left[\frac{r_z}{s}\left(1+\frac{r_{\tau kl2}}{r_z}\right){k}_{kz2r}-{x}_{cz\sigma}\left(1+\frac{x_{\Pi z}}{x_{cz\sigma}}+\frac{x_{\tau kl2}}{x_{cz\sigma}}\right){k}_{kz2x}\right]\hfill \\ {}\kern.59em +j\left[{x}_{cz\sigma}\left({k}_{kz2r}+\frac{r_z/s}{x_{cz\sigma}}{k}_{kz2x}\right)+{x}_{\Pi z}{k}_{kz2r}+{x}_{\tau kl2}\left({k}_{kz2r}+\frac{r_{\tau kl2}/s}{x_{\tau kl2}}{k}_{kz2x}\right)\right]\hfill \\ {}=\frac{r_z}{s}\left[\left(1+\frac{r_{\tau kl2}}{r_z}\right){k}_{kz2r}-\frac{x_{cz\sigma}}{r_z/s}\left(1+\frac{x_{\Pi z}}{x_{cz\sigma}}+\frac{x_{\tau kl2}}{x_{cz\sigma}}\right){k}_{kz2x}\right]\hfill \\ {}\kern.59em +j\left({x}_{cz\sigma}{k}_{cxk2}+{x}_{\Pi z}{k}_{kz2r}+{x}_{\tau kl2}{k}_{\tau cxk}\right)=\frac{r_{kz}}{s}+j\left({x}_{cz\sigma}^{\prime }+{x}_{\Pi z}^{\prime }+{x}_{\tau kl2}^{\prime}\right)=\frac{r_{kz}}{s}+j{x}_{kz}\hfill \end{array} $$

(A.21.14)

where $\begin{array}{l}\hfill \frac{r_{kz}}{s}=\frac{r_z}{s}\left[\left(1+\frac{r_{\tau kl2}}{r_z}\right){k}_{kz2r}-\frac{x_{cz\sigma }}{r_z/s}\left(1+\frac{x_{\Pi z}}{x_{cz\sigma }}+\frac{x_{\tau kl2}}{x_{cz\sigma }}\right){k}_{kz2x}\right]\hfill \\ {}\hfill {x}_{kz}={x}_{cz\sigma}^{\prime }+{x}_{\Pi z}^{\prime }+{x}_{\tau kl2}^{\prime };{x}_{cz\sigma}^{\prime }={x}_{cz\sigma }{k}_{cxk2};{x}_{\Pi z}^{\prime }={x}_{\Pi z}{k}_{kz2r};{x}_{\tau kl2}^{\prime }={x}_{\tau kl2}{k}_{\tau cxk}\hfill \\ {}\hfill \kern-.8em {k}_{cxk2}={k}_{kz2r}+\frac{r_z/s}{x_{cz\sigma }}{k}_{kz2x};{k}_{\tau cxk}={k}_{kz2r}+\frac{r_{\tau kl2}/s}{x_{\tau kl2}}{k}_{kz2x}\hfill \end{array}$

Considering that Z _kl2 = r _kl2/s + jx _kl2 and (Z _τkl2 + Z _Πz )c ² _kl2  = r _kz /s + jx _kz , we can use in (A.21.13) the following non-dimensional values:

$$ {\alpha}_{kl2}=\frac{r_{kz}}{r_{kl2}};{\beta}_{kl2}=\frac{x_{kl2}}{r_{kl2}/s};{\gamma}_{kl2}=\frac{x_{kz}}{r_{kl2}/s} $$

(A.21.15)

Then, it follows

$$ \begin{array}{l}\frac{Z_{kl2}\left({Z}_{\tau kl2}+{Z}_{\Pi z}\right){c}_{kl2}^2}{Z_{kl2}+\left({Z}_{\tau kl2}+{Z}_{\Pi z}\right){c}_{kl2}^2}=\frac{\left({r}_{kl2}/s+j{x}_{kl2}\right)\left({r}_{kz}/s+j{x}_{kz}\right)}{\left({r}_{kl2}/s+{r}_{kz}/s\right)+j\left({x}_{kl2}+j{x}_{kz}\right)}\hfill \\ {}\kern9.49em =\frac{r_{kl2}}{s}\kern.18em \frac{\left(1+j{\beta}_{kl2}\right)\left({\alpha}_{kl2}+j{\gamma}_{kl2}\right)}{\left(1+{\alpha}_{kl2}\right)+j\left({\beta}_{kl2}+{\gamma}_{kl2}\right)}\hfill \\ {}\kern9.49em =\frac{r_{kl2}}{s}\kern.18em \frac{\alpha_{kl2}\left(1+{\beta}_{kl2}^2\right)+{\alpha}_{kl2}^2+{\gamma}_{kl2}^2}{{\left(1+{\alpha}_{kl2}\right)}^2+{\left({\beta}_{kl2}+{\gamma}_{kl2}\right)}^2}\hfill \\ {}\kern10.49em +j{x}_{kl2\sigma}\frac{\left({\gamma}_{kl2}/{\beta}_{kl2}\right)\left(1+{\beta}_{kl2}^2\right)+{\alpha}_{kl2}^2+{\gamma}_{kl2}^2}{{\left(1+{\alpha}_{kl2}\right)}^2+{\left({\beta}_{kl2}+{\gamma}_{kl2}\right)}^2}\hfill \\ {}\kern9.49em =\frac{r_{kl2}}{s}{k}_{kl2r}^{{\prime\prime} }+j{x}_{kl2}{k}_{kl2x}^{{\prime\prime}}\hfill \end{array} $$

(A.21.16)

where ${k}_{kl2r}^{{\prime\prime} }=\frac{\alpha_{kl2}\left(1+{\beta}_{kl2}^2\right)+{\alpha}_{kl2}^2+{\gamma}_{kl2}^2}{{\left(1+{\alpha}_{kl2}\right)}^2+{\left({\beta}_{kl2}+{\gamma}_{kl2}\right)}^2};{k}_{kl2x}^{{\prime\prime} }=\frac{\left({\gamma}_{kl2}/{\beta}_{kl2}\right)\left(1+{\beta}_{kl2}^2\right)+{\alpha}_{kl2}^2+{\gamma}_{kl2}^2}{{\left(1+{\alpha}_{kl2}\right)}^2+{\left({\beta}_{kl2}+{\gamma}_{kl2}\right)}^2}$.