Capacitive Reactance

AC Circuits · 10 min read

A capacitor that laughs at steady DC turns cooperative at AC. The higher the frequency, the easier the cap is to push current through — it acts like an ohmic opposition that shrinks as frequency rises. The engineering name for that opposition is capacitive reactance.

The formula

For a pure capacitor carrying a sinusoidal current, the opposition to that current at frequency $f$ is:

X_C = \frac{1}{\omega C} = \frac{1}{2 \pi f C} \quad [\Omega]

$X_C$ has units of ohms and describes, at a given frequency, how much the capacitor throttles the current for a given RMS voltage. Like a resistor, it ties current and voltage via a version of Ohm's Law:

I_{rms} = \frac{V_{rms}}{X_C}

But unlike a resistor, $X_C$ is frequency-dependent and introduces a 90° phase shift between V and I. Both differences come from the cap's core behaviour — current flows in response to a change in voltage, not the voltage itself.

On log-log axes, X_C = 1/(ωC) is a straight line of slope −1. A decade rise in frequency drops the reactance by a decade.

Frequency dependence — the key intuition

Double the frequency and $X_C$ halves. A decade rise in frequency drops reactance by a decade. Two handy limits:

DC ( $f = 0$ ): $X_C \to \infty$ . The cap looks like an open circuit. Steady current can't flow.
High frequency: $X_C \to 0$ . The cap looks like a short circuit. Any AC passes through freely.

This is why a cap works as a coupling element between stages of an amplifier (blocks the DC bias, passes the AC signal) and as a bypass element to ground (shorts out ripple at high frequencies while leaving low-frequency behaviour alone).

Phase relationship — I leads V by 90°

A capacitor's defining equation is $i = C \, dv/dt$ — current scales with the rate of change of voltage. For a sinusoidal voltage $v = V_p \sin(\omega t)$ , the derivative is $V_p \omega \cos(\omega t)$ : the same frequency, with a 90° phase shift.

i = C \frac{dv}{dt} = C V_p \omega \cos(\omega t) = I_p \sin(\omega t + 90°)

So the current hits its peaks a quarter-cycle before the voltage does. Engineers remember this with “ICE” — in a Capacitor, I leads E (an older symbol for voltage). The inductor is the reverse, and we'll meet its mnemonic in Topic 6.

In a pure capacitor, the current phasor sits 90° ahead of the voltage phasor — so the current sine peaks a quarter-period before the voltage does. Mnemonic: “ICE” — I leads V through a C.

Capacitive reactance is the AC opposition of a capacitor:

X_C = 1/(\omega C)

in ohms, down by a decade for every decade of frequency. Current leads voltage by 90° through a pure cap — ICE.

Common applications

Coupling caps sit in series between two stages of an amplifier. They block the DC operating point of stage 1 ( $X_C = \infty$ at DC) while passing the audio / RF content ( $X_C$ small at the signal frequencies).
Bypass / decoupling caps tie a supply rail to ground. At low frequency $X_C$ is large — the cap is invisible to DC. At high frequency it's a short — it shunts ripple and switching noise straight to ground before it can contaminate the rest of the circuit.
High-pass filters put a cap in series with the signal path: at low f the cap blocks, at high f it passes. We'll derive the cut-off formula in Topic 9.

Every modern PCB is peppered with bypass caps — often dozens near each IC. Understanding $X_C$ is understanding why they're there.

Capacitive Reactance

The formula

Frequency dependence — the key intuition

Phase relationship — I leads V by 90°

Common applications

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