AC Impedance and Reactance

Definition

When you drive a circuit with a steady direct current, a single real number — the resistance R — relates voltage to current through Ohm's law, V = IR. Alternating current is richer: capacitors and inductors respond to the rate of change of the signal, so at any instant the current can be ahead of or behind the voltage. Impedance packages the amplitude ratio and this timing offset into one complex number.

Z = R + jX      (ohms, Ω)

Here R is the resistance (the real part, energy dissipated as heat), X is the reactance (the imaginary part, energy stored and returned by fields), and j = √(−1) is the imaginary unit. Electrical engineers write j instead of i so it does not collide with the symbol for current. Both R and X are measured in ohms.

The reactance itself is the sum of an inductive and a capacitive contribution:

X = X_L + X_C = ωL − 1/(ωC)

where ω = 2πf is the angular frequency in radians per second, f is the ordinary frequency in hertz, L is the inductance in henries (H), and C is the capacitance in farads (F). Inductive reactance X_L = ωL is positive and grows with frequency; capacitive reactance X_C = −1/(ωC) is negative and shrinks in magnitude with frequency.

Phasors: why the number is complex

Represent a sinusoidal voltage as the real part of a rotating vector — a phasor: v(t) = Re[V·e^{jωt}], where V is a complex amplitude carrying both size and starting phase. The trick is that differentiation of e^{jωt} just multiplies by jω. The three defining element laws become algebra:

Element	Time-domain law	Impedance Z	Voltage–current phase
Resistor R	v = R i	R	in phase (0°)
Inductor L	v = L di/dt	jωL	voltage leads current by 90°
Capacitor C	i = C dv/dt	1/(jωC) = −j/(ωC)	current leads voltage by 90°

The factor of j is not bookkeeping — it is the 90° phase shift. Multiplying a phasor by j rotates it a quarter turn counter-clockwise in the complex plane. So the moment you write an inductor's impedance as jωL, you have said "the voltage across it leads the current through it by 90°," and you can read amplitude ratios and phase offsets straight off the complex arithmetic.

Magnitude and phase

Because Z is complex, it has a magnitude and an angle. Writing Z = R + jX in polar form:

|Z| = √(R² + X²)          (impedance magnitude, Ω)
φ   = arctan(X / R)        (phase angle, radians or degrees)

The magnitude |Z| is the ratio of voltage amplitude to current amplitude — the AC analog of resistance. The phase angle φ tells you the timing: if φ > 0 the load is net inductive and the current lags the voltage; if φ < 0 the load is net capacitive and the current leads. A pure resistor has φ = 0 (in phase); a pure inductor has φ = +90°; a pure capacitor has φ = −90°. Two mnemonics capture the sign convention: ELI (in an inductor, voltage E leads current I) and ICE (in a capacitor, current I leads voltage E).

Impedance generalizes Ohm's law

Everything you know about DC circuits survives if you replace resistances with impedances and treat voltages and currents as complex phasors:

V = I Z        (Ohm's law for AC)
Z_series   = Z₁ + Z₂ + Z₃ + …
1/Z_parallel = 1/Z₁ + 1/Z₂ + 1/Z₃ + …

Kirchhoff's voltage and current laws, the voltage divider, Thévenin and Norton equivalents, and superposition all carry over unchanged. This is why the phasor/impedance method is the workhorse of AC circuit analysis: a hard set of coupled differential equations collapses into the same linear algebra you already use for resistor networks. The only bookkeeping is that the answers are complex, so at the end you convert back to amplitude and phase.

Frequency dependence and filters

The two reactances pull in opposite directions with frequency, and that opposition is the entire basis of frequency-selective circuits.

Element	Reactance	At DC (f → 0)	At high f (f → ∞)
Inductor L	X_L = ωL	0 Ω — a short circuit	∞ — blocks the signal
Capacitor C	X_C = 1/(ωC)	∞ — blocks DC	0 Ω — a short circuit

Because an inductor passes low frequencies and a capacitor passes high frequencies, pairing them makes high-pass, low-pass, band-pass, and notch filters. The corner (half-power) frequency of a simple RC filter is f_c = 1/(2πRC), where |X_C| = R; for an RL filter it is f_c = R/(2πL). A radio's tuning knob sweeps the resonance of an LC circuit across the band so that only one station's carrier sees a large response.

