Question

Derive the expression for the root mean square value of alternating current, in terms of its peak value.

Answer 1

An alternating current (AC) varies sinusoidally as:

I = I₀sinωt

Where:

I₀ = Peak current,

ω = Angular frequency,

t = Time

The root mean square (RMS) value of current is given by:

I_rms = `sqrt(1/T int_0^T I^2 dt)`

Where T is the time period of the AC cycle. 

Substituting I = I₀sinωt

I_rms = `sqrt(1/T int_0^T I_0^2 sin^2 omega t dt)`

Since `I_0^2` is constant, it can be taken outside the integral:

I_rms = `sqrt(1/T int_0^T I_0^2 sin^2 omega t dt)`

The time-averaged value of sin²ωt over one full cycle is:

`1/T int_0^T sin^2 omega t dt = 1/2`

Thus, I_rms = `sqrt(I_0^2 xx 1/2)`

= `I_0/sqrt 2`

= 0.707 I₀