A capacitor has impedance Z_C = −j/(ωC). An engineer says 'voltage lags current by 90° in this capacitor.' Which explanation correctly connects the complex impedance to the phase relationship?
AThe imaginary part of Z_C is negative, meaning the voltage phasor is rotated −90° relative to the current phasor, so voltage lags current by 90°
BBecause the capacitor stores charge, it delays the voltage by exactly half a period relative to the current
CThe impedance magnitude |Z_C| decreases with frequency, so the phase lag also decreases as frequency rises
DCapacitors block DC, which means at all AC frequencies, the voltage is exactly 90° behind in time
In phasor analysis, V = IZ. If Z = −j/(ωC) = |Z_C|∠(−90°), then multiplying the current phasor by Z rotates it by −90°. The voltage phasor is 90° behind the current phasor in the complex plane, meaning voltage lags current by 90°. Option B conflates a time-domain intuition with a phase-domain statement; option C confuses magnitude with phase (|Z_C| decreases with frequency, but the phase is always −90° for an ideal capacitor regardless of frequency). Reading the phase angle directly from the angle of Z is the correct phasor technique.
Question 2 Multiple Choice
Which statement correctly identifies the core advantage of representing AC circuits with complex impedances?
AImpedances allow you to ignore the frequency dependence of circuit elements, simplifying analysis
BAll DC circuit analysis techniques — Kirchhoff's laws, series/parallel combination rules, voltage and current dividers — apply directly to phasors using complex impedances, converting differential equations into algebraic ones
CComplex impedances eliminate the need to know the amplitude of signals; only phase relationships matter
DImpedance analysis is only valid for sinusoidal signals at a single fixed frequency; it cannot handle circuits with multiple frequency components
The central payoff of phasor analysis is the transfer of the entire DC toolkit to AC. KVL holds for phasors (sum of phasor voltages around a loop = 0), KCL holds for phasors (sum of phasor currents at a node = 0), series impedances sum, parallel impedances combine reciprocally, and voltage/current dividers use the identical formulas as for resistors. This works because replacing d/dt with jω converts the underlying differential equations into linear algebraic equations with complex coefficients — the same structure as DC circuits with real resistances. Option D notes a real limitation (impedance analysis applies at a single frequency), but this does not negate the advantage stated in option B.
Question 3 True / False
The magnitude of a complex impedance |Z| gives the ratio of the voltage amplitude to the current amplitude in an AC circuit.
TTrue
FFalse
Answer: True
From Ohm's law in phasor form, V = IZ. The phasor V has magnitude |V| (voltage amplitude) and the phasor I has magnitude |I| (current amplitude). Since |V| = |I| × |Z|, we have |Z| = |V|/|I|. The angle of Z gives the phase of V relative to I. So a complex impedance Z = R + jX encodes both the amplitude ratio (|Z| = √(R² + X²)) and the phase relationship (∠Z = arctan(X/R)) between voltage and current. This is why reading impedance in polar form — magnitude and angle — immediately gives you both pieces of AC circuit information you care about.
Question 4 True / False
An inductor (Z_L = jωL) and a capacitor (Z_C = −j/(ωC)) in series have impedances that usually cancel, giving zero total reactance for any AC circuit containing both components.
TTrue
FFalse
Answer: False
The total impedance of an inductor and capacitor in series is Z = jωL − j/(ωC) = j(ωL − 1/(ωC)). This is zero only when ωL = 1/(ωC), i.e., at the resonant frequency ω₀ = 1/√(LC). At other frequencies, the net reactance is nonzero — inductive above resonance (net positive imaginary) and capacitive below resonance (net negative imaginary). The cancellation is frequency-specific, not universal. This is the basis of resonant circuits: at exactly one frequency, the series LC combination has zero reactance (minimum impedance), which is why series resonant circuits are used as frequency-selective filters.
Question 5 Short Answer
Explain why replacing resistance R with complex impedance Z allows all DC circuit analysis techniques to work directly for AC circuits, even though AC circuits involve time-varying signals and differential equations.
Think about your answer, then reveal below.
Model answer: In AC steady state, all voltages and currents are sinusoids at the same frequency ω. Representing them as phasors (complex amplitudes) and replacing d/dt with multiplication by jω converts the circuit's differential equations into linear algebraic equations with complex coefficients. For a resistor, v = Ri becomes V = RI. For an inductor, v = L di/dt becomes V = jωL·I = Z_L·I. For a capacitor, i = C dv/dt becomes I = jωC·V, so V = I/(jωC) = Z_C·I. In every case, the element law takes the algebraic form V = ZI — exactly Ohm's law with Z replacing R. Since KVL and KCL are linear (they sum voltages or currents at nodes), they hold equally for the complex phasor equations. The algebraic structure is identical to DC circuits, so all the DC solution techniques follow immediately.
The key move is recognizing that for sinusoidal excitation, d/dt ↔ ×jω is an exact substitution that converts differential operators into algebraic constants. This is essentially the circuit-theory version of Fourier or Laplace analysis restricted to steady-state sinusoids. The power of the substitution is that it makes the multi-element AC circuit problem as straightforward as the DC resistor circuit problem — once you accept complex arithmetic.