What is the correct expression of ω?

Question

Examoogle · Accepted Answer

$\omega = 2\pi f$

Explanation: The parameter $ \omega $ represents angular frequency in radians per second (rad/s), which is directly related to the cyclic frequency $ f $ measured in Hertz (cycles per second). Since one complete cycle corresponds to an angle of $ 2\pi $ radians, the relationship is defined by $ \omega = 2\pi f $.

Working Principle: In an AC sinusoidal signal, the waveform completes one full cycle over a time period $ T $. Because one full rotation around a unit circle is $ 2\pi $ radians, the rate of change of the phase angle with respect to time gives the angular velocity $ \omega $. Mathematically, $ \omega = \frac{2\pi}{T} $. Substituting the relation between frequency and period, $ f = \frac{1}{T} $, we derive the formula $ \omega = 2\pi f $.

Key Formulas: $\omega = 2\pi f$ — Angular frequency formula; $\omega = \frac{2\pi}{T}$ — Relation to time period

Key Points: The SI unit for angular frequency is radians per second (rad/s).; Cyclic frequency (f) is measured in Hertz (Hz) or cycles per second (cps).; In the Indian power system, the grid frequency is 50 Hz, corresponding to an angular frequency of approximately 314.16 rad/s.

Advantages: Simplifies the mathematical analysis of sinusoidal AC circuits.; Allows for easy calculation of inductive and capacitive reactance.

Applications: Calculation of Inductive Reactance: $X_L = 2\pi f L$.; Calculation of Capacitive Reactance: $X_C = \frac{1}{2\pi f C}$.; Phasor diagram analysis in power systems.

Additional Info: Frequency (f) is a scalar quantity, whereas angular frequency ($\omega$) is a vector-related parameter representing the rate of rotation of the phasor.. Option A is incorrect because it lacks the frequency component. Option C and D provide incorrect coefficients.

Answer: B is correct — The angular frequency $ \omega $ is equal to $ 2\pi $ times the cyclic frequency $ f $.

Exam Tip: Always verify the units when working with frequency; using $ f $ instead of $ \omega $ in reactance formulas is a common source of calculation errors in AC circuit exams.

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