For AC circuits, what is the complex impedance of a resistor, an inductor, and a capacitor, and what is the total impedance for series R-L-C?

For AC circuits, what is the complex impedance of a resistor, an inductor, and a capacitor, and what is the total impedance for series R-L-C?

• A. Z_R = R; Z_L = i ω L; Z_C = 1/(i ω C) = −i/(ω C); Z_total = R + i(ωL − 1/(ωC))
• B. Z_R = R; Z_L = −i ω L; Z_C = i/(ω C); Z_total = R − i(ωL − 1/(ωC))
• C. Z_R = R; Z_L = i ω L; Z_C = 1/(−i ω C) = i/(ω C); Z_total = R − i(ωL − 1/(ωC))
• D. Z_R = R; Z_L = i ω L; Z_C = 1/(i ω C); Z_total = R + i(ωL + 1/(ωC))

Answer: (A)

In AC circuits, impedance combines resistance and reactance into a complex quantity. A resistor contributes a real impedance Z_R = R; an inductor contributes Z_L = j ω L, which is purely imaginary and positive; a capacitor contributes Z_C = 1/(j ω C) = -j/(ω C), which is purely imaginary and negative. When components are in series, their impedances add: Z_total = Z_R + Z_L + Z_C = R + j ω L − j/(ω C) = R + j(ω L − 1/(ω C)). This matches the correct form, with a real part R and an imaginary part ω L − 1/(ω C). The sign of the imaginary part tells whether the net reactance is inductive (positive) or capacitive (negative).

In AC circuits, impedance combines resistance and reactance into a complex quantity. A resistor contributes a real impedance Z_R = R; an inductor contributes Z_L = j ω L, which is purely imaginary and positive; a capacitor contributes Z_C = 1/(j ω C) = -j/(ω C), which is purely imaginary and negative. When components are in series, their impedances add: Z_total = Z_R + Z_L + Z_C = R + j ω L − j/(ω C) = R + j(ω L − 1/(ω C)). This matches the correct form, with a real part R and an imaginary part ω L − 1/(ω C). The sign of the imaginary part tells whether the net reactance is inductive (positive) or capacitive (negative).