By Lipi Gupta Updated Mar 24, 2022
The sine function represents the ratio of the y‑coordinate of a point on a unit circle to its radius. Its cosine counterpart does the same for the x‑coordinate.
In AC circuits, the voltage and current follow a sinusoidal waveform. Calculating average or RMS values of these periodic signals is essential for circuit design.
A sine wave, defined as sin(θ), has a unit amplitude, a period of 2π, and no phase shift unless explicitly added. While a phase offset changes the starting point of the waveform, it does not affect the average amplitude or power.
Power in a resistive circuit is given by P = I V, and because V = I R, we have P = I²R.
For a time‑varying current I(t) = I₀ sin(ωt), the instantaneous power is:
P(t) = I₀² R sin²(ωt)
To find the average power, integrate P(t) over one full period T and divide by T:
⟨P⟩ = (1/T) ∫₀ᵀ I₀² R sin²(ωt) dt = (I₀² R)/2
Note that the average value of sin² over a complete cycle is ½, which simplifies the calculation.
Root‑mean‑square (RMS) is obtained by squaring the quantity, averaging it, and then taking the square root. For a sine wave, the RMS value is 1/√2 (≈0.707) of its peak.
Thus, for a sinusoidal current, the RMS current is I₀/√2 and the RMS voltage is V₀/√2, where V₀ = I₀ R.
In practice, you can estimate the average as peak / 2 and the RMS as peak / √2.
