Let's break down the steps to calculate the total current phasor and its magnitude and phase:

1. Convert Phase Angles to Radians: I_L_{\text{phase\_rad}} = \text{radians}(70^\circ) = \frac{70 \times \pi}{180} \approx 1.2217 \text{ radians} I_C_{\text{phase\_rad}} = \text{radians}(-90^\circ) = \frac{-90 \times \pi}{180} \approx -1.5708 \text{ radians}

2. Convert Currents into Complex Form (Phasors): $I_L = 7 \angle 70^\circ = 7 \times (\cos(1.2217) + j \sin(1.2217)) \approx 2.39 + 6.57j \, \text{A}$ $I_C = 7.23 \angle -90^\circ = 7.23 \times (\cos(-1.5708) + j \sin(-1.5708)) \approx 0 - 7.23j \, \text{A}$

3. Given $I_R$ is purely real: $I_R = 4 \text{ A (Real part only, no imaginary component)}$ Hence, in complex form: I_R_{\text{complex}} = 4 + 0j

4. Calculate the Total Current (Phasor Sum): I_{\text{total}} = I_R_{\text{complex}} + I_L + I_C = (4 + 0j) + (2.39 + 6.57j) + (0 - 7.23j) Combine the real and imaginary parts: $I_{\text{total}} = (4 + 2.39) + (6.57 - 7.23)j = 6.39 - 0.66j \, \text{A}$

5. Get the Magnitude and Phase of the Total Current:

   • Magnitude: $|I_{\text{total}}| = \sqrt{(6.39)^2 + (-0.66)^2} \approx 6.42 \, \text{A}$
   • Phase: $\text{Phase of } I_{\text{total}} = \text{atan2}(-0.66, 6.39) \approx -5.89^\circ$

Final Answer:

The magnitude of the total current is approximately 6.42 A, and the phase angle is approximately -5.89°.

Would you like more details or have any further questions?

Here are some related questions:

1. How would the total current change if $I_R$ had a phase angle?
2. What would be the effect of increasing the magnitude of $I_L$?
3. How can you find the power factor given these currents?
4. What does the negative phase angle of the total current signify?
5. How would the total current be affected if $I_C$ had a different phase angle?

Tip: When dealing with AC circuits, always convert phase angles to radians before performing trigonometric calculations.