Solid Rotor with Conducting Slot Wedges: Circuit Loops at the Weak Skin Effect

Abstract

Rotor slot wedges of large Conducting Slot Wedges: Circuit Loops…?>electric machines are usually composed of conductive material. Solid rotors constructed in this fashion are mainly used in turbo-generators and large high-speed (two-pole) synchronous motors. While using turbo-generators and large high-speed synchronous motors, the operating conditions can take place, when the rotor winding is open or closed on the large external resistance. In this case, electromagnetic processes in electric machines are determined by eddy currents induced in the conducting slot wedges and currents flowing in the body of the solid rotor. In this chapter, circuit loops of these eddy currents and their impedances are considered at the weak skin effect.

Appendix A.20 Transformations

20.1.1 A.20.1 Factors c _kl ²c_kz and c_kl ²c_Πz ²c_kz ²: Real and Imaginary Components

The products of factors c ² _kl c _kz and c ² _kl c ²_Πz c ² _kz used in (20.87) can be represented as the sum of the real and imaginary components. From (20.85) and (20.89), we have for the product of factors c ² _kl c _kz the equation

$$ \begin{array}{l}{c}_{kl}^2{c}_{kz}=\left(1+j2{k}_{kl}{\varepsilon}_{kl}^2\right)\left(1+j2{k}_{kz}{\varepsilon}_{k\Pi z}^2\right)=1+j2\left({k}_{kl}{\varepsilon}_{kl}^2+{k}_{kz}{\varepsilon}_{k\Pi z}^2\right)\hfill \\ {}\kern2.1em =1+j2{\varepsilon}_{kl}^2\left({k}_{kl}+{k}_{kz}\frac{\varepsilon_{k\Pi z}^2}{\varepsilon_{kl}^2}\right)=1+j2{k}_{kkz}{\varepsilon}_{kl}^2\hfill \end{array} $$

(A.20.1)

where ${k}_{kkz}={k}_{kl}+{k}_{kz}\frac{\varepsilon_{k\Pi z}^2}{\varepsilon_{kl}^2}$.

From (16.79), (20.85) and (20.90), the product of factors c ² _kl c ²_Πz c ² _kz acquires the form

$$ \begin{array}{l}{c}_{kl}^2{c}_{\Pi z}^2{c}_{kz}^2=\left(1+j2{k}_{kl}{\varepsilon}_{kl}^2\right)\left(1+j2{\varepsilon}_{\Pi z}^2\right)\left(1+j4{k}_{kz}{\varepsilon}_{k\Pi z}^2\right)\hfill \\ {}\kern3.3em =\left[1+j2\left({k}_{kl}{\varepsilon}_{kl}^2+{\varepsilon}_{\Pi z}^2\right)\right]\left(1+j4{k}_{kz}{\varepsilon}_{k\Pi z}^2\right)=1+j2\left({k}_{kl}{\varepsilon}_{kl}^2+{\varepsilon}_{\Pi z}^2+2{k}_{kz}{\varepsilon}_{k\Pi z}^2\right)\hfill \\ {}\kern3.3em =1+j2{\varepsilon}_{kl}^2\left({k}_{kl}+\frac{\varepsilon_{\Pi z}^2}{\varepsilon_{kl}^2}+2{k}_{kz}\frac{\varepsilon_{k\Pi z}^2}{\varepsilon_{kl}^2}\right)=1+j2{k}_{k\Pi z}{\varepsilon}_{kl}^2\hfill \end{array} $$

(A.20.2)

where ${k}_{k\Pi z}={k}_{kl}+\frac{\varepsilon_{\Pi z}^2}{\varepsilon_{kl}^2}+2{k}_{kz}\frac{\varepsilon_{k\Pi z}^2}{\varepsilon_{kl}^2}$.