Question

If two resistors are connected in parallel, the total impedance is given by:

Z_total = (Z1 * Z2) / (Z1 + Z2)

Find the total impedance assuming that the resistors Z1 = 5 + 4i ohms and Z2 = 9 - 3i ohms are connected in parallel.

Answer 1

To find the total impedance of two resistors connected in parallel, apply the formula with the given complex impedances, and then simplify. The resulting total impedance for the resistors Z1 and Z2 is 2.26 + 1.34i ohms.

Step-by-step explanation:

To calculate the total impedance of two resistors connected in parallel, you can use the formula Z_total = (Z1 * Z2) / (Z1 + Z2). For resistors Z1 = 5 + 4i ohms and Z2 = 9 - 3i ohms, we can plug these values into the formula:

Z_total = (5 + 4i) * (9 - 3i) / ((5 + 4i) + (9 - 3i))
Z_total = (45 - 15i + 36i - 12) / (14 + i)
Z_total = (33 + 21i) / (14 + i)

To simplify, we must multiply the numerator and denominator by the complex conjugate of the denominator:

Z_total = (33 + 21i) * (14 - i) / (14 + i) * (14 - i)
Z_total = (462 - 33i + 294i - 21) / (196 + i^2)
Since i^2 = -1, we have:

Z_total = (441 + 261i) / 195
Z_total = 441/195 + 261i/195
Z_total = 2.26 + 1.34i ohms

The total impedance of the parallel-connected resistors Z1 and Z2 is 2.26 + 1.34i ohms.