Expressions R L 0882F4

1. **State the problem:** We are given two equations: $$|U_b| = \frac{\sqrt{(\omega L)^2 + R^2}}{\sqrt{(\omega L)^2 + (R_b + R_{cu} + R)^2}} \frac{l_{p0}R_b}{N_s}$$ and $$\varphi = \arctg \frac{\omega L(R_b + R_{cu})}{(\omega L)^2 + R(R + R_b + R_{cu})}$$ We want to find independent expressions for $R$ and $L$. 2. **Rewrite variables for clarity:** Let $A = R_b + R_{cu}$ and $K = \frac{l_{p0} R_b}{N_s}$. Then the equations become: $$|U_b| = K \frac{\sqrt{(\omega L)^2 + R^2}}{\sqrt{(\omega L)^2 + (A + R)^2}}$$ $$\varphi = \arctg \frac{\omega L A}{(\omega L)^2 + R(A + R)}$$ 3. **From the phase angle equation, express $\tan \varphi$:** $$\tan \varphi = \frac{\omega L A}{(\omega L)^2 + R(A + R)}$$ Rearranged: $$\tan \varphi \left((\omega L)^2 + R(A + R)\right) = \omega L A$$ 4. **Express $R$ in terms of $L$ and known quantities:** Expand: $$\tan \varphi (\omega^2 L^2) + \tan \varphi R (A + R) = \omega L A$$ Rewrite: $$\tan \varphi R (A + R) = \omega L A - \tan \varphi \omega^2 L^2$$ This is a quadratic in $R$: $$\tan \varphi R^2 + \tan \varphi A R - \omega L A + \tan \varphi \omega^2 L^2 = 0$$ 5. **Solve quadratic for $R$:** $$R = \frac{-\tan \varphi A \pm \sqrt{(\tan \varphi A)^2 - 4 \tan \varphi (\tan \varphi \omega^2 L^2 - \omega L A)}}{2 \tan \varphi}$$ Simplify inside the square root: $$= \frac{-\tan \varphi A \pm \sqrt{\tan^2 \varphi A^2 - 4 \tan^2 \varphi \omega^2 L^2 + 4 \tan \varphi \omega L A}}{2 \tan \varphi}$$ 6. **From the magnitude equation, isolate $\sqrt{(\omega L)^2 + R^2}$:** $$|U_b| = K \frac{\sqrt{(\omega L)^2 + R^2}}{\sqrt{(\omega L)^2 + (A + R)^2}}$$ Multiply both sides by denominator: $$|U_b| \sqrt{(\omega L)^2 + (A + R)^2} = K \sqrt{(\omega L)^2 + R^2}$$ Square both sides: $$|U_b|^2 ((\omega L)^2 + (A + R)^2) = K^2 ((\omega L)^2 + R^2)$$ 7. **Expand and rearrange:** $$|U_b|^2 (\omega^2 L^2 + A^2 + 2 A R + R^2) = K^2 (\omega^2 L^2 + R^2)$$ Rewrite: $$|U_b|^2 \omega^2 L^2 + |U_b|^2 A^2 + 2 |U_b|^2 A R + |U_b|^2 R^2 = K^2 \omega^2 L^2 + K^2 R^2$$ Group terms: $$|U_b|^2 A^2 + 2 |U_b|^2 A R + (|U_b|^2 - K^2) R^2 + (|U_b|^2 - K^2) \omega^2 L^2 = 0$$ 8. **Express $L^2$ in terms of $R$ and known quantities:** $$\omega^2 L^2 = \frac{-|U_b|^2 A^2 - 2 |U_b|^2 A R - (|U_b|^2 - K^2) R^2}{|U_b|^2 - K^2}$$ 9. **Summary:** - $R$ is given by the quadratic formula in step 5 as a function of $L$ and known constants. - $L^2$ is expressed in step 8 as a function of $R$ and known constants. 10. **To find independent explicit expressions for $R$ and $L$, one can use these two equations simultaneously, or numerically solve for $R$ and $L$ given $|U_b|$, $\varphi$, $\omega$, $A$, and $K$.