Given a complex impedance Z = R + jX, which statement correctly identifies its complex conjugate and how to compute the power factor from Z?

Given a complex impedance Z = R + jX, which statement correctly identifies its complex conjugate and how to compute the power factor from Z?

• A. Z* = R + jX; φ = atan(X/R); PF = cos φ
• B. Z* = R − jX; φ = atan(-R/X); PF = cos φ
• C. Z* = −R + jX; φ = atan(X/R); PF = cos φ
• D. Z* = R − jX; φ = atan(X/R); PF = cos φ

Answer: (D)

The key idea is how a complex impedance behaves in the complex plane and how its phase relates to power factor. For Z = R + jX, the complex conjugate flips the sign of the imaginary part, so Z* = R − jX. The phase angle of Z, φ, comes from tan φ = X/R, so φ = arctan(X/R). The power factor is defined as the cosine of that angle, PF = cos φ. This also means PF = R/√(R^2 + X^2), since cos φ = adjacent over hypotenuse in the impedance’s right triangle. That’s why the statement Z* = R − jX; φ = atan(X/R); PF = cos φ is correct: it uses the conjugate’s imaginary sign flip and the correct expression for the impedance angle, with the power factor derived from the same angle. Note that the angle of Z* is −φ, but cos(−φ) = cos φ, so the power factor remains the same.

The key idea is how a complex impedance behaves in the complex plane and how its phase relates to power factor. For Z = R + jX, the complex conjugate flips the sign of the imaginary part, so Z* = R − jX. The phase angle of Z, φ, comes from tan φ = X/R, so φ = arctan(X/R). The power factor is defined as the cosine of that angle, PF = cos φ. This also means PF = R/√(R^2 + X^2), since cos φ = adjacent over hypotenuse in the impedance’s right triangle.

That’s why the statement Z* = R − jX; φ = atan(X/R); PF = cos φ is correct: it uses the conjugate’s imaginary sign flip and the correct expression for the impedance angle, with the power factor derived from the same angle. Note that the angle of Z* is −φ, but cos(−φ) = cos φ, so the power factor remains the same.