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Inductive Reactance: Relationship of XL with Frequency and Inductance In AC circuit analysis, the inductive reactance XL of an inductor is said to vary with the signal frequency and with the inductance value. Evaluate the statement: “XL is directly proportional to frequency and inductance.”

Difficulty: Easy

Correct Answer: True

Explanation:


Introduction / Context:
Inductive reactance is a cornerstone concept in AC circuit theory. It quantifies how strongly an inductor resists changes in alternating current. The statement under review claims that inductive reactance depends linearly on both the excitation frequency and the inductance value. Understanding this helps with filter design, impedance matching, and predicting current draw in power electronics and signal-processing applications.


Given Data / Assumptions:

  • An ideal inductor with inductance L (henry).
  • A sinusoidal source with frequency f (hertz).
  • Steady-state AC conditions (phasor analysis).
  • No core nonlinearity, parasitics, or saturation effects considered.


Concept / Approach:

For an ideal inductor, the voltage-current relation in the time domain is v(t) = L * di/dt. In sinusoidal steady state, the phasor form relates voltage and current by an imaginary impedance jXL, where XL is the inductive reactance. By definition, XL rises with frequency because faster current changes induce larger opposing voltages. It also scales directly with L because a larger inductance produces a larger induced voltage for the same rate of current change.


Step-by-Step Solution:

Start from v(t) = L * di/dt for a sinusoidal current i(t) = I_max * sin(2π f t).Differentiate: di/dt = I_max * 2π f * cos(2π f t).Thus v(t) = L * 2π f * I_max * cos(2π f t).In phasor form, V = j 2π f L * I, so the impedance Z_L = j XL with XL = 2π f L.Therefore XL is directly proportional to f and to L (linear dependence).


Verification / Alternative check:

Units confirm consistency: L has units of henry, and f is in hertz. Multiplying 2π f L yields ohms (Ω), the correct unit of reactance. Experimental Bode plots show a +20 dB/decade rise of |Z_L| with frequency for an ideal inductor, aligning with XL ∝ f.


Why Other Options Are Wrong:

“False” contradicts the standard formula XL = 2π f L. “True only at resonance” is irrelevant; resonance concerns LC combinations, not a lone inductor. “Depends only on core material” ignores frequency dependence. “True only for DC” is incorrect because XL at DC (f = 0) is 0 Ω, not meaningful for proportionality claims.


Common Pitfalls:

Confusing reactance XL with resistance R, or misapplying the formula to non-sinusoidal signals without proper harmonic analysis. Real inductors also have winding resistance and parasitic capacitance that modify behavior at high frequency, but the first-order relationship XL = 2π f L remains fundamental.


Final Answer:

True

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