RL circuit phasor notation problem

I am asked to find the voltage across the resistor and the inductor(in phasor notation).

These are the steps I took

$$v_{R}=\frac{R}{R+jX_{L}}v$$

$$v_{R}=\frac{vR}{R+jX_{L}}\cdot \frac{R-jX_{L}}{R-jX_{L}}=\frac{vR^{2}}{R^{2}+X_{L}^{2}}-j\frac{vRX_{L}}{R^{2}+X_{L}^{2}}$$

$$\sqrt{(\frac{vR^{2}}{R^{2}+X_{L}^{2}})^{2}+(\frac{vRX_{L}}{R^{2}+X_{L}^{2}})^{2}}\angle tan^{-1}(-\frac{X_{L}}{R})$$

I used the same method to get the voltage across the inductor. Is this correct? I am confused as to whether sqrt(2) should be included in the answer.

• Is $v$ a phasor or something else? Commented Apr 20, 2023 at 2:53
• I'm actually not sure. I used it to describe the input ac voltage (as noted in the circuit) Commented Apr 20, 2023 at 3:12

Yes. You can go on with your calculations to remove a few terms from the square root though: $$\sqrt{\frac{(VR^2)^2+(VRX_L)^2}{(R^2+X_L^2)^2}}=\sqrt{V^2R^2(R^2+X_L^2)^{-1}}=\frac{VR}{\sqrt{R^2+X_L^2}}$$ I am assuming that both $$\v\$$ and $$\v_r\$$ in your equations are phasors ($$\V\$$ and $$\V_r\$$), in which case they can be represented as vectors in the complex plane.