Question

Choose the correct statement

• A. Capacitor behaves like a short circuit at very high frequency and inductor behaves like a short circuit at very low frequency
• B. Capacitor behaves like an open circuit at very high frequency and inductor behaves like a short circuit at very low frequency
• C. Capacitor behaves like a short circuit at very high frequency and inductor behaves like an open circuit at very low frequency
• D. Capacitor behaves like an open circuit at very high frequency and inductor behaves like an open circuit at very low frequency

Hint:

To intuitively recall this behavior during circuit analysis: - A capacitor blocks DC ($f=0$, Open Circuit) and passes high-frequency AC ($f\to\infty$, Short Circuit). - An inductor passes DC ($f=0$, Short Circuit) and blocks high-frequency AC ($f\to\infty$, Open Circuit).

Answer 1

The Correct Option is A

Concept: The electrical behavior of reactive elements like capacitors and inductors in alternating current (AC) networks is entirely dictated by their frequency-dependent reactances. Reactance represents the opposition offered by these components to the flow of alternating current. Let us establish the exact mathematical formulas for both elements:
• Inductive Reactance ($X_L$): The reactance of an inductor with inductance $L$ at an operating frequency $f$ (or angular frequency $\omega = 2\pi f$) is expressed as: \[ X_L = \omega L = 2\pi f L \] This formula shows that inductive reactance is directly proportional to the frequency ($X_L \propto f$).
• Capacitive Reactance ($X_C$): The reactance of a capacitor with capacitance $C$ at an operating frequency $f$ is expressed as: \[ X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} \] This formula shows that capacitive reactance is inversely proportional to the frequency ($X_C \propto \frac{1}{f}$). Now, let us rigorously evaluate the limiting behavior of these expressions under extreme frequency thresholds:

Step 1: Behavior at Very High Frequencies ($f \to \infty$)
• For the capacitor: \[ \lim_{f \to \infty} X_C = \lim_{f \to \infty} \frac{1}{2\pi f C} = 0 \] An electrical element offering zero opposition ($X_C = 0$) behaves mathematically and physically as an ideal short circuit.
• For the inductor: \[ \lim_{f \to \infty} X_L = \lim_{f \to \infty} (2\pi f L) = \infty \] An electrical element offering infinite opposition ($X_L = \infty$) completely blocks current flow, behaving as an open circuit.

Step 2: Behavior at Very Low Frequencies Direct Current ($f \to 0$)
• For the inductor: \[ \lim_{f \to 0} X_L = \lim_{f \to 0} (2\pi f L) = 0 \] At zero frequency (DC condition), a pure inductor offers absolutely zero reactance, behaving precisely as an ideal short circuit.
• For the capacitor: \[ \lim_{f \to 0} X_C = \lim_{f \to 0} \frac{1}{2\pi f C} = \infty \] At zero frequency, the capacitor offers infinite reactance, completely blocking any steady-state direct current, which corresponds to an open circuit. Synthesizing these analytical findings: At very high frequencies, the capacitor is a short circuit. At very low frequencies, the inductor is a short circuit. This perfectly matches the statement in Option (1).