To analyze the given system, we can use phasor notation and apply the principles of balanced three-phase circuits. Let's calculate the line-to-line voltage at the load (Vab).
Given data:
- Impedance of the load per phase (Zload) = 24 + j19 ohm
- Impedance of the line per phase (Zline) = 0.6 + j0.7 ohm
- Phase voltage of the source (Van) = 120∠30° V
First, let's find the line current (Iab) using the formula:
Iab = Van / (Zload + Zline)
Iab = 120∠30° V / (24 + j19 + 0.6 + j0.7) ohm
= 120∠30° V / (24.6 + j19.7) ohm
To simplify the calculation, let's convert the polar form of the impedance to rectangular form:
Zload = 24 + j19 = 24 + j19 = 31.5∠39.805° ohm
Zline = 0.6 + j0.7 = 0.6 + j0.7 = 0.939∠49.825° ohm
Now, we can substitute the values into the equation:
Iab = 120∠30° V / (31.5∠39.805° + 0.939∠49.825°) ohm
To divide complex numbers, we multiply the numerator and denominator by the complex conjugate of the denominator:
Iab = (120∠30° V / (31.5∠39.805° + 0.939∠49.825°) ohm) * (31.5∠39.805° - 0.939∠49.825°) / (31.5∠39.805° - 0.939∠49.825°) ohm
Simplifying the numerator:
Iab = (120 * 31.5∠30°∠39.805° - 120 * 0.939∠30°∠49.825°) V*ohm / (31.5^2 - 0.939^2∠39.805°+49.825°) ohm
Iab = (3780∠69.805° - 112.68∠79.825°) V*ohm / (985.515 + j61.101) ohm
Now we can divide the real and imaginary parts separately:
Iab = [(3780 / (985.515 + j61.101)) * cos(69.805° - 79.825°) + j * (3780 / (985.515 + j61.101)) * sin(69.805° - 79.825°)] V*ohm
Finally, calculate the real and imaginary parts to find the line current Iab.
Iab ≈ [(3780 / (985.515 + j61.101)) * cos(-10.02°) + j * (3780 / (985.515 + j61.101)) * sin(-10.02°)] V*ohm
Iab ≈ [3.823 - j0.12] A
Therefore, the line current Iab is approximately 3.823 - j0.12 A.