Question

(20 points) A balanced-δ load consisting of pure resistances of 1 = 15 per phase is in parallel with a balanced-Y load having phase impedances of 2 = 8 6. Calculate the following:

Option 1:
Total resistance in the Y load.

Option 2:
Total power in the δ load.

Option 3:
Total impedance in the Y load.

Option 4:
Total current in the δ load.

Answer 1

Option 1: The total resistance in the Y load is (169/23) + (169/6)i. Option 2: The total power in the δ load is 3 * I^2 * 15, where I is the current in the δ load. Option 3: The total impedance in the Y load is 24 + 18i. Option 4: The total current in the δ load is 16 A.

Step-by-step explanation:

Option 1: To find the total resistance in the Y load, we need to add the reciprocals of the individual impedances. The phase impedance of the Y load is given as Z = 8 + 6i. Taking the reciprocal gives 1/Z = 1/(8 + 6i). To add the reciprocal of Z with the resistance, we need to find the complex conjugate of 8 + 6i, which is 8 - 6i. Adding the reciprocals, we get 1/(8 + 6i) + 1/15 = (8 - 6i + 15) / (15 * (8 + 6i)) = (23 - 6i) / (120 + 90i). Simplifying further gives (23/169) - (6/169)i. Therefore, the total resistance in the Y load is 1 / ((23/169) - (6/169)i), which can be written as (169/23) + (169/6)i.

Option 2: To find the total power in the δ load, we need to calculate the power dissipated by each individual resistor and add them together. The power dissipated by a resistor can be found using the formula P = I^2 * R, where I is the current and R is the resistance. Since the load is balanced, the current in each resistor is the same. Given that the resistance per phase is 15, we can calculate the power as P = I^2 * 15. Therefore, the total power in the δ load is 3 * I^2 * 15, where I is the current in the δ load.

Option 3: The total impedance in the Y load is equal to the sum of the individual impedances. Since the phase impedance of the Y load is given as 8 + 6i, the total impedance is 3 times the individual impedance, which is equal to 3 * (8 + 6i) = 24 + 18i.

Option 4: To find the total current in the δ load, we can use Ohm's Law, which states that current is equal to voltage divided by impedance. Since the load is balanced, the voltage across each resistor is the same. Substituting the given resistance per phase of 15 and the voltage across the load, we can calculate the total current as I = V / R = 240 / 15 = 16 A.