Resonance: when the reactances cancel

Inductive reactance grows with frequency and capacitive reactance shrinks, so somewhere they are equal in magnitude. Setting ωL = 1/(ωC) gives the resonant angular frequency:

ω₀ = 1/√(LC)          f₀ = 1/(2π√(LC))

At ω₀ the net reactance X = ωL − 1/(ωC) is zero, so Z = R is purely real. In a series RLC circuit this is the impedance minimum, so current peaks and the circuit draws maximum power at unity power factor. In a parallel RLC ("tank") circuit it is the impedance maximum, so the line current dips to a minimum. The sharpness of the peak is the quality factor:

Q = ω₀L / R = 1/(ω₀CR) = (1/R)·√(L/C)
bandwidth  Δf = f₀ / Q

A high-Q circuit resonates in a narrow band (a selective radio front end); a low-Q circuit has a broad, gentle peak. Near resonance the voltage across the inductor or capacitor can be Q times the source voltage — a real hazard in high-Q power circuits.

Worked example: a series RLC circuit

Take R = 100 Ω, L = 50 mH, and C = 2 µF, driven at f = 500 Hz. First the angular frequency:

ω = 2πf = 2π(500) ≈ 3141.6 rad/s

X_L = ωL = 3141.6 × 0.050 ≈ 157.1 Ω
X_C = 1/(ωC) = 1/(3141.6 × 2×10⁻⁶) ≈ 159.2 Ω
X   = X_L − X_C ≈ 157.1 − 159.2 = −2.1 Ω   (slightly capacitive)

|Z| = √(R² + X²) = √(100² + (−2.1)²) ≈ 100.0 Ω
φ   = arctan(X/R) = arctan(−2.1/100) ≈ −1.2°

We are just below resonance: the resonant frequency is f₀ = 1/(2π√(LC)) = 1/(2π√(0.050 × 2×10⁻⁶)) ≈ 503 Hz, so at 500 Hz the reactances almost cancel, |Z| ≈ R, and the current lags essentially not at all. With a 10 V amplitude source the current amplitude is I = V/|Z| ≈ 0.1 A, in phase to within about one degree. The quality factor here is Q = (1/R)√(L/C) = (1/100)√(0.050/2×10⁻⁶) ≈ 1.58 — a broad resonance.

Power, the power factor, and RMS values

Only the resistive part of the impedance dissipates energy. Writing the load phase angle as φ, the average (real) power is

P = V_rms · I_rms · cos φ        (watts, W)  — real power
S = V_rms · I_rms                (volt-amperes, VA) — apparent power
Q = V_rms · I_rms · sin φ        (volt-amperes reactive, VAR) — reactive power

The power factor cos φ is the fraction of the apparent power that does useful work. Purely reactive elements have φ = ±90°, so cos φ = 0 and they consume no net power over a cycle — they merely borrow and return energy. Industrial loads dominated by motors are inductive, so utilities charge for poor power factor; adding parallel capacitors cancels the inductive reactance (power-factor correction) and reduces the line current and the resulting I²R losses. Note that in AC engineering, RMS values (V_rms = V_peak/√2 for a sinusoid) are used so that P = I_rms²R matches the DC heating formula.

A short history

The complex-impedance method grew out of work by Oliver Heaviside in the 1880s, whose operational calculus turned circuit differential equations into algebra, and Charles Proteus Steinmetz, who in an 1893 paper before the AIEE (and a widely read 1897 book) systematized the phasor representation of AC quantities. Steinmetz's method let engineers design the polyphase power systems that electrified the industrial world without solving differential equations by hand. The term "impedance" was coined earlier by Heaviside in 1886. Together they converted alternating-current engineering from a specialist art into routine algebra.

Common misconceptions

• "Reactance dissipates power like resistance." No — a pure reactance stores energy in a field and returns all of it over each cycle. Average power in a reactance is zero; only R converts electrical energy to heat.
• Confusing impedance magnitude with impedance. |Z| = √(R²+X²) is only half the story; the phase angle φ is equally physical. Two loads with the same |Z| can behave very differently.
• Adding reactances arithmetically the wrong way. X_C is negative in the convention X = ωL − 1/(ωC). You add impedances as complex numbers (real and imaginary parts separately), not magnitudes: |Z₁+Z₂| ≠ |Z₁|+|Z₂| in general.
• Thinking resonance always minimizes impedance. It minimizes |Z| in a series RLC but maximizes it in a parallel (tank) circuit. Which one you have determines whether current peaks or dips.
• Using peak instead of RMS in power formulas. P = V_rms·I_rms·cos φ needs RMS amplitudes. Mixing peak and RMS introduces a factor of 2 error in the power.
• Assuming voltage and current always peak together. Except in a pure resistor (or exactly at resonance), they are offset by the phase angle φ — that offset is the defining feature of AC circuits.