$\rho=\sqrt{\frac{R+j \omega L}{G+j \omega C}}$, find the condition to have $Re\{\rho\}\ge0$ and $Im\{\rho\}\ge0$

Question

$$\rho=\sqrt{\frac{R+j \omega L}{G+j \omega C}} \qquad R,G \ge 0 \qquad \omega,L,C>0 $$

$$\rho=\sqrt{\frac{(R+j \omega L) \ (G-j \omega C)}{G^2+\omega^2C^2}}$$

$$\rho=\sqrt{\frac{RG+j \omega LG-j\omega RC+\omega^2 LC}{G^2+\omega^2C^2}}$$

$$0 \le \arctan\Big( \frac{\omega LG-\omega RC}{RG+\omega^2LC} \Big) \le \frac{\pi}{2}$$

Is it correct?

Thanks!

Answer 1

Hint.

$$ |\rho| = \sqrt{\frac{L^2 \omega ^2+R^2}{C^2 \omega ^2+G^2}} $$

and

$$ \angle \rho = \frac 12\arctan{\left(\frac{(GL-CR)\omega}{CL \omega^2+GR}\right)} $$

then $\Re(\rho) \ge 0 $ and $\Im(\rho) \ge 0 $ implies on

$$ 0 \le \frac 12\arctan{\left(\frac{(GL-CR)\omega}{CL \omega^2+GR}\right)}\le \frac{\pi}{2} $$

NOTE

$$ z = \rho e^{i\phi}\to \sqrt z = \rho^{\frac 12} e^{\frac i2 \phi} $$