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Impedance of a Capacitor
A capacitor's impedance is purely imaginary and decreases with increasing angular frequency.
Interpretation
Impedance of a capacitor is Z_C = 1/(jωC) = −j/(ωC). Its magnitude decreases with increasing frequency.
Capacitors oppose changes in voltage, allowing high‑frequency signals to pass more easily.
Example: C=100µF, f=50Hz (ω≈314) → Z_C = −j/(314 × 100e-6) ≈ −j31.8Ω.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| ZC | Capacitive impedance | Ω |
| j | Imaginary unit (√−1) | — |
| ω | Angular frequency (2πf) | rad/s |
| C | Capacitance | F |
What it means
The impedance of a capacitor is Z_C = 1/(jωC) = −j/(ωC). It is purely imaginary, with a negative sign indicating that the voltage lags the current by 90°. The magnitude |Z_C| = 1/(ωC) decreases with increasing frequency. At DC (ω=0), the impedance is infinite (open circuit), and at high frequencies, it approaches zero (short circuit). This behaviour is used in coupling capacitors, decoupling capacitors, and filters. Capacitors pass AC while blocking DC. They are used in signal processing, power supplies, and timing circuits. Example: A capacitor C=100µF at f=50Hz (ω≈314) has |Z_C| = 1/(314*100e-6) = 1/0.0314 = 31.8Ω. At 1kHz (ω=6283), |Z_C| = 1/(6283*100e-6) = 1/0.6283 = 1.59Ω.
Worked example
Impedance of a Capacitor – Practical Example
Real‑World| Parameter | Value |
|---|---|
| C | 100 µF = 100×10⁻⁶ F |
| f | 50 Hz |
| Formula | ZC = -j / (ω·C), ω = 2πf |
Common mistakes
- j in denominator: 1/(jωC) = −j/(ωC) – the negative sign indicates phase shift.
- Phase: Current leads voltage by 90° (capacitor).
- Magnitude: |Z_C| = 1/(ωC) – decreases with frequency.
- Units: Ohms.
- DC: At DC (ω=0), impedance is infinite (open circuit).
Applications
The impedance of a capacitor is Z_C = −j/(ωC), which decreases with increasing frequency. Capacitors oppose changes in voltage, so they pass high‑frequency signals and block DC. Engineers use this formula to design high‑pass and band‑pass filters, to couple AC signals between stages, and to bypass noise to ground. In power supplies, capacitors smooth the output voltage. By understanding the frequency dependence, professionals can select appropriate capacitors for filtering, coupling, and timing applications. This impedance is fundamental to AC circuit analysis and is used in many practical circuits.
- Filter design (high‑pass, band‑pass, notch)
- AC coupling between amplifier stages
- Decoupling and bypass capacitors for noise reduction
- Power supply smoothing and ripple reduction
- Educational understanding of capacitive reactance
Frequently Asked Questions
The impedance of a capacitor is Z_C = 1/(jωC), purely imaginary and inversely proportional to frequency.
−90° (current leads voltage by 90°).
Z_C decreases with frequency (Z_C ∝ 1/f). At DC (f=0), Z_C = ∞ (open).
|Z_C| = 1/(ωC) = 1/(2πfC).
Q = V² / (1/(ωC)) = V² ωC (negative VAR for leading power factor).
Series: 1/C_total = 1/C₁ + 1/C₂; Parallel: C_total = C₁ + C₂.
I = V / (1/(jωC)) = jωC V, leading voltage by 90°.
Filters, coupling/decoupling, power factor correction, and timing circuits.
Real capacitors have equivalent series resistance (ESR), leakage, and parasitic inductance.
Common errors include: 1) forgetting the j term, 2) using the wrong frequency, 3) applying to DC (where it's infinite), 4) combining with other impedances incorrectly, and 5) using the wrong units.