Solve those questions

Tunatumia $\cos \phi$ kupata pembe $\phi$.
Kwa awamu Nyekundu (unity power factor): $\cos \phi_R = 1 \implies \phi_R = 0^\circ$.
Kwa awamu ya Njano (0.8 lagging power factor): $\cos \phi_Y = 0.8 \implies \phi_Y = -\arccos(0.8) \approx -36.87^\circ$.
Kwa awamu ya Bluu (0.9 leading power factor): $\cos \phi_B = 0.9 \implies \phi_B = \arccos(0.9) \approx 25.84^\circ$.

• Mkondo wa Awamu Nyekundu: $$I_R = 20 \angle 0^\circ = 20 \cos(0^\circ) + j20 \sin(0^\circ) = 20 + j0 \text{ A}$$
• Mkondo wa Awamu ya Njano: $$I_Y = 15 \angle -36.87^\circ = 15 \cos(-36.87^\circ) + j15 \sin(-36.87^\circ) = 15(0.8) + j15(-0.6) = 12 - j9 \text{ A}$$
• Mkondo wa Awamu ya Bluu: $$I_B = 10 \angle 25.84^\circ = 10 \cos(25.84^\circ) + j10 \sin(25.84^\circ) = 10(0.9) + j10(0.4359) = 9 + j4.359 \text{ A}$$

Mkondo wa neutral ni:

Volti ya laini ($V_L$) ni $400 \text{ V}$.

Nguvu kwa awamu $P = V_p I \cos \phi$.

• Nguvu ya Awamu Nyekundu ($P_R$): $$P_R = V_p I_R \cos \phi_R = \frac{400}{\sqrt{3}} \times 20 \times 1 = \frac{8000}{\sqrt{3}} \approx 4618.80 \text{ W}$$
• Nguvu ya Awamu ya Njano ($P_Y$): $$P_Y = V_p I_Y \cos \phi_Y = \frac{400}{\sqrt{3}} \times 15 \times 0.8 = \frac{4800}{\sqrt{3}} \approx 2771.28 \text{ W}$$
• Nguvu ya Awamu ya Bluu ($P_B$): $$P_B = V_p I_B \cos \phi_B = \frac{400}{\sqrt{3}} \times 10 \times 0.9 = \frac{3600}{\sqrt{3}} \approx 2078.46 \text{ W}$$

Jumla ya nguvu $P_{total} = P_R + P_Y + P_B$.

Jumla ya nguvu inayotumika ni: