What is the complex impedance Z in an AC circuit?

What is the complex impedance Z in an AC circuit?

• A. Z = R + X, with magnitude |Z| = sqrt(R^2 + X^2)
• B. Z = R + jX, with magnitude |Z| = sqrt(R^2 + X^2).
• C. Z = R - jX, with magnitude |Z| = sqrt(R^2 + X^2)
• D. Z = R + jX, with magnitude |Z| = R^2 + X^2

Answer: (B)

In AC circuits, impedance is a complex quantity that combines resistance and reactance. It is written as Z = R + jX, where R is the real part (resistance) and X is the reactance (the imaginary part’s coefficient). The magnitude is |Z| = sqrt(R^2 + X^2) because the magnitude of a + jb is sqrt(a^2 + b^2). This form matches the correct choice: Z = R + jX with the magnitude sqrt(R^2 + X^2). The other forms aren’t correct as written: removing the imaginary unit loses the representation of phase shift; using Z = R - jX changes the sign of the imaginary part (which can be accommodated by letting X change sign, but the standard convention shown uses +jX); and presenting the magnitude without the square root is incorrect because the magnitude is the square root of the sum of squares.

In AC circuits, impedance is a complex quantity that combines resistance and reactance. It is written as Z = R + jX, where R is the real part (resistance) and X is the reactance (the imaginary part’s coefficient). The magnitude is |Z| = sqrt(R^2 + X^2) because the magnitude of a + jb is sqrt(a^2 + b^2). This form matches the correct choice: Z = R + jX with the magnitude sqrt(R^2 + X^2).

The other forms aren’t correct as written: removing the imaginary unit loses the representation of phase shift; using Z = R - jX changes the sign of the imaginary part (which can be accommodated by letting X change sign, but the standard convention shown uses +jX); and presenting the magnitude without the square root is incorrect because the magnitude is the square root of the sum of squares.