In AC circuits, how are frequency, angular frequency, and period related?

In AC circuits, how are frequency, angular frequency, and period related?

• A. ω = 2π f; T = 1/f; ω = 2π / T.
• B. ω = f / 2π; T = 2π f; ω = 2π T.
• C. f = ω / T; T = ω / f; ω = f T.
• D. ω = π f; T = f/2.

Answer: (A)

In AC circuits, the speed of the oscillation can be described in three related ways: how many cycles occur each second (frequency f), how many radians the waveform sweeps each second (angular frequency ω), and how long one complete cycle takes (the period T). One full cycle corresponds to a change of 2π radians, so over one second you sweep ω radians and you have f cycles per second. From that, the standard relations are: ω = 2π f because each cycle adds 2π radians; T = 1/f because the period is the reciprocal of cycles per second; and ω T = 2π, which rearranges to ω = 2π / T. These combined give the familiar set: ω = 2π f, T = 1/f, and ω = 2π / T. Other forms would misplace the factors or mix units (for example, ω = f / 2π or T = 2π f would not be dimensionally consistent or physically correct).

In AC circuits, the speed of the oscillation can be described in three related ways: how many cycles occur each second (frequency f), how many radians the waveform sweeps each second (angular frequency ω), and how long one complete cycle takes (the period T). One full cycle corresponds to a change of 2π radians, so over one second you sweep ω radians and you have f cycles per second.

From that, the standard relations are: ω = 2π f because each cycle adds 2π radians; T = 1/f because the period is the reciprocal of cycles per second; and ω T = 2π, which rearranges to ω = 2π / T. These combined give the familiar set: ω = 2π f, T = 1/f, and ω = 2π / T.

Other forms would misplace the factors or mix units (for example, ω = f / 2π or T = 2π f would not be dimensionally consistent or physically correct).