Impedance and AC circuit analysis
Impedance generalises resistance to AC. With it, capacitors and inductors become complex resistances, and every DC technique carries over.
Impedance
Impedance Z is the ratio of phasor voltage to phasor current: \(\mathbf{V} = \mathbf{I}Z\), Ohm's law for AC. It is measured in ohms and is generally complex: \(Z = R + jX\), with resistance R and reactance X. Its magnitude gives the ratio of amplitudes, and its angle the phase shift between voltage and current.
Impedance of R, L, and C
A resistor's impedance is R, with voltage and current in phase. An inductor's is \(Z_L = j\omega L\): its reactance \(X_L = \omega L\) grows with frequency, and voltage leads current by 90 degrees. A capacitor's is \(Z_C = \frac{1}{j\omega C} = -\frac{j}{\omega C}\): its reactance shrinks with frequency, and current leads voltage by 90 degrees.
Frequency behaviour
At low frequency, capacitors approach open circuits and inductors short circuits, matching their DC behaviour. At high frequency, capacitors approach short circuits and inductors open circuits. This frequency dependence is what lets circuits filter signals, passing some frequencies and blocking others.
Combining impedances
Impedances combine exactly like resistances: in series they add, in parallel their reciprocals (admittances, Y = 1/Z, in siemens) add. Voltage and current dividers, nodal and mesh analysis, superposition, and Thevenin and Norton equivalents all work with complex impedances, at a single frequency.
Series RLC and resonance
A series RLC circuit has \(Z = R + j(\omega L - \frac{1}{\omega C})\). At the resonant frequency \(\omega_0 = \frac{1}{\sqrt{LC}}\), or \(f_0 = \frac{1}{2\pi\sqrt{LC}}\), the inductive and capacitive reactances cancel, the impedance is purely R and at its minimum, and current is at its maximum. Resonance lets radios select one station among many.
Quality factor and bandwidth
The quality factor Q measures how sharp a resonance is: for a series RLC circuit, \(Q = \frac{\omega_0 L}{R}\). High Q means a narrow bandwidth, approximately \(\frac{f_0}{Q}\), and strong frequency selectivity. At resonance, voltages across L and C can each be Q times the source voltage, far larger than the input.
Mixed AC and DC
Circuits often carry a DC bias plus an AC signal. Superposition handles this: analyse the DC part with capacitors open and inductors short, analyse the AC part with phasors and impedances, and add the time-domain results. Signals at different frequencies must be analysed separately, since each frequency gives different impedances.
Worked example
A series circuit has R = 30 \(\Omega\) and an inductor with \(X_L = 40\) \(\Omega\) at the source frequency, driven by 100 V RMS. \(Z = 30 + j40 = 50\angle 53.1^\circ\) \(\Omega\). The current is \(\frac{100}{50} = 2\) A RMS, lagging the voltage by 53.1 degrees. The resistor drops 60 V and the inductor 80 V; these add as phasors, not arithmetically, to 100 V.
Key takeaways
Impedance Z = V / I is AC Ohm's law, complex: Z = R + jX. Resistors have Z = R; inductors Z = j omega L, with voltage leading current by 90 degrees; capacitors Z = 1/(j omega C), with current leading by 90 degrees. Capacitors block low frequencies and inductors block high frequencies, enabling filters. Impedances combine like resistances, and all DC techniques apply at each frequency. Series RLC resonance at f0 = 1/(2 pi root(LC)) cancels reactance, giving minimum impedance and maximum current; Q sets sharpness and bandwidth about f0/Q. Mixed AC and DC are analysed separately and superposed.
Flashcard
Complex ratio of phasor voltage to current
Check your understanding
What happens to a capacitor's reactance as frequency increases?