A capacitor has impedance Z_C = 1/(jωC). As frequency increases from 100 Hz to 10,000 Hz, what happens to the capacitor's impedance magnitude, and what does this mean for current flow?
AImpedance increases — the capacitor becomes harder to drive at higher frequencies, limiting current
BImpedance decreases — the capacitor passes high-frequency signals more easily, as |Z_C| = 1/(ωC) shrinks with increasing ω
CImpedance stays constant — frequency does not affect the capacitor's opposition to current
DImpedance becomes purely resistive at high frequencies as the imaginary part cancels
The magnitude of capacitive impedance is |Z_C| = 1/(ωC). As ω increases, the denominator grows, so |Z_C| decreases. Physically, at higher frequencies the capacitor charges and discharges more rapidly, presenting less opposition to current flow. This is why capacitors block DC (ω = 0, |Z_C| → ∞) but pass high-frequency AC — the opposite behavior from inductors. This frequency dependence is what makes capacitors and inductors useful as filters.
Question 2 Multiple Choice
A series RC circuit has R = 3 Ω and capacitive reactance X_C = −4 Ω. What are the impedance magnitude and the phase angle between voltage and current?
A|Z| = 7 Ω, phase angle = −53° (current leads voltage by 53°)
B|Z| = 5 Ω, phase angle = −53° (current leads voltage by 53°)
C|Z| = 5 Ω, phase angle = +53° (voltage leads current by 53°)
D|Z| = 1 Ω, phase angle = −53° (current leads voltage by 53°)
Z = R + jX = 3 + j(−4) = 3 − 4j. The magnitude is |Z| = √(3² + 4²) = √25 = 5 Ω. The phase angle is ∠Z = arctan(−4/3) ≈ −53°. A negative phase angle on Z means voltage lags current (or equivalently, current leads voltage) — expected for a capacitive circuit. Series impedances add directly: 3 Ω + (−4j) Ω = (3 − 4j) Ω. This is the same arithmetic as adding DC resistors, but complex.
Question 3 True / False
In AC circuit analysis, working with admittance Y = 1/Z is useful for parallel combinations because parallel admittances add, just as parallel conductances add in DC circuits.
TTrue
FFalse
Answer: True
For parallel branches, the total admittance Y_total = Y₁ + Y₂ + ···, then Z_total = 1/Y_total. This mirrors DC: for parallel resistors, G_total = G₁ + G₂ (conductances add) and R_total = 1/G_total. Admittance Y = G + jB has a real part (conductance G = R/|Z|²) and an imaginary part (susceptance B). The parallel admittance rule lets you analyze complex AC networks using the same step-by-step reduction that works for DC resistor networks.
Question 4 True / False
An inductor and a capacitor in series usually have zero total impedance because their reactances have opposite signs and cancel substantially.
TTrue
FFalse
Answer: False
Inductive reactance X_L = ωL and capacitive reactance X_C = −1/(ωC) are opposite in sign, so they partially cancel in series. But they are equal in magnitude only at the resonant frequency ω₀ = 1/√(LC). At that specific frequency, X_L + X_C = 0 and total reactance is zero (series resonance). At any other frequency, the magnitudes differ and a net reactance remains. Well above resonance, the inductor dominates (Z ≈ jωL); well below resonance, the capacitor dominates (Z ≈ 1/jωC). Zero total impedance is a special case, not the general rule.
Question 5 Short Answer
A DC-trained engineer says 'For AC circuits, I just replace every R with Z and use all the same DC formulas.' Explain why this works, and what the complex nature of Z adds that DC analysis cannot capture.
Think about your answer, then reveal below.
Model answer: It works because Kirchhoff's voltage and current laws are linear equations, and linearity is the only property that the DC derivations of Ohm's law, voltage dividers, current dividers, Thevenin equivalents, and superposition actually require. These derivations never assumed V and I were constant — only that V = IR. Replacing R with Z (and V, I with phasors V̅, I̅) is valid because phasors are complex representations of sinusoidal signals, and impedance is the generalized proportionality constant. What complex Z adds: it encodes both amplitude ratio (|Z| = peak voltage / peak current) and phase relationship (∠Z = phase lead of voltage over current) in a single quantity. DC analysis captures only magnitude; AC with complex impedance captures magnitude and phase simultaneously, which is essential for power factor, filter design, and resonance analysis.
This replacement principle — that all DC techniques extend to AC by replacing R with Z — is one of the most powerful tools in circuit analysis. It is not a trick or approximation; it is mathematically exact for linear circuits in sinusoidal steady state. The complex arithmetic handles the phase bookkeeping automatically.