Linearity of an air-cored inductor at fixed frequency Consider an ideal air-cored inductor operated at a fixed frequency with sinusoidal excitation. How should it be classified regarding linearity, and why?

Introduction / Context:
Linearity of passive elements is judged by whether current is proportional to voltage (or vice versa) for a given frequency, independent of amplitude. Air-cored inductors are widely used where magnetic linearity is required (RF chokes, filters).

Given Data / Assumptions:

• No ferromagnetic core; the core is air (μ ≈ μ0) and does not saturate.
• Sinusoidal steady state at a fixed angular frequency ω.
• Ideal inductor behavior (negligible series resistance and stray capacitance for the conceptual argument).

Concept / Approach:
For an ideal inductor, v(t) = L * di/dt. With sinusoidal excitation of angular frequency ω, the phasor relation is V = j ω L I, giving I = V / (j ω L). At fixed ω and fixed L (no saturation), current magnitude is proportional to voltage magnitude, demonstrating linearity.

Step-by-Step Solution:
Identify reactance: X_L = ω L (independent of current and voltage amplitude).At fixed ω, I = V / X_L → I ∝ V.Absence of ferromagnetic core → no saturation or hysteresis → L is constant.Therefore, the element is linear under stated conditions.

Verification / Alternative check:
Small-signal measurements show a constant inductance for air coils across a wide current range. In contrast, ferromagnetic-core inductors show L varying with current due to B–H nonlinearity.

Why Other Options Are Wrong:
(a) invokes saturation, which does not occur with an air core. (c) incorrectly states reactance depends on voltage; it does not. (d) presents a wrong voltage–current relation. (e) cites hysteresis, absent in air cores.

Common Pitfalls:

• Confusing frequency dependence (X_L ∝ ω) with amplitude dependence (not present here).
• Assuming all inductors are nonlinear because many use ferromagnetic cores.

Final Answer:
Linear because its current varies linearly with applied voltage at fixed frequency