AC behavior of a practical (air-cored) capacitor: If a capacitor of capacitance C has a series parasitic inductance L, the apparent capacitance at an angular frequency ω is given by which expression?

Introduction / Context:
Real capacitors are not ideal. Their leads and electrode geometry introduce a small series inductance L. At high frequencies, this parasitic inductance alters the net reactance, and it is convenient to describe the combined series L–C as an equivalent “apparent capacitance” C_app that reproduces the actual reactance at the operating frequency ω. This concept is vital in RF design and high-speed electronics.

Given Data / Assumptions:

• Ideal capacitor C in series with small inductance L (air-cored, negligible loss).
• Angular frequency ω, sinusoidal steady state.
• Define C_app so that the series combination has the same capacitive reactance magnitude as an ideal capacitor of value C_app (when the net reactance is capacitive).

Concept / Approach:

The series reactance is X_series = ωL − 1/(ωC). For an equivalent single capacitor, X_eq = −1/(ω C_app). Equate X_eq to X_series to solve for C_app. This yields a frequency-dependent capacitance that diverges at series resonance (ω^2 L C = 1), beyond which the branch appears inductive (no meaningful positive C_app).

Step-by-Step Solution:

Verification / Alternative check:

Check limits: as ω → 0, ω^2 L C → 0 ⇒ C_app → C (correct). As ω approaches 1/√(LC), denominator → 0 and |C_app| → ∞ (series resonance), matching the expected behavior of a minimum impedance point.

Why Other Options Are Wrong:

• Forms with (1 + ω^2 L C) arise for different equivalent definitions (e.g., parallel forms) or are simply incorrect for the series case.
• Frequency-independent C contradicts observed high-frequency behavior.

Common Pitfalls:

• Confusing series and parallel equivalences; be explicit about the chosen model.
• Using the expression beyond resonance where the branch is effectively inductive.

Final Answer:

C_app = C / (1 - ω^2 L C)