Capacitive reactance

Capacitive reactance is the frequency-dependent opposition a capacitor presents to an AC signal. Unlike a resistor, a capacitor’s “resistance” to current depends on how fast the voltage across it is changing — i.e. on frequency. This single fact is what makes RC filters, coupling, and amplifier bandwidth work.

Impedance of a capacitor

In the phasor / time-harmonic framework (signals written as $e^{j\omega t}$ — see Phasor), the complex impedance of a capacitor $C$ is

$Z_C=\frac{1}{j\omega C}$

where $\omega =2\pi f$ is the angular frequency in rad/s, $C$ is the Capacitance in farads, and $j=\sqrt{-1}$. This comes straight from the capacitor’s current–voltage law $i=Cdv/dt$: for $v=V{e}^{j\omega t}$, differentiating brings down a factor $j\omega$, so $i=j\omega Cv$, hence $v/i=1/(j\omega C)$.

The factor of $j$ in the denominator means the impedance is purely reactive — it stores and returns energy rather than dissipating it, and it shifts the current $90°$ ahead of the voltage. The magnitude of this impedance is the reactance:

$X_C=|Z_C|=\frac{1}{\omega C}=\frac{1}{2\pi fC}$

with units of ohms. Writing a general impedance as $Z=R+jX$, a capacitor contributes a negative reactance $X=-1/(\omega C)$; the magnitude $X_C$ is what you compare against a resistance.

(A handwritten note on the course slide writes $X_C=1/\sqrt{2\pi fC}$ with a stray square-root sign. This is a slide typo acknowledged in the notes — the correct formula has no square root: $X_C=1/(2\pi fC)$.)

Frequency behaviour: open at DC, short at high frequency

Everything about RC circuits follows from the two limits of $X_C=1/(2\pi fC)$:

• At DC ($f\to 0$): $X_C\to \infty$. A capacitor is an open circuit to DC. This is exactly why a Coupling capacitor blocks the DC bias of one amplifier stage from reaching the next.
• At high frequency ($f\to \infty$): $X_C\to 0$. A capacitor looks like a short circuit (a plain wire) to a fast-changing signal. This is why a bypass capacitor can shunt unwanted high-frequency content straight to ground.

Between these extremes the reactance falls smoothly as $1/f$: doubling the frequency halves the reactance. Worked example: a $0.1\mu \text{F}$ capacitor at $f=1\text{kHz}$ has

$X_C=\frac{1}{2\pi (10^3)(10^{-7})}\approx 1.59\text{k}\Omega ,$

but the same capacitor at $1\text{MHz}$ has $X_C\approx 1.59\Omega$ — practically a short. The capacitor’s value is fixed; its effective opposition is set entirely by the signal frequency.

Why it matters

Because $X_C$ depends on frequency, any circuit mixing a capacitor with a resistor treats different frequencies differently — that is the entire mechanism of an RC lowpass filter and RC highpass filter. The crossover happens where $X_C$ equals the resistance in the circuit, which is precisely how the Cutoff frequency $f_c=1/(2\pi RC)$ arises. The same reactance idea generalises to inductors and underlies Impedance matching and all of AC electric circuits analysis; the phasor rules used here are collected in Phasor relationships for circuit elements